id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
dict |
|---|---|---|---|---|---|
1208.4626
|
Families of Lagrangian fibrations on hyperkähler manifolds
Ljudmila Kamenova, Misha Verbitsky111Partially supported by by RFBR grants
12-01-00944- , 10-01-93113-NCNIL-a, and AG Laboratory NRI-HSE, RF government
grant, ag. 11.G34.31.0023.
Abstract
A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is
a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that
a given compact manifold admits only finitely many holomorphic symplectic
structures, up to deformation. We prove that a given compact manifold with
$b_{2}\geqslant 7$ admits only finitely many deformation types of holomorphic
Lagrangian fibrations. We also prove that all known hyperkähler manifolds are
never Kobayashi hyperbolic.
###### Contents
1. 1 Introduction
1. 1.1 Hyperkähler manifolds
2. 1.2 The Bogomolov-Beauville-Fujiki form
3. 1.3 The hyperkähler SYZ conjecture
2. 2 Hyperkähler geometry: preliminary results
1. 2.1 Teichmüller space and the moduli space
2. 2.2 The polarized Teichmüller space
3. 3 Main results
1. 3.1 The moduli of manifolds with Lagrangian fibrations
2. 3.2 Kobayashi hyperbolicity in hyperkähler geometry
## 1 Introduction
Irreducible compact hyperkähler manifolds, or irreducible holomorphic
symplectic manifolds, are a natural generalization of K3 surfaces in higher
dimensions. The geometry of K3 surfaces is well studied. In particular, it is
known that any two K3 surfaces are deformation equivalent to each other, i.e.,
there is only one deformation type of K3 surfaces.
A natural question to ask is whether the same is true in higher dimensions.
The answer is negative due to Beauville’s examples. In every possible complex
dimension $2n$ there are at least the Hilbert scheme of $n$ points on a K3
surface $S$, $Hilb^{n}(S)$, and the generalized Kummer varieties $K^{n+1}(A)$,
where $A$ is an Abelian surface. These two examples are not deformation
equivalent since they have different Betti numbers. There are two more
exceptional examples due to K. O’Grady in dimensions $6$ and $10$.
It is conjectured that in every fixed dimension there are finitely many
deformation types of irreducible compact hyperkähler manifolds. It is also
conjectured that every hyperkähler manifold can be deformed to one that admits
a holomorphic Lagrangian fibration. It would be interesting to classify the
deformation types of tha pairs $(M,L)$ of a hyperkähler manifold together with
a Lagrangian fibration on it. In the present paper, we show that the number of
deformational classes of such pairs is finite, if one fixes the smooth
manifold undelying $M$.
In [Hu2] Huybrechts proved that for a fixed compact manifold there are at most
finitely many deformation types of hyperkähler structures on it. Therefore, to
prove that the number of deformation classes of pairs $(M,L)$ is finite, it
would suffice to prove it when a deformational class of $M$ is fixed.
Let $M\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}X$ be a Lagrangian
fibration. Then $X$ is known to be projective, with $H^{2}(X)={\mathbb{C}}$,
hence $\operatorname{rk}\operatorname{Pic}(X)=1$. Therefore, the primitive
ample bundle $L_{X}$ on $X$ is unique (up to torsion). Denote by $L_{M}$ the
semiample bundle $\pi^{*}(L_{X})$ on $M$. Clearly,
$c_{1}(L_{M})^{\operatorname{rk}M}=0$; a (1,1)-class satisfying this equation
is called parabolic. The Lagrangian fibration
$M\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}X$ is uniquely
determined by a class $[c_{1}(L_{M})]\in\operatorname{Pic}(M)$ which is
parabolic and semiample (this is due to D. Matsushita, [Ma1]; see [Saw] for a
detailed exposition of an early work on Lagrangian fibrations). Therefore, to
classify the Lagrangian fibrations it would suffice to classify pairs
$(M,L_{M})$, where $L_{M}$ is a parabolic semiample line bundle.
We prove that in the Teichmüller space of hyperkähler manifolds with a fixed
parabolic class the pairs admitting a Lagrangian fibration form a dense and
open subset. The other main result is that the action of the monodromy group
has finitely many orbits. As a corollary of these results we obtain that for a
fixed compact manifold, there are only finitely many deformation types of
hyperkähler structures equipped with a Lagrangian fibration.
### 1.1 Hyperkähler manifolds
Definition 1.1: A hyperkähler manifold is a compact, Kähler, holomorphically
symplectic manifold.
Definition 1.2: A hyperkähler manifold $M$ is called simple if $H^{1}(M)=0$,
$H^{2,0}(M)={\mathbb{C}}$.
Theorem 1.3: (Bogomolov’s Decomposition Theorem, [Bo1], [Bes]). Any
hyperkähler manifold admits a finite covering, which is a product of a torus
and several simple hyperkähler manifolds.
Remark 1.4: Further on, all hyperkähler manifolds are assumed to be simple.
A note on terminology. Speaking of hyperkähler manifolds, people usually mean
one of two different notions. One either speaks of holomorphically symplectic
Kähler manifold, or of a manifold with a hyperkähler structure, that is, a
triple of complex structures satisfying quaternionic relations and parallel
with respect to the Levi-Civita connection. The equivalence (in compact case)
between these two notions is provided by the Yau’s solution of Calabi’s
conjecture ([Bes]). Throughout this paper, we use the complex algebraic
geometry point of view, where “hyperkähler” is synonymous with “Kähler
holomorphically symplectic”, in lieu of the differential-geometric approach.
The reader may check [Bes] for an introduction to hyperkähler geometry from
the differential-geometric point of view.
Notice also that we included compactness in our definition of a hyperkähler
manifold. In the differential-geometric setting, one does not usually assume
that the manifold is compact.
### 1.2 The Bogomolov-Beauville-Fujiki form
Theorem 1.5: ([F]) Let $\eta\in H^{2}(M)$, and $\dim M=2n$, where $M$ is
hyperkähler. Then $\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, for some integer
quadratic form $q$ on $H^{2}(M)$ and a constant $c>0$.
Definition 1.6: This form is called Bogomolov-Beauville-Fujiki form. It is
defined by this relation uniquely, up to a sign. The sign is determined from
the following formula (Bogomolov, Beauville; [Bea], [Hu1], 23.5)
$\displaystyle\lambda q(\eta,\eta)$
$\displaystyle=(n/2)\int_{X}\eta\wedge\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}+$
$\displaystyle+(1-n)\frac{\left(\int_{X}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\right)\left(\int_{X}\eta\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\right)}{\int_{M}\Omega^{n}\wedge\overline{\Omega}^{n}}$
where $\Omega$ is the holomorphic symplectic form, and $\lambda$ a positive
constant.
Remark 1.7: The form $q$ has signature $(3,b_{2}-3)$. It is negative definite
on primitive forms, and positive definite on the space
$\langle\Omega,\overline{\Omega},\omega\rangle$ where $\omega$ is a Kähler
form, as seen from the following formula
$\mu q(\eta_{1},\eta_{2})=\\\
\int_{X}\omega^{2n-2}\wedge\eta_{1}\wedge\eta_{2}-\frac{2n-2}{(2n-1)^{2}}\frac{\int_{X}\omega^{2n-1}\wedge\eta_{1}\cdot\int_{X}\omega^{2n-1}\wedge\eta_{2}}{\int_{M}\omega^{2n}},\
\ \mu>0$ (1.1)
(see e. g. [V2], Theorem 6.1, or [Hu1], Corollary 23.9).
Definition 1.8: Let $[\eta]\in H^{1,1}(M)$ be a real (1,1)-class on a
hyperkähler manifold $M$. We say that $[\eta]$ is parabolic if
$q([\eta],[\eta])=0$. A line bundle $L$ is called parabolic if $c_{1}(L)$ is
parabolic.
### 1.3 The hyperkähler SYZ conjecture
Theorem 1.9: (D. Matsushita, see [Ma1]). Let $\pi:\;M{\>\longrightarrow\>}X$
be a surjective holomorphic map from a hyperkähler manifold $M$ to $X$, with
$0<\dim X<\dim M$. Then $\dim X=1/2\dim M$, and the fibers of $\pi$ are
holomorphic Lagrangian tori (this means that the symplectic form vanishes on
the fibers).111Here, as elsewhere, we assume that the hyperkähler manifold $M$
is simple.
Definition 1.10: Such a map is called a holomorphic Lagrangian fibration.
Remark 1.11: The base of $\pi$ is conjectured to be rational. J.-M. Hwang
([Hw]) proved that $X\cong{\mathbb{C}}P^{n}$, if it is smooth. D. Matsushita
([Ma2]) proved that it has the same rational cohomology as
${\mathbb{C}}P^{n}$.
Remark 1.12: The base of $\pi$ has a natural flat connection on the smooth
locus of $\pi$. The combinatorics of this connection can be used to determine
the topology of $M$ ([KZ], [G]),
Definition 1.13: Let $(M,\omega)$ be a Calabi-Yau manifold, $\Omega$ the
holomorphic volume form, and $Z\subset M$ a real analytic subvariety,
Lagrangian with respect to $\omega$. If
$\Omega{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ is proportional to the
Riemannian volume form, $Z$ is called special Lagrangian (SpLag).
The special Lagrangian varieties were defined in [HL] by Harvey and Lawson,
who proved that they minimize the Riemannian volume in their cohomology class.
This implies, in particular, that their moduli are finite-dimensional. In
[McL], McLean studied deformations of non-singular special Lagrangian
subvarieties and showed that they are unobstructed.
In [SYZ], Strominger-Yau-Zaslow tried to explain the mirror symmetry
phenomenon using the special Lagrangian fibrations. They conjectured that any
Calabi-Yau manifold admits a Lagrangian fibration with special Lagrangian
fibers. Taking its dual fibration, one obtains “the mirror dual” Calabi-Yau
manifold.
Definition 1.14: A line bundle is called semiample if $L^{N}$ is generated by
its holomorphic sections, which have no common zeros.
Remark 1.15: From semiampleness it obviously follows that $L$ is nef. Indeed,
let $\pi:\;M{\>\longrightarrow\>}{\mathbb{P}}H^{0}(L^{N})^{*}$ be the standard
map. Since the sections of $L$ have no common zeros, $\pi$ is holomorphic.
Then $L\cong\pi^{*}{\cal O}(1)$, and the curvature of $L$ is a pullback of the
Kähler form on ${\mathbb{C}}P^{n}$. However, the converse is false: a nef
bundle is not necessarily semiample (see e.g. [DPS1, Example 1.7]).
Remark 1.16: Let $\pi:\;M{\>\longrightarrow\>}X$ be a holomorphic Lagrangian
fibration, and $\omega_{X}$ a Kähler class on $X$. Then
$\eta:=\pi^{*}\omega_{X}$ is semiample and parabolic. The converse is also
true, by Matsushita’s theorem: if $L$ is semiample and parabolic, $L$ induces
a Lagrangian fibration. This is the only known source of non-trivial special
Lagrangian fibrations.
Conjecture 1.17: (Hyperkähler SYZ conjecture) Let $L$ be a parabolic nef line
bundle on a hyperkähler manifold. Then $L$ is semiample.
Remark 1.18: This conjecture was stated by many people (Tyurin, Bogomolov,
Hassett-Tschinkel, Huybrechts, Sawon); please see [Saw] for an interesting and
historically important discussion, and [V3] for details and references.
Remark 1.19: The SYZ conjecture can be seen as a hyperkähler version of the
“abundance conjecture” (see e.g. [DPS2], 2.7.2).
Claim 1.20: Let $M$ be an irreducible hyperkähler manifold in one of 4 known
classes, that is, a deformation of a Hilbert scheme of points on K3, a
deformation of generalized Kummer variety, or a deformation of one of two
examples by O’Grady. Then $M$ admits a deformation equipped with a holomorphic
Lagrangian fibration.
Proof: When $S$ is an elliptic K3 surface, the Hilbert scheme of points
$\operatorname{Hilb}^{n}(S)$ has an induced Lagrangian fibration with smooth
fibers that are products of $n$ elliptic curves:
$\operatorname{Hilb}^{n}(S)\rightarrow
Sym^{n}({\mathbb{P}}^{1})\simeq{\mathbb{P}}^{n}$. Similarly, when $A$ is an
elliptic Abelian surface, the generalized Kummer variety $K^{n}(A)$ admits a
Lagrangian fibration. Another construction gives Lagrangian fibrations on
$\operatorname{Hilb}^{n}(S)$ and on $K^{n}(A)$ if $S$ contains a smooth genus
$n$ curve and if $A$ contains a smooth genus $n+2$ curve (see examples 3.6 and
3.8 in [Saw]).
O’Grady’s examples are deformation equivalent to Lagrangian fibrations, as
follows from Corollary 1.1.10 in [R].
## 2 Hyperkähler geometry: preliminary results
### 2.1 Teichmüller space and the moduli space
Here we cite the relevant result from the deformation theory of hyperkähler
manifolds. We follow [V1].
Let $M$ be a hyperkähler manifold (compact and simple, as usual), and
$\operatorname{Comp}_{0}$ be the Frèchet manifold of all complex structures of
hyperkähler type on $M$. The quotient
$\operatorname{Teich}:=\operatorname{Comp}_{0}/\operatorname{Diff}^{0}$ of
$\operatorname{Comp}_{0}$ by isotopies is a finite-dimensional complex
analytic space ([Cat]). This quotient is called the Teichmüller space of $M$.
When $M$ is a complex curve, the quotient
$\operatorname{Comp}_{0}/\operatorname{Diff}^{0}$ is the Teichmüller space of
this curve.
The mapping class group
$\Gamma=\operatorname{Diff}^{+}/\operatorname{Diff}^{0}$ acts on
$\operatorname{Teich}$ in the usual way, and its quotient $\operatorname{Mod}$
is the moduli space of $M$.
As shown in [Hu2], $\operatorname{Teich}$ has a finite number of connected
components. Take a connected component $\operatorname{Teich}^{I}$ containing a
given complex structure $I$, and let $\Gamma^{I}\subset\Gamma$ be the set of
elements of $\Gamma$ fixing this component. Since $\operatorname{Teich}$ has
only a finite number of connected components, $\Gamma^{I}$ has finite index in
$\Gamma$. On the other hand, as shown in [V1], the image of the group $\Gamma$
is commensurable to ${O}\big{(}H^{2}(M,\mathbb{Z}),q\big{)}$.
In [V1, Lemma 2.6] it was proved that any hyperkähler structure on a given
simple hyperkähler manifold is also simple. Therefore,
$H^{2,0}(M,I^{\prime})=\mathbb{C}$ for all $I^{\prime}\in\operatorname{Comp}$.
This trivial observation is a key to the following well-known definition.
Definition 2.1: Let $(M,I)$ be a hyperkähler manifold, and
$\operatorname{Teich}$ its Teichmüller space. Consider a map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\mathbb{P}H^{2}(M,\mathbb{C})$,
sending $J$ to the line $H^{2,0}(M,J)\in\mathbb{P}H^{2}(M,\mathbb{C})$. It is
easy to see that $\operatorname{\sf Per}$ maps $\operatorname{Teich}$ into the
open subset of a quadric, defined by
$\operatorname{{\mathbb{P}}\sf
er}:=\big{\\{}l\in\mathbb{P}H^{2}(M,\mathbb{C})\ \big{|}\ q(l,l)=0,\
q(l,\overline{l})>0\big{\\}}.$
The map $\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is called the period map, and the set $\operatorname{{\mathbb{P}}\sf er}$
the period space.
The following fundamental theorem is due to F. Bogomolov [Bo2].
Theorem 2.2: Let $M$ be a simple hyperkähler manifold, and
$\operatorname{Teich}$ its Teichmüller space. Then the period map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is a local diffeomorphism (that is, an etale map). Moreover, it is
holomorphic.
Remark 2.3: Bogomolov’s theorem implies that $\operatorname{Teich}$ is smooth.
However, it is not necessarily Hausdorff (and it is non-Hausdorff even in the
simplest examples).
### 2.2 The polarized Teichmüller space
In [V4, Corollary 2.6], the following proposition was deduced from [Bou] and
[DP].
Theorem 2.4: Let $M$ be a simple hyperkähler manifold, such that all integer
$(1,1)$-classes satisfy $q(\nu,\nu)\geqslant 0$. Then its Kähler cone is one
of the two connected components of the set $K:=\big{\\{}\nu\in
H^{1,1}(M,\mathbb{R})\ \big{|}\ q(\nu,\nu)>0\big{\\}}$.
Remark 2.5: From 2.2 it follows that on a hyperkähler manifold with
$\operatorname{Pic}(M)={\mathbb{Z}}$, for any rational class $\eta\in
H^{1,1}(M)$ with $q(\eta,\eta)\geqslant 0$, either $\eta$ or $-\eta$ is nef.
Remark 2.6: Consider an integer vector $\eta\in H^{2}(M)$ which is positive,
that is, satisfies $q(\eta,\eta)>0$. Denote by $\operatorname{Teich}^{\eta}$
the set of all $I\in\operatorname{Teich}$ such that $\eta$ is of type $(1,1)$
on $(M,I)$. The space $\operatorname{Teich}^{\eta}$ is a closed divisor in
$\operatorname{Teich}$. Indeed, by Bogomolov’s theorem, the period map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is etale, but the image of $\operatorname{Teich}^{\eta}$ is the set of
all $l\in\operatorname{{\mathbb{P}}\sf er}$ which are orthogonal to $\eta$;
this condition defines a closed divisor $C_{\eta}$ in
$\operatorname{{\mathbb{P}}\sf er}$, hence
$\operatorname{Teich}^{\eta}=\operatorname{\sf Per}^{-1}(C_{\eta})$ is also a
closed divisor.
Remark 2.7: When $I\in\operatorname{Teich}^{\eta}$ is generic, Bogomolov’s
theorem implies that the space of rational $(1,1)$-classes
$H^{1,1}(M,\mathbb{Q})$ is one-dimensional and generated by $\eta$. This is
seen from the following argument. Locally around a given point $I$ the period
map
$\operatorname{Teich}^{\eta}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is surjective on the set $\operatorname{{\mathbb{P}}\sf er}^{\eta}$ of
all $I\in\operatorname{{\mathbb{P}}\sf er}$ for which $\eta\in H^{1,1}(M,I)$.
However, the Hodge-Riemann relations give
$\operatorname{{\mathbb{P}}\sf
er}^{\eta}=\big{\\{}l\in\operatorname{{\mathbb{P}}\sf er}\ \big{|}\
q(\eta,l)=0\big{\\}}.$ (2.1)
Denote the set of such points of $\operatorname{Teich}^{\eta}$ by
$\operatorname{Teich}^{\eta}_{\text{gen}}$. It follows from 2.2 that, for any
$I\in\operatorname{Teich}^{\eta}_{\text{gen}}$, either $\eta$ or $-\eta$ is a
Kähler class on $(M,I)$.
Consider a connected component $\operatorname{Teich}^{\eta,I}$ of
$\operatorname{Teich}^{\eta}$. Changing the sign of $\eta$ if necessary, we
may assume that $\eta$ is Kähler on $(M,I)$. By Kodaira’s theorem about
stability of Kähler classes, $\eta$ is Kähler in some neighbourhood
$U\subset\operatorname{Teich}^{\eta,I}$ of $I$. Therefore, the sets
$V_{+}:=\big{\\{}I\in\operatorname{Teich}^{\eta}_{\text{gen}}\ \big{|}\
\eta\text{ is K\"{a}hler on }(M,I)\big{\\}}$
and
$V_{-}:=\big{\\{}I\in\operatorname{Teich}^{\eta}_{\text{gen}}\ \big{|}\
-\eta\text{ is K\"{a}hler on }(M,I)\big{\\}}$
are open in $\operatorname{Teich}^{\eta}_{\text{gen}}$. It is easy to see that
$\operatorname{Teich}^{\eta}_{\text{gen}}$ is a complement to a union of
countably many divisors in $\operatorname{Teich}^{\eta}$ corresponding to the
points $I^{\prime}\in\operatorname{Teich}^{\eta}$ with
$\operatorname{rk}\operatorname{Pic}(M,I^{\prime})>1$. Therefore, for any
connected open subset $U\subset\operatorname{Teich}^{\eta}$, the intersection
$U\cap\operatorname{Teich}^{\eta}_{\text{gen}}$ is connected. Since
$\operatorname{Teich}^{\eta}_{\text{gen}}$ is represented as a disjoint union
of open sets $V_{+}\sqcup V_{-}$, every connected component of
$\operatorname{Teich}^{\eta}_{\text{gen}}$ and of
$\operatorname{Teich}^{\eta}$ is contained in $V_{+}$ or in $V_{-}$. We
obtained the following corollary.
Corollary 2.8: Let $\eta\in H^{2}(M)$ be a positive integer vector,
$\operatorname{Teich}^{\eta}$ the corresponding divisor in the Teichmüller
space, and $\operatorname{Teich}^{\eta,I}$ a connected component of
$\operatorname{Teich}^{\eta}$ containing a complex structure $I$. Assume that
$\eta$ is Kähler on $(M,I)$. Then $\eta$ is Kähler for all
$I^{\prime}\in\operatorname{Teich}^{\eta,I}$ which satisfy
$\operatorname{rk}H^{1,1}(M,\mathbb{Q})=1$.
We call the set $\operatorname{Teich}^{\eta}_{\text{pol}}$ of all
$I\in\operatorname{Teich}^{\eta}$ for which $\eta$ is Kähler the polarized
Teichmüller space, and $\eta$ its polarization. From the above arguments it is
clear that the polarized Teichmüller space
$\operatorname{Teich}^{\eta}_{\text{pol}}$ is open and dense in
$\operatorname{Teich}^{\eta}$.
The quotient ${\cal M}_{\eta}$ of $\operatorname{Teich}^{\eta}_{\text{pol}}$
by the subgroup of the mapping class group fixing $\eta$ is called the moduli
of polarized hyperkähler manifolds. It is known (due to the general theory
which goes back to Viehweg and Grothendieck) that ${\cal M}_{\eta}$ is
Hausdorff and quasiprojective (see e.g. [Vi] and [GHS]).
Remark 2.9: We conclude that there are countably many quasiprojective
divisors ${\cal M}_{\eta}$ immersed in the moduli space $\operatorname{Mod}$
of hyperkähler manifolds. Moreover, every algebraic complex structure belongs
to one of these divisors. However, these divisors need not to be closed.
Indeed, as proven in [AV], each of ${\cal M}_{\eta}$ is dense in
$\operatorname{Mod}$.
In [AV, Theorem 1.7], the following theorem was proven.
Theorem 2.10: Let $M$ be a compact, simple hyperkähler manifold,
$\operatorname{Teich}^{I}$ a connected component of its Teichmüller space, and
$\operatorname{Teich}^{I}\stackrel{{\scriptstyle\Psi}}{{{\>\longrightarrow\>}}}\operatorname{Teich}^{I}/\Gamma^{I}=\operatorname{Mod}$
its projection to the moduli space of complex structures. Consider a positive
or negative vector $\eta\in H^{2}(M,\mathbb{Z})$, and let
$\operatorname{Teich}^{I,\eta}$ be the corresponding connected component of
the polarized Teichmüller space. Assume that $b_{2}(M)>3$. Then the image
$\Psi(\operatorname{Teich}^{I,\eta})$ is dense in $\operatorname{Mod}$.
The proof relies on a more general proposition about lattices.
Proposition 2.11: [AV, Proposition 3.2, Remark 3.12] Let $V$ be an
$\mathbb{R}$-vector space equipped with a non-degenerate symmetric form of
signature $(s_{+},s_{-})$ with $s_{+}\geq 3$ and $s_{-}\geq 1$. Consider a
lattice $L\subset V$. Let $\Gamma$ be a subgroup of finite index in
$\mathop{\text{\rm O}}(L)$, and $l\in L$. Then $\Gamma\cdot\mathop{\text{\rm
Gr}}_{++}(l^{\perp})$ is dense in $\mathop{\text{\rm Gr}}_{++}(V)$.
Remark 2.12: Since the proof of this statement is symmetric in $s_{+}$ and
$s_{-}$, the same proposition is valid if we assume that $s_{+}\geq 1$ and
$s_{-}\geq 3$.
## 3 Main results
### 3.1 The moduli of manifolds with Lagrangian fibrations
Here we assume that $b_{2}(M)\geqslant 7$ as we need it for our proof of 3.1.
The authors conjecture that the result is valid for smaller Betti numbers as
well.
Definition 3.1: Let $L$ be a holomorphic line bundle on a hyperkähler
manifold. We call $L$ Lagrangian if it is parabolic and semiample.
Definition 3.2: Let $M$ be a hyperkähler manifold. Fix a parabolic class $L\in
H^{2}(M,{\mathbb{Z}})$. We denote by $\operatorname{Teich}_{L}$ the
Teichmüller space of all complex structures $I$ of hyperkähler type on $M$
such that $L$ is of type $(1,1)$ on $(M,I)$. Clearly,
$\operatorname{Teich}_{L}$ is a divisor in the whole Teichmüller space of $M$.
The space $\operatorname{Teich}_{L}$ is called the Teichmüller space of
hyperkähler manifolds with parabolic class.
Matsushita proves the following openness result in [Ma3, Theorem 1.1]:
Theorem 3.3: Let
$\operatorname{Teich}_{L}^{\circ}\subset\operatorname{Teich}_{L}$ be the set
of all $I\in\operatorname{Teich}_{L}$ for which $L$ is Lagrangian. Then
$\operatorname{Teich}_{L}^{\circ}$ is open in $\operatorname{Teich}_{L}$.
The main results of the present paper are the following two theorems.
Theorem 3.4: The subspace
$\operatorname{Teich}_{L}^{\circ}\subset\operatorname{Teich}_{L}$ is dense and
open in $\operatorname{Teich}_{L}$.
Proof: Fix a positive class $\eta\in H^{2}(M,{\mathbb{Z}})$ and define
$\operatorname{Teich}_{L,\eta}^{\circ}$ to be the open subset of
$\operatorname{Teich}_{L}^{\circ}$ for which $\eta$ is a polarization.
Consider the projection $\Psi$ to the moduli space $\operatorname{Mod}$ as
defined in 2.2. Since ${\cal M}_{\eta}$ is quasiprojective (see [Vi]), then
$\Psi(\operatorname{Teich}_{L,\eta}^{\circ})$ is Zariski open, and therefore
dense in $\Psi(\operatorname{Teich}_{L,\eta})$.
Fix a negative vector $L^{\prime}\in H^{2}(M,{\mathbb{Z}})$ such that the
sublattice $<L,L^{\prime}>$ is of rank 2. Notice that
$\Psi(\operatorname{Teich}_{L})=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{Z}})|q(l,l)=0,q(l,\overline{l})>0,q(L,l)=0\\}/\Gamma_{L}$
and
$\Psi(\operatorname{Teich}_{L,\eta})=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{Z}})|q(l,l)=0,q(l,\overline{l})>0,q(L,l)=0,q(\eta,l)=0\\}/\Gamma_{L,\eta}$.
Applying 2.2 to the quotient $H^{2}(M,{\mathbb{Z}})/<L,L^{\prime}>$, we see
that $\Psi(\operatorname{Teich}_{L,L^{\prime},\eta})$ is dense in
$\Psi(\operatorname{Teich}_{L,L^{\prime}})$ for any $L^{\prime}$. Here we
needed to assume $b_{2}\geqslant 7$, because $H^{2}(M,{\mathbb{Z}})$ is of
signature $(3,b_{2}-3)$ and the quotient
$H^{2}(M,{\mathbb{Z}})/<L,L^{\prime}>$ is of signature $(2,b_{2}-4)$. This
satisfies the conditions of 2.2 since $b_{2}-4\geqslant 3$.
However, $\bigcup_{L^{\prime}}\Psi(\operatorname{Teich}_{L,L^{\prime}})$ is
dense in $\Psi(\operatorname{Teich}_{L})$, and
$\bigcup_{L^{\prime}}\Psi(\operatorname{Teich}_{L,L^{\prime},\eta})$ is dense
in $\Psi(\operatorname{Teich}_{L,\eta})$. Therefore,
$\Psi(\operatorname{Teich}_{L,\eta})$ is dense in
$\Psi(\operatorname{Teich}_{L})$ and $\operatorname{Teich}_{L}^{\circ}$ is
dense in $\operatorname{Teich}_{L}$.
Remark 3.5: Together with 2.2, 3.1 implies that the set of manifolds with
Lagrangian fibrations is dense within the deformation space of a hyperkähler
manifold $M$, if $M$ admits a Lagrangian fibration.
Theorem 3.6: Consider the action of the monodromy group $\Gamma_{I}$ on
$H^{2}(M,{\mathbb{Z}})$, and let $S\subset H^{2}(M,{\mathbb{Z}})$ be the set
of all classes which are parabolic and primitive. Then there are only finitely
many orbits of $\Gamma_{I}$ on $S$.
Proof: In the proof we use Nikulin’s technique of discriminant-forms
described in [Ni].
Denote by $\Lambda$ the lattice $(H^{2}(M,{\mathbb{Z}}),q)$. It is a free
${\mathbb{Z}}$-module of finite rank together with a non-degenerate symmetric
bilinear form $q$ with values in ${\mathbb{Z}}$. If $\\{e_{i}\\}_{i\in I}$ is
a basis of the lattice $\Lambda$, its discriminant is defined to be
$\text{discr}(\Lambda)=\text{det}(e_{i}\cdot e_{j})$. There is a canonical
embedding
$\Lambda\hookrightarrow\Lambda^{\ast}=\text{Hom}(\Lambda,\mathbb{Z})$ using
the bilinear form of $\Lambda$. The discriminant group
$A_{\Lambda}=\Lambda^{\ast}/\Lambda$ is a finite Abelian group of order
$|\text{discr}(\Lambda)|$. One can extend the bilinear form to
$\Lambda^{\ast}$ with values in ${\mathbb{Q}}$ and define the discriminant-
bilinear form of the lattice $b_{\Lambda}:A_{\Lambda}\times
A_{\Lambda}\rightarrow{\mathbb{Q}}/{\mathbb{Z}}$. It is a finite non-
degenerate form. A subgroup $H\subset A_{\Lambda}$ is isotropic if
$q_{\Lambda}|_{H}=0$, where $q_{\Lambda}$ is the quadratic form corresponding
to $b_{\Lambda}$. Given any subset $K\subset\Lambda$, its orthogonal
complement is $K^{\bot}=\\{v\in\Lambda|(v,K)=0\\}$.
An embedding of lattices $\Lambda_{1}\hookrightarrow\Lambda_{2}$ is primitive
if $\Lambda_{2}/\Lambda_{1}$ is a free ${\mathbb{Z}}$-module. Take a primitive
vector $v\in\Lambda$ with $q(v)=0$. We can choose a vector $f\in\Lambda$ with
minimal positive quadratic intersection $\alpha=q(v,f)$. Then
$0<\alpha\leqslant|\text{discr}(\Lambda)|$. It is implied by the following
lemma:
Lemma 3.7: The minimal positive intersection $\alpha$ divides
$\text{discr}(\Lambda)$.
Proof: Since $v$ is primitive, we can choose a free ${\mathbb{Z}}$-basis
$\\{v_{1}=v,v_{2},\dots,v_{n}\\}$ of $\Lambda$, where $n=\text{rk}(\Lambda)$.
If $\alpha=\text{min}\\{q(v,f)|f\in{\mathbb{Z}}^{n}\\}$, then
$\alpha{\mathbb{Z}}$ is an ideal generated by $\\{q(v,v_{i}),i=1,\dots,n\\}$.
For every $i=1,\dots,n,~{}~{}q(v,v_{i})=\alpha\cdot a_{i}$ for some
$a_{i}\in{\mathbb{Z}}$. Thus the matrix $[q(v_{j},v_{i})]$ has first column
divisible by $\alpha$. Then $\text{det}[q(v_{j},v_{i})]=\text{discr}(\Lambda)$
is divisible by $\alpha$.
Let $K$ be the primitive sublattice of $\Lambda$ spanned by $v$ and $f$. The
intersection matrix of $\text{Span}(v,f)$ has determinant
$q(v,v)q(f,f)-q(v,f)^{2}=-\alpha^{2}$ which is bounded:
$-|\text{discr}(\Lambda)|^{2}\leqslant-\alpha^{2}<0$. Since
$\operatorname{rk}(K)=2$, $K$ has at most four primitive isotropic vectors
($2\operatorname{rk}(K)=4$).
An overlattice of $\Lambda$ is a lattice embedding
$i:\Lambda\rightarrow\Lambda^{\prime}$ with $\Lambda$ and $\Lambda^{\prime}$
of the same rank, or equivalently, such that
$H_{\Lambda^{\prime}}=\Lambda^{\prime}/\Lambda$ is a finite Abelian group.
Note that we have the inclusions:
$\Lambda\hookrightarrow\Lambda^{\prime}\hookrightarrow\Lambda^{\prime\ast}\hookrightarrow\Lambda^{\ast}$.
Therefore,
$H_{\Lambda^{\prime}}\subset\Lambda^{\prime\ast}/\Lambda\subset\Lambda^{\ast}/\Lambda=A_{\Lambda}$.
Proposition 3.8: [Ni, Proposition 1.4.1] The correspondence
$\Lambda^{\prime}\rightarrow H_{\Lambda^{\prime}}$ determines a bijection
between overlattices of $\Lambda$ and isotropic subgroups of $A_{\Lambda}$.
Furthermore, $H_{\Lambda^{\prime}}^{\bot}=\Lambda^{\prime\ast}/\Lambda$ and
$H_{\Lambda^{\prime}}^{\bot}/H_{\Lambda^{\prime}}=A_{\Lambda^{\prime}}$.
Let $L=K^{\bot}$ be the orthogonal complement of $K$ in $\Lambda$. Then
$K\oplus L\subset\Lambda\subset K^{\ast}\oplus L^{\ast}$. Since $\det(L)$ is
bounded, in view of 3.1, there are finitely many ways of expressing $\Lambda$
as an overlattice of $\Lambda_{K}\doteq K\oplus K^{\bot}$ because
$A_{\Lambda}$ is finite of order $|\text{discr}(\Lambda)|$ and there are
finitely many isotropic subgroups.
Define the lattices $\Lambda$ and $\Lambda^{\prime}$ to be stably equivalent
if there exists a lattice $M$ such that $\Lambda\oplus
M\simeq\Lambda^{\prime}\oplus M$. The following proposition is a reformulation
of Theorem 1.1 in Chapter 9 of Cassels’ book [Cas].
Proposition 3.9: There exist only a finite number of lattices stably
equivalent to $\Lambda$.
If we assume that there are infinitely many orbits of $\Gamma_{I}$, this would
imply that there exist infinitely many non-isomorphic pairs of lattices
$(K,K^{\bot})$. Then for infinitely many of them $K^{\bot}$ would be stably
equivalent to $K_{1}^{\bot}$ for another $K_{1}$ since there are only finitely
many choices for $K$. This contradicts 3.1 and the result follows.
Corollary 3.10: For any hyperkähler manifold, there are only finitely many
orbits of $\Gamma_{I}$ on the set of all divisors $\operatorname{Teich}_{L}$
with a parabolic class.
Combining 3.1 and 3.1, we obtain the following result.
Corollary 3.11: Let $M$ be a hyperkähler manifold. Then there are only
finitely many deformation types of Lagrangian fibrations
$(M,I){\>\longrightarrow\>}S$, for all complex structures on $M$.
Proof: By 2.2 we can assume that $H^{1,1}(M,\mathbb{Q})$ is one-dimensional
and generated by a parabolic class $L$. Since either $L$ or $-L$ is nef, we
can assume $L$ to be nef. From 3.1 it follows that for each pair $(M,L)$ there
exists a unique deformation type of a fibration structure. We conclude
finiteness of the deformation types of Lagrangian fibrations since there are
finitely many orbits of $\Gamma_{I}$ on the set $\operatorname{Teich}_{L}$.
### 3.2 Kobayashi hyperbolicity in hyperkähler geometry
Definition 3.12: A compact manifold $M$ is called Kobayashi hyperbolic if any
holomorphic map ${\mathbb{C}}{\>\longrightarrow\>}M$ is constant.
For an introduction to the hyperbolic geometry, please see [L].
As an application of 3.1, we obtain the following result.
Theorem 3.13: Let $M$ be an irreducible holomorphic symplectic manifold in one
of 4 known classes known, that is, a deformation of a Hilbert scheme of points
on K3, a deformation of generalized Kummer variety, or a deformation of one of
two examples by O’Grady. Then $M$ is not Kobayashi hyperbolic.
Proof: From Brody’s lemma [L] it follows that a limit of non-hyperbolic
manifolds is again non-hyperbolic. Therefore, it would suffice to find a dense
set of non-hyperbolic manifolds within the moduli space. A hyperkähler
manifold admitting a holomorphic Lagrangian fibration is non-hyperbolic,
because it contains abelian subvarieties. As follows from 1.3, all known types
of hyperkähler manifolds admit a deformation which has a Lagrangian fibration.
By 3.1, such deformations are dense in the moduli.
It is conjectured that all hyperkähler and Calabi-Yau manifolds are not
hyperbolic. The strongest result about non-hyperbolicity of hyperkähler
manifolds so far was due to F. Campana, who proved in [Cam] that any twistor
family of a hyperkähler manifold has at least one fiber which is non-
hyperbolic.
Acknowledgments. We are grateful to V. Nikulin for many conversations on
bilinear forms, Valery Gritsenko for a helpful email, and to F. Bogomolov for
his remarks. The first named author thanks the Laboratory of Algebraic
Geometry and its Applications for their kind invitation and hospitality during
her stay in Moscow in June 2012. We are thankful to the Simons Center for
Geometry and Physics and to John Morgan for inviting the second named author
and for making our work more enjoyable. Many thanks to Frédéric Campana,
Simone Diverio, Klaus Hulek and Daniel Huybrechts for a discussion of non-
hyperbolicity of hyperkähler manifolds.
## References
* [AV] Sasha Anan’in, Misha Verbitsky, Any component of moduli of polarized hyperkaehler manifolds is dense in its deformation space, arXiv:1008.2480, 17 pages, 4 figures.
* [Bea] Beauville, A. Varietes Kähleriennes dont la première classe de Chern est nulle. J. Diff. Geom. 18 (1983) 755 - 782.
* [Bes] Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987)
* [Bo1] Bogomolov, F. A., On the decomposition of Kähler manifolds with trivial canonical class, Math. USSR-Sb. 22 (1974) 580 - 583.
* [Bo2] F. Bogomolov, Hamiltonian Kähler manifolds, Sov. Math. Dokl. 19 (1978) 1462 - 1465.
* [Bou] Boucksom, S., Higher dimensional Zariski decompositions, Ann. Sci. Ecole Norm. Sup. (4) 37 (2004) no. 1, 45 - 76, arXiv:math/0204336
* [Cam] F. Campana, An application of twistor theory to the nonhyperbolicity of certain compact symplectic Kähler manifolds J. Reine Angew. Math., 425:1-7, 1992.
* [Cas] J. W. S. Cassels, Rational Quadratic Forms, Dover Publications (2008).
* [Cat] F. Catanese, A Superficial Working Guide to Deformations and Moduli, arXiv:1106.1368, 56 pages.
* [DP] Demailly, J.-P., Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, math.AG/0105176 also in Annals of Mathematics, 159 (2004) 1247 - 1274.
* [DPS1] Jean-Pierre Demailly, Thomas Peternell, Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geometry 3 (1994) 295 - 345.
* [DPS2] Jean-Pierre Demailly, Thomas Peternell, Michael Schneider, Pseudo-effective line bundles on compact Kähhler manifolds, International Journal of Math. 6 (2001) 689 - 741.
* [F] Fujiki, A. On the de Rham Cohomology Group of a Compact Kähler Symplectic Manifold, Adv. Stud. Pure Math. 10 (1987) 105-165.
* [GHS] Gritsenko, V., Hulek, K., Sankaran, G. K., Moduli spaces of irreducible symplectic manifolds, Compositio Mathematica 146 (2010) no. 2, 404 - 434, arXiv:0802.2078
* [G] Mark Gross, The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations, arXiv:0802.3407, 44 pages.
* [HL] R. Harvey, B. Lawson, Calibrated geometries, Acta Math. 148 (1982) 47 - 157.
* [Hu1] Huybrechts, Daniel, Compact hyperkähler manifolds, Calabi-Yau manifolds and related geometries, Universitext, Springer-Verlag, Berlin, 2003, Lectures from the Summer School held in Nordfjordeid, June 2001, pp. 161-225.
* [Hu2] Huybrechts, D., Finiteness results for hyperkähler manifolds, J. Reine Angew. Math. 558 (2003) 15 - 22, arXiv:math/0109024
* [Hw] Jun-Muk Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds, arXiv:0711.3224, Inventiones mathematicae, vol. 174, issue 3, pp. 625 - 644.
* [KZ] Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrations, arXiv:math/0011041, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 203 - 263.
* [L] S. Lang, Introduction to complex hyperbolic spaces, Springer, New York, 1987
* [McL] McLean, R.C. Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998) 705 - 747.
* [Ma1] D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, alg-geom/9709033, math.AG/9903045, also in Topology 38 (1999), No. 1, 79-83. Addendum, Topology 40 (2001) No. 2, 431 - 432.
* [Ma2] Matsushita, D., Higher direct images of Lagrangian fibrations, Amer. J. Math. 127 (2005) arXiv:math/0010283.
* [Ma3] D. Matsushita, On deformations of Lagrangian fibrations, arXiv: 0903.2098v1 [math.AG].
* [Ni] V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR, Izv. 14 (1980) 103 - 167.
* [R] A. Rapagnetta, Topological invariants of O’Grady’s six dimensional irreducible symplectic variety, Math. Zeitsch., 256, No. 1 (2007) 1-34.
* [Saw] Sawon, J., Abelian fibred holomorphic symplectic manifolds, Turkish Jour. Math. 27 (2003) no. 1, 197 - 230, math.AG/0404362.
* [Ser] Serre, J.-P., Cours d’arithmétique, Presses Universitaires de France (1970).
* [SYZ] A. Strominger, S.-T. Yau, and E. Zaslow, Mirror Symmetry is T -duality, Nucl. Phys. B479 (1996) 243 - 259.
* [V1] Verbitsky, M., A global Torelli theorem for hyperkähler manifolds, arXiv: 0908.4121, 47 pages.
* [V2] Verbitsky, M., Cohomology of compact hyperkähler manifolds. alg-geom electronic preprint 9501001, 89 pages, LaTeX.
* [V3] Verbitsky, M., Hyperkahler SYZ conjecture and semipositive line bundles, arXiv:0811.0639, 21 pages, GAFA 19, No. 5 (2010) 1481-1493.
* [V4] Misha Verbitsky, Parabolic nef currents on hyperkaehler manifolds, arXiv:0907.4217, 22 pages.
* [Vi] Viehweg, E., Quasi-projective Moduli for Polarized Manifolds, Springer-Verlag, Berlin, Heidelberg, New York, 1995, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 30, also available at http://www.uni-due.de/$\widetilde{\phantom{a}}$mat903/books.html
Ljudmila Kamenova
Department of Mathematics, 3-115
Stony Brook University
Stony Brook, NY 11794-3651, USA
Misha Verbitsky
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University HSE,
7 Vavilova Str. Moscow, Russia
|
arxiv-papers
| 2012-08-22T21:08:49 |
2024-09-04T02:49:34.478930
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ljudmila Kamenova, Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1208.4626"
}
|
1208.4665
|
arxiv-papers
| 2012-08-23T04:37:01 |
2024-09-04T02:49:34.485437
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "De-Zi Liu, Shuo Yuan, Yu Lu and Tong-Jie Zhang",
"submitter": "Dezi Liu",
"url": "https://arxiv.org/abs/1208.4665"
}
|
|
1208.4788
|
# Boundedness of sublinear operators on weighted Morrey spaces and
applications
Zunwei Fu , Shanzhen Lu and Shaoguang Shi∗ Zunwei Fu
Department of Mathematics
Linyi University
Linyi 276005
P. R. China [email protected] Shanzhen Lu
School of Mathematical Sciences
Beijing Normal University
Beijing, 100875
P. R. China [email protected] Shaoguang Shi
Department of Mathematics
Linyi University
Linyi 276005
P. R. China [email protected]
###### Abstract.
We study the boundedness of some sublinear operators on weighted Morrey spaces
under certain size conditions. These conditions are satisfied by most of the
operators in harmonic analysis, such as the Hardy-Littlewood maximal operator,
Calderón-Zygmund singular integral operator, Bochner-Riesz means at the
critical index, oscillatory singular operators, singular integral operators
with oscillating kernels and so on. As applications, the regularity in
weighted Morrey spaces of strong solutions to nondivergence elliptic equations
with VMO coefficients are established.
###### Key words and phrases:
Weighted Morrey space; singular integral operator; commutator.
###### 2000 Mathematics Subject Classification:
Primary 42B20; Secondary 42B25.
This work was partially supported by NSF of China (Grant Nos. 10901076,
10931001 and 11171345), NSF of Shandong Province (Grant No. ZR2012AQ026),
Beijing Natural Science Foundation (Grant No. 1102023), Program for Changjiang
Scholars and Innovative Research Team in University. This work was also
supported by the Key Laboratory of Mathematics and Complex System(Beijing
Normal University), Ministry of Education, China.
∗Corresponding author.
## 1\. Introduction and main results
As is well known that Morrey [34] introduced the classical Morrey spaces to
investigate the local behavior of solutions to second order elliptic partial
differential equations(PDE). We recall its definition as
$M_{p,q}(\mathbb{R}^{n})=\left\\{f:\|f\|_{M_{p,q}(\mathbb{R}^{n})}=\sup_{B\subset\mathbb{R}^{n}}\left(\frac{1}{|B|^{1-\frac{p}{q}}}\int_{B}|f(x)|^{p}dx\right)^{\frac{1}{p}}<\infty\right\\},$
where $f\in L_{loc}^{p}(\mathbb{R}^{n})$ and $1\leq p\leq q<\infty.$ Here and
after, $B$ denotes any balls in $\mathbb{R}^{n}$. $M_{p,q}(\mathbb{R}^{n})$
was an expansion of $L^{p}(\mathbb{R}^{n})$ in the sense that
$M_{p,p}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$. Morrey found that many
properties of solutions to PDE can be attributed to the boundedness of some
operators on Morrey spaces.
Maximal functions and singular integrals play a key role in harmonic analysis
since maximal functions could control crucial quantitative information
concerning the given functions, despite their larger size, while singular
integrals, Hilbert transform as it’s prototype, nowadays intimately connected
with PDE, operator theory and other fields.
Let $f\in L_{loc}(\mathbb{R}^{n})$. The Hardy-Littlewood(H-L) maximal function
of $f$ is defined by
$Mf(x)=\sup_{B\ni x}\frac{1}{|B|}\int_{B}|f(y)|dy.$
The Calderón-Zygmund(C-Z) singular integral operator is defined by
$Tf(x)=p.v.\int_{\mathbb{R}^{n}}K(x-y)f(y)\,dy,$
where $K$ is a C-Z kernel [18]. Chiarenza and Frasca [11] obtained the
boundedness of H-L maximal function $Mf(x)$ and C-Z singular integral operator
$T$ on $M_{p,q}(\mathbb{R}^{n})$. For some works on the boundedness for the
multilinear C-Z singular integral operators on Morrey type spaces, see e.g
[19].
Let $0<\alpha<n$. The fractional integral is defined by
$I_{\alpha}f(x)=\int_{\mathbb{R}^{n}}\frac{f(y)}{|y-x|^{n-\alpha}}dy.$
An early impetus to the study of fractional integrals originated from the
problem of fractional derivation, see e.g. [3] and [38]. Besides it’s
contributions to harmonic analysis, fractional integrals also play an
essential role in many other fields. The Hardy-Littlewood-Soblev inequality
about fractional integral is still an indispensable tool to establish time-
space estimates for the heat semigroup of nonlinear evolution equations, for
some of this work, see e.g. [22]. In recent times, the applications to Chaos
and Fractal have become another motivation to study fractional integrals, see
e.g. [25] and [28]. The boundedness of $I_{\alpha}$ on
$M_{p,q}(\mathbb{R}^{n})$ was first established by Adams in [1].
On the other hand, it is very important to study weighted estimates for these
operators in harmonic analysis. It is well known that $M$ is a bounded
operator on $L^{p}(w)$[35] with $w\in A_{p}$, $1<p<\infty$. For any non-
negative locally functions $w$ and any Lebesgue measurable function $f$, we
set
$\|f\|_{L^{p}(w)}=\left(\int_{\mathbb{R}^{n}}|f(x)|^{p}w(x)dx\right)^{1/p}$
and if $w\equiv 1$, we denote $\|f\|_{L^{p}(w)}$ simply by
$\|f\|_{L^{p}(\mathbb{R}^{n})}$. For the weighted $L^{p}(w)$ estimates and
weighted weak (1.1) type estimates for $T$, see [18]. In [36], the authors
obtained the corresponding weighted boundedness on weighted $L^{p}$ spaces for
$I_{\alpha}$ with $w\in A_{(p,q)}(1\leq p,q<\infty)$. Here and after,
$A_{p}(1\leq p<\infty)$ and $A_{(p,q)}(1\leq p,q<\infty)$ denote the
Muckenhoupt classes [35].
In [26], Komori and Shirai introduced a weighted Morrey space, which is a
natural generalization of weighted Lebesgue space, and investigated the
boundedness of classical operators in harmonic analysis, that is, the H-L
maximal operator $M$, the C-Z singular integral operator $T$ and the
fractional integral $I_{\alpha}$. Let $1\leq p<\infty,$ $0<\lambda<1$ and $w$
be a function. Then the weighted Morrey space $M_{p,\lambda}(w)$ is defined by
$M_{p,\lambda}(w)=\left\\{f:\|f\|_{M_{p,\lambda}(w)}=\sup_{B}\left(\frac{1}{w(B)^{\lambda}}\int_{B}|f(x)|^{p}w(x)dx\right)^{\frac{1}{p}}<\infty\right\\}.$
It is obviously that if $w=1,\lambda=1-\frac{p}{q}$, then
$M_{p,\lambda}(w)=M_{p,q}(\mathbb{R}^{n})$. For $w\in A_{p}(1\leq p<\infty)$,
if $\lambda=0,$ then $M_{p,0}(w)=L^{p}(w)$ and if $\lambda=1,$
$M_{p,1}(w)=L^{\infty}(w)$.
In the fractional case, we need to consider a weighted Morrey space with two
weights which also introduced by Komori and Shirai in [26]. Let $1\leq
p<\infty$, $0<\lambda<1$. For two weights $w_{1}$ and $w_{2}$,
$M_{p,\lambda}(w_{1},w_{2})=\left\\{f:\|f\|_{M_{p,k}(w_{1},w_{2})}=\sup_{B}\left(\frac{1}{w_{2}(B)^{\lambda}}\int_{B}|f(x)|^{p}w_{1}(x)dx\right)^{\frac{1}{p}}<\infty\right\\}.$
If $w_{1}=w_{2}=w$, then we denote
$M_{p,\lambda}(w_{1},w_{1})=M_{p,\lambda}(w_{2},w_{2})=M_{p,\lambda}(w)$.
In [49], Wang obtained some estimates for Bochner-Riesz means operators on
$M_{p,\lambda}(w)$ by the similar method as in [26]. In this paper, we shall
establish some boundedness for some sublinear operators on $M_{p,\lambda}(w)$
and $M_{p,\lambda}(w_{1},w_{2})$, which includes, as particular cases, the
known results in [26] and [49]. Applications to the strong solutions of
nondivergence elliptic equations with VMO coefficients are also given.
Let $D_{k}=\\{x\in\mathbb{R}^{n}:|x|\leq 2^{k}\\}$ and $A_{k}=D_{k}/D_{k-1}$
for $k\in Z$. Let $\chi_{k}=\chi_{A_{k}}$ for $k\in Z$, where $\chi_{E}$ is
the characteristic function of the set $E$. Our mean results are as follows:
###### Theorem 1.1.
Suppose that a sublinear operator $\mathcal{T}$ satisfies the size conditions
(1.1) $|\mathcal{T}f(x)|\leq C\|f\|_{L^{1}(\mathbb{R}^{n})}/|x|^{n},$
when $\operatorname{supp}f\subseteq A_{k}$ and $|x|\geq 2^{k+1}$ with $k\in Z$
and
(1.2) $|\mathcal{T}f(x)|\leq C2^{-kn}\|f\|_{L^{1}(\mathbb{R}^{n})},$
when $\operatorname{supp}f\subseteq A_{k}$ and $|x|\leq 2^{k-1}$ with $k\in
Z$. Then we have
$(a)$ If $\mathcal{T}$ is bounded on $L^{p}(w)$ with $w\in A_{p}(1<p<\infty)$,
then $\mathcal{T}$ is bounded on $M_{p,\lambda}(w)$.
$(b)$ If $\mathcal{T}$ is bounded from $L^{1}(w)$ to $L^{1,\infty}(w)$ with
$w\in A_{1}$, then there exist constant $C>0$ such that for all $\mu>0$ and
all $B$,
$w(\\{x\in
B:\mathcal{T}f(x)>\mu\\})\leq{C}/{\mu}\|f\|_{M_{1,\lambda}(w)}w(B)^{\lambda}.$
It is easy to check that the H-L maximal function $M(f)$ satisfies the
hypotheses of Theorem 1.1.
We say $b$ is a $BMO$ function, which means
$\|b\|_{BMO}=\|b^{\sharp}\|_{L^{\infty}}<\infty$, where $b^{\sharp}(x)$ is
sharp maximal function
$b^{\sharp}(x)=\sup_{B}\frac{1}{|B|}\int_{B}\left|f(y)-f_{B}\right|dy,$
where the supreme is taken over all balls $B\subset\mathbb{R}^{n}$ and
$f_{B}=\frac{1}{|B|}\int_{B}f(y)dy.$ For $1<p<\infty$, there is a close
relation between $BMO$ and $A_{p}$ weights
$BMO=\left\\{\alpha\log w:w\in A_{p},\alpha\geq 0\right\\}.$
Given a operator $N$ acting on functions and given a function $b$, the
commutator $[b,N]$ is formally defined as
$N_{b}f=[b,N]f=bN(f)-N(bf).$
There is a great amount of works that deal with the topic of commutators of
different operators with $BMO$ functions on Lesbugue spaces. The first results
on this commutator were obtained by Coifman, Rochberg and Weiss [15] in their
study of certain factorization theorems for generalized Hardy spaces. They
show that $N_{b}f$ is bounded on $L^{p}(\mathbb{R}^{n}),1<p<\infty$, if and
only if $b\in BMO$ when $N$ is a classical singular integral operator with
smooth kernel. For some classical weighted boundedness of $N_{b}$ on
$L^{p}(w)$ with $w\in A_{p},1<p<\infty$, see e.g. [6], [7]. It is well known
that the commutators formed by $BMO$ functions and the fractional integral
$I_{\alpha}$, the C-Z singular integral $T$ are all bounded on weighted
$L^{p}$ spaces [44]. In [26], the authors also established the weighted
boundedness for $T_{b}$ and $I_{\alpha,b}$ on weighted Morrey spaces. In this
paper, we extend the results of [26] and obtain
###### Theorem 1.2.
Let $1<p<\infty$, $w\in A_{p}$ and a sublinear operator
$\mathcal{\overline{T}}$ satisfies the conditions
(1.3) $|\mathcal{\overline{T}}f(x)|\leq
C\int_{\mathbb{R}^{n}}\frac{|f(y)|}{|x-y|^{n}}dy,\,\,\,x\notin\emph{supp}f$
for any integral function $f$ with compact support. Then we have
$(a)$ If $\mathcal{\overline{T}}$ is bounded on $L^{p}(w)$, then
$\mathcal{\overline{T}}$ is bounded on $M_{p,\lambda}(w)$.
$(b)$ If $\mathcal{\overline{T}}_{b}$ is bounded $L^{p}(w)$ with $b\in
BMO(\mathbb{R}^{n})$, then $\mathcal{\overline{T}}_{b}$ is bounded on
$M_{p,\lambda}(w)$
It is worth pointing out that (1.3), which implies the size conditions in
Theorem 1.1, is satisfied by many operators in harmonic analysis, such as C-Z
singular integral operator, the Carleson maximal operator, C. Fefferman’s
singular multiplier operator, R. Fefferman’s singular integral operator and so
on, see e.g. [40] and [45].
Besides the H-L maximal operators and C-Z singular integral operators,
oscillatory integral operators have been an essential part of harmonic
analysis; three chapters are devoted to them in the celebrated Stein’s book
[46]. Many important operators in harmonic analysis are some versions of
oscillatory integrals, such as the Fourier transform, the Bochner-Riesz means,
the Radon transform [39] in CT technology and so on. For a more complete
account on oscillatory integrals in classical harmonic analysis, we would like
to refer the interested reader to [29], [30], [32] and references therein.
Another early impetus for the study of oscillatory integrals came with their
application to number theory [5]. In more recent times, the operators
fashioned from oscillatory integrals, such as pseudo-differential operator in
PDE become another motivation to study them. Based on the estimates of some
kinds of oscillatory integrals, one can establish the well-posedness theory of
a class of dispersive equations, for some of this works, we refer to [14],
[23] and [24].
In 1987, Ricci and Stein [40] introduced one class of oscillatory integrals
$T_{O}f(x)=\mathrm{p.v.}\int_{\mathbb{R}^{n}}e^{iP(x,y)}K(x-y)f(y)\,dy,$
which initially defined for smooth function $f$ with compact support. Here
$P(x,y)$ is a real valued polynomial defined on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$, and $K\in
C^{1}(\mathbb{R}^{n}\setminus\\{0\\})$ is a C-Z kernel. In [40], Ricci and
Stein established the boundedness of $T_{O}$ on
$L^{p}(\mathbb{R}^{n})(1<p<\infty)$. In 1992, Lu and Zhang [33] gave the
boundedness of $T_{O}$ on $L^{p}(w)(1<p<\infty)$ with $w\in A_{p}$. For the
case $p=1$, Chanillo and Christ [8] gave a weak (1,1) type estimates while
Sato obtained the weighted version of [8] in [42]. It is easy to see that
$T_{O}$ satisfies Theorem 1.2.
The Bochner-Riesz mean operators of order $\delta>0$ in $\mathbb{R}^{n}(n\geq
2)$ are defined initially for Schwartz functions in terms of Fourier
transforms by
$(T_{R}^{\delta}f)^{\wedge}(\xi)=(1-\frac{|\xi|^{2}}{R^{2}})_{+}^{\delta}\hat{f}(\xi),$
where $\hat{f}$ denotes the Fourier transform of $f$. These operators were
first introduced by Bochner [4] in connection with summation of multiple
Fourier series and played an important role in harmonic analysis.
$T_{R}^{\delta}$ can be expressed as convolution operators
$T_{R}^{\delta}f(x)=(f\ast\phi_{\frac{1}{R}})(x)$, where
$\phi(x)=[(1+|\cdot|^{2})_{+}^{\delta}]^{\wedge}(x)$. It is well know that the
kernel $\phi$ can be represented as [31]:
$\phi(x)=\pi^{-\delta}\Gamma(\delta+1)|x|^{-(n/2+\delta)}J_{n/2+\delta}(2\pi|x|),$
where $J_{\mu}(t)$ is the Bessel function. In [43], Shi and Sun obtained the
weighted $L^{p}$ boundedness of $T_{R}^{\delta}$ while Vargas given the
weighted weak (1,1) type estimates in [48]. By the well known boundedness
criterion for the commutators of linear operators, which was obtained by
Alvarez, Bagby, Kurtz and Pérez [2], we see that the commutator
$T_{R,b}^{\delta}$ is also bounded on $L^{p}(w)$ with $w\in A_{p}$ for all
$1<p<\infty.$ From the asymptotic properties of the Bessel function, we can
deduce that when $\delta=(n-1)/2$, the critical index,
$|\phi(x)|\leq\frac{C}{(1+|x|)^{n}}$, which implies that $T_{R}^{\delta}$
satisfies the condition of Theorem 1.2.
Given a positive real number $0<a\neq 1$, the oscillating kernel $K_{a}$ is
defined by [10]:
$K_{a}(x)=(1+|x|)^{-1}e^{i|x|^{a}}.$
The convolution operator $T_{a}=K_{a}\ast f$ and closely related weakly
singular operators and multiplier operators have been studied by many authors,
see e.g. [9], [16] and [21]. In [10], Chanillo, Kurtz and Sampson had given
the weighted weak (1,1) type estimates and weighted $L^{p}(1<p<\infty)$
estimates for $T_{a}$. It is easy to see that $T_{a}$ satisfies Theorem 1.2.
###### Theorem 1.3.
Let $0<\alpha<n$, $1<p<\frac{n}{\alpha}$
,$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and $1<p\leq q\leq\infty$. Suppose
that a sublinear operator $\mathcal{T}_{\alpha}$ satisfies the size conditions
(1.4) $|\mathcal{T}_{\alpha}f(x)|\leq
C|x|^{-(n-\alpha)}\|f\|_{L^{1}(\mathbb{R}^{n})}$
when $\operatorname{supp}f\subseteq A_{k}$ and $|x|\geq 2^{k+1}$ with $k\in Z$
and
(1.5) $|\mathcal{T}_{\alpha}f(x)|\leq
C2^{-k(n-\alpha)}\|f\|_{L^{1}(\mathbb{R}^{n})}$
when $\operatorname{supp}f\subseteq A_{k}$ and $|x|\leq 2^{k-1}$ with $k\in
Z$. Then we have
$(a)$ If $\mathcal{T}_{\alpha}$ maps $L^{p}(w^{p})$ into $L^{q}(w^{q})$ with
$w\in A_{(p,q)}$, then $\mathcal{T}_{\alpha}$ is bounded from
$M_{p,\lambda}(w^{p},w^{q})$ to $M_{q,q\lambda/p}(w^{q})$.
$(b)$ If $\mathcal{T}_{\alpha}$ is bounded from $L^{1}(w)$ to
$L^{q,\infty}(w^{q})$ with $w\in A_{(1,q)}$, then there exist constant $C>0$
such that for all $\mu>0$ and all ball $B$,
$w^{q}(\\{x\in
B:\mathcal{T}_{\alpha}f(x)>\mu\\})\leq{C}/{\mu^{q}}\|f\|_{M_{1,\lambda}(w,w^{q})}^{q}w^{q}(B)^{q\lambda}.$
The fractional maximal operator $M_{\alpha}$ is defined by
$M_{\alpha}f(x)=\sup_{B\ni
x}\frac{1}{|B|^{1-\alpha/n}}\int_{B}|f(y)|dy,\,\,\,\,0<\alpha<n.$
$M_{\alpha}$ satisfies the hypotheses of Theorem 1.3 since the pointwise
inequality $M_{\alpha}f(x)\leq I_{\alpha}(|f|)(x)$ for $0<\alpha<n$.
###### Theorem 1.4.
Let $p,q,\alpha,w$ be as the same as that of Theorem 1.3 and a sublinear
operator $\mathcal{\overline{T}}_{\alpha}$ satisfies the conditions
(1.6) $|\mathcal{T}_{\alpha}f(x)|\leq
C\int_{\mathbb{R}^{n}}\frac{|f(y)|}{|x-y|^{n-\alpha}}dy,\,\,\,x\notin\emph{supp}f$
for any integral function $f$ with compact support.
$(a)$ If $\mathcal{\overline{T}}_{\alpha}$ maps $L^{p}(w^{p})$ into
$L^{q}(w^{q})$, then $\mathcal{\overline{T}}_{\alpha}$ is bounded from
$M_{p,\lambda}(w^{p},w^{q})$ to $M_{q,q\lambda/p}(w^{q})$.
$(b)$ If $\mathcal{\overline{T}}_{\alpha,b}$ maps $L^{p}(w^{p})$ into
$L^{q}(w^{q})$ with $b\in BMO(\mathbb{R}^{n})$, then
$\mathcal{\overline{T}}_{\alpha,b}$ is bounded from
$M_{p,\lambda}(w^{p},w^{q})$ to $M_{q,q\lambda/p}(w^{q})$.
We remarks that fractional integral $I_{\alpha}$ and oscillatory fractional
integral of Ricci and Stein’s [40] are all examples of operators which
satisfies (1.6). For the corresponding boundedness in unweighted cases of the
sublinear operators on Herz space, we refer to [20] and [27].
We end this section with the outline of this paper. In Section 2, we give the
proofs of Theorem 1.1-Theorem 1.4. Section 3 contains some applications of
Theorem 1.1-Theorem 1.4. Throughout this paper, the letter $C$ is used for
various constants, and may change from one occurrence to another. All balls
are assumed to have their sides paralled to the coordinate axes.
$B=B(x_{0},r)$ denotes the ball centered at $x_{0}$ and with radius $r$ and
$\lambda B=B(x_{0},\lambda r)$.
## 2\. Proofs of the main results
Our methods are adopted from [17] in the case of the Lebesgue measure and from
[26] dealing with the classical operators. Before the proof of Theorem 1.1, we
give some properties of $A_{p}$ weights.
###### Lemma 2.1.
[18] Let $1\leq p<\infty$, and $w\in A_{p}$. Then the following statements are
true
$\mathrm{(a)}$ There exists a constant $C$ such that
(2.1) $w(2B)\leq Cw(B).$
When $w$ satisfies this condition, we say $w$ satisfies doubling condition.
$\mathrm{(b)}$ There exists a constant $C>1$ such that
(2.2) $w(2B)\geq Cw(B).$
When $w$ satisfies this condition, we say $w$ satisfies reverse doubling
condition.
$\mathrm{(c)}$ There exist two constant $C$ and $r>1$ such that the following
reverse Hölder inequality holds for every ball $B\subset\mathbb{R}^{n}$
(2.3) $\left(\frac{1}{|B|}\int_{B}w(x)^{r}dx\right)^{\frac{1}{r}}\leq
C\left(\frac{1}{|B|}\int_{B}w(x)dx\right).$
$\mathrm{(d)}$ For all $\lambda>1,$ we have
(2.4) $w(\lambda B)\leq C\lambda^{np}w(B).$
$\mathrm{(e)}$ There exist two constant $C$ and $\delta>0$ such that for any
measurable set $Q\subset B$
(2.5) $\frac{w(Q)}{w(B)}\leq C\left(\frac{|Q|}{|B|}\right)^{\delta}.$
If $w$ satisfies (2.5), we say $w\in A_{\infty}$.
### 2.1. Proof of Theorem 1.1
Let $1<p<\infty$, $w\in A_{p}$ and $0<\lambda<1$. We first give the proof of
(a), which suffices to show that
(2.6) $\frac{1}{w(B)^{\lambda}}\int_{B}|\mathcal{T}f(x)|^{p}w(x)dx\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}.$
For a fixed ball $B=B(x_{0},r)$, there is no loss of generality in assuming
$r=1$. We decompose $f=f\chi_{2B}+f\chi_{(2B)^{c}}:=f_{1}+f_{2}$. Since
$\mathcal{T}$ is a sublinear operator, so we get
$\frac{1}{w(B)^{\lambda}}\int_{B}|\mathcal{T}f(x)|^{p}w(x)dx\leq\frac{1}{w(B)^{\lambda}}\int_{B}(|\mathcal{T}f_{1}(x)|^{p}+|\mathcal{T}f_{2}(x)|^{p})w(x)dx:=I+II.$
Using the fact that $\mathcal{T}$ is bounded on $L^{p}(w)$, we can easily get
(2.7)
$I\leq\frac{1}{w(B)^{\lambda}}\int_{\mathbb{R}^{n}}|\mathcal{T}f_{1}(x)|^{p}w(x)dx\leq\frac{C}{w(B)^{\lambda}}\int_{2B}|f(x)|^{p}w(x)dx\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}.$
We are now in a position to estimate the term $II$. We conclude from $w\in
A_{p}$ that
$\displaystyle\int_{(2B)^{c}}|f(y)|dy$ $\displaystyle\leq
C\sum_{k=1}^{\infty}\int_{2^{k+1}B/2^{k}B}|f(y)|dy$ $\displaystyle\leq
C\sum_{k=1}^{\infty}\left(\int_{2^{k+1}B}|f(y)|^{p}w(y)dy\right)^{1/p}\left(\int_{2^{k+1}B}w(y)^{-p^{\prime}/p}dy\right)^{1/p^{\prime}}$
$\displaystyle\leq
C\|f\|_{M_{p,\lambda}(w)}\sum_{k=1}^{\infty}\frac{|2^{k+1}B|}{w(2^{k+1}B)^{1-\lambda/p}}.$
By (1.2), we have
(2.8)
$\begin{split}II&\leq\frac{C}{w(B)^{\lambda}}2^{-knp}\int_{B}\|f_{2}\|_{L^{1}(\mathbb{R}^{n})}^{p}w(x)dx\\\
&\leq\frac{C}{w(B)^{\lambda-1}}2^{-knp}\left(\int_{(2B)^{c}}|f(y)|dy\right)^{p}\\\
&\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}\left(\sum_{k=1}^{\infty}\frac{w(B)^{(1-\lambda)/p}}{w(2^{k+1}B)^{(1-\lambda)/p}}\right)^{p}\\\
&\leq C\|f\|_{M_{p,\lambda}(w)}^{p}.\end{split}$
Here we use (2.1) in the last inequality above. Combing (2.7) with (2.8), we
get (2.6), which yields the proof of (a).
Now, we have a position to give the proof of (b), which is similar to that of
(a). We want to set up the following inequality
$\sup_{\mu>0}\frac{\mu}{w(B)^{\lambda}}w\left(\\{x\in
B:|\mathcal{T}f(x)|>\mu\\}\right)\leq C\|f\|_{M_{1,\lambda}(w)}^{p}.$
Decompose $f=f\chi_{2B}+f\chi_{(2B)^{c}}:=f_{1}+f_{2}$ with $B$ as that of
(a). For any given $\mu>0$, we write
$\displaystyle w\left(\\{x\in B:|\mathcal{T}f(x)|>\lambda\\}\right)$
$\displaystyle\leq w\left(\left\\{x\in
B:|\mathcal{T}f_{1}(x)|>{\mu}/{2}\right\\}\right)+w\left(\left\\{x\in
B:|\mathcal{T}f_{2}(x)|>{\mu}/{2}\right\\}\right)$ $\displaystyle:=J+JJ.$
An application of (2.1) and the weighted weak (1,1) type estimates for
$\mathcal{T}$ yield that
$J\leq
w\left(\left\\{x\in\mathbb{R}^{n}:|\mathcal{T}f_{1}(x)|>{\mu}/{2}\right\\}\right)\leq{C}/{\mu}\|f\|_{M_{1,\lambda}}(w)w(B)^{\lambda}.$
Next we turn to deal with the term $JJ.$ An elementary estimate shows
$JJ\leq\frac{C}{\mu}\int_{\left\\{x\in
B:|\mathcal{T}f(x)|>\frac{\mu}{2}\right\\}}|\mathcal{T}f_{2}(x)|w(x)dx.$
Applying $(\ref{(1.2)})$, we conclude that
$|\mathcal{T}f_{2}(x)|\leq C2^{-kn}\int_{(2B)^{c}}|f(y)|dy\leq
C\sum_{k=1}^{\infty}2^{-kn}\int_{2^{k+1}B}|f(y)|dy.$
Hölder’s inequality and the $A_{1}$ condition imply that
$\displaystyle JJ$
$\displaystyle\leq\frac{C}{\mu}\sum_{k=1}^{\infty}2^{-kn}\int_{2^{k+1}B}|f(y)|w(y)dy$
$\displaystyle\leq\frac{C}{\mu}\|f\|_{M_{1,k}}(w)\sum_{k=1}^{\infty}2^{kn(\lambda-1)}w(B)^{\lambda}$
$\displaystyle\leq\frac{C}{\mu}\|f\|_{M_{1,k}}(w)w(B)^{\lambda}.$
Then, we have completed the proof of (b). ∎
### 2.2. Proof of Theorem 1.2
The proof of Theorem 1.2 depend heavily on the following remarks about $BMO$
functions.
###### Lemma 2.2.
[47] Let $1\leq p<\infty$, $b\in BMO(\mathbb{R}^{n})$. Then for any ball
$B\subset\mathbb{R}^{n}$, the following statements are true
$\mathrm{(a)}$ There exist constants $C_{1}$, $C_{2}$ such that for all
$\alpha>0$
(2.9) $\left|\\{x\in B:|b(x)-b_{B}|>\alpha\\}\right|\leq
C_{1}|B|e^{-C_{2}\alpha/\|b\|_{BMO(\mathbb{R}^{n})}}.$
The inequality $(\ref{3.9})$ is also called John-Nirenberg inequality.
$\mathrm{(b)}$
(2.10) $|b_{2^{\lambda}B}-b_{B}|\leq 2^{n}\lambda\|b\|_{BMO(\mathbb{R}^{n})}.$
###### Lemma 2.3.
[37] Let $w\in A_{\infty}$. Then the following statements are equivalent
$\mathrm{(a)}$
(2.11)
$\|b\|_{BMO(\mathbb{R}^{n})}\sim\sup_{B}\left(\frac{1}{|B|}\int_{B}|b(x)-b_{B}|^{p}dx\right)^{\frac{1}{p}}.$
$\mathrm{(b)}$
(2.12)
$\|b\|_{BMO(\mathbb{R}^{n})}\sim\sup_{B}\inf_{a\in\mathbb{R}}\frac{1}{|B|}\int_{B}|b(x)-a|dx.$
$\mathrm{(c)}$
(2.13) $\|b\|_{BMO(w)}=\sup_{B}\frac{1}{w(B)}\int_{B}|b(x)-b_{B,w}|w(x)dx.$
where $BMO(w)=\\{b:\|b\|_{BMO(w)}<\infty\\}$ and
$b_{B,w}=\frac{1}{w(B)}\int_{B}b(y)w(y)dy.$
###### Lemma 2.4.
Let $b\in BMO(\mathbb{R}^{n})$, $B=B(x_{0},r)$, $0<\lambda<1$ and
$1<p<\infty$. Then the inequality
(2.14)
$\left(\int_{|x_{0}-y|>2r}\frac{|f(y)|}{|x_{0}-y|^{n}}|b_{B,w}-b(y)|dy\right)^{p}w(B)^{1-\lambda}\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}\|b\|_{BMO(\mathbb{R}^{n})}^{p}.$
holds for every $y\in(2B)^{c}$, where $(2B)^{c}=\mathbb{R}^{n}/2B.$
###### Proof.
Using Hölder’s inequality to the left-hand-side of $(\ref{3.14})$, we have
$\displaystyle\left(\int_{|x_{0}-y|>2r}\frac{|f(y)|}{|x_{0}-y|^{n}}|b_{B,w}-b(y)|dy\right)^{p}w(B)^{1-\lambda}$
$\displaystyle\leq\left(\sum_{j=1}^{\infty}\int_{2^{j}r<|x_{0}-y|<2^{j+1}r}\frac{|f(y)|}{|x_{0}-y|^{n}}|b_{B,w}-b(y)|dy\right)^{p}w(B)^{1-\lambda}$
$\displaystyle\leq\left(\sum_{j=1}^{\infty}\frac{1}{|2^{j}B|}\int_{2^{j+1}B}|f(y)||b_{B,w}-b(y)|dy\right)^{p}w(B)^{1-\lambda}$
$\displaystyle\leq
C\left[\sum_{j=1}^{\infty}\frac{1}{|2^{j}B|}\left(\int_{2^{j+1}B}|f(y)|^{p}w(y)dy\right)^{\frac{1}{p}}\left(\int_{2^{j+1}B}|b_{B,w}-b(y)|^{p^{\prime}}w(y)^{1-p^{\prime}}dy\right)^{\frac{1}{p^{\prime}}}\right]^{p}w(B)^{1-\lambda}$
$\displaystyle\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}\left[\sum_{j=1}^{\infty}\frac{w(2^{j+1}B)^{\frac{\lambda}{p}}}{|2^{j}B|}\left(\int_{2^{j+1}B}|b_{B,w}-b(y)|^{p^{\prime}}w(y)^{1-p^{\prime}}dy\right)^{\frac{1}{p^{\prime}}}\right]^{p}w(B)^{1-\lambda}.$
For the simplicity of analysis, we denote $A$ as
$\left(\int_{2^{j+1}B}|b_{B,w}-b(y)|^{p^{\prime}}w(y)^{1-p^{\prime}}dy\right)^{\frac{1}{p^{\prime}}}.$
By an elementary estimate, we have
$\displaystyle A$
$\displaystyle\leq\left(\int_{2^{j+1}B}(|b_{2^{j+1}B,w^{1-p^{\prime}}}-b(y)|+|b_{2^{j+1}B,w^{1-p^{\prime}}}-b_{B,w}|)^{p^{\prime}}w(y)^{1-p^{\prime}}dy\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq\left\|\frac{|b_{2^{j+1}B,w^{1-p^{\prime}}}-b(\cdot)|+|b_{2^{j+1}B,w^{1-p^{\prime}}}-b_{B,w}|}{w(\cdot)}\right\|_{L^{p^{\prime}}(w)}$
$\displaystyle\leq\left(\int_{2^{j+1}B}|b_{2^{j+1}B,w^{1-p^{\prime}}}-b(y)|w(y)^{1-p^{\prime}}dy\right)^{\frac{1}{p^{\prime}}}+|b_{2^{j+1}B,w^{1-p^{\prime}}}-b_{B,w}|w^{1-p^{\prime}}(2^{j+1}B)^{\frac{1}{p^{\prime}}}$
$\displaystyle=:A_{1}+A_{2}.$
For the term $A_{1}$, Lemma 2.3 implies
(2.15) $A_{1}\leq
C\|b\|_{BMO(w^{1-p^{\prime}})}w^{1-p^{\prime}}(2^{j+1}B)^{\frac{1}{p^{\prime}}}\leq
Cw^{1-p^{\prime}}(2^{j+1}B)^{\frac{1}{p^{\prime}}}.$
To deal with $A_{2}$, by $(\ref{3.10})$, we have
$\begin{split}|b_{2^{j+1}B,w^{1-p^{\prime}}}-b_{B,w}|&\leq|b_{2^{j+1}B,w^{1-p^{\prime}}}-b_{2^{j+1}B}|+|b_{2^{j+1}B}-b_{B}|+|b_{B}-b_{B,w}|\\\
&\leq\frac{1}{w^{1-p^{\prime}}(2^{j+1}B)}\int_{2^{j+1}B}|b(y)-b_{2^{j+1}B}|w(y)^{1-p^{\prime}}dy+2^{n}(j+1)\|b\|_{BMO(\mathbb{R}^{n})}\\\
&\quad\quad\quad\quad+\frac{1}{w(B)}\int_{B}|b(y)-b_{B}|w(y)dy\\\
&:=A_{21}+A_{22}+A_{23}.\end{split}$
Combining $(\ref{3.5})$ with $(\ref{3.9})$, we have
$\displaystyle A_{23}$ $\displaystyle=\frac{1}{w(B)}\int_{0}^{\infty}w(\\{x\in
B:|b(y)-b_{B}|>\alpha\\})d\alpha$ $\displaystyle\leq
C\int_{0}^{\infty}e^{-C_{2}\alpha\delta/\|b\|_{BMO(\mathbb{R}^{n})}}d\alpha$
$\displaystyle\leq C.$
In the same manner we can see that
$A_{21}\leq C.$
It follows immediately that
(2.16) $A_{2}\leq
C(2^{n}(j+1)+2)w^{1-p^{\prime}}(2^{j+1}B)^{\frac{1}{p^{\prime}}}.$
As a by-product of $(\ref{3.15})$ and $(\ref{3.16})$, we have
$A\leq C(j+1)w^{1-p^{\prime}}(2^{j+1}B)^{\frac{1}{p^{\prime}}}.$
Then, applying $(\ref{3.2})$, the proof of $(\ref{3.14})$ based on the
following observation
$\displaystyle\left[\sum_{j=1}^{\infty}\frac{w(2^{j+1}B)^{\frac{k}{p}}}{|2^{j}B|}\left(\int_{2^{j+1}B}|b(y)-b_{B,w}|^{p^{\prime}}w(y)^{1-p^{\prime}}dy\right)^{\frac{1}{p^{\prime}}}\right]^{p}w(B)^{1-k}$
$\displaystyle\leq
C\left[\sum_{j=1}^{\infty}\frac{w(B)^{\frac{1-k}{p}(j+1)}}{w(2^{j+1}B)^{\frac{1-k}{p}}}\right]^{p}=C.$
∎
Now, we come back to the proof of Theorem 1.2. (a) is trivial since (1.3)
satisfies Theorem 1.1. We only need to give the proof of (b). The task is now
to find a constant $C$ such that for fixed ball $B=B(x_{0},1)$, we can obtain
(2.17)
$\frac{1}{w(B)^{\lambda}}\int_{B}\left|\mathcal{\overline{T}}_{b}f(x)\right|^{p}w(x)dx\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}.$
We decompose $f=f\chi_{2B}+f\chi_{(2B)^{c}}:=f_{1}+f_{2},$ and consider the
corresponding splitting
$\displaystyle\int_{B}\left|\mathcal{\overline{T}}_{b}f(x)\right|^{p}w(x)dx$
$\displaystyle\leq
C\left(\int_{B}|\mathcal{\overline{T}}_{b}f_{1}(x)|^{p}w(x)dx+\int_{B}|\mathcal{\overline{T}}_{b}f_{2}(x)|^{p}w(x)dx\right)$
$\displaystyle=:K+KK.$
It follows from the $L^{p}(w)$ boundedness of $\mathcal{\overline{T}}_{b}$ and
$w\in A_{p}$ that
(2.18) $K\leq C\int_{2B}|f(x)|^{p}w(x)dx\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}w(B)^{\lambda}.$
Then a further use of $(\ref{(1.3)})$ derives that
$\displaystyle\left|\mathcal{\overline{T}}_{b}f_{2}(x)\right|^{p}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{n}}\frac{|f_{2}(y)||b(x)-b(y)|}{|x-y|^{n}}dy\right)^{p}$
$\displaystyle\leq
C\left(\int_{|x_{0}-y|>2}\frac{|f(y)|}{|x_{0}-y|^{n}}\\{|b(x)-b_{B,w}|+|b_{B,w}-b(y)|\\}dy\right)^{p}.$
where $b_{B,w}=\frac{1}{w(B)}\int_{B}b(x)w(x)dx$. Then, we have
$\displaystyle KK$ $\displaystyle\leq
C\left(\int_{|x_{0}-y|>2}\frac{|f(y)|}{|x_{0}-y|^{n}}dy\right)^{p}\int_{B}|b(x)-b_{B,w}|^{p}w(x)dx$
$\displaystyle\quad\quad\quad\quad+C\left(\int_{|x_{0}-y|>2}\frac{|f(y)|}{|x_{0}-y|^{n}}|b(y)-b_{B,w}|dy\right)^{p}w(B)$
$\displaystyle:=KK_{1}+KK_{2}.$
A further use of Lemma 2.4, we get
$KK_{2}\leq C\|f\|_{M_{p,\lambda}(w)}^{p}w(B)^{\lambda}.$
To get the desired estimate, we are led to estimate the term $KK_{1}$. This
estimate will be done via (2.1), (2.3) and Lemma 2.3. In fact,
$\begin{split}KK_{1}&=\left(\sum_{j=1}^{\infty}\int_{2^{j}<|x_{0}-y|<2^{j+1}}\frac{|f(y)|}{|x_{0}-y|^{n}}dy\right)^{p}\int_{B}|b(x)-b_{B,w}|^{p}w(x)dx\\\
&\leq\left(\sum_{j=1}^{\infty}\frac{1}{|2^{j}B|}\int_{2^{j+1}B}|f(y)|dy\right)^{p}\int_{B}|b(x)-b_{B,w}|^{p}w(x)dx\\\
&\leq
C\sum_{j=1}^{\infty}\frac{1}{|2^{j}B|}\left(\frac{1}{w(2^{j+1}B)^{\lambda}}\int_{2^{j+1}B}|f(y)|^{p}w(y)dy\right)^{{1}/{p}}\\\
&\quad\quad\quad\quad\times
w(2^{j+1}B)^{{\lambda}/{p}}\left(\int_{2^{j+1}B}w(y)^{-{1}/{p-1}}dy\right)^{\frac{p-1}{p}}\int_{B}|b(x)-b_{B,w}|^{p}w(x)dx\\\
&\leq
C\|f\|_{M_{p,\lambda}(w)}\left(\sum_{j=1}^{\infty}\frac{|2^{j+1}B|^{-\frac{1}{p}}}{|2^{j}B|}\left(\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}w(y)dy\right)^{-{1}/{p}}w(2^{j+1}B)^{{\lambda}/{p}}\right)^{p}\\\
&\quad\quad\quad\quad\times\int_{B}|b(x)-b_{B,w}|^{p}w(x)dx\\\ &\leq
C\|f\|_{M_{p,\lambda}(w)}^{p}\|b\|_{BMO(\mathbb{R}^{n})}^{p}\sum_{j=1}^{\infty}\left(\frac{w(B)^{{(1-\lambda)}/{p}}}{w(2^{j+1}B)^{{(1-k)}/{p}}}\right)^{p}w(B)^{\lambda}\\\
&\leq C\|f\|_{M_{p,\lambda}(w)}^{p}w(B)^{\lambda}\end{split}$
Hence
(2.19) $KK\leq C\|f\|_{M^{p,\lambda}(w)}^{p}w(B)^{\lambda}.$
Combing (2.18), (2.19), we obtain (2.17), which is the desired conclusion.∎
### 2.3. Proof of Theorem 1.3
We can use the similar argument as the proof of Theorem 1.1. For the proof of
(a), it suffices to show that
(2.20)
$\frac{1}{w^{q}(B)^{q\lambda/p}}\int_{B}|\mathcal{T}_{\alpha}f(x)|^{q}w(x)^{q}dx\leq
C\|f\|_{M_{p,\lambda}(w^{p},w^{q})}^{q}.$
For a fixed ball $B=B(x_{0},1)$, we decompose
$f=f\chi_{2B}+f\chi_{(2B)^{c}}:=f_{1}+f_{2}$. Since $\mathcal{T}_{\alpha}$ is
a sublinear operator, so we get
$\frac{1}{w^{q}(B)^{q\lambda/p}}\int_{B}|\mathcal{T}_{\alpha}f(x)|^{q}w(x)^{q}dx\leq\frac{1}{w^{q}(B)^{q\lambda/p}}\int_{B}(|\mathcal{T}_{\alpha}f_{1}(x)|^{q}+|\mathcal{T}_{\alpha}f_{2}(x)|^{q})w^{q}(x)dx:=L+LL.$
To estimate the term $L$, using the fact that $\mathcal{T}_{\alpha}$ is
bounded from $L^{p}(w^{p})$ to $L^{q}(w^{q})$ with $w\in A_{(p,q)}$, we can
get
$\int_{B}|\mathcal{T}_{\alpha}f_{1}(x)|^{q}w^{q}(x)dx\leq
C\|f\|_{M_{p,\lambda}(w^{p},w^{q})}^{q}w^{q}(B)^{q\lambda/p},$
which implies that
$L\leq C\|f\|_{M_{p,\lambda}(w^{p},w^{q})}.$
For the term $LL$. By the similar argument as that of Theorem 1.1, we obtain
$\displaystyle LL$ $\displaystyle\leq
C\sum_{k}\left(2^{-k(n-\alpha)}\int_{A_{k}}|f(y)|dy\right)^{q}w^{q}(B)^{1-q\lambda/p}$
$\displaystyle\leq
C\sum_{k}\left(2^{-k(n-\alpha)}\|f\|_{M_{p,\lambda}(w^{p},w^{q})}|2^{k+1}B|^{1-\alpha/n}\frac{1}{w^{q}(2^{k+1}B)^{1/q-\lambda/p}}\right)^{q}w^{q}(B)^{1-q\lambda/p}$
$\displaystyle\leq
C\|f\|_{M_{p,\lambda}(w^{p},w^{q})}^{q}\left(\sum_{k=1}^{\infty}\frac{w^{q}(B)^{(1/q-\lambda/p)}}{w^{q}(2^{k+1}B)^{(1/q-\lambda/p)}}\right)^{q}$
$\displaystyle\leq C\|f\|_{M_{p,\lambda}(w^{p},w^{q})}^{q}.$
We have completed the proof of (a).
We shall omit the proof of (b) since we can prove by using $A_{(1,q)}$
condition and the weak type estimates of $\mathcal{T}_{\alpha}$. ∎
### 2.4. Proof of Theorem 1.4
The proof of Theorem 1.4 is similar to that of Theorem 1.2, except using $w\in
A_{(p,q)}$.
## 3\. Applications
In this section, we shall give some applications of our main results to
nondivergence elliptic equations. Dirichlet problem on the second order
elliptic equation in nondivergence form is
(3.1)
$\left\\{\begin{array}[]{ll}&Lu=\sum_{i,j}^{n}a_{ij}(x)u_{x_{i}}u_{x_{j}}=f\,\quad
a.e\quad in\,\,\,\,\Omega,\\\ &u=0\,\quad
on\quad\partial\Omega.\end{array}\right.$
Here $x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$, $\Omega$ is a bounded domain
of $\mathbb{R}^{n}$. The coefficients $(a_{ij})_{i,j=1}^{n}$ of $L$ are
symmetric and uniformly elliptic, i.e., for some $\nu\geq 1$ and every
$\xi\in\mathbb{R}^{n}$, $a_{ij}(x)=a_{ji}(x)$ and
$\nu^{-1}|\xi|^{2}\leq\sum_{i,j=1}^{n}a_{ij}(x)\xi_{i}\xi_{j}\leq\nu|\xi|^{2}$
with a.e.$x\in\Omega$. In [17], Fan, Lu and Yang investigate the regularity in
$M_{p,\lambda}(\Omega)$ of the strong solution to (3.1) with $a_{ij}\in
VMO(\Omega)$, the space of the functions of vanishing mean oscillation
introduced by Sarason in [41]. The main method of [17] is based on integral
representation formulas established in [12] and [13] for the second
derivatives of the solution $u$ to (3.1), and on the theories of singular
integrals and sublinear commutators in Morrey spaces.
By extending some theorems of [17] to weighted versions, we can establish
regularity in weighted Morrey spaces of strong solutions to nondivergence
elliptic equations with $VMO$ coefficients. Theorem 1.2 is just the weighted
version of the important theorem-Theorem 2.1 in [17]. For the complement of
our paper, we take another important theorem-Theorem 2.3 of [17] to state
this. All other proofs of the corresponding theorems are straightforward.
Let
$\mathbb{R}_{+}^{n}=\\{x=(x^{\prime},x_{n}):x^{\prime}=(x_{1},\cdots,x_{n-1})\in\mathbb{R}^{n-1},x_{n}>0\\}$,
$L^{p}_{+}(w)=L^{p}(w,\mathbb{R}_{+}^{n})$ and
$M_{p,\lambda}^{+}=M_{p,\lambda}(w,\mathbb{R}_{+}^{n})$. To establish the
boundary estimates of the solutions to (3.1), we need the following general
theorem for sublinear operators.
###### Theorem 3.1.
Let $1<p<\infty$, $0<\lambda<1$, $w\in A_{p}$, $\tilde{x}=(x^{\prime},-x_{n})$
for $x=(x^{\prime},x_{n})\in\mathbb{R}_{+}^{n}$. If a sublinear operator
$\mathfrak{T}$ is bounded on $L^{p}_{+}(w)$ for any $f\in L^{1}_{+}(w)$ with
compact support and satisfies
(3.2) $|\mathfrak{T}f(x)|\leq
C\int_{\mathbb{R}_{+}^{n}}\frac{|f(y)|}{|\tilde{x}-y|^{n}}dy,$
then $\mathfrak{T}$ is bounded on $M_{p,\lambda}^{+}(w)$.
###### Proof.
Let $z\in\mathbb{R}_{+}^{n}$ and $\delta>0$. Set
$B_{\delta}^{+}(z)=B_{\delta}(z)\cap\mathbb{R}_{+}^{n}$, where
$B_{\delta}(z)=\\{y\in\mathbb{R}^{n}:|z-y|<\delta\\}$. We consider two cases
Case 1. $0\leq z_{n}<2\delta.$ In this case, we write
$f(y)=f(y)\chi_{B_{2^{4}\delta}^{+}(z)}(y)+\sum_{i=4}^{\infty}f(y)\chi_{B_{2^{i+1}\delta}^{+}(z)/B_{2^{i}\delta}^{+}(z)}(y)\equiv\sum_{i=3}^{\infty}f_{i}(y).$
Therefore, by the $L^{p}_{+}(w)$ boundedness of $\mathfrak{T}$ and (3.2), we
obtain
$\displaystyle\frac{1}{w(B_{\delta}^{+})^{\lambda/p}}\left(\int_{B_{\delta}^{+}}|\mathfrak{T}f(x)|^{p}w(x)dx\right)^{1/p}$
$\displaystyle\leq\frac{1}{w(B_{\delta}^{+})^{\lambda/p}}\sum_{i=3}^{\infty}\left(\int_{B_{\delta}^{+}}|\mathfrak{T}f_{i}(x)|^{p}w(x)dx\right)^{1/p}$
$\displaystyle\leq\frac{C}{w(B_{\delta}^{+})^{\lambda/p}}\|f_{3}\|_{L^{p}_{+}(w)}+\frac{C}{w(B_{\delta}^{+})^{\lambda/p}}\sum_{i=4}^{\infty}\left(\int_{B_{\delta}^{+}}\left(\int_{B_{2^{i+1}\delta}^{+}(z)/B_{2^{i}\delta}^{+}(z)}\frac{|f(y)|}{|\tilde{x}-y|^{n}}dy\right)^{p}w(x)dx\right)^{1/p}$
$\displaystyle\leq
C\|f\|_{M_{p,\lambda}^{+}(w)}+C\sum_{i=4}^{\infty}\frac{1}{(2^{i}\delta)^{n}}\left(\int_{B_{2^{i+1}\delta}^{+}}|f(y)|dy\right)w(B_{\delta}^{+})^{1-\lambda/p}$
$\displaystyle\leq
C\|f\|_{M_{p,\lambda}^{+}(w)}\left(1+\sum_{i=4}^{\infty}\frac{w(B_{\delta}^{+})^{1-\lambda/p}}{w(B_{2^{i+1}\delta}^{+})^{1-\lambda/p}}\right)$
$\displaystyle\leq C\|f\|_{M_{p,\lambda}^{+}(w)}.$
In the last inequality, we use Lemma 2.1 in Section 2.
Case 2. There exists $i\in\mathbb{N}$ such that $2^{i}\delta\leq
z_{n}<2^{i+1}\delta$. In this case, we write
$f(y)=f(y)\chi_{B_{2^{i+1}\delta}^{+}(z)}(y)+\sum_{j=1}^{\infty}f(y)\chi_{B_{2^{i+j+4}\delta}^{+}(z)}(y)\equiv\sum_{j=0}^{\infty}f_{j}(y).$
By (3.2) and Lemma 2.1, we have
$\displaystyle\frac{1}{w(B_{\delta}^{+})^{\lambda/p}}\left(\int_{B_{\delta}^{+}}|\mathfrak{T}f(x)|^{p}w(x)dx\right)^{1/p}$
$\displaystyle\leq\frac{C}{w(B_{\delta}^{+})^{\lambda/p}}\left(\int_{B_{\delta}^{+}}\left(\int_{B_{2^{i+4}\delta}^{+}(z)}\frac{|f(y)|}{|\tilde{x}-y|^{n}}dy\right)^{p}w(x)dx\right)^{1/p}$
$\displaystyle\quad+\frac{C}{w(B_{\delta}^{+})^{\lambda/p}}\sum_{j=1}^{\infty}\left(\int_{B_{\delta}^{+}}\left(\int_{B_{2^{i+j+4}\delta}^{+}(z)/B_{2^{i+j+3}\delta}^{+}(z)}\frac{|f(y)|}{|\tilde{x}-y|^{n}}dy\right)^{p}w(x)dx\right)^{1/p}$
$\displaystyle\leq\frac{C}{w(B_{\delta}^{+})^{\lambda/p}}\frac{1}{(2^{i}\delta)^{n}}\left(\int_{B_{\delta}^{+}}\left(\int_{B_{2^{i+4}\delta}^{+}(z)}|f(y)|dy\right)^{p}w(x)dx\right)^{1/p}$
$\displaystyle\quad+\frac{C}{w(B_{\delta}^{+})^{\lambda/p}}\sum_{j=1}^{\infty}\frac{1}{(2^{i+j}\delta)^{n}}\left(\int_{B_{\delta}^{+}}\left(\int_{B_{2^{i+j+4}\delta}^{+}(z)}|f(y)|dy\right)^{p}w(x)dx\right)^{1/p}$
$\displaystyle\leq
C\|f\|_{M_{p,\lambda}^{+}(w)}\left(\frac{w(B_{\delta}^{+})^{1-\lambda/p}}{w(B_{2^{i+4}\delta}^{+})^{1-\lambda/p}}+\sum_{j=1}^{\infty}\frac{w(B_{\delta}^{+})^{1-\lambda/p}}{w(B_{2^{i+j}\delta}^{+})^{1-\lambda/p}}\right)$
$\displaystyle\leq C\|f\|_{M_{p,\lambda}^{+}(w)}.$
∎
### Acknowledgements
The authors thank Professor Heping Liu for the valuable suggestions.
## References
* [1] D. Adams, A note on Riesz potentials, _Duke Math. J._ , 42(1975), 765–778.
* [2] J. Acrarez, R. Bagby, D. Kurtz and C. Perez, Weighted estimates for commutators of linear operators, _Studia Math._ , 104(2)(1993), 195–209.
* [3] K. Andersen and E. Sawyer, Weighted norm inequalities for the Riemann- Liouville and Weyl fractional integral operators, _Trans. Amer. Math. Soc._ , 308(2)(1988), 547–558.
* [4] S. Bochner, Summation of multiple Fourier series by spherical means, _Trans. Amer. Math. Soc._ , 40(1936), 175–207.
* [5] J. Bourgain, Fourier retsriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I. Schrödinger equations; II. The KdV-equation, _Geom. Funct. Anal._ , 3(1993), 107-156; 209–262.
* [6] S. Chanillo, A note on commutators, _Indiana Univ. Math. J._ , 31(1982), 7–16.
* [7] S. Chanillo, Remarks on commutators of pseudo-differential operators, multidimensional complex analysis and partial differential equations (Serra Negra 1995) Contemporary Mathematics of the AMS, v. 205, Providence RI, 1997, 33–37.
* [8] S. Chanillo and M. Christ, Weak (1,1) bounds for oscillatory integrals, _Duke. Math. J._ , 55(1987), 141–155.
* [9] S. Chanillo, D. Kurtz and G. Sampson, Weighted $L^{p}$ estimates for oscillating kernels, _Ark. Math._ , 21(1983), 233–257.
* [10] S. Chanillo, D. Kurtz and G. Sampson, Weighted weak (1,1) and weighted $L^{p}$ estimates for oscillating kernels, _Trans. Amer. Math. Soc._ , 295(1986), 127–145.
* [11] F. Chiarenza and M. Frasca , Morrey spaces and Hardy-Littlewood maximal function, _Rend. Math. Appl._ , 7(7)(1987), 273–279.
* [12] F. Chiarenza, M. Frasca and P.Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinous coefficients, _Ricerche Mat._ , XL(1991), 149–168.
* [13] F. Chiarenza, M. Frasca and P.Longo, $W^{2,p}$ solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, _Trans. Amer. Math. Soc._ , 336(1993), 841–853.
* [14] R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-difiréntiels, Astérisque 57, 1978.
* [15] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, _Ann. of Math._ , 103(1976), 611–635.
* [16] V. Drobot, A. Naparstek and G. Sampson, $(L^{p},L^{q})$ mapping properties of convolution transforms, _Studia Math._ , 55(1976), 41–70.
* [17] D. Fan, S. Lu and D. Yang, Regularity in Morry spaces of strong solutions to nondivergence elliptic equations with VMO coefficients, _Georgian Math. J._ , 5(5)(1998), 425–440.
* [18] L. Grafakos, Classical and mordern Fourier analysis, Dearson Education, Inc, Upper saddle river, NJ, 2004.
* [19] L. Grafakos and R. Torres, Multilinear Calderón-Zygmund theory, _Adv. Math._ , 165(2002), 124-164.
* [20] E. Hernandez and D. Yang, Interpolation of Herz spaces and applications, _Math. Nachr_ , 205(1999), 69-87.
* [21] W. Jurkat and G. Sampson, The complete solution to the $(L^{p},L^{q})$ mapping problem for a class of oscillating kernels, _Indiana Univ. Math. J._ , 30(1981), 403–413.
* [22] T. Kato, Strong $L^{p}$ solutions of the Navier-Stokes equations in $\mathbb{R}^{m}$ with applications to weak solutions, _Math. Z._ , 187(1984), 471–480.
* [23] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, _Indiana Univ. Math. J._ , 40(1991), 33–69.
* [24] C. Kenig, G. Ponce and L. Vega, Well-Posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, _Comm. Pure Appl. Math._ , 4(1993), 527–620.
* [25] K. Kolwankar and A. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, _Chaos_ , 6(4)(1996), 505–513.
* [26] Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, _Math. Nachr._ , 282(2009), 219–231.
* [27] X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, _Illinois J. of Math._ , 40(1996), 484–501.
* [28] Y. Liang and W. Su, The relationship between fractal dimensions of a type of fractal functions and the order of their fractional calculus, _Chaos, Solitions and Fractals_ , 34(2007), 682–692.
* [29] S. Lu, Multilinar oscillatory integrals with Calderón-Zygmund kernel, _Sci. China(Ser.A),_ 42(1999), 1039–1046.
* [30] S. Lu, A class of oscillatory Integrals, _Int. J. Appl. Math. Sci._ , 2(1)(2005), 42–58.
* [31] S. Lu and K. Wang, Bochner-Risez means(in Chinese), Beijing Normal Univ Press, Beijing, 1988.
* [32] S. Lu, Y. Ding and D. Yan, Singular integrals and Related Topics, World Scientific Publishing, Singapore, 2007.
* [33] S. Lu and Y. Zhang, Weighted norm inequality of a class of oscillatory singular operators, _Chin. Sci. Bull._ , 37(1992), 9–13.
* [34] C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, _Trans. Amer. Math. Soc.,_ 43(1938), 126–166.
* [35] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal functions, _Trans. Amer. Math. Soc._ , 165(1972), 207–226.
* [36] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, _Trans. Amer. Math. Soc._ , 192(1974), 261–274.
* [37] B. Muckenhoupt and R. Wheeden , Weighted bounded mean oscillation and the Hilbert transform, _Studia Math._ , 54(1976), 221–237.
* [38] K. Oldham and J. Spanier, The fractional calculus, New York, Academic press, 1974.
* [39] D. Phong and E. Stein, Singular integrals related to the Radon transform and boundary value problems, _Proc. Nat. Acad. USA._ , 80(1983),7697–7701.
* [40] F. Ricci and E. Stein, Harmonic analysis on nilpotant groups and singular integrals I: Oscillatory Integrals, _J. Funct. Anal._ , 73(1987), 179–194.
* [41] D. Sarason, On functions of vanishing mean oscillation, _Trans. Amer. Math. Soc._ , 207(1975), 391–405.
* [42] S. Sato, Weighted weak type (1,1) estimates for oscillatory singular integrals, _Studia Math._ , 47(2000), 1–14.
* [43] X. Shi and Q. Sun, Weighted norm ineauqlities for Bochner-Risez operators and singular integral operators, _Proc. Amer. Math. Soc._ , 116(1992), 665–673.
* [44] C. Segovia and J. Tottra, Weighted inequalities for commutators of fractional and singular integrals, _Publ. Math._ , 35(1991), 209–235.
* [45] F. Soria and G. Weiss, A remark on singular integrals and power weights, _Indiana Univ. Math. J._ , 43(1994), 187–204.
* [46] E. Stein, Harmonic Analysis (real-variable methods, orthogonality, and oscillatory integrals), Princeton Univ. Press, Princeton, 1993.
* [47] A. Torchinsky, Real variable Methods in Harmonic Analysis Academic Press, San DIego, 1986.
* [48] A. Varges, Weighted weak type (1,1) bound for rough operators, _J. London Math. Soc._ , 54(1996), 297–310.
* [49] H. Wang, Some estimates for Bochner-Risez operators on weighted Morrey spaces, arXiv:1011.1622v1.
|
arxiv-papers
| 2012-08-23T16:05:30 |
2024-09-04T02:49:34.491878
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zunwei Fu, Shanzhen Lu and Shaoguang Shi",
"submitter": "Fu Zunwei",
"url": "https://arxiv.org/abs/1208.4788"
}
|
1208.4833
|
# Classification of factorial generalized down-up algebras
Stéphane Launois and Samuel A. Lopes I am grateful for the full financial
support of EPSRC first grant EP/I018549/1.Research funded by the European
Regional Development Fund through the programme COMPETE and by the Portuguese
Government through the FCT – Funda o para a Ci ncia e a Tecnologia under the
project PEst-C/MAT/UI0144/2011.
###### Abstract
We determine when a generalized down-up algebra is a Noetherian unique
factorisation domain or a Noetherian unique factorisation ring.
Keywords: generalized down-up algebra; Noetherian unique factorisation domain;
Noetherian unique factorisation ring.
2010 Mathematics Subject Classification: 16U30; 16S30.
## Introduction
Down-up algebras were introduced by Benkart and Roby in [6], motivated by the
study of certain “down” and “up” operators on posets. In this seminal paper,
the highest weight theory for a down-up algebra was developed and a parallel
was drawn between down-up algebras and enveloping algebras of Lie algebras,
based on the apparent similarity between their respective representation
theories and structural properties. Later, in [11], Cassidy and Shelton
introduced a larger class of algebras which, when defined over an
algebraically closed field, contains all down-up algebras.
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Fix
$r,s,\gamma\in\mathbb{K}$ and $f\in\mathbb{K}[x]$. The generalized down-up
algebra $L=L(f,r,s,\gamma)$ is the unital associative $\mathbb{K}$-algebra
generated by $d$, $u$ and $h$, subject to the relations:
$[d,h]_{r}+\gamma d=0,\quad[h,u]_{r}+\gamma
u=0\quad\mbox{and}\quad[d,u]_{s}+f(h)=0,$
where $[a,b]_{\lambda}:=ab-\lambda ba$. A down-up algebra can be seen as a
generalized down-up algebra, as above, with $\deg(f)=1$.
Noteworthy examples of generalized down-up algebras are the enveloping algebra
of the semisimple Lie algebra $\mathfrak{sl}_{2}$, of traceless matrices of
size $2$, which is isomorphic to $L(x,1,1,1)$, and the enveloping algebra of
the $3$-dimensional Heisenberg Lie algebra $\mathfrak{h}$, which is isomorphic
to $L(x,1,1,0)$. Another example is the quantum Heisenberg Lie algebra
$U_{q}(\mathfrak{sl}^{+}_{3})$, where $q\in\mathbb{K}^{*}$, which can be seen
as $L(x,q,q^{-1},0)$. Under a mild restriction on the parameters, the algebra
of regular functions on quantum affine $3$-space,
$\mathcal{O}_{Q}(\mathbb{K}^{*})$, is a generalized down-up algebra of the
form $L(0,r,s,0)$, with $rs\neq 0$. In [34], Smith defined a class of algebras
similar to the enveloping algebra of $\mathfrak{sl}_{2}$. Subsequently, Rueda
considered in [33] a larger family of algebras, including Smith’s algebras.
The algebras in Rueda’s family are generalized down-up algebras of the form
$L(f,1,s,1)$, and by setting $s=1$ we retrieve Smith’s algebras. Other
examples of generalized down-up algebras can be found in [4, Secs. 5 and 6].
Like down-up algebras, generalized down-up algebras display several features
of the structure and representation theory of a semisimple Lie algebra, but
their defining parameters allow enough freedom to obtain a variety of
different behaviours. An example of this is the global dimension of a
generalized down-up algebra, which can be $1$, $2$ or $3$, by [11, Thm. 3.1]
(for a down-up algebra, the global dimension is always $3$). Similarly, in
some cases the centre is reduced to the scalars, but in others it can be
large, and there are cases in which the generalized down-up algebra is finite
over its centre. Other examples of properties which hold in some generalized
down-up algebras and do not in others are: being Noetherian, being primitive,
having all finite-dimensional modules semisimple, and having a Hopf algebra
structure.
Generalized down-up algebras have been studied mostly from the point of view
of representation theory (see [6], [10], [24], [19], [11] and [29]); their
primitive ideals have been determined in [21], [28], [31], [30] and [32]. In
this paper we study generalized down-up algebras from the point of view of
noncommutative algebraic geometry, namely, we provide a complete
classification of those generalized down-up algebras which are
(noncommutative) Noetherian unique factorisation rings (resp. domains), as
defined by Chatters and Jordan in [12] and [13].
An element $p$ of a Noetherian domain $R$ is normal if $pR=Rp$. In our case, a
Noetherian domain $R$ is said to be a unique factorisation ring, Noetherian
UFR for short, if $R$ has at least one height one prime ideal, and every
height one prime ideal is generated by a normal element. If, in addition,
every height one prime factor of $R$ is a domain, then $R$ is called a unique
factorisation domain, Noetherian UFD for short. As well as the usual
commutative Noetherian UFDs, examples of Noetherian UFDs include certain group
algebras of polycyclic-by-finite groups [8] and various quantum algebras [26,
25] such as quantised coordinate rings of semisimple groups. Unfortunately,
the notion of a Noetherian UFD is not closed under polynomial extensions. To
the opposite, the notion of a Noetherian UFR is closed under polynomial
extensions. Moreover, Chatters and Jordan proved general results for skew
polynomial extensions of the type $R[x;\sigma]$ and $R[x;\delta]$. The general
case of skew polynomial extensions of type $R[x;\sigma,\delta]$ is more
intricate and only partial results have been obtained for a class of “quantum”
algebras called CGL extensions [26], which includes (generic) quantum
matrices, positive parts of quantum enveloping algebras of semisimple Lie
algebras, etc.
Going back to enveloping algebras, it follows from results of Conze in [14]
that, over the complex numbers, the universal enveloping algebra of a finite-
dimensional semisimple Lie algebra is a Noetherian UFD, and an analogous
result holds for a finite-dimensional solvable Lie algebra, by [12]. It is
thus natural to investigate the factorial properties of generalized down-up
algebras. Moreover, by considering cases in which the parameters $r$ and $s$
are roots of unity, we hope to get some insight into the behaviour of
enveloping algebras over fields of finite characteristic (see [7] and
references therein). Indeed, our analysis yields the following result, which
shows that, for generalized down-up algebras, the distinction between a
Noetherian UFR and a Noetherian UFD depends only on the existence of torsion
in the multiplicative subgroup of $\mathbb{K}^{*}$ generated by $r$ and $s$.
###### Theorem A.
Let $L=L(f,r,s,\gamma)$ be a generalized down-up algebra with $rs\neq 0$. Then
$L$ is a Noetherian UFD if and only if $L$ is a Noetherian UFR and $\langle
r,s\rangle$ is torsionfree.
Noetherian generalized down-up algebras can be viewed as iterated skew
polynomial rings as well as generalized Weyl algebras (see [22] and [11]).
They also can be described as ambiskew polynomial rings (see [21]). In his
paper [20], Jordan determined the height one prime ideals of ambiskew
polynomial rings under two additional conditions:
* •
conformality; recall that $f$ is conformal in $L(f,r,s,\gamma)$ if there
exists $g\in\mathbb{K}[h]$ such that $f(h)=sg(h)-g(rh-\gamma)$;
* •
$\sigma$-simplicity (see below for the definition of $\sigma$-simplicity).
He then applied these results in [21, Sec. 6] to determine the height one
prime ideals of down-up algebras, under certain technical restrictions arising
from [20]. Here we consider any Noetherian generalized down-up algebra and
obtain the following classification:
###### Theorem B.
Let $L=L(f,r,s,\gamma)$ be a generalized down-up algebra with $rs\neq 0$. Then
$L$ is a Noetherian UFR except if $f\neq 0$ and one of the following
conditions is satisfied:
1. (a)
$f$ is not conformal, $r$ is not a root of unity, and there exists
$\zeta\neq\gamma/(r-1)$ such that $f(\zeta)=0$;
2. (b)
$f$ is conformal, $\langle r,s\rangle$ is a free abelian group of rank $2$,
and there exists $\zeta\neq\gamma/(r-1)$ such that $f(\zeta)=0$;
3. (c)
$\gamma\neq 0$, $r=1$, $s$ is not a root of unity, and $f\notin\mathbb{K}$.
Acknowledgments. The authors would like to thank support from the _Treaty of
Windsor Programme_. They also wish to thank Christian Lomp and Paula Carvalho
for helpful discussions concerning the topics of this paper, and David Jordan
for the reference [14].
## 1 Generalized down-up algebras and Factoriality
Throughout this paper, $\mathbb{N}$ is the set of nonnegative integers,
$\mathbb{K}$ denotes an algebraically closed field of characteristic $0$ and
$\mathbb{K}^{*}$ is the multiplicative group of units of $\mathbb{K}$. If $X$
is a subset of the ring $L$ then the two-sided ideal of $L$ generated by $X$
is denoted by $\langle X\rangle$; we also write $\langle
x_{1},\ldots,x_{n}\rangle$ in place of
$\langle\\{x_{1},\ldots,x_{n}\\}\rangle$. Moreover, we denote by
$\mathcal{Z}(L)$ the centre of $L$.
Given a polynomial $f=a_{0}+a_{1}x+\cdots+a_{n}x^{n}\in\mathbb{K}[x]$, with
all $a_{i}\in\mathbb{K}$, we define the support of $f$ to be the set
$\mathrm{supp}\,(f)=\\{i\mid a_{i}\neq 0\\}$ and the degree of $f$, denoted
$\deg(f)$, as the supremum of $\mathrm{supp}\,(f)$. In particular, the zero
polynomial has degree $-\infty$. In the context of this paper, a monomial in
the variable $x$ is a (nonzero) polynomial of the form $\lambda x^{k}$, for
some $\lambda\in\mathbb{K}^{*}$ and some $k\geq 0$.
### 1.1 Noetherian generalized down-up algebras
Let $f\in\mathbb{K}[x]$ be a polynomial and fix scalars
$r,s,\gamma\in\mathbb{K}$. The generalized down-up algebra $L=L(f,r,s,\gamma)$
was defined in [11] as the unital associative $\mathbb{K}$-algebra generated
by $d$, $u$ and $h$, subject to the relations:
$\displaystyle dh-rhd+\gamma d$ $\displaystyle=0,$ (1.1) $\displaystyle hu-
ruh+\gamma u$ $\displaystyle=0,$ (1.2) $\displaystyle du-sud+f(h)$
$\displaystyle=0.$ (1.3)
When $f$ has degree one, we retrieve all down-up algebras
$A(\alpha,\beta,\gamma)$, $\alpha$, $\beta$, $\gamma\in\mathbb{K}$, for
suitable choices of the parameters of $L$.
It is well known that $L$ is Noetherian $\iff$ $L$ is a domain $\iff$ $rs\neq
0$. Thus, from now on, we will always assume $rs\neq 0$. Moreover we can view
$L$ as an iterated skew polynomial ring,
$L=\mathbb{K}[h][d;\sigma][u;\sigma^{-1},\delta],$ (1.4)
where $\sigma(h)=rh-\gamma$, $\sigma(d)=sd$, $\delta(h)=0$,
$\delta(d)=s^{-1}f(h)$. (See [11] for more details.)
To finish this section, we describe the $\mathbb{Z}$-graduation of $L$
obtained by assigning to the generators the following degrees (see [11, Sec.
4]):
$\deg(u)=1,~{}\deg(d)=-1,~{}\deg(h)=0.$ (1.5)
The decomposition $L=\oplus_{i\in\mathbb{Z}}L_{i}$ of $L$ into homogeneous
components has been described in [11, Prop. 4.1]:
$L_{0}=\mathbb{K}[h,ud]\mbox{ is the commutative polynomial algebra generated
by $h$ and $ud$},$
and
$L_{-i}=L_{0}d^{i}=d^{i}L_{0},\quad L_{i}=L_{0}u^{i}=u^{i}L_{0},\quad\mbox{
for }i>0.$ (1.6)
### 1.2 Conformality and isomorphisms
When we consider two generalized down-up algebras, say $L=L(f,r,s,\gamma)$ and
$\tilde{L}=L(\tilde{f},\tilde{r},\tilde{s},\tilde{\gamma})$, we may denote
their canonical generators by $d$, $u$, $h$ and $\tilde{d}$, $\tilde{u}$,
$\tilde{h}$, respectively, if any confusion could arise regarding which
algebra we are referring to.
###### Lemma 1.1.
The sets $\left\\{d^{i}\right\\}_{i\geq 0}$ and $\left\\{u^{i}\right\\}_{i\geq
0}$ are right and left denominator sets in $L$.
###### Proof.
See [20, 1.5]. It follows from [16, Lem. 1.4] that
$\left\\{d^{i}\right\\}_{i\geq 0}$ is a right and left denominator set in $L$.
Using the anti-automorphism that fixes $h$ and interchanges $d$ and $u$ we
obtain the corresponding statement for $\left\\{u^{i}\right\\}_{i\geq 0}$. ∎
Fix the parameters $r,s\in\mathbb{K}^{*}$, $\gamma\in\mathbb{K}$, and consider
the linear transformation $s\cdot\mathrm{1}-\sigma$ of $\mathbb{K}[h]$. We
denote the image of $p\in\mathbb{K}[h]$ under this transformation by $p^{*}$.
Specifically, $p^{*}(h)=sp(h)-p(rh-\gamma)$.
###### Lemma 1.2.
Let $L=L(f,r,s,\gamma)$, $p\in\mathbb{K}[h]$ and
$\tilde{L}=L(f-p^{*},r,s,\gamma)$. Consider the denominator sets
$D=\left\\{d^{i}\right\\}_{i\geq 0}$ in $L$,
$\tilde{D}=\\{\tilde{d}^{i}\\}_{i\geq 0}$ in $\tilde{L}$ and the corresponding
localisations $LD^{-1}$ and $\tilde{L}\tilde{D}^{-1}$.
There is an isomorphism $\phi:LD^{-1}\rightarrow\tilde{L}\tilde{D}^{-1}$,
determined by $\phi(d)=\tilde{d}$, $\phi(h)=\tilde{h}$,
$\phi(u)=\tilde{u}+p(\tilde{h})\tilde{d}^{-1}$.
###### Proof.
To show the existence of an algebra endomorphism
$\phi:L\rightarrow\tilde{L}\tilde{D}^{-1}$ as stated, the following relations
need to be checked in $L(f-p^{*},r,s,\gamma)\tilde{D}^{-1}$:
$\displaystyle\tilde{d}\tilde{h}-r\tilde{h}\tilde{d}+\gamma\tilde{d}$
$\displaystyle=0;$ (1.7)
$\displaystyle\tilde{h}\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)-r\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)\tilde{h}+\gamma\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)$
$\displaystyle=0;$ (1.8)
$\displaystyle\tilde{d}\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)-s\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)\tilde{d}+f(\tilde{h})$
$\displaystyle=0.$ (1.9)
As the first two of these relations are immediately checked, we show only
(1.9):
$\displaystyle\tilde{d}\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)$
$\displaystyle=\tilde{d}\tilde{u}+\tilde{d}p(\tilde{h})\tilde{d}^{-1}$
$\displaystyle=\tilde{d}\tilde{u}+p(r\tilde{h}-\gamma)$
$\displaystyle=s\tilde{u}\tilde{d}-(f-p^{*})(\tilde{h})+\left(sp(\tilde{h})-p^{*}(\tilde{h})\right)$
$\displaystyle=s\left(\tilde{u}+p(\tilde{h})\tilde{d}^{-1}\right)\tilde{d}-f(\tilde{h}).$
As $\phi(d)$ is a unit in $\tilde{L}\tilde{D}^{-1}$, the map $\phi$ above
extends (uniquely) to a map $\phi:LD^{-1}\rightarrow\tilde{L}\tilde{D}^{-1}$.
Now, similar considerations show the existence of an inverse map
$\psi:\tilde{L}\tilde{D}^{-1}\rightarrow LD^{-1}$, such that
$\psi(\tilde{d})=d$, $\psi(\tilde{h})=h$, $\psi(\tilde{u})=u-p(h)d^{-1}$.
Hence, $\phi$ is bijective.
∎
Given $r,s,\gamma\in\mathbb{K}$, we say that $f\in\mathbb{K}[h]$ is conformal
if there is $g$ such that $f=g^{*}$. We also say, somewhat abusively, that $f$
is conformal in $L(f,r,s,\gamma)$. Thus, if $f$ is conformal, then $LD^{-1}$
is isomorphic to $L(0,r,s,\gamma)\tilde{D}^{-1}$. In this case, in particular,
the nonzero element $z:=ud-g(h)$ is normal and satisfies the relations
$zh=hz$, $dz=szd$ and $zu=suz$.
The following results from [9] determine when a polynomial $f$ is conformal in
$L(f,r,s,\gamma)$.
###### Lemma 1.3 ([9, Lem. 1.6]).
Let $f=\sum a_{i}h^{i}$. Then $f$ is conformal in $L(f,r,s,0)$ if and only if
$s\neq r^{i}$ for all $i\in\mathrm{supp}\,(f)$. In that case, a polynomial $g$
satisfying $f(h)=sg(h)-g(rh)$ exists and is unique if we impose the additional
condition that $\mathrm{supp}\,(f)=\mathrm{supp}\,(g)$; in particular, $g$ can
be chosen so that $\deg(g)=\deg(f)$.
###### Proposition 1.4 ([9, Prop. 1.7]).
If $r\neq 1$ then $L(f,r,s,\gamma)\simeq L(\tilde{f},r,s,0)$ for some
polynomial $\tilde{f}$ of the same degree as $f$. Furthermore, $f$ is
conformal in $L(f,r,s,\gamma)$ if and only if $\tilde{f}$ is conformal in
$L(\tilde{f},r,s,0)$.
###### Proposition 1.5 ([9, Prop. 1.8]).
$f$ is conformal in $L(f,1,s,\gamma)$ except if $s=1$, $\gamma=0$ and $f\neq
0$.
### 1.3 Noetherian unique factorisation rings and domains
In this section, we recall the notions of Noetherian unique factorisation
rings and Noetherian unique factorisation domains introduced by Chatters and
Jordan (see [12, 13]).
An ideal $I$ in a ring $L$ is called principal if there exists a normal
element $x$ in $L$ such that $I=\langle x\rangle=xL=Lx$.
###### Definition 1.6.
A ring $L$ is called a Noetherian unique factorisation ring (Noetherian UFR
for short) if the following two conditions are satisfied:
1. (a)
$L$ is a prime Noetherian ring;
2. (b)
Any nonzero prime ideal in $L$ contains a nonzero principal prime ideal.
###### Definition 1.7.
A Noetherian UFR $L$ is said to be a _Noetherian unique factorisation domain_
(Noetherian UFD for short) if $L$ is a domain and each height one prime ideal
$P$ of $L$ is completely prime; that is, $L/P$ is a domain for each height one
prime ideal $P$ of $L$.
Note that the generalized down-up algebra $L=L(f,r,s,\gamma)$, with $rs\neq
0$, is Noetherian and has finite Gelfand-Kirillov dimension; so, it satisfies
the descending chain condition for prime ideals, see for example, [23, Cor.
3.16]. As, moreover, $L$ is a prime Noetherian ring, we deduce from [13] the
following result.
###### Proposition 1.8.
Let $L=L(f,r,s,\gamma)$ be a generalized down-up algebra with $rs\neq 0$. Then
$L$ is a Noetherian UFR if and only if all of its height one prime ideals are
principal.
To end this section, we recall a noncommutative analogue of Nagata’s Lemma (in
the commutative case, see [15, 19.20 p. 487]) that allows one to prove that a
ring is a Noetherian UFR or a Noetherian UFD by proving this property for
certain localisations of the ring under consideration.
If $L$ is a prime Noetherian ring and $x$ is a nonzero normal element of $L$,
we denote by $L_{x}$ the (right) localisation of $L$ with respect to the
powers of $x$.
###### Lemma 1.9 ([26, Lem. 1.4]).
Let $L$ be a prime Noetherian ring and $x$ a nonzero, nonunit, normal element
of $L$ such that $\langle x\rangle$ is a completely prime ideal of $L$.
1. (a)
If $P$ is a prime ideal of $L$ not containing $x$ and such that the prime
ideal $PL_{x}$ of $L_{x}$ is principal, then $P$ is principal.
2. (b)
If $L_{x}$ is a Noetherian UFR, then so is $L$.
3. (c)
If $L_{x}$ is a Noetherian UFD, then so is $L$.
### 1.4 Some prime ideals of $L$
In [20, 2.10], Jordan defines prime ideals $Q(P)$ which depend on certain
prime ideals $P$ of a subalgebra which, in our setting, is $\mathbb{K}[h]$. It
is easy to generalize that construction so as to include the case when $f$ is
not conformal in $L$, which we will do below. We give the details only for the
prime ideals of $\mathbb{K}[h]$ of the form $\langle h-\lambda\rangle$, with
$\lambda\in\mathbb{K}$. The only case remaining concerns $\langle 0\rangle$,
the zero ideal of $\mathbb{K}[h]$, which will not be necessary for our
discussion and carries additional technical issues.
###### Lemma 1.10.
Let $L=L(f,r,s,0)$ and write $f=\sum a_{i}h^{i}$. Then, for every $k\geq 0$,
$du^{k}=s^{k}u^{k}d-P_{k}(h)u^{k-1},$ (1.10)
where:
1. (a)
$P_{k}(h)=\sum_{i=0}^{k-1}s^{i}f(r^{-i}h)$;
2. (b)
If $f=g^{*}$ then $P_{k}(h)=s^{k}g(r^{1-k}h)-g(rh)$;
3. (c)
The coefficient of $h^{m}$ in $P_{k}(h)$ is $a_{m}k$, if $s=r^{m}$, and
$a_{m}\frac{(sr^{-m})^{k}-1}{sr^{-m}-1}$ if $s\neq r^{m}$.
In particular, if $f$ is not conformal then $P_{k}\neq 0$ for all $k>0$.
###### Proof.
Equation (1.10) along with parts (a) and (b) follow readily by induction on
$k\geq 0$. Part (c) follows from (a). Finally, if $f$ is not conformal, recall
from [9, Lem. 1.6] that there is $m$ such that $a_{m}\neq 0$ and $s=r^{m}$.
Thus, the coefficient of $h^{m}$ in $P_{k}(h)$ is nonzero, for $k>0$. ∎
Let $L=L(f,r,s,0)$. Fix $\lambda\in\mathbb{K}$ and define the $L$-module
$V_{\lambda}$ as follows. As a $\mathbb{K}$-vector space,
$V_{\lambda}=\bigoplus_{i\geq 0}\mathbb{K}v_{i}$
and the $L$-action is given by:
$\displaystyle h.v_{k}$ $\displaystyle=r^{k}\lambda v_{k}$ $\displaystyle
u.v_{k}$ $\displaystyle=v_{k+1}$ $\displaystyle d.v_{k}$
$\displaystyle=-P_{k}(r^{k-1}\lambda)v_{k-1},\ \text{for $k\geq 1$},\
\text{and}\quad d.v_{0}=0.$
Assume there is $k>0$ such that $P_{k}(r^{k-1}\lambda)=0$. Then
$\bigoplus_{i\geq k}\mathbb{K}v_{i}$ is a proper submodule of $V_{\lambda}$.
Let $k>0$ be minimal with this property, and define
$M_{\lambda}=\bigoplus_{i\geq k}\mathbb{K}v_{i}$. Thus,
$L_{\lambda}:=V_{\lambda}/M_{\lambda}$ is a finite-dimensional representation
of $L$. Let $Q_{\lambda}:=\mathrm{ann}_{L}L_{\lambda}$. By the minimality of
$k$ it is straightforward to see that $L_{\lambda}$ is simple. Thus,
$Q_{\lambda}$ is a primitive ideal; in particular, it is prime.
###### Remark 1.11.
1. (a)
$M_{\lambda}$, $L_{\lambda}$ and $Q_{\lambda}$ are defined only if there
exists $k>0$ such that $P_{k}(r^{k-1}\lambda)=0$.
2. (b)
When $f$ is conformal, this construction is a special case of the construction
in [20, 2.10], where $P$ is the ideal of $\mathbb{K}[h]$ generated by
$h-\lambda$ and $Q_{\lambda}=Q(\langle h-\lambda\rangle)$.
###### Theorem 1.12.
Let $L=L(f,r,s,0)$. Suppose $\lambda\in\mathbb{K}$ is such that
$P_{k}(r^{k-1}\lambda)=0$ for some $k>0$. Then $Q_{\lambda}$ is a non-
principal maximal ideal of $L$ containing $d^{k}$ and $u^{k}$.
Furthermore, if $P_{k}\neq 0$ for all $k>0$ (e.g., if $f$ is not conformal)
and $Q$ is any prime ideal of $L$ containing $d^{k}$ and $u^{k}$ for some
$k>0$, then there exists $\lambda\in\mathbb{K}$ such that $Q_{\lambda}$ is
defined and $Q=Q_{\lambda}$.
###### Proof.
This follows essentially as in [20, Thm. 2.12]. We give details for
completeness.
Let $\rho:L\rightarrow\mathrm{End}_{\mathbb{K}}(L_{\lambda})$ be the map which
defines the representation. Since $L_{\lambda}$ is finite-dimensional and
simple, and $\mathbb{K}$ is algebraically closed, Schur’s Lemma implies that
$\mathrm{End}_{L}(L_{\lambda})$, the centraliser algebra of $L_{\lambda}$, is
just $\mathbb{K}$. Thus, by the Jacobson Density Theorem, $\rho$ is onto and
induces an algebra isomorphism
$L/Q_{\lambda}\simeq\mathrm{End}_{\mathbb{K}}(L_{\lambda})$. As
$\mathrm{End}_{\mathbb{K}}(L_{\lambda})$ is simple, the ideal $Q_{\lambda}$ is
maximal.
If $Q$ is any prime ideal of $L$ containing $d^{k}$ and $u^{k}$ for some
$k>0$, then the proof of [20, Thm. 2.12] shows that there is $k>0$ and a prime
ideal $P$ of $\mathbb{K}[h]$ such that $P_{k}(r^{k-1}h)\in P$. As we are
assuming $P_{k}\neq 0$ for all $k>0$, and $\mathbb{K}$ is algebraically
closed, it follows that there is $\lambda\in\mathbb{K}$ such that
$P_{k}(r^{k-1}\lambda)=0$. Then, as in the proof of [20, Thm. 2.12], we have
$Q_{\lambda}\subseteq Q$, and hence, by the maximality of $Q_{\lambda}$, we
obtain $Q=Q_{\lambda}$.
Finally, $Q_{\lambda}$ is not principal because, by the definition of
$L_{\lambda}$, we have $d^{k},u^{k}\in Q_{\lambda}$. This is a general fact
concerning any generalized Weyl algebra $D(\phi,a)$ over a commutative domain
$D$ such that $0\neq a\in D$ is not a unit. (Recall, e.g. [11, Lem. 2.7], that
$L$ is a generalized Weyl algebra, where $D$ is the polynomial algebra in the
variables $h$ and $a=ud$.) Nevertheless, we give the specific details for $L$.
Assume $\xi L$ is a principal ideal of $L$ containing $u^{k}$, for some $k>0$.
Then, the equation $\xi x=u^{k}$, for $x\in L$, implies that both $\xi$ and
$x$ must be homogeneous, with respect to the $\mathbb{Z}$-grading defined in
(1.5). Assume $\xi$ has degree $n<0$. Then we can write $\xi=td^{-n}$ and
$x=t^{\prime}u^{k-n}$, for some $t,t^{\prime}\in\mathbb{K}[h,ud]$. We have:
$u^{k}=(td^{-n})(t^{\prime}u^{k-n})=t\phi^{-n}(t^{\prime})d^{-n}u^{-n}u^{k}=t\phi^{-n}(t^{\prime})\left(\prod_{i=1}^{-n}\phi^{i}(ud)\right)u^{k},$
where $\phi$ is the automorphism of $\mathbb{K}[h,ud]$ defined by $\phi(h)=rh$
and $\phi(ud)=sud-f(h)$. The above equation implies that $ud$ is a unit in
$\mathbb{K}[h,ud]$, which is a contradiction. Hence, $\xi$ has degree $n\geq
0$. Similarly, assuming that $d^{k}\in\xi L$, we conclude that $\xi$ has
degree $n\leq 0$. It follows that , if $\xi L$ contains both $u^{k}$ and
$d^{k}$, then $\xi\in\mathbb{K}[h,ud]$. But then the equation
$\xi(tu^{k})=u^{k}$, for $t\in\mathbb{K}[h,ud]$, implies that $\xi$ is a unit
and $\xi L=L$. Thus, no proper ideal of $L$ containing $u^{k}$ and $d^{k}$ can
be principal. ∎
We end this section by pointing out some principal height one prime ideals
which will also be of interest later.
###### Lemma 1.13.
Let $L=L(f,r,s,0)$. Then the normal element $h$ generates a height one,
completely prime ideal of $L$. Furthermore, if $r$ is a primitive root of
unity of order $l\geq 1$ then, for any $\lambda\in\mathbb{K}^{*}$, the central
element $h^{l}-\lambda$ generates a height one prime ideal of $L$ which is
completely prime if and only if $r=1$.
###### Proof.
First, notice that $h$ is normal, as $\gamma=0$, and generates a completely
prime ideal, as the factor algebra $L/\langle h\rangle$ is either a quantum
plane or a quantum Weyl algebra, or one of their classical analogues, in case
$s=1$. By the Principal Ideal Theorem (see [27, 4.1.11]), $\langle h\rangle$
has height one.
If $r$ is a primitive root of unity of order $l\geq 1$ then $h^{l}$ is
central. Consider the presentation
$L=\mathbb{K}[h][d;\sigma][u;\sigma^{-1},\delta]$ of $L$ as an iterated skew
polynomial ring, as given in (1.4) above, with $\sigma(h)=rh$ and
$\delta(h)=0$. It is easy to see that $(h^{l}-\lambda)\mathbb{K}[h]$ is a
$\sigma$-prime ideal of $\mathbb{K}[h]$ (i.e., it is a prime ideal in the
lattice of $\sigma$-stable ideals of $\mathbb{K}[h]$). It follows, e.g. by [5,
Prop. 2.1], that $h^{l}-\lambda$ generates a prime ideal of
$\mathbb{K}[h][d;\sigma]$. In particular, this ideal is $\sigma$-prime and
$\delta$-stable, so it follows by [5, Prop. 2.1] that $\langle
h^{l}-\lambda\rangle$ is a prime ideal of
$L=\mathbb{K}[h][d;\sigma][u;\sigma^{-1},\delta]$. Again by the Principal
Ideal Theorem, this ideal has height one.
If $l\geq 2$, then $h^{l}-\lambda$ factors nontrivially, as $\mathbb{K}$ is
algebraically closed, so $\langle h^{l}-\lambda\rangle$ is not completely
prime, by simple degree arguments. Otherwise, if $l=1$ then $r=1$ and the
factor algebra $L/\langle h-\lambda\rangle$ is again a quantum plane or a
quantum Weyl algebra, or one of their classical analogues, so in this case the
ideal $\langle h-\lambda\rangle$ is completely prime. ∎
## 2 The case $f$ not conformal
Assume $f=\sum a_{i}h^{i}$ is not conformal. Then, by Propositions 1.4 and
1.5, we can assume $\gamma=0$. By Lemma 1.3, we can write $f=f_{c}+f_{nc}$,
where $f_{c}=g^{*}$ is conformal and $f_{nc}$ is such that $s=r^{i}$ for all
$i\in\mathrm{supp}\,(f_{nc})$. Such a decomposition $f=f_{c}+f_{nc}$ is
unique, and $f_{nc}\neq 0$, as $f$ is not conformal.
###### Lemma 2.1.
Let $L=L(f,r,s,0)$ with $f$ not conformal. There is $j\in\mathrm{supp}\,(f)$
such that $s=r^{j}$ and $f_{nc}=h^{j}F$, where $0\neq
F\in\mathbb{K}[h]\cap\mathcal{Z}(L)$. Furthermore:
1. (a)
If $r$ is not a root of unity, then $F\in\mathbb{K}^{*}$;
2. (b)
If $r$ is a root of unity of order $l\geq 1$, then $F(h)=G(h^{l})$ is a
polynomial in the central indeterminate $h^{l}$.
###### Proof.
Let us write $f_{nc}=\sum_{i\in\mathrm{supp}\,(f_{nc})}a_{i}h^{i}$. Let
$j=\min\mathrm{supp}\,(f_{nc})$. Thus, $s=r^{j}$ and we can write
$f_{nc}=h^{j}F(h)$, where
$F(h)=\sum_{i\in\mathrm{supp}\,(f_{nc})}a_{i}h^{i-j}$. Given
$i\in\mathrm{supp}\,(f_{nc})$, we have $i-j\geq 0$ and $r^{i}=s=r^{j}$, so
$r^{i-j}=1$.
If $r$ is not a root of unity, then $i=j$ and $F(h)\in\mathbb{K}^{*}$.
Otherwise, if $r$ is a primitive $l$-th root of unity, then $l$ divides $i-j$
and $F(h)$ is a polynomial in $h^{l}$, which is thus central, as $\gamma=0$. ∎
###### Proposition 2.2.
Let $L=L(f,r,s,0)$, with $f$ not conformal, and consider the localisation
$LD^{-1}$, where $D=\left\\{d^{i}\right\\}_{i\geq 0}$. Then
$\left\\{h^{i}\right\\}_{i\geq 0}$ is a right and left denominator set in
$LD^{-1}$ and the localisation at this set is isomorphic to
$\hat{L}=L(F,r,1,0)$ localised at the multiplicative set generated by the
corresponding elements $\hat{d}$ and $\hat{h}$ in $\hat{L}$, where
$f=f_{c}+f_{nc}$ and $f_{nc}=h^{j}F$, as in the previous lemma.
###### Proof.
The first statement follows from the normality of $h$ in $L$. We have already
seen in Lemma 1.2 that $LD^{-1}$ is isomorphic to $\tilde{L}\tilde{D}^{-1}$,
where $\tilde{L}=L(f_{nc},r,s,0)$, under an isomorphism that maps $h$ to
$\tilde{h}$, $d$ to $\tilde{d}$ and $u$ to
$\tilde{u}+g(\tilde{h})\tilde{d}^{-1}$, where $f_{c}=g^{*}$. So it suffices to
show that $\tilde{L}$ localised at the multiplicative set generated by
$\tilde{d}$ and $\tilde{h}$ is isomorphic to the corresponding localisation of
$\hat{L}=L(F,r,1,0)$ at the multiplicative set generated by $\hat{d}$ and
$\hat{h}$.
It is easy to see that there is an algebra homomorphism
$\Phi:L(f_{nc},r,s,0)\longrightarrow L(F,r,1,0)$ such that
$\Phi(\tilde{d})=\hat{d}$, $\Phi(\tilde{h})=\hat{h}$ and
$\Phi(\tilde{u})=\hat{u}\hat{h}^{j}$. This homomorphism clearly extends to an
isomorphism when we pass to the localisation under consideration. ∎
Next, we define a (left and right) denominator set $X$ in $L(f,r,s,0)$, which
depends on $r$:
1. (a)
If $r$ is not a root of unity, then $X$ is the multiplicative set generated by
$d$ and $h$.
2. (b)
If $r$ is a root of unity of order $l\geq 1$, then $X$ is the multiplicative
set generated by $d$, $h$ and the central elements of the form
$h^{l}-\lambda$, for $\lambda\in\mathbb{K}^{*}$.
###### Proposition 2.3.
Let $L=L(f,r,s,0)$ and assume $f$ is not conformal. Then the localisation of
$L$ at the denominator set $X$ defined above is a simple algebra.
###### Proof.
By the previous result, it is enough to assume $L=L(F,r,1,0)$, where $F\neq 0$
is either a scalar (if $r$ is not a root of unity) or a polynomial in the
central indeterminate $h^{l}$ (if $r$ has order $l\geq 1$). Furthermore, since
$F$ is central and invertible in the localisation under consideration
($\mathbb{K}$ is algebraically closed), we can assume $F=1$, on replacing u by
$uF^{-1}$. The result then follows from the description below of the prime
ideals of $L(1,r,1,0)$. ∎
###### Theorem 2.4.
Let $L=L(1,r,1,0)$.
1. (a)
Assume that $r$ is not a root of unity. Then $\mathrm{Spec}(L)=\\{\langle
0\rangle,\langle h\rangle\\}$.
2. (b)
Assume that $r$ is a primitive $l$-th root of unity, for $l\geq 1$. Then
$\mathrm{Spec}(L)=\\{\langle 0\rangle,\langle h\rangle\\}\cup\\{\langle
h^{l}-\lambda\rangle~{}|~{}\lambda\in\mathbb{K}^{*}\\}.$
###### Proof.
This follows from the isomorphism
$L(1,r,1,0)\simeq\mathbb{A}_{1}(\mathbb{K})[h;\phi]$, where
$\mathbb{A}_{1}(\mathbb{K})$ denotes the first Weyl algebra over $\mathbb{K}$,
generated by $d$ and $u$, subject to the relation $ud-du=1$, and $\phi$ is the
automorphism of $\mathbb{A}_{1}(\mathbb{K})$ defined by $\phi(d)=r^{-1}d$ and
$\phi(u)=ru$. Thus, we identify the algebras $L$ and
$\mathbb{A}_{1}(\mathbb{K})[h;\phi]$. Below we sketch the proof, which relies
on the simplicity of $\mathbb{A}_{1}(\mathbb{K})$ (recall that $\mathbb{K}$
has characteristic $0$).
Firstly, all ideals listed in the statement are prime, e.g. by Lemma 1.13. We
will show that there are no other prime ideals. There is an
$\mathbb{N}$-grading on $\mathbb{A}_{1}(\mathbb{K})[h;\phi]$ such that the
homogenous component of degree $n\geq 0$ is
$\mathbb{A}_{1}(\mathbb{K})h^{n}=h^{n}\mathbb{A}_{1}(\mathbb{K})$. This
grading is, of course, different from the usual $\mathbb{Z}$-grading of $L$ we
consider in the paper, but for the remainder of this proof, this is the
grading we will consider. As usual, the degree of an element in
$\mathbb{A}_{1}(\mathbb{K})[h;\phi]$ is the maximum of the degrees of its
nonzero homogeneous components, i.e., its degree as a polynomial in $h$.
Assume $P\neq\langle 0\rangle$ is a prime ideal of
$\mathbb{A}_{1}(\mathbb{K})[h;\phi]$. Let $0\neq\xi\in P$ be a (not
necessarily homogeneous) element of minimum degree, say $n\geq 0$. Then the
set of leading coefficients of nonzero elements of $P$ of degree $n$, adjoined
with $0$, is easily seen to be an ideal of $\mathbb{A}_{1}(\mathbb{K})$. As
the latter is simple and $\xi\neq 0$, it follows that this ideal contains $1$.
Therefore, we can assume that $\xi$ is monic. By the minimality of the degree
of $\xi$ and the fact that its leading coefficient is a unit, we can use right
and left division algorithms to conclude that $P$ is principal and generated
by $\xi$, both on the right and on the left. In particular, $\xi$ is normal.
Since $\mathbb{A}_{1}(\mathbb{K})[h;\phi]$ is $\mathbb{N}$-graded, every
homogeneous constituent of $\xi$ is normal, so we will first determine the
homogeneous elements of $\mathbb{A}_{1}(\mathbb{K})[h;\phi]$ which are normal.
Assume $ah^{i}$ is normal, where $a\in\mathbb{A}_{1}(\mathbb{K})$ and $i\geq
0$. Then, as $h^{i}$ is itself normal, it follows that $a$ is normal in
$\mathbb{A}_{1}(\mathbb{K})$. Thus, $a\in\mathbb{K}$. This shows, in
particular, that the normal elements of $\mathbb{A}_{1}(\mathbb{K})[h;\phi]$
are polynomials in $h$ with coefficients in $\mathbb{K}$, but not every such
polynomial is normal, except if $r=1$. Indeed, suppose $0\neq\xi=\sum_{i\geq
0}\lambda_{i}h^{i}$ is normal, where $\lambda_{i}\in\mathbb{K}$. Then there is
$a$ such that $d\xi=\xi a$. It must then be that
$a\in\mathbb{A}_{1}(\mathbb{K})$, by degree considerations, and
$\lambda_{i}d=\lambda_{i}\phi^{i}(a)$, for all $i$. If $\lambda_{i}$ and
$\lambda_{j}$ are nonzero, then $r^{i}d=\phi^{-i}(d)=a=\phi^{-j}(d)=r^{j}d$,
so $r^{i-j}=1$. This implies that we can write $\xi=h^{k}G$, where $k\geq 0$
and either $G$ is a (nonzero) scalar, if $r$ is not a root of unity, or $G$ is
a polynomial in $h^{l}$ with scalar coefficients and nonzero constant term, if
$r$ is a primitive $l$-th root of unity.
As $h$ and $G$ are normal, and $P=\langle\xi\rangle$ is prime, either $h\in P$
or $G\in P$. If the former occurs, then $P=\langle h\rangle$. Otherwise,
$k=0$, $r$ is a primitive $l$-th root of unity, and $P=\langle
h^{l}-\lambda\rangle$, for some $\lambda\in\mathbb{K}^{*}$, as $\mathbb{K}$ is
algebraically closed and, up to a scalar, $G$ can be factored into central
polynomials of the form $h^{l}-\mu$, for $\mu\in\mathbb{K}^{*}$. This
establishes the claim. ∎
Similarly, we can define a (left and right) denominator set $Y$ in
$L(f,r,s,0)$, which can be obtained from $X$ by replacing $d$ by $u$.
Specifically:
1. (a)
If $r$ is not a root of unity, then $Y$ is the multiplicative set generated by
$u$ and $h$.
2. (b)
If $r$ is a root of unity of order $l\geq 1$ then $Y$ is the multiplicative
set generated by $u$, $h$ and the central elements of the form
$h^{l}-\lambda$, for $\lambda\in\mathbb{K}^{*}$.
###### Proposition 2.5.
Let $L=L(f,r,s,0)$ and assume $f$ is not conformal. Then the localisation of
$L$ at the denominator set $Y$ defined above is a simple algebra.
###### Proof.
Consider the isomorphism $L(f,r,s,0)\longrightarrow L(f,r^{-1},s^{-1},0)$,
defined by the correspondence $d\mapsto-s^{-1}u$, $u\mapsto d$, $h\mapsto h$.
Notice that $f$ is conformal in $L(f,r,s,0)$ if and only if $f$ is conformal
in $L(f,r^{-1},s^{-1},0)$ (if $f(h)=sg(h)-g(rh)$ then
$f(h)=s^{-1}G(h)-G(r^{-1}h)$, for $G(h)=-sg(rh)$). Thus, our claim follows
from applying our previous result to $L(f,r^{-1},s^{-1},0)$ and the
denominator set $X$, and using this isomorphism. ∎
We can now determine when $L(f,r,s,0)$ is a Noetherian UFR or a Noetherian
UFD, assuming $f$ is not conformal.
###### Theorem 2.6.
Let $L=L(f,r,s,0)$ and assume $f$ is not conformal. Then $L$ is a Noetherian
UFR, except in the case that $f$ is not a monomial and $r$ is not a root of
unity. Moreover, $L$ is a Noetherian UFD if and only if either $r=1$ or if $r$
is not a root of unity and $f$ is a monomial.
###### Proof.
Let us first identify the possible height one primes.
Let $P$ be a height one prime ideal of $L$. If $P$ does not contain any power
of $d$ or if $P$ does not contain any power of $u$ then, by Propositions 2.3
and 2.5, $P$ must contain either $h$ or $h^{l}-\lambda$, for some
$\lambda\in\mathbb{K}^{*}$ (the latter can occur only when $r$ is a primitive
$l$-th root of unity, for some $l\geq 1$), as these elements are normal. But
both $h$ and $h^{l}-\lambda$ generate prime ideals, by Lemma 1.13, so it
follows that either $P=\langle h\rangle$ or $P=\langle h^{l}-\lambda\rangle$.
Otherwise, $P$ must contain both a power of $d$ and a power of $u$. Thus,
$P=Q_{\lambda}$, for some $\lambda\in\mathbb{K}$, by Theorem 1.12. In
particular, $P_{k}(r^{k-1}\lambda)=0$ for some $k>0$ (with the notation of
Lemma 1.10). Assume that $\lambda=0$. Then $h\in Q_{0}=P$ and $P=\langle
h\rangle$, which is a contradiction, as $\langle h\rangle$ does not contain
any power of $d$. So $P=Q_{\lambda}$ for some $\lambda\in\mathbb{K}^{*}$.
To summarise, the possible height one primes of $L$ are: $\langle h\rangle$;
$\langle h^{l}-\lambda\rangle$ with $\lambda\in\mathbb{K}^{*}$ and
$Q_{\lambda}$ for some $\lambda\in\mathbb{K}^{*}$ such that
$P_{k}(r^{k-1}\lambda)=0$, for some $k>0$.
We now distinguish between three different cases.
Let us first consider the case that $f$ is a monomial. Then $P_{k}\neq 0$ is
also a monomial and hence the only possibility for $\lambda$ to satisfy
$P_{k}(r^{k-1}\lambda)=0$ is $\lambda=0$, which is a contradiction. So the
only possible height one primes of $L$ are $\langle h\rangle$ and $\langle
h^{l}-\lambda\rangle$ with $\lambda\in\mathbb{K}^{*}$. They are all principal
so that $L$ is a Noetherian UFR. Furthermore, it follows from Lemma 1.13 that
$L$ is a Noetherian UFD except if $r$ is a primitive root of unity of order
$l\geq 2$.
If $f$ is not a monomial, then we consider two cases:
Case 1: $r$ is a primitive $l$-th root of unity. In this case,
$h^{l}-\lambda^{l}$ annihilates $L_{\lambda}$, so $h^{l}-\lambda^{l}\in
Q_{\lambda}$. It follows that $Q_{\lambda}$ is not a height one prime. So, as
above, the only height one primes of $L$ are $\langle h\rangle$ and $\langle
h^{l}-\lambda\rangle$ with $\lambda\in\mathbb{K}^{*}$, whence the final
statement follows.
Case 2: $r$ is not a root of unity. As $f$ is not a monomial, there is
$\eta\in\mathbb{K}^{*}$ such that $P_{1}(\eta)=f(\eta)=0$. Assume $Q_{\eta}$
does not have height 1. Then it properly contains a nonzero prime ideal $Q$.
This ideal $Q$ cannot be of the form $Q_{\lambda}$, as these ideals are
maximal, hence either $Q$ does not contain a power of $d$ or $Q$ does not
contain a power of $u$. By the first part of our argument, as $r$ is not a
root of unity, $Q$ must contain $h$. In particular, $h$ annihilates the module
$L_{\eta}$, which is a contradiction, as $\eta\neq 0$. Thus, $Q_{\eta}$ indeed
has height one and is not principal, so $L$ is not a Noetherian UFR in this
case. ∎
## 3 The case $f=0$
We will consider separately the cases $\gamma=0$ and $\gamma\neq 0$.
### 3.1 The case $f=0$ and $\gamma=0$
In this case, the defining relations of $L=L(0,r,s,\gamma)$ are:
$dh=rhd,\quad hu=ruh,\quad du=sud,$
and $L$ is the so-called quantum coordinate ring of affine $3$-space over
$\mathbb{K}$. The normal elements $d$, $u$ and $h$ generate pairwise distinct
completely prime ideals so, by [26, Prop. 1.6], it is enough to show that the
localisation $T$ of $L$ with respect to the Ore set generated by these three
elements is a Noetherian UFR. Well, by [17, 1.3(i) and Cor. 1.5], the height
one prime ideals of $T$ are generated by a single central element, so $T$ is a
Noetherian UFR. Thus, $L$ is a Noetherian UFR. We record this result below.
###### Proposition 3.1.
Assume $f=0$ and $\gamma=0$. Then $L(0,r,s,0)$ is a Noetherian UFR.
We conclude this section by studying for which values of $r$ and $s$ the
generalized down-up algebra $L(0,r,s,0)$ is a Noetherian UFD.
###### Proposition 3.2.
Assume $f=0$ and $\gamma=0$. Then $L=L(0,r,s,0)$ is a Noetherian UFD if and
only if $\langle r,s\rangle$ is torsionfree.
###### Proof.
As was observed above, $L$ is just the quantum coordinate ring of an affine
3-space. So we deduce from [18, Thm. 2.1] that, if $\langle r,s\rangle$ is
torsionfree, then all prime ideals of $L$ are completely prime. Thus, the
result is proved in this case.
Now assume that $\langle r,s\rangle$ is not torsionfree. First, if $r$ is a
root of unity of order $l\geq 2$, then the result follows from Lemma 1.13. So
we are left with the cases where $r$ is either 1 or not a root of unity.
Before distinguishing between different cases, let us describe our strategy to
prove that $L$ is not a Noetherian UFD in these cases.
If $L$ were a Noetherian UFD, then so would be the localisation $T$ of $L$ at
the Ore set generated by the normal elements $h$, $d$, $u$. (Note that this is
due to the fact that we are localising at elements that are “$q$-central” -
see also Proposition 4.1.) This localised algebra $T$ is a quantum torus. More
precisely, it is the quantum torus generated by the three indeterminates $h$,
$d$ and $u$, and their inverses $h^{-1}$, $d^{-1}$ and $u^{-1}$, subject to
the relations
$dh=rhd,~{}hu=ruh,~{}du=sud.$
Now it follows from [18] that extension and contraction provide mutually
inverse bijections between the prime spectrum of $T$ and the prime spectrum of
the centre $\mathcal{Z}(T)$ of $T$, and that $\mathcal{Z}(T)$ is a
(commutative) Laurent polynomial algebra over $\mathbb{K}$. Moreover, we can
compute this centre explicitly, using [18, 1.3]. So, to prove that $L$ is not
a Noetherian UFD in the remaining cases, we will construct a height one prime
ideal of $T$ which is not completely prime. This is achieved by computing the
centre of $T$ in each case.
We distinguish between three cases:
Case 1: $r=1$ and $s$ is a root of unity of order $\beta\geq 2$. In this case,
we get
$\mathcal{Z}(T)=\mathbb{K}[u^{\pm\beta},h^{\pm 1},d^{\pm\beta}].$
Hence, $u^{\beta}-1$ generates a height one prime ideal in $T$ which is not
completely prime.
Case 2: $r$ is not a root of unity and $s$ is a root of unity of order
$\beta\geq 2$. In this case, we get
$\mathcal{Z}(T)=\mathbb{K}[(ud)^{\pm\beta}].$
Hence, $(ud)^{\beta}-1$ generates a height one prime ideal in $T$ which is not
completely prime.
Case 3: $r$ and $s$ are not roots of unity. Hence, there exists
$(\alpha_{0},\beta_{0})$ with $\beta_{0}>0$ minimal such that
$r^{\alpha_{0}}s^{\beta_{0}}=1$. In this case, we deduce from [18, 1.3] that
$\mathcal{Z}(T)=\mathbb{K}[(u^{\beta_{0}}h^{\alpha_{0}}d^{\beta_{0}})^{\pm
1}].$
Now observe that the fact that $\langle r,s\rangle$ is not torsionfree imposes
that $\gcd(\alpha_{0},\beta_{0})>1$. Hence,
$u^{\beta_{0}}h^{\alpha_{0}}d^{\beta_{0}}-1$ generates a height one prime
ideal in $T$ which is not completely prime. ∎
### 3.2 The case $f=0$ and $\gamma\neq 0$
If $r\neq 1$, then on replacing $h$ by $\tilde{h}=h+\gamma/(1-r)$ we can
reduce to the case $\gamma=0$ studied above. So we can assume $r=1$. We can
further assume $\gamma=1$, by replacing the generator $h$ by $\gamma^{-1}h$.
Let $Q$ be the subalgebra of $L$ generated by $d$ and $u$. Then $Q$ is the
quantum plane with relation $du=sud$ and $L=Q[h;\partial]$, where $\partial$
is the derivation of $Q$ determined by $\partial(d)=d$, $\partial(u)=-u$. By
the arguments of Section 3.1, $Q$ is a Noetherian UFR.
###### Proposition 3.3.
Assume $f=0$, $r=1$ and $\gamma\neq 0$. Then $L=L(0,1,s,\gamma)$ is a
Noetherian UFR.
###### Proof.
Without loss of generality, we assume $\gamma=1$. By [13, Thm. 5.5], it is
enough to show that every non-zero $\partial$-prime ideal of $Q$ contains a
non-zero principal $\partial$-ideal, for the derivation $\partial$ of $Q$
defined above.
Let $0\neq I\leq Q$ be a $\partial$-prime ideal of $Q$. Choose $p=p(d,u)\in
I\setminus\\{0\\}$ with minimal support, i.e., $0\neq p=\sum
a_{ij}d^{i}u^{j}\in I$ such that $a_{ij}\in\mathbb{K}$ and the set
$\\{(i,j)\mid a_{ij}\neq 0\\}$ has minimal cardinality. Fix $(\alpha,\beta)$
such that $a_{\alpha\beta}\neq 0$. It is straightforward to check that
$\partial(d^{i}u^{j})=(i-j)d^{i}u^{j}$, for all nonnegative integers $i,j$.
Then,
$\displaystyle I\ni(\alpha-\beta)p-\partial(p)$ $\displaystyle=\sum
a_{ij}(\alpha-\beta)d^{i}u^{j}-\sum a_{ij}(i-j)d^{i}u^{j}$ $\displaystyle=\sum
a_{ij}(\alpha+j-\beta-i)d^{i}u^{j}.$
Furthermore, $(\alpha-\beta)p-\partial(p)$ has a smaller support than $p$, as
its coefficient of $d^{\alpha}u^{\beta}$ is $0$. By the minimality assumption,
it must be that $(\alpha-\beta)p-\partial(p)=0$. Thus,
$\partial(p)=(\alpha-\beta)p$. In particular, $i-j$ is constant for all
$(i,j)$ such that $a_{ij}\neq 0$, and we can write $p=\sum_{i\geq
0}a_{i}d^{i}u^{j(i)}$.
Choose $\alpha$ such that $a_{\alpha}\neq 0$. Then,
$up-s^{-\alpha}pu=\sum_{i\geq 0}a_{i}(s^{-i}-s^{-\alpha})d^{i}u^{j(i)+1}$
and this is still an element of $I$, with smaller support than $p$. Thus,
$up-s^{-\alpha}pu=0$ and $up=s^{-\alpha}pu$. Similarly, $dp=s^{\beta}pd$ for
some $\beta\in\mathbb{Z}$, which shows that $p$ is normal. In particular, $I$
contains the nonzero principal ideal generated by $p$, which is a
$\partial$-ideal, as $\partial(p)=\lambda p$ for some integer $\lambda$. This
proves our claim. ∎
We now inquire when $L$ is a Noetherian UFD. For that purpose, we will
determine the height one prime ideals of $L$ explicitly and check which are
completely prime. Since $L$ is a Noetherian UFR, we know that all height one
prime ideals are principal. So we start out by determining the normal elements
of $L$.
Recall the $\mathbb{Z}$-graduation of $L$ described in (1.5).
###### Lemma 3.4.
Assume $f=0$, $r=1$ and $\gamma\neq 0$. Let $L=L(0,1,s,\gamma)$ and consider
the normal element $z=ud$. The normal elements of $L$ are homogeneous and of
the form $p(z)u^{i}$ or $p(z)d^{i}$, for $i\in\mathbb{N}$ and
$p(z)\in\mathbb{K}[z]$. Furthermore:
1. (a)
If $s$ is not a root of unity, then $p(z)=\lambda z^{c}$ for some
$\lambda\in\mathbb{K}$ and some integer $c\geq 0$;
2. (b)
If $s$ is a primitive $l$-th root of unity ($l\geq 1$), then
$p(z)=z^{c}\tilde{p}(z^{l})$ for some integer $0\leq c<l$ and some polynomial
$\tilde{p}(z^{l})$ in the central element $z^{l}$.
###### Proof.
We again assume $\gamma=1$. Let $\nu=\sum_{i\in\mathbb{Z}}\nu_{i}$ be a
nonzero normal element of $L$, with $\nu_{i}$ homogeneous of degree $i$.
Notice that $hu^{i}d^{j}=u^{i}d^{j}(h+j-i)$. In particular,
$h\nu_{i}=\nu_{i}(h-i)$ for all $i\in\mathbb{Z}$. By degree considerations,
there is $\xi(h,z)\in\mathbb{K}[h,z]$ such that $h\nu=\nu\xi(h,z)$. Thus,
$\sum\nu_{i}\xi(h,z)=h\sum\nu_{i}=\sum\nu_{i}(h-i).$
Therefore, equating homogeneous components and factoring out the nonzero
$\nu_{i}$, we get that $h-i=\xi(h,z)$ for all $i$ such that $\nu_{i}\neq 0$,
which proves that $\nu$ is homogeneous.
Assume $\nu\neq 0$ has degree $i\geq 0$. We can write $\nu=p(h,z)u^{i}$ (see
(1.6)). As $u^{i}d^{j}$ is clearly normal, for all $i,j\in\mathbb{N}$, and $L$
is a domain, $p(h,z)u^{i}$ is normal if and only if $p(h,z)$ is normal. Write
$p(h,z)=\sum_{j\geq 0}p_{j}(h)z^{j}$, with $p_{j}(h)\in\mathbb{K}[h]$. As
before, there must exist $\xi(h,z)\in\mathbb{K}[h,z]$ such that
$up(h,z)=p(h,z)\xi(h,z)u$. Using the commutation relations
$up_{j}(h)=p_{j}(h+1)u$ and $uz^{j}=s^{-j}z^{j}u$, and factoring out $u$ on
the right from both terms of that equation, we obtain
$\sum_{j\geq 0}p_{j}(h+1)s^{-j}z^{j}=\xi(h,z)\sum_{j\geq 0}p_{j}(h)z^{j}.$
From the above equation we readily conclude that $\xi(h,z)=\xi\in\mathbb{K}$,
as we are assuming $p(h,z)\neq 0$. Next, equating coefficients of $z^{j}$, we
get $p_{j}(h+1)s^{-j}=\xi p_{j}(h)$ for all $j$. This implies that $p_{j}(h)$
is a constant polynomial, for all $j$. Thus, we conclude that $\nu=p(z)u^{i}$,
for some $p(z)\in\mathbb{K}[z]$. The case $i\leq 0$ is symmetric.
It remains to determine when a nonzero element $p(z)\in\mathbb{K}[z]$ is
normal. Write $p(z)=\sum_{i}a_{i}z^{i}$, with $a_{i}\in\mathbb{K}$. Since
$up(z)=\sum_{i}s^{-i}a_{i}z^{i}u$, it is easy to conclude that $p(z)$ is
normal if and only if there is $\lambda\in\mathbb{K}$ such that
$s^{-i}=\lambda$ for all $i$ such that $a_{i}\neq 0$. Let $c\geq 0$ be the
first index for which $a_{c}\neq 0$. It follows that $p(z)=a_{c}z^{c}$ if $s$
is not a root of unity. In case $s$ is a primitive $l$-th root of unity, with
$l\geq 1$, then $p(z)=z^{c}p^{\prime}(z^{l})$, where $p^{\prime}(z^{l})$ is a
polynomial in $z^{l}$, and the result follows.
∎
We can now list all height one prime ideals of $L$ and check when $L$ is a
Noetherian UFD.
###### Theorem 3.5.
Assume $f=0$, $r=1$ and $\gamma\neq 0$. Let $L=L(0,1,s,\gamma)$ and $z=ud$.
Then $L$ is a Noetherian UFD if and only if either $s$ is not a root of unity
or $s=1$. The height one prime ideals of $L$ are:
1. (a)
$\langle d\rangle$ and $\langle u\rangle$, if $s$ is not a root of unity.
These ideals are completely prime.
2. (b)
$\langle d\rangle$, $\langle u\rangle$ and $\langle z-\lambda\rangle$, for
$\lambda\in\mathbb{K}^{*}$, if $s=1$. These ideals are completely prime.
3. (c)
$\langle d\rangle$, $\langle u\rangle$ and $\langle z^{l}-\lambda\rangle$, for
$\lambda\in\mathbb{K}^{*}$, if $s$ is a primitive $l$-th root of unity, with
$l>1$. The ideals $\langle d\rangle$ and $\langle u\rangle$ are completely
prime but those of the form $\langle z^{l}-\lambda\rangle$ are not.
###### Proof.
Once more, we assume $\gamma=1$. Let $P$ be a height one prime ideal of $L$.
By Proposition 3.3, there exists a normal element $\nu$ such that
$P=\langle\nu\rangle$. Furthermore, as $P$ is prime, $\nu$ cannot be the
product of two nonzero, nonunit normal elements. Thus, by Lemma 3.4, the only
possibilities are, up to nonzero scalars, $\nu=u$, $\nu=d$ or $\nu=p(z)$. The
ideals generated by either $u$ or $d$ are indeed completely prime, as the
corresponding factor algebra is isomorphic to the enveloping algebra of the
two-dimensional nonabelian Lie algebra. By the Principal Ideal Theorem, they
have height one. So it remains to consider the case $\nu=p(z)$. Note first
that $z=ud$, and since $u$ and $d$ are nonzero nonunit normal elements, the
ideal generated by $z$ is not prime.
Assume first that $s$ is not a root of unity. Then, by Lemma 3.4(a) and the
above, there is no other possibility for $P$. This proves part (a). Now assume
$s$ is a primitive $l$-th root of unity, with $l\geq 1$. Since $\mathbb{K}$ is
algebraically closed, the only other possibility for the generator $\nu$ of
$P$ is $\nu=z^{l}-\lambda$, for some $\lambda\in\mathbb{K}^{*}$. If $l=1$,
i.e., in case $s=1$, then $\langle z-\lambda\rangle$ is completely prime, for
$\lambda\neq 0$, as the factor algebra is isomorphic to the differential
operator ring $\mathbb{K}[u^{\pm 1}][h;\partial]$, where
$\partial(u^{i})=-iu^{i}$, for all $i\in\mathbb{Z}$. This proves part (b).
Finally, assume $l>1$. Recall that $L$ can be presented as the differential
operator ring $Q[h;\partial]$, where $Q$ is the quantum plane with relation
$du=sud$ and $\partial$ is the derivation of $Q$ determined by
$\partial(d)=d$, $\partial(u)=-u$. The centre of $Q$ is the polynomial algebra
$\mathbb{K}[d^{l},u^{l}]$, by [17, 1.3(i)], and the element $z^{l}-\lambda$ is
irreducible in this polynomial algebra. Thus, $z^{l}-\lambda$ generates a
prime ideal of $\mathbb{K}[d^{l},u^{l}]$. By [17, Cor. 1.5], $z^{l}-\lambda$
also generates a prime ideal of $Q$. Furthermore, $\partial(z^{l}-\lambda)=0$,
as $\partial(z)=0$. Hence, by [27, Prop. 14.2.5], $z^{l}-\lambda$ generates a
prime ideal of $L$. This ideal has height one, by the Principal Ideal Theorem.
Since $l>1$ and $\mathbb{K}$ is algebraically closed, $z^{l}-\lambda$ factors
nontrivially as a polynomial in $z$, so the ideal $\langle
z^{l}-\lambda\rangle$ is not completely prime, which proves part (c). ∎
## 4 The case $f$ conformal and $r$ not a root of unity
In this case, as $r\neq 1$, we can and will assume that $\gamma=0$. Thus, the
defining relations for $L=L(f,r,s,0)$ are:
$\displaystyle dh-rhd$ $\displaystyle=0,$ (4.11) $\displaystyle hu-ruh$
$\displaystyle=0,$ (4.12) $\displaystyle du-sud+f(h)$ $\displaystyle=0.$
(4.13)
In particular, $h$ is a nonzero, nonunit normal element of $L$ which generates
a completely prime ideal of $L$.
Let $L_{h}$ be the localisation of $L$ with respect to the powers of $h$. It
is clear that
$L_{h}=\mathbb{K}[h^{\pm 1}][d;\sigma][u;\sigma^{-1},\delta],$
where $\sigma$ and $\delta$ are extended by setting
$\sigma(h^{-1})=r^{-1}h^{-1}$ and $\delta(h^{-1})=0$.
###### Proposition 4.1.
Assume $\gamma=0$. Then $L_{h}$ is a Noetherian UFR (resp. UFD) if and only if
$L$ is a Noetherian UFR (resp. UFD).
###### Proof.
The direct implication follows from Lemma 1.9. The converse follows from
standard arguments in localisation theory, provided we can show that any
normal element of $L$ is still normal in $L_{h}$.
Let $\nu\in L$ be normal. We can assume $\nu\neq 0$. Write
$\nu=\sum_{i\in\mathbb{Z}}\nu_{i}$ with $\nu_{i}$ homogeneous of degree $i$.
As $h\nu_{i}=\nu_{i}\sigma^{i}(h)=r^{i}\nu_{i}h$, it follows that
$h\nu=\sum_{i\in\mathbb{Z}}r^{i}\nu_{i}h$. On the other hand, by the normality
of $\nu$, there is $h^{\prime}\in L$ such that $h\nu=\nu h^{\prime}$. The
$\mathbb{Z}$-grading implies that $h^{\prime}$ has degree $0$. Hence, the
degree $i$ component of $\nu h^{\prime}$ is $\nu_{i}h^{\prime}$. Equating
elements of the same degree we conclude that
$r^{i}\nu_{i}h=\nu_{i}h^{\prime}$, for all $i$. Choose $\alpha$ with
$\nu_{\alpha}\neq 0$. We must have $h^{\prime}=r^{\alpha}h$ and thus
$h\nu=r^{\alpha}\nu h$. Hence, in $L_{h}$, we have $\nu
h^{-1}=r^{\alpha}h^{-1}\nu$, which is enough to show that $\nu$ is normal in
$L_{h}$. This concludes the proof. ∎
Let $g\in\mathbb{K}[h]$ be such that $f=g^{*}$, i.e., $f(h)=sg(h)-g(rh)$. We
will assume $f\neq 0$, as the case $f=0$ has already been dealt with in
Section 3. In particular, $g\neq 0$.
The algebra $L_{h}$ is in the scope of the algebras studied by Jordan in [20],
where our polynomial $g$ plays the role of the element $u$ of [20]. We will
start out with a few technical observations which will allow us to apply the
results of [20] to $L_{h}$. We will remind the reader of the necessary
definitions as they are needed.
###### Lemma 4.2.
Assume $\gamma=0$ and $r$ is not a root of unity. Then the Laurent polynomial
algebra $\mathbb{K}[h^{\pm 1}]$ is $\sigma$-simple, i.e., its only
$\sigma$-invariant ideals are itself and the zero ideal.
###### Proof.
This is worked out in Example 1.2.(i) of [20]. ∎
We recall Definition 1.7 of [20], applied in our context. Suppose there exists
$0\neq p\in\mathbb{K}[h^{\pm 1}]$ such that $\sigma(p)=s^{-n}p$ for some
positive integer $n$. Let $n\geq 1$ be minimal with respect to the existence
of such an element $p$. Then, any $0\neq p\in\mathbb{K}[h^{\pm 1}]$ satisfying
$\sigma(p)=s^{-n}p$ will be called a principal eigenvector, and $n$ will be
its degree.
In order to discuss the existence of principal eigenvectors, we will make use
of two integers $\epsilon\in\mathbb{Z}$ and $\tau\in\mathbb{N}$, which have
been defined in [9], as follows:
$\tau=\min\\{i>0\mid s^{i}=r^{j}\quad\text{for some
$j\in\mathbb{Z}$}\\}\quad\quad\text{and}\quad\quad r^{\epsilon}=s^{\tau},$
if $\\{i>0\mid s^{i}=r^{j}\quad\text{for some
$j\in\mathbb{Z}$}\\}\neq\emptyset$, and $\tau=0=\epsilon$, otherwise. As long
as $r$ is not a root of unity, $\epsilon$ is uniquely defined. Furthermore, by
[9, Lem. 2.1], if $\delta,\eta\in\mathbb{Z}$ then $r^{\delta}s^{\eta}=1$ if
and only if there is $\lambda\in\mathbb{Z}$ such that
$(\delta,\eta)=\lambda(-\epsilon,\tau)$.
###### Lemma 4.3.
Assume $\gamma=0$. There exist principal eigenvectors in $L_{h}$ if and only
if $\tau>0$, i.e., if and only if there are integers $\alpha,\beta$, with
$\alpha\neq 0$, such that $s^{\alpha}r^{\beta}=1$.
###### Proof.
Assume $s^{\alpha}r^{\beta}=1$, with $\alpha\neq 0$. We can thus assume
$\alpha\geq 1$. Then
$\sigma(h^{\beta})=r^{\beta}h^{\beta}=s^{-\alpha}h^{\beta}$, so there are
principal eigenvectors.
Conversely, assume $\sigma(p)=s^{-n}p$ for some $n\geq 1$ and some
$p=\sum_{i=\alpha}^{\beta}a_{i}h^{i}$, with $\alpha\leq\beta$ and
$a_{\alpha}a_{\beta}\neq 0$. Then,
$0=\sigma(p)-s^{-n}p=\sum_{i=\alpha}^{\beta}a_{i}(r^{i}-s^{-n})h^{i}$. In
particular, $r^{\beta}-s^{-n}=0$ and $s^{n}r^{\beta}=1$. ∎
###### Theorem 4.4.
Assume $\gamma=0$, $r$ is not a root of unity and $f\neq 0$ is conformal. Then
$L$ is a Noetherian UFR if and only if either $\tau>0$ or $f$ is a monomial.
###### Proof.
By Proposition 4.1, we can work over the localisation $L_{h}$. Then, the
result for $L_{h}$ follows from Example 2.21 of [20]. ∎
It remains to establish when $L$ is a Noetherian UFD, which we do next.
###### Theorem 4.5.
Assume $\gamma=0$, $r$ is not a root of unity and $f\neq 0$ is conformal. Then
$L$ is a Noetherian UFD if and only if either one of the following two
conditions holds:
1. (a)
$\langle r,s\rangle$ is a free abelian group of rank $2$ and $f$ is a
monomial, or
2. (b)
$\langle r,s\rangle$ is a free abelian group of rank $1$.
###### Proof.
Once again, we can work over the localisation $L_{h}$, by virtue of
Proposition 4.1. Notice that, since $r$ is not a root of unity, $\langle
r,s\rangle$ is a free abelian group of rank $2$ if and only if $\tau=0$, and
$\langle r,s\rangle$ is a free abelian group of rank $1$ if and only if
$\tau\geq 1$ and $\gcd(\tau,\epsilon)=1$.
Assume first that $\tau=0$. Then, by the above, $L_{h}$ is a Noetherian UFR if
and only if $f$ is a monomial. When this is the case, $\langle
z:=ud-g(h)\rangle$ is the unique height one prime ideal of $L_{h}$, and it is
clearly completely prime, as the factor algebra is a quantum torus in two
variables (namely, the cosets of $u$ and $h$).
Let us now suppose $\tau\geq 1$. Then, by the proof of Lemma 4.3, there is a
principal eigenvector, $h^{-\epsilon}$, and it has degree $\tau$. Furthermore,
this principal eigenvector is unique, up to nonzero scalar multiples. It
follows that $\langle h^{-\epsilon}z^{\tau}-\lambda\rangle$ is a height one
prime ideal of $L_{h}$, for all $\lambda\in\mathbb{K}^{*}$ such that $\lambda
h^{\epsilon}\neq(-g(h))^{\tau}$, by [20, Cor. 2.9.(ii)]. Note that $\lambda
h^{\epsilon}=(-g(h))^{\tau}$ can occur for at most one value of
$\lambda\in\mathbb{K}^{*}$. By [20, Thm. 2.24], it is easy to conclude that,
for $\lambda\in\mathbb{K}^{*}$, $\langle h^{-\epsilon}z^{\tau}-\lambda\rangle$
is completely prime if and only if $\gcd(\tau,\epsilon)=1$. In particular,
under the current hypotheses, if $L_{h}$ is a Noetherian UFD, then $\langle
r,s\rangle$ is free abelian of rank $1$.
Conversely, assume $\gcd(\tau,\epsilon)=1$, with $\tau\geq 1$. Then, by [20,
2.17 and Prop. 2.18], the height one prime ideals of $L_{h}$ include $\langle
z\rangle$, which is completely prime, and the ideals of the form $\langle
h^{-\epsilon}z^{\tau}-\lambda\rangle$, for $\lambda\in\mathbb{K}^{*}$ such
that $\lambda h^{\epsilon}\neq(-g(h))^{\tau}$, which are all completely prime,
as $\gcd(\tau,\epsilon)=1$. In case
$h^{-\epsilon}(-g(h))^{\tau}\notin\mathbb{K}$, then this is the complete list
of height one prime ideals of $L_{h}$, and it follows that $L_{h}$ is a
Noetherian UFD. Suppose that $h^{-\epsilon}(-g(h))^{\tau}\in\mathbb{K}$. Then
$g(h)$ is a unit, say $g(h)=\mu h^{a}$, and it follows that $\epsilon=\tau a$.
As we are assuming $\tau$ and $\epsilon$ to be coprime, it must be that
$\tau=1$ and $\epsilon=a$. Thus,
$f(h)=sg(h)-g(rh)=\mu(s-r^{a})h^{a}=\mu(s^{\tau}-r^{\epsilon})h^{a}=0$, which
contradicts our hypothesis on $f$. Therefore, it is always the case that
$h^{-\epsilon}(-g(h))^{\tau}\notin\mathbb{K}$ and the proof is complete. ∎
## 5 The case $f$ conformal and $r=1$
When $r=1$, we cannot assume that $\gamma=0$, so we will consider separately
the cases $\gamma\neq 0$ and $\gamma=0$. The defining relations of
$L=L(f,1,s,\gamma)$ are:
$\displaystyle dh-hd+\gamma d$ $\displaystyle=0,$ (5.14) $\displaystyle hu-
uh+\gamma u$ $\displaystyle=0,$ (5.15) $\displaystyle du-sud+f(h)$
$\displaystyle=0.$ (5.16)
Note that if $r=1$ and $\gamma\neq 0$ we retrieve the algebras studied by
Rueda in [33]. The latter include Smith’s algebras [34], which occur as
generalized down-up algebras when $r=s=1$ and $\gamma\neq 0$. We assume
throughout that $f\neq 0$.
### 5.1 The case $f$ conformal, $r=1$ and $\gamma\neq 0$
Let $g$ be such that $f(h)=sg(h)-g(h-\gamma)$. In particular, $g\neq 0$.
Recall, from Section 4, the definition of a principal eigenvector.
###### Lemma 5.1.
Assume $r=1$ and $\gamma\neq 0$. If $p\in\mathbb{K}[h]$ is such that
$\sigma(p)=\mu p$ for some $\mu\in\mathbb{K}$ then $p\in\mathbb{K}$. In
particular, the only nonzero $\sigma$-invariant ideal of $\mathbb{K}[h]$ is
$\mathbb{K}[h]$ and there are principal eigenvectors if and only if $s$ is a
root of unity.
###### Proof.
Suppose that $\sigma(p)=\mu p$ for some
$p\in\mathbb{K}[h]\setminus\mathbb{K}$. Then, since $\mathbb{K}$ is
algebraically closed, there is $\alpha\in\mathbb{K}$ such that $p(\alpha)=0$.
It follows that $0=\mu p(\alpha)=\sigma(p)(\alpha)=p(\alpha-\gamma),$ and
hence $\alpha-\gamma$ is also a root of $p$. Since $\alpha$ was an arbitrary
root of $p$ and $\gamma\neq 0$, this is impossible. Thus, $p\in\mathbb{K}$.
Let $I$ be a $\sigma$-invariant ideal of $\mathbb{K}[h]$. Then $I=\langle
p\rangle$, for some $p\in\mathbb{K}[h]$, and $\sigma(p)\in\mathbb{K}^{*}p$, so
either $I=\langle 0\rangle$ or $I=\mathbb{K}[h]$.
Finally, assume there is a principal eigenvector $0\neq p\in\mathbb{K}[h]$.
Then there is $n\geq 1$ so that $\sigma(p)=s^{-n}p$. In particular, by the
above, it follows that $p\in\mathbb{K}^{*}$ and $s^{n}=1$. Conversely, if $s$
is a primitive $n$-th root of unity, then $1$ is a principal eigenvector of
degree $n$. ∎
###### Proposition 5.2.
Assume $s$ is not a root of unity and $\gamma\neq 0$. Take
$p\in\mathbb{K}[h]$. Then $p$ satisfies
$\forall\lambda\in\mathbb{K}\ \ \forall n\geq 1\quad
s^{n}p(\lambda)=p(\lambda-n\gamma)\implies p(\lambda)=0$ (5.17)
if and only if $p\in\mathbb{K}$.
###### Proof.
Let $p\in\mathbb{K}[h]$ and assume, by way of contradiction, that $p$ is not
constant. Then the set of roots of $p$ is finite and nonempty. Let
$\Delta=\\{\alpha-\beta\mid\alpha\ \mbox{and}\ \beta\ \mbox{are roots of}\
p\\}$ be the set of differences of (not necessarily distinct) roots of $p$.
Since $\Delta$ is finite, there exists an integer $n\geq 1$ such that
$n\gamma\notin\Delta$. Consider the polynomial
$p_{n}(h)=s^{n}p(h)-p(h-n\gamma)$. Since $s$ is not a root of unity, $p_{n}$
has the same degree as $p$. In particular, $p_{n}$ has some root, say
$\alpha\in\mathbb{K}$. By (5.17), $\alpha$ is a root of $p$, which in turn
implies that $\alpha-n\gamma$ is a root of $p$, as well. Hence,
$n\gamma=\alpha-\left(\alpha-n\gamma\right)\in\Delta$, which contradicts our
choice of $n$. So indeed, $p\in\mathbb{K}$. The converse implication is
trivial. ∎
We are now ready to say when $L$ is a Noetherian UFR.
###### Theorem 5.3.
Assume $f$ and $\gamma$ are nonzero and $r=1$. Then $L=L(f,1,s,\gamma)$ is a
Noetherian UFR if and only if one of the following conditions hold:
1. (a)
$s$ is a root of unity, or
2. (b)
$s$ is not a root of unity and $f\in\mathbb{K}$.
###### Proof.
The first and second parts follow from [20, Prop. 2.18] and [20, Prop. 2.20],
respectively, and the results in this section. Note that, since $s\neq 1$,
then $f\in\mathbb{K}\iff g\in\mathbb{K}$. ∎
Next, we deduce from [20, 2.22 and Remark 2.25] the cases where $L$ is a
Noetherian UFD.
###### Theorem 5.4.
Assume $f$ and $\gamma$ are nonzero and $r=1$. Then $L=L(f,1,s,\gamma)$ is a
Noetherian UFD if and only if one of the following conditions hold:
1. (a)
$s=1$, or
2. (b)
$s$ is not a root of unity and $f\in\mathbb{K}$.
### 5.2 The case $f$ conformal, $r=1$ and $\gamma=0$
In this case, $h$ is central in $L$. Note also that since $f$ is conformal in
$L$, then $s\neq 1$. Nevertheless, the conformality condition will not be used
in this section, as we will not refer to [20]. We will consider separately the
cases $s$ not a root of unity and $s$ a root of unity of order $l\geq 2$.
#### 5.2.1 The case $f$ conformal, $r=1$, $\gamma=0$ and $s$ not a root of
unity
###### Theorem 5.5.
Assume $f$ is nonzero, $r=1$, $\gamma=0$ and $s$ is not a root of unity. Then
$L=L(f,1,s,0)$ is a Noetherian UFD. In fact, the height one prime ideals of
$L$ are $\langle h-\lambda\rangle$, for $\lambda\in\mathbb{K}$, and $\langle
du-ud\rangle$.
###### Proof.
Let $\lambda\in\mathbb{K}$. Then $h-\lambda$ is central and the factor algebra
$L/\langle h-\lambda\rangle$ is either the quantum plane or the quantum Weyl
algebra, depending on whether $\lambda$ is a root of $f$ or not. In either
case, $L/\langle h-\lambda\rangle$ is a domain, and thus the ideal $\langle
h-\lambda\rangle$ is completely prime.
The element $du-ud=(s-1)z$ is normal and the factor algebra $L/\langle du-
ud\rangle$ is a commutative algebra generated by $h$, $d$ and $u$, subject to
the relation $ud-\frac{1}{s-1}f(h)=0$. It is easy to see that the element
$ud-\frac{1}{s-1}f(h)$, viewed as an element of the polynomial algebra in the
$3$ commuting variables $h$, $d$ and $u$, is irreducible, provided that $f\neq
0$. This shows that $L/\langle du-ud\rangle$ is a domain. Another way of
reaching this conclusion is by realising this factor algebra as the
generalized Weyl algebra
$\mathbb{K}[h]\left(\mathrm{id}_{\mathbb{K}[h]},\frac{1}{s-1}f(h)\right)$.
(The reader is referred to [3] for more details on generalized Weyl algebras.)
The above shows that all ideals of the form $\langle h-\lambda\rangle$, for
$\lambda\in\mathbb{K}$, and $\langle du-ud\rangle$ are completely prime and
principal. By the Principal Ideal Theorem, they have height one. To finish the
proof, we need only show that any nonzero prime ideal of $L$ must contain one
of these ideals. We do so in the next proposition. ∎
###### Proposition 5.6.
Assume $f$ is nonzero, $r=1$, $\gamma=0$ and $s$ is not a root of unity. Then
any nonzero prime ideal of $L=L(f,1,s,0)$ must contain either $du-ud$ or
$h-\lambda$, for some $\lambda\in\mathbb{K}$.
###### Proof.
Let $P$ be a nonzero prime ideal of $L$ and assume $h-\lambda$ is not in $P$,
for any scalar $\lambda$. We will show that $du-ud\in P$.
Since $\mathbb{K}$ is algebraically closed and $\mathbb{K}[h]$ is a central
subalgebra, it follows that $P\cap\mathbb{K}[h]=\langle 0\rangle$. Let
$\widetilde{L}$ be the localisation of $L$ at the central multiplicative set
of nonzero elements of $\mathbb{K}[h]$. Let $\mathbb{F}=\mathbb{K}(h)$ be the
field of fractions of $\mathbb{K}[h]$. Then $\widetilde{L}$ can be seen as the
first quantised Weyl algebra $\mathbb{A}^{s}_{1}(\mathbb{F})$, generated over
$\mathbb{F}$ by $X$ and $Y$, and subject to the relation $XY-sYX=1$. In fact,
it is easy to check that there are mutually inverse $\mathbb{F}$-algebra maps,
$\Phi:\widetilde{L}\rightarrow\mathbb{A}^{s}_{1}(\mathbb{F})$ and
$\Psi:\mathbb{A}^{s}_{1}(\mathbb{F})\rightarrow\widetilde{L}$, such that
$\Phi(d)=X$, $\Phi(u)=-f(h)Y$, $\Psi(X)=d$ and
$\Psi(Y)=-u\left(f(h)\right)^{-1}$.
Thus, $P$ extends to a nonzero prime ideal $\widetilde{P}$ of $\widetilde{L}$,
which we identify, via the map $\Phi$ above, with a prime ideal of
$\mathbb{A}^{s}_{1}(\mathbb{F})$. The element $Z:=XY-YX$ of
$\mathbb{A}^{s}_{1}(\mathbb{F})$ is normal, nonzero and not a unit. In fact,
$Z$ corresponds, via $\Psi$, to the element $(ud-du)\left(f(h)\right)^{-1}$.
Let $\mathbb{B}^{s}_{1}(\mathbb{F})$ be the localisation of
$\mathbb{A}^{s}_{1}(\mathbb{F})$ at the powers of $Z$. Since $s$ is not a root
of unity, $\mathbb{B}^{s}_{1}(\mathbb{F})$ is simple, by [2, Lem. 2.2] (note
that this result does not depend on the base field being algebraically
closed). Since $Z$ is normal, this means that every nonzero prime ideal of
$\mathbb{A}^{s}_{1}(\mathbb{F})$ contains $Z$. In particular,
$Z\in\Phi(\widetilde{P})$, i.e., $(ud-
du)\left(f(h)\right)^{-1}\in\widetilde{P}$. Thus, $du-ud\in
P=\widetilde{P}\cap L$. ∎
#### 5.2.2 The case $f$ conformal, $r=1$, $\gamma=0$ and $s\neq 1$ a root of
unity
We finally tackle the case in which $s$ is a primitive $l$-th root of unity,
for some $l\geq 2$. It is straightforward to see that, in this case, both
$d^{l}$ and $u^{l}$ are central. Our aim is to prove that, in this case, $L$
is a Noetherian UFR.
Let $\widetilde{L}$ be the localisation of $L$ with respect to the
multiplicative set generated by the central elements of the form $h-\lambda$,
where $\lambda$ runs through the roots of $f$. In case $f$ is a (nonzero)
constant polynomial, we have $\widetilde{L}=L$.
Since, for $\lambda$ a root of $f$, $L/\langle h-\lambda\rangle$ is a quantum
plane, the ideals of the form $\langle h-\lambda\rangle$, with $f(\lambda)=0$,
are completely prime as well as pairwise distinct. Thus, by [26, Prop. 1.6],
it will be enough to show that $\widetilde{L}$ is a Noetherian UFR.
Let $S$ be the localisation of $\mathbb{K}[h]$ at the multiplicative set
generated by the $h-\lambda$, with $\lambda$ running through the roots of $f$.
Since $\mathbb{K}$ is algebraically closed, $f$ is a product of linear
factors, and thus is invertible in $S$. The localised algebra $\widetilde{L}$
can be seen as the algebra over $S$, generated by elements $D$ and $U$,
subject to the relation
$DU-sUD=1,$
where $D=d$ and $U=-u\left(f(h)\right)^{-1}$. Consider the nonzero normal
element $Z=DU-UD$, of $\widetilde{L}$. It satisfies $ZU=sUZ$ and
$ZD=s^{-1}DZ$. In particular, $Z^{l}$ is central in $\widetilde{L}$. The
algebra $\widetilde{L}/\langle Z\rangle$ is isomorphic to the commutative
Laurent polynomial algebra $S[U^{\pm 1}]$, and hence $Z$ generates a
completely prime ideal of $\widetilde{L}$. Therefore, it will suffice to show
that the localisation $\widehat{L}$ of $\widetilde{L}$ at the multiplicative
set generated by $Z$ is a Noetherian UFR, by [26, Prop. 1.6]. The latter is a
consequence of the result that follows.
###### Proposition 5.7.
Under the above assumptions, $\widehat{L}$ is an Azumaya algebra over its
centre $\mathcal{Z}(\widehat{L})$, with
$[\widehat{L}:\mathcal{Z}(\widehat{L})]=l^{2}$. Moreover, the centre
$\mathcal{Z}(\widehat{L})$ of $\widehat{L}$ is the localisation of
$S[U^{l},D^{l}]$ at the powers of $Z^{l}$.
###### Proof.
The proof is entirely analogous to that of [1, Prop. 1.3]. We give details for
completeness.
First, it is easy to see that the centre of $\widetilde{L}$ is
$S[U^{l},D^{l}]$, and it must contain $Z^{l}$, as this element commutes with
$D$ and $U$.
Let $b=aZ^{n}$ be an element of $\widehat{L}$ with $a\in\widetilde{L}$ and
$n\in\mathbb{Z}$. Take $q\in\mathbb{Z}$ and $0\leq r<l$ such that $n=ql+r$. As
$Z^{ql}$ is central in $\widehat{L}$, we get that $b=aZ^{ql}Z^{r}$ is central
in $\widehat{L}$ if and only if $aZ^{r}$ is central in $\widetilde{L}$. Hence,
$\mathcal{Z}(\widehat{L})=\\{cZ^{ql}~{}|~{}c\in
S[U^{l},D^{l}],~{}q\in\mathbb{Z}\\}$ is the localisation of $S[U^{l},D^{l}]$
at the powers of $Z^{l}$.
By [1, Lem. 1.2], $\left\\{U^{i}D^{j}\right\\}_{0\leq i,j\leq l-1}$ is a basis
for $\widehat{L}$ over its centre. So
$[\widehat{L}:\mathcal{Z}(\widehat{L})]=l^{2}$.
To conclude, it is enough to show that the irreducible finite dimensional
representations of $\widehat{L}$ over $\mathbb{K}$ all have dimension $l$, by
the Artin-Procesi Theorem. Let
$\rho:\widehat{L}\rightarrow\mathrm{End}_{\mathbb{K}}(V)$ be such an
irreducible representation, with $\dim_{\mathbb{K}}V=m$. Since $\mathbb{K}$ is
algebraically closed, and $V$ is finite-dimensional, it follows by Schur’s
Lemma that the centre of $\widehat{L}$ acts on $V$ as scalars. Thus,
$\dim_{\mathbb{K}}\rho({\widehat{L}})\leq l^{2}$. By the Jacobson Density
Theorem, $\rho$ is surjective. Therefore,
$m^{2}=\dim_{\mathbb{K}}\mathrm{End}_{\mathbb{K}}(V)=\dim_{\mathbb{K}}\rho({\widehat{L}})\leq
l^{2},$
and $m\leq l$.
On the other hand, let $X=\rho(D)$, $Y=\rho(U)$. Then, $XY-sYX=\rho(DU-
sUD)=\rho(1)=1\in\mathrm{End}_{\mathbb{K}}(V)$. Furthermore, as $Z$ is
invertible in $\widehat{L}$, the same is true of $\rho(Z)=XY-
YX\in\mathrm{End}_{\mathbb{K}}(V)$. Thus,
$\dim_{\mathbb{K}}\mathrm{End}_{\mathbb{K}}(V)\geq l^{2}$, again by [1, Lem.
1.2]. So $m\geq l$ and $m=l$. ∎
###### Theorem 5.8.
Assume $f$ is nonzero, $\gamma=0$, $r=1$ and $s$ is a primitive $l$-th root of
unity, for some $l\geq 2$. Then $L=L(f,1,s,0)$ is a Noetherian UFR, but not a
Noetherian UFD.
###### Proof.
Since the algebra $\widehat{L}$ is Azumaya over its centre, it follows that
all ideals of $\widehat{L}$ are centrally generated. Hence, as
$\mathcal{Z}(\widehat{L})$ is a (commutative) UFR, by Proposition 5.7, we
deduce from the Principal Ideal Theorem that $\widehat{L}$ is a Noetherian
UFR. We can thus conclude that $L$ is a Noetherian UFR, by [26, Prop. 1.6].
We will now observe that the ideal $\langle d^{l}-1\rangle$ of $L$ is prime.
To see this, notice that $L$ is an Ore extension over the commutative
polynomial algebra $\mathbb{K}[h,d]$. So, by [5, Prop. 2.1], it will be enough
to prove that $d^{l}-1$ generates a $\delta$-stable, $\sigma$-prime ideal of
this polynomial algebra, where $\sigma$ and $\delta$ are as in (1.4). In
particular, $\sigma(d)=sd$. This ideal is stable under $\delta$ and $\sigma$
because $d^{l}-1$ is central in $L=\mathbb{K}[h,d][u;\sigma^{-1},\delta]$.
Consider the prime ideal $I$ of $\mathbb{K}[h,d]$ generated by $d-1$. Since
$s$ is a primitive root of unity of order $l$, it follows that
$\bigcap_{i\in\mathbb{Z}}\sigma^{i}(I)=\prod_{0\leq i\leq
l-1}(d-s^{i})\mathbb{K}[h,d]=(d^{l}-1)\mathbb{K}[h,d],$
so $(d^{l}-1)\mathbb{K}[h,d]$ is indeed a $\sigma$-prime ideal of
$\mathbb{K}[h,d]$, as it is the intersection of a $\sigma$-orbit of a prime
ideal. Thus, $\langle d^{l}-1\rangle$ is a prime ideal of $L$.
By the Principal Ideal Theorem, $\langle d^{l}-1\rangle$ has height one. Yet,
it is not completely prime, as $l\geq 2$ and hence the central element
$d^{l}-1$ factors non-trivially. So $L$ is not a Noetherian UFD. ∎
## 6 The case $f$ conformal and $r\neq 1$ a root of unity
The final part of our discussion concerns the case when $f$ is conformal and
$r$ is a primitive root of unity of order $l\geq 2$. Since $r\neq 1$ we will
assume, without loss of generality, that $\gamma=0$, by Proposition 1.4.
We start with a negative result, which follows immediately from Lemma 1.13.
###### Corollary 6.1.
Let $L=L(f,r,s,0)$ and assume $r\neq 1$ is a root of unity. Then $L$ is not a
Noetherian UFD.
The remainder of this section is devoted to establishing that, under the
current assumptions, $L=L(f,r,s,0)$ is a Noetherian UFR. The following general
result will play, in this section, the role of Propositions 2.3 and 2.5.
We consider a Noetherian ring $R$, with a subring $A$, which is a domain, and
such that $R$ is free both as a left and as a right $A$-module, with basis
$S:=\\{X^{i}~{}|~{}i\geq 0\\}$. Assume the multiplicative system $S$ satisfies
the Ore condition on both sides, and let $\widehat{R}:=RS^{-1}$ be the
corresponding localisation.
###### Lemma 6.2.
Let $P$ be a nonzero prime ideal of $R$ such that $P\cap S=\emptyset$, and
assume that there exists $b\in\widehat{R}$ such that:
1. (a)
$PS^{-1}=\widehat{R}b=b\widehat{R}$;
2. (b)
$Xb=\eta bX$, for some central unit $\eta$ of $A$.
Then $P=xR=Rx$, where $e\in\mathbb{Z}$ is minimal such that $bX^{e}\in R$, and
$x=bX^{e}$.
###### Proof.
Observe that, since $b\neq 0$, a minimal $e\in\mathbb{Z}$ such that $bX^{e}\in
R$ exists; also, $Xx=\eta xX$. We will prove that $P=Rx$. As $Xb=\eta bX$, $e$
is also minimal such that $X^{e}b\in R$, and $X^{e}b=\eta^{e}x$, so a similar
argument will show that $P=xR$, using the fact that $\\{X^{i}~{}|~{}i\geq
0\\}$ is a free basis for $R$ as a right $A$-module.
By construction, it is clear that $Rx\subseteq P$, as $x\in PS^{-1}\cap R=P$.
Let $y\in P$. Then $y\in
PS^{-1}=\widehat{R}b=\widehat{R}xX^{-e}=\widehat{R}x$, as $x$ and $X$
$\eta$-commute. Hence, there exists $u\in\widehat{R}$ such that $y=ux$.
Moreover, there exists $t\geq 0$ such that $uX^{t}\in R$. Therefore,
$yX^{t}=uxX^{t}=\eta^{-t}uX^{t}x$, i.e., there exist $t\geq 0$ and $r\in R$
such that $yX^{t}=rx$. We choose a minimal such $t$.
Assume that $t\geq 1$. Write
$r=\sum_{i=0}^{k}r_{i}X^{i},~{}~{}y=\sum_{i=0}^{k}y_{i}X^{i},~{}~{}x=\sum_{i=0}^{k}x_{i}X^{i},$
where $r_{i},y_{i},x_{i}\in A$. Note that $x_{0}\neq 0$, as otherwise
$xX^{-1}\in R$, so that $bX^{e-1}\in R$, contradicting the minimality of $e$.
On the other hand, as $Xx=\eta xX$, the equality $yX^{t}=rx$ can be written as
follows:
$\sum_{i=0}^{k}y_{i}X^{i+t}=\sum_{i=0}^{k}r_{i}X^{i}bX^{e}=\sum_{i=0}^{k}r_{i}\eta^{i}bX^{e+i}=\sum_{i=0}^{k}r_{i}\eta^{i}xX^{i}=\sum_{i,j=0}^{k}r_{i}\eta^{i}x_{j}X^{i+j}.$
As $t\geq 1$, identifying the degree 0 coefficients yields $0=r_{0}x_{0}$. As
$x_{0}\neq 0$ and $A$ is a domain, this forces $r_{0}=0$. Hence, $rX^{-1}\in
R$ and $yX^{t-1}=rxX^{-1}=\eta rX^{-1}x$. This contradicts the minimality of
$t$. Thus, $t=0$ and $y=rx\in Rx$, as desired. ∎
###### Proposition 6.3.
Let $L=L(f,r,s,0)$, with $f$ conformal. If $P$ is a prime ideal of $L$ of
height one, which either does not contain any power of $d$ or does not contain
any power of $u$, then $P$ is a principal ideal, generated by a normal element
of $L$.
###### Proof.
By Lemma 1.2, the localisation $\widehat{L}$ of $L$ at the denominator set
$D=\\{d^{i}\\}_{i\geq 0}$ is isomorphic to a quantum coordinate ring of affine
$3$-space over $\mathbb{K}$, localised at the powers of one of its canonical
generators. As in Section 3.1, it follows that $\widehat{L}$ is a Noetherian
UFR.
If $P$ is a height one prime ideal of $L$ which is disjoint from $D$, then
$PD^{-1}$ is a height one prime ideal of $\widehat{L}$, so it is generated by
a normal element $b\in\widehat{L}$. It is easy to see that in a quantum
coordinate ring the normal elements are $q$-central, so there is
$\eta\in\mathbb{K}^{*}$ such that $db=\eta bd$. Thus, by Lemma 6.2, $P$ is a
principal ideal, generated by some normal element $x\in L$.
The statement regarding $u$ follows similarly. ∎
So it remains to consider the prime ideals that contain both a power of $d$
and a power of $u$. We start by discussing the simpler case where $s$ is not a
root of unity.
###### Proposition 6.4.
Let $L=L(f,r,s,0)$, with $f\neq 0$ conformal and $r\neq 1$ a root of unity. If
$s$ is not a root of unity, then $L$ is a Noetherian UFR, but not a Noetherian
UFD.
###### Proof.
In view of Corollary 6.1 and Proposition 6.3, it is enough to show that the
height one prime ideals of $L$ either do not contain any power of $d$ or do
not contain any power of $u$.
Let $P$ be a prime ideal of $L$ which contains a power of $d$ and a power of
$u$. Since $r$ is a root of unity and $s$ is not, it follows that
$(s/r^{m})^{k}\neq 1$, for all $k>0$. Thus, by Lemma 1.10, the polynomials
$P_{k}$ are all nonzero, for $k>0$. Hence, $P=Q_{\lambda}$, for some
$\lambda\in\mathbb{K}$, by Theorem 1.12.
If $\lambda=0$, then $h\in P$; otherwise $h^{l}-\lambda^{l}\in P$, where
$l\geq 2$ is the order of $r$. Therefore, either $\langle h\rangle\subsetneq
P$ or $\langle h^{l}-\lambda^{l}\rangle\subsetneq P$, as $P=Q_{\lambda}$ is
not principal, so $P$ has height at least two, by Lemma 1.13, thus proving our
claim. ∎
In the next lemma we deal with the case in which $s$ is a root of unity. Note
that if $r$ and $s$ are roots of unity and $f$ is conformal, then Lemma 1.10
guarantees the existence of a positive integer $k$ such that $P_{k}=0$. For
any such $k$, the elements $u^{k}$ and $d^{k}$ are normal.
###### Lemma 6.5.
Let $L=L(f,r,s,0)$, with $f\neq 0$ conformal, and assume $r$ and $s$ are roots
of unity. Take $k>0$ minimal such that $P_{k}=0$. Then, $u^{k}$ and $d^{k}$
are normal and each generates a height one prime ideal of $L$.
###### Proof.
We will prove the statement for $u^{k}$; the result for $d^{k}$ will thus
follow, by symmetry.
Consider the Ore set $D=\left\\{d^{i}\right\\}_{i\geq 0}$ in $L$ and the
localisation $\widehat{L}=LD^{-1}$. Recall that $z:=ud-g(h)$ is normal and
satisfies $zh=hz$, $dz=szd$ and $zu=suz$ (see Section 1.2). It is easy to see
that $h$ and $z$ generate a (commutative) polynomial algebra in two variables,
$\mathbb{K}[h,z]$, and $\widehat{L}=\mathbb{K}[h,z][d^{\pm 1};\tau]$, where
$\tau(h)=rh$, $\tau(z)=sz$, with $u=(z+g(h))d^{-1}$.
Let $\xi=z+g(h)\in\mathbb{K}[h,z]$. This is an irreducible polynomial in the
polynomial algebra $\mathbb{K}[h,z]$, hence it generates a prime ideal
$P=\xi\mathbb{K}[h,z]$. Furthermore, $\tau^{i}(\xi)$ and $\tau^{j}(\xi)$ are
associated irreducible polynomials if and only if $k$ divides $i-j$. The
latter follows from the minimality of $k$, as $P_{i}(h)=0\iff
s^{i}g(h)=g(r^{i}h)\iff$ $k$ divides $i$.
Thus,
$I:=\bigcap_{i\in\mathbb{Z}}\tau^{i}(P)=\bigcap_{i\in\mathbb{Z}}\tau^{i}(\xi)\mathbb{K}[h,z]=\bigcap_{1-k\leq
i\leq 0}\tau^{i}(\xi)\mathbb{K}[h,z]=\prod_{1-k\leq i\leq
0}\tau^{i}(\xi)\mathbb{K}[h,z]$
is a $\tau$-prime ideal of $\mathbb{K}[h,z]$. It follows (e.g. by [5, Prop.
2.1]) that $Q:=I\widehat{L}$ is a prime ideal of $\widehat{L}$.
Claim: $\displaystyle{\prod_{1-n\leq i\leq 0}\tau^{i}(\xi)=u^{n}d^{n}}$, for
all $n\geq 0$.
The claim above can be readily established by induction. In particular,
$Q=u^{k}d^{k}\widehat{L}=u^{k}\widehat{L}$.
It remains to show that the prime ideal that $Q$ contracts to in $L$ is
generated by $u^{k}$. This follows by applying Lemma 6.2 to the contraction of
$Q$ to $L$, and noting that:
* •
$du^{k}=s^{k}u^{k}d$, and
* •
for $n\in\mathbb{Z}$, $u^{k}d^{n}\in L\iff n\geq 0.$
Finally, the height of $\langle u^{k}\rangle$ is one, by the Principal Ideal
Theorem. ∎
Our final result finishes the classification of which generalized down-up
algebras are Noetherian UFR’s.
###### Theorem 6.6.
Let $L=L(f,r,s,0)$, with $f\neq 0$ conformal and $r\neq 1$ a root of unity.
Then $L$ is a Noetherian UFR but not a Noetherian UFD.
###### Proof.
By Proposition 6.4, it remains to consider the case where $s$ is a root of
unity (possibly equal to $1$), and by Corollary 6.1 and Proposition 6.3, it
will be enough to show that there are no height one prime ideals of $L$ which
contain both a power of $d$ and a power of $u$.
Let $P$ be a prime ideal of $L$ which contains a power of $d$ and a power of
$u$. Let $k>0$ be minimal such that $P_{k}=0$. Since $u^{k}$ is normal, we
must have $u^{k}\in P$, so P contains the height one prime ideal $\langle
u^{k}\rangle$, by Lemma 6.5. So $P$ does not have height one, as $\langle
u^{k}\rangle$ contains no power of $d$. ∎
## 7 Proofs of Theorems A and B
In this final section, we start by proving Theorem B, which gives a complete
classification of the generalized down-up algebras which are a Noetherian UFR,
and then we prove Theorem A. We also specialise our results to down-up
algebras, as introduced by Benkart and Roby in [6].
###### Proof of Theorem B.
Assume first that $\gamma=0$. Then the condition _there exists
$\zeta\neq\gamma/(r-1)$ such that $f(\zeta)=0$_ is equivalent to the condition
_$f$ is not a monomial_, and the condition _$\langle r,s\rangle$ is a free
abelian group of rank $2$_ is equivalent to the condition _$r$ is not a root
of unity and $\tau=0$_. Thus, in this case, the result follows from Theorem
2.6, Proposition 3.1, Theorem 4.4, Theorem 5.5, Theorem 5.8 and Theorem 6.6.
Now assume that $\gamma\neq 0$ and $r=1$. Then, by Proposition 1.5, $f$ is
conformal, and the result follows from Proposition 3.3 and Theorem 5.3.
Finally, if $\gamma\neq 0$ and $r\neq 1$, then Proposition 1.4 asserts that
$L$ is isomorphic to a generalized down-up algebra $L(\tilde{f},r,s,0)$, such
that $f$ is conformal in $L(f,r,s,\gamma)$ if and only if $\tilde{f}$ is
conformal in $L(\tilde{f},r,s,0)$. Furthermore, by the proof of this result
(see [9, Prop. 1.7]), we can take $\tilde{f}(h)=f(\frac{h+\gamma}{r-1})$.
Hence, in this case, the result follows from applying our previously
established criteria to $L(\tilde{f},r,s,0)$. ∎
To finish the classification, we just need to determine the generalized down-
up algebras which are a Noetherian UFD, and prove Theorem A.
###### Proof of Theorem A.
It will be enough to establish this result in the case $\gamma=0$, and the
case $\gamma\neq 0$, $r=1$, by Proposition 1.4, as the statement does not
involve $f$ or $\gamma$. So we assume that either $\gamma=0$ or $r=1$.
* •
If $f$ is not conformal then $\gamma=0$, by Proposition 1.5, and thus, by
Lemma 1.3, $\langle r,s\rangle=\langle r\rangle$. Then Theorem 2.6 establishes
the result.
* •
If $f=0$ and $\gamma=0$, then the result follows from Propositions 3.1 and
3.2.
* •
If $f=0$ and $\gamma\neq 0$, then we assume $r=1$ and the result follows from
Proposition 3.3 and Theorem 3.5.
* •
If $f\neq 0$ is conformal and $r$ is not a root of unity, then we assume
$\gamma=0$ and the result follows from Theorems 4.4 and 4.5.
* •
If $f\neq 0$ is conformal, $r=1$ and $\gamma\neq 0$, then Theorems 5.3 and 5.4
establish the result.
* •
If $f\neq 0$ is conformal, $r=1$ and $\gamma=0$, then Proposition 1.5 implies
that $s\neq 1$. Thus, Theorems 5.5 and 5.8 imply the result.
* •
If $r\neq 1$ is a root of unity, then we can assume that $\gamma=0$, and the
result follows directly from Corollary 6.1.
∎
We note that the hypothesis that $L$ be a Noetherian UFR, in Theorem A, is
essential, as the following example illustrates. Let $r\in\mathbb{K}^{*}$ be a
non-root of unity, $s\in\\{1,r\\}$ and $f=h\in\mathbb{K}[h]$. Then
$L=L(h,r,s,1)$ is not a Noetherian UFD, by Proposition 1.4 and Theorem B(a).
Yet, $\langle r,s\rangle\simeq\mathbb{Z}$ is torsionfree. Notice that
$L(h,r,s,1)$ is isomorphic to the down-up algebra $A(r+s,-rs,1)$.
In general, the down-up algebra $A(\alpha,\beta,\gamma)$, as defined in [6],
can be viewed as the generalized down-up algebra $L(h,r,s,\gamma)$, where
$\alpha=r+s$ and $\beta=-rs$ (see [9, Lem. 1.1] for more details). So we have:
###### Corollary 7.1.
Let $A=A(\alpha,\beta,\gamma)$ be a down-up algebra over $\mathbb{K}$ with
$\beta\neq 0$. Let $r,s\in\mathbb{K}$ be the roots of $h^{2}-\alpha h-\beta$.
Then $A$ is a Noetherian UFR except if $\gamma\neq 0$, $\beta$ is not a root
of unity and one of the following conditions is satisfied:
1. (a)
$\alpha+\beta=1$;
2. (b)
$\alpha^{2}+4\beta=0$;
3. (c)
$\langle r,s\rangle$ is a free abelian group of rank $2$.
Furthermore, $A$ is a Noetherian UFD if and only if $A$ is a Noetherian UFR
and $\langle r,s\rangle$ is torsionfree.
###### Proof.
We use the isomorphism $A(\alpha,\beta,\gamma)\simeq L(h,r,s,\gamma)$. First,
by Proposition 1.4, Lemma 1.3 and Proposition 1.5, we conclude that $f(h)=h$
is conformal in $L(h,r,s,\gamma)$ if and only if one of the following
conditions holds:
* •
$\gamma=0$ and $r\neq s$;
* •
$\gamma\neq 0$, $r\neq 1$, $s\neq 1$ and $r\neq s$;
* •
$\gamma\neq 0$ and $r=1$.
Thus, we can apply Theorem B to conclude that $A$ is a Noetherian UFR except
in the cases listed below:
* •
$\gamma\neq 0$, $s=1$ and $r$ is not a root of unity;
* •
$\gamma\neq 0$, $s=r$ and $r$ is not a root of unity;
* •
$\gamma\neq 0$ and $\langle r,s\rangle$ is a free abelian group of rank $2$;
* •
$\gamma\neq 0$, $r=1$ and $s$ is not a root of unity.
Notice that, in all of these cases, $\gamma\neq 0$ and $\beta=-rs$ is not a
root of unity. Also, $\alpha+\beta=1\iff r=1$ or $s=1$, and
$\alpha^{2}+4\beta=0\iff r=s$. The first part of the theorem thus follows. The
second part is a direct consequence of Theorem A. ∎
Two down-up algebras of particular interest are the enveloping algebra of the
Lie algebra $\mathfrak{sl}_{2}$ and the enveloping algebra of the
$3$-dimensional Heisenberg Lie algebra, which occur as $A(2,-1,1)$ and
$A(2,-1,0)$, respectively. Using Corollary 7.1, we retrieve the well-known
fact that each of these two algebras is a Noetherian UFD (see [14] and [12,
Prop. 3.1]).
Generalized down-up algebras also include other classes of algebras, such as
Smith’s algebras [34] and Rueda’s algebras [33]. In the case of Smith’s
algebras, the result is quite straightforward. Let $f\in\mathbb{K}[H]$. Recall
that the Smith algebra $S(f)$ is the $\mathbb{K}$-algebra generated by $A,B,H$
with relations:
$[H,A]=A,~{}[H,B]=-B\mbox{ and }[A,B]=f(H).$
It is well known that $S(f)\simeq L(f,1,1,1)$. Hence, we deduce from Theorems
A and B the following result.
###### Corollary 7.2.
Let $S(f)$ be a Smith algebra with $f\in\mathbb{K}[H]$. Then, $S(f)$ is a
Noetherian UFD.
## References
* [1] J. Alev and F. Dumas, Rigidité des plongements des quotients primitifs minimaux de $U_{q}({\rm sl}(2))$ dans l’algèbre quantique de Weyl-Hayashi, Nagoya Math. J. 143 (1996), 119–146.
* [2] M. Awami, M. Van den Bergh and F. Van Oystaeyen, Note on derivations of graded rings and classification of differential polynomial rings, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 175–183.
* [3] V. Bavula, Generalized Weyl algebras and their representations, (Russian) Algebra i Analiz 4 (1992), no. 1, 75–97; translation in St. Petersburg Math. J. 4 (1993), no. 1, 71–92.
* [4] V.V. Bavula and D.A. Jordan, Isomorphism problems and groups of automorphisms for generalized Weyl algebras, Trans. Am. Math. Soc. 353 (2001), no. 2, 769–794.
* [5] A. Bell, When are all prime ideals in an Ore extension Goldie?, Comm. Algebra 13 (1985), no. 8, 1743–1762.
* [6] G. Benkart and T. Roby, Down-up algebras, J. Algebra 209 (1998), no. 1, 305–344.
* [7] A. Braun, Factorial properties of the enveloping algebra of a nilpotent Lie algebra in prime characteristic, J. Algebra 308 (2007), no. 1, 1–11.
* [8] K.A. Brown, Height one primes of polycyclic group rings, J. London Math. Soc. (2) 32 (1985), no. 3, 426–438.
* [9] P.A.A.B. Carvalho and S.A. Lopes, Automorphisms of generalized down-up algebras, Comm. Algebra 37 (2009), no. 5, 1622–1646.
* [10] P.A.A.B. Carvalho and I.M. Musson, Down-up algebras and their representation theory, J. Algebra 228 (2000), no. 1, 286–310.
* [11] T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279 (2004), no. 1, 402–421.
* [12] A.W. Chatters, Non-commutative unique factorisation domains, Math. Proc. Camb. Phil. Soc. 95 (1984), 49–54.
* [13] A.W. Chatters and D.A. Jordan, Noncommutative unique factorisation rings, J. London Math. Soc. (2) 33 (1986), no. 1, 22–32.
* [14] N. Conze, Algèbres d’opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France 102 (1974), 379–415.
* [15] D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New-York, 1995.
* [16] K.R. Goodearl, Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), 324–377.
* [17] K.R. Goodearl and E.S. Letzter, Prime and primitive spectra of multiparameter quantum affine spaces, in Trends in ring theory (Proc. Miskolc Conf. 1996), Canad. Math. Soc. Conf. Proc. Series 22 (1986), 22–32.
* [18] K.R. Goodearl and E.S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994), 1017–1025.
* [19] J. Hildebrand, Irreducible finite-dimensional modules of prime characteristic down-up algebras, J. Algebra 273 (2004), no. 1 295–319.
* [20] D.A. Jordan, Height one prime ideals of certain iterated skew polynomial rings, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 3, 407–425.
* [21] D.A. Jordan, Down-up algebras and ambiskew polynomial rings, J. Algebra 228 (2000), no. 1, 311–346.
* [22] E.E. Kirkman, I.M. Musson and D.S. Passman, Noetherian down-up algebras, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3161–3167.
* [23] G.R. Krause and T.H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, revised ed., Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000.
* [24] R.S. Kulkarni, Down-up algebras and their representations, J. Algebra 245 (2001), no. 2, 431–462.
* [25] S. Launois and T.H. Lenagan, Quantised coordinate rings of semisimple groups are unique factorisation domains, Bull. Lond. Math. Soc. 39 (2007), no. 3, 439–446.
* [26] S. Launois, T.H. Lenagan and L. Rigal, Quantum unique factorisation domains, J. London Math. Soc. (2) 74 (2006), no. 2, 321–340.
* [27] J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings, revised ed., Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001\.
* [28] I. Praton, Primitive ideals of Noetherian down-up algebras, Comm. Algebra 32 (2004), no. 2, 443–471.
* [29] I. Praton, Simple weight modules of non-Noetherian generalized down-up algebras, Comm. Algebra 35 (2007), no. 1, 325–337.
* [30] I. Praton, Simple modules and primitive ideals of non-Noetherian generalized down-up algebras, Comm. Algebra 37 (2009), no. 3, 811–839.
* [31] I. Praton and S. May, Primitive ideals of non-Noetherian down-up algebras, Comm. Algebra 33 (2005), no. 2, 605–622.
* [32] I. Praton, Primitive ideals of Noetherian generalized down-up algebras, arXiv:1003.2361v1, preprint.
* [33] S. Rueda, Some algebras similar to the enveloping algebra of $sl(2)$, Commun. Algebra 30 (2002), no. 3, 1127–1152.
* [34] S.P. Smith, A class of algebras similar to the enveloping algebra of sl(2), Trans. Am. Math. Soc. 322 (1990), no. 1, 285–314.
Stéphane Launois
School of Mathematics, Statistics & Actuarial Science,
University of Kent, Canterbury, Kent CT2 7NF, United Kingdom
E-mail: [email protected]
Samuel A. Lopes
Centro de Matemática da Universidade do Porto,
Universidade do Porto,
Rua do Campo Alegre 687, 4169-007 Porto, Portugal
E-mail: [email protected]
|
arxiv-papers
| 2012-08-23T19:27:07 |
2024-09-04T02:49:34.502153
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "St\\'ephane Launois and Samuel A. Lopes",
"submitter": "Samuel Lopes",
"url": "https://arxiv.org/abs/1208.4833"
}
|
1208.4844
|
# An Off-center Density Peak in the Milky Way’s Dark Matter Halo?
Michael Kuhlen11affiliationmark: , Javiera Guedes22affiliationmark: ,
Annalisa Pillepich33affiliationmark: , Piero Madau33affiliationmark: , and
Lucio Mayer44affiliationmark: 1Theoretical Astrophysics Center, University of
California Berkeley, Hearst Field Annex, Berkeley, CA 94720
2ETH Zurich, Institute for Astronomy, Wolfgang-Pauli-Strasse 27, Zurich 8049,
Switzerland
3Department of Astronomy & Astrophysics, University of California Santa Cruz,
1156 High St., Santa Cruz, CA 95064
4University of Zurich, Institute for Theoretical Physics, Zurich 8057,
Switzerland [email protected]
###### Abstract
We show that the position of the central dark matter density peak may be
expected to differ from the dynamical center of the Galaxy by several hundred
parsec. In Eris, a high resolution cosmological hydrodynamics simulation of a
realistic Milky-Way-analog disk galaxy, this offset is 300 - 400 pc ($\sim 3$
gravitational softening lengths) after $z=1$. In its dissipationless dark-
matter-only twin simulation ErisDark, as well as in the Via Lactea II and
GHalo simulations, the offset remains below one softening length for most of
its evolution. The growth of the DM offset coincides with a flattening of the
central DM density profile in Eris inwards of $\sim 1$ kpc, and the direction
from the dynamical center to the point of maximum DM density is correlated
with the orientation of the stellar bar, suggesting a bar-halo interaction as
a possible explanation. A dark matter density offset of several hundred parsec
greatly affects expectations of the dark matter annihilation signals from the
Galactic Center. It may also support a dark matter annihilation interpretation
of recent reports by Weniger (2012) and Su & Finkbeiner (2012) of highly
significant 130 GeV gamma-ray line emission from a region $1.5^{\circ}$ ($\sim
200$ parsec projected) away from Sgr A* in the Galactic plane.
## 1\. Introduction
Dissipationless (dark matter only) N-body simulations predict a nearly
universal density profile of dark matter (DM) halos, the so-called NFW profile
(Navarro et al., 1997), which features a central $1/r$ density cusp. If the
cooling and condensation of gas inside these halos (White & Rees, 1978; Fall &
Efstathiou, 1980) is gradual, then so-called “adiabatic contraction”
(Blumenthal et al., 1986; Gnedin et al., 2004) will pull DM into the central
regions, thereby increasing the central DM density and further steepening the
slope of its density profile. It is thus natural to expect the maximum of the
DM density to occur at the dynamical center of a galaxy, the lowest point of
its gravitational potential. This expectation has made the Galactic Center
(GC) a preferred target for indirect DM detection efforts searching for an
annihilation signal (Bergström et al., 1998; Gondolo & Silk, 1999; Aharonian
et al., 2006; Baltz et al., 2008; Danninger & the IceCube collaboration,
2012).
As data from the Fermi Gamma-ray Space Telescope (Atwood et al., 2009) has
been accumulating, the number of studies reporting gamma-ray “excesses” or
“anomalies” from the GC, that could be interpreted as a DM annihilation
signal, has steadily grown (see e.g. Goodenough & Hooper, 2009; Hooper &
Goodenough, 2011; Hooper & Linden, 2011; Abazajian & Kaplinghat, 2012). These
analyses have searched for the broad gamma-ray continuum signal thought to
arise from the decay and hadronization of DM annihilation products (Bergström
et al., 1998). While this is expected to be the dominant DM annihilation
signature, it is unfortunately difficult to distinguish it from conventional
sources (e.g. milli-second pulsars, Abazajian 2011, or cosmic-ray interactions
with molecular clouds, Yusef-Zadeh et al. 2012), and at present a purely
astrophysical explanation of these signals cannot be excluded.
More recently, however, there have been surprising reports of a highly
statistically significant line-like feature at $\sim 130$ GeV in Fermi data
from the GC (Bringmann et al., 2012; Weniger, 2012; Su & Finkbeiner, 2012).
Although such a DM annihilation line should be loop-suppressed by a factor
$\sim 10^{2}-10^{4}$ compared to the expected continuum gamma-ray production
in typical DM models, the fact that it is difficult to produce such a high
energy line with astrophysical processes (however, see Aharonian et al., 2012)
has motivated further exploration of the DM annihilation explanation. In fact,
DM particle physics models do exist in which the continuum is suppressed with
respect to the line emission (e.g. Cline, 2012; Buckley & Hooper, 2012;
Bergström, 2012; Dudas et al., 2012; Chalons et al., 2012). In the analysis of
Su & Finkbeiner (2012), the line’s significance is maximized at Galactic
coordinates $(\ell,b)=(1.5^{\circ},0^{\circ})$, i.e. displaced from Sgr A*,
the presumed dynamical center of the Galaxy, by about 200 projected parsec in
the disk plane (see also Tempel et al., 2012). This offset has been viewed as
a strike against a DM annihilation interpretation of the line signal. It is
possible that the offset is simply due to small number statistics (Yang et
al., 2012), but nevertheless it is commonly viewed as a strike against a DM
annihilation interpretation of the line signal.
On the other hand, the GC is a dynamically and energetically complex region
(Genzel et al., 2010). It harbors a supermassive black hole (SMBH) (Ghez et
al., 2008; Gillessen et al., 2009), which may have been active as recently as
10 Myr ago, if the giant Fermi bubbles (Dobler et al., 2010; Su et al., 2010)
are interpreted as resulting from a black hole accretion event; it is rich in
massive young stars (Bartko et al., 2009), which likely formed in a single
star burst event $6\pm 2$ Myr ago (Paumard et al., 2006); it hosts plenty of
highly energetic compact objects radiating in X-rays and gamma-rays (Muno et
al., 2003, 2005; Abazajian, 2011); and on $\sim$kpc scales, its gravitational
potential is non-axisymmetric due to the presence of a stellar bar and a boxy
bulge (Blitz & Spergel, 1991; Martinez-Valpuesta & Gerhard, 2011). Given that
our Galaxy is baryon-dominated inwards of $\sim 5-10$ kpc (Klypin et al.,
2002), one might expect astrophysical processes to modify the underlying DM
distribution in significant ways. In principle these processes may even
displace the maximum of the DM density away from the dynamical center, thus
greatly affecting the expected DM annihilation signal from the GC.
In isolated galaxy simulations, resonant interactions between the stellar bar
and the DM halo have been shown to alter the shape of DM halos reducing their
triaxiality (Berentzen et al., 2006; Machado & Athanassoula, 2010), and to
flatten a central cusp into a core (Weinberg & Katz, 2002; Athanassoula, 2003;
Holley-Bockelmann et al., 2005; Weinberg & Katz, 2007). The latter results
remain controversial, however, since other numerical studies do not see such
strong effects (Sellwood, 2003; Valenzuela & Klypin, 2003; Colín et al.,
2006). Regarding an off-center DM density peak, these interactions are
interesting because they may also induce a “dark bar” and other non-
axisymmetric perturbations in the DM (Athanassoula, 2002; Ceverino & Klypin,
2007; McMillan & Dehnen, 2005).
External gravitational perturbations, for example during a merger or a near
passage of a satellite galaxy, could displace the tightly-bound baryonic
component from the center of the overall mass distribution (the DM halo). Such
offsets have been measured in galaxy clusters (Allen, 1998), in which
separations between the center of the X-ray emitting gas and the gravitational
center determined from strong lensing can be as large as $\sim 30$ kpc for
relaxed clusters (Shan et al., 2010). A study of THINGS galaxies by
Trachternach et al. (2008) found that offsets between the photometric and
dynamical centers were less than one radio beam width ($\sim
10^{\prime\prime}$ or $150-700$ pc) for 13 out of 15 galaxies with well-
constrained photometric centers. However, two galaxies in their sample, NGC
3627 (a barred Sb galaxy showing signs of a recent interaction) and NGC 6946
(a barred Scd galaxy), exhibit moderate offsets between one and two beam
widths.
An offset DM density peak may be a reflection of an intrinsic lopsidedness in
the Galaxy (Saha et al., 2009). In fact, disk galaxies commonly exhibit
substantial asymmetry in the central regions of their light distribution (for
a review, see Jog & Combes, 2009). The origin of these asymmetries is not
fully understood, with tidal encounters, gas accretion, and a global
gravitational instability being some of the physical mechanisms under
consideration. The central regions of advanced galaxy mergers often show long-
lived unsettled sloshing behavior (Schweizer, 1996; Jog & Maybhate, 2006), and
even in isolated galaxies the dynamical center can remain unrelaxed for many
dynamical times (Miller & Smith, 1992), especially in systems with a cored
mass distribution.
Supernova- or AGN-driven gas outflows may rapidly and non-adiabatically alter
the potential in the central regions of proto-galaxies, prior to the formation
of the bulk of their stars. Repeated episodes of such impulsive outflows,
followed by slow adiabatic re-accretion of gas, may irreversibly transfer
energy to the DM, flattening the central cusp in the process (Read & Gilmore,
2005; Pontzen & Governato, 2012). Although this effect may not by itself
produce an off-center DM density peak, the resulting cored density profile
will be more susceptible to perturbations.
Lastly, the presence of a SMBH has been argued to lead to the formation of a
steep cusp of DM (Gondolo & Silk, 1999; Gnedin & Primack, 2004) in the inner
parsec centered on the SMBH. This would obviously preclude any significant
offset between the peak in DM annihilation signal and Sgr A*.
In this work we report on the search for the presence of an offset between the
dynamical center and the maximum DM density in the Eris simulation (Guedes et
al., 2011), one of the highest resolution and most realistic cosmological
simulations of the formation of a Milky-Way-like barred spiral galaxy. We find
evidence for such an offset, at a scale of $300-400$ pc, consistent with the
offset seen by Su & Finkbeiner (2012). Since such an offset is only seen in
the dissipational hydrodynamic simulations, and not in our collisionless pure-
DM runs, we suggest that baryonic physics is in some way responsible for its
formation. We examine a number of different physical mechanisms, but the
limited resolution of this study does not allow us to conclusively settle on a
single preferred explanation. At this stage, we wish to draw attention to the
possibility of the maximum DM density (and hence annihilation luminosity) not
being coincident with the dynamical center of our Galaxy, commonly associated
with Sgr A*. We hope that our results will stimulate future work, examining
other high resolution hydrodynamic galaxy formation simulations, and
investigating in more detail the physical mechanisms that can give rise to an
offset DM density peak.
The remainder of this paper is organized as follows. In Section 2 we describe
the numerical simulations that we have analyzed. In Section 3 we present the
evidence for a DM offset in the Eris simulation. In Section 4 we go over
several possible formation mechanisms and confront each of them with data from
the simulations. In Section 5 we discuss implications for indirect detection
searches towards the GC, and finally in Section 6 we present our conclusions.
## 2\. Simulations
Table 1Four Cosmological Zoom-in Simulations.
Name | Code | $m_{\rm DM}\;[{\rm M_{\odot}}]$ | $\epsilon_{\rm soft}\;[{\rm pc}]$ | $N_{\rm vir}$ | $M_{\rm vir}\;[{\rm M_{\odot}}]$ | $R_{\rm vir}\;[{\rm kpc}]$ | $(\alpha,\,r_{-2}\;[{\rm kpc}],\,\rho_{-2}\;[\,{\rm M}_{\odot}\,{\rm kpc}^{-3}])$
---|---|---|---|---|---|---|---
Eris | Gasoline | $9.8\times 10^{4}$ | 124 | $1.9\times 10^{7}$ | $7.9\times 10^{11}$ | 239 | $(0.0984,\;18.8,\;3.84\times 10^{6})$
ErisDark | Gasoline | $1.2\times 10^{5}$ | 124 | $7.6\times 10^{6}$ | $9.1\times 10^{11}$ | 247 | $\;\;(0.173,\;42.5,\;8.73\times 10^{5})$
Via Lactea II | PKDGRAV2 | $4.1\times 10^{3}$ | 40 | $4.1\times 10^{8}$ | $1.7\times 10^{12}$ | 309 | $\;\;(0.144,\;53.0,\;7.78\times 10^{5})$
GHalo | PKDGRAV2 | $1.0\times 10^{3}$ | 60 | $1.1\times 10^{9}$ | $1.1\times 10^{12}$ | 267 | $\;\;(0.137,\;61.1,\;3.53\times 10^{5})$
Note. — $m_{\rm DM}$ is the mass of high-resolution DM particles,
$\epsilon_{\rm soft}$ their gravitational softening. $N_{\rm vir}$ and $M_{\rm
vir}$ are the total number and mass of all particles (including gas and star
particles in Eris) within the virial radius $R_{\rm vir}$, defined as the
radius enclosing a density of 92.5 times the critical density. The last column
lists the parameters of the best-fit Einasto profile
($\ln(\rho/\rho_{-2})=-2/\alpha\,[(r/r_{-2})^{\alpha}-1]$) for the mean
enclosed density $\langle\rho\rangle(<\\!r)=$ $M(<\\!r)/(4\pi/3\,r^{3})$ (see
Fig. 1).
Our analysis makes use of four cosmological zoom-in simulations of the
formation and evolution of a Milky-Way-analog galaxy: one DM+hydrodynamics
simulation, “Eris” (Guedes et al., 2011); and three pure-DM dissipationless
simulations, “ErisDark” (Pillepich et al., in prep.), “Via Lactea II” (Diemand
et al., 2008), and “GHalo” (Stadel et al., 2009). Details of these simulations
are presented in the listed references, and we only briefly summarize their
salient features here. Eris and its DM-only twin ErisDark were run with the
N-body+SPH code Gasoline (Wadsley et al., 2004), but only Eris utilized the
SPH hydrodynamics. Eris resolves the formation and evolution of a galaxy with
$1.3\times 10^{7}$ high resolution DM particles of mass $9.8\times
10^{4}\,{\rm M}_{\odot}$ and a similar number of SPH gas particles with an
initial mass of $1.2.1\times 10^{4}\,{\rm M}_{\odot}$. Gravitational
interactions are softened using a cubic spline density kernel of length
$\epsilon_{\rm soft}=124$ proper pc111Forces become Newtonian at 2
$\epsilon_{\rm soft}$.. Throughout the course of the simulation, star
particles are created from dense gas ($>5$ atoms cm-3) according to a
heuristic star formation recipe with a 10% star-formation efficiency. These
star particles affect surrounding gas through a supernova blastwave feedback
prescription that injects thermal energy, mass, and metals. The simulation
results in a realistic looking barred late-type spiral disk galaxy, that
matches many observational constraints on the structure of the Milky Way. For
example, it has a low bulge-to-disk ratio of 0.35, falls on the Tully-Fisher
relation, has a stellar-to-total mass ratio of 0.04, and a star formation rate
of 1.1 M⊙ yr-1. Eris is the most realistic such simulation available today.
ErisDark is a DM-only twin to Eris, meaning that it was initialized with the
same phases of the Gaussian random density field, but all of the matter is
treated as DM, while in Eris 17% is baryonic. ErisDark thus has a slightly
higher DM particle mass of $1.2\times 10^{5}\,{\rm M}_{\odot}$, but employs
the same $\epsilon_{\rm soft}$. A detailed comparison of the two simulations
will be presented in Pillepich et al. (2012, in preparation). Lastly, we also
compare results with two of the highest resolution pure-DM simulations ever
performed, Via Lactea II (VL2, Diemand et al., 2008) with a particle mass of
$m_{p}=4,100\,{\rm M}_{\odot}$ and a force softening of $\epsilon_{\rm
soft}=40$ pc and GHalo (Stadel et al., 2009) with $m_{p}=1,000\,{\rm
M}_{\odot}$ and $\epsilon_{\rm soft}=60$ pc. Both VL2 and GHalo were run with
the purely collisionless N-body code PKDGRAV2. The relevant parameters of all
simulations are summarized in Table 1.
## 3\. An Off-Center DM Density Peak
We define the dynamical center of the halo to be the location of the minimum
of the total gravitational potential. The gravitational potential is
calculated during the simulation by solving the Poisson equation, with source
terms contributed by all DM, star, and gas particles (for details, see Wadsley
et al., 2004). In the Eris simulation the potential minimum is typically set
by the stars, which dominate the potential in the center of the galaxy. The
location of the potential minimum is coincident with the center of mass of the
stellar disk (and with the point of maximum stellar and gas density) to within
10 pc. We select for our analysis all particles within a radius of 2.5 kpc
from the dynamical center. At $z=0$ there are 104,781 DM, 13,792 gas, and
4,597,762 star particles in this region in Eris, contributing $1.03\times
10^{10}$, $2.07\times 10^{8}$, and $1.97\times 10^{10}\,{\rm M}_{\odot}$ in
mass. In ErisDark, VL2, and GHalo there are 42,611, 1,718,223, and 4,041,566
DM particles in this region, for a mass of $5.06\times 10^{9}$, $7.04\times
10^{9}$, and $4.12\times 10^{9}\,{\rm M}_{\odot}$, respectively.
### 3.1. Enclosed density profiles
Figure 1.— Radial profiles of the average enclosed dark matter density
$\langle\rho\rangle(<\\!r)$ centered on the location of the potential minimum,
starting at the radius enclosing at least 20 particles. The vertical dotted
lines indicate the softening lengths, $\epsilon_{\rm soft}$ = 124, 124, 40,
and 60 pc for Eris, ErisDark, VL2, and GHalo, respectively. The location of
the maximum enclosed density (for $r>\epsilon_{\rm soft}$) is denoted with a
filled circle. Dotted lines show the best-fit Einasto profiles (fitted to 1
kpc $<r<R_{\rm vir}$), whose parameters are given in Table 1.
Fig. 1 shows radial profiles of the averaged enclosed DM density,
$\langle\rho(<\\!r)\rangle=M(<\\!r)/(4\pi/3r^{3})$, for the four simulations.
The densities are higher in VL2 than in GHalo and ErisDark owing to the larger
virial mass and concentration of the VL2 host halo, but in Eris even higher DM
densities are reached at $r>500$ pc due to the baryonic contraction. Compared
to ErisDark the density profile is slightly steeper in Eris, and this is also
reflected in a much higher halo concentration. At even smaller radii, baryonic
physics produces a roughly constant mean density core.222These density
profiles are centered on the potential minimum. Centering on the point of
maximum DM density slightly changes the central slope. The radii at which the
enclosed density is maximized are denoted with filled circles in Fig. 1, and
it is clear that in Eris this maximum is significantly offset from the
dynamical center, while in the dissipationless simulations it occurs at close
to one $\epsilon_{\rm soft}$.
### 3.2. Three-dimensional localization of the density peak
Figure 2.— Slices of the central 2 kpc $\times$ 2 kpc DM density field at
$z=0$ through the location of maximum DM density. From left to right: x-y,
x-z, and y-z planes; from top to bottom: Eris, ErisDark, VL2, and GHalo
simulations. The position of the slice in the perpendicular direction is given
in the top left corner of every panel. The density field has been smoothed
with a Gaussian kernel of width $\sigma$ equal to one gravitational softening
length $\epsilon_{\rm soft}$ = 124, 124, 40, and 60 pc, respectively. The
z-axis coincides with the disk normal in Eris, and with the major axis of the
prolate DM density ellipsoid in the other cases. The slice thickness is 10pc
for Eris and ErisDark and 5pc for VL2 and GHalo. The images are centered on
the minimum of the total potential (marked with a white cross), while the
location of the maximum density is indicated with a red ’x’. In the three
dissipationless simulations (ErisDark, VL2, and GHalo) the offset between the
maximum DM density and the total potential minimum is less than $\epsilon_{\rm
soft}$, but it is 2.3 $\epsilon_{\rm soft}$ in Eris.
To study the DM density offset in more detail, we deposit DM particles using a
cloud-in-cell algorithm onto a three-dimensional grid with cell width equal to
10 pc for Eris and ErisDark ($500^{3}$ grid) and 5 pc for VL2 and GHalo
($1000^{3}$ grid). The grid is oriented in space such that the z-axis is
aligned with the disk normal in Eris, and with the major axis of the prolate
density ellipsoid for the other cases. We smooth the discretized 3D density
fields by applying a Gaussian smoothing kernel of width $\sigma$. Fig. 2 shows
slices through this grid with $\sigma=\epsilon_{\rm soft}$. The position of
the cell with the maximum DM density is indicated with a red cross, and again
it is clear that it is significantly displaced from the dynamical center is
Eris. The maximum occurs at $(x,y,z)=(260,30,-90)$ pc, corresponding to an
offset distance $D_{\rm off}=280$ pc, which is about 2.3 $\epsilon_{\rm
soft}$. In the dissipationless simulations on the other hand, $D_{\rm
off}=78$, 7.1, and 7.1 pc, all well within one $\epsilon_{\rm soft}$.
Figure 3.— The dependence of $D_{\rm off}$ and $\rho_{\rm max}$ on the width
$\sigma$ of the Gaussian density smoothing kernel. The gravitational softening
scale $\epsilon_{\rm soft}=124$ pc for Eris and ErisDark is indicated with the
dotted line. In the dissipationless simulations (ErisDark, VL2, and GHalo) the
DM offset remains well below $\epsilon_{\rm soft}$ for all $\sigma$. In Eris
the offset is larger than $\epsilon_{\rm soft}$ out to $\sigma=260$ pc (about
2 $\epsilon_{\rm soft}$). The maximum density is about a factor of two higher
in Eris than in ErisDark, and depends only mildly on $\sigma$ for
$\sigma>\epsilon_{\rm soft}$. $\rho_{\rm max}$ is larger in VL2 due to its
higher halo mass.
We have repeated the determination of $D_{\rm off}$ for different choices of
$\sigma$, and the results are shown in the top panel of Fig. 3. The values of
$D_{\rm off}$ for $\sigma\ll\epsilon_{\rm soft}$ are not meaningful, since
shot noise can lead to unphysical density spikes on very small scales. But for
$\sigma\gtrsim\epsilon_{\rm soft}$ the conclusions are robust: the DM offset
remains below one $\epsilon_{\rm soft}$, i.e. is consistent with no offset,
for all three DM-only simulations regardless of smoothing, while for Eris the
offset is greater than $\epsilon_{\rm soft}$ up to $\sigma\approx 260$ pc.
That $D_{\rm off}$ drops to zero for even higher $\sigma$ is not surprising,
since any density distribution will appear centered when smoothed on
sufficiently large scales.
In the bottom panel of Fig. 3 we show the maximum value of the density,
$\rho_{\rm max}$, for our four simulations. Of course $\rho_{\rm max}$ is much
more sensitive to $\sigma$ than $D_{\rm off}$, since larger smoothing includes
contributions from surrounding lower density material. At
$\sigma=\epsilon_{\rm soft}$ the density peaks a value of $0.84\,{\rm
M}_{\odot}\,{\rm pc}^{-3}$ in Eris, about 25% higher than the value
($0.67\,{\rm M}_{\odot}\,{\rm pc}^{-3}$) at its dynamical center. In ErisDark
$\rho_{\rm max}=0.40\,{\rm M}_{\odot}\,{\rm pc}^{-3}$, less than half of the
peak density in Eris. We caution, that the resolution of our numerical
simulations is not sufficient to resolve the internal density structure of the
offset peak. With higher resolution run the peak-to-center density contrast
may well be higher.
### 3.3. Evolution of the DM offset
Figure 4.— The time dependence of the DM offset from the total potential
minimum in Eris (blue) and ErisDark (red). The gravitational softening
$\epsilon_{\rm soft}(z)=124$ pc is indicated with a dashed line. In ErisDark
the DM offset remains around or below 1 $\epsilon_{\rm soft}$ for almost the
entire simulation, while in Eris it begins to significantly exceed
$\epsilon_{\rm soft}$ around $z=1.5$ and remains at $\approx 3\epsilon_{\rm
soft}$ afterwards.
We have performed the same analysis described in the previous section on all
400 outputs of the Eris and ErisDark simulations. Fig. 4 shows that the offset
measured in Eris at $z=0$ is no fluke, but persists over cosmological time
scales. At very early times ($z\gtrsim 2$) there is no offset in either Eris
or ErisDark. Starting at $z\approx 1.5$, however, the DM density maximum in
Eris starts to depart from the dynamical center. Over a period of about two
Gyr the DM offset grows to $D_{\rm off}\approx 340$ pc (almost 3
$\epsilon_{\rm soft}$), where it remains for the remainder of the simulation.
In contrast, in the ErisDark simulation $D_{\rm off}$ remains below 1
$\epsilon_{\rm soft}$ for almost its entire evolution, albeit with occasional
spikes up to $\sim 200$ pc. These results are qualitatively quite similar to
those reported by Macciò et al. (2012, see their Fig.4) in a similar, albeit 8
times lower resolution, simulation.
In Eris, $D_{\rm off}$ fluctuates around its time average (over the last 4
Gyr) of 340 pc with a root mean square (rms) dispersion of 51 pc. The closest
the peak comes to the center over this time is $D_{\rm off}=180$ pc. The peak
preferentially lies near the disk plane; its mean vertical (perpendicular to
the disk plane) displacement is only $\langle|z|\rangle=64$ pc, with an rms
dispersion of 46 pc. The maximum density varies around a mean value of
$\langle\rho_{\rm max}\rangle=0.84\,{\rm M}_{\odot}\ {\rm pc}^{-3}$ with an
rms dispersion of $0.02\,{\rm M}_{\odot}\ {\rm pc}^{-3}$, and reaches minimum
and maximum values of 0.79 and 0.92 $\,{\rm M}_{\odot}\ {\rm pc}^{-3}$.
Outputs in the Eris simulation are spaced $\sim 35$ Myr apart, which is too
long to resolve the dynamics of the offset peak, given the local dynamical
time of $\sim 15$ Myr. Nevertheless it is already clear from looking at this
coarsely time-sampled data that the peak locations are not randomly
distributed throughout the central region. We defer further discussion of the
temporal evolution of the DM offset and implications for its physical nature
to Section 4.
Figure 5.— Evolution of the inner DM density profiles. Top: Density profiles
at $z=$ 3, 2, 1.5, 1, 0.5, and 0 for Eris (left) and ErisDark (right). All
quantities are plotted in proper units. The black dotted lines indicate the
best-fitting modified Burkert profile (see Eq. 1). The location of $r_{5000}$,
the radius enclosing 5000 DM particles, is indicated with squares. $r_{5000}$
corresponds to the density profile convergence radius in ErisDark at $z=0$.
For Eris the density profile can be trusted to smaller radii due to its larger
particle counts, and so we additionally mark $r_{1000}$ with a circle. The
vertical bars indicate $r=3\epsilon_{\rm soft}$. Bottom: The evolution of the
logarithmic slope ($d\\!\ln\rho/d\\!\ln r$) measured at different radii: the
symbols and thick lines correspond to the slopes at $r_{5000}$, and for Eris
we also show the slope at $r_{1000}$. All lines are boxcar averages over 10
outputs. This plot shows that baryonic physics in Eris leads to the flattening
of the central density profile. This flattening appears to be correlated with
the growth of the DM offset (cf. Fig. 4). We emphasize that these slopes are
not asymptotic slopes, and a value of less than $-1$ does not imply a strongly
cusped profile all the way to the center, but merely indicates the local slope
at $r_{5000}$, which may lie outside of the scale radius at early times.
### 3.4. Correlation with DM density profile flattening
The growth of the DM density offset appears to be well correlated with a
flattening of the central DM density profile in Eris, as demonstrated in Fig.
5. The top panels show mean enclosed density profiles in the inner region
($r<5$ proper kpc) at several output times from $z=3$ to $z=0$. We see two
notable differences between Eris (left panel) and ErisDark (right). First, the
DM densities (plotted in proper units) tend to be higher in Eris than in
ErisDark, which is indicative of “adiabatic contraction”. Secondly, while the
enclosed density in ErisDark continues to increase towards smaller radii, the
profiles flatten out in Eris, indicating the formation of a core. Note that we
use the term “core” loosely, indicating a substantial flattening of the
density profile, but not necessarily implying a constant density.
Inner density profiles are notoriously difficult to properly resolve in N-body
simulations, requiring high force resolution, large particle counts, and
accurate time integration with sufficiently small timesteps (Power et al.,
2003; Zemp et al., 2007; Diemand et al., 2008; Dubinski et al., 2009). In
Pillepich et al. (in preparation) we investigate the influence of baryons on
the full shape of the density profiles in more detail, including convergence
studies. We find a convergence radius for the $z=0$ density profile in
ErisDark of 0.9 kpc, corresponding to $\sim 5000$ enclosed particles. The
square symbols in the top panel of Fig. 5 indicate $r_{5000}$ (the radius
enclosing 5000 DM particles) for the different outputs. Note that $r_{5000}$
is significantly farther out than $3\,\epsilon_{\rm soft}$ (indicated with the
vertical bars), a commonly used “rule of thumb” for how far in density
profiles can be trusted. For Eris, $r_{5000}$ is probably overly conservative,
since the potential is dominated by baryons and many more than 5000 gas and
star particles are found inside of $r_{5000}$. As an intermediate case we also
mark with circles $r_{1000}$, the radius enclosing 1000 DM particles.333At
$z=0$, there are over a million star particles within $r_{5000}$ and more than
400,000 within $r_{1000}$.
In the bottom panel of Fig. 5 we show the evolution of the logarithmic slope
$d\\!\ln\rho/d\\!\ln r$, measured at these radii. We obtained these slopes by
first fitting the enclosed density profiles inwards of 5 kpc to a modified
Burkert profile,
$\langle\rho\rangle(r)=\frac{\rho_{0}\,r_{c}^{\beta}}{r^{\alpha}\,\left(r^{2}+r_{c}^{2}\right)^{(\beta-\alpha)/2}},$
(1)
and then evaluating its logarithmic slope at the radii of interest,
$\frac{{\rm d}\\!\ln\langle\rho\rangle}{{\rm d}\\!\ln
r}=-\alpha+\frac{\alpha-\beta}{1+(r_{c}/r)^{2}}.$ (2)
This functional form describes the central enclosed density profile in Eris
and ErisDark much better than a constant power law or Einasto profile. Note,
however, that this fitting function is designed to track the enclosed density
profile all the way down to $\epsilon_{\rm soft}$, so way beyond the
convergence radius. We wish to emphasize that we merely use this fit to
evaluate the logarithmic slope at radii outside of 3 $\epsilon_{\rm soft}$,
and do not attach any physical meaning to the asymptotic inner slopes $\alpha$
preferred by the fits.
Fig. 5 clearly shows that Eris develops a cored density profile, in the sense
that it has a much shallower logarithmic slope at $r_{5000}$ than ErisDark.
Measured at $r_{5000}$ the log slope in Eris becomes progressively shallower
until it stabilizes at a value of $\sim-0.3$ at $z\approx 1$. Note that
$r_{5000}$ is larger than $D_{\rm off}(z)$ at all redshifts. Measured at
$r_{1000}$ the profile even flattens out completely with a log slope of $\sim
0$. In ErisDark, on the other hand, the slope at $r_{5000}$ remains close to
the NFW value of $-1$ after $z=1.5$. Comparison with Fig. 4 shows that the
onset and time scale of the core formation in Eris is remarkably well
correlated with the growth of the DM density offset. This is a strong hint
that whatever baryonic process drives the core formation may also be
responsible for the offset DM density peak.
## 4\. Nature and Origin of the Offset Peak
The existence of a well defined offset to the peak of the DM distribution in
Eris is unexpected, and its physical nature not immediately apparent. Some
possibilities that we have considered are: (i) statistical fluctuations, (ii)
an incompletely dissolved subhalo core, and (iii) excitation of a DM density
wave by the stellar bar or other perturber. In the following sections we
address these possibilities in turn.
As we have seen, the output cadence of the original Eris simulation is not
sufficient to follow the dynamics of the peak. To help with understanding the
physical nature of the offset, much higher cadence outputs are desirable. To
this end we have restarted the Eris simulation from output number 393
($z=0.0151$) with a $\sim 20$ times finer temporal resolution ($\Delta t=1.43$
Myr) and evolved the simulation for 140 time steps (206 Myr). Visualizations
of nine consecutive high cadence outputs, spanning 11.5 Myr, are shown in Fig.
6, and a movie of the full evolution can be viewed at
http://vimeo.com/45114776.
Visual inspection of these much higher cadence outputs shows that the offset
peak appears to sometimes jump around somewhat discontinuously between two
subsequent outputs. Occasionally there are multiple peaks of roughly equal
density. As already seen with the coarser time resolution, the point of
maximum density appears to avoid the very center and remains close to the disk
plane.
Figure 6.— 3D contour plots of the DM density in the central $(2\,{\rm
kpc})^{3}$ volume of Eris. The outer contour (light blue) corresponds to
$\rho=0.45\,\,{\rm M}_{\odot}\ {\rm pc}^{-3}$, the middle contour (light
orange) to $0.8\,\,{\rm M}_{\odot}\ {\rm pc}^{-3}$, and the opaque innermost
contour (red) to the 99th percentile of the DM density in the volume. The
dynamical center is marked with a black dot, and the stellar disk lies in the
X-Y plane. The images show the time evolution over 11.5 Myr ($\Delta t=1.43$
Myr) starting from $z=0.0151$. In the last two frames we have plotted the
positions of all 78 particles located within 1 $\epsilon_{\rm soft}$ of the
maximum density at the second-to-last output shown. Only 1.43 Myr later (last
frame) they have already dispersed throughout the plotted volume, indicating
that the density offset is not a bound structure.
Figure 7.— Probability distribution functions of the maximum density
$\rho_{\rm max}$ (top panel) and its distance to the potential minimum $D_{\rm
off}$ (bottom), compared between the Eris simulation (blue) and random samples
(orange). The Eris sample consists of the last $\sim 124$ outputs spanning 4
Gyr of evolution. The random sample is comprised of 500 realizations of a
randomly drawn particle distributions with the same uniform mean density as
the Eris core inwards of 1 kpc.
### 4.1. Statistical fluctuations?
As shown in Sec. 3.3, the growth of the DM offset coincides in time with the
formation of a core in the Eris DM distribution. In a constant density core,
sampled with a small number of N-body particles, Poisson fluctuations can give
rise to a spurious density peak offset from the center. The mean DM density in
the Eris’ core (inwards of 1 kpc) is $0.51\,{\rm M}_{\odot}\,{\rm pc}^{-3}$,
corresponding to an N-body particle density of 5,300 particles kpc-3. At $z=0$
there are 509 particles within 2 $\epsilon_{\rm soft}$ of the location of
maximum density, much higher than the expected 306 for the average core
density and well beyond what Poissonian fluctuations can produce. We have
numerically confirmed this by constructing 500 randomly drawn particle
distributions with the same mean particle density as the Eris core, and
analyzed these samples in the same manner as described above.
In Fig. 7 we compare the distributions of $\rho_{\rm max}$ and $D_{\rm off}$
between the random samples and the last $\sim 124$ Eris outputs spanning 4 Gyr
of evolution. Eris’ $\rho_{\rm max}$ distribution is peaked at $0.85\,{\rm
M}_{\odot}\,{\rm pc}^{-3}$, way beyond the random distribution, which peaks at
$0.61\,{\rm M}_{\odot}\,{\rm pc}^{-3}$ and has an rms dispersion of only
$0.01\,{\rm M}_{\odot}\,{\rm pc}^{-3}$. Similarly, the $D_{\rm off}$
probability distribution function for Eris is peaked around 350 pc and looks
nothing like that of the random samples, which slowly increases towards larger
$D_{\rm off}$, in accordance with a simple volumetric $D_{\rm off}^{1/3}$
scaling. These comparisons very clearly demonstrate that the DM offsets we
have found in Eris are not due to statistical fluctuations.
### 4.2. Incompletely disrupted subhalo core?
In this potential explanation, the offset density peak would be identified
with the tightly bound central regions of an incompletely disrupted subhalo.
The subhalo would have to have been massive enough to have experienced
substantial baryonic condensation and associated contraction of its DM, making
it more resilient to complete tidal disruption than its DM-only counterpart in
ErisDark. If such a halo fell into the host at $z\gtrsim 1.5$, dynamical
friction would quickly drag it in to the center, where tidal interactions
would have stripped most of the weakly bound material in its outskirt. The
inspiral would have stalled once its mass dropped to the point where dynamical
friction becomes unimportant. After that point it would continue to orbit
around the dynamical center, its temporal cohesion possibly aided by the
stabilizing effect of a harmonic potential (Kleyna et al., 2003; Read et al.,
2006).
We can rule out this possibility, based on several observations: (i) as
mentioned above there is no stellar counterpart to the DM offset, and (ii) the
peak persists for hundreds of dynamical times, which seems too long for an
orbiting subhalo core to survive. The final nail in the coffin (iii) is the
fact that the offset DM peak does not appear to be a bound feature. This is
demonstrated in the last two panels of Fig. 6, in which we show the positions
of all 78 particles located within 1 $\epsilon_{\rm soft}$ of the maximum
density peak at $t=10$ Myr (after $z=0.015$). A mere 1.4 Myr later, these same
particles have dispersed throughout the central volume. This implies that the
offset density peak is comprised of different particles at different times.
### 4.3. External perturber?
Figure 8.— Visualizations of the motion of the potential minimum in Eris in
absolute (code) units, color coded by redshift. Top: From $z=1.5$ to $z=1.1$.
At $z=1.3$ a satellite passes near to the center, but it does not noticeably
perturb the dynamical center. Bottom: the last 40 outputs, from $z=0.1$ down
to $z=0$. The location of the maximum DM density is marked with a small black
sphere connected with a thin line to the position of the potential minimum at
that time.
Next we consider an external perturber as a possible mechanism to give rise to
the DM offset. The stellar disk in the center is self-bound, in the sense that
it does not require the DM halo to be held together gravitationally. An
external perturber, for example in the form of a passing satellite, could then
impart a kick to the stellar disk that might cause it to slosh around in the
underlying stationary DM halo. In this situation the DM density peak would
correspond to the cuspy center of the DM halo, and the offset would simply
reflect the displacement of the potential minimum (the stars). One might
expect dynamical friction to quickly damp out any such sloshing, but the
efficiency of dynamical friction is strongly reduced in a constant density
(harmonic) core (Read et al., 2006), and thus the center of the self-bound
stellar disk may be able to “orbit” around the DM density maximum for many
dynamical times.
Some fraction of the DM (the low velocity tail) would also be bound to the
disk and would move with the stars. In this picture, the existence of a DM
offset may be very sensitive to the stellar mass of the galaxy. For a DM halo
of a given mass, a higher stellar mass galaxy would bind more of the DM to it,
and reduce the strength of the DM offset or eliminated it altogether.
Hydrodynamic galaxy formation simulations suffering from a strong baryonic
overcooling problem may thus not be able to observe DM offsets.
In Eris we have indeed identified a satellite passing close to the center
($D\approx 65$ kpc) around $z=1.3$, which may be a good candidate for an
external perturber, since the DM offset begins to grow around that time. The
satellite’s mass is $1.8\times 10^{10}\,{\rm M}_{\odot}$ at infall ($z=2.7$)
and $2.8\times 10^{9}\,{\rm M}_{\odot}$ at the time of close passage. We have
examined the motion of the potential minimum in absolute code coordinates (top
panel of Fig. 8), and do not see any evidence for a sharp kink or abrupt
displacement at the time of the satellite’s passage. Furthermore, it appears
that the potential minimum is moving smoothly in code coordinates, and it is
the DM offset that moves around the potential minimum, not the other way
around (bottom panel of Fig. 8).
### 4.4. Density wave excitation by the stellar bar?
Figure 9.— Probability distribution functions of the cosine of the angle
between the direction towards the DM offset and the orientation of the stellar
bar, defined as the major axis of the stellar density ellipsoid inwards of 1
kpc. The three panels show the distributions split by cosmic time: early
times, prior to the appearance of the DM offset (top panel), an intermediate
time period over which the DM offset grows in magnitude (middle), and late
times, during which the DM offset remains more or less constant at $\sim 340$
pc (bottom). The early distribution is consistent with uniform in
$|\\!\cos\,\theta|$, whereas once the DM offset becomes pronounced the
distributions prefer large values of $|\\!\cos\,\theta|$, indicating alignment
between the offset peak and the stellar bar.
Another possibility is that the DM offset is the result of a density wave
excitation of some kind. In this explanation the peak would be contributed by
different particles at different times, just as observed, its spatial
evolution would be set by whatever external source is providing the
excitation, and it may be long lived, provided a continuous excitation
mechanism exists. A natural candidate for such an external source would be the
stellar bar in Eris. The potential is completely dominated by the stars in the
central region, and so it seems plausible that the DM distribution may be
affected by departures from axisymmetry in the stellar component. Indeed,
resonant interactions between the DM halo and a stellar bar have been invoked
to explain the transformation of an initially cuspy DM density profile into a
cored one (Weinberg & Katz, 2002, 2007). In related work, McMillan & Dehnen
(2005) identified an instability in which a rotating stellar bar pinned to the
origin causes the DM cusp to move away from the origin. While they used this
effect to argue that the observed flattening of the DM density profile may be
artificial, their work provides a proof-of-concept for an off-center DM
density peak.
In isolated galaxy simulations it has been possible to identify DM halo
particles trapped in different resonances (Athanassoula, 2002; Ceverino &
Klypin, 2007; Dubinski et al., 2009). The Inner Lindblad Resonance may lead to
the formation of a “dark bar” (Colín et al., 2006; Ceverino & Klypin, 2007),
which may be showing up in Eris as a DM offset. The ring-like morphology of
particles trapped in the co-rotation resonance (see e.g. Fig.11 of Ceverino &
Klypin, 2007) is reminiscent of the distribution of DM in the plane of Eris
(see Fig. 2). A detailed investigation of the resonant structure of DM
particle orbits in Eris is beyond the scope of this paper, but will be pursued
in future work.
Figure 10.— Top: Frequency scan of the angular evolution of the direction
towards the DM offset. $R^{2}$ is the sum of the squared differences between
the best-fit sinusoid of frequency $\omega$ and the angular position of the DM
offset. Lower values of $R^{2}$ indicate a better fit, and the deep spike to
$R^{2}<1$ at $\sim 15\,{\rm Gyr}^{-1}$ corresponds to a periodicity of 69 Myr
in the angular evolution. Bottom: The best-fitting sinusoid (red) overplotted
on angular position of the DM offset in the last 22 outputs.
Arguments in favor of the bar-driven explanation are that the growth of the
offset coincides with the formation of a DM core, that the offset peak
preferentially lies close to the stellar disk plane, and that the direction
from the dynamical center towards the offset density peak appears to be
correlated with the orientation of the stellar bar. This last point is
demonstrated in Fig. 9, which shows distributions of $|\\!\cos\,\theta|$, the
angle between the offset peak and the major axis of the stellar density
ellipsoid inwards of 1 kpc (i.e. the orientation of the stellar bar).
We have split the distributions up by time epoch: during early times ($<4$
Gyr), prior to the appearance of a significant DM density offset, the
distribution is consistent with uniform in $|\\!\cos\,\theta|$, corresponding
to a completely random placement of the density peak. Between 4 and 8 Gyr the
DM offset first begins to become significantly larger than $\epsilon_{\rm
soft}$, and the $|\\!\cos\,\theta|$ PDF is strongly peaked towards unity. Its
mean value of $\langle|\\!\cos\,\theta|\rangle=0.83$ corresponds to an angle
of $34^{\circ}$. At later times, once the DM offset has become established,
the correlation weakens slightly, with $\langle|\\!\cos\,\theta|\rangle$
dropping to 0.77, but it still remains preferentially aligned with the bar.
The top panel of Fig. 10 shows the results of a frequency scan from
$\omega/2\pi=0.1$ to $10^{3}\,{\rm Gyr}^{-1}$ of the angular evolution of the
direction towards the offset. For every frequency $\omega$, we performed a
least squares fit to a sinusoid for all coarsely sampled outputs after 8 Gyr.
The plot of $R^{2}=\sum_{i}[\theta_{\rm
off}(t_{i})-\pi/2\cos(\omega(t_{i}-t_{o})]^{2}$ exhibits a distinct low
$R^{2}$ spike at a frequency of $\omega/2\pi=15\,{\rm Gyr}^{-1}$,
corresponding to a period of 69 Myr, as well as at several of its harmonics
($30,45,60\,{\rm Gyr}^{-1}$, etc.). These spikes indicate the presence of a
periodic signal in the offset angle evolution, which is consistent with the
alignment of the offset with the orientation of the stellar bar. In the lower
panel of Fig. 10 we show the best-fit sinusoid overplotted on the offset angle
evolution for the 22 outputs since 13 Gyr.
For completeness we should also mention two problematic aspects of the stellar
bar-driven explanation. One is that the strength of stellar bars typically
grows and fades with time (Bournaud & Combes, 2002), modulated by gas
accretion. Indeed this is the case in Eris as well: the amplitude of the $m=2$
mode in its stellar disk is strongest at $z>3$, then declines somewhat only to
grow again around $z\approx 2$, after which it gradually decreases in strength
towards $z=0$ (see Fig. 4 of Guedes et al., 2012). It is not clear then why
the DM offset would saturate at around $300-400$ pc in the last 8 Gyr, when
the strength of the bar is gradually decreasing. The $\sim 340$ pc extent of
the DM offset is also surprisingly small, since Eris’ stellar bar is several
kpc in length, and its co-rotation radius occurs at 2.7 kpc. It is clear that
more work is needed to fully elucidate the role of a stellar bar interaction
in explaining the DM offset.
### 4.5. Numerical Resolution
At $D_{\rm off}\approx 3\epsilon_{\rm soft}$, the scale of the offset is
worrisomely close to the force resolution of the simulation. A detailed study
of the internal density structure of the offset peak must certainly await much
higher resolution hydrodynamic simulations. Nevertheless, the fact that no
offset is observed in ErisDark, which has the same force resolution as Eris,
gives us some confidence that the existence of an offset DM density peak is no
numerical artifact, and indicates that baryonic physics appears to be
responsible in some way.
We would like to be able to check whether the offset remains at the same
physical scale ($\sim 340$ pc) even in a simulation with a smaller
gravitational softening length. At the moment we do not have access to such a
simulation444We are in the process of re-running the Eris simulation with
$\epsilon_{\rm soft}$=50 pc, but this is an expensive simulation that will not
finish for several months., but we can go in the opposite direction and look
in a lower resolution run. We have run ErisLores and ErisDarkLores at eight
times poorer mass resolution ($7.8\times 10^{5}\,{\rm M}_{\odot}$ and
$9.5\times 10^{5}\,{\rm M}_{\odot}$, respectively) and with a four times
larger gravitational softening length, $\epsilon_{\rm soft}$=495 pc. We have
analyzed these simulations in the same way as described above, except that all
length scales have been increased by a factor of four to account for the
larger $\epsilon_{\rm soft}$. We CIC-deposit particles onto a grid with cell
width of 40 pc and smooth the density field with a Gaussian kernel with
$\sigma=\epsilon_{\rm soft}=495$ pc.
As with the higher resolution simulations, we find that the point of maximum
density is displaced from the minimum of the potential in ErisLores: $D_{\rm
off}=430$ pc at $z=0.15$ and 650 pc at $z=0$. And again the DM-only
counterpart ErisDarkLores has a much smaller offset, $D_{\rm off}$=60 pc.
Although $D_{\rm off}$ is somewhat larger in ErisLores than in Eris in
absolute terms, it is clear that the offset is not scaling linearly with
$\epsilon_{\rm soft}$: while $D_{\rm off}\approx 3\epsilon_{\rm soft}$ in
Eris, it is only 1 - 1.5 $\epsilon_{\rm soft}$ in ErisLores.
We have also checked time steps and energy conservation in our simulations, as
insufficiently short time steps can lead to the formation of a spurious
density core (Zemp et al., 2007; Dubinski et al., 2009). The Eris simulation
suite uses adaptive time steps that scale as $\sqrt{\epsilon_{\rm soft}/a}$,
where $a$ is the acceleration of a particle, as recommended by Power et al.
(2003). This results in time steps as small as 0.16 Myr at low redshift, which
is three times smaller than the fixed time step that Dubinski et al. (2009)
employed in a simulation with comparable $\epsilon_{\rm soft}$ (their model
m100K) and which they found to be short enough to avoid the formation of an
artificial core.
These checks give us further confidence that the DM offset we have observed in
Eris cannot be attributed solely to insufficient numerical resolution or
timestepping. On the other hand, we cannot claim to have fully resolved
$D_{\rm off}$ either – higher resolution studies are needed.
### 4.6. The role of the central DM core
It is clear that the central DM core plays an important role for the formation
of an off-center DM density peak. Without a flattened central density profile
it may be difficult to excite an offset peak, since any non-central density
enhancement would be overwhelmed by the steeply rising central cusp. Indeed,
as we have seen above, the growth of the DM offset temporally coincides with
the formation of such a core. We have not yet determined what physical
processes led to the formation of this core in Eris.
A core formation mechanism that has recently been suggested is the repeated
removal of large amounts of gas from the central regions through the action of
violent supernovae explosions (Read & Gilmore, 2005; Pontzen & Governato,
2012). If these baryonic outflows abruptly alter the potential, the DM may
respond by flowing out of the center, transforming the cusp into a core in the
process. Perhaps this DM redistribution could also create an offcenter DM
density peak. Note, however, that the central potential in the Eris galaxy is
already dominated by its stellar component by the time the core formation
begins at $z\sim 2$. Even though Eris does experience a star burst triggered
by a merger at $z\simeq 1$, the associated supernovae feedback is therefore
unlikely to substantially alter the central potential. Furthermore, this is a
single star formation event, while the mechanism of Pontzen & Governato (2012)
requires multiple application of violent outflows followed by gradual re-
accretion of gas.
To further check the role of SN feedback on the formation of a DM density core
and peak offset, we have analyzed the last output ($z=0.7$) of the ErisLT run,
which is identical to Eris except that it uses a lower star formation
threshold (0.1 cm-3). This lower SF threshold reduces the efficiency of the
supernova-driven gas blowout and results in a disk galaxy with a larger bulge-
to-disk ratio and a more compact stellar configuration (see Guedes et al.,
2011). The DM distribution, however, appears not to have been affected as
dramatically. The DM density profile in ErisLT is almost identical to the one
in Eris at $z=0.7$, both exhibiting a core with radius $\sim 1$ kpc, with
$D_{\rm off}$=350 pc in ErisLT and $D_{\rm off}$=360 pc in Eris.
The absence of a dependence in the DM distribution on the star formation
threshold suggests that SN-driven outflows are not the only core formation
mechanism, highlighting a possible important difference between the evolution
of Milky-Way sized galaxies and that of the dwarf galaxies described by
Governato et al. (2010, 2012). Instead, perhaps the stellar bar may be
responsible for this transformation, through resonant angular momentum
exchange between the bar and the DM as suggested by Weinberg & Katz (2002,
2007). Clearly, a more detailed investigation of the DM core formation process
in Eris is needed, and will be the topic of a future study.
## 5\. Discussion
Provided the central DM distribution in the Eris simulation is representative
of that in our Milky Way, the results we have presented here have several
important consequences for indirect DM detection efforts aimed at the GC.
The first concerns the shape of the DM density profile used to predict the
annihilation luminosity from the central regions of the Galaxy. Outside of
$\sim 1$ kpc the DM density profile in both Eris and ErisDark (as well as VL2
and GHalo) is well described by an NFW or Einasto profile. However, cooling
and condensation of gas in Eris has dragged DM towards the center, such that
the normalization of the density profile in Eris is larger than in ErisDark:
at 1 kpc the mean enclosed DM density is 2.7 times higher in Eris ($0.52\,{\rm
M}_{\odot}\,{\rm pc}^{-3}$) than in ErisDark ($0.20\,{\rm M}_{\odot}\,{\rm
pc}^{-3}$). At angles greater than $\sim 7$ degrees from the GC, one may thus
expect baryonic processes to lead to a significant enhancement of the surface
brightness of diffuse radiation arising from DM annihilation.
As discussed in Sec. 3.4, at even smaller radii baryonic physics leads to the
formation of a DM core in Eris, and as a result the mean enclosed DM density
in Eris never rises to more than $0.73\,{\rm M}_{\odot}\,{\rm pc}^{-3}$. For
comparison, an extrapolation of the best-fitting Einasto profile for ErisDark
reaches 1 $\,{\rm M}_{\odot}\,{\rm pc}^{-3}$ at 200 pc, 10 $\,{\rm
M}_{\odot}\,{\rm pc}^{-3}$ at 3 pc, and asymptotes to 90 $\,{\rm
M}_{\odot}\,{\rm pc}^{-3}$. We thus caution against extrapolating density
profiles determined from dissipationless DM-only simulations (such as Via
Lactea II, GHalo, and Aquarius) all the way in to the GC in order to infer
central annihilation luminosities.
Figure 11.— DM annihilation luminosity “surface density” ($\int\\!\rho_{\rm
DM}^{2}\,{\rm d}\ell$) in the central $2\,{\rm kpc}\times 2\,{\rm kpc}$ region
of Eris. The contrast between the dynamical center (white plus) and the peak
(red cross) is $\sim 11\%$. The vertical axis corresponds to the disk normal,
and the coordinate system has been rotated to maximize the angular offset of
the peak.
Secondly, a maximum in the DM density that is not coincident with the
dynamical center of the Galaxy significantly modifies expectations for where
the central surface brightness of DM annihilation radiation should peak. The
time-averaged value of $D_{\rm off}$ in Eris is 340 pc, which, if seen edge
on, would correspond to an angular offset of 2.4 degrees. For comparison, the
search for a DM annihilation signal by the H.E.S.S. imaging Atmospheric
Cherenkov Telescope was restricted to a region of $45-150$ pc in projected
distance from the GC (Abramowski et al., 2011). In this context, the recent
report of a highly significant detection of a gamma-ray line at $127\pm 2$ GeV
in Fermi data from the GC (Su & Finkbeiner, 2012) is particularly intriguing.
Su & Finkbeiner (2012) report that the signal is maximized at a Galactic
longitude of $\ell\approx 1.5$ degrees. While this offset may have initially
been viewed as a strike against a DM annihilation interpretation of the line,
our work demonstrates as a proof-of-principle that just such an offset should
perhaps be expected.
In Fig. 11 we show the DM annihilation luminosity “surface density”,
$\int\\!\rho_{\rm DM}^{2}{\rm d}\ell$, calculated from the central $(2\,{\rm
kpc})^{3}$ region in Eris. The view is through the disk plane (the vertical
axis corresponds to the disk normal) and the coordinate system has been
rotated such that the offset peak is viewed edge-on, i.e. maximizing the
angular offset. As before, the 3D density grid was first smoothed with a
Gaussian kernel of width $\sigma=\epsilon_{\rm soft}$. We then squared it,
multiplied the cells by their volume to get a luminosity, summed them up along
the Y-axis, and finally divided the resulting map by the area of each cell to
get a surface density555To further clarify, the sum of all $200^{2}$ “pixels”
multiplied by the total area ($4\times 10^{6}\,{\rm pc}^{2}$) is equal to the
total luminosity emitted from the $(2\,{\rm kpc})^{3}$ cube. in units of ${\rm
M}_{\odot}^{2}\,{\rm pc}^{-5}$. This map does not take into account any
luminosity produced by DM along the line of sight outside of the central cube.
The contrast between the maximum of the map ($870\,{\rm M}_{\odot}^{2}\,{\rm
pc}^{-5}$) and the center ($780\,{\rm M}_{\odot}^{2}\,{\rm pc}^{-5}$) is only
about 11% at $z=0$. Furthermore there is a second sub-dominant peak to the
left of the center with a luminosity surface density of $830\,{\rm
M}_{\odot}^{2}\,{\rm pc}^{-5}$, only 5% less than the global maximum. If
anything the contrast at $z=0$ is a bit of an outlier, as the mean contrast
between offset peak and GC over the past 4 Gyr is only $\sim 5\%$ with a 3%
standard deviation. The greatest contrast measured over all outputs is 15%.
Such small contrasts may not be compatible with the measurement of Su &
Finkbeiner (2012), but it is important to keep in mind that the internal
structure of the peak is certainly not resolved in the simulation. It is
conceivable that the contrast would increase with higher resolution,
especially if resonances with the stellar bar are responsible, since those
would be spread out artificially at low resolution.
## 6\. Conclusions
We have analyzed the distribution of DM in the central regions of the
hydrodynamical galaxy formation simulation Eris, one of the highest resolution
and most realistic simulations to date of the formation of a barred spiral
galaxy like our own Milky Way. Surprisingly, we find that the peak of the DM
density in Eris is typically offset from its dynamical center by several
hundred parsec. No such offset is observed in its DM-only twin simulation
ErisDark, nor in the much higher resolution DM-only Via Lactea II and GHalo
simulations.
The DM offset in Eris begins to appear around $z=1.5$ and grows over a period
of 2 Gyr to a stable value of $\langle D_{\rm off}\rangle=340$ pc (almost
three gravitational softening lengths), with a dispersion of 50 pc. The onset
and duration of the DM offset appears to be well correlated with the formation
of a nearly constant DM density core. The distributions of $\rho_{\rm max}$
and $D_{\rm off}$ over the past 4 Gyr are inconsistent with a statistical
fluctuation. Neither is the density peak a gravitationally bound structure,
which rules out an incompletely disrupted subhalo core as an explanation. The
most likely explanation may be a density-wave-like excitation by the stellar
bar, possibly related to the resonant mechanism proposed by Weinberg & Katz
(2002, 2007) to explain the transformation of a central DM cusp into a core.
Arguments in favor of this explanation are the fact that the DM offset appears
preferentially near the disk plane, that it is aligned to $\sim 30$ degrees
with the orientation of the stellar bar, and that is shows a periodicity of
$\sim 70$ Myr.
A central DM offset is of particularly interest in the context of the recent
report by Su & Finkbeiner (2012) of a highly significant detection of gamma-
ray line emission from a region $\sim 1.5$ degrees ($\sim 200$ pc projected)
away from the Galactic Center. At first impression such a large angular offset
would seem to argue against a DM annihilation interpretation of this signal.
Our work demonstrates that in fact just such an offset should perhaps be
expected. We note, however, that at the current resolution of our numerical
simulations the low contrast ($5-15\%$) of the annihilation surface brightness
between the offset peak and the GC may be too small to accommodate a DM
annihilation explanation.
We conclude by acknowledging that properly resolving the effects of baryonic
physics on the central DM distribution, in particular those involving resonant
bar-halo interactions, requires much higher resolution than we have been able
to afford so far in cosmological hydrodynamics simulations. Further studies at
higher resolution and exploring different baryonic physics implementation are
sorely needed. Of particular importance are clarifying the role of the star
formation threshold parameter and supernovae feedback for the formation of a
DM core, as well as the influence of the supermassive black hole at the GC on
the DM distribution.
MK thanks Doug Finkbeiner for initially suggesting this analysis and several
encouraging exchanges afterwards, and Andrey Kravtsov, Daniel Ceverino,
Anatoly Klypin, Eliot Quataert, and Justin Read for valuable discussion of
these results. JG was partially funded by the ETH Zurich Postdoctoral
Fellowship and the Marie Curie Actions for People COFUND Program. The high
time resolution re-reun of Eris was performed on the Cray XE6 Monte Rosa at
the Swiss National Supercomputing Centre (CSCS). This work was supported in
part by a grant from the Swiss National Supercomputing Centre (CSCS) under
project IDs s205 and s352, and in part by the U.S. National Science
Foundation, grants OIA-1124453 (PI P. Madau) and OIA-1124403 (PI A. Szalay).
## References
* Abazajian (2011) Abazajian, K. N. 2011, Journal of Cosmology and Astro-Particle Physics, 03, 010
* Abazajian & Kaplinghat (2012) Abazajian, K. N., & Kaplinghat, M. 2012, ArXiv e-print 1207.6047
* Abramowski et al. (2011) Abramowski, A., Acero, F., Aharonian, F., et al. 2011, Physical Review Letters, 106, 161301
* Aharonian et al. (2012) Aharonian, F., Khangulyan, D., & Malyshev, D. 2012, ArXiv e-print 1207.0458
* Aharonian et al. (2006) Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006, Physical Review Letters, 97, 221102
* Allen (1998) Allen, S. W. 1998, Monthly Notices of the Royal Astronomical Society, 296, 392
* Athanassoula (2002) Athanassoula, E. 2002, The Astrophysical Journal Letters, 569, L83
* Athanassoula (2003) —. 2003, Monthly Notices of the Royal Astronomical Society, 341, 1179
* Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, The Astrophysical Journal, 697, 1071
* Baltz et al. (2008) Baltz, E. A., Berenji, B., Bertone, G., et al. 2008, Journal of Cosmology and Astro-Particle Physics, 07, 013
* Bartko et al. (2009) Bartko, H., Martins, F., Fritz, T. K., et al. 2009, The Astrophysical Journal, 697, 1741
* Berentzen et al. (2006) Berentzen, I., Shlosman, I., & Jogee, S. 2006, The Astrophysical Journal, 637, 582
* Bergström (2012) Bergström, L. 2012, Physical Review D, 86, 103514
* Bergström et al. (1998) Bergström, L., Ullio, P., & Buckley, J. H. 1998, Astroparticle Physics, 9, 137
* Blitz & Spergel (1991) Blitz, L., & Spergel, D. N. 1991, The Astrophysical Journal, 379, 631
* Blumenthal et al. (1986) Blumenthal, G. R., Faber, S. M., Flores, R., & Primack, J. R. 1986, The Astrophysical Journal, 301, 27
* Bournaud & Combes (2002) Bournaud, F., & Combes, F. 2002, Astronomy and Astrophysics, 392, 83
* Bringmann et al. (2012) Bringmann, T., Huang, X., Ibarra, A., Vogl, S., & Weniger, C. 2012, Journal of Cosmology and Astro-Particle Physics, 07, 054
* Buckley & Hooper (2012) Buckley, M. R., & Hooper, D. 2012, Physical Review D, 86, 43524
* Ceverino & Klypin (2007) Ceverino, D., & Klypin, A. 2007, Monthly Notices of the Royal Astronomical Society, 379, 1155
* Chalons et al. (2012) Chalons, G., Dolan, M. J., & McCabe, C. 2012, ArXiv e-prints, 1211, 5154
* Cline (2012) Cline, J. M. 2012, Physical Review D, 86, 15016
* Colín et al. (2006) Colín, P., Valenzuela, O., & Klypin, A. 2006, The Astrophysical Journal, 644, 687
* Danninger & the IceCube collaboration (2012) Danninger, M., & the IceCube collaboration. 2012, Journal of Physics: Conference Series, 375, 012038
* Diemand et al. (2008) Diemand, J., Kuhlen, M., Madau, P., et al. 2008, Nature, 454, 735
* Dobler et al. (2010) Dobler, G., Finkbeiner, D. P., Cholis, I., Slatyer, T., & Weiner, N. 2010, The Astrophysical Journal, 717, 825
* Dubinski et al. (2009) Dubinski, J., Berentzen, I., & Shlosman, I. 2009, The Astrophysical Journal, 697, 293
* Dudas et al. (2012) Dudas, E., Mambrini, Y., Pokorski, S., & Romagnoni, A. 2012, Journal of High Energy Physics, 10, 123
* Fall & Efstathiou (1980) Fall, S. M., & Efstathiou, G. 1980, Monthly Notices of the Royal Astronomical Society, 193, 189
* Genzel et al. (2010) Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121
* Ghez et al. (2008) Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, The Astrophysical Journal, 689, 1044
* Gillessen et al. (2009) Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, The Astrophysical Journal, 692, 1075
* Gnedin et al. (2004) Gnedin, O. Y., Kravtsov, A. V., Klypin, A. A., & Nagai, D. 2004, The Astrophysical Journal, 616, 16
* Gnedin & Primack (2004) Gnedin, O. Y., & Primack, J. R. 2004, Physical Review Letters, 93, 61302
* Gondolo & Silk (1999) Gondolo, P., & Silk, J. 1999, Physical Review Letters, 83, 1719
* Goodenough & Hooper (2009) Goodenough, L., & Hooper, D. 2009, ArXiv e-print 0910.2998
* Governato et al. (2010) Governato, F., Brook, C., Mayer, L., et al. 2010, Nature, 463, 203
* Governato et al. (2012) Governato, F., Zolotov, A., Pontzen, A., et al. 2012, Monthly Notices of the Royal Astronomical Society, 422, 1231
* Guedes et al. (2011) Guedes, J., Callegari, S., Madau, P., & Mayer, L. 2011, The Astrophysical Journal, 742, 76
* Guedes et al. (2012) Guedes, J., Mayer, L., Carollo, M., & Madau, P. 2012, ArXiv e-prints 1211.1713
* Holley-Bockelmann et al. (2005) Holley-Bockelmann, K., Weinberg, M., & Katz, N. 2005, Monthly Notices of the Royal Astronomical Society, 363, 991
* Hooper & Goodenough (2011) Hooper, D., & Goodenough, L. 2011, Physics Letters B, 697, 412
* Hooper & Linden (2011) Hooper, D., & Linden, T. 2011, Physical Review D, 84, 123005
* Jog & Combes (2009) Jog, C. J., & Combes, F. 2009, Physics Reports, 471, 75
* Jog & Maybhate (2006) Jog, C. J., & Maybhate, A. 2006, Monthly Notices of the Royal Astronomical Society, 370, 891
* Kleyna et al. (2003) Kleyna, J. T., Wilkinson, M. I., Gilmore, G., & Evans, N. W. 2003, The Astrophysical Journal Letters, 588, L21
* Klypin et al. (2002) Klypin, A., Zhao, H., & Somerville, R. S. 2002, Astrophysical Journal, 573, 597
* Macciò et al. (2012) Macciò, A. V., Stinson, G., Brook, C. B., et al. 2012, The Astrophysical Journal Letters, 744, L9
* Machado & Athanassoula (2010) Machado, R. E. G., & Athanassoula, E. 2010, Monthly Notices of the Royal Astronomical Society, 406, 2386
* Martinez-Valpuesta & Gerhard (2011) Martinez-Valpuesta, I., & Gerhard, O. 2011, The Astrophysical Journal Letters, 734, L20
* McMillan & Dehnen (2005) McMillan, P. J., & Dehnen, W. 2005, Monthly Notices of the Royal Astronomical Society, 363, 1205
* Miller & Smith (1992) Miller, R. H., & Smith, B. F. 1992, The Astrophysical Journal, 393, 508
* Muno et al. (2005) Muno, M. P., Pfahl, E., Baganoff, F. K., et al. 2005, The Astrophysical Journal Letters, 622, L113
* Muno et al. (2003) Muno, M. P., Baganoff, F. K., Bautz, M. W., et al. 2003, The Astrophysical Journal, 589, 225
* Navarro et al. (1997) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, The Astrophysical Journal, 490, 493
* Paumard et al. (2006) Paumard, T., Genzel, R., Martins, F., et al. 2006, The Astrophysical Journal, 643, 1011
* Pontzen & Governato (2012) Pontzen, A., & Governato, F. 2012, Monthly Notices of the Royal Astronomical Society, 421, 3464
* Power et al. (2003) Power, C., Navarro, J. F., Jenkins, A., et al. 2003, Monthly Notices of the Royal Astronomical Society, 338, 14
* Read & Gilmore (2005) Read, J. I., & Gilmore, G. 2005, Monthly Notices of the Royal Astronomical Society, 356, 107
* Read et al. (2006) Read, J. I., Goerdt, T., Moore, B., et al. 2006, Monthly Notices of the Royal Astronomical Society, 373, 1451
* Saha et al. (2009) Saha, K., Levine, E. S., Jog, C. J., & Blitz, L. 2009, The Astrophysical Journal, 697, 2015
* Schweizer (1996) Schweizer, F. 1996, The Astronomical Journal, 111, 109
* Sellwood (2003) Sellwood, J. A. 2003, The Astrophysical Journal, 587, 638
* Shan et al. (2010) Shan, H., Qin, B., Fort, B., et al. 2010, Monthly Notices of the Royal Astronomical Society, 406, 1134
* Stadel et al. (2009) Stadel, J., Potter, D., Moore, B., et al. 2009, Monthly Notices of the Royal Astronomical Society, 398, L21
* Su & Finkbeiner (2012) Su, M., & Finkbeiner, D. P. 2012, ArXiv e-print 1206.1616
* Su et al. (2010) Su, M., Slatyer, T. R., & Finkbeiner, D. P. 2010, The Astrophysical Journal, 724, 1044
* Tempel et al. (2012) Tempel, E., Hektor, A., & Raidal, M. 2012, ArXiv e-print 1205.1045
* Trachternach et al. (2008) Trachternach, C., de Blok, W. J. G., Walter, F., Brinks, E., & Kennicutt, R. C. 2008, The Astronomical Journal, 136, 2720
* Valenzuela & Klypin (2003) Valenzuela, O., & Klypin, A. 2003, Monthly Notices of the Royal Astronomical Society, 345, 406
* Wadsley et al. (2004) Wadsley, J. W., Stadel, J., & Quinn, T. 2004, New Astronomy, 9, 137
* Weinberg & Katz (2002) Weinberg, M. D., & Katz, N. 2002, The Astrophysical Journal, 580, 627
* Weinberg & Katz (2007) —. 2007, Monthly Notices of the Royal Astronomical Society, 375, 425
* Weniger (2012) Weniger, C. 2012, ArXiv e-print 1204.2797
* White & Rees (1978) White, S. D. M., & Rees, M. J. 1978, Monthly Notices of the Royal Astronomical Society, 183, 341
* Yang et al. (2012) Yang, R., Yuan, Q., Feng, L., Fan, Y., & Chang, J. 2012, ArXiv e-print 1207.1621
* Yusef-Zadeh et al. (2012) Yusef-Zadeh, F., Hewitt, J. W., Wardle, M., et al. 2012, ArXiv e-print 1206.6882
* Zemp et al. (2007) Zemp, M., Stadel, J., Moore, B., & Carollo, C. M. 2007, Monthly Notices of the Royal Astronomical Society, 376, 273
|
arxiv-papers
| 2012-08-23T20:00:01 |
2024-09-04T02:49:34.513736
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Kuhlen, Javiera Guedes, Annalisa Pillepich, Piero Madau, Lucio\n Mayer",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/1208.4844"
}
|
1208.4970
|
# Can Resonant Oscillations of the Earth Ionosphere Influence the Human Brain
Biorhythm?
V.D. Rusov1, K.A. Lukin2, T.N. Zelentsova1, E.P. Linnik1, M.E. Beglaryan1,
V.P. Smolyar1, M. Filippov1 and B. Vachev3 Cooresponding author e-mail:
[email protected]
Within the frames of Alfvén sweep maser theory the description of
morphological features of geomagnetic pulsations in the ionosphere with
frequencies (0.1-10 Hz) in the vicinity of Schumann resonance (7.83 Hz) is
obtained. It is shown that the related regular spectral shapes of geomagnetic
pulsations in the ionosphere determined by ”viscosity” and ”elasticity” of
magneto-plasma medium that control the nonlinear relaxation of energy and
deviation of Alfvén wave energy around its equilibrium value. Due to the fact
that the frequency bands of Alfvén maser resonant structures practically
coincide with the frequency band delta\- and partially theta-rhythms of human
brain, the problem of degree of possible impact of electromagnetic ”pearl”
type resonant structures (0.1-5 Hz) onto the brain bio-rhythms stability is
discussed.
Keywords: Ionospheric Alfvén resonator (IAR); ELF waves; cosmic rays;
magnetobiological effect (MBE) in cell; brain diseases statistics
PACS: 87.50.C-, 87.53.-j, 94.20.-y, 94.30.-d
## 1 Introduction
Lately Nobelist L. Montagnier’s group has published three articles deeply
challenging the standard views about genetic code and providing strong support
for the notion of water memory [1, 2, 3]. In a series of delicate experiments
[1, 2] they demonstrated the possibility of the emission of low-frequency
electromagnetic Extremely Low Frequency (ELF) waves from bacterial DNA
sequences and the apparent ability of these waves to organize nucleotides (the
”building” material of DNA) into new bacterial DNA by mediation of structures
within water [4].
Without going into details of physical justification of quantum-field
interpretation of these results111New Scientist reacted by sharp article
”Scorn over claim of teleported DNA” [5]., let us emphasize one significant
experimental result of this group. This result is related to stable detection
of ELF waves ($<$7 Hz) from bacterial DNA sequences. Obviously, if the result
is reproduced in similar experiments of another research groups, the unique
importance of this fact is difficult to overestimate in understanding of
living matter essence.
First of all, it concerns not only research of drastically new fundamental
properties of spatial-temporal structure of eukaryotic genome, but also refers
to studying of equally important issues that are related to exogenous nature
of $<$7 Hz ELF waves impact directly onto a human brain and its biorhythms.
It is well known that the brain neurons constitute different types of networks
that interact by means of electrical signals. Neuron networks configurations
comprise electrical circuits of oscillatory type. Electrical oscillations with
different frequencies correspond to different states of brain. These
oscillations could be detected by brain electroencephalogram.
Numerous investigations have shown that electrical oscillations of different
frequencies dominate in a healthy human being brain at its different states
[6, 7]. Transitions between brain activities happen not continuously, but only
in discrete steps, from one level to another. The rest state corresponds to
the steadiest alpha-rhythm with frequencies laying within the frequency band
from 8 to 13 Hz. beta-rhythm with boundary frequencies 14-35 Hz corresponds to
brain work. The slowest oscillations at frequencies 0.5-4 Hz are typical for
delta-rhythm which corresponds to deep sleep. At last, theta-rhythm with
frequencies from 4 to 7 Hz dominates in the brain if the state of nuisance or
danger appears.
At the same time it is known [8, 9, 10, 11, 12] that the weak magnetic fields
impact on biological systems is a subject of the biophysics section called
magnetobiology. It studies the biological reactions and mechanisms of the weak
fields action. Magnetobiology is a part of a general fundamental problem of
the biological efficiency of the weak and ultraweak physicochemical factors,
which operate below the biological defense mechanisms threshold, and so may be
accumulated on a subcelluar level.
It is necessary to note here that there is no acceptable physical
understanding of the way the weak magnetic fields cause the living systems
reaction [9] so far, although it has been experimentally found that such
fields may change the biochemical reactions rate sharply in a resonance-like
way [9, 13]. The physical nature of this phenomenon is still unclear, and it
forms one of the most important, if not a general, problem of magnetobiology
which includes the co-called ”$kT$ problem”.
The problem consists in the fact that the weak magnetic field energy (say,
geomagnetic filed) of the same order as the $kT$ heat energy is distributed
over the volume 12 orders of magnitude larger (which approximately corresponds
to a cell size). In such form the problem of the biological impact of the low-
frequency magnetic oscillations has two aspects[9]:
* •
what is a mechanism of the weak low-frequency magnetic signal transformation
that causes the changes on the biochemical processes level of $kT$ order?
* •
what is a mechanism of such stability, i.e. how do such small impacts not get
lost on the heat disturbances of $kT$ order background?
Not going into details of this complex and fundamental problem, the essence of
which is expounded in review by [9], let us note that in spite of the stated
magnetobiology difficulties, there are serious reasons to believe that the
main features of the magnetobiological effect are reliably established in
numerous experiments and tests and are reproducible on different experimental
models and under different magnetic conditions. On the other hand, the answers
to the above-mentioned questions ”lie” in the nonequilibrium thermodynamics
field. ”It is generally known that metabolism in living systems is a
combination of primary non-equilibrium processes. The origin and breakdown of
biophysical structures at time smaller than the time of thermalization of all
degrees of freedom in these structures provide a good example of systems that
are far from equilibrium where even weak field quanta can be manifested in
system’s breakdown parameters. In other words, if the life (thermalization)
time of certain degrees of freedom interacting with field quanta is larger
than the system’s characteristic time of life, then such degrees of freedom
exist in the absence of temperature proper. Therefore, a comparison of their
energy changes due to field quanta absorption with $kT$ has no sense” [14].
The candidates for the solution of this problem today are the mechanism of the
molecule quantum states interference for the idealized protein cavity [14, 9]
and the mechanism of the molecular gyroscope interference [15, 9].
Turning back to experiment, let us examine the possible physical causes of the
low-frequency geomagnetic fields generation and the consequences of their
impact on the eucaryotic cells in short.
It is well known that Earth’s atmosphere between dense ionized shell called
ionosphere (at an altitude of 100 km) and Earth’s surface, possesses the
electromagnetic resonant properties (Fig. 1 [16]). Hence, resonances of the
spherical cavity ”Earth’s surface - ionosphere” manifest themselves in
electromagnetic quasi-monochromatic signals that permanently present nearby
Earth’s surface and has certain impact onto the Earth’s biosphere. Among
resonances of this type in the frequency band between (0.1-10) Hz the most
known and studied is the so called Schumann resonance at the frequency of 7.83
Hz. This resonance is observed for electromagnetic waves with the wavelength
exactly equal to the Earth’s circle. Schumann resonance has drawn attention of
physicians practically immediately after its discovery in connection with
studying of impact of electromagnetic radiation onto the alpha-rhythms of
human brain which lie within the 8-13 Hz frequencies band.
Figure 1: Earth and surrounding electromagnetic resonant objects (adapted from
[16, 17]). a) Air gap at the altitudes of 0-100 km is the global Schumann
spherical resonator with 7.83 Hz resonant frequency; the altitude region of
100-1000 km is a dense ionized shell (ionosphere). Inside its mass ionosphere
Alfvén resonator with the first resonant frequency that varies in time within
the limits of 0.5-3.0 Hz is located. Geomagnetic field lines lie above the
ionosphere and are shown in red. High-energy protons cross this tubes. This
Geomagnetic field line rests upon magneto-conjugated regions of the Earth’s
ionosphere which all together form the resonator of, so called, magnetosphere
Alfvén maser which generates ”pearl” type electromagnetic signals. Traffic
diagram for the particles in radiation belt is also depicted in the figure.
Particles with velocities being inside the loss cone (green line), possess
small transversal velocities and fall into dense layers of the atmosphere,
whereas particles outside the loss cone (blue line with arrow) possess bigger
transversal velocities and are captured by geomagnetic tube-trap due to their
reflections from the magnetic mirrors of the ionosphere; b) Density of charged
particles in plasma of Earth’s ionosphere versus altitude.
Thunderstorms feed the Schumann resonator eternally. Initial frequency
spectrum of electrical discharges (lightnings) during thunderstorms represents
practically white noise. The resonant systems of near-Earth space filter out
corresponding parts of the spectrum which is shown in Fig. 2 [17].
Figure 2: Electromagnetic noise spectrum structure for middle latitudes has a
pronounced resonant structure (adapted from [17]). In the daytime (c) and d))
the spectrum has a peak associated with Schuman resonance at 7.83 Hz, while at
night time (a) and b)) electromagnetic noise produced by lightning’s radiation
at the frequencies below Schumann resonance is filtered out by Ionosphere
Alfvén Resonator (IAR). In the years of solar activity maximum the noise
spectrum at night time is similar to that of daytime.
Ionospheric Alfvén Resonator (IAR) is being considered along with Schumann
resonator as a near-Earth resonant system, as well. In particular, with the
help of IAR it was possible to explain new resonant radiation in the frequency
band of 0.1-10 Hz [18]. This radiation was discovered in 1985 and
characterized by quasi-periodic modulation (within frequency range of 0.5-3
Hz) of the oscillations. This modulation appears above the background noise
electromagnetic spectrum of the atmosphere and has regular daily variation
(Fig. 2). It is not difficult to show [18], that within the frame of IAR model
the resonant frequency $f_{res}$ of these oscillations is defined by
ionosphere layer thickness $l$, Earth’s magnetic field strength $H_{Earth}$,
and concentration $n$ of particles with mass $M$
$f_{res}=\frac{v_{A}}{2l},~{}~{}v_{A}=\frac{H_{Earth}}{\sqrt{4\pi Mn}},$ (1)
where $v_{A}$ is Alfvén velocity. According to [18] the spectrum structure
shown in Fig. 2 is defined by resonant frequency $f_{res}$ and its harmonics.
For typical values of $H_{Earth}\sim 0.4~{}E$, $M\sim 1.5\cdot 10^{-23}~{}g$,
$n\sim 10^{5}~{}cm^{-3}$ and $l\sim 500~{}km$ this estimation gives
$f_{res}\sim 2~{}Hz$. Apparently, this estimation is in a good agreement with
experimental frequencies estimations of the detected radiations 0.5-3 Hz [18].
Here it is interesting to mention another important role of IAR properties in
affecting dynamics of larger scale resonator for Alfvén waves: magnetospheric
resonator for Alfvén waves – Alfvén Resonator (AR), formed by geomagnetic
field line resting upon magneto-conjugated regions of Earth’s surface. High-
energy protons may cross geomagnetic field lines of the resonator and excite
ultra-low frequency (ULF) electromagnetic oscillations practically in the same
frequency band: 0.2-5 Hz , due to maser effect for the trapped protons and
self-oscillatory mode of this resonator [19]. This generator was called as
magnetospheric Alfvén maser [16, 17, 20, 21, 22], which is schematically shown
in Fig. 1. The signals generated via this mechanism are often referred to as
”pearls”. The spectral dynamical characteristics of the ”pearls” and their
temporal dependencies had been a mystery for researchers until recent times. a
remarkable fact had been discovered recently consisting in the strong negative
correlation between intensity level of low-frequency resonant lines in the
atmospheric noise radiation spectrum and solar activity [17], and,
correspondingly, between intensity of resonant radiations of ”pearl” type and
solar activity [23, 24, 25], that directly correlate with predictions for IAR
models [17] and Alfvén maser ones [22].
Due to the fact that frequency bands of Alfvén maser resonances practically
coincide with the frequency band of delta\- and partially theta-rhythms of
human brain, the problem of possible impact of electromagnetic fields of
”pearl” type onto stability of mentioned brain bio-rhythms arises.
Thus, investigation of possible direct correlation between the values of
average annual frequencies of resonant electromagnetic signals of ”pearl” type
appearance, which have certain impact onto brain biorhythms, and rate of
average annual mortality because of diseases due to various abnormal
functioning of human brain was the subject of this paper. Let us describe the
basic laws of the electromagnetic fields generation and oscillations in the
abovementioned resonators.
## 2 Types of relaxation oscillations in Alfvén maser
Below we present a brief analysis of solutions for the equation describing
small oscillations of the wave energy around its equilibrium state in the
Alfvén sweep maser [20]:
$\ddot{\text{w}}+2v\dot{\text{w}}+\Omega_{R}^{2}\text{w}=0,$ (2)
where
w $\displaystyle=\frac{E-E_{0}}{E_{0}},$ $\displaystyle 2v$
$\displaystyle=2v_{R}\left(1-\frac{\gamma_{0}\tau_{rec}\chi
N_{0}}{1+\tau_{rec}^{2}\Omega_{R}^{2}}\cdot\frac{\partial}{\partial
n_{is}}\ln{\left(\delta_{m}\gamma_{0}\right)}\right)$ (3)
and
$2v\ll\Omega_{R}.$ (4)
Here $\Omega_{R}$ and $2v_{R}$ are characteristic frequency and oscillations
decrement in Alfvén resonator, respectively; w is the oscillation of the
Alfvén wave energy with respect to its equilibrium value $E_{0}$. The latter
is defined for the case of no changes in the Earth’s ionosphere. These changes
define reflection coefficients of Alfvén waves and, consequently, their
attenuation in the resonator under consideration; $\tau_{rec}$ is
characteristic recombination time in ionosphere plasma; $N_{0}$ is total
number of fast particles in Geomagnetic field line having unit cross section
at the ionosphere level; $n_{is}$ is electron concentration in the ionosphere;
$\chi=(\vec{k}\hat{,}\vec{v}_{g})$, $k$ is wave vector; $v_{g}$ is Alfvén
waves group velocity; $\gamma_{0}$ corresponds to the steady state value of
attenuation factor
$\gamma=\left|\ln R(\omega)\right|/\tau_{g},$ (5)
where $R(\omega)$ is a coefficient of Alfvén wave reflection from the magnetic
mirrors, that lasts over ionosphere and planet surface; $\tau_{g}$ is
propagation time of electromagnetic signal along geomagnetic field line
between magneto-conjugated regions of the ionosphere;
$\delta_{m}=\phi(\omega_{m})\delta$, $\phi(\omega_{m})$ is normalized to unity
Alfvén wave amplification for one pass along the radiation belt (RB), while
coefficient $\delta$ equals to:
$\delta=\frac{4\pi
e^{2}\beta_{0}}{m_{e}n_{A}\Omega_{L}W_{0}},~{}~{}\beta_{0}=\frac{v_{0}}{c},~{}~{}W_{0}=\frac{1}{2}m_{e}v_{0}^{2},~{}~{}n_{A}=\frac{\Omega_{pL}}{\Omega_{L}},$
(6)
where $v_{0}$ is typical velocity of particles, $\Omega_{pL}$ is the ion
plasma frequency of background plasma in magnetosphere RB equatorial cross-
section; $\Omega_{L}$ is gyro frequency in magnetosphere equatorial cross-
section. It should be noted here, that the following approximation for
coefficient $\delta_{m}$ is used for taking into account the effect of the
generated frequency sweep (drift):
$\delta_{m}=\delta_{0}+\frac{\partial\delta_{m}}{\partial n_{is}}\Delta
n_{is},$ (7)
where $\delta_{0}$ corresponds to equilibrium value of coefficient $\delta$,
which is obtained for the case of stationary solution for differential
equations which describe the dynamics of cyclotron instability in case of
equilibrium electron density in the ionosphere.
It is convenient for analysis of typical forms of relaxation oscillations in
Alfvén sweep-maser to rewrite the equations (2) in the following way:
$\ddot{\text{w}}+\lambda\dot{\text{w}}+\Gamma\lambda\text{w}=0,$ (8)
with initial conditions
$\text{w}(0)=\text{w}_{0},~{}~{}\dot{\text{w}}(0)=0,$ (9)
where $\Gamma=\Omega_{R}^{2}/2v$, $\lambda=2v$ and
$\text{w}_{0}=\left[E(0)-E_{0}\right]/E_{0}$ is a normalized initial energy of
Alfvén wave.
The advantages of such representation of the Alfvén waves energy relaxation
oscillations become apparent during the study of the physical reasons for the
time evolution of dispersion and, consequently, the morphology of such
oscillations. For instance, it is easy to show that the Polyakov-Rappoport-
Trakhtengerts equation (2) is equivalent to the following integro-differential
equation:
$\dot{\text{w}}+\left(\frac{\Omega_{R}^{2}}{2v}\right)\cdot
2v\int\limits_{0}^{t}e^{-2v(t-t^{\prime})}\text{w}(t^{\prime})dt^{\prime}=0,~{}~{}\text{w}(0)=\text{w}_{0}.$
(10)
As follows from (10), the medium memory function which characterizes its
”elastic” properties, has the following form:
$f(t-t^{\prime})=u(t-t^{\prime})\cdot 2v\cdot
e^{-2v(t-t^{\prime})},~{}~{}\tau_{\lambda}=1/2v,$ (11)
where $u(t-t^{\prime})$ is the unit Heaviside function. Obviously, when
$v\to\infty$, equation (11) gains the $\delta$-asymptotycs
$f(t-t^{\prime})\rightarrow\delta(t-t^{\prime}),$ (12)
while the equation (10) and, consequently, equation (2) as well, turns into a
trivial relaxation equation of exponential type with the initial conditions:
$\text{w}=\text{w}_{0}e^{-\left(\Omega_{R}^{2}/2v\right)t}=\text{w}_{0}e^{-\Gamma
t},$ (13)
where $1/\Gamma=2v/\Omega_{R}^{2}$ is the time of w function ”viscosity”
relaxation to an equilibrium value $\text{w}_{0}$, i.e. it is the relaxation
time $\tau_{M}$ of the Maxwellian energy distribution w to the equilibrium.
Willing to preserve the properties of ”viscosity” ($\tau_{M}$) and ”quasi-
elasiticity” ($\tau_{\lambda}$) of the medium in equation (2) let us
hereinafter consider the equation (2) in the form of (8) with any finite
$\tau_{\lambda}=1/\lambda$.
So the characteristic equation, corresponding to (8) has the roots
$k_{1,2}=\frac{\lambda}{2}\left(1\mp\sqrt{1-4\eta}\right),~{}~{}\eta=\frac{\Gamma}{\lambda}=\Gamma\tau_{\lambda},$
(14)
which are real when $\eta<1/4$ so that the effective time of the medium
aftereffect $\tau_{\lambda}<\tau_{M}/4$, where $\tau_{M}=1/\Gamma$ is the time
of the Maxwellian energy distribution settling. Particularly, in the case of a
very short medium memory ($\eta\ll 1$) we have
$\displaystyle k_{1}$
$\displaystyle=\lambda\eta(1+\eta+\dots)\cong\Gamma(1+\eta),$ $\displaystyle
k_{2}$ $\displaystyle=\lambda(1-\eta+\dots)\cong\Gamma(1-\eta)/\eta.$ (15)
In the case of $\eta>1/4$, which corresponds to $\Omega_{R}\gg 2v_{R}$, or
$\tau_{M}<4\tau_{\lambda}$ (medium with a significant ”elasticity”)
$k_{1,2}=\frac{1}{2}\lambda\mp
i\omega,~{}~{}\omega=\frac{1}{2}\lambda\sqrt{4\eta-1}.$ (16)
Then the general solution satisfying the initial conditions has the form
$\displaystyle\text{w}=\text{w}_{0}\frac{1}{k_{1}+k_{2}}\left(k_{2}e^{-k_{1}t}-k_{1}e^{-k_{2}t}\right),$
$\displaystyle\text{w}_{0}=\frac{E(0)-E_{0}}{E_{0}}.$ (17)
In the case of $\eta<1/4$ the solution (17) takes on the following form:
$\text{w}\cong\text{w}_{0}\left[(1+\eta)e^{-\Gamma(1+\eta)t}-\eta e^{-\lambda
t}\right],~{}~{}\eta\ll 1,$ (18)
which describes the exponential relaxation which is qualitatively different
from $e^{-\Gamma t}$ (a case of $\tau_{\lambda}=0$) only in the range
$0<t<\tau_{\lambda}=1/\lambda$ (Fig. 3a). Meanwhile for the case of $\eta>1/4$
the solution (17) describes a new kind of mode
$\text{w}=\text{w}_{0}e^{-\lambda t/2}\frac{\sin(\omega
t+\varphi)}{\sin\varphi},~{}~{}\eta>\frac{1}{4},~{}~{}\sin\varphi=\frac{\sqrt{4\eta-1}}{2\sqrt{\eta}}$
(19)
in a form of a attenuated periodic relaxation (Fig. 3b).
Figure 3: Exponential (a) and oscillatory (b) types of relaxations of Alfvén
wave energy perturbations
Then the expression (19), allowing for (3) and (17), may be represented in the
following form
$E(t)=\left[E(0)-E_{0}\right]\cdot e^{-\lambda t/2}\frac{\sin(\omega
t+\varphi)}{\sin\varphi}+E_{0},~{}~{}E(0)\geqslant E,$ (20)
where $E(t)$ is the Alfvén waves energy.
It is known that the spectral analysis of experimental data corresponding to
registration of magnetosphere radiation of ”pearl” type or, in other words,
geomagnetic pulsations Pc1, allows one to reveal their internal frequency
structure [2], and the frequency inside of each ”pearl” (separate packet of
Alfvén waves) increases from its beginning to the end [17]. Quantitative
estimates of ”pearl” parameters following from the theory above are in a good
agreement with the related experiments [17, 20]. Relying on that theory and
experiment correspondence we will show below how the mentioned above
morphological features find their explanations (within the framework of Alfvén
sweep-maser theory) on the basis of evolution of damping periodic oscillations
(19) or (20) with taking into account dispersion relaxation of Alfvén wave
energy to its equilibrium value.
## 3 Maxwell distribution and relaxation of Alfvén wave energy dispersion
toward equilibrium value
According to Eq. (17), deviation of Alfvén wave energy fluctuation from
average value is given by
$\displaystyle\Delta\text{w}(t)=\text{w}(t)-\overline{\text{w}(t)}=$
$\displaystyle=\int\limits_{0}^{t}\frac{1}{k_{2}-k_{1}}\left(k_{2}e^{-k_{1}(t-t^{\prime})}-k_{1}e^{-k_{2}(t-t^{\prime})}\right)\xi(t^{\prime})dt^{\prime},$
(21)
where, according to (17)
$\overline{\text{w}(t)}=\text{w}_{0}\frac{1}{k_{2}-k_{1}}\left(k_{2}e^{-k_{1}t}-k_{1}e^{-k_{2}t}\right)$
(22)
and $\xi(t)$ is normalized Gaussian noise with the following moments:
$\overline{\xi(t)}=0,~{}~{}\overline{\xi(t)\xi(t^{\prime})}=\phi(t-t^{\prime})=\phi\tau\delta(t-t^{\prime}).$
(23)
Let us consider the standard deviation
$\displaystyle\overline{(\Delta\text{w})^{2}}=\int\limits_{0}^{t}dt_{1}\int\limits_{0}^{t}dt_{2}$
$\displaystyle\prod\limits_{t=1}^{2}\frac{1}{k_{2}-k_{1}}\left(k_{2}e^{-k_{1}(t-t_{i})}-\right.$
$\displaystyle\left.-k_{1}e^{-k_{2}(t-t_{i})}\right)\overline{\xi(t_{1})\xi(t_{2})}.$
(24)
Then, after integration (24) and taking into account (21) and (23) we obtain
the following general expression for dispersion of Alfvén wave energy:
$\displaystyle\overline{(\Delta\text{w})^{2}}=\frac{\phi\tau}{(k_{2}-k_{1})^{2}}\left[\frac{k_{2}^{2}}{2k_{1}}\left(1-e^{-2k_{1}t}\right)-\right.$
$\displaystyle\left.-\frac{2k_{1}k_{2}}{k_{1}+k_{2}}\left(1-e^{-(k_{1}+k_{2})t}\right)+\frac{k_{1}^{2}}{2k_{2}}\left(1-e^{-2k_{2}t}\right)\right].$
(25)
Assuming that for $t\gg\tau_{M}=1/\Gamma$ the energy distribution of Alfvén
wave is relaxing to Maxwell distribution [26, 27]:
$\left.\overline{(\Delta\text{w})^{2}}\right|_{t\gg
1/\Gamma}=\overline{\text{w}^{2}}=\theta^{2}\cdot\frac{\partial\overline{\text{w}}}{\partial\theta},~{}~{}\theta=\frac{kT}{E_{0}},$
(26)
we have
$\frac{\phi\tau}{(k_{2}-k_{1})^{3}}=\frac{\theta^{2}(\partial\overline{\text{w}}/\partial\theta)}{k_{2}^{2}/2k_{1}-2k_{1}k_{2}/(k_{1}+k_{2})+k_{1}^{2}/2k_{2}}.$
(27)
In the case of $\eta\ll 1$ ($\tau_{\lambda}\ll\tau_{M}$) the dispersion
relaxation to its equilibrium value (26) is schematically shown in Fig. 4a. It
is characterized by three relaxation times:
$\frac{1}{2k_{2}}=\frac{\tau_{\lambda}}{2}(1+\eta),~{}~{}\frac{1}{k_{1}+k_{2}}=\tau_{\lambda},~{}~{}\frac{1}{2k_{1}}=\frac{1}{2\Gamma}(1-\eta).$
(28)
In the case of $\eta>1/4$, when according to Eq. (16), $k_{1,2}=\lambda/2\pm
i\omega$ and the relaxation character of dispersion of Alfvén wave energy to
its equilibrium value (26) becomes an oscillatory one (Fig. 4b):
$\overline{(\Delta\text{w})^{2}}=\theta^{2}\frac{\partial\overline{\text{w}}}{\partial\theta}\cdot\left[1-e^{-\lambda
t}\frac{1-(1/\sqrt{4\eta}\cos(2\omega t+3\varphi)}{1-(1/\sqrt{4\eta})\cos
3\varphi}\right],$ (29)
where value $\varphi$ is defined according to formula after Eq. (19)
$\varphi=\arctan\sqrt{4\eta-1},$ (30)
and the thermodynamical function $\partial\overline{\text{w}}/\partial\theta$
by definition is the thermal capacity $C_{V}$, which in the simplest case for
plasma has the following form [26]:
$C_{V}=\left(\frac{\partial\overline{\text{w}}}{\partial\theta}\right)_{V}=\left(C_{ideal}\right)_{V}+\frac{1}{2}\frac{A}{T^{3/2}V^{1/2}},~{}~{}A=const,$
(31)
where $C_{ideal}$ is the thermal capacity of ideal gas.
Figure 4: Relaxation of Alfvén wave energy dispersion $var(\text{w})$ to
equilibrium value for aperiodic (a) and oscillatory (b) character of
relaxation in the medium with memory.
Let us remind that the expression (29) taking into account (3) and (26) can be
presented in the following form:
$\overline{(\Delta
E)^{2}}=\theta^{2}_{T}\cdot\frac{\partial\overline{E}}{\partial\theta_{T}}\cdot\left[1-e^{-\lambda
t}\frac{1-(1/\sqrt{4\eta}\cos(2\omega t+3\varphi)}{1-(1/\sqrt{4\eta})\cos
3\varphi}\right],$ (32)
where $\theta_{T}=kT$.
Temporal behavior of average energy $\overline{E}$ (20) and energy root-mean-
square deviation $\left(\overline{\Delta E}\right)^{1/2}$ (32) of relaxation
oscillations of Alfvén waves in magnetoplasma medium (medium with memory) are
presented in Fig. 5.
Figure 5: Oscillatory relaxation of Alfvén wave average energy $\bar{E}$ and
its dispersion $var(E)$ in magnetoplasma medium (medium with memory).
It is worth noting here, that such approach opens up nontrivial possibility
for experimental numerical estimations of some important parameters of Alfvén
sweep-maser relaxation oscillations. For instance, the limit width of
distribution (26) allows determining the plasma thermal capacity $C_{V}$ for
the given temperature $\theta$. In combination with the measured frequency of
oscillations $\Omega_{R}$ it allows successively finding the values
$\lambda$(at $t=\pi/\omega$), $\eta$ (for any $t<\tau_{\lambda}=1/\lambda$)
and $\tau_{m}=1/\Gamma$.
In other words, analysis of energy dispersion evolution of Alfvén sweep-maser
relaxation oscillations allows finding experimentally (see Fig. 5) the values
of plasma thermal capacity $C_{V}$, decrement $\lambda$, relaxation time
(”elasticity”) $\tau_{\lambda}$ of magnetoplasma medium and settling time
(”viscosity”) $\tau_{M}$ of Maxwellian energy distribution (see Fig. 3).
And finally, folowing [20], let us give some quantitative estimations.
Accrording to [25], the recurrence period of the elements in ”pearls” is about
50-300 s (Fig. 2), the bandwidth $\Delta f$ fits into the range 0.05-0.3 Hz,
and the dynamic spectrum tilt is
$\frac{df}{dt}\cong 2\cdot 10^{-3}~{}[Hz].$ (33)
In order to estimate the radiation parameters following from the sweep-maser
theory [20], let us consider a magnetic flux tube on the morning side of a
magnetosphere222Geomagnetic pulsations of ”pearl” type are known [25] to
appear primarily on the morning side of magnetosphere at mid latitudes at
magnetically calm times. at a distance of $R\approx 3R_{Earth}$ from the Earth
center, where $R_{Earth}$ is the Earth radius. In the framework of the sweep-
maser theory it has been shown that the relaxation oscillations period, which
characterizes the recurrence period of the elements in pulsations of the
”pearl” type is
$T_{R}=\frac{2\pi}{\Omega_{R}}=2\pi\sqrt{\frac{\sigma\cdot
l}{W_{0}\cdot\delta\cdot 2S_{0}}},$ (34)
where $\Omega_{R}$ is the characteristic frequency of relaxation oscillations
in Alfvén resonator, $\sigma=B_{m}/B_{L}$ is the mirror ratio for the Earth’s
radiation belt, $B_{m}$ is the magnetic field at the ends of the magnetic
trap, $B_{L}$ is a magnetic field in equatorial section of magnetosphere, $l$
is the effective length of a resonator, $S_{0}$ is the equilibrium
precipitating protons flux density in the Earth’s radiation belt.
According to [20], the amplification curve $\phi(\omega_{m})$ (see. (3))
reaches its maximum when
$\frac{\Omega_{L}^{2}}{\Omega_{pL}\beta_{0}\omega_{m}}\cong 1.$ (35)
If we take into account the experimental values for $\Omega_{L}\approx
10^{2}~{}s^{-1}$, $n_{A}\approx 10$ and $\beta_{0}\approx 2\cdot 10^{-2}$ (for
the particles with energy $W_{0}\approx 200~{}keV$) and allow for (6) and
(35), we find that
$\omega_{m}\sim 5s^{-1}\Leftrightarrow f_{m}\sim 1.25~{}Hz.$ (36)
On the other hand, from (6), (35) and (36) it is not hard to find the value of
$W_{0}\cdot\delta$, which (if we assume that $n_{isL}\approx 3\cdot
10^{2}~{}cm^{-3}$) would be equal
$W_{0}\delta\sim\frac{10\Omega_{L}^{2}}{\omega_{m}n_{isL}}\approx
10^{2}~{}~{}\left[s^{-1}\right].$ (37)
where the ionospheric plasma density $n_{isL}$ is defined by the so-called
plasma frequency
$\Omega_{pL}=\sqrt{4\pi e^{2}n_{isL}/m_{e}},$
which determines the characteristic time scale of the plasma oscillations.
Consequently, taking into account (34), (37) and the experimental data for
$\sigma=27$ and $l\approx R$ we derive
$T_{R}=\frac{2\pi}{\Omega_{R}}=2\pi\frac{10^{4}}{S_{0}^{1/2}}.$ (38)
The period $T_{R}$ obviously hits the experimentally observed range $50\div
300~{}s$ with the experimentally justified value $S_{0}\sim
10^{5}~{}cm^{-2}s^{-1}$.
Let us pass on to the dynamic spectrum tilt estimation:
$\frac{df}{dt}\approx\frac{df}{dn_{is}}\cdot\frac{dn_{is}}{dt}\cong\chi\cdot
S_{0}\cdot\frac{df}{dn_{is}},$ (39)
where $n_{is}$ is the ionospheric plasma density, $cm^{-3}$.
According to [20], the numerical calculation of $df/dn_{is}$ for the calm
morning gives the value $\sim 3\cdot 10^{-6}~{}cm^{3}\cdot s^{-1}$. On the
other hand there are reasons to believe that under weak magnetic storminess
the protons with energy about 200 keV (and $\chi\approx 10^{-2}~{}cm^{-1}$)
flux density is $S_{0}\sim 10^{5}~{}cm^{-2}s^{-1}$. From this it follows that
the dynamic spectrum tilt is
$\frac{df}{dt}\approx\chi\cdot S_{0}\cdot\frac{df}{dn_{is}}\sim 3\cdot
10^{-3}~{}~{}\left[Hz\cdot s^{-1}\right],$ (40)
which corresponds to the experimental data [25] with a satisfiable accuracy.
Therefore we may conclude that the Polyakov-Rappoport-Trakhtengerts sweep-
maser theory [16, 17, 20, 21, 22, 18, 19] makes it possible to build a closed
theory of generation of a wide range of geomagnetic pulsations of the ”pearl”
type, which reside in the 0.1$\div$10 Hz band and are observed primarily at
the mid latitudes under the conditions of a weak magnetic storminess on the
morning side of magnetosphere. In other words, it is shown that all the
morphological features of geomagnetic pulsations of the ”pearl” type mentioned
above find their adequate explanation in the framework of the Alfvén sweep-
maser theory.
## 4 On connections between variations of geomagnetic pulsations, solar
cycles and brain diseases mortality rate
As mentioned above, the theory of formation of the ”pearl” wave packets
(geomagnetic pulsations Pc1) in Alfvén maser may help in solving another
problem. This problem lies in the fact of strong inverse correlation between
the appearance frequency of geomagnetic pulsations Pc1 and 11-year solar
cycle. It has been found by means of long-term observations that geomagnetic
pulsations Pc1 activity is more intense (by factor of 10) during the periods
of solar minima rather than in its maxima (Fig.6). Below we will try to
clarify this dependency within the frame of Alfvén maser theory.
Figure 6: Solar cycle (yellow) variations of geomagnetic pulsations Pc1
(green) activity for about four (from 19th through 21st) cycles [25].
Experimental observations of ionosphere have shown that steepness of electron
concentration profile at the attitudes $\sim$1000 km (see Fig. 1b) is
considerably decreasing in the years of solar activity maxima [28]. This
factor leads to decrease in the Alfvén waves reflection coefficient from the
upper layer of the IAR and, hence, to decrease in Q-factor of AR. Fig. 7 shows
experimental dependence of the reflection coefficient $R$ of Alfvén waves from
IAR. It was obtained using ionosphere data for the minimum and maximum of
solar activity [28]. One can see that appreciable decrease in the reflection
coefficient R (and, therefore, worsening of the conditions for ”pearl”
generation in magnetospheric Alfvén resonator (AR)) is observed for maximum of
solar activity compared to its minimum. The explanation of such behavior of
the reflection coefficient $R$ and, consequently, behavior of the ”pearl”
generation rate is rather straightforward and is given below.
The criterion for the wave generation in Alfvén maser according to (5) has the
form [16, 17]
$R(\omega)\cdot\exp\Gamma_{0}>1,$ (41)
where $\Gamma_{0}=\gamma\tau_{g}$ is the logarithmic wave amplification at a
single passing of AR (Fig. 1a), $R(\omega)$ is a coefficient of Alfvén wave
reflection from the magnetic mirrors, that lasts over ionosphere and planet
surface. The ”pearl” amplification changes little during the solar activity
cycle and has its value considerably less than unity. Whereas the maximal
value of reflection coefficient $R$ in the typical for the ”pearls” frequency
band of 0.2-5 Hz changes considerably according to Fig. 7 and decreases in the
years of solar activity maxima. So, it is follows from the (41) that the
temporal variations of IAR Q-factor affect the appearance rate of ”pearls”
generation. In other words, reflection coefficient R behavior clearly explains
(via the criterion of wave generation in Alfvén maser (41)) the dynamics of
”pearls” appearance rate, which, in turn, explains the reason for strong
anticorrelation between ”pearls” appearance and solar activity (Fig. 6).
Figure 7: Frequency dependence of Alfvén wave reflection coefficient R from
ionosphere containing IAR [17]: 1 - for solar activity minimum, 2 - for solar
activity maxim. Drop of plasma concentration at altitudes of 250-1000 km is
much more pronounced in the solar activity minimum. It leads to the greater
value of reflection coefficient $R$ (regions of $R$ maximal values are
indicated with arrows), where high rate generation of ”pearls” takes place.
Using this dependence and having temporal evolution of solar activity or, that
the same, the temporal evolution of Sun’s magnetic field we may conclude about
our principal knowledge of temporal evolution of ”pearls” appearance over the
past 100 years, at the least. Temporal dynamics of Sun’s magnetic field is
depicted in Fig. 8. Existence of strict anticorrelation between Sun’s magnetic
field and terrestrial magnetic field333Note that the strong (inverse)
correlation between the temporal variations of magnetic flux in the tachocline
zone and the Earth magnetic field (Y-component) are observed only for
experimental data obtained at that observatories where the temporal variations
of declination ($\partial D/\partial t$) or the closely associated east
component ($\partial Y/\partial t$) are directly proportional to the westward
drift of magnetic features [29]. This condition is very important for
understanding of physical nature of indicated above correlation, so far as it
is known that just motions of the top layers of the Earth’s core are
responsible for most magnetic variations and, in particular, for the westward
drift of magnetic features seen on the Earth’s surface on the decade time
scale. Europe and Australia are geographical places, where this condition is
fulfilled (see Fig. 2 in [29]). (Y-component)[30] is seen from this figure, as
well.
Due to the fact that the frequency band of Alfvén maser resonant structures
practically coincides with the frequency band of delta-rhythms and, partially,
theta-rhythms of human brain (see Fig. 7), the question naturally arises about
the rate of possible influence of global geomagnetic pulsations of ”pearl”
type onto stability of the above brain biorhythms. If such effect really
exists, then one would expect positive correlation between variation of
geomagnetic pulsations of ”pearl” type Pc1 in the frequency band of 0.1-5 Hz
and the death rate from disruption of brain diseases. Obviously, the choice in
this case should concern only the currently incurable brain diseases so that
their statistics were close to the real one and not being masked by intensive
treatment. This fully applies to such diseases as malignant neoplasm of brain
[31]. Their temporal dynamics in West Germany [31] is shown in Fig. 8. It is
of interest that for our purposes the male statistics of infectious diseases
(incl. Tuberculosis) in France [32] also applicable, that reflects,
apparently, features of local spatial-temporal dynamics of magnetic field in
Europe.
Figure 8: Time evolution (a) the variations of magnetic flux at the bottom
(tachocline zone) of the Sun convective zone (see Fig. 7f in [33]), (b)
fractional change in female breast cancer mortality for birth cohort in US
(see Fig. 3b in [34]), (c) fractional change in female breast cancer mortality
for birth cohort in UK (see Fig. 2b in [34]), (d) geomagnetic field secular
variations (Y-component, nT/year) as observed at the Eskdalemuir observatory
(England) [35], where the variations ($\partial Y/\partial t$) are directly
proportional to the westward drift of magnetic features, (e) Malignant brain
tumor (brain stem) [36], (f) the number of deaths from ICD9 item n∘191
Malignant neoplasm of brain [31], (g) Brain lymphoma incidences in US [37] and
(h) the mortality rates from infectious diseases (incl. Tuberculosis) at ages
15-34 in France [32]. The curves (a) and (d) are smoothed by the sliding
intervals of 5 and 11 years.
It is easy to show, that degree of anti-correlation between temporal
variations of Sun’s magnetic field or, that the same, degree of direct
correlation between frequency of ”pearls” appearance and the number of deaths
from considered diseases is high enough. This result is based on the
experimental data on malignant neoplasm of brain [31], malignant brain tumor
[36], brain lymphomas [37] and infectious diseases (incl. Tuberculosis) [32].
In this way, according to Fig. 8, time arranged statistics of these diseases
are lagging behind the variations of the Solar and Earth magnetic fields 22-27
years and 10-15 years respectively. This delay effect on the one hand can be a
consequence of the long-time hidden disease incubation period, but on the
other hand opens up possibilities for prediction (at the time lag length) of
behavior variations of indicated diseases by means of experimental observation
of the geomagnetic field temporal variations.
It is interesting to note here that a strong correlation between the galactic
cosmic ray variations and cancer mortality birth cohorts has been discovered
recently [38, 39]. It was observed for population cohorts in five countries on
the three continents. Previous evidence [39] has implicated a role for cosmic
rays in US female cancer, involving a possible cross-generational foetal
effect (grandmother effects). According to the assumption of the authors [38,
39], it may provide in-sight into the exploration of the role of germ cells as
a possible target of this radiation and genetic or epigenetic sources of
cancer predisposition that could be used to identify individuals carrying the
radiation damage. And the conclusion about the galactic cosmic rays as a
direct physical cause of the cancer mortality birth cohorts is based on a
similar dependence of the total cancer age-standardized incidence rates and
cosmic ray rigidity from geomagnetic latitude (see Fig. 8 in [38]).
Not reducing the role of the physical mechanisms of radiation-induced effects
formation and non-linear cell response in low doses of ionizing radiation
(e.g. [40]),the study of which is a fundamental basis for the contemporary
microdosimetry [41], let us consider the possibility of indirect impact of
electromagnetic resonance structures (in 0.1-5 Hz band) of ionospheric Alfvén
maser on the cells of the birth cohorts through their direct impact on the
germ cells of their parents. Fig. 8b,c shows a high level of inverse
correlation between the temporal variations of the solar magnetic field (or
direct correlation between the ”pearls” appearance frequency) and cancer
mortality rate of birth cohorts.
Time lag between the inverse solar magnetic field and cancer mortality birth
cohorts is $\sim$6 years for UK-data and $\sim$10 years for USA-data, as
follows from Fig. 8b,c. At first sight it may seem to contradict the 28-year
lag between the galactic cosmic rays variations and cancer mortality birth
cohorts, established in the paper by [38] basing on the data [42, 43, 44].
However, it may be explained by the known and hard-to-remove effect of the
time shift in ice core data accompanying any 10Be measurements (proxy of
galactic cosmic rays) in ice cores of Greenland and Antarctica.On the other
hand, theoretical verification of the actual 10Be-data [45] and their
comparison with the analogous data obtained from [42, 43, 44] indicates that
the time lag between the galactic cosmic rays variations and cancer mortality
birth cohorts is $\sim$6-10 years.
Figure 9: Long-term cosmic rays reconstruction (from [45]). Calculated (grey
curve) and actual annual 10Be content in Greenland ice (dotted curve). Open
circles represent the 8-year data from Antarctica [44]. Red line represents
the 33-year moving average of the grey curve [45].
There is also another more trivial justification of the 10-year time lag on
the Fig. 8c. The variations of galactic cosmic rays are obviously a
consequence of their modulation by the solar magnetic field. It means that
magnetic fields of the solar wind deflect the primary flux of charged cosmic
particles, which leads to a reduction of cosmogenic nuclide (e.g. 10Be and
14C) production in the Earth’s atmosphere. In other words, cosmogenic nuclides
(e.g. 10Be and 14C) are a kind of a ”shadow” of galactic cosmic rays on the
Earth playing the role of a proxy for the solar magnetic variability.
Therefore the variations of the solar magnetic field and galactic cosmic rays
(or 10Be-proxy) must inversely coincide which visually demonstrates the
experimentally justified result of the 1-year lag between 10Be and sunspot
originally detected by Beer et al. [42]. After all, one could not expect
anything else because the galactic cosmic rays variations are caused by the
solar magnetic field variations and not vice versa.
Turning back to a direct physical cause of the cancer mortality birth cohorts,
it should be noted that it is practically impossible to separate the possible
radiative effect on germ cells (according to [34]) from the magnetobiological
effect induced by such electromagnetic radiation as ”pearls”, since:
1. a)
electromagnetic radiation of the ”pearls” type is generated as a result of the
protons (a dominant component of the cosmic rays) passage through the Alfvén
maser resonator, which is a magnetospheric magnetic flux tube resting upon the
parts of ionosphere in the conjugate hemispheres of the Earth;
2. b)
the intensity variations of electromagnetic radiation of the ”pearls” type –
because of their origin – not only is correlated with the galactic cosmic rays
variations, but also display a similar latitude dependence;
3. c)
magnetic field of the ”pearls” can freely penetrate the human body just like
any other magnetic field, because the human body tissues almost do not
decrease their intensity; indeed, the harmonic amplitude of the field with
frequency $\omega$ in a oscillatory circuit on the depth $h$ inside the body
is decreased $f_{s}$ times
$f_{s}(\omega,h,\sigma,\mu)=\exp(-h/\delta),$ (42)
where the path till absorption $\delta$ depends, according to [46], on the
permeability $\mu~{}(\sim 1)$ and conductivity $\sigma$, and is defined as
follows: $\delta=c(2\pi\mu\omega\sigma)^{-1/2}$. Since for
$\omega<10^{6}~{}s^{-1}$ we have $\delta>10^{3}~{}cm$, for $h\leqslant
10~{}cm$ from (42) we obtain the value $f_{s}\sim 1$.
Taking into account the stated properties and the known fact (e.g. [47, 48])
that the magnetic field may be a kind of an agent that amplifies the original
cause (chemical impact or exposure to ionizing radiation) of the
carcinogenesis, we may assume that the magnetobiological effect induced by the
electromagnetic radiation of the ”pearls” type amplifies the cosmic rays
radiative effect in the germ cells of the parents [34].
As one can easily see, the level of inverse correlation between the solar
magnetic field variations (or direct correlation between the frequency of the
”pearls” appearance) and cancer mortality rate of birth cohorts is high
enough. Also, according to Fig. 8c, the time series of this effect lag
approximately 10 years behind the temporal variations of the Earth magnetic
field. On the one hand, such delay effect may be a consequence of the cross-
generational foetal effect [34], and on the other hand, it makes it possible
to predict the discussed variations by experimental observation of the
geomagnetic field variations.
## 5 Conclusions
In the frames of Alfvén maser theory the description of morphological features
of relaxation oscillations in the mode of geomagnetic pulsations of ”pearl”
type (Pc1) in the ionosphere is obtained. These features are determined by
”viscosity” and ”elasticity” of magnetoplasma medium that control the
nonlinear relaxation of energy and dispersion of Alfvén wave energy to the
equilibrium values.
On the basis of analysis of the ”pearls” generation criterion in Alfvén maser
(41) the physical reasons for strong anticorrelation of the appearance of
”pearls” relatively to solar activity are discussed. Obviously, that a priori
knowledge of the temporal evolution of solar activity or, that the same, the
temporal evolution of Sun’s magnetic field gives possibility to build the
temporal evolution of ”pearls” appearance over the past 100 years, at least.
The latter opens up a possibility for studying of positive correlation between
”pearls” appearance rate and temporal variations of death rate from various
disruptions of brain diseases. The anti-correlation rate between temporal
variations of Sun’s magnetic field or, that the same, direct correlation rate
between of ”pearls” appearance rate and the number of deaths from considered
diseases was demonstrated to have the high enough value. This result is
supported by the experimental data on malignant neoplasm of brain [31],
malignant brain tumor [36], brain lymphomas [37] and infectious diseases
(incl. Tuberculosis) [32].
The analysis of the known correlation between the galactic cosmic rays
variations and cancer mortality birth cohorts observed for population cohorts
in five countries on the three continents [34] let us suggest a hypothesis of
a cooperative action of the cosmic rays and electromagnetic radiation of the
”pearls” type on the germ cells of the parents which is responsible for the
so-called cross-generational foetal effect [34] with a lag of $\sim$6-10
years.
In conclusion, we have obtained results clearly showing the possible impact of
electromagnetic resonant radiations generated in ionospheric Alfvén maser onto
stability of human brain biorhythms, such as delta-rhythms and, partially,
theta-rhythms.
## References
* Montagnier et al. [2009a] L. Montagnier, .J Aissa, S. Ferris, J.-L. Montagnier, and C. Lavallee. Electromagnetic signals are produced by aqueous nanostructures derived from bacterial DNA sequences. _Interdiscip. Sci. Comput. Life Sci_ , 1:81–90, 2009a.
* Montagnier et al. [2009b] L. Montagnier, J. Aissa, C. Lavallee, M. Mbamy, J. Varon, and H. Chenal. Electromagnetic detection of HIV DNA in the blood of AIDS patients treated by antiretroviral therapy. _Interdiscip. Sci. Comput. Life Sci_ , 1:245–253, 2009b.
* Montagnier et al. [2010] L. Montagnier, J. Aissa, E. Del Giudice, C. Lavallee, A. Tedeschi, and G. Vittielo. DNA waves and water. arXiv:1012.5166v1 [q-bio.OT], 2010.
* Hecht [2011] L. Hecht. New evidence for a non-particle view of life. _EIR Science_ , pages 72–77, February 11 2011.
* Coghlan [2011] Andy Coghlan. Scorn over claim of teleported DNA. _New Scientist_ , (2795), 15 January 2011.
* Ptitsyna et al. [1998] N. G. Ptitsyna, G. Villoresi, L. I. Dorman, N. Iucci, and M. I. Tyasto. Natural and man-made low frequency magnetic fields as a potential health hazard. _Physics-Uspekhi_ , 41:687–710, 1998.
* Kholodov [1982] Yu. A. Kholodov. _Brain in Electromagnetic Fields_. Nauka, 1982. (in Russian).
* Binhi [2002] V. N. Binhi. _Magnetobiology: Underlying Physical Problems_. Academic Press, San Diego, 2002.
* Binhi and Savin [2003] V. N. Binhi and A. V. Savin. Effects of weak magnetic fields on biological systems: physical aspects. _Physics-Uspekhi_ , 46:259–291, 2003.
* Adair [2000] R. K. Adair. Static and low-frequency magnetic field effects: health risks and therapies. _Rep. Prog. Phys._ , 63:415–454, 2000.
* Nittby et al. [2008] H. Nittby, G. Grastrom, J. L. Eberhardt, L. Malmgten, A. Brun, R. P. Persson, and L. G. Sarford. Radiofrequency and extremely low-frequency electromagnetic fields effects on the blood-brain barrier. _Electromagnetic Biology and Medicine_ , 27:103–126, 2008\.
* Santini et al. [2009] M. T. Santini, G. Rainaldi, and P. Indovina. Cellular effects of extremely low frequency electromagnetic fields. _Int. J. Radiat. Biology_ , 85:294–313, 2009.
* Liboff [2009] A. R. Liboff. Electric polarization and the vialibility of living systems: Ion cyclotron resonance-like interactions. _Electromagnetic Biology and Medicine_ , 28:124–134, 2009\.
* Binhi [1997] V. N. Binhi. The mechanism of magnetosensitive binding of ions by some proteins. _Biophysics_ , 42:317, 1997.
* Binhi and Savin [2002] V. N. Binhi and A. V. Savin. Molecular gyroscopes and biological effects of weak extremely low-frequence magnetic fields. _Phys. Rev. E_ , 65:051912, 2002.
* Trakhtengerts and Demekhov [2002] V. Yu. Trakhtengerts and A. G. Demekhov. Space cyclotron masers. _Nature (in Russian)_ , (4), 2002.
* Belyaev et al. [1997] P. P. Belyaev, A. G. Demekhov, E. N. Ermakova, S. V. Isaev, S. V. Polyakov, and V. Yu. Trakhtengerts. New electromagnetic pulses of near space. In _Russian science: withstand and be reborn_ , International Science Foundation. Russian Foundation for Basic Research, page 368. Nauka, Moscow, 1997. (in Russian).
* Belyaev et al. [1987a] P. P. Belyaev, S. V. Polyakov, V. O. Rappoport, and V. Yu. Trakhtengerts. Detection of resonant structure of spectrum of atmosphere electromagnetic noise background in the band of short-period geomagnetic pulsations. _DAN SSSR_ , 297:840–843, 1987a. (in Russian).
* Belyaev et al. [1987b] P. P. Belyaev, S. V. Polyakov, V. O. Rappoport, and V. Yu. Trakhtengerts. Formation of dynamical spectra of geomagnetic pulsations in the Pc1 band. _Geomagnetism and Aeronomy_ , 27:652–656, 1987b. (in Russian).
* Polyakov et al. [1983] S. V. Polyakov, V. O. Rappoport, and V. Yu. Trakhtengerts. Alfvén sweep-maser. _Plasma Physics_ , 9(2):371–377, 1983. (in Russian).
* Bespalov and Trakhtengerts [1986] P. A. Bespalov and V. Yu. Trakhtengerts. _Alfvén masers_. Inst. Of Applied Physics, USSR academy of Sciences, Gorkiy, 1986. (in Russian).
* Trakhtengerts [1995] V. Yu. Trakhtengerts. Magnetosphere cyclotron maser: backward wave oscillator generation regime. _J. Geophys. Res._ , 100:17205–17210, 1995.
* Mursula et al. [1991] K. Mursula, J. Kangas, T. Pikkarainen, and M. Kivinen. Micropulsations at a high-letitude station: A study over nearly four solar cycles. _J. Geophys. Res._ , 96:17651–17661, 1991.
* Kangas et al. [1999] J. Kangas, J. Kultima, T. Pikkarainen, R. Kerttula, and K. Mursula. Solar cycle variation in the occurence rate of Pc1 and IPDP type magnetic pulsations at sodakyla. _Geophysica_ , 35:23–31, 1999.
* Guglielmi et al. [2006] A. Guglielmi, A. Potapov, E. Matveyeva, T. Polyushkina, and J. Kangas. Temporal and spatial characteristic of Pc1 geomagnetic pulsations. _Advanced Space Res._ , 8:1572–1575, 2006.
* Rumer and Ryvkin [2001] Yu. B. Rumer and M. Sh. Ryvkin. _Thermodynamics, statistical physics and kinetics: Textbook_. Novosibirsk University publishers, Novosibirsk, 3rd edition edition, 2001. (in Russian).
* Kvasnikov [2003] I. A. Kvasnikov. _Thermodynamics and statistical physics. Vol. 3: The theory of non-equilibrium systems: Textbook_. URSS, Moscow, $2^{nd}$ extended edition edition, 2003. (in Russian).
* Buonsanto [1989] M. J. Buonsanto. Comparison of incoherent scatter observations of electron density, and electron and ion temperature at millstone hill with the international reference ionosphere. _J. Atmos. Solar-Terr. Phys._ , 51:441–468, 1989.
* Le Mouel et al. [1981] J.-L. Le Mouel, T. R. Madden, J. Ducruix, and V. Courtillot. Decade fluctuations in geomagnetic westward drift and earth rotation. _Nature_ , 290:763, 1981.
* Rusov et al. [2010] V. D. Rusov, E. P. Linnik, K. Kudela, S. Cht. Mavrodiev, I. V. Sharph, T. N. Zelentsova, V. P. Smolyar, and K. K. Merkotan. Axion mechanism of the sun luminosity and solar dynamo - geodynamo connection. arXiv:1009.3340v3 [astro-ph.SR], 2010.
* Pechholdova [2008] M. Pechholdova. Reconstruction of continuous series of mortality by case of death in West Germany for the years 1968-1997. working paper wp2008-009, Max Planck Institute for Demographic Research, February 2008.
* Mesle [2006] F. Mesle. Recent improvements in life expectancy in France: Men and starting to catch up. _Population-E_ , 61:365–388, 2006.
* Dikpati et al. [2008] M. Dikpati, G. de Toma, and P. A. Gilman. Polar flux, cross-equatorial flux, and dynamo-generated tachocline toroidal flux as predictors of solar cycle. _The Astrophysics J._ , 675:920, 2008.
* Juckett [2009] David A. Juckett. A 17-year oscillation in cancer mortality birth cohorts on three continents – synchrony to cosmic ray modulations one generation earlier. _Int. J. Biometeorol._ , 53:487–499, 2009.
* BGS [2010] BGS. Data of the observatory eskdalemuir (england). world data centre for geomagnetic (edinburg) 2007 worldwide observatory annual means. http://www.geomag.bgs.ac.uk./gifs/annual_means.shtml, 2010.
* Legler et al. [1999] Julie M. Legler, Lynn A. Gloeckler Ries, Malcolm A. Smith, Joan L. Warren, Ellen F. Heineman, Richard S. Kaplan, and Martha S. Linet. Brain and other central nervous system cancers: Recent trends in incidence and mortality. _Journal of the National Cancer Institute_ , 91(16):1382, August 18 1999.
* Hoffman et al. [2006] Sara Hoffman, Jennifer M. Propp, and Bridget J. McCarthy. Temporal trends in incidence of primary brain tumors in the united states, 1985–1999. _Neuro-Oncology_ , 8:27–37, 2006.
* Juckett [2007] D. A. Juckett. Correlation of a 140-year global time signature in cancer mortality birth cohorts with galactic cosmic rays variation. _Int. J. Astrobiology_ , 6:307–319, 2007. DOI:10.1017/S1473550407003928.
* Juckett and Rosenberg [1997] D. A. Juckett and B. Rosenberg. Time series analysis supporting the hypothesis that enchanced cosmic radiation during germ cell formation can increase breast cancer mortality in germ cell cohorts. _Int. J. Biometeorology_ , 40:206–222, 1997.
* Rusov et al. [2001] V.D. Rusov, T.N. Zelentsova, I. Melenchuk, and M. Beglaryan. About some physical mechanisms of statistics of radiation-induced effects formation and non-linear cell response in low doses area. _Rad. Meas._ , 34:105–108, 2001.
* Committee to Assess Health Risks from Exposure to Low Levels of Ionizing Radiation [2006] National Research Council Committee to Assess Health Risks from Exposure to Low Levels of Ionizing Radiation. _Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2_. The National Academies Press, Washington, DC, 2006.
* Beer et al. [1990] J. Beer, A. Blinov, G. Bonani, R. C. Finkel, H. J. Hofmann, B. Lehmann, H. Oeschger, A. Sigg, J. Schwander, T. Staffelbach, B. Stauffer, M. Suter, and W. Wolf. Use of 10Be in polar ice to trace the 11-year cycle of solar activity. _Nature_ , 347:164–166, 1990.
* Beer et al. [1994] J. Beer, S. T. Baumgartner, B. Dittrich-Hannen, J. Hauenstein, P. Kubik, C. H. Lukaschuk, W. Mende, R. Stellmacher, and M. Suter. Solar variability traced by cosmogenic isotopes. In J. M. Pap, C. Frölich, and S. K. Solanki, editors, _The Sun As a Variable Star: Solar and Stellar Irradiance Variations_ , pages 291–300, Cambridge, 1994. Cambridge University Press.
* Bard et al. [2000] E. Bard, G. Raisbeck, F Yiou, and J. Jouzel. Solar irradiance during the last 1200 years based on cosmogenic nuclides. _Tellus Ser. B: Chem. Phys. Meteor._ , 52:985–992, 2000\.
* Usoskin [2004] I. G. Usoskin. Long-term variation and terrestrial enviroment. In _Frontiers of Cosmic Ray Science_. Universal Academy Press, Inc., 2004.
* Jackson [1975] J. D. Jackson. _Classical Electrodynamics_. Wiley&Sons, New York, 1975.
* Nakagawa [1997] M. Nakagawa. A study on extremely low-frequency electric and magnetic fields and cancer: Discussion of EMF safety limits. _J. Occup. Health_ , 39:18–28, 1997.
* Simon [1992] N. J. Simon. Biological effects of static magnetic fields: A review. Technical report, Boulder, CO: Intern. Cryogenic Materials Commission, 1992.
|
arxiv-papers
| 2012-08-23T18:41:05 |
2024-09-04T02:49:34.525219
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.D. Rusov, K.A. Lukin, T.N. Zelentsova, E.P. Linnik, M.E. Beglaryan,\n V.P. Smolyar, M. Filippov and B. Vachev",
"submitter": "Vladimir Smolyar",
"url": "https://arxiv.org/abs/1208.4970"
}
|
1208.5012
|
# Precoder Design for Orthogonal Space-Time Block Coding based Cognitive Radio
with Polarized Antennas
Abdelwaheb Marzouki, Xin Jin
Institut Mines-Telecom, Telecom SudParis, CNRS Samovar UMR 5157, France
{abdelwaheb.marzouki, xin.jin}@it-sudparis.eu
###### Abstract
The spectrum sharing has recently passed into a mainstream Cognitive Radio
(CR) strategy. We investigate the core issue in this strategy: interference
mitigation at Primary Receiver (PR). We propose a linear precoder design which
aims at alleviating the interference caused by Secondary User (SU) from the
source for Orthogonal Space-Time Block Coding (OSTBC) based CR. We resort to
Minimum Variance (MV) approach to contrive the precoding matrix at Secondary
Transmitter (ST) in order to maximize the Signal to Noise Ratio (SNR) at
Secondary Receiver (SR) on the premise that the orthogonality of OSTBC is
kept, the interference introduced to Primary Link (PL) by Secondary Link (SL)
is maintained under a tolerable level and the total transmitted power
constraint at ST is satisfied. Moreover, the selection of polarization mode
for SL is incorporated in the precoder design. In order to provide an analytic
solution with low computational cost, we put forward an original precoder
design algorithm which exploits an auxiliary variable to treat the
optimization problem with a mixture of linear and quadratic constraints.
Numerical results demonstrate that our proposed precoder design enable SR to
have an agreeable SNR on the prerequisite that the interference at PR is
maintained below the threshold.
###### Index Terms:
Cognitive radio, precoder design, orthogonal space-time block coding,
polarized antennas.
## I Introduction
Cognitive Radio (CR) is an encouraging technology to combat the spectrum
scarcity. In order to further enhance the spectrum utilization, the spectrum
sharing strategy that Primary Users (PUs) and Secondary Users (SUs) coexist in
licensed bands as long as PUs are preserved from the interference caused by
SUs attracts much research efforts. Such a strategy is tantamount to a multi-
user system in which the inter-user interference mitigation is the core.
Various inter-user interference mitigation techniques for spectrum sharing CR
systems have been put forward. They can be roughly grouped into two
categories: power allocation [1]-[3] and precoding in Multiple-Input Multiple-
Output (MIMO) CR systems [4]-[7].
Space Time Block Coding (STBC) exploits time and space diversity in MIMO
systems so as to heighten the reliability of the message signal. Orthogonal
STBC (OSTBC) are contrived in such a fashion that the vectors of coding matrix
are orthogonal in both time and space dimensions. This feature yields a simple
linear decoding at the receiver side so that no complex matrix
manipulation—Singular Value Decomposition (SVD), for instance, is required for
recovering the information bit from the gathered received symbols. Numerous
precoding techniques have been mooted for unstructured codes. However, these
techniques cannot be applied to OSTBC which should forcibly preserve a special
space-time structure. The precoding design for OSTBC CR systems attracts less
attention in previous work. Such previous work in [7] was based on the Maximum
Likelihood (ML) space-time decoder, whereas the ML decoder is a nonlinear
method. Inspired by Minimum Variance (MV) receiver applied for OSTBC multi-
access systems [8] which used a weight matrix at the receiver side to quell
the inter-user interference, we make use of MV approach to design a precoding
matrix at Secondary Transmitter (ST).
The precoding matrix at ST is designed to comply with the needs in our CR
system: maximizing the Signal to Noise Ratio (SNR) at Secondary Receiver (SR)
on the premise that the orthogonality of OSTBC is kept, the interference
introduced to Primary Link (PL) by Secondary Link (SL) is maintained under a
tolerable level and the total transmitted power constraint at ST is satisfied.
The classic MV beamforming [9], [10] built an optimization problem which
includes only one linear constraint, that cannot administer to the needs in
our CR system. On the other hand, some precoder designs for CR systems [6]
introduced a mixture of linear and quadratic constraints to the optimization
problem which leads to iterative solutions with high computational complexity.
For the purpose of contriving a precoder that applies to our CR system and
provides an analytic solution with low computational cost, we moot an original
precoder design algorithm: we first take advantage of an optimization problem
which includes one linear constraint with the objective of preserving the
orthogonality of OSTBC and making SL introduce minimal interference to PL for
different combinations of the polarization mode at ST and SR. This
optimization problem provides an analytic solution in terms of an auxiliary
variable which is the system gain on SL. Then we regulate this auxiliary
variable using the quadratic constraints evoked by the transmitted power
budget at ST and the maximum tolerable interference at Primary Receiver (PR).
The polarization mode at ST and SR are conclusively settled on based upon the
maximization criteria of SNR at SR.
The rest of the paper is organized as follows. The system model and OSTBC are
presented in Section II. In Section III, we introduce the proposed precoder
design for OSTBC based CR with polarized antennas. We report the numerical
results and provide insights on the expected performance in Section IV.
Finally, we give the conclusion in Section V.
## II System Descriptions
We consider a CR system that consists of one SL which exploits OSTBC and one
PL. ST and PT are only allowed to communicate with their peers. ST or PT is
equipped with $N_{t}$ antennas and SR or PR is equipped with $N_{r}$ antennas.
The antennas in the same array have identical polarization mode. On each link,
the transmit antenna array or the receive antenna array is able to switch its
polarization mode between vertical mode $V$ and horizontal mode $H$. We denote
by $qt$ and $qr$, respectively, the transmit antenna array’s polarization mode
and the receive antenna array’s polarization mode.
### II-A System Model
In this paper, we exploit 3GPP Spatial Channel Model (SCM) [11]. The space
channel impulse response between a pair of antennas $u$ and $s$ of path $n$
can be expressed as a function in terms of the polarization channel response
and the geometric configuration of the antennas at both sides of the link:
$H_{u,s,n}\left(\chi_{BS}^{\left(v\right)},\chi_{BS}^{\left(h\right)},\chi_{MS}^{\left(v\right)},\chi_{MS}^{\left(h\right)},\theta_{n,m,AoD},\theta_{n,m,AoA}\right)$
(1)
where $\chi_{BS}^{\left(v\right)}$ is the BS antenna complex response for the
V-pol component, $\chi_{BS}^{\left(h\right)}$ is the BS antenna complex
response for the H-pol component, $\chi_{MS}^{\left(v\right)}$ is the MS
antenna complex response for the V-pol component, $\chi_{MS}^{\left(h\right)}$
is the MS antenna complex response for the H-pol component, $\theta_{n,m,AoD}$
is the Angle of Departure (AOD) for the $m$th subpath of the $n$th path and
$\theta_{n,m,AoA}$ is the Angle of Arrival (AOA) for the $m$th subpath of the
$n$th path.
We assume that the system is operated over a frequency-flat channel with
$N_{path}$ paths and each path contains only one subpath. For a point to point
communication link, the baseband input-output relationship at time-slot $t$ is
expressed as:
$\mathbf{y}\left(t\right)=\sqrt{\frac{\rho}{N_{t}}}\mathbf{H}^{qt,qr}\mathbf{x}\left(t\right)+\mathbf{n}\left(t\right)$
(2)
where $\rho$ is the SNR at each receive antenna, $\mathbf{x}\left(t\right)$ is
a $N_{t}\times 1$ size transmitted signal vector which satisfies
$E\left\\{\mathbf{x}\left(t\right)\mathbf{x^{\mathit{H}}}\left(t\right)\right\\}=N_{t}$,
$\mathbf{n}_{j}(t)$ is a $N_{r}\times 1$ size complex Gaussian noise vector at
receiver with zero-mean and unit-variance and $\mathbf{H}^{qt,qr}$ is the
$N_{r}\times N_{t}$ channel matrix for the specified $qt$ and $qr$ with the
entry
$H_{u,s}^{qt,qr}=\sum_{n=1}^{N_{path}}H_{u,s,n}\left(\chi_{BS}^{\left(x\neq
qt\right)}=0,\,\chi_{MS}^{\left(y\neq qr\right)}=0\right)$ (3)
where $x,y\in\left\\{V,\,H\right\\}$. $\mathbf{H}^{qt,qr}$ has unit variance
and satisfies
$E\left\\{\mathrm{tr\left(\mathbf{\mathbf{H}^{\mathit{qt,qr}}}\mathbf{\mathbf{H}^{\mathit{qt,qr}}}^{\mathrm{\mathit{H}}}\right)}\right\\}=N_{t}N_{r}$.
Assuming that the channel is constant from $t=1$ to $t=T$, then Equation (2)
can be extended into:
$\mathbf{Y}=\sqrt{\frac{\rho}{N_{t}}}\mathbf{\mathbf{H}^{\mathit{qt,qr}}}\mathbf{X}+\mathbf{N}$
(4)
where $\mathbf{Y}=\left[\mathbf{y}(1),\ldots,\mathbf{y}(T)\right]$,
$\mathbf{X}=\left[\mathbf{x}(1),\ldots,\mathbf{x}(T)\right]$ and
$\mathbf{N}=\left[\mathbf{n}\left(1\right),\ldots,\mathbf{n}\left(T\right)\right]$.
### II-B Orthogonal Space-Time Block Coding
If $\mathbf{X}$ is OSTBC matrix, then $\mathbf{X}$ has a linear representation
in terms of complex information symbols prior to space-time encoding
$s_{k},\,k=1,\ldots,K$ [12]:
$\mathbf{X}=\sum_{k=1}^{K}\left(\mathbf{C}_{k}\mathrm{Re}\left\\{s_{k}\right\\}+\mathbf{D}_{k}\mathrm{Im}\left\\{s_{k}\right\\}\right)$
(5)
where $\mathbf{C}_{k}$ and $\mathbf{D}_{k}$ are $N_{t}\times T$ code matrices
[13].
OSTBC matrix has the following unitary property:
$\mathbf{X}\mathbf{X}^{H}=\left(\sum_{k=1}^{K}\left|s_{k}\right|^{2}\right)\mathbf{I}_{N_{t}\times
N_{t}}$ (6)
In order to represent the relationship between the original symbols and the
received signal by multiplication of matrices, we introduce the “underline”
operator [13] to rewrite Equation (2) as:
$\mathbf{\underline{Y}}=\mathbf{\mathcal{\mathbb{\mathcal{H}}}}^{qt,qr}\mathbf{A}\mathbf{\mathrm{\mathbf{\underline{s}}}}+\underline{\mathbf{N}}$
(7)
where $\mathbf{s}=\left[s_{1},\ldots,s_{K}\right]$ is the data stream which is
QPSK modulated in this paper,
$\mathbb{\mathcal{H}}^{qt,qr}=\left[\begin{array}[]{cc}\mathrm{Re}\left\\{\mathbf{I}_{T}\otimes\mathbf{H}^{qt,qr}\right\\}&\mathrm{-Im}\left\\{\mathbf{I}_{T}\otimes\mathbf{H}^{qt,qr}\right\\}\\\
\mathrm{Im}\left\\{\mathbf{I}_{T}\otimes\mathbf{H}^{qt,qr}\right\\}&\mathrm{Re}\left\\{\mathbf{I}_{T}\otimes\mathbf{H}^{qt,qr}\right\\}\end{array}\right]$
is the equivalent channel matrix with the specified polarization mode,
$\mathbf{A}=\left[\underline{\mathbf{C}_{1}},\ldots,\underline{\mathbf{C}_{k}},\underline{\mathbf{D}_{1}},\ldots,\underline{\mathbf{D}_{k}}\right]$
is the OSTBC compact dispersion matrix and the “underline” operator for any
matrix $\mathbf{P}$ is defined as:
$\underline{\mathbf{P}}\triangleq\left[\begin{array}[]{c}\mathrm{vec}\left\\{\mathrm{Re}\left(\mathbf{P}\right)\right\\}\\\
\mathrm{vec}\left\\{\mathrm{Im}\left(\mathbf{P}\right)\right\\}\end{array}\right]$
(8)
where $\mathrm{vec}\left\\{\bullet\right\\}$ is the vectorization operator
stacking all columns of a matrix on top of each other.
The earliest OSTBC scheme which is well known as Alamouti’s code was proposed
in [14]. Alamouti’s code gives full diversity in the spatial dimension without
data rate loss. The transmission matrix of Alamouti’s code $C_{2}$ is given
as:
$C_{2}=\left[\begin{array}[]{cc}s_{1}&s_{2}\\\
-s_{2}^{*}&s_{1}^{*}\end{array}\right]$ (9)
In [15], Alamouti’s code was extended for more antennas. For instance, four
antennas, the transmission matrix of the half rate code $C_{4}$ is given as:
$C_{4}=\left[\begin{array}[]{cccc}s_{1}&s_{2}&s_{3}&s_{4}\\\
-s_{2}&s_{1}&-s_{4}&s_{3}\\\ -s_{3}&s_{4}&s_{1}&-s_{2}\\\
-s_{4}&-s_{3}&s_{2}&s_{1}\\\ s_{1}^{*}&s_{2}^{*}&s_{3}^{*}&s_{4}^{*}\\\
-s_{2}^{*}&s_{1}^{*}&-s_{4}^{*}&s_{3}^{*}\\\
-s_{3}^{*}&s_{4}^{*}&s_{1}^{*}&-s_{2}^{*}\\\
-s_{4}^{*}&-s_{3}^{*}&s_{2}^{*}&s_{1}^{*}\end{array}\right]$ (10)
## III Precoder for OSTBC based CR with Polarized Antennas
We design a precoding matrix at ST which acts on the entry of the OSTBC
compact dispersion matrix and has no influence on the codes’ structure. Our
precoder design relies on the equivalent transmit correlation matrix on the
link between ST and PR (SPL). This matrix can be estimated easily by SU in the
sensing step and enables our precoder design to regulate the interference
introduced by SL to PL.
### III-A Constraints from SL
With the precoding operation, the received signal at SR for the specified
polarization mode at ST and SR can be expressed as:
$\underline{\mathbf{Y_{\mathit{ST,SR}}^{\mathit{qt,qr}}}}=\sqrt{\frac{\rho_{SR}}{N_{t}}}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr}\mathbf{W}^{qt,qr}\mathbf{A}\mathbf{\underline{s}}+\underline{\mathbf{N}}$
(11)
where $\rho_{SR}$ is the SNR at each receive antenna of SR,
$\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr}$ is the SL equivalent channel matrix
with the specified polarization mode at ST and SR,
$\mathbf{\mathbf{W}^{\mathit{qt,qr}}}$ is the precoding matrix for the
specified polarization mode at ST and SR.
A straightforward approach to estimate the transmitted signal from ST is using
the following soft output detector:
$\displaystyle\mathbf{\underline{\hat{s}}}$ $\displaystyle=$
$\displaystyle\mathbf{A}^{T}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr^{T}}\underline{\mathbf{Y_{\mathit{ST,SR}}^{\mathit{qt,qr}}}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\rho_{SR}}{N_{t}}}\mathbf{A}^{T}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr^{T}}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr}\mathbf{\mathbf{W}^{\mathit{qt,qr}}}\mathbf{A}\mathbf{\underline{s}}+\mathbf{A}^{T}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr^{T}}\underline{\mathbf{N}}$
The OSTBC structure conservation puts forward the following constraint:
$\mathbf{A}^{T}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr^{T}}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr}\mathbf{\mathbf{W}^{\mathit{qt,qr}}}\mathbf{A}=\alpha^{qt,qr}\mathbf{I}_{2K}$
(13)
where $\alpha^{qt,qr}$ is the system gain on SL for the specified polarization
mode at ST and SR which will be adjusted to satisfy the other constraints.
Additionally, the transmitted power budget at ST induces another constraint:
$P_{t}^{qt,qr}\leq P_{tmax}$ (14)
where
$P_{t}^{qt,qr}=\frac{\rho_{SR}}{N_{t}}\mathrm{tr}\left(\mathbf{W^{\mathit{qt,qr}}}^{T}\mathbf{W}^{qt,qr}\right)$
and $P_{tmax}$ are, respectively, the transmitted power for the specified
polarization mode at ST and SR and the maximum transmitted power at ST.
### III-B Constraints from PL
The received signal at PR from ST is deemed as baleful signal by PL and can be
expressed as:
$\underline{\mathbf{Y_{\mathit{ST,PR}}^{\mathit{qt,qr^{\prime}}}}}=\sqrt{\frac{\rho_{PR}}{N_{t}}}\mathbb{\mathcal{H}}_{ST,PR}^{qt,qr^{\prime}}\mathbf{W}^{qt,qr}\mathbf{A}\mathbf{\underline{s}}+\underline{\mathbf{N}}$
(15)
where $\rho_{PR}$ is the SNR at each receive antenna of PR and
$\mathbb{\mathcal{H}}_{ST,PR}^{qt,qr^{\prime}}$ is the equivalent channel
matrix for the specified polarization mode at ST and PR.
The interference power introduced by SL to PL for the specified polarization
mode at ST and PR can be calculated as:
$\displaystyle P_{ST,PR}^{qt,qr^{\prime}}$ $\displaystyle=$
$\displaystyle\mathrm{tr}\left[E\left(\mathbf{\underline{\mathbf{\mathbf{Y_{\mathit{ST,PR}}^{\mathit{qt,qr^{\prime}}}}}}}\,\mathbf{\underline{\mathbf{\mathbf{Y_{\mathit{ST,PR}}^{\mathit{qt,qr^{\prime}}}}}}}^{H}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\rho_{SR}}{N_{t}}\mathrm{tr}\left(\mathbf{W^{\mathit{qt,qr}}}^{T}R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}\mathbf{W}^{qt,qr}\right)$
where
$\mathcal{R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}}=E\left(\mathbb{\mathcal{H}}_{PR,ST}^{qt,qr^{\prime}T}\mathbb{\mathcal{H}}_{PR,ST}^{qt,qr^{\prime}*}\right)$
is the equivalent transmit correlation matrix on SPL for the specified
polarization mode at ST and PR. The maximum tolerable interference power
$\eta$ at PR evokes the following constraint:
$P_{ST,PR}^{qt,qr^{\prime}}\leq\eta$ (17)
### III-C Minimum Variance Algorithm
SU can dominate the configuration of the precoding matrix and the polarization
mode on SL, while SU has no eligibility to select the polarization mode on PL.
Our algorithm is based on an optimization problem which includes one linear
constraint with the objective of preserving the orthogonality of OSTBC and
making SL introduce minimal interference to PL for different combinations of
the polarization mode at ST and SR. This optimization problem provides an
analytic solution in terms of an auxiliary variable which is the system gain
on SL. Then this auxiliary variable is regulated by using the quadratic
constraints evoked by the transmitted power budget at ST and the maximum
tolerable interference at PR. The polarization mode at ST and SR are
conclusively settled on based upon the maximization criteria of SNR at SR.
Such an optimization problem that includes one linear constraint is described
as follow:
$\left(\widehat{\mathbf{W^{\mathit{qt,qr}}}},\,\widehat{qt},\,\widehat{qr}\right)=\arg\min_{\mathbf{W^{\mathit{qt,qr}}},\,qt,\,qr}\frac{\rho_{SR}}{N_{t}}\mathrm{tr}\left(\mathbf{W^{\mathit{qt,qr}}}^{T}\mathcal{R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}}\mathbf{W^{\mathit{qt,qr}}}\right)$
(18)
$\mathrm{subject}\;\mathrm{to}:\;\mathrm{tr}\left(\mathbf{A}^{T}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr^{T}}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr}\mathbf{\mathbf{W^{\mathit{qt,qr}}}}\mathbf{A}\mathit{\mathrm{-}}\alpha^{qt,qr}\mathbf{I}_{2K}\right)=0$
(19)
We exploit the method of Lagrange multipliers to find
$\widehat{\mathbf{W^{\mathit{qt,qr}}}}$ for each combination of the
polarization mode at ST and SR. The Lagrangian function can be written as:
$\displaystyle
L\left(\mathbf{\mathbf{W^{\mathit{qt,qr}}}},\,\mathbf{\mathbf{\boldsymbol{\Lambda}}}\right)$
$\displaystyle=$
$\displaystyle\frac{\rho_{SR}}{N_{t}}\mathrm{tr}\left(\mathbf{\mathbf{W^{\mathit{qt,qr}}}}^{T}\mathcal{R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}}\mathbf{\mathbf{W^{\mathit{qt,qr}}}}\right)$
$\displaystyle-\mathrm{tr}\left(\mathbf{\mathbf{\boldsymbol{\Lambda}}}^{T}\left(\mathbf{A}^{T}\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}\mathbf{\mathbf{W^{\mathit{qt,qr}}}}\mathbf{A}\mathit{\mathrm{-}}\alpha^{qt,qr}\mathbf{I}_{2K}\right)\right)$
(20)
where
$\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}=\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr^{T}}\mathbb{\mathcal{H}}_{ST,SR}^{qt,qr}$
and $\boldsymbol{\Lambda}$ is a $2K\times 2K$ size matrix of Lagrange
multipliers.
By differentiating the Lagrange function with respect to
$\mathbf{\mathbf{W^{\mathit{qt,qr}}}}$ and equating it to zero, we obtain an
analytic solution in terms of $\alpha^{qt,qr}$ which is expressed as:
$\widehat{\mathbf{W^{\mathit{qt,qr}}}}=\alpha^{qt,qr}\mathcal{R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}}^{-1}\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}\mathbf{A}\mathbf{Q^{\mathit{qt,qr}}}\mathbf{A}^{T}$
(21)
where
$\mathbf{Q^{\mathit{qt,qr}}}=\left(\mathbf{A}^{T}\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}\left(R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}\right)^{-1}\mathbf{A}\right)^{-1}$.
The estimated interference power at PR can be expressed in terms of
$\alpha^{qt,qr}$ as:
$\widehat{P_{ST,PR}^{qt,qr}}=\frac{\rho_{SR}\left(\alpha^{qt,qr}\right)^{2}\mathrm{tr}\left(\mathbf{Q^{\mathit{qt,qr}}}\right)}{N_{t}}$
(22)
The estimated SNR at SR can be written in terms of $\alpha^{qt,qr}$ as:
$\widehat{SNR_{ST,PR}^{qt,qr}}=\frac{\rho_{SR}\left(\alpha^{qt,qr}\right)^{2}\gamma^{qt,qr}}{N_{t}}$
(23)
where
$\displaystyle\gamma^{qt,qr}=$
$\displaystyle\mathrm{tr}\left(\mathbf{Q^{\mathit{qt,qr}}}\mathbf{A}^{T}\left(\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}{}^{-1}\right)^{2}\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}\mathbf{AQ^{\mathit{qt,qr}}}\right)$
(24)
.
The estimated transmit power at ST in terms of $\alpha^{qt,qr}$ is given by:
$\widehat{P_{t}^{qt,qr}}=\frac{\rho_{SR}\left(\alpha^{qt,qr}\right)^{2}\delta^{qt,qr}}{N_{t}}$
(25)
where
$\delta^{qt,qr}=\mathrm{tr}\left(\mathbf{Q}^{qt,qr}\mathbf{A}^{T}\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}\left(R_{\mathit{PR,ST}}^{\mathit{qt,qr^{\prime}}}\right)^{-2}\mathcal{R_{\mathrm{\mathit{ST,SR}}}^{\mathit{qt,qr}}}\mathbf{AQ}^{qt,qr}\right)$
(26)
We derive $\alpha^{qt,qr}$ by substituting $\widehat{P_{t}^{qt,qr}}$ and
$\widehat{P_{ST,PR}^{qt,qr}}$ into Equation (14) and Equation (17) which
indicate the transmitted power budget constraint and the maximum tolerable
interference constraint:
$\alpha^{qt,qr}=\min\left(\sqrt{\frac{N_{t}}{\delta^{qt,qr}}},\,\sqrt{\frac{N_{t}\eta}{\rho_{SR}\mathrm{tr}\left(\mathbf{Q^{\mathit{qt,qr}}}\right)}}\right)$
(27)
Therefore the estimated SNR at SR can be determined as:
$\widehat{SNR_{ST,PR}^{qt,qr}}=\min\left(\frac{\rho_{SR}}{\delta^{qt,qr}},\frac{\eta}{\mathrm{tr}\left(\mathbf{Q}^{\mathit{qt,qr}}\right)}\right)\gamma^{qt,qr}$
(28)
Based upon the maximization criteria of SNR at SR, Finally, we destine the
estimated polarization mode of ST and SR as:
$\left(\widehat{qt},\widehat{qr}\right)=\arg\max_{qt,qr}\left[\min\left(\frac{\rho_{SR}}{\delta^{qt,qr}},\frac{\eta}{\mathrm{tr}\left(\mathbf{Q}^{qt,qr}\right)}\right)\gamma^{qt,qr}\right]$
(29)
## IV Numerical Results
For the purpose of validating our proposed precoding design algorithm, we
simulated our CR system using the proposed precoder design algorithm and
measure the SNR at SR by using varying maximum transmitted power at ST and a
reasonable interference threshold at PR.
We firstly carried out our simulation with Alamouti’s code at ST for different
combinations of $qt$ and $qr$ on SL under different multipath scenarios. Then,
we executed our simulation with different codes for different number of
transmit antennas at ST based upon a determinate combination of $qt$ and $qr$
on SL and multipath scenario. In both simulation scenarios, the Signal to
Interference plus Noise Ratio (SINR) threshold to perceive the received signal
at PR was chosen equal to $0\mathrm{dB}$ and the Cross-polar Discrimination
(XPD) was set to $8\mathrm{dB}$. The channel matrix on each link was modeled
according to 3GPP SCM. Since the status of polarization at PR $qr^{\prime}$ is
normally unidentified for SU, the equivalent transmit correlation matrix on
SPL becomes random. This thereby results in a random SNR at SR. In our
simulation, we calculated the SNR at SR in terms of the polarization tilt
angle at PR by introducing a rotation matrix to the equivalent transmit
correlation matrix on SPL. We assumed that the polarization tilt angle at PR
follows a continuous uniform distribution between $0$ and $\frac{\pi}{2}$.
Then we sampled uniformly over the range of the polarization tilt angle at PR
and calculated the SNR at SR for each sample of tilt angle. Finally, we worked
out an average the SNR at SR to evaluate the system performance.
### IV-A Performance Analysis of Polarization Diversity
We simulated a CR system, where ST is equipped with 2 antennas, SR is equipped
with 1 antenna and PR is equipped with 2 antennas. We observe the variation of
the average SNR at SR for different combinations of $qt$ and $qr$ on SL as the
transmit power at ST increases. First, we set SL channel as a 2-path frequency
flat fading channel and SPL channel as a single path frequency flat fading
channel. The variation tendencies in this scenario were depicted in Fig.1.
Then we reset SPL channel as a 4-path frequency flat fading channel and the
corresponding variation tendencies were shown in Fig.2. The average SNR at SR
for a large number of samples leads to the smooth curves. As the transmit
power at ST increases, the average SNR at SR of all different combinations of
$qt$ and $qr$ on SL exhibit uptrend in both scenarios and linear increase is
obtained when $P_{maxSU}/P_{noise}$ are below $15\mathrm{dB}$ in both
scenarios, where $P_{noise}$ denotes the noise power at SR. The mismatch of
$qt$ and $qr$ on SL induces a $15\mathrm{dB}$ gap between the matched modes
and the mismatched modes when the average SNR at SR has linear increase in the
first scenario. When we enhanced the number of paths in SPL channel, the
average SNR at SR for the mismatched modes was declined by $6\mathrm{dB}$ and
the gap was enlarged in the second scenario.
Figure 1: Average SNR at SR versus $P_{maxSU}/P_{noise}$ for different
polarization mode on SL under single path scenario Figure 2: Average SNR at SR
versus $P_{maxSU}/P_{noise}$ for different polarization mode under four paths
scenario
### IV-B Performance Analysis of Transmit Antennas Diversity
In the second simulation, we aimed to observe the average SNR at SR by using
different number of transmit antennas. In the first circumstance, 2 transmit
antennas and Alamouti’s code $C_{2}$ were utilized at ST. In the second
circumstance, 4 transmit antennas and the half rate code $C_{4}$ were utilized
at ST. In both circumstances, we set $qt=V$ and $qr=V$. SR is equipped with 1
antenna and PR is equipped with 4 antennas. The number of paths is chosen
equal to 2 on SL and 6 on SPL.
For the case of 2 transmit antennas at ST, the SNR at SR reaches the
saturation point at $20\mathrm{dB}$ when $P_{maxSU}/P_{noise}$ achieves
$40\mathrm{dB}$. Compare to the previous results in Fig. 1 and 2, the SNR at
SR reaches the saturation point faster due to the increase in number of paths
on the SPL. However, the increase in number of antennas will significantly
delay the arrival of the saturation point even the number of paths on the SPL
is also increased. For the case of 4 transmit antennas at ST, the SNR at SR
reaches the saturation point at $65\mathrm{dB}$ when $P_{maxSU}/P_{noise}$
achieves $100\mathrm{dB}$.
Figure 3: Average SNR at SR versus $P_{maxSU}/P_{noise}$ for different number
of transmit antennas at ST
## V Conclusions
A linear precoder design which aims at alleviating the interference at PR for
OSTBC based CR has been introduced. One of the principal contributions is to
endow the conventional prefiltering technique with the excellent features of
OSTBC in the context of CR. The prefiltering technique has been optimized for
the purpose of maximizing the SNR at SR on the premise that the orthogonality
of OSTBC is kept, the interference introduced to PL by SL is maintained under
a tolerable level and the total transmitted power constraint is satisfied.
Numeral Results have shown that polarization diversity contributes to achieve
better SNR at SR, moreover, the increase in number of antennas will
significantly delay the arrival of the saturation point for the SNR at SR.
## Acknowledgements
This research is supported by SACRA project (FP7-ICT-2007-1.1, European
Commission-249060).
## References
* [1] Y.-C. Liang, K.-C. Chen, G.-Y. Li, P. Mahonen, ”Cognitive Radio Networking and Communications: An Overview,” IEEE Trans. Vehicular Technology, vol. 60, no. 7, pp. 3386-3407, Sept. 2011.
* [2] A. Hoang, Y. Liang, and M. Islam, “Power control and channel allocation in cognitive radio networks with primary users’ cooperation,” IEEE Trans. Mobile Comput., vol. 9, no. 3, pp. 348–360, Mar. 2010.
* [3] R.-C. Xie, F.-R. Yu, H. Ji, “Dynamic Resource Allocation for Heterogeneous Services in Cognitive Radio Networks With Imperfect Channel Sensing,” IEEE Trans. Vehicular Technology, vol. 61, no. 2, pp. 770 - 780, Feb. 2012.
* [4] R. Prasad and A. Chockalingam, “Precoder Optimization in Cognitive Radio with Interference Constraints,” in Proc. IEEE ICC 2011.
* [5] M. Jung, K. Hwang, and S. Choi, “Interference Minimization Approach to Precoding Scheme in MIMO-Based Cognitive Radio Networks,” IEEE Commun. Lett., vol. 15, no. 8, pp. 789-791, Aug. 2011.
* [6] K. T. Phan, S. A. Vorobyov, N. D. Sidiropoulos, and C. Tellambura, “Spectrum sharing in wireless networks: A QoS-aware secondary multicast approach with worst user performance optimization,” in Proc. IEEE SAM’08, Jul. 2008, pp. 23–27.
* [7] A. Punchihewa, V.-K. Bhargava, C. Despins, “Linear Precoding for Orthogonal Space-Time Block Coded MIMO-OFDM Cognitive Radio,” IEEE Trans. Commun., vol. 59, no. 3, pp. 767-779, Mar. 2011.
* [8] S. Shahbazpanahi, M. Beheshti, A. B. Gershman, M. GharaviAlkhansari, and K. M. Wong, “Minimum variance linear receivers for multiaccess MIMO wireless systems with space–time block coding,” IEEE Trans. Signal Process., vol. 52, no. 12, pp. 3306–3313, Dec. 2004\.
* [9] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beamforming,” IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1684–1696, May 2005.
* [10] C. D. Richmond, “Capon algorithm mean squared error threshold SNR prediction and probability of resolution,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 2748–2764, Aug. 2005
* [11] 3GPP TR 25.996 V10.0.0, “Spatial channel model for MIMO simulations,” www.3gpp.org, Mar. 2011.
* [12] R. L. G. Cavalcante and I. Yamada, “Multiaccess interference suppression in OSTBC-MIMO systems by adaptive projected subgradient method,” IEEE Trans. Signal Process., vol. 56, no. 3, pp. 1028–1042, Mar. 2008.
* [13] M. Gharavi-Alkhansari and A. B. Gershman, “Constellation space invariance of orthogonal space–time block codes,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 331–334, Jan. 2005.
* [14] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998.
* [15] V. Tarokh, H. Javarkhani and Calderbank, “Space-time block codes from orthogonal designs,” IEEE Transactions on Information Theory, vol. 45, no. 5, July 1999, pp. 1456-1467.
|
arxiv-papers
| 2012-08-24T17:22:57 |
2024-09-04T02:49:34.532575
|
{
"license": "Public Domain",
"authors": "Abdelwaheb Marzouki and Xin Jin",
"submitter": "Xin Jin",
"url": "https://arxiv.org/abs/1208.5012"
}
|
1208.5193
|
# Categorical aspects of compact quantum groups
Alexandru Chirvasitu111UC Berkeley, [email protected]
###### Abstract
We show that either of the two reasonable choices for the category of compact
quantum groups is nice enough to allow for a plethora of universal
constructions, all obtained “by abstract nonsense” via the adjoint functor
theorem. This approach both recovers constructions which have appeared in the
literature, such as the quantum Bohr compactification of a locally compact
semigroup, and provides new ones, such as the coproduct of a family of compact
quantum groups, and the compact quantum group freely generated by a locally
compact quantum space. In addition, we characterize epimorphisms and
monomorphisms in the category of compact quantum groups.
Keywords: compact quantum group, CQG algebra, presentable category, SAFT
category, adjoint functor theorem
###### Contents
1. 1 Preliminaries
1. 1.1 Compact quantum groups
2. 1.2 CQG algebras
3. 1.3 Locally compact and algebraic quantum spaces and semigroups
4. 1.4 SAFT categories and the adjoint functor theorem
5. 1.5 Finitely presentable categories
2. 2 The category of CQG algebras is finitely presentable
3. 3 $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is SAFT
4. 4 Applications
1. 4.1 Limits in $\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
2. 4.2 Quantum groups generated by quantum spaces
3. 4.3 Variations on the Bohr compactification theme
4. 4.4 Kac quotients
5. 5 Monomorphisms
## Introduction
Compact quantum groups were introduced in essentially their present form in
[Wor87] (albeit under a different name), and the area has been expanding
rapidly ever since. The subject can be viewed as part of Connes’ general
program [Con94] to make “classical” notions (spaces, topology, differential
geometry) non-commutative: One recasts compact groups as $C^{*}$-algebras via
their algebras of continuous functions, and then removes the commutativity
assumption from the definition (see § 1.1 for details). Starting with the
relatively simple resulting definition, all manner of compact-group-related
notions and constructions can then be generalized to the non-commutative
setting: Peter-Weyl theory ([Wor87, Wor98]), Tannaka-Krein duality and
reconstruction ([Wor88, Wan97]), Pontryagin duality ([PW90]), actions on
operator algebras ([Boc95, Wan98, Wan99]) and other structures, such as
(classical or quantum) metric spaces ([Ban05, QS10]) or graphs ([Bic03]), and
so on. This list is not (and cannot be) exhaustive.
The goal of this paper is to analyze compact quantum groups from a category-
theoretic perspective, with a view towards universal constructions.
Bits and pieces appear in the literature: In [Wan95], Wang constructs
coproducts in the category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (see §
1.1), opposite to that of compact quantum groups (it consists of
$C^{*}$-algebras, which are morally algebras of functions on the non-existent
quantum groups). More generally, he constructs other types of colimits (e.g.
pushouts) of diagrams with one-to-one connecting morphisms. The fact that this
coproduct can be constructed simply at the level of $C^{*}$-algebras
(forgetting about comultiplications) parallels the fact that classically, the
underlying space of a categorical product $\prod G_{i}$ of compact groups
$G_{i}$ is, as a set, just the ordinary Cartesian product. The category of
compact groups, however, also admits coproducts, and they are slightly more
difficult to construct: One endows the ordinary, discrete coproduct $\coprod
G_{i}$ (i.e. coproduct in the category of discrete groups, also known as the
free product of the $G_{i}$) with the finest topology making the canonical
inclusions $G_{j}\to\coprod G_{i}$ continuous, and then takes the Bohr
compactification of the resulting topological group. It is natural, then, to
ask whether or not coproducts of compact quantum groups exist, or
equivalently, whether the category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
opposite to that of compact quantum groups has products. We will see in § 4.1
that this is indeed the case, and moreover, the category is complete (i.e. it
has all small limits).
Another example of universal construction that fits well within the framework
of this paper is the notion of quantum Bohr compactification [Soł05]. One of
the main results of that paper is, essentially, that the forgetful functor
from compact quantum groups to locally compact quantum semigroups has a left
adjoint; remembering that we are always passing from (semi)groups to algebras
of functions and hence reversing arrows, this amounts to the existence of a
certain right adjoint ([Soł05, 3.1,3.2]). Section 4 recovers this as one among
several right-adjoint-type constructions, such as compact quantum groups
“freely generated by a quantum space” (as opposed to quantum semigroup; see §
4.2).
Most compact-quantum-group-related universal constructions in the literature
seem to be of a “left adjoint flavor”: the already-mentioned colimits in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, the quantum automorphism groups
of, say, [Wan98], which are basically initial objects in the category of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ objects endowed with a coaction on
a fixed $C^{*}$-algebra, etc. By contrast, apart from the Bohr
compactification mentioned in the previous paragraph, universal constructions
of the right adjoint flavor (limits in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, or right adjoints to functors with
domain $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) appear not to have received
much attention. This is all the more surprising for at least two reasons.
First, they seem to be more likely to exist than the other kind of universal
construction; example: a (unital, say) $C^{*}$-algebra $A$ endowed with a
coassociative map $A\to A\otimes A$ into its minimal tensor square (this would
be the object dual to a compact quantum semigroup) always has a compact
quantum group (meaning its dual object, as in Definition 1.1.1) mapping into
it universally, but does not, in general, have a compact quantum group
receiving a universal arrow from it (Remark 4.3.3). Secondly, the
representation-theoretic interpretation of limits in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is often simpler than that of
colimits; see Proposition 4.1.3 (especially part (a)) and surrounding
discussion.
The structure of the paper is as follows:
Section 1 recalls the machinery that will be used in the sequel and fixes
notations and conventions, introducing the two versions of the category
opposite that of compact quantum groups: $\operatorname{\mathrm{CQG}}$,
consisting of so-called CQG algebras (these are like the algebra of
representative functions on a compact group; see Definition 1.2.2), and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, whose objects are analogous to
algebras of continuous functions on compact groups (Definition 1.1.1).
In Section 2, Theorem 2.0.1 shows that the category
$\operatorname{\mathrm{CQG}}$ is finitely presentable (§ 1.5). This technical
property will later allow us to reduce the existence of right adjoints for
functors defined on $\operatorname{\mathrm{CQG}}$ to checking that these
functors are cocontinuous, i.e. preserve colimits. This is typically an easy
task, as the routine nature of most proofs in Section 4 shows.
Section 3 proves a property slightly weaker than finite presentability for the
category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (Theorem 3.0.1). The nice
features from the previous section are preserved however, and the same types
of results (existence of right adjoints to various functors defined on
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) follow.
In Section 4 we list some of the consequences of the previous two sections.
These include the automatic existence of limits in
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
(§ 4.1), compact quantum groups freely generated by quantum spaces and
semigroups (§§ 4.2 and 4.3 respectively), and universal Kac type compact
quantum groups associated to any given compact quantum group (§ 4.4).
Finally, in Section 5 we characterize monomorphisms in the categories
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
It turns out that in the former they have to be one-to-one, whereas in the
latter being mono is slightly weaker than injectivity (Proposition 5.0.1). The
results are analogous to the fact ([Rei70, Proposition 9]) that epimorphisms
of compact groups are surjective.
### Acknowledgements
This work is part of my PhD dissertation. I would like to thank my advisor
Vera Serganova for all the support, and Piotr Sołtan for helpful discussions
on the contents of [Soł05].
## 1 Preliminaries
All algebraic entities in this paper (algebras, coalgebras, bialgebras, etc.)
are complex. A $*$-algebra is, as usual, a complex algebra endowed with a
conjugate linear, involutive, algebra anti-automorphism ‘$*$’. Unless we are
dealing with non-unital $C^{*}$-algebras as in § 1.3 below, in which case the
reader will be warned, algebras are assumed to be unital (and coalgebras are
always counital).
Our main references for the necessary basics on coalgebra, bialgebra and Hopf
algebra theory are [Abe80, Mon93, Swe69]. The notation pertaining to
coalgebras is standard: $\Delta$ for antipodes and $\varepsilon$ for the
counit, perhaps adorned with the name of the coalgebra if we want to be more
precise (example: $\Delta_{C}$, $\varepsilon_{C}$). The same applies to
antipodes for Hopf algebras, which are usually denoted by $S$. We use Sweedler
notation both for comultiplication, as in $\Delta(c)=c_{1}\otimes c_{2}$, and
for comodule structures: If $\rho:M\to M\otimes C$ is a right $C$-comodule
structure, it will be written as $\rho(m)=m_{0}\otimes m_{1}$. All comodules
are right, and the category of right comodules over a coalgebra $C$ is denoted
by $\mathcal{M}^{C}$.
For any comodule $V$ over any coalgebra $H$ (the notation suggests that it
will become a Hopf algebra soon), there is a largest subcoalgebra $H(V)$ over
which $V$ is a comodule. If the comodule structure map is $\rho:V\to V\otimes
H$ and $(e_{i})_{i\in I}$ is a basis for $V$,then $H(V)$ is simply the span of
the elements $u_{ij}$ defined by
$\rho(e_{j})=\sum_{i}e_{i}\otimes u_{ij}.$
We refer to $u_{ij}$ as the coefficients of the basis $(e_{i})$, to $(u_{ij})$
as the coefficient matrix of the basis, and to $H(V)$ as the coefficient
coalgebra of $V$. The coalgebra structure is particularly simple on
coefficients:
$\Delta(u_{ij})=\sum_{k}u_{ik}\otimes
u_{kj},\quad\varepsilon(u_{ij})=\delta_{ij}.$
Henceforth, the standing assumption whenever we mention coefficients and
coefficient coalgebras is that the comodule in question is finite-dimensional
(that is, $I$ is finite). When $V$ is simple, the coefficients $u_{ij}$ with
respect to some basis are linearly independent, and the coefficient coalgebra
is a matrix coalgebra, in the sense that its dual is a matrix algebra.
###### 1.0.1 Remark.
Note that maps $V\to V\otimes C$ are the same as elements of $V\otimes
V^{*}\otimes C=\operatorname{End}(V)\otimes C$. If $u=(u_{ij})$ is the
coefficient matrix of a basis $e_{i}$, $i=\overline{1,n}$ for $V$ and
$\operatorname{End}(V)$ is identified with $M_{n}$ via the same basis $e_{i}$,
then the element of $\operatorname{End}(V)\otimes C\cong M_{n}(C)$
corresponding to the coaction is exactly the coefficient matrix $u$. We will
often blur the distinction between these two points of view, and might refer
to $u$ itself as the comodule structure. $\blacklozenge$
If in the above discussion $H$ is a Hopf algebra, more can be said: The matrix
$S(u_{ij})_{i,j}$ is inverse to $(u_{ij})_{i,j}$. Moreover, giving the dual
$V^{*}$ the usual right $H$-comodule structure
$\langle f_{0},v\rangle f_{1}=\langle f,v_{0}\rangle S(v_{1}),\quad v\in V,\
f\in V^{*},$
the coefficient matrix of the basis dual to $(e_{i})$ is precisely
$(S(u_{ji}))_{i,j}$ (note the flipped indices).
A word on tensor products: In this paper, the symbol ‘$\otimes$’ means at
least three things. When appearing between purely algebraic objects, such as
algebras or just vector spaces, it is the usual, algebraic tensor product.
Between $C^{*}$-algebras it always means the minimal, or injective tensor
product ([Tak02, IV.4]). Finally, on rare occasions, we use the so-called
spatial tensor product (referred to as $W^{*}$-tensor product in [Tak02,
IV.5]) between von Neumann (or $W^{*}$) algebras. It will always be made clear
what the nature of the tensored objects is, so that no confusion is likely to
arise.
### 1.1 Compact quantum groups
This is by now a very rich and well-referenced theory, so we will be very
brief, and will refer the reader to one of the many excellent sources (e.g.
the papers and book cited below and the references therein) for details on the
topic.
No list of references would be complete without mentioning the seminal papers
[Wor87, Wor88], where Woronowicz laid the foundation of the subject,
introducing the main characters under the name “compact matrix pseudogroups”,
while an exposition of the main features of the theory is given by the same
author in [Wor98]. Other good references are the survey paper [KT99], and
[KS97, 11.4].
As mentioned in the introduction, the main idea is that since one can study
compact groups by means of the algebras of continuous functions on them, which
are commutative, unital $C^{*}$-algebras with some additional structure,
dropping the commutativity assumption but retaining the extra structure should
still lead to interesting objects, which are trying to be “continuous
functions on a quantum group”. The additional structure just alluded to is
captured in the following definition ([KT99, 3.1.1]):
###### 1.1.1 Definition.
A compact quantum group is a pair $(A,\Delta)$, where $A$ is a unital
$C^{*}$-algebra, and $\Delta:A\to A\otimes A$ is a morphism of unital
$C^{*}$-algebras satisfying the conditions
1. 1.
(Coassociativity)
$(\Delta\otimes\operatorname{id})\circ\Delta=(\operatorname{id}\otimes\Delta)\circ\Delta$;
2. 2.
(Antipode) The subspaces $\Delta(A)(1\otimes A)$ and $\Delta(A)(A\otimes 1)$
are dense in $A\otimes A$.
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is the category whose objects are
compact quantum groups, and whose morphisms $f:A\to B$ are unital
$C^{*}$-algebra maps preserving the comultiplication in the sense that
$(f\otimes f)\circ\Delta_{A}=\Delta_{B}\circ f$. $\blacklozenge$
The second condition needs some explanation. The space $\Delta(A)(1\otimes A)$
is defined as the linear span of products of the form $\Delta(a)(1\otimes
b)\in A\otimes A$, and similarly for $\Delta(A)(A\otimes 1)$. The condition is
named ‘antipode’ because it has to do with the demand that $A$, regarded as a
kind of bialgebra, have something like an antipode. For comparison, consider
the case of an ordinary, purely algebraic bialgebra $B$. The condition that it
have an antipode, i.e. that it be a Hopf algebra, is equivalent to the
requirement that the map
$B\otimes B\to B\otimes B,\quad x\otimes y\mapsto x_{1}\otimes x_{2}y$
be a bijection.
Since compact quantum groups as defined above are morally functions of
algebras, representations of the group must be comodules of some sort over the
corresponding algebras (endowed with their comultiplication). Some care must
be taken, as Definition 1.1.1 makes no mention of a counit, and so the usual
definition of comodule has to be modified slightly. The solution is (see
[KT99, discussion before Proposition 3.2.1]):
###### 1.1.2 Definition.
Let $(A,\Delta)$ be a compact quantum group. A finite-dimensional comodule
over $A$ is a finite-dimensional complex vector space $V$ together with a
coassociative coaction $\rho:V\to V\otimes A$ such that the corresponding
element of $\operatorname{End}(V)\otimes A$ is invertible. $\blacklozenge$
We will often drop the adjective ‘finite-dimensional’. Remark 1.0.1 applies,
and we will often refer to the coefficient comatrix of some basis as being the
comodule structure. [KS97, 11.4.3, Lemma 45] says that any comodule is
unitarizable, in the sense that there is an inner product on $V$ making the
coefficient matrix $u\in M_{n}(A)$ of an orthonormal basis unitary (cf.
Definition 1.2.1).
### 1.2 CQG algebras
These are the algebraic counterparts of compact quantum groups. More
precisely, if a compact quantum group as in Definition 1.1.1 plays the role of
the algebra of continuous functions on a “quantum group”, then the associated
CQG algebra wants to be the algebra of representative functions of the quantum
group, i.e. matrix coefficients of finite-dimensional unitary representations.
The main reference for this subsection is [KS97, 11.1-4].
Recall that a Hopf $*$-algebra $H$ is a Hopf algebra with a $*$-structure
making $H$ into a $*$-algebra, and such that the comultiplication and counit
are morphisms of $*$-algebras. This is the kind of structure that allows one
to define what it means for a representations of a quantum group (i.e. a
comodule over the corresponding “function algebra”) to be unitary.
Let $V$ be an $n$-dimensional comodule over a Hopf $*$-algebra $H$.
###### 1.2.1 Definition.
If $(\ \mid\ )$ is an inner product on $V$, the pair $(V,(\ \mid\ ))$ is said
to be unitary provided the coefficients $u_{ij}$ of an orthonormal basis
$e_{i}$, $i=\overline{1,n}$ form a unitary matrix in $H$.
A comodule $V$ is said to be unitarizable if there exists an inner product
making it unitary. This is equivalent to saying that for any basis $(e_{i})$,
the coefficient matrix $(u_{ij})_{i,j}$ can be made unitary by conjugating it
with a scalar $n\times n$ matrix. $\blacklozenge$
This is [KS97, Definition 5], and it is the correct compatibility condition
for a comodule structure and an inner product. See also [KS97, 11.1.5,
Proposition 11] for alternative characterizations of unitary comodules. We are
now ready to recall the main definition of this subsection ([KS97, 11.3.1,
Definition 9]):
###### 1.2.2 Definition.
A CQG algebra is a Hopf $*$-algebra which is the linear span of the
coefficient matrices of its unitarizable (or equivalently, unitary) finite-
dimensional comodules.
The category having CQG algebras as objects and Hopf $*$-algebra morphisms as
arrows will be denoted by $\operatorname{\mathrm{CQG}}$. $\blacklozenge$
###### 1.2.3 Remark.
It is a simple but useful observation that a quotient Hopf $*$-algebra of a
CQG algebra is automatically CQG. Indeed, a morphism of Hopf $*$-algebras will
turn a unitary coefficient matrix into another such. $\blacklozenge$
Let us recall that CQG algebras are automatically cosemisimple [KS97, 11.2],
i.e. their categories of comodules are semisimple. Another way to say this is
that a CQG algebra is the direct sum of its matrix subcoalgebras.
The following example will play an important role in Section 2. It is a family
of “universal” CQG algebras, in a sense that will be made precise below (see
[KS97, 11.3.1, Example 6], or [VDW96], where these objects were introduced in
their $C^{*}$-algebraic incarnation).
###### 1.2.4 Example.
Let $Q\in GL_{n}(\mathbb{C})$ be a positive operator, and denote by $A_{u}(Q)$
the $*$-algebra freely generated by elements $u_{ij}$, $i,j=\overline{1,n}$
subject to the relations making both $u=(u_{ij})_{i,j}$ and $\displaystyle
Q^{\frac{1}{2}}\overline{u}Q^{-\frac{1}{2}}$ unitary, where
$\overline{u}=(u^{*}_{ij})_{i,j}$. Strictly speaking, the main character here
is the pair $(A_{u}(Q),u)$ rather than just $A_{u}(Q)$: We always assume the
$u_{ij}$ are fixed as part of the structure, and refer to them as the standard
generators of $A_{u}(Q)$.
One way to state the universality property mentioned above is: For any CQG
algebra $A$ and any unitary coefficient matrix $v=(v_{ij})$ satisfying
$S^{2}(v)=QvQ^{-1}$, the map $u_{ij}\mapsto v_{ij}$ lifts to a unique CQG
algebra morphism $A_{u}(Q)\to A$.
Note that $A_{u}(Q)$ has a standard $n$-dimensional unitary comodule with
orthonormal basis $(e_{i})_{i=1}^{n}$, with the obvious structure
$e_{j}\mapsto\sum_{i}e_{i}\otimes u_{ij}$. $\blacklozenge$
There are various functors going back and forth between
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
First, since a CQG algebra is generated by elements of unitary matrices, there
is, for any element of the algebra, a uniform bound on the norm that element
can have when acting on any Hilbert space. It follows that any CQG algebra $A$
has an enveloping $C^{*}$-algebra $\overline{A}$. The fact that the
comultiplication and counit of the CQG algebra lift to give $\overline{A}$ a
compact quantum group structure follows from the universality property of this
envelope, as does the functoriality of this construction ([KS97, 11.3.3]).
This functor will be denoted by
$\operatorname{\textsc{univ}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
On the other hand, for any compact quantum group $B$, the coefficients of all
comodules (Definition 1.1.2) span a sub-Hopf $*$-algebra of $B$ in the obvious
sense, and again, the construction is easily seen to be functorial. We denote
this functor by
$\operatorname{\textsc{alg}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{CQG}}$.
Moreover, $\operatorname{\textsc{alg}}(B)$ is the only dense sub-Hopf
$*$-algebra of $B$ ([BMT01, A.1]).
Functors constructed in some natural way and going in opposite directions are
in the habit of being adjoints, and this situation is no different:
$\operatorname{\textsc{univ}}$ is the left adjoint. As it happens,
$\operatorname{\textsc{alg}}$ almost has right adjoint too. ‘Almost’ because
only its restriction to the category of CQG algebras and one-to-one morphisms
has a right adjoint, $\operatorname{\textsc{red}}$, associating to each CQG
algebra $A$ the so-called reduced [BMT01, $\S$2] compact quantum group having
$A$ as its dense sub-Hopf $*$-algebra. It is the “smallest” such object, in
the sense that any compact quantum group $A$ admits a unique surjective
morphism $A\to\operatorname{\textsc{red}}(\operatorname{\textsc{alg}}(A))$
which restricts to the identity on $\operatorname{\textsc{alg}}(A)$. A
detailed discussion on the interplay between the three functors mentioned in
this paragraph can be found in [BKQ11, 6.2].
It is probably clear by now that (the opposites of)
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are “the two reasonable choices for the category of compact quantum groups” of
the abstract. Which one is most convenient in any given case depends on which
aspects of the theory one wishes to focus on. For representation-theoretic
purposes, $\operatorname{\mathrm{CQG}}$ seems to be the correct choice, since
the CQG algebra $\operatorname{\textsc{alg}}(B)$ discussed above is tailor-
made to capture all information about unitary $B$-comodules. On the other
hand, there are purely analytic concepts (coamenability [BMT01]) whose very
definition requires the use of $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
In this paper, the $\operatorname{\mathrm{CQG}}$ vs.
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ distinction is a matter of
technical necessity. For various reasons having to do with the topological
aspect of being a $C^{*}$ rather than a $*$-algebra,
$\operatorname{\mathrm{CQG}}$ is the easier category to work with, as should
be apparent from the announced results once we review the necessary category
theory: The title of Section 2 (see § 1.5) is stronger than that of Section 3
(§ 1.4).
### 1.3 Locally compact and algebraic quantum spaces and semigroups
One of the themes that will be explored in Section 4 is, very roughly, the
existence of “compact quantum groups freely generated by quantum objects”.
Here, ‘objects’ can be things like ‘semigroups’ or ‘spaces’. Keeping in mind
that we are placing ourselves in the dual picture, where spaces are explored
through functions on them, we recall in this subsection how non-unital
$C^{*}$-algebras or plain $*$-algebras allow one to formalize such notions.
A good, brief account of more or less everything we need for the locally
compact side of the picture can be found in the ‘Notations and conventions’
section of [KV00] (assuming rudiments on multiplier algebras of
$C^{*}$-algebras [Tak02, III.6]).
Recall that for not-necessarily-unital $C^{*}$-algebras $A$ and $B$, a
morphism from $A$ to $B$ is by definition a continuous $*$-algebra
homomorphism $f:A\to M(B)$ into the multiplier algebra of $B$ which is non-
degenerate, meaning that the space $f(A)B$ is dense in $B$. It is then
explained in [KV00] how two such creatures can be composed, meaning that non-
unital $C^{*}$-algebras together with morphisms as defined above constitute a
category denoted here by $\operatorname{\mathrm{C}^{*}_{0}}$. It is to be
thought of as the category dual to that of locally compact quantum spaces.
$\operatorname{\mathrm{C}^{*}}$ is the subcategory consisting of unital
$C^{*}$-algebras. Note that the non-degeneracy condition on morphisms
automatically makes a $\operatorname{\mathrm{C}^{*}_{0}}$ arrow between
objects of $\operatorname{\mathrm{C}^{*}}$ unital.
For the definition of locally compact quantum semigroups we follow [Soł05],
referring again to the preliminary section of [KV00] for the missing details
on compositions of morphisms in $\operatorname{\mathrm{C}^{*}_{0}}$ (needed to
make sense of the coassociativity condition below).
###### 1.3.1 Definition.
A locally compact quantum semigroup is a pair $(A,\Delta)$, where $\Delta:A\to
A\otimes A$ is a morphism in $\operatorname{\mathrm{C}^{*}_{0}}$,
coassociative in the obvious sense.
The category $\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ has locally
compact quantum semigroups as objects, and
$\operatorname{\mathrm{C}^{*}_{0}}$-morphisms compatible with
comultiplications as arrows. $\blacklozenge$
Turning now to the algebraic side, everything just said has a natural
analogue. We again have to deal with multiplier algebras, this time of not-
necessarily-unital $*$-algebras; [VD94, Appendix] provides sufficient
background, and we will freely use the results and terminology therein. All
$*$-algebras are assumed to be non-degenerate, in the sense that $ab=0$,
$\forall b$ implies $a=0$ (the $*$-structure makes this condition symmetric).
For $*$-algebras $A$ and $B$, a morphism $A\to B$ is by definition a
$*$-homomorphism $f:A\to M(B)$, non-degenerate in the sense that $f(A)B$ spans
$B$. Composition goes through essentially as in the $C^{*}$ case, and we thus
get a category $\operatorname{\mathrm{Alg}^{*}_{0}}$. As before, the full
subcategory $\operatorname{\mathrm{Alg}^{*}}$ on unital $*$-algebras has only
unital morphisms as arrows. The algebraic counterpart to Definition 1.3.1 is
###### 1.3.2 Definition.
An algebraic quantum semigroup is a pair $(A,\Delta)$, where $\Delta:A\to
A\otimes A$ is a coassociative morphism in
$\operatorname{\mathrm{Alg}^{*}_{0}}$.
The category $\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ has algebraic
quantum semigroups as objects and and
$\operatorname{\mathrm{Alg}^{*}_{0}}$-morphisms compatible with
comultiplications as arrows. $\blacklozenge$
###### 1.3.3 Remark.
Whatever results we prove below within the framework of Definition 1.3.2,
close analogues exist for plain complex algebras rather than $*$-algebras. I
believe this one example is sufficient to illustrate how the universal
constructions of Section 4 go through in the algebraic, as well as the
$C^{*}$-algebraic setting. $\blacklozenge$
### 1.4 SAFT categories and the adjoint functor theorem
As Section 4 below is all about showing that certain functors have adjoints,
in this subsection and the next we recall the categorical machinery involved
in this. The main reference here is [ML98]. The set of morphisms $x\to y$ in a
category $\mathcal{C}$ will be denoted by $\mathcal{C}(x,y)$. Recall that
categories with all (co)limits (always small in this paper) are said to be
(co)complete, and functors preserving those (co)limits are called
(co)continuous (so ‘complete’ here means the same thing as Mac Lane’s ‘small-
complete’ [ML98, V]).
A class $S$ of objects in a category is said to be a generator (or a
generating class) if any two distinct parallel arrows $f\neq g:y\to z$ stay
distinct upon composition with an arrow $S\ni x\to y$ ([ML98, V.7]). We call
category generated if there is a generating set (as opposed to a proper
class).
An arrow $f:x\to y$ in a category $\mathcal{C}$ is an epimorphism if arrows
out of $y$ are uniquely determined by their “restriction to $x$” via
composition with $f$ ([ML98, I.5]). The quotient objects of $x$ are the
epimorphisms with source $x$, identified up to isomorphism in the comma
category $x\downarrow\mathcal{C}$ of arrows with source $x$ [ML98, II.6].
Finally, $\mathcal{C}$ is said to be co-wellpowered if for every object $x$,
the class of quotient objects of $x$ is actually a set.
We explained above how the aim is to construct things like “the compact
quantum group freely generated by a quantum semigroup”. What this means,
precisely, remembering that we are working with algebra-of-functions-type
objects, is that we want a right adjoint to, say, the inclusion functor
$\iota:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$
(this is just one example; there is also a $C^{*}$ version). Typically, when
trying to show that a functor $\iota$ is a left adjoint, one needs to check
(1) that $\iota$ is cocontinuous (this is certainly necessary, as left
adjoints are always cocontinuous) and (2) that some kind of solution set
condition is satisfied [ML98, V.6.2]. For some categories, however, (2) is
unnecessary: they are such that any cocontinuous functor out of them is
automatically a left adjoint. One sufficient set of conditions that will
ensure this is provided by the following result, due to Freyd and referred to
in the literature as the special adjoint functor theorem (dual to [ML98,
V.8.2]):
###### 1.4.1 Theorem.
Let $\mathcal{C}$ be a cocomplete, generated, and co-wellpowered category.
Then, any cocontinuous functor with domain $\mathcal{C}$ is a left adjoint.
In view of this result, it is natural to isolate the hypotheses:
###### 1.4.2 Definition.
A category is SAFT if it is cocomplete, generated, and co-wellpowered.
$\blacklozenge$
###### 1.4.3 Remark.
Not-necessarily-cocomplete categories satisfying the adjoint functor theorem
in the sense that functors are only required to preserve those colimits which
exist are called ‘compact’ in [Kel86]. One important property of compact (and
hence SAFT) categories is that they are automatically complete. This is, for
example, the implication (ii) $\Rightarrow$ (v) in [Kel86, Theorem 5.6].
$\blacklozenge$
### 1.5 Finitely presentable categories
One way to be SAFT is to be what in the literature is called ‘locally
presentable’. We review the main features of the theory here, and refer mainly
to [AR94] for details. For brevity, in this paper we drop the word ‘locally’.
A poset $(J,\leq)$ is said to be filtered if every finite subset is majorized
by some element (this is [AR94, 1.4], restricted to posets as opposed to
arbitrary categories). In the sequel, a filtered diagram in a category
$\mathcal{C}$ is a functor a $(J,\leq)\to\mathcal{C}$, and a filtered colimit
is a colimit of such a functor. Then, [AR94, 1.9] is (essentially, via [AR94,
1.5]):
###### 1.5.1 Definition.
An object $x\in\mathcal{C}$ is finitely presentable if the functor
$\mathcal{C}(x,-)$ preserves filtered colimits.
The category $\mathcal{C}$ is finitely presentable if it is cocomplete, and
there is a set $S$ of finitely presentable objects such that every object in
$\mathcal{C}$ is a filtered colimit of objects in $S$. $\blacklozenge$
More rigorously, the last condition says that every object is the colimit of a
functor $F:J\to\mathcal{C}$ taking values in $S$, with $(J,\leq)$ filtered.
###### 1.5.2 Example.
All categories familiar from algebra, of the form ‘set with this or that kind
of structure’, such as groups, abelian groups, monoids, semigroups, algebras,
$*$-algebras, modules over a ring, etc. are finitely presentable. These are
the so-called finitary varieties of algebras [AR94, 3.A], ‘finitary’ having to
do with ‘finitely presentable’.
The terminology is also inspired by such examples. In the category of modules
over a ring, say, an object is finitely presentable in the above abstract
sense if and only if it has a finite presentation in the usual sense (in full
generality, the result is [AR94, 3.11]). $\blacklozenge$
What matters here is that as mentioned above, finitely presentable implies
SAFT. Indeed, cocompleteness is part of Definition 1.5.1, and the generating
set required for SAFT-ness almost is: $S$ is easily seen to be a generator.
Co-wellpowered-ness, on the other hand, is the difficult result [AR94, 1.58].
The following proposition is the criterion of finite presentability we use in
the proof of the main result of Section 2. It is a consequence of [AR94, 1.11]
(via 0.5, 0.6 of op. cit.), and in order to state it, one more piece of
terminology is needed.
###### 1.5.3 Definition.
We will say that a generator $S$ of a cocomplete category $\mathcal{C}$ is
regular if every object of $\mathcal{C}$ is the coequalizer of two parallel
arrows $f,g:y\to z$, where $y$ and $z$ are coproducts of objects in $S$.
$\blacklozenge$
###### 1.5.4 Proposition.
A cocomplete category with a regular generator consisting of finitely
presentable objects is finitely presentable.
## 2 The category of CQG algebras is finitely presentable
This section is devoted to proving the result in the title:
###### 2.0.1 Theorem.
The category $\operatorname{\mathrm{CQG}}$ is finitely presentable.
The main tool in the proof is Proposition 1.5.4, according to which
cocompleteness (by now well known) is first on the agenda.
###### 2.0.2 Proposition.
$\operatorname{\mathrm{CQG}}$ is cocomplete.
###### Proof.
It is enough to show that the category has coproducts and coequalizers of
parallel pairs of arrows [ML98, V.2.1]. Both will be constructed as simply the
colimits of the underlying diagrams of $*$-algebras (i.e. coalgebra structures
play no role in the construction of colimits).
Coproducts are essentially constructed in [Wan95, Theorem 1.1]. That result is
concerned with the $C^{*}$-algebraic version (constructing coproducts in the
category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$), but as remarked by Wang
at the end of [Wan95, $\S$1], the algebraic version holds as well. Given a set
of CQG agebras, the universal property of the coproduct of underlying algebras
gives this coproduct the extra structure that will make it into a CQG algebra;
we leave the details to the reader.
To construct coequalizers, let $f,g:A\to B$ be morphisms of CQG algebras. The
ideal $I$ generated by the elements $f(a)-g(a)$, $a\in A$ is in fact a
coideal, as well as invariant under $*$. The latter assertion is trivial, so
let us focus on $I$ being a coideal. Compatibiity of $f$ and $g$ with counits
says that $f(a)-g(a)$ is annihilated by $\varepsilon_{B}$. On the other hand,
the familiar computation
$\displaystyle\Delta_{B}(f(a)-g(a))=$ $\displaystyle\ (f\otimes
f)(\Delta_{A}(a))-(g\otimes g)(\Delta_{A}(a))$ $\displaystyle=$
$\displaystyle\ f(a_{1})\otimes f(a_{2})-g(a_{1})\otimes g(a_{2})$
$\displaystyle=$ $\displaystyle\
f(a_{1})\otimes(f-g)(a_{2})+(f-g)(a_{1})\otimes g(a_{2})\in B\otimes
I+I\otimes B$
shows that $I$ plays well with the comultiplication. It follows that the
coequalizer of $f$ and $g$ in $\operatorname{\mathrm{Alg}^{*}}$ is a quotient
Hopf $*$-algebra of $B$, and hence a CQG algebra by Remark 1.2.3.
$\blacksquare$
The plan now is to show that the set $S$ consisting of the CQG algebras
$A_{u}(Q)$ of Example 1.2.4 (for all possible positive operators $Q$, of all
possible sizes) satisfies the hypotheses of Proposition 1.5.4: Every
$A_{u}(Q)$ is finitely presentable in $\operatorname{\mathrm{CQG}}$ in the
sense of Definition 1.5.1, and $S$ is a regular generator. We start with the
former.
###### 2.0.3 Proposition.
For any positive $Q\in GL_{n}(\mathbb{C})$, the object
$A=A_{u}(Q)\in\operatorname{\mathrm{CQG}}$ is finitely presentable.
###### Proof.
Let $(J,\leq)$ be a filtered poset, and let $A_{j}$, $j\in J$ implement a
functor $J\to\operatorname{\mathrm{CQG}}$ by means of CQG algebra morphisms
$\iota_{j^{\prime}j}:A_{j}\to A_{j^{\prime}}$ for $j\leq j^{\prime}$. Denote
also by $\displaystyle\iota_{i}:A_{i}\to B=\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}} {}{}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}A_{j}$ the
structural morphisms into the colimit. We have to show that the canonical map
$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}\operatorname{\mathrm{CQG}}(A,A_{j})\to\operatorname{\mathrm{CQG}}(A,B)$
(1)
is a bijection.
It is clear from the description in Example 1.2.4 that $A$ is finitely
presented as a $*$-algebra, and is hence a finitely presentable object in
$\operatorname{\mathrm{Alg}^{*}}$ ([AR94, 3.11] with $\lambda=\aleph_{0}$). In
conclusion, the canonical map
$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}\operatorname{\mathrm{Alg}^{*}}(A,A_{j})\to\operatorname{\mathrm{Alg}^{*}}(A,B)$
(2)
is bijective, and the injectivity of 1 follows from the commutative square
$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-3.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-0.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@moveto{7.14445pt}{-0.2pt}\pgfsys@moveto{6.14445pt}{-3.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@lineto{5.68446pt}{-3.2pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-3.2pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\operatorname{\mathrm{CQG}}(A,A_{j})$$\operatorname{\mathrm{CQG}}(A,B)$$\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-3.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-0.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@moveto{7.14445pt}{-0.2pt}\pgfsys@moveto{6.14445pt}{-3.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@lineto{5.68446pt}{-3.2pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-3.2pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\operatorname{\mathrm{Alg}^{*}}(A,A_{j})$$\operatorname{\mathrm{Alg}^{*}}(A,B)$,
where the vertical arrows are the obvious inclusions.
To prove that 1 is surjective, fix a CQG algebra morphism $f:A\to
B=\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}A_{j}$. It is, in
particular, a morphism in $\operatorname{\mathrm{Alg}^{*}}$, so by the
surjectivity of 2, it factors through a unital $*$-algebra morphism
$f_{i}:A\to A_{i}$ for some $i\in J$, and hence through $f_{j}=\iota_{ji}\circ
f_{i}:A\to A_{j}$ for $j\geq i$. For such $j$, consider the diagram
$A$$A_{j}$$B$$B\otimes B$,$A_{j}\otimes A_{j}$$A\otimes A$$\scriptstyle
f_{j}$$\scriptstyle\iota_{j}$$\scriptstyle f_{j}\otimes
f_{j}$$\scriptstyle\iota_{j}\otimes\iota_{j}$
where the vertical maps are comultiplications. The commutativity of the outer
rectangle is nothing but the preservation of coproducts by $f=\iota_{j}\circ
f_{j}$, while the right hand square commutes because $\iota_{j}:A_{j}\to B$ is
by definition a colimit in $\operatorname{\mathrm{CQG}}$. It follows that the
two $J$-indexed systems of morphisms $\Delta_{A_{j}}\circ f_{j}$ and
$(f_{j}\otimes f_{j})\circ\Delta_{A}$ become equal upon composing further with
$\iota_{j}\otimes\iota_{j}$. The fact that they must then be equal for
sufficiently large $j$ follows from the next lemma, which says essentially
that $\iota_{j}\otimes\iota_{j}$ make $B\otimes B$ the colimit of the diagram
consisting of the maps $\iota_{j^{\prime}j}\otimes\iota_{j^{\prime}j}$,
together with the injectivity of
$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}\operatorname{\mathrm{Alg}^{*}}(A,A_{j}\otimes
A_{j})\longrightarrow\operatorname{\mathrm{Alg}^{*}}(A,\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}A_{j}\otimes
A_{j})$
resulting from the finite presentability of $A$ in
$\operatorname{\mathrm{Alg}^{*}}$. $\blacksquare$
###### 2.0.4 Lemma.
The tensor square endofunctor $A\mapsto A\otimes A$ on
$\operatorname{\mathrm{Alg}^{*}}$ preserves filtered colimits.
###### Proof.
Working with $*$-algebras is of little importance here: The forgetful functor
from any finitary variety of algebras to the category Set of sets creates
filtered colimits. We refer again to [AR94, 3.A] for background on varieties
of algebras. The claim just made can be proven either by realizing the variety
as the Eilenberg-Moore category of a finitary monad (i.e. one which preserves
filtered colimits) on Set [AR94, 3.18], or directly. It follows that it is
enough to prove the analogous statement in the category Vec of complex vector
spaces.
Let $(J,\leq)$ be a filtered poset, $\iota_{j^{\prime}j}:V_{j}\to
V_{j^{\prime}}$ a functor from it to Vec, and $\displaystyle
V=\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}V_{j}$. We have
to show that the canonical map $\displaystyle\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}(V_{j}\otimes
V_{j})\to V\otimes V$ is an isomorphism.
Let $(J\times J,\leq)$ be the cartesian square of the category $(J,\leq)$; it
is simply the poset structure on the set $J\times J$ defined by
$(i,j)\leq(i^{\prime},j^{\prime})$ iff $i\leq i^{\prime}$ and $j\leq
j^{\prime}$. Consider the functor $F:(J\times J,\leq)\to\textsc{Vec}$ given by
$(i,j)\mapsto V_{i}\otimes V_{j}$ (with the obvious action on morphisms). For
any vector space $W$, the endofunctor
$W\otimes\bullet:\textsc{Vec}\to\textsc{Vec}$ is left adjoint to
$\operatorname{Hom}(W,\bullet)$, and hence cocontinuous. Applying this
observation first to $V_{i}$ and then to $V$, we get the last two isomorphisms
in the chain
$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i,j}(V_{i}\otimes
V_{j})\cong\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i}\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}(V_{i}\otimes
V_{j})\cong\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i}(V_{i}\otimes
V)\cong V\otimes V.$ (3)
The first one, on the other hand, is the usual Fubini-type separation of
variables for colimits ([ML98, IX.8]).
The original poset $J$ sits diagonally inside $J\times J$ as the set of pairs
$(j,j)$. Moreover, the fact that $J$ is filtered translates to $J$ being
cofinal in $J\times J$ in the sense that everyone in the latter is majorized
by someone in the former. But it then follows [ML98, IX.3.1] that the
canonical map
$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}(V_{j}\otimes
V_{j})=\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}F|_{J}\longrightarrow\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}F=\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i,j}(V_{i}\otimes
V_{j})$
is an isomorphism. Composing it with 3 finishes the proof. $\blacksquare$
###### 2.0.5 Remark.
We mentioned briefly at the end of § 1.2 that the category
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is not as friendly as
$\operatorname{\mathrm{CQG}}$. Lemma 2.0.4, for example, is somewhat
problematic. The problem with the above proof is that it hinges on functors of
the form
$A\otimes\bullet:\operatorname{\mathrm{Alg}^{*}}\to\operatorname{\mathrm{Alg}^{*}}$
preserving filtered colimits; I do not know whether the analogous result holds
for the category $\operatorname{\mathrm{C}^{*}}$ of unital $C^{*}$-algebras
with the minimal tensor product (which is what would be needed to make the
proof work verbatim for $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$).
Some partial results (which we do not prove here) are that (a) for a
$C^{*}$-algebra $A$, the functor $A\otimes\bullet$ does preserve filtered
colimits of injections, (b) the same functor preserves all filtered colimits
provided $A$ is an exact $C^{*}$-algebra in the sense of [Was94] (this simply
means that minimal tensoring with $A$ preserves short exactness of sequences
in $\operatorname{\mathrm{C}^{*}_{0}}$), and (c) the maximal tensor product
with $A$ does preserve all filtered colimits. This suggests that trying to
adapt Lemma 2.0.4 to $\operatorname{\mathrm{C}^{*}}$ adds a layer of
difficulty, in that one has to deal with issues like nuclearity and exactness.
$\blacklozenge$
The last piece of the puzzle is
###### 2.0.6 Proposition.
The set $S$ of all $A_{u}(Q)$ is a regular generator in
$\operatorname{\mathrm{CQG}}$.
###### Proof.
According to Definition 1.5.3, we have to show that an arbitrary CQG algebra
$A$ is the coequalizer of two arrows $f,g:Y\to Z$ between coproducts of
$A_{u}(Q)$’s.
Let $V^{\alpha}$ be representatives for the set $\widehat{A}$ of unitary
simple comodules of $A$. Then, $A$ is the direct sum of the matrix coalgebras
$C^{\alpha}$ spanned by the unitary coefficient matrices
$v^{\alpha}=(v^{\alpha}_{ij})$ with respect to orthonormal bases
$(e^{\alpha}_{i})$ of $V^{\alpha}$.
The squared antipode $S^{2}$ conjugates every matrix $v^{\alpha}$ by some
positive operator $Q^{\alpha}$ [KS97, 11.2.3, Lemma 30], and hence, by the
universality property of the $A_{u}(Q)$’s as cited in Example 1.2.4, the
assignment $u_{ij}\mapsto v^{\alpha}_{ij}$ defines a CQG algebra morphisms
$A_{u}(Q^{\alpha})\to A$. Moreover, the resulting map
$\pi:\coprod_{\widehat{A}}A_{u}(Q^{\alpha})\to A$ (4)
is surjective. The left hand side of this expression will be our $Z$. For
$\alpha\in\widehat{A}$, we denote by $u^{\alpha}$ the standard coefficient
matrix in $A_{u}(Q^{\alpha})$ (earlier in this paragraph, where we reasoned
one $\alpha$ at a time, it was denoted simply by $u$).
The two morphisms $f,g:Y\to Z$ that we are looking for (and whose coequalizer
4 should be) ought to somehow recover the relations of $A$, i.e. the
multiplication table with respect to the basis $(v^{\alpha}_{ij})$ of $A$.
To define $f$, fix $\alpha,\beta\in\widehat{A}$. A simple calculation shows
that $u^{\alpha}u^{\beta}=(u^{\alpha}_{ij}u^{\beta}_{kl})_{ik,jl}$ is a
unitary coefficient matrix of $Z=\coprod A_{u}(Q^{\alpha})$, which the squared
antipode of this CQG algebra conjugates by $Q^{\alpha\beta}=Q^{\alpha}\otimes
Q^{\beta}$ (this is just notation). It follows that there is a unique morphism
$A_{u}(Q^{\alpha\beta})\to Z$ defined by $u_{ik,jl}\mapsto
u^{\alpha}_{ij}u^{\beta}_{kl}$. Putting all of these together for all pairs of
comodules, we get
$f:Y=\coprod_{\widehat{A}\times\widehat{A}}A_{u}(Q^{\alpha\beta})\to\coprod
A_{u}(Q^{\alpha})=Z.$
As before, since we need to distinguish between the various coefficient
matrices in $Y$, we denote them by $u^{\alpha\beta}$ in the obvious way.
We now start on our way towards constructing $g:Y\to Z$. The same game as in
the previous paragraph can be played in $A$: For any
$\alpha,\beta\in\widehat{A}$,
$v^{\alpha}v^{\beta}=(v^{\alpha}_{ij}v^{\beta}_{kl})_{ik,jl}$ is the unitary
coefficient matrix with respect to the tensor product basis
$e^{\alpha}_{i}\otimes e^{\beta}_{k}$ of the tensor product Hilbert space
$V^{\alpha}\otimes V^{\beta}$. But now, since we are in $A$, the elements of
this matrix can be expressed as linear combinations of $v^{\gamma}_{ij}$’s. In
order to avoid cumbersome indices on the coefficients of such linear
combinations, we simply write
$v^{\alpha}_{ij}v^{\beta}_{kl}=\ell^{\alpha\beta}_{ik,jl}\in\bigoplus_{\gamma}C^{\gamma},$
where $C^{\gamma}$ is the matrix coalgebra corresponding to
$\gamma\in\widehat{A}$, and $\gamma$ ranges over the simple comodules
appearing in the decomposition of $V^{\alpha}\otimes V^{\beta}$.
Now, because the restriction of $\pi$ to the direct sum $C\leq Z$ of matrix
coalgebras spanned by $u^{\alpha}\subset Z$ is one-to-one (in fact, this
restriction is by definition an isomorphism onto $A$), the elements
$\ell^{\alpha\beta}_{ik,jl}$ defined above lift uniquely to elements of $C$,
and we slightly abusively denote these lifts by
$\pi^{-1}(\ell^{\alpha\beta}_{ik,jl})$. I claim that for fixed $\alpha$ and
$\beta$, these elements form a unitary coefficient matrix which $S^{2}$
conjugates by $Q^{\alpha}\otimes Q^{\beta}$. Indeed, all of these properties
can be stated inside $C$ (without appealing to multiplication), using only the
antipode and the $*$-structure (being unitary, for example, amounts to the
antipode turning $\pi^{-1}(\ell^{\alpha\beta}_{ik,jl})$ into
$\pi^{-1}(\ell^{\alpha\beta}_{jl,ik})^{*}$); since $\pi$ preserves both the
antipode and the $*$ structure and its restriction to $C$ is a coalgebra
isomorphism, the properties all lift from the $\ell$’s to the
$\pi^{-1}(\ell)$’s.
Finally, the claim just proven allows us to construct $g:Y\to Z$ by sending
$u^{\alpha\beta}_{ik,jl}$ to $\pi^{-1}(\ell^{\alpha\beta}_{ik,jl})$. The
coequalizer of $f$ and $g$ is the quotient of $Z$ by the relations imposing on
$u^{\alpha}_{ij}$ the same multiplication table as that of the
$v^{\alpha}_{ij}$’s, so it is now clear that this coequalizer is precisely
$\pi:Z\to A$. $\blacksquare$
We can now put the last few results together:
###### Proof of Theorem 2.0.1.
We know from Proposition 2.0.2 that $\operatorname{\mathrm{CQG}}$ is
cocomplete, and from Propositions 2.0.3 and 2.0.6 that a set of finitely
presentable objects forms a regular generator. The conclusion follows from
Proposition 1.5.4. $\blacksquare$
###### 2.0.7 Remark.
In essentially the same way, we can show that the category
$\operatorname{\mathrm{CQG}_{ab}}$ of commutative CQG algebras is finitely
presentable. In this case, all distinctions between the algebraic and the
$C^{*}$-algbraic vanish: The restriction of $\operatorname{\textsc{univ}}$ to
$\operatorname{\mathrm{CQG}_{ab}}$ is an equivalence onto the full subcategory
$\operatorname{\mathrm{C}^{*}\mathrm{QG}_{ab}}$ of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ consisting of commutative algebras.
Moreover, $\operatorname{\mathrm{CQG}_{ab}}$ (or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}_{ab}}$) is nothing but the opposite
of the category of compact groups, with a compact group $G$ corresponding to
the CQG algebra of representative functions on it.
The only changes we need to make to the proofs in order to adapt the
presentability result to $\operatorname{\mathrm{CQG}_{ab}}$ are (a) substitute
tensor products (of perhaps infinite families) for coproducts, and (b) use the
set of CQG algebras associated to all unitary groups $U_{n}$ for a generator,
instead of the $A_{u}(Q)$’s. $\blacklozenge$
###### 2.0.8 Remark.
Although, strictly speaking, SAFT-ness would have sufficed for the purposes of
Section 4, the finite presentability of $\operatorname{\mathrm{CQG}}$ is
interesting in its own right, as it is somewhat surprising: Given the close
relationship between $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, discussed a little in § 1.2 above,
one might think that the former category should look more or less like “unital
$C^{*}$-algebras with a lot of extra structure”, and hence should be at least
as reluctant to be finitely presentable as the category
$\operatorname{\mathrm{C}^{*}}$ of unital $C^{*}$-algebras. However, this is
not the case.
There is a more general notion of presentability for categories (local
presentability in the literature, e.g. [AR94]) parametrized by a regular
cardinal number, so that the technical term for ‘finitely presentable’ is
‘$\aleph_{0}$-presentable’; the larger the cardinal, the weaker the notion.
Now, it can be shown that $\operatorname{\mathrm{C}^{*}}$ is
$\aleph_{1}$-presentable but not finitely presentable. Worse still, the same
is true in the commutative setting: Although the previous remark notes that
$\operatorname{\mathrm{CQG}_{ab}}$ is finitely presentable, the category of
commutative unital $C^{*}$-algebras is $\aleph_{1}$, but not finitely
presentable. $\blacklozenge$
## 3 $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is SAFT
The main result of the section is the one just stated:
###### 3.0.1 Theorem.
The cateory $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is SAFT.
We prove the three properties required for SAFT-ness (Definition 1.4.2)
separately.
###### 3.0.2 Proposition.
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is cocomplete.
###### Proof.
This parallels the proof of Proposition 2.0.2 by constructing coequalizers and
coproducts, so we will be brief.
As noted in the proof just mentioned, coproducts are constructed in [Wan95,
Theorem 1.1].
As for coequalizers, they are constructed as before. The coequalizer of two
morphisms $f,g:A\to B$ in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is the
quotient of $B$ by the closed ideal $I$ generated by $f(a)-g(a)$, and the
argument from the proof of Proposition 2.0.2 can be paraphrased to show this.
Although Sweedler notation is not available anymore (because we are working
with $C^{*}$ tensor products rather than algebraic ones), the computation
carried out there can be written down in a Sweedler-notation-free manner as
saying that $\Delta_{B}\circ(f-g)$ equals $(f\otimes(f-g)+(f-g)\otimes
g)\circ\Delta_{A}$. It follows that $B/I$ inherits a coassociative
comultiplication and $B\to B/I$ respects it, while the (Antipode) condition of
Definition 1.1.1 follows immediately from that of $B$. In conclusion, the
quotient $B\to B/I$ is naturally a map in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$. $\blacksquare$
###### 3.0.3 Remark.
Filtered colimits and pushouts in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are constructed in [Wan95, 3.1,3.4] in the case when the morphisms in the
diagram are one-to-one. According to (the proof of) Proposition 3.0.2,
injectivity is not necessary in order to conclude that the colimit of a
diagram in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ in the category of
unital $C^{*}$-algebras is automatically endowed with a compact quantum group
structure. $\blacklozenge$
Next in line is the generation condition of Definition 1.4.2. It turns out
that $A_{u}(Q)$ will once more come in handy. We need them as objects of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, so recall the enveloping
$C^{*}$-algebra functor
$\operatorname{\textsc{univ}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
###### 3.0.4 Proposition.
The set $\operatorname{\textsc{univ}}(A_{u}(Q))$ for $Q$ ranging over all
positive matrices generates $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
###### Proof.
That the $A_{u}(Q)$ form a generator in $\operatorname{\mathrm{CQG}}$ is part
of the statement of Theorem 2.0.1. It is a simple exercise that left adjoints,
such as $\operatorname{\textsc{univ}}$, turn generators into generators
provided their right adjoints are faithful. In our case, the faithfulness of
the right adjoint $\operatorname{\textsc{alg}}$ to
$\operatorname{\textsc{univ}}$ follows from the density of the inclusion
$\operatorname{\textsc{alg}}(A)\subset A$: Any arrow $f:A\to B$ in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is the extension by continuity of
$\operatorname{\textsc{alg}}(f):\operatorname{\textsc{alg}}(A)\to\operatorname{\textsc{alg}}(B)$,
and hence $\operatorname{\textsc{alg}}(f)=\operatorname{\textsc{alg}}(g)$
implies $f=g$. $\blacksquare$
The only ingredient of Definition 1.4.2 still to be addressed is co-
wellpoweredness. Recall (§ 1.4) that this meant that every object has only a
set of quotient objects. It will help, then, to know exactly which morphisms
in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ are epimorphisms; this is what
the following result does.
###### 3.0.5 Proposition.
A morphism $f:A\to B$ in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is an
epimorphism if and only if it is surjective.
###### Proof.
As usual in categories where objects are sets with some kind of structure and
morphisms are maps preserving that structure, the implication surjective
$\Rightarrow$ epimorphism is immediate.
To prove the other implication, we will show that if $f$ is an epimorphism,
then $\operatorname{\textsc{alg}}(f)$ is surjective (the conclusion follows
from the denseness of $\operatorname{\textsc{alg}}(B)\subset B$). Since we can
always substitute the image of $f$ for $A$, we can (and will) assume that $f$
is injective.
First, recall the construction $\operatorname{\mathrm{CQG}}\ni
X\mapsto\operatorname{\textsc{red}}(X)\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
mentioned in § 1.2. It is functorial when restricted to the category
$\operatorname{\mathrm{CQG}}_{\mathrm{inj}}$ of CQG algebras and injective
morphisms, (this is the essence of [BKQ11, 6.2.4]). In order to keep the
notation manageable, indicate the functors $\operatorname{\textsc{alg}}$ and
$\operatorname{\textsc{red}}$ by superscripts, as in $X^{a}$ for
$\operatorname{\textsc{alg}}(X)$, $X^{r}$ for
$\operatorname{\textsc{red}}(X)$, $X^{ar}$ for
$\operatorname{\textsc{red}}(\operatorname{\textsc{alg}}(X))$, etc.; the same
conventions are in place for morphisms.
Let $\displaystyle\iota:B\to B\coprod_{A}B$ be the left hand canonical
inclusion into the pushout of $f$ along itself in the category
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, or equivalently (by the proof of
Proposition 3.0.2), in the category $\operatorname{\mathrm{C}^{*}}$ of unital
$C^{*}$-algebras. One condition equivalent to $f$ being epimorphic is that
$\iota$ be an isomorphism. Similarly, we denote by $\iota^{\prime}$ the left
hand inclusion $\displaystyle B^{a}\to B^{a}\coprod_{A^{a}}B^{a}$ into the
pushout in $\operatorname{\mathrm{Alg}^{*}}$. Note that $\iota$ and
$\iota^{\prime}$ are both injective, as they have left inverses by the
universality property of pushouts.
Assume now that $f^{a}$ is not surjective. Then, I claim that (a)
$\iota^{\prime}$ cannot be surjective (equivalently, an isomorphism), and
hence (b) neither can $(\iota^{\prime})^{r}$. That (a) does indeed imply (b)
follows from the fact [BKQ11, 6.2.12] that the functor
$\operatorname{\textsc{red}}:\operatorname{\mathrm{CQG}}_{\mathrm{inj}}$
reflects isomorphisms.
To prove (a), recall that an inclusion $K\subseteq H$ of cosemisimple Hopf
algebras always splits as a $K$-$K$-bimodule map (e.g. as argued in the proof
of [Chi11, 2.0.4]). Applied to the inclusion $f^{a}:A^{a}\to B^{a}$, this
observation yields a direct sum decomposition $B^{a}=A^{a}\oplus M$ as
$A^{a}$-$A^{a}$-bimodules for some non-zero $M$, and the pushout
$\displaystyle B^{a}\coprod_{A^{a}}B^{a}$ breaks up as a direct sum of $2^{n}$
copies of $M^{\otimes n}$ for $n\geq 0$ (tensor product of
$A^{a}$-$A^{a}$-bimodules). Moreover, $\iota^{\prime}$ identifies $B^{a}$ with
the summand $A^{a}\oplus M$ therein.
Now consider the commutative diagram
$B$$B\coprod_{A}B$$B^{ar}\coprod_{A^{ar}}B^{ar}$$B^{ar}$$(B^{a}\coprod_{A^{a}}B^{a})^{r}$$\scriptstyle\iota$$\scriptstyle(\iota^{\prime})^{r}$
where the right hand vertical map comes from the universality property of the
pushout applied to the two inclusions $\displaystyle B^{a}\to
B^{a}\coprod_{A^{a}}B^{a}$, and the other two unnamed maps are induced by the
surjection $B\to B^{ar}$. We have just argued that if $f^{a}$ is not
surjective, then the lower left corner path is not surjective. But then the
upper right corner path isn’t either. However, the right hand upper horizontal
arrow is surjective, as is the right hand vertical arrow. In conclusion, the
only morphism in this path which can fail to be surjective (under the
assumption that $f^{a}$ is not surjective) is $\iota$. $\blacksquare$
###### 3.0.6 Remark.
The proof makes it clear that the analogous result is true for
$\operatorname{\mathrm{CQG}}$, i.e. epimorphisms are surjective.
$\blacklozenge$
Since for any compact quantum group $A$ there is only a set of quotients of
$\operatorname{\textsc{alg}}(A)$ and hence only a set of compact quantum
groups having such quotients as dense CQG subalgebras, the previous result
implies:
###### 3.0.7 Proposition.
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is co-wellpowered.
###### Proof of Theorem 3.0.1.
Propositions 3.0.2, 3.0.4 and 3.0.7 are precisely what is required by
Definition 1.4.2. $\blacksquare$
## 4 Applications
The goal of this section is to apply Theorems 2.0.1 and 3.0.1, together with
the adjoint functor theorem and abstract properties of presentable or SAFT
categories, to the construction of compact quantum groups with various
universal properties. These constructions fall roughly into two categories:
right adjoints to functors defined on $\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, as direct applications of Theorem
1.4.1, and left adjoints arising in a slightly more roundabout way in § 4.4.
Note that limits in the categories $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, discussed in the next subsection,
fit in this framework as right adjoints: Given a small category $J$ and a
category $\mathcal{C}$, the limits of functors $J\to\mathcal{C}$, if they
exist, can be obtained as images of a right adjoint to the diagonal functor
$\Delta:\mathcal{C}\to\mathcal{C}^{J}$ (the latter is notation for the
category of all functors $J\to\mathcal{C}$) sending an object
$c\in\mathcal{C}$ to the functor $\Delta(c):J\to\mathcal{C}$ constant at $c$.
### 4.1 Limits in $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
We now know from Theorems 2.0.1 and 3.0.1 that both
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are SAFT. Remembering that SAFT-ness implies completeness (Remark 1.4.3), we
get:
###### 4.1.1 Theorem.
The categories $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ are complete.
Limits in these categories are quantum analogues of colimits of compact
groups. It is natural to ask whether functors
$J\to\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ taking commutative values have
commutative limits, or in other words, whether Theorem 4.1.1 recovers ordinary
colimits of compact groups. Since $\operatorname{\mathrm{CQG}_{ab}}$ is
complete (by Remark 2.0.7 or simply constructing coequalizers and coproducts
in the category of compact groups), the next result confirms this:
###### 4.1.2 Proposition.
The inclusions
$\operatorname{\mathrm{CQG}_{ab}}\to\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}_{ab}}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are right adjoints.
###### Proof.
In both cases the left adjoint is abelianization, associating to an object
$A\in\operatorname{\mathrm{CQG}}$ (or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) its largest commutative quotient
$*$-algebra (resp. $C^{*}$-algebra) $A_{ab}$. This is all rather routine, so
we omit most of the details.
The universality property of the canonical quotient map $\pi:A\to A_{ab}$
ensures that the composition
$A$$A\otimes A$$A_{ab}\otimes
A_{ab}$$\scriptstyle\Delta$$\scriptstyle\pi\otimes\pi$
factors through a map $\Delta_{ab}:A_{ab}\to A_{ab}\otimes A_{ab}$. The
coassociativity of $\Delta_{ab}$ follows from the uniqueness of the
factorization of $\pi^{\otimes
3}\circ(\Delta\otimes\operatorname{id})\circ\Delta:A\to A_{ab}^{\otimes 3}$
through $\pi$. By construction, we get a commutative square
$A$$A_{ab}$$A\otimes A$$A_{ab}\otimes
A_{ab}$$\scriptstyle\pi$$\scriptstyle\pi\otimes\pi$$\scriptstyle\Delta$$\scriptstyle\Delta_{ab}$
The rest of the structure and properties (e.g. counit
$\varepsilon_{ab}:A_{ab}\to\mathbb{C}$ and counitality of $\Delta_{ab}$ in the
algebraic case) follow similarly, as do the functoriality and the desired
universality property of $A\mapsto A_{ab}$. To see the latter, for example,
let $f:A\to B$ be a morphism in $\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ with $B$ commutative. Then, $f$
factors uniquely as $f_{ab}\circ\pi$ for an algebra map $f_{ab}:A_{ab}\to B$.
The outer rectangle of
$A$$A_{ab}$$B$$A\otimes A$$A_{ab}\otimes A_{ab}$$B\otimes
B$$\scriptstyle\pi$$\scriptstyle
f_{ab}$$\scriptstyle\pi\otimes\pi$$\scriptstyle f_{ab}\otimes f_{ab}$
is commutative because $f=f_{ab}\circ\pi$ is compatible with
comultiplications, and we have just observed that the left hand square
commutes. It follows that the precomposition of the right hand square with
$\pi$ commutes also, and since $\pi$ is onto, the right hand square must be
commutative. We again skip the entirely similar arguments for compatibility of
$f_{ab}$ with counits and antipodes in the algebraic case. $\blacksquare$
Outside of the general categorical framework provided by Theorems 2.0.1 and
3.0.1, one can also arrive at limits in the categories
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
by means of the Tannakian formalism introduced in [Wor88]
($\operatorname{\mathrm{CQG}}$ is better suited for this, so we focus on it).
There, Woronowicz associates a compact quantum group (or rather a compact
quantum group of the form $\operatorname{\textsc{univ}}(A)$ for
$A\in\operatorname{\mathrm{CQG}}$, so effectively, he recovers a CQG algebra)
from a so-called concrete monoidal $W^{*}$-category with complex conjugation.
These are basically just rigid, monoidal $W^{*}$-categories endowed with a
faithful, monoidal $*$-functor [GLR85] to the category Hilb of finite-
dimensional Hilbert spaces.
This is a version of Tannaka duality for Hopf algebras (e.g. as in [Sch92,
Ver12] and the many references therein): The CQG algebra constructed in
[Wor88, 1.3] given a concrete monoidal $W^{*}$-category $\mathcal{C}$ is what
in [Sch92] would be called the coendomorphism Hopf algebra of the functor
$\mathcal{C}\to\textsc{Hilb}$ that is implicitly part of Woronowicz’s
definition.
Now, if one starts with the category of unitary comodules of a CQG algebra $A$
and performs the above construction, the resulting CQG algebra is again $A$.
In other words, unitary comodules know all there is to know about a CQG
algebra (hence the name ‘Tannaka reconstruction’). It follows from this that
the construction of new CQG algebras out of old (such as, say, the limit of
some functor $F:J\to\operatorname{\mathrm{CQG}}$ out of the values of $F$) has
a chance of being carried out categorically: Identify the category of unitary
comodules, and you know the CQG algebra.
To get some insight into what limits in $\operatorname{\mathrm{CQG}}$ look
like, we do what the previous paragraph suggests, for products (but also state
the result for pullbacks, as it will be useful in Section 5): Given a family
$A_{i}$, $i\in I$ of objects in $\operatorname{\mathrm{CQG}}$, what does the
category of finite-dimensional, unitary comodules of the product
$\displaystyle A=\prod_{i}A_{i}$ in $\operatorname{\mathrm{CQG}}$ look like in
terms of the categories of comodules of the individual $A_{i}$? The answer
turns out to be quite simple; arguably simpler, in fact, than the description
of the category of unitary comodules for the (more familiar, in the
literature) coproduct $\displaystyle\coprod A_{i}$ from [Wan95]. All comodules
below are understood to be finite-dimensional and unitary.
Putting an $A$-comodule structure on an $n$-dimensional Hilbert space $V$ is
the same as giving a CQG algebra morphism $f:A_{u}(Q)\to A$ for some positive
$Q\in GL_{n}(\mathbb{C})$: In one direction, the comodule structure induces a
morphism by sending the standard generators $u_{ij}\in A_{u}(Q)$ (for an
appropriate $Q$) to the coefficients $v_{ij}$ with respect to an orthonormal
basis of $V$; in the opposite direction, make $A$ coact on the standard
$A_{u}(Q)$-comodule by “scalar corestriction” via the coalgebra morphism $f$.
In turn, by the defining property of the categorical product, a morphism
$f:A_{u}(Q)\to A$ means a family of morphisms $f_{i}:A_{u}(Q)\to A_{i}$, $i\in
I$. Going through this comodule structure - morphism correspondence in reverse
for each $i$, the data consisting of the $f_{i}$’s is equivalent to putting an
$A_{i}$-comodule structure on the canonical $n$-dimensional comodule of
$A_{u}(Q)$ for every $i$. A moment’s thought will show how do modify this
argument to take care pullbacks, and all in all, we have the following result:
###### 4.1.3 Proposition.
Let $A_{i}\in\operatorname{\mathrm{CQG}}$, $i\in I$ be a set of objects, and
$A=\prod A_{i}$ the product in $\operatorname{\mathrm{CQG}}$. Then, the
category of $A$-comodules has as objects finite-dimensional Hilbert spaces
admitting an $A_{i}$-comodule structure for each $A_{i}$, and as morphisms
linear maps respecting all of these structures.
Let $f:B\to C$ and $f^{\prime}:B^{\prime}\to C$ be morphisms in
$\operatorname{\mathrm{CQG}}$, and $\displaystyle A=B\times_{C}B^{\prime}$ the
pullback in $\operatorname{\mathrm{CQG}}$. The category of $A$-comodules has
1. (a)
as objects, triples $(V,V^{\prime},\varphi)$ where $V$ and $V^{\prime}$ are
$B$ and $B^{\prime}$-comodules respectively, and $\varphi:V\to V^{\prime}$ is
a unitary map identifying $V$ and $V^{\prime}$ as $C$-comodules;
2. (b)
as morphisms from $(V,V^{\prime},\varphi)$ to $(W,W^{\prime},\psi)$, pairs
$(\xi,\xi^{\prime})$, where $\xi:V\to W$ and $\xi^{\prime}:V^{\prime}\to
W^{\prime}$ are morphisms in $\mathcal{M}^{B}$ and $\mathcal{M}^{B^{\prime}}$
respectively, making the diagram
$V$$V^{\prime}$$W$$W^{\prime}$$\scriptstyle\varphi$$\scriptstyle\psi$$\scriptstyle\xi$$\scriptstyle\xi^{\prime}$
commutative.
###### 4.1.4 Remark.
This statement describes the sought-after categories of unitary comodules very
explicitly. There is a more abstract, but also more elegant way to phrase all
of this. We need some basic 2-categorical notions to say it all (as in
[Lac10]).
Rigid, monoidal $W^{*}$-categories $\mathcal{C}$ endowed with monoidal
$*$-functors $\mathcal{C}\to\textsc{Hilb}$ form a bicategory in a natural way,
while $\operatorname{\mathrm{CQG}}$ can be regarded as a bicategory with only
identity 2-cells. Then, sending a CQG algebra to its category of finite-
dimensional, unitary comodules (together with its forgetful functor to Hilb)
is a pseudofunctor from the latter to the former. The essence of Proposition
4.1.3 is that this pseudofunctor preserves limits. This is a familiar theme in
Tannaka duality: Woronowicz’s construction of a CQG algebra out of a functor
$\mathcal{C}\to\textsc{Hilb}$ is in fact nothing but the left adjoint of the
pseudofunctor mentioned above. This sort of situation is treated in [Sch11],
with a biadjunction analogous to the one just discussed appearing in Theorem
3.1.1. $\blacklozenge$
###### 4.1.5 Remark.
The references to [Wor88] in the above discussion are somewhat of a
paraphrase, as Woronowicz works with what are called compact matrix quantum
groups (or CMQG algebras on the algebraic side [DK94, 2.5]). These are
basically compact quantum groups finitely generated as $C^{*}$-algebras, and
are analogous to compact Lie groups (the latter being precisely those compact
groups which embed in some unitary group). He also distinguishes a comodule
whose coefficients generate the algebra, and so works with pairs $(A,u)$,
where $u\in M_{n}(A)$ is a unitary coefficient matrix. Adapting the results of
that paper to the general setting is straightforward.
A CMQG algebra always has a countable set of (isomorphism classes of) simple
comodules. As we will see in the next example, abandoning this restriction is
absolutely necessary if we are going to discuss limits in
$\operatorname{\mathrm{CQG}}$, since products, for example, are very unlikely
to satisfy this property. $\blacklozenge$
###### 4.1.6 Example.
One does not even have to go “quantum” to give an example of a very large
(that is, non-CMQG) product in $\operatorname{\mathrm{CQG}}$. Indeed, the
smallest possible example will do: a coproduct of two copies of $\mathbb{Z}/2$
in the category of compact groups.
Denote this coproduct by $G$. According to the first part of Proposition
4.1.3, a unitary representation of $G$ consists of a finite-dimensional
Hilbert space $V$ and two involutive unitary operators $x$ and $y$ on $V$. The
projections $p=\frac{1+x}{2}$ and $q=\frac{1+y}{2}$ provide precisely the same
information, so we work with them instead. $V$ is irreducible preciely when
$p$ and $q$ have no common proper, non-zero invariant subspace. By the
discussion carried out prior to the statement of [Tak02, Theorem 1.41], this
is equivalent to the four pairwise infima $p\wedge q$, $p\wedge(1-q)$ etc. all
being zero (where ‘$\wedge$’ means orthogonal projection on the intersection
of the ranges of the two projections).
If $\dim V\geq 2$, the vanishing of all wedges $p\wedge q$, etc. makes it
necessary that $V$ be even-dimensional and that $p$ and $q$ both have rank
$\frac{\dim V}{2}$, but this is it: The set of pairs of projections satisfying
the requirements is open dense in the set of all pairs of projections of rank
$\frac{\dim V}{2}$ on $V$. Hence, if $\dim V=2n$ for some $n\geq 1$, the set
of pairs $(p,q)$ that will make $V$ into an irreducible unitary
$G$-representation is a manifold of dimension $4n^{2}$ (twice the dimension of
the Grassmannian variety of $n$-dimensional subspaces of $V$ as a real
manifold). We now have to quotienting by the equivalence relation
$(p,q)\sim(upu^{*},uqu^{*})$ for unitary $u$, which accounts for isomorphic
$G$-representations induced by different pairs of projections. Since the
unitary group of $V$ has dimension $n(2n-1)$, this still leaves continuum many
isomorphism classes of simples. $\blacklozenge$
We end this subsection with a note on terminology. As observed in [Sol10,
$\S$3], the name ‘free product of compact quantum groups’ from [Wan95] is
somewhat inconsistent with the prevailing point of view that compact quantum
groups form a category opposite to $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
The problem is that in universal algebra, ‘free product’ is often synonymous
to ‘coproduct’. Even though Wang’s construction is a coproduct in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ rather than its opposite, and hence
‘product of compact quantum groups’ would perhaps be a better fit, ‘free
product’ seems to have been established through use (besides, ‘product’ would
clash with the interpretation of $A\otimes A$ as the Cartesian square of a
compact quantum group, implicit in Definition 1.1.1). On the other hand,
products in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (whose existence
Theorem 4.1.1 proves) are probably best referred to as ‘coproducts of compact
quantum groups’.
### 4.2 Quantum groups generated by quantum spaces
The idea here is that functors of the form “forget the comultiplication”,
regarded as quantum analogues of forgetting the multiplication on a group,
have right adjoints. As most of the previous discussion, all of this works
both algebraically and $C^{*}$-algebraically:
###### 4.2.1 Theorem.
The functors
1. (a)
$\operatorname{\textsc{forget}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{Alg}^{*}_{0}}$
sending a CQG algebra to its underlying $*$-algebra and
2. (b)
$\operatorname{\textsc{forget}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}_{0}}$
sending a compact quantum group to its underlying $C^{*}$-algebra
are left adjoints.
###### Proof.
We already know from the proofs of Propositions 2.0.2 and 3.0.2 that the
forgetful functors from $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ to unital $*$-algebras and
$C^{*}$-algebras respectively are cocontinuous. I claim that so are the
inclusions
$\operatorname{\mathrm{Alg}^{*}}\to\operatorname{\mathrm{Alg}^{*}_{0}}$ and
$\operatorname{\mathrm{C}^{*}}\to\operatorname{\mathrm{C}^{*}_{0}}$. Assuming
this for now, (a) and (b) are cocontinuous; that they are left adjoints then
follows from the SAFT-ness of $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (Theorems 2.0.1 and 3.0.1) and the
adjoint functor theorem.
Finally, to prove the claim made above that the two inclusions
$\operatorname{\mathrm{Alg}^{*}}\to\operatorname{\mathrm{Alg}^{*}_{0}}$ and
$\operatorname{\mathrm{C}^{*}}\to\operatorname{\mathrm{C}^{*}_{0}}$ are
cocontinuous, note that they are in fact left adjoints: In both cases, the
right adjoint is the multiplier algebra construction $A\mapsto M(A)$.
$\blacksquare$
###### 4.2.2 Remark.
Von Neumann algebras would be an alternative way to formalize the idea of
“quantum space”. This is the point of view espoused in [Kor12], where the
category $\operatorname{\mathrm{W}^{*}}$ of unital von Neumann algebras and
unital normal homomorphisms is opposite to the category of so-called quantum
collections. The idea here is that a von Neumann algebra is a quantum analogue
of a set, ordinary sets $X$ corresponding to $\ell^{\infty}(X)$.
Adopting this perspective, the enveloping $W^{*}$-algebra functor
$\operatorname{\textsc{env}}:\operatorname{\mathrm{C}^{*}}\to\operatorname{\mathrm{W}^{*}}$
is a kind of forgetful functor, disregarding the topological side of a compact
quantum space and remembering only the underlying quantum collection;
similarly, the forgetful functor
$\operatorname{\mathrm{W}^{*}}\to\operatorname{\mathrm{C}^{*}}$ (which is
right adjoint to $\operatorname{\textsc{env}}$) is a kind of quantum Stone-
Cech compactification.
Composing (b) of Theorem 4.2.1 further with $\operatorname{\textsc{env}}$ is
again a left adjoint (the composition of two left adjoints), and its right
adjoint could be thought of as the functor associating to every quantum
ollection the compact quantum group freely generated by it. $\blacklozenge$
We refer to the image of a $*$ or $C^{*}$-algebra $A$ through the respective
right adjoint as the cofree CQG algebra or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ object on $A$ (‘co’ because it
universally maps into $A$, as opposed to being mapped into). The notation
$\operatorname{\textsc{cofree}}$ stands for either of the two functors, and we
rely on context to distinguish between the possibilities.
It is to be expected in such cofree-Hopf-algebra-on-an-algebra type
constructions that commutativity will be preserved (as explained, for
instance, in the introduction of [Por11]). In other words, one would like the
right adjoint from part (b), say, when applied to the algebra of functions
vanishing at infinity on a locally compact space $X$, to yield precisely the
compact group freely generated by $X$: Construct the abstract group $G$ freely
generated by $X$, endow it with the strongest topology making the canonical
map $X\to G$ continuous, and take the Bohr compactification of the resulting
topological group. That this is indeed the case is essentially the content of
the following proposition:
###### 4.2.3 Proposition.
If $A$ is a commutative $*$ or $C^{*}$-algebra, then
$\operatorname{\textsc{cofree}}(A)$ is commutative.
###### Proof.
To fix ideas, we prove the algebraic statement regarding $*$-algebras, and
leave the simple modifications that will adapt the proof to the other cases to
the reader.
Let $H=\operatorname{\textsc{cofree}}(A)$, and recall the CQG algebra
structure on $H_{ab}$ from Proposition 4.1.2. The universality property of the
abelianization factorizes the left hand diagonal arrow in the following
diagram through the right hand one, while the cofree-ness gives the other
commutative triangle, passing through $\iota$:
$H$$H_{ab}$$A$$\pi$$\iota$
By cofree-ness again, the loop $\iota\circ\pi$ must be the identity; since
$\pi$ was a surjection, it must be an isomorphism. $\blacksquare$
### 4.3 Variations on the Bohr compactification theme
A right adjoint to the inclusion functor
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$
is constructed directly in [Soł05], and this construction is referred to as
the quantum Bohr compactification. For any
$A\in\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$, it provides an object
$H\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ mapping universally into $A$ so
as to preserve the comultiplication; remembering the arrow reversal inherent
to passing from spaces to functions on them, this should indeed be thought of
as a compact quantum group into which the locally compact quantum semigroup
maps universally. Moreover, when $A$ is commutative and hence the algebra of
functions vanishing at infinity on a locally compact semigroup $X$, the
construction returns precisely the algebra of functions on the ordinary Bohr
compactification of $X$ ([Soł05, 4.1]).
We recover these results and their algebraic counterparts below (Theorem 4.3.1
and Proposition 4.3.4), as applications of the categorical machinery already
in place.
###### 4.3.1 Theorem.
The inclusion functors
1. (a)
$\operatorname{\textsc{forget}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$
sending a CQG algebra to its underlying algebraic quantum semigroup and
2. (b)
$\operatorname{\textsc{forget}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$
sending a compact quantum group to its underlying compact quantum group
are left adjoints.
###### Proof.
As before, Theorems 2.0.1 and 3.0.1 and the adjoint functor theorem ensure
that we only need to prove the two functors cocontinuous. Equivalently, this
means showing they preserve coequalizers of pairs and coproducts. The four
arguments (coequalizers and coproducts, (a) and (b)) follow essentially the
same path, so let us focus on coproducts for part (a).
Let $I$ be a set, and $A_{i}$, $i\in I$ objects in
$\operatorname{\mathrm{CQG}}$. Let also $f_{i}:A_{i}\to B$ be morphisms in
$\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ (strictly speaking, they are
morphisms $\operatorname{\textsc{forget}}(A_{i})\to B$, but since
$\operatorname{\textsc{forget}}$ really is just an inclusion, we omit it in
the rest of the proof). Since forgetting further to
$\operatorname{\mathrm{Alg}^{*}_{0}}$ (i.e. disregarding comultiplications)
is, according to part (a) of Theorem 4.2.1, cocontinuous, the $f_{i}$
aggregate into a unique $*$-algebra morphism $f:A=\coprod A_{i}\to B$. We are
done if we can show that $f$ preserves comultiplications. To see this,
consider the diagram
$A_{i}$$A$$B$$A_{i}\otimes A_{i}$$A\otimes A$$B\otimes
B$$\scriptstyle\iota_{i}$$\scriptstyle
f$$\scriptstyle\iota_{i}\otimes\iota_{i}$$\scriptstyle f\otimes f$
where the vertical arrows are comultiplications, and $\iota_{i}:A_{i}\to A$
are the structure maps of the coproduct. The commutativity of the outer
rectangle is the preservation of comultiplications by the
$f_{i}=f\circ\iota_{i}$, while the left hand square commutes because $A$ was
defined as the coproduct of the $A_{i}$ in $\operatorname{\mathrm{CQG}}$. It
follows that precomposing the two possible ways to get from $A$ to $B\otimes
B$ with $\iota_{i}$ yields the same morphism $A_{i}\to B\otimes B$; then, by
the universality in $\operatorname{\mathrm{Alg}^{*}_{0}}$ of the coproduct
$A$, the right hand square must also be commutative. $\blacksquare$
###### 4.3.2 Remark.
Part (b) of the proposition can be tweaked slightly in the spirit of Remark
4.2.2.
The enveloping von Neumann algebra functor
$\operatorname{\textsc{env}}:\operatorname{\mathrm{C}^{*}_{0}}\to\operatorname{\mathrm{W}^{*}}$
can be lifted to a functor (again called $\operatorname{\textsc{env}}$) from
$\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ to the category
$\operatorname{\mathrm{W}^{*}\mathrm{QS}}$ of von Neumann quantum semigroups,
consisting of von neumann algebras $M$ endowed with a coassociative morphism
$\Delta:M\to M\otimes M$ in $\operatorname{\mathrm{W}^{*}}$ (remember that the
tensor product of von Neumann algebras here is the spatial one) and with
$\operatorname{\mathrm{W}^{*}}$ maps which preserve these comultiplications as
morphisms. It is to be thought of as a kind of forgetful functor, ignoring the
topology of a locally compact quantum semigroup and remembering only the
underlying quantum collection, together with the “multiplication”.
It can be shown further that
$\operatorname{\textsc{env}}\circ\operatorname{\textsc{forget}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{W}^{*}\mathrm{QS}}$
is cocontinuous, and hence a left adjoint. In other words, every $W^{*}$
quantum semigroup has a quantum Bohr compactification. $\blacklozenge$
###### 4.3.3 Remark.
In a $C^{*}$-algebraic variant of the Tambara construction [Tam90], (a
particular case of) [Soł09, Theorem 3.3] constructs, for every finite-
dimensional $C^{*}$-algebra $A$, an object $B$ of
$\operatorname{\mathrm{C}^{*}\mathrm{QS}}$ (the full subcategory of
$\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ consisting of unital algebras)
coacting universally on $A$. In other words, there is a coassociative
$C^{*}$-algebra map $A\to A\otimes B$ making $B$ an initial object in the
category of objects of $\operatorname{\mathrm{C}^{*}\mathrm{QS}}$ endowed with
such maps.
If $B$ were to map universally into an object
$B^{\prime}\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, the latter would be
the quantum automorphism group of $A$ in the sense of [Wan98]. However, we
know from [Wan98, Theorem 6.1 (1)] that finite-dimensional $C^{*}$-algebras do
not have compact quantum automorphism groups, in general. In conclusion,
although
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}\mathrm{QS}}$
is a left adjoint (by a variant of Theorem 4.3.1), it is not a right adjoint.
$\blacklozenge$
As in § 4.2, we denote the right adjoints of Theorem 4.3.1 by
$\operatorname{\textsc{cofree}}$. Once more, it turns out that they preserve
commutativity.
###### 4.3.4 Proposition.
If the object $A$ of $\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ or
$\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ is commutative, so is
$\operatorname{\textsc{cofree}}(A)$.
###### Proof.
We focus on the $\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ case.
Setting $H=\operatorname{\textsc{cofree}}(A)$, the canonical map $f:H\to A$
factors as
$H$$H_{ab}$$A$$\scriptstyle\pi$$\scriptstyle f$$\scriptstyle f_{ab}$
for some morphism $f_{ab}$ in $\operatorname{\mathrm{Alg}^{*}_{0}}$.
Essentially the same argument as the one used in the proof of Proposition
4.1.2 shows that $f_{ab}$ is actually a morphism in
$\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$. Finally, we can now repeat the
proof of Proposition 4.2.3 to conclude that $\pi$ is in fact an isomorphism,
and hence $H$ is commutative. $\blacksquare$
###### 4.3.5 Remark.
Proposition 4.3.4 goes through in the setting of Remark 4.3.2: If $M$ is a
commutative von Neumann algebra, then it can be shown in much the same way as
above that the quantum Bohr compactification is commutative. $\blacklozenge$
### 4.4 Kac quotients
Recall that a compact quantum group
$A\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is said to be of Kac type if
the antipode on $\operatorname{\textsc{alg}}(A)$ lifts to a continuous map
$A\to A$. Equivalently, the antipode of $\operatorname{\textsc{alg}}(A)$ is
involutive ($S^{2}=\operatorname{id}$), or commutes with the $*$ operation.
This definition extends in the obvious way to CQG algebras; in that case, ‘of
Kac type’ or simply ‘Kac’ will be synonymous to ‘having involutive antipode’.
Alternate terms are ‘Kac algebra’ (under which these objects and their
relatives were introduced; e.g. [ES92] and the references therein) or
sometimes ‘Woronowicz-Kac algebra’ (as in [Ban99]). For brevity, we will
sometimes simply use ‘Kac’ as an adjective. A $k$ subscript on either
$\operatorname{\mathrm{CQG}}$ of $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
indicates the full subcategory on objects of Kac type.
In [Soł05, Appendix], Sołtan constructs what in that paper is called the
canonical Kac quotient of an object
$A\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (notion originally due to of
Stefaan Vaes). It is obtained by quotienting out all elements of $A$ killed by
some trace (meaning, as usual, that the trace sends $x^{*}x$ to zero). This,
however, seems to be a bit of a misnomer: While it is shown in [Soł05, A.1]
that the result is indeed a compact quantum group of Kac type, it is not clear
that a Kac compact quantum group is its own canonical Kac quotient222I am
grateful to Piotr Sołtan for pointing this out.! A more appropriate term might
be, perhaps, canonical tracial quotient: One quotients out as much as one
needs to in order to ensure that the result has enough traces.
The notion has also received attention in [Tom07], where Tomatsu shows in
Theorem 4.8 that a compact quantum group $A$ has a largest quotient of Kac
type (he uses dual phrasing, thinking of the latter as a largest compact
quantum subgroup of Kac type) provided $A$ is coamenable and the Grothendieck
ring of its category of comodules (its so-called fusion algebra) is
commutative. Regarding the terminology problem from the previous paragraph,
note that for the reason pointed out there, it is not clear, a priori, that
Tomatsu’s quotient is the same as Sołtan’s. The two do coincide, however, if
the Haar measure of the compact quantum group is faithful (which is the
standing assumption of [Tom07]), hence [Tom07, Remark 4.9].
In conclusion, the question of whether or not every
$A\in\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ has a largest Kac quotient seems to
be an interesting one. One of the aims of this subsection is to show that this
is indeed the case: The desired quotient map is precisely the reflection of
$A$ in the subcategory $\operatorname{\mathrm{CQG}}_{k}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}$ (i.e. the image of $A$ through
the left adjoint to the inclusion of the subcategory into
$\operatorname{\mathrm{CQG}}$ or $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$,
respectively).
###### 4.4.1 Theorem.
The inclusions $\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}$
and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
each have a left, as well as a right adjoint.
###### Proof.
That the inclusions are both left adjoints is shown in much the same way in
which we have proven all results asserting the existence of various right
adjoints so far. The arguments of Sections 2 and 3 can be repeated to show
that $\operatorname{\mathrm{CQG}}_{k}$ is finitely presentable and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}$ is SAFT. The only difference
that is even remotely significant is the fact that now the regular generator
to go into an analogue of Proposition 2.0.6 consists of $A_{u}(I_{n})$, $n\geq
1$ rather than all $A_{u}(Q)$. Colimits are again constructed simply at the
level of $*$ or $C^{*}$-algebras, making it clear that the inclusions are
cocontinuous and hence left adjoints.
The interesting problem, then, is the one discussed before the statement of
the theorem: constructing left adjoints to the two inclusions. For each
$A\in\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ we want an arrow $\kappa:A\to
A_{k}$ into a Kac type object, universal in the sense that any morphism from
$A$ into a Kac object factors uniquely through $\kappa$. In other words, we
have to show that the comma category $A\downarrow\operatorname{\mathrm{CQG}}$
(resp. $A\downarrow\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) consisting of
arrows from $A$ into Kac objects has an initial object. To do this, we apply
Freyd’s initial object theorem [ML98, V.6.1]. It says that a complete category
has an initial object as soon as it has a weakly initial set of objects; this
simply means a set $S$ of objects such that any object $y$ admits at least one
arrow $S\ni x\to y$ (not necessarily unique).
In our case, a weakly initial set is easy to come by: All surjections $A\to B$
for Kac type $B$ will do, since any map of $A$ into a Kac type object will
certainly factor through the image of that map. On the other hand completeness
follows quickly if we show that $\operatorname{\mathrm{CQG}}_{k}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}$ are closed under limits in
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
respectively: Limits would then be created by the forgetful functor
$A\downarrow\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}$
(and similarly for $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$).
In conclusion, it is enough to show that products of Kac objects in
$\operatorname{\mathrm{CQG}}$ (or $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$)
are again Kac, and similarly, equalizers of parallel pairs of arrows between
Kac objects are Kac. Since (a)
$\operatorname{\textsc{alg}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{CQG}}$
is a right adjoint and hence preserves limits, and (b) by definition, an
object $B\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is Kac if and only if
$\operatorname{\textsc{alg}}(B)$ is, it is enough to restrict ourselves to
$\operatorname{\mathrm{CQG}}$.
Equalizers are easy: If $f,g:B\to C$ are arrows between Kac CQG algebras, the
equalizer injects into $B$, so it is again Kac. To prove the statement about
products, let $B_{i}$, $i\in I$ be a set of Kac CQG algebras, and denote the
structure maps of their product in $\operatorname{\mathrm{CQG}}$ by
$\pi_{i}:B\to B_{i}$. Throughout the rest of this proof, for a CQG algebra
$C$, we denote by $C^{\prime}$ the CQG algebra with the same underlying set,
but reversed multiplication and comultiplication. Note that the product of the
objects $B_{i}^{\prime}$ is precisely $B^{\prime}$. Let $S_{i}$ be the
antipodes of $B_{i}$, and $S=\prod S_{i}:B\to B^{\prime}$ the map obtained
from the functoriality of products. By this same functoriality, $S$ is
involutive (strictly speaking, this means $S^{\prime}\circ
S=\operatorname{id}$, where $S^{\prime}:B^{\prime}\to B$ is $S$ as a map, but
we have switched the domain and codomain). If we show that $S$ is the antipode
$S_{B}$ of $B$, we are done (Kac means involutive). To prove this, note that
the two squares in the diagram
$B$$B^{\prime}$$B_{i}$$B_{i}^{\prime}$$\scriptstyle S$$\scriptstyle
S_{B}$$\scriptstyle S_{i}$$\scriptstyle\pi_{i}$$\scriptstyle\pi_{i}$
are both commutative (the $S$-square by the definition of $S$ as the product
of the $S_{i}$, and the $S_{B}$-square because $\pi_{i}$ are Hopf algebra
morphisms and hence preserve antipodes). $\blacksquare$
Denote by $A\mapsto A_{k}$ the left adjoint to either of the inclusions
$\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
It is a simple observation now that the canonical map $A\to A_{k}$ is a
surjection, and hence warrants the name ‘Kac quotient’:
###### 4.4.2 Proposition.
Let $A\in\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ and $\kappa:A\to A_{k}$ the
universal arrow resulting from the unit of the adjunction that the Kac
quotient functor is part of. Then, $\kappa$ is onto.
###### Proof.
To keep things streamlined, let us focus on $\operatorname{\mathrm{CQG}}$. We
have seen this sort of argument before, in a dual form, in Proposition 4.2.3.
Let $\iota:A_{k}^{\prime}\to A_{k}$ be the inclusion of
$A_{k}^{\prime}=\mathrm{Im}(\kappa)$, and denote the corestriction by
$\kappa^{\prime}:A\to A_{k}^{\prime}$. We have the diagram
$A_{k}^{\prime}$$A_{k}$$A$$\displaystyle\iota$$\displaystyle
f$$\displaystyle\kappa^{\prime}$$\displaystyle\kappa$
where $f$ is the unique arrow $\kappa\to\kappa^{\prime}$ in
$A\downarrow\operatorname{\mathrm{CQG}}_{k}$. Both triangles are commutative,
and by the universality of $\kappa$, the loop $\iota\circ f$ must be the
identity. Since $\iota$ was by definition one-to-one, it must be an
isomorphism. $\blacksquare$
## 5 Monomorphisms
One subject is conspicuously absent from Proposition 3.0.5: what about
monomorphisms? The main result of this section is just such a
characterization:
###### 5.0.1 Proposition.
A morphism in $\operatorname{\mathrm{CQG}}$ is a monomorphism if and only if
it is one-to-one. Similarly, a morphism $f$ in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is mono if and only if
$\operatorname{\textsc{alg}}(f)$ is one-to-one.
###### 5.0.2 Remark.
Morphisms in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ that are injective at
the algebraic level play an important role in [BKQ11, 6.2]. The proposition
gives a nice interpretation: They are precisely the monomorphisms of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$. $\blacklozenge$
###### 5.0.3 Remark.
In the commutative setting, where all distinctions between the algebraic and
the $C^{*}$-algebraic sides of the picture disappear (Remark 2.0.7), the
analogous result would be that epimorphisms of compact groups are surjective.
This is [Rei70, Proposition 9], and in fact, the proof below is a paraphrase
of Reid’s. $\blacklozenge$
###### Proof.
First, let’s reduce the second part of the statement to the first. Recall that
right adjoints (such as $\operatorname{\textsc{alg}}$) send monomorphisms to
monomorphisms, so a morphism $f$ in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
can only be mono if $\operatorname{\textsc{alg}}(f)$ is. On the other hand,
the faithfulness of $\operatorname{\textsc{alg}}$ (a consequence of the
density of $\operatorname{\textsc{alg}}(A)\subset A$ for
$A\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) implies the converse. Indeed,
if $\operatorname{\textsc{alg}}(f)$ is mono and $f\circ g=f\circ h$, the
series
$\operatorname{\textsc{alg}}(f)\circ\operatorname{\textsc{alg}}(g)=\operatorname{\textsc{alg}}(f)\circ\operatorname{\textsc{alg}}(h)\quad\Rightarrow\quad\operatorname{\textsc{alg}}(g)=\operatorname{\textsc{alg}}(h)\quad\Rightarrow\quad
g=h$
of equalities does the trick (the first implication says that
$\operatorname{\textsc{alg}}(f)$ is a monomorphism, while the second one is
faithfulness).
We are now left with the first statement, on CQG algebras. Just as in the case
of epimorphisms treated in Proposition 3.0.5, the implication injective
$\Rightarrow$ mono is the easy part. Focusing on the converse, let $f:A\to B$
be a monomorphism in $\operatorname{\mathrm{CQG}}$. Let also
$\pi:P=A\times_{B}A\to A$ be one of the defining projections of the pullback
(in the category $\operatorname{\mathrm{CQG}}$). The functor
$\operatorname{\mathrm{CQG}}(\bullet,P)$ represents the functor sending
$C\in\operatorname{\mathrm{CQG}}$ to the set of pairs of morphisms $g,h:C\to
A$ satisfying the condition $fg=fh$. Since the latter implies $g=h$ by $f$
being a monomorphism, this functor is isomorphic to
$\operatorname{\mathrm{CQG}}(\bullet,A)$, and it follows that the natural
transformation
$\operatorname{\mathrm{CQG}}(\bullet,P)\to\operatorname{\mathrm{CQG}}(\bullet,A)$
induced by $\pi$ (and hence $\pi$ itself) is an isomorphism; we have made use
of this sort of result, in its dual form having to do with epimorphisms, in
Proposition 3.0.5. It will actually be more convenient to say it like this:
The diagonal map $d:A\to A\times_{B}A=P$ is an isomorphism; indeed, its very
definition implies that $\pi d=\operatorname{id}$.
Now transport all of the above to the level of (finite-dimensional, unitary)
comodules. Recalling what the category of $P$-comodules looks like from part
(b) of Proposition 4.1.3, the fact $d$ is an isomorphism says that
$V\mapsto(V,V,\operatorname{id})$ is an equivalence from $A$-comodules to
$P$-comodules. In particular, the essential surjectivity of this functor
implies that for any $P$-comodule $(V,V^{\prime},\varphi)$, the a priori
$B$-comodule isomorphism $\varphi:V\to V^{\prime}$ is actually an $A$-comodule
map.
According to the previous paragraph, it is enough, assuming that $f$ is not
one-to-one, to find finite-dimensional, unitary $A$-comodules $V$ and
$V^{\prime}$ together with a unitary isomorphism $\varphi:V\to V^{\prime}$ as
$B$-comodules which does not preserve the $A$-comodule structures. Let us
simplify the situation further. Suppose $V$ is a non-trivial, simple, unitary
$A$-comodule (non-trivial meaning not isomorphic to the monoidal unit of the
cateory of comodules) which has trivial components when regarded as a
$B$-comodule via scalar corestriction through $f:A\to B$. This means that
there is some non-zero vector $v\in V$ fixed by $B$ in the sense that
$v_{1}\otimes f(v_{2})=v\otimes 1$, but not fixed by $A$. The unitary
reflection across the orthogonal complement of $v$ would then be a morphism in
$\mathcal{M}^{B}$ but not in $\mathcal{M}^{A}$, and we would be done.
In conclusion, it suffices to find $V$ as above. Since the Peter-Weyl theorem
for CQG algebras expresses each as a direct sum of $W^{\oplus\dim W}$ for $W$
ranging over its set of unitary simple comodules, the only ways in which $f$
can be non-injective are if (a) some simple $A$-comodule becomes non-simple as
a $B$-comodule, or (b) there are two distinct simples over $A$ which become
isomorphic over $B$ (we will see soon that in fact (a) always happens).
In case (a), choose some such simple $W\in\widehat{A}$. Then, the trivial
comodule has multiplicity one in $W^{*}\otimes W$ as an $A$-comodule, but
strictly larger than one over $B$. Hence, there is a simple $V\leq
W^{*}\otimes W$ that will satisfy the sought-for conditions. In case (b), let
$W$ and $W^{\prime}$ be non-isomorphic, unitary simple $A$-comodules which
bcome isomorphic over $B$. Then, $W^{*}\otimes W^{\prime}$ does not contain
the trivial comodule over $A$, but it does over $B$. So as before, we can find
our desired $V$ among the simple summands of $W^{*}\otimes W^{\prime}$.
$\blacksquare$
## References
* [Abe80] Eiichi Abe. Hopf algebras, volume 74 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka.
* [AR94] Jiří Adámek and Jiří Rosický. Locally presentable and accessible categories, volume 189 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1994.
* [Ban99] Teodor Banica. Fusion rules for representations of compact quantum groups. Exposition. Math., 17(4):313–337, 1999.
* [Ban05] Teodor Banica. Quantum automorphism groups of small metric spaces. Pacific J. Math., 219(1):27–51, 2005.
* [Bic03] Julien Bichon. Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc., 131(3):665–673 (electronic), 2003.
* [BKQ11] Erik Bédos, S. Kaliszewski, and John Quigg. Reflective-coreflective equivalence. Theory Appl. Categ., 25:No. 6, 142–179, 2011.
* [BMT01] E. Bédos, G. J. Murphy, and L. Tuset. Co-amenability of compact quantum groups. J. Geom. Phys., 40(2):130–153, 2001.
* [Boc95] Florin P. Boca. Ergodic actions of compact matrix pseudogroups on $C^{*}$-algebras. Astérisque, (232):93–109, 1995. Recent advances in operator algebras (Orléans, 1992).
* [Chi11] A. Chirvasitu. Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. ArXiv e-prints, October 2011.
* [Con94] Alain Connes. Noncommutative Geometry. Academic Press, San Diego, CA, 1994. available for download at http://www.alainconnes.org/en/.
* [DK94] Mathijs S. Dijkhuizen and Tom H. Koornwinder. CQG algebras: a direct algebraic approach to compact quantum groups. Lett. Math. Phys., 32(4):315–330, 1994.
* [ES92] Michel Enock and Jean-Marie Schwartz. Kac algebras and duality of locally compact groups. Springer-Verlag, Berlin, 1992. With a preface by Alain Connes, With a postface by Adrian Ocneanu.
* [GLR85] P. Ghez, R. Lima, and J. E. Roberts. $W^{\ast}$-categories. Pacific J. Math., 120(1):79–109, 1985.
* [Kel86] G. M. Kelly. A survey of totality for enriched and ordinary categories. Cahiers Topologie Géom. Différentielle Catég., 27(2):109–132, 1986.
* [Kor12] A. Kornell. Quantum Collections. ArXiv e-prints, February 2012.
* [KS97] Anatoli Klimyk and Konrad Schmüdgen. Quantum groups and their representations. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997.
* [KT99] Johan Kustermans and Lars Tuset. A survey of $C^{*}$-algebraic quantum groups. I. Irish Math. Soc. Bull., (43):8–63, 1999.
* [KV00] Johan Kustermans and Stefaan Vaes. Locally compact quantum groups. Ann. Sci. École Norm. Sup. (4), 33(6):837–934, 2000.
* [Lac10] Stephen Lack. A 2-categories companion. In Towards higher categories, volume 152 of IMA Vol. Math. Appl., pages 105–191. Springer, New York, 2010.
* [ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998.
* [Mon93] Susan Montgomery. Hopf algebras and their actions on rings, volume 82 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1993.
* [Por11] Hans-E. Porst. Limits and colimits of Hopf algebras. J. Algebra, 328:254–267, 2011.
* [PW90] P. Podleś and S. L. Woronowicz. Quantum deformation of Lorentz group. Comm. Math. Phys., 130(2):381–431, 1990.
* [QS10] J. Quaegebeur and M. Sabbe. Isometric coactions of compact quantum groups on compact quantum metric spaces. ArXiv e-prints, July 2010.
* [Rei70] G. A. Reid. Epimorphisms and surjectivity. Invent. Math., 9:295–307, 1969/1970.
* [Sch92] Peter Schauenburg. Tannaka duality for arbitrary Hopf algebras. 66:ii+57, 1992.
* [Sch11] D. Schäppi. The formal theory of Tannaka duality. ArXiv e-prints, December 2011.
* [Soł05] Piotr M. Sołtan. Quantum Bohr compactification. Illinois J. Math., 49(4):1245–1270 (electronic), 2005.
* [Soł09] Piotr M. Sołtan. Quantum families of maps and quantum semigroups on finite quantum spaces. J. Geom. Phys., 59(3):354–368, 2009.
* [Sol10] P. M. Soltan. On quantum maps into quantum semigroups. ArXiv e-prints, October 2010.
* [Swe69] Moss E. Sweedler. Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York, 1969\.
* [Tak02] M. Takesaki. Theory of operator algebras. I, volume 124 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5.
* [Tam90] D. Tambara. The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 37(2):425–456, 1990.
* [Tom07] Reiji Tomatsu. A characterization of right coideals of quotient type and its application to classification of Poisson boundaries. Comm. Math. Phys., 275(1):271–296, 2007.
* [VD94] A. Van Daele. Multiplier Hopf algebras. Trans. Amer. Math. Soc., 342(2):917–932, 1994.
* [VDW96] Alfons Van Daele and Shuzhou Wang. Universal quantum groups. Internat. J. Math., 7(2):255–263, 1996.
* [Ver12] J. Vercruysse. Hopf algebras—Variant notions and reconstruction theorems. ArXiv e-prints, February 2012.
* [Wan95] Shuzhou Wang. Free products of compact quantum groups. Comm. Math. Phys., 167(3):671–692, 1995.
* [Wan97] Shuzhou Wang. Krein duality for compact quantum groups. J. Math. Phys., 38(1):524–534, 1997.
* [Wan98] Shuzhou Wang. Quantum symmetry groups of finite spaces. Comm. Math. Phys., 195(1):195–211, 1998.
* [Wan99] Shuzhou Wang. Ergodic actions of universal quantum groups on operator algebras. Comm. Math. Phys., 203(2):481–498, 1999.
* [Was94] Simon Wassermann. Exact $C^{*}$-algebras and related topics, volume 19 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1994.
* [Wor87] S. L. Woronowicz. Compact matrix pseudogroups. Comm. Math. Phys., 111(4):613–665, 1987.
* [Wor88] S. L. Woronowicz. Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted ${\rm SU}(N)$ groups. Invent. Math., 93(1):35–76, 1988.
* [Wor98] S. L. Woronowicz. Compact quantum groups. In Symétries quantiques (Les Houches, 1995), pages 845–884. North-Holland, Amsterdam, 1998.
|
arxiv-papers
| 2012-08-26T04:50:00 |
2024-09-04T02:49:34.546539
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandru Chirvasitu",
"submitter": "Alexandru Chirv\\u{a}situ L.",
"url": "https://arxiv.org/abs/1208.5193"
}
|
1208.5278
|
Low dimensional cohomology of Hom-Lie algebras
and $q$-deformed $W(2,2)$ algebra111Corresponding author:
[email protected]
Lamei Yuan${}^{\,{\ddagger}}$, Hong You${}^{\,{\ddagger}\,{\dagger}}$
‡Academy of Fundamental and Interdisciplinary Sciences,
Harbin Institute of Technology, Harbin 150080, China
†Department of Mathematics, Suzhou University, Suzhou 200092, China
E-mail: [email protected], [email protected]
Abstract. This paper aims to study the low dimensional cohomology of Hom-Lie
algebras and $q$-deformed $W(2,2)$ algebra. We show that the $q$-deformed
$W(2,2)$ algebra is a Hom-Lie algebra. Also, we establish a one-to-one
correspondence between the equivalence classes of one dimensional central
extensions of a Hom-Lie algebra and its second cohomology group, leading us to
determine the second cohomology group of the $q$-deformed $W(2,2)$ algebra. In
addition, we generalize some results of derivations of finitely generated Lie
algebras with values in graded modules to Hom-Lie algebras. As application we
compute all $\alpha^{k}$-derivations and in particular the first cohomology
group of the $q$-deformed $W(2,2)$ algebra.
Key words: Hom-Lie algebras, $q$-deformed $W(2,2)$ algebra, derivation, second
cohomology group, first cohomology group.
Mathematics Subject Classification (2000): 17A30, 17A60, 17B68, 17B70.
1\. Introduction
The notion of Hom-Lie algebras was initially introduced in [3] motivated by
examples of deformed Lie algebras coming from twisted discretizations of
vector fields. In this paper we will follow the slightly more general
definition of Hom-Lie algebras given by Makhlouf and Silvestrov in [6].
Precisely, a Hom-Lie algebra is a triple $({\cal L},[\cdot,\cdot],\alpha)$
consisting of a vector space ${\cal L}$, a bilinear map $[\cdot,\cdot]:{\cal
L}\times{\cal L}\rightarrow{\cal L}$ and a linear map $\alpha:{\cal
L}\rightarrow{\cal L}$ such that
$\displaystyle[x,y]$ $\displaystyle=$ $\displaystyle-[y,x],\ \ (\mbox{skew-
symmetry})$ $\displaystyle\circlearrowleft_{x,y,z}[[x,y],\alpha(z)]$
$\displaystyle=$ $\displaystyle 0,\ \ \ (\mbox{Hom-Jacobi identity})$
for all $x,y,z\in{\cal L}$, and where the symble $\circlearrowleft_{x,y,z}$
denotes summation over the cyclic permutation on $x,y,z.$ One sees that the
classical Lie algebras recover from Hom-Lie algebras if the twisting map
$\alpha$ is the identity map. The Hom-Lie algebras were discussed intensively
in [7, 8, 9, 10] while the graded cases were considered in [1, 5, 11]. But the
cohomology with values in graded Hom-modules is not very clear. Therefore, one
of the aims of the present paper is to fill this gap.
The $W(2,2)$ Lie algebra was introduced in [13] for the study of
classification of vertex operator algebras generated by vectors of weight $2$.
It is an extension of the Virasoro algebra. In the following we denote by
$\mathcal{W}$ the centerless $W(2,2)$ Lie algebra, which is an infinite
dimensional Lie algebra generated by $L_{n}$ and $M_{n}$ ($n\in\mathbb{Z}$)
satisfying the following Lie brackets
$\displaystyle\ \ \ [L_{m},L_{n}]=(n-m)L_{m+n},\
{[L_{m},M_{n}]}=(n-m)M_{m+n},\ {[M_{m},M_{n}]}=0,\ \mbox{for}\
m,n\in\mathbb{Z}.$
In [12] we presented a realization of the centerless $W(2,2)$ Lie algebra
$\mathcal{W}$ by using bosonic and fermionic oscillators. The bosonic
oscillator $a$ and its hermitian conjugate $a^{+}$ obey the commutation
relations:
$\displaystyle[a,a^{+}]=aa^{+}-a^{+}a=1,\ \ \ [1,a^{+}]=[1,a]=0.$ (1.1)
It follows by induction on $n$ that
$[a,(a^{+})^{n}]=n(a^{+})^{n-1},\ \mbox{for all}\ n\in\mathbb{Z}.$
The fermionic oscillators $b$ and $b^{+}$ satisfy the anticommutators
$\displaystyle\\{b,b^{+}\\}=bb^{+}+b^{+}b=1,\ \ \ b^{2}=(b^{+})^{2}=0.$ (1.2)
Moreover, we set $[a,b]=[a,b^{+}]=[a^{+},b]=[a^{+},b^{+}]=0.$
###### Lemma 1.1
([12]) With notations above. The generators of the form
$\displaystyle\ \ \ \ \ \ \ \ L_{n}\equiv(a^{+})^{n+1}a,\ \ \
M_{n}\equiv(a^{+})^{n+1}b^{+}a,\ \ \mbox{for all}\ n\in\mathbb{Z},$ (1.3)
realize the centerless $W(2,2)$ Lie algebra $\mathcal{W}$ under the commutator
$[A,B]=AB-BA,\ \ \mbox{ for all}\ A,B\in\mathcal{W}.$
Now fix a nonzero $q\in\mathbb{C}$ such that $q$ is not a root of unity. We
introduce the following notation
$\displaystyle[A,B]_{(\alpha,\beta)}=\alpha AB-\beta BA,$
and the $q$-number
$\displaystyle[n]_{q}=\frac{q^{n}-q^{-n}}{q-q^{-1}}.$
It is clear to see that $[-n]_{q}=-[n]_{q}.$ Furthermore, one can deduce that
$\displaystyle q^{n}[m]_{q}-q^{m}[n]_{q}=[m-n]_{q},\ \ \
q^{-n}[m]_{q}+q^{m}[n]_{q}=[m+n]_{q}.$ (1.4)
Then the generators $L_{n}$ and $M_{n}$ ($n\in\mathbb{Z}$) satisfy the
following $q$-brackets:
$\displaystyle[L_{n},L_{m}]_{(q^{n-m},\,q^{m-n})}$ $\displaystyle=$
$\displaystyle[m-n]_{q}L_{m+n},$
$\displaystyle{[L_{n},M_{m}]_{(q^{n-m},\,q^{m-n})}}$ $\displaystyle=$
$\displaystyle[m-n]_{q}M_{m+n},$
$\displaystyle{[M_{n},M_{m}]_{(q^{n-m},q^{m-n})}}$ $\displaystyle=$
$\displaystyle 0,$
for all $m,n\in\mathbb{Z}$. We call this algebra the $q$-deformed $W(2,2)$
algebra, which is the second object considered in this paper. In the following
we will denote $q$-deformed $W(2,2)$ algebra by $\mathcal{W}_{q}$ and simply
write the $q$-bracket as $[\cdot,\cdot]_{q}$. In [12] we determined quantum
groups and one dimensional central extensions of $\mathcal{W}_{q}$. In this
paper, we will study its low dimensional cohomology theory. That is the second
aim of this paper.
Throughout this paper, $\mathbb{C}$ denotes the field of complex number and
$\mathbb{Z}$ denotes the set of all integers. All vector spaces and algebras
are assumed to be over $\mathbb{C}$.
2\. Second cohomology group
In this section, we first recall some basic definitions and in particular
central extension of Hom-Lie algebras. Then we establish a one-to-one
correspondence between the equivalence classes of one dimensional central
extensions of a Hom-Lie algebra and its second cohomology group with
coefficients in $\mathbb{C}$. As application we determine the second
cohomology group of the $q$-deformed $W(2,2)$ algebra which is considered as a
Hom-Lie algebra.
In the sequel we will often simply write a Hom-Lie algebra $({\cal
L},[\cdot,\cdot],\alpha)$ as $({\cal L},\alpha)$. A Hom-Lie algebra $({\cal
L},\alpha)$ is said to be multiplicative if the twisting map $\alpha$ is an
endomorphism. Let $G$ be an abelian group. A Hom-Lie algebra $({\cal
L},\alpha)$ is said to be $G$-graded, if its underlying vector space is
$G$-graded (i.e., ${\cal L}=\oplus_{g\in G}{\cal L}_{g}$) satisfying $[{\cal
L}_{g},{\cal L}_{h}]\subseteq{\cal L}_{g+h}$, and if $\alpha$ is an even map,
i.e., $\alpha({\cal L}_{g})\subseteq{\cal L}_{g}$, for all $g,h\in G$.
The theory of central extensions of Hom-Lie algebras was studied in [3, 4]. An
extension of a Hom-Lie algebra $({\cal L},\zeta)$ by an abelian Hom-Lie
algebra $({\mathfrak{a}},\zeta_{\mathfrak{a}})$ is a commutative diagram with
exact rows
$\begin{CD}0@>{}>{}>{\mathfrak{a}}@>{\rm\iota}>{\rm}>{\hat{\cal L}}@>{\rm
pr}>{}>{\cal L}@>{}>{}>0\\\
@V{\zeta_{\mathfrak{a}}}V{}V@V{\hat{\zeta}}V{}V@V{\zeta}V{}V\\\
0@>{}>{}>{\mathfrak{a}}@>{\rm\iota}>{\rm}>{\hat{\cal L}}@>{\rm pr}>{}>{\cal
L}@>{}>{}>0\end{CD}$
where $({\hat{\cal L}},{\hat{\zeta}})$ is a Hom-Lie algebra. The extension is
central if
$\iota({\mathfrak{a}})\subseteq Z({\hat{\cal L}})=\\{x\in{\hat{\cal
L}}\,|\,[x,{\hat{\cal L}}]_{\hat{\cal L}}=0\\}.$
In the following we focus on the central extension of $({\cal L},\alpha)$ by a
one-dimensional center $\mathbb{C}c$, where ${c}=\iota(1)$. Note that the
center $\mathbb{C}c$ can be considered as the one-dimensional trivial Hom-Lie
algebra with the identity map.
###### Definition 2.1
Let $({\cal L},\alpha)$ be a Hom-Lie algebra. A bilinear map $\psi:{\cal
L}\times{\cal L}\rightarrow\mathbb{C}$ is called a $2$-cocycle on ${\cal L}$
if the following conditions are satisfied
$\displaystyle\psi(x,y)=-\psi(y,x),$ (2.1)
$\displaystyle\psi(\alpha(x),[y,z])+\psi(\alpha(y),[z,x])+\psi(\alpha(z),[x,y])=0,$
(2.2)
for all $x,y,z\in L$.
Now we have the following theorem:
###### Theorem 2.2
Let $({\cal L},\alpha)$ be a Hom-Lie algebra and $\psi:{\cal L}\times{\cal
L}\rightarrow\mathbb{C}$ be a bilinear map. Define on the vector space
${\hat{\cal L}}={\cal L}\oplus\mathbb{C}$ the following bracket and linear map
by
$\displaystyle[x+c,y+b]_{\hat{\cal L}}$ $\displaystyle=$
$\displaystyle[x,y]_{{\cal L}}+\psi(x,y),$ (2.3)
$\displaystyle{\hat{\alpha}}(x+c)$ $\displaystyle=$
$\displaystyle\alpha(x)+c,$ (2.4)
for all $x,y\in{\cal L}$ and $c,b\in\mathbb{C}$. Then $({\hat{\cal
L}},[\cdot,\cdot]_{\hat{\cal L}},{\hat{\alpha}})$ is a Hom-Lie algebra one
dimensional central extension of $({\cal L},\alpha)$ if and only if $\psi$ is
a $2$-cocycle on $({\cal L},\alpha)$. If, in addition, $({\cal L},\alpha)$ is
multiplicative and $\psi$ satisfies $\psi(\alpha(x),\alpha(y))=\psi(x,y)$, for
all $x,y\in{\cal L}$, then the Hom-Lie algebra $({\hat{\cal
L}},{\hat{\alpha}})$ is also multiplicative.
Proof. Since $[\cdot,\cdot]_{{\cal L}}$ is skew-symmetric, the new bracket
$[\cdot,\cdot]_{\hat{\cal L}}$ is skew-symmetric if and only if the map $\psi$
is skew-symmetric. For any $x,y,z\in{\cal L}$ and $a,b,c\in\mathbb{C}$, we
have
$\displaystyle[{\hat{\alpha}}(x+a),[y+b,z+c]_{\hat{\cal L}}]_{\hat{\cal L}}$
$\displaystyle=$ $\displaystyle[\alpha(x)+a,[y,z]_{{\cal
L}}+\psi(y,z)]_{\hat{\cal L}}$ $\displaystyle=$
$\displaystyle[\alpha(x),[y,z]_{{\cal L}}]_{{\cal
L}}+\psi(\alpha(x),[y,z]_{{\cal L}}).$
Consequently,
$\displaystyle[{\hat{\alpha}}(x+a),[y+b,z+c]_{\hat{\cal L}}]_{\hat{\cal
L}}+[{\hat{\alpha}}(y+b),[z+c,x+a]_{\hat{\cal L}}]_{\hat{\cal
L}}+[{\hat{\alpha}}(z+c),[x+a,y+b]_{\hat{\cal L}}]_{\hat{\cal L}}=0$
if and only if
$\displaystyle\psi(\alpha(x),[y,z]_{{\cal L}})+\psi(\alpha(y),[z,x]_{{\cal
L}})+\psi(\alpha(z),[x,y]_{{\cal L}})=0,$
which proves $({\hat{\cal L}},[\cdot,\cdot]_{\hat{\cal L}},{\hat{\alpha}})$ is
a Hom-Lie algebra if and only if $\psi$ is a $2$-cocycle on $({\cal
L},\alpha)$.
If $({\cal L},\alpha)$ is multiplicative, then we have
$\displaystyle{\hat{\alpha}}([x+a,y+b]_{\hat{\cal L}})$ $\displaystyle=$
$\displaystyle{\hat{\alpha}}([x,y]_{\cal L}+\psi(x,y))$ $\displaystyle=$
$\displaystyle\alpha([x,y]_{\cal L})+\psi(x,y)$ $\displaystyle=$
$\displaystyle[\alpha(x),\alpha(y)]_{\cal L}+\psi(x,y).$
On the other hand, we have
$\displaystyle[{\hat{\alpha}}(x+a),{\hat{\alpha}}(y+b)]_{\hat{\cal L}}$
$\displaystyle=$ $\displaystyle[\alpha(x)+a,\alpha(y)+b]_{\hat{\cal L}}$
$\displaystyle=$ $\displaystyle[\alpha(x),\alpha(y)]_{\cal
L}+\psi(\alpha(x),\alpha(y)).$
According to the hypothesis that $\psi(\alpha(x),\alpha(y))=\psi(x,y)$ for all
$x,y\in{\cal L}$, we have
$\displaystyle{\hat{\alpha}}([x+a,y+b]_{\hat{\cal
L}})=[{\hat{\alpha}}(x+a),{\hat{\alpha}}(y+b)]_{\hat{\cal L}},\ \mbox{for\
all}\ x,y\in{\cal L},\ a,b\in\mathbb{C},$
which shows that $({\hat{\cal L}},{\hat{\alpha}})$ is multiplicative.
Finally, we define $\rm pr$ and $\iota$ as the natural projection and
inclusion respectively by
$\displaystyle{\rm pr}:\hat{\cal L}\rightarrow{\cal L},\qquad{\rm pr}(x+a)=x;$
$\displaystyle\iota:\mathbb{C}\rightarrow\hat{\cal L},\qquad\iota(a)=0+a.$
Then it is easy to show that $({\hat{\cal L}},{\hat{\alpha}})$ is a one-
dimensional central extension of $({\cal L},\alpha)$. $\Box$
Denote by $Z^{2}({\cal L},\mathbb{C})$ the vector space of all $2$-cocycles on
a Hom-Lie algebra $({\cal L},\alpha)$. For any linear map $f:{\cal
L}\rightarrow\mathbb{C}$, we can define a $2$-cocycle $\psi_{f}$ by
$\psi_{f}(x,y)=f([x,y]),\ \ \mbox{for\ all}\ x,y\in{\cal L}.$ (2.5)
Such a $2$-cocycle is called a $2$-coboundary or a trivial $2$-cocycle on
${\cal L}$. Let $B^{2}({\cal L},\mathbb{C})$ denote the vector space of all
$2$-coboundaries on ${\cal L}$. The quotient space
$H^{2}({\cal L},\mathbb{C})=Z^{2}({\cal L},\mathbb{C})/B^{2}({\cal
L},\mathbb{C})$
is called the second cohomology group of ${\cal L}$ with trivial coefficients
$\mathbb{C}$. A $2$-cocycle $\psi$ is said to be equivalent to another
$2$-cocycle $\phi$ if $\psi-\phi$ is trivial. For a $2$-cocycle $\psi$, let
$[\psi]$ be the equivalent class of $\psi$. Then we have the following
corollary:
###### Corollary 2.3
For any Hom-Lie algebra $({\cal L},\alpha)$, there exists a one-to-one
correspondence between the equivalence classes of one dimensional central
extensions of $({\cal L},\alpha)$ and its second cohomology group $H^{2}({\cal
L},\mathbb{C})$.
In the following, we consider the $q$-deformed $W(2,2)$ algebra
$\mathcal{W}_{q}$. Note that $\mathcal{W}_{q}$ is not a Lie algebra, because
the classical Jacobi identity does not hold (but the antisymmetry is true). By
straightforward calculations, we have
$\displaystyle(q^{l}+q^{-l})[\,[\,L_{m},L_{n}]_{(q^{m-n},\,q^{n-m})},L_{l}\,]_{(q^{m+n-l},\,q^{l-m-n})}+\mbox{cyclic\
permutations}=0,\ \ \ \ \ \ \ \ $ (2.6)
$\displaystyle(q^{l}+q^{-l})[\,[\,L_{m},L_{n}]_{(q^{m-n},\,q^{n-m})},M_{l}\,]_{(q^{m+n-l},\,q^{l-m-n})}+\mbox{cyclic\
permutations}=0.$ (2.7)
Define on $\mathcal{W}_{q}$ a linear map $\alpha$ by
$\displaystyle\alpha(L_{n})=(q^{n}+q^{-n})L_{n},\ \
\alpha(M_{n})=(q^{n}+q^{-n})M_{n}.$
Then, using the $q$-deformed Jacobi identities (2.6) and (2.7), we obtain the
following result.
###### Theorem 2.4
The triple ($\mathcal{W}_{q},[\cdot,\cdot]_{q},\alpha$) forms a Hom-Lie
algebra.
In [12] we provided a computation of one-dimensional central extensions of
$\mathcal{W}_{q}$. Hence, according to Corollary 2.3, we can determine the
second cohomology group of the $q$-deformed $W(2,2)$ algebra $\mathcal{W}_{q}$
as follows:
###### Proposition 2.5
$H^{2}(\mathcal{W}_{q},\mathbb{C})=\mathbb{C}\beta\oplus\mathbb{C}\gamma$,
where
$\displaystyle\beta(L_{m},L_{n})$ $\displaystyle=$
$\displaystyle\delta_{m,-n}\frac{[m-1]_{q}[m]_{q}[m+1]_{q}}{[2]_{q}[3]_{q}\langle
m\rangle_{q}},\ \ \ \beta(L_{m},M_{n})=\beta(M_{m},M_{n})=0,$
$\displaystyle\gamma(L_{m},M_{n})$ $\displaystyle=$
$\displaystyle\delta_{m,-n}\frac{[m-1]_{q}[m]_{q}[m+1]_{q}}{[2]_{q}[3]_{q}\langle
m\rangle_{q}},\ \ \ \gamma(L_{m},L_{n})=\gamma(M_{m},M_{n})=0,$
and where $\langle m\rangle_{q}=q^{m}+q^{-m}$, for all $m,n\in\mathbb{Z}.$
3\. Derivations of Hom-Lie algebras and $q$-deformed $W(2,2)$ Lie algebra
This section is devoted to discuss derivations of graded Hom-Lie algebras. We
extend to Hom-Lie algebras some concepts and results of derivations of
finitely generated Lie algebras with values in graded modules studied in [2].
As application we compute all $\alpha^{k}$-derivations and particularly the
first cohomology group of the $q$-deformed $W(2,2)$ algebra.
###### Definition 3.1
Let $({\cal L},\alpha)$ be a Hom-Lie algebra. A representation of ${\cal L}$
is a triple $(V,\rho,\beta)$, where $V$ is a $\mathbb{C}$-vector space,
$\beta\in End(V)$ and $\rho:{\cal L}\rightarrow End(V)$ is a
$\mathbb{C}$-linear map satisfying
$\displaystyle\rho([x,y])\circ\beta=\rho(\alpha(x))\circ\rho(y)-\rho(\alpha(y))\circ\rho(x),$
for all $x,y\in{\cal L}.$ $V$ is also called a Hom-${\cal L}$-module, denoted
by $(V,\beta)$ for convenience.
One recovers the definition of a representation in the case of Lie algebras by
setting $\alpha={\rm id}_{\cal L}$ and $\beta={\rm id}_{V}$. For any
$x\in{\cal L},$ define ${\rm ad}:{\cal L}\rightarrow End({\cal L})$ by ${\rm
ad}_{x}(y)=[x,y]$ for all $y\in{\cal L}$. Then $({\cal L},{\rm ad},\alpha)$ is
a representation of ${\cal L}$, which is called the adjoint representation of
${\cal L}$.
###### Definition 3.2
Let $(V,\beta_{V})$ and $(W,\beta_{W})$ be two Hom-${\cal L}$-modules. A
linear map $f:V\rightarrow W$ is called a morphism of Hom-${\cal L}$-modules
if it satisfies
$\displaystyle f\circ\beta_{V}$ $\displaystyle=$ $\displaystyle\beta_{W}\circ
f,$ $\displaystyle f(x\cdot v)$ $\displaystyle=$ $\displaystyle x\cdot f(v),$
for all $x\in{\cal L},$ $v\in V.$
Let $G$ be an abelian group, $({\cal L}=\oplus_{g\in G}{\cal
L}_{g},[\cdot,\cdot],\alpha)$ be a $G$-graded Hom-Lie algebra. An Hom-${\cal
L}$-module $V$ is said to be $G$-graded if $V=\oplus_{g\in G}V_{g}$ and ${\cal
L}_{g}V_{h}\subseteq V_{g+h}$ for all $g,h\in G$. For any nonnegative integer
$k$, denote by $\alpha^{k}$ the $k$-times composition of $\alpha$, i.e.,
$\displaystyle\alpha^{k}=\alpha\circ\alpha\circ\cdots\circ\alpha.$ (3.1)
In particular, $\alpha^{0}={\rm id}$ and $\alpha^{1}=\alpha.$ Then we can
define $\alpha^{k}$-derivations of ${\cal L}$ with values in its Hom-${\cal
L}$-modules.
###### Definition 3.3
A linear map $D:{\cal L}\rightarrow V$ is called an $\alpha^{k}$-derivation if
it satisfies
$\displaystyle D\circ\alpha$ $\displaystyle=$ $\displaystyle\alpha\circ D,$
$\displaystyle D[x,y]$ $\displaystyle=$ $\displaystyle\alpha^{k}(x)\cdot
D(y)-\alpha^{k}(y)\cdot D(x),$
for all $x,y\in{\cal L}$
We recover the definition of a derivation by setting $k=0$ in the definition
above. Hence, an $\alpha^{0}$-derivation is often simply called a derivation
in the present paper. We say that an $\alpha^{k}$-derivation $D$ has degree
$g$ (denoted by ${\rm deg}(D)=g$) if $D\neq 0$ and $D({\cal L}_{h})\subseteq
V_{g+h}$ for any $h\in G$. Let $D$ be an $\alpha^{k}$-derivation. If there
exists $v\in V$ such that $D(x)=\alpha^{k}(x)\cdot v$ for all $x\in{\cal L}$,
then $D$ is called an inner $\alpha^{k}$-derivation. Denote by
$Der_{\alpha^{k}}({\cal L},V)$ and $Inn_{\alpha^{k}}({\cal L},V)$ the space of
$\alpha^{k}$-derivations and the space of inner $\alpha^{k}$-derivations,
respectively. In particular, let $Der({\cal L},V)$ and $Inn({\cal L},V)$
denote the space of derivations and the space of inner derivations,
respectively, and write $Der({\cal L},V)_{g}:=\\{D\in Der({\cal
L},V)\,\big{|}\,{\rm deg}(D)=g\\}\cup\\{0\\}$. The first cohomology group of
${\cal L}$ with coefficients in $V$ is defined by
$\displaystyle H^{1}({\cal L},V):=Der({\cal L},V)/Inn({\cal L},V).$ (3.2)
###### Remark 3.4
The set $Der_{\alpha^{k}}({\cal L},V)$ (resp. $Inn_{\alpha^{k}}({\cal L},V)$)
is not close under map composition or commutator bracket. But the space of all
such $\alpha^{k}$-derivations $\oplus_{k\geq 0}Der_{\alpha^{k}}({\cal L},V)$
(resp. $\oplus_{k\geq 0}Inn_{\alpha^{k}}({\cal L},V)$) form an Lie algebra via
commutator bracket.
Now let ${\cal L}$ be a $G$-graded Hom-Lie algebra which is finitely
generated. In the following we present two results, which can be seen as Hom
versions of those obtained in [2].
###### Proposition 3.5
Let $V$ be a $G$-graded Hom-${\cal L}$-module. For every $D\in Der({\cal
L},V)$, we have
$\displaystyle D=\mbox{$\sum_{g\in G}D_{g}$},$ (3.3)
where $D_{g}\in Der({\cal L},V)_{g}$ and where there are only finitely many
$D_{g}(u)\neq 0$ in the equation $D(u)=\sum_{g\in G}D_{g}(u)$, for any
$u\in{\cal L}$.
Proof. For any $g\in G$, define a homogeneous linear map $D_{g}:{\cal
L}\rightarrow V$ as follows: for any $u\in{\cal L}_{h}$ with $h\in G$, write
$D(u)=\sum_{p\in G}u_{p}$ with $u_{p}\in V$, then set $D_{g}(u)=u_{g+h}$.
Clearly, $D_{g}$ is well defined and $D_{g}\in Der({\cal L},V)_{g}$. Also,
(3.3) is true. $\Box$
###### Proposition 3.6
Let $V$ be a $G$-graded Hom-${\cal L}$-module such that
* (a)
$H^{1}({\cal L}_{0},V_{g})=0$, for $g\in G\setminus\\{0\\}$.
* (b)
${\rm Hom}_{L_{0}}({\cal L}_{g},V_{h})=0$, for $g\neq h$.
Then
$Der({\cal L},V)=Der({\cal L},V)_{0}+Inn({\cal L},V).$
Proof. Let $D$ be a derivation from ${\cal L}$ into its Hom-${\cal L}$-module
$V$. According to (3.3) we can decompose $D$ into its homogeneous components
$D=\sum_{g\in G}D_{g}$ with $D_{g}\in Der({\cal L},V)_{g}.$ Suppose that
$g\neq 0$. Then $D_{g}|_{{\cal L}_{0}}$ is a derivation from ${\cal L}_{0}$
into the Hom-${\cal L}_{0}$-module $V_{g}$. By virtue of (a), $D_{g}|_{{\cal
L}_{0}}$ is inner, i.e., there exists $v_{g}\in V_{g}$ such that
$D_{g}(u)=u\cdot v_{g}$ for all $u\in{\cal L}_{0}.$ Consider $\psi_{g}:{\cal
L}\rightarrow V$ defined by $\psi_{g}(x):=D_{g}(x)-x\cdot v_{g}$ for all
$x\in{\cal L}$. Then $\psi_{g}$ is a derivation of degree $g$ which vanishes
on ${\cal L}_{0}$. Hence $\psi_{g}$ is a morphism of Hom-${\cal
L}_{0}$-modules and condition (b) entails the vanishing of $\psi_{g}$ on
${\cal L}_{h}$ for every $h\in G$. Consequently, $D_{g}\in Inn({\cal L},V)$,
which completes the proof. $\Box$
In the following we focus on the $q$-deformed W(2,2) algebra $\mathcal{W}_{q}$
as a Hom-Lie algebra ($\mathcal{W}_{q},[\cdot,\cdot]_{q},\alpha$) defined in
Theorem 2.4. Obviously, $\mathcal{W}_{q}$ is $\mathbb{Z}$-graded by
$\mathcal{W}_{q}=\oplus_{n\in\mathbb{Z}}\mathcal{W}_{q}^{n},\ \mbox{where}\
\mathcal{W}_{q}^{n}=span_{\mathbb{C}}\\{L_{n},M_{n}\\}.$
Note that $\mathcal{M}:=span_{\mathbb{C}}\\{M_{n}\\}$ is an ideal of
$(\mathcal{W}_{q},\alpha)$, or in other words, $\mathcal{M}$ is an adjoint
Hom-$\mathcal{W}_{q}$-module. In addition, $\mathcal{W}_{q}$ is finitely
generated by $\\{L_{1},L_{-1},M_{1}\\}$. Let $D$ be an $\alpha^{k}$-derivation
of $\mathcal{W}_{q}$. For all $m,n\in\mathbb{Z}$, we have
$\displaystyle(q^{m}+q^{-m})^{k}[D(L_{n}),L_{m}]_{q}+(q^{n}+q^{-n})^{k}[L_{n},D(L_{m})]_{q}$
$\displaystyle=$ $\displaystyle[m-n]_{q}D(L_{m+n}),$ (3.4)
$\displaystyle(q^{m}+q^{-m})^{k}[D(L_{n}),M_{m}]_{q}+(q^{n}+q^{-n})^{k}[L_{n},D(M_{m})]_{q}$
$\displaystyle=$ $\displaystyle[m-n]_{q}D(M_{m+n}).$ (3.5)
Now we aim to determine all $\alpha^{k}$-derivation of $\mathcal{W}_{q}$.
First, we compute the ($\alpha^{0}$-)derivations of $\mathcal{W}_{q}$. Denote
by $Der(\mathcal{W}_{q})$ and $Inn(\mathcal{W}_{q})$ the set of all
derivations and the set of all inner derivations, respectively. Let
$Der(\mathcal{W}_{q})_{m}$ be the set of derivations of degree $m$.
###### Lemma 3.7
$H^{1}(\mathcal{W}_{q}^{0},\mathcal{W}_{q}^{n})=0$ for any nonzero integer
$n$.
Proof. Note that $[L_{0},X_{0}]_{q}=0$, for any
$X_{0}\in\mathcal{W}_{q}^{0}=span\\{L_{0},M_{0}\\}$. Let $D$ be any element in
$Der(\mathcal{W}_{q}^{0},\mathcal{W}_{q}^{n})$. Then it follows
$D(L_{0})\in\mathcal{W}_{q}^{n}.$ Applying $D$ to $[L_{0},X_{0}]_{q}=0$, we
have $[n]_{q}D(X_{0})=[L_{0},D(X_{0})]=[X_{0},D(L_{0})]$. Consequently,
$D(X_{0})=[X_{0},v]$ with $v=\frac{1}{[n]_{q}}D(L_{0})$ in
$\mathcal{W}_{q}^{n}$. In other words, $D$ is an inner derivation from
$\mathcal{W}_{q}^{0}$ into its adjoint module $\mathcal{W}_{q}^{n}$. $\Box$
###### Lemma 3.8
${\rm Hom}_{\mathcal{W}_{q}^{0}}(\mathcal{W}_{q}^{m},\mathcal{W}_{q}^{n})=0$
for $m\neq n$.
Proof. Let $f\in{\rm
Hom}_{\mathcal{W}_{q}^{0}}(\mathcal{W}_{q}^{m},\mathcal{W}_{q}^{n})$ with
$m\neq n$. For any $X_{m}\in\mathcal{W}_{q}^{m}$, we have
$(q^{n}+q^{-n})\big{(}f(X_{m})\big{)}=\alpha\big{(}f(X_{m})\big{)}=f\big{(}\alpha(X_{m})\big{)}=(q^{m}+q^{-m})f(X_{m}),$
leading to $f(X_{m})=0$, since $m\neq n$. Hence, $f=0$. $\Box$
Now according to Proposition 3.6, we have the following result:
###### Proposition 3.9
$Der(\mathcal{W}_{q})=Der(\mathcal{W}_{q})_{0}+Inn(\mathcal{W}_{q})$.
Thanks to Proposition 3.9, the study of $Der(\mathcal{W}_{q})$ reduces to that
of its constitute of degree zero. Let $D$ be an element of
$Der(\mathcal{W}_{q})_{0}$. For any integer $n$, assume that
$\displaystyle D(L_{n})=a_{n}L_{n}+b_{n}M_{n},\ \
D(M_{n})=c_{n}L_{n}+d_{n}M_{n},$ (3.6)
where the coefficients are complex numbers. Applying $D$ to
$[L_{0},L_{n}]_{q}=[n]_{q}L_{n}$, one can obtain $D(L_{0})=0,$ i.e.,
$a_{0}=b_{0}=0$. Using (3.4), we have
$\displaystyle a_{m+n}=a_{m}+a_{n},\ \ b_{m+n}=b_{m}+b_{n},\ \mbox{for}\ m\neq
n.$
Let $m=-n$. Then we have
$\displaystyle a_{-m}=-a_{m},\ \ b_{-m}=-b_{m},\ \mbox{for}\ m\neq 0.$
Furthermore, we have
$\displaystyle a_{m}=ma_{1},\ \ b_{m}=mb_{1},\ \ \mbox{for all}\
m\in\mathbb{Z}.$
Similarly, using (3.5) we have
$\displaystyle c_{m+n}=c_{n},\ \ d_{m+n}=a_{m}+d_{n},\ \ \mbox{for}\ m\neq n,$
from which it follows
$\displaystyle c_{m}=c_{0},\ \ d_{m}=ma_{1}+d_{0},\ \ \mbox{for\ all}\
m\in\mathbb{Z}.$
Applying $D$ to $[M_{1},M_{0}]=0$, we have $c_{0}=0$. It follows that
$c_{m}=0$ for all $m\in\mathbb{Z}$. Hence, there exist $a,b,d\in\mathbb{C}$
such that
$\displaystyle D(L_{n})=n(aL_{n}+bM_{n}),\ \ D(M_{n})=(na+d)M_{n},\ \
\mbox{for\ all}\ n\in\mathbb{Z}.$ (3.7)
From the discussions above we obtain the following result:
###### Proposition 3.10
All the derivations of $\mathcal{W}_{q}$ is
$Der(\mathcal{W}_{q})=span_{\mathbb{C}}\\{D\\}\oplus Inn(\mathcal{W}_{q}),$
where $D$ is defined by (3.7).
###### Corollary 3.11
The first cohomology group of $\,\mathcal{W}_{q}$ with values in its adjoint
module is one-dimensional.
Next we compute the $\alpha^{1}$-derivations of $\mathcal{W}_{q}$. Let $D$ be
an $\alpha^{1}$-derivation of degree $s$. Assume that
$\displaystyle D(L_{n})=a_{s,n}L_{n+s}+b_{s,n}M_{n+s},\ \
D(M_{n})=c_{s,n}L_{n+s}+d_{s,n}M_{n+s},$
where the coefficients are complex numbers. Then from equation (3.4) we obtain
$\displaystyle[m-n]_{q}a_{s,m+n}$ $\displaystyle=$
$\displaystyle(q^{m}+q^{-m})[m-s-n]_{q}a_{s,n}+(q^{n}+q^{-n})[m+s-n]_{q}a_{s,m},$
(3.8) $\displaystyle{[m-n]_{q}b_{s,m+n}}$ $\displaystyle=$
$\displaystyle(q^{m}+q^{-m})[m-s-n]_{q}b_{s,n}+(q^{n}+q^{-n})[m+s-n]_{q}b_{s,m}.$
(3.9)
We first consider the case of $s\neq 0$. Taking $m=0$ in (3.8), we have
$\displaystyle(2[s+n]_{q}-[n]_{q})a_{s,n}=(q^{n}+q^{-n})[s-n]_{q}a_{s,0}.$
Furthermore,
$\displaystyle
a_{s,n}=\frac{(q^{n}+q^{-n})[s-n]_{q}}{(2[s+n]_{q}-[n]_{q})}a_{s,0}.$ (3.10)
Plugging (3.10) into (3.8), we have
$\displaystyle\frac{(q^{m+n}+q^{-m-n})[m-n]_{q}[s-m-n]_{q}}{2[s+m+n]_{q}-[m+n]_{q}}a_{s,0}$
$\displaystyle=$
$\displaystyle\frac{(q^{m}+q^{-m})(q^{n}+q^{-n})[m-s-n]_{q}[s-n]_{q}}{2[s+n]_{q}-[n]_{q}}a_{s,0}$
$\displaystyle+$
$\displaystyle\frac{(q^{m}+q^{-m})(q^{n}+q^{-n})[m+s-n]_{q}[s-m]_{q}}{2[s+m]_{q}-[m]_{q}}a_{s,0}.$
Let $m=s$ in the equation above, we have
$\displaystyle\frac{(q^{s+n}+q^{-s-n})[s-n]_{q}[-n]_{q}}{2[2s+n]_{q}-[s+n]_{q}}a_{s,0}=\frac{(q^{s}+q^{-s})(q^{n}+q^{-n})[s-n]_{q}[-n]_{q}}{2[s+n]_{q}-[n]_{q}}a_{s,0}.$
(3.11)
Then taking $n=-s$ in (3.11), we get
$\displaystyle\frac{[2s]_{q}[s]_{q}}{[s]_{q}}a_{s,0}=\frac{(q^{s}+q^{-s})^{2}[2s]_{q}[s]_{q}}{[s]_{q}}a_{s,0}.$
It follows $a_{s,0}=0$ since $s\neq 0$. Then we have $a_{s,n}=0$ for
$n\in\mathbb{Z}$ and $s\neq 0$ by (3.10). Similarly, from (3.9) we can deduce
that $b_{s,n}=0$ for $s\neq 0$ and $n\in\mathbb{Z}$.
In the case of $s=0$, we simply write $a_{0,n}$ as $a_{n}$. Then it can be
deduced from (3.8) that
$\displaystyle a_{m+n}=(q^{m}+q^{-m})a_{n}+(q^{n}+q^{-n})a_{m},\ \mbox{for}\
m\neq n.$ (3.12)
Let $m=0$ in (3.12), we have
$\displaystyle a_{n}=-(q^{n}+q^{-n})a_{0},$ (3.13)
which implies $a_{n}=a_{-n}$ for $n>0$. Taking $m=-n$ in (3.12), we have
$\displaystyle a_{0}=(q^{n}+q^{-n})(a_{n}+a_{-n}).$ (3.14)
Substituting (3.13) into (3.14), we have $a_{0}=0$ and $a_{n}=0$ for all
$n\in\mathbb{Z}$. Similarly, we can deduce that $b_{0,m}=0$ for all
$m\in\mathbb{Z}$ by using (3.9) where $s=0$. Hence, we have proved that
$\displaystyle a_{s,m}=b_{s,m}=0,\ \mbox{for all}\ m,s\in\mathbb{Z},$
or, in other words, we get $D(L_{m})=0$ for $m\in\mathbb{Z}.$
It remains to determine $D(M_{n})$ for all $n\in\mathbb{Z}.$ Using
$D(L_{n})=0$, we can deduce from (3.5) that
$\displaystyle[m-n]_{q}c_{s,m+n}$ $\displaystyle=$
$\displaystyle(q^{n}+q^{-n})[m+s-n]_{q}c_{s,m},$ (3.15)
$\displaystyle{[m-n]_{q}d_{s,m+n}}$ $\displaystyle=$
$\displaystyle(q^{n}+q^{-n})[m+s-n]_{q}d_{s,m}.$ (3.16)
Let $m=0$ in (3.15) and (3.16), respectively. We have
$\displaystyle-[n]_{q}c_{s,n}$ $\displaystyle=$
$\displaystyle(q^{n}+q^{-n})[s-n]_{q}c_{s,0},$ (3.17)
$\displaystyle-[n]_{q}d_{s,n}$ $\displaystyle=$
$\displaystyle(q^{n}+q^{-n})[s-n]_{q}d_{s,0}.$ (3.18)
Taking $n=0$ in (3.15) and (3.16), respectively, one has
$\displaystyle[m]_{q}c_{s,m}$ $\displaystyle=$ $\displaystyle
2[m+s]_{q}c_{s,m}.$ (3.19) $\displaystyle{[m]_{q}d_{s,m}}$ $\displaystyle=$
$\displaystyle 2[m+s]_{q}d_{s,m}.$ (3.20)
Taking $m=0$ in (3.19) and (3.20), respectively, we have $c_{s,0}=d_{s,0}=0$
for $s\neq 0$. Then it follows from (3.17) (resp. (3.18)) that $c_{s,n}=0$
(resp. $d_{s,n}=0$) for $n\in\mathbb{Z}$ and $s\neq 0.$
If $s=0$, then it follows from (3.15) that
$\displaystyle[m-n]_{q}c_{0,m+n}$ $\displaystyle=$
$\displaystyle(q^{n}+q^{-n})[m-n]_{q}c_{0,m}.$ (3.21)
Let $n=0$ in (3.21), we have $[m]_{q}c_{0,m}=2[m]_{q}c_{0,m}.$ It follows that
$c_{0,m}=0$ for $m\neq 0.$ Taking $n=-m$ in (3.21), we have
$[2m]_{q}c_{0,0}=(q^{m}+q^{-m})[2m]_{q}c_{0,m}$, leading us to $c_{0,0}=0$.
Similarly, we can deduce from (3.16) that $d_{0,m}=0$ for all
$m\in\mathbb{Z}.$ Thereby, the following proposition is proved.
###### Proposition 3.12
If $D$ is an $\alpha^{1}$-derivation of $\mathcal{W}_{q}$, then $D=0$.
With the similar discussions as above, we can compute $\alpha^{k}$-derivations
of $\mathcal{W}_{q}$ for $k>1$ and thus we have all the
$\alpha^{k}$-derivations of $\mathcal{W}_{q}$ for $k>0$ determined.
###### Proposition 3.13
For $k>0$, all the $\alpha^{k}$-derivations of $\mathcal{W}_{q}$ are zero.
## References
* [1] F. Ammar, A. Makhlouf, Hom-Lie superalgebras and Hom-Lie admissible superalgebras, J. Algebra, 324(7) (2010), 1513–1528.
* [2] R. Farnsteiner, Derivations ans central extensions of finitely generated graded Lie algebras, J. Algebra, 118 (1988), 33–45.
* [3] J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivation, J. Algebra, 295(2006), 314–361.
* [4] D. Larsson, S.D. Silvestrov, Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra, 288(2005), 321–344.
* [5] D. Larsson, S.D. Silvestrov, Graded quasi-Lie agebras, Czechoslovak J. Phys., 55 (2005), 1473–1478.
* [6] A. Makhlouf, S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2(2) (2008), 51–64.
* [7] A. Makhlouf, S.D. Silvestrov, Notes on $1$-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math., 22 (2010), 715–739.
* [8] Y. Sheng, Reprensentations of Hom-Lie algebras, Algebras and Representation Theory, (2010), 1–18.
* [9] D.Yau, Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl. 2 (2008), 95–108
* [10] D.Yau, Hom-algebras and homology, J. Lie Theory, 19 (2009), 409–421.
* [11] Lamei Yuan, Hom-Lie color algebra structures,Comm. Algebra, 40(2)(2012), 575–592.
* [12] Lamei Yuan, $q$-Deformation of $W(2,2)$ Lie algebra associated with quantum groups, Acta Mathematica Sinica, English Series, (2012), DOI: 10.1007/s10114-012-0544-y.
* [13] W. Zhang, C. Dong, $W$-Algebra $W(2,2)$ and the Vertex Operator Algebra $L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)$, Comm. Math. Phys., 285 (2009), 991–1004.
|
arxiv-papers
| 2012-08-27T02:29:03 |
2024-09-04T02:49:34.559316
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lamei Yuan and Hong You",
"submitter": "Lamei Yuan",
"url": "https://arxiv.org/abs/1208.5278"
}
|
1208.5301
|
aainstitutetext: HEP Division, Argonne National Laboratory,
9700 South Cass Avenue, Argonne, IL 60439, U.S.A.bbinstitutetext: Department
of Physics, Korea University,
Seoul 136-713, Korea.
# Higher-order relativistic corrections to gluon fragmentation into spin-
triplet $\bm{S}$-wave quarkonium
Geoffrey T. Bodwin b U-Rae Kim b and Jungil Lee [email protected]
[email protected] [email protected]
###### Abstract
We compute the relative-order-$v^{4}$ contribution to gluon fragmentation into
quarkonium in the ${}^{3}S_{1}$ color-singlet channel, using the
nonrelativistic QCD (NRQCD) factorization approach. The QCD fragmentation
process contains infrared divergences that produce single and double poles in
$\epsilon$ in $4-2\epsilon$ dimensions. We devise subtractions that isolate
the pole contributions, which ultimately are absorbed into long-distance NRQCD
matrix elements in the NRQCD matching procedure. The matching procedure
involves two-loop renormalizations of the NRQCD operators. The subtractions
are integrated over the phase space analytically in $4-2\epsilon$ dimensions,
and the remainder is integrated over the phase-space numerically. We find that
the order-$v^{4}$ contribution is enhanced relative to the order-$v^{0}$
contribution. However, the order-$v^{4}$ contribution is not important
numerically at the current level of precision of quarkonium-hadroproduction
phenomenology. We also estimate the contribution to hadroproduction from gluon
fragmentation into quarkonium in the ${}^{3}P_{J}$ color-octet channel and
find that it is significant in comparison to the complete next-to-leading-
order-in-$\alpha_{s}$ contribution in that channel.
###### Keywords:
quarkonium, fragmentation, NRQCD, relativistic corrections
††arxiv: 1208.5301††preprint: ANL-HEP-PR-12-65
## 1 Introduction
Heavy-quarkonium production in hard-scattering collisions has a long and rich
history of experimental measurements and theoretical calculations
Brambilla:2010cs . Intense efforts in this area are expected to continue as
the Large Hadron Collider (LHC) makes available data with unprecedented
momentum transfers and statistics.
In recent years, a great deal of theoretical effort has been focused on the
nonrelativistic QCD (NRQCD) factorization approach BBL to calculations of
quarkonium production rates. In this approach, it is conjectured that the
inclusive quarkonium production cross section at large transverse momentum
(${{p}}_{T}$) can be written as a sum of products of short-distance
coefficients and long-distance matrix elements (LDMEs):
$\sigma(H)=\sum_{n}F_{n}(\mu_{\Lambda})\langle 0|{\cal
O}_{n}^{H}(\mu_{\Lambda})|0\rangle.$ (1)
Here, $\mu_{\Lambda}$ is the factorization scale, which is the cutoff of the
effective field theory NRQCD. A short-distance coefficient
$F_{n}(\mu_{\Lambda})$ is, in essence, the partonic cross section to produce a
heavy-quark-antiquark ($Q\bar{Q}$) pair with certain quantum numbers,
convolved with parton distribution functions. The short-distance coefficients
can be calculated as perturbation series in the strong-coupling constant
$\alpha_{s}$. A production LDME ${\langle 0|\cal
O}_{n}^{H}(\mu_{\Lambda})|0\rangle$ is the probability for a $Q\bar{Q}$ pair
with certain quantum numbers to evolve into a particular heavy-quarkonium
state. It is expressed as the vacuum expectation value of a four-fermion
operator
${\cal O}_{n}^{H}(\mu_{\Lambda})=\langle 0|\chi^{\dagger}\kappa_{n}\psi{\cal
P}_{H(P)}\psi^{\dagger}\kappa^{\prime}_{n}\chi|0\rangle,$ (2)
where $\psi^{\dagger}$ and $\chi$ are two-component (Pauli) fields that create
a heavy quark and a heavy antiquark, respectively, and $\kappa_{n}$ and
$\kappa_{n}^{\prime}$ are combinations of Pauli and color matrices.111 It was
pointed out by Nayak, Qiu, and Sterman that gauge invariance requires that the
definitions of the NRQCD LDMEs include Wilson lines that run from the quark
and antiquark fields to infinity Nayak:2005rw ; Nayak:2005rt . For simplicity,
we have omitted these Wilson lines here.
${\cal P}_{H(P)}=\sum_{X}|H(P)+X\rangle\langle H(P)+X|$ (3)
is a projection onto a state consisting of a quarkonium $H$, with four-
momentum $P$, plus anything. ${\cal P}_{H(P)}$ contains a sum over any
quarkonium polarization quantum numbers that are not specified explicitly. The
NRQCD LDMEs are evaluated in the rest frame of the quarkonium, in which
$P=(M,\bm{0})$, where $M$ is the quarkonium mass. In the remainder of this
paper, we suppress the momentum argument of ${\cal P}_{H(P)}$ in NRQCD LDMEs.
The production LDMEs for the evolution of color-singlet $Q\bar{Q}$ pairs into
a quarkonium state are related to the color-singlet quarkonium decay LDMEs.
These color-singlet production LDMEs can be determined from comparison of
theory with quarkonium production or decay data or from lattice QCD
calculations. However, the production LDMEs for the evolution of color-octet
$Q\bar{Q}$ pairs into a quarkonium state can be determined, at least at
present, only through comparison of theory with experimental quarkonium-
production data.
Complete calculations of short-distance coefficients in the NRQCD
factorization approach now exist through next-to-leading order (NLO) in
$\alpha_{s}$ for production of the $J/\psi$ and the $\psi(2S)$ in $e^{+}e^{-}$
collisions, in $ep$ collisions, and in $p\bar{p}$ and $pp$ collisions
Brambilla:2010cs ; Butenschoen:2009zy ; Ma:2010yw ; Butenschoen:2010rq ;
Butenschoen:2010px ; Ma:2010jj ; Butenschoen:2011yh ; Butenschoen:2011ks ;
Butenschoen:2012px ; Chao:2012iv ; Butenschoen:2012qh ; Gong:2012ug . These
calculations include the contributions from all of the color-octet channels
through relative order $v^{4}$, as well as the contribution of the color-
singlet channel at leading order (LO) in $v$. Specifically, the calculations
include the contributions of the ${}^{3}S_{1}$, ${}^{1}S_{0}$ and
${}^{3}P_{J}$ color-octet channels and the ${}^{3}S_{1}$ color-singlet channel
at the leading nontrivial order in $v$ in each channel. Here, $v$ is half the
relative velocity of the heavy quark and the heavy antiquark in the quarkonium
rest frame. $v^{2}\approx 0.22$ for the $J/\psi$, and $v^{2}\approx 0.1$ for
the $\Upsilon$. These theoretical results are generally compatible with
experimental measurements of quarkonium production cross sections. However,
significant discrepancies remain between theoretical predictions for
quarkonium polarization and experimental measurements Chao:2012iv ;
Butenschoen:2012qh ; Gong:2012ug . These discrepancies might point to as-yet-
uncalculated theoretical contributions, to experimental difficulties, to a
failure of convergence of the NRQCD series in $\alpha_{s}$ or $v$, or to a
failure of the NRQCD factorization conjecture itself.
The calculations at NLO in $\alpha_{s}$ have revealed very large corrections
in that order to the ${}^{3}P_{J}$ color-octet channel and the ${}^{3}S_{1}$
color-singlet channel. Large corrections have also been found in a calculation
of the real-emission contributions to $\Upsilon$ hadroproduction at next-to-
next-to-leading order in $\alpha_{s}$ Artoisenet:2008fc . The large
corrections are the result of kinematic enhancements of the higher-order cross
sections at high ${{p}}_{T}$ relative to the LO cross sections. The sizes of
these corrections have cast some doubt on the convergence of the perturbation
series.
It has been suggested recently that the large higher-order corrections to
quarkonium production can be brought under control by re-organizing the
perturbation series according to the ${{p}}_{T}$ behavior of the various
contributions Kang:2011zza . In this approach, the cross section can be shown
to factorize into convolutions of hard-scattering cross sections with
fragmentation functions. The factorization holds up to corrections of relative
order $m_{c}^{4}/{{p}}_{T}^{4}$, where $m_{c}$ is the charm-quark mass. The
factorized cross section consists of a leading contribution, which arises from
single-particle fragmentation into a quarkonium and falls as $1/{{p}}_{T}^{4}$
in the partonic cross section and a first subleading contribution, which
arises from two-particle fragmentation into a quarkonium and falls as
$1/{{p}}_{T}^{6}$ in the partonic cross section. If NRQCD factorization holds,
then the various fragmentation functions can be expressed in terms of a sum of
products of short-distance coefficients and NRQCD LDMEs. This picture has been
shown to account for the large corrections at NLO in $\alpha_{s}$ in the
${}^{3}S_{1}$ color-singlet channel Kang:2011mg .
In this paper, we compute the NRQCD short-distance coefficient for gluon
fragmentation into a ${}^{3}S_{1}$ color-singlet $Q\bar{Q}$ pair in relative
order $v^{4}$. The short-distance coefficients for gluon fragmentation in this
channel have been computed in relative order $v^{0}$ Braaten:1993rw ;
Braaten:1995cj and relative order $v^{2}$ Bodwin:2003wh . In both cases, the
contributions are not important phenomenologically. Nevertheless, it is
worthwhile to consider the order-$v^{4}$ contribution for two reasons. First,
this contribution is interesting theoretically because it is in order $v^{4}$
that the ${}^{3}S_{1}$ color-singlet fragmentation channel first develops soft
divergences in full QCD. As we shall see, these soft divergences in full QCD
correspond to soft divergences in the LDMEs for the ${}^{3}S_{1}$ and
${}^{3}P_{J}$ color-octet channels and cancel in the short-distance
coefficients, as is required by NRQCD factorization. A second motivation for
examining the order-$v^{4}$ contribution is that it is potentially large.
Contributions from gluon fragmentation into the ${}^{3}S_{1}$ and
${}^{3}P_{J}$ color-octet channels are known to be important
phenomenologically. The ${}^{3}S_{1}$ color-singlet channel mixes with these
channels in order $v^{4}$, and the partitioning of the various contributions
is controlled by single and double logarithms of the factorization scale.
Therefore, it is plausible that the order-$v^{4}$ contributions to the color-
singlet channel could be large.
Our method of calculation is based on the Collins-Soper definition
Collins:1981uw of the fragmentation function for a gluon fragmenting into a
quarkonium. We assume that NRQCD factorization holds, that is, that the
fragmentation function can be decomposed into a sum of products of short-
distance coefficients and NRQCD LDMEs. We then compute the full-QCD
fragmentation functions for a gluon fragmenting into free $Q\bar{Q}$ states
with various quantum numbers. The ultimate aim is to match these full-QCD
fragmentation functions to the corresponding NRQCD fragmentation functions in
order to determine the NRQCD short-distance coefficients. Some additional
details of this approach can be found in Ref. Bodwin:2003wh .
This paper is organized as follows. We give the Collins-Soper definition of
the fragmentation function in Sec. 2. Section 3 contains the NRQCD
factorization formula for the fragmentation function and also contains a
discussion of the NRQCD LDMEs and short-distance coefficients that are
relevant through relative order $v^{4}$. The kinematics and variables that we
use in our calculation are described in Sec. 4. In Sec. 5, we discuss the
calculation of the fragmentation processes in full QCD. As we have mentioned,
an important feature of the present calculation is that soft divergences arise
in the ${}^{3}S_{1}$ color-singlet channel in both full QCD and NRQCD. These
divergences ultimately cancel in the short-distance coefficients when we carry
out the matching between full QCD and NRQCD. In both full QCD and NRQCD, we
regulate the divergences dimensionally. In the case of full QCD, we devise
subtractions that remove the divergent terms from the integrand, and we
compute the subtraction contributions analytically. This computation is
described in Sec. 5.3. After we remove the subtraction terms, we calculate the
remainder of the full-QCD contribution in four dimensions, carrying out the
integration numerically. We compute the relevant NRQCD LDMEs for free
$Q\bar{Q}$ states analytically in dimensional regularization. These
calculations are described in Sec. 6. We also determine the evolution
equations for the LDMEs and find a discrepancy with a result in Ref.
Gremm:1997dq . In Sec. 7, we match the NRQCD and full-QCD fragmentation
functions to obtain the short-distance coefficients, and we present numerical
results for them in Sec. 8. Finally, in Sec. 9, we summarize our results.
## 2 Collins-Soper definition of the fragmentation function
Here, and throughout this paper, we use the following light-cone coordinates
for a four-vector $V$:
$\displaystyle V$ $\displaystyle=$
$\displaystyle(V^{+},V^{-},\bm{V}_{\bot})=(V^{+},V^{-},V^{1},V^{2}),$ (4a)
$\displaystyle V^{+}$ $\displaystyle=$ $\displaystyle(V^{0}+V^{3})/\sqrt{2},$
(4b) $\displaystyle V^{-}$ $\displaystyle=$
$\displaystyle(V^{0}-V^{3})/\sqrt{2}.$ (4c)
The scalar product of two four-vectors $V$ and $W$ is then
$V\cdot W=V^{+}W^{-}+V^{-}W^{+}-\bm{V}_{\bot}\cdot\bm{W}_{\bot}.$ (5)
The Collins-Soper definition for the fragmentation function for a gluon
fragmenting into a hadron (quarkonium) $H$ Collins:1981uw is
$\displaystyle D[g\to H](z,\mu_{\Lambda})$ $\displaystyle=$
$\displaystyle\frac{-g_{\mu\nu}z^{d-3}}{2\pi
k^{+}(N_{c}^{2}-1)(d-2)}\int_{-\infty}^{+\infty}dx^{-}e^{-ik^{+}x^{-}}$
$\displaystyle\times\langle
0|G^{+\mu}_{c}(0)\mathcal{E}^{\dagger}(0,0,\bm{0}_{\perp})_{cb}\;\mathcal{P}_{H(P)}\;\mathcal{E}(0,x^{-},\bm{0}_{\perp})_{ba}G^{+\nu}_{a}(0,x^{-},\bm{0}_{\perp})|0\rangle\,.$
Here, $z$ is the fraction of the gluon’s $+$ component of momentum that is
carried by the hadron, $G_{\mu\nu}$ is the gluon field-strength operator, $k$
is the momentum of the field-strength operator, $\mu_{\Lambda}$ is the
factorization scale, and $d=4-2\epsilon$ is the number of space-time
dimensions. There is an implicit average over the color and polarization
states of the initial gluon. The projection $\mathcal{P}_{H(P)}$ is given in
Eq. (3). The fragmentation function is evaluated in the frame in which the
hadron has zero transverse momentum:
$P=[zk^{+},M^{2}/(2zk^{+}),\bm{0}_{\perp}]$. The operator
$\mathcal{E}(0,x^{-},\bm{0}_{\perp})$ is a path-ordered exponential of the
gluon field:
$\mathcal{E}(0,x^{-},\bm{0}_{\perp})_{ba}\;=\;\textrm{P}\exp\left[+ig_{s}\int_{x^{-}}^{\infty}dz^{-}A^{+}(0,z^{-},\bm{0}_{\perp})\right]_{ba},$
(7)
where $g_{s}=\sqrt{4\pi\alpha_{s}}$ is the QCD coupling constant and
$A^{\mu}(x)$ is the gluon field. Both $A_{\mu}$ and $G_{\mu\nu}$ are SU(3)
matrices in the adjoint representation. The expression (LABEL:eq:D-def) is
manifestly gauge invariant. We use the Feynman gauge in our calculation.
The Feynman rules for the perturbative expansion of Eq. (LABEL:eq:D-def) are
given in Ref. Collins:1981uw . The quantity
$\mathcal{E}(0,x^{-},\bm{0}_{\perp})$ appears in the Feynman rules as an
eikonal line. Owing to the charge-conjugation properties of the $Q\bar{Q}$
states that we consider and the Landau-Yang theorem landau-thm ; Yang:1950rg ,
gluon attachments to the eikonal lines from
$\mathcal{E}(0,x^{-},\bm{0}_{\perp})$ do not appear in our calculation. Hence,
we need only the standard QCD Feynman rules, an overall factor
$C_{\rm frag}=\frac{z^{d-3}k^{+}}{2\pi(N_{c}^{2}-1)(d-2)}$ (8)
from Eq. (LABEL:eq:D-def), and the special Feynman rule for the vertex that
creates a gluon and an eikonal line. That vertex is shown in Fig. 1. Its
Feynman rule, in momentum space, is a factor
$+i\left(g^{\nu\alpha}-\frac{Q^{\nu}n^{\alpha}}{k^{+}}\right)\delta_{ab},$ (9)
where $k$ is the sum of the momenta of the gluon and the eikonal line, $Q$ is
the momentum of the gluon, $\alpha$ is the polarization index of the gluon,
and $a$ and $b$ are the color indices, respectively, of the gluon and the
eikonal line. In the absence of interactions with the eikonal lines, $k=Q$.
$n$ is a light-like vector whose components are given by
$n=(0,1,\bm{0}_{\perp})$.
Figure 1: Feynman diagram for the vertex that creates a gluon and an eikonal
line. The circle represents the operator $G_{a}^{+\nu}$, which creates a gluon
with momentum $Q$ and polarization and color indices $\alpha$ and $a$,
respectively. $b$ is the color index for the eikonal line. The operator
momentum $k$ is the sum of $Q$ and the momentum of the eikonal line.
The final-state phase space that is implied by Eq. (LABEL:eq:D-def) is
$d\Phi_{n}=\frac{4\pi
M}{S}\;\delta\left(k^{+}-P^{+}-\sum_{i=1}^{n}k_{i}^{+}\right)\theta(k^{+})\prod_{i=1}^{n}\frac{dk_{i}^{+}}{4\pi
k^{+}_{i}}\,\frac{d^{d-2}\bm{k}_{i\perp}}{(2\pi)^{d-2}}\,\theta(k_{i}^{+}),$
(10)
where $S$ is the statistical factor for identical particles in the final
state, $k_{i}$ is the momentum of the $i$th final-state particle, and the
product is over all of the final-state particles except $H$. We use
nonrelativistic normalization for the state $H$, and so a factor $2M$ appears
in the phase space in order to cancel the relativistic normalization of $H$ in
the definition (LABEL:eq:D-def). We use relativistic normalization for all
particles other than $H$.
## 3 NRQCD factorization
We assume that the fragmentation function for a gluon fragmenting into a
quarkonium $H$ satisfies NRQCD factorization. Then, in analogy with Eq. (1),
we have
$D[g\to H](z)=\sum_{n}d_{n}(z)\langle 0|\mathcal{O}_{n}^{H}|0\rangle,$ (11)
where the $\langle 0|\mathcal{O}_{n}^{H}|0\rangle$ are NRQCD LDMEs and the
$d_{n}(z)$ are the fragmentation short-distance coefficients. We have
suppressed the dependences of $d_{n}(z)$ and $\langle
0|\mathcal{O}_{n}^{H}|0\rangle$ on the factorization scale $\mu_{\Lambda}$. In
discussing specific cases, we use the notation $\langle
0|\mathcal{O}_{n_{v},n^{\prime}}^{H}({}^{2s+1}l_{j}^{[c]})|0\rangle$ for the
LDMEs and the notation $d_{n_{v},n^{\prime}}({}^{2s+1}l_{j}^{[c]})(z)$ for the
short-distance coefficients, where ${}^{2s+1}l_{j}$ is the standard
spectroscopic notation for the angular-momentum quantum numbers of the
corresponding NRQCD operator, and $c$ is the color quantum number of the NRQCD
operator ($1$ or $8$). $n_{v}$ is the order in $v$, relative to the leading
order, of the field operators and derivatives in
$\mathcal{O}_{n_{v},n^{\prime}}^{H}$, excluding factors of $v$ from the
projection onto the final state $H$. $n^{\prime}$ is an integer that is used
to distinguish operators that have the same quantum numbers and order in $v$.
We denote the contribution of order $v^{k}$ to $D[g\to H]$ by $D_{k}[g\to H]$.
We can write the fragmentation functions for gluon fragmentation into free
$Q\bar{Q}$ states as
$D[g\to Q\bar{Q}](z)=\sum_{n}d_{n}(z)\langle
0|\mathcal{O}_{n}^{Q\bar{Q}}|0\rangle.$ (12)
Since the short-distance coefficients $d_{n}(z)$ are independent of the
specifics of the hadronic states, the $d_{n}(z)$ in Eq. (12) are identical to
the $d_{n}(z)$ in Eq. (11). We determine the $d_{n}(z)$ by computing the left
side of Eq. (12) in full QCD and comparing it with the right side, in which
the free $Q\bar{Q}$ LDMEs are computed in NRQCD. Since we choose a
factorization scale $\mu_{\Lambda}$ of order the heavy-quark mass $m$, we can
carry out this computation in perturbation theory. We will denote the
contribution of order $v^{k}$ to $D[g\to Q\bar{Q}]$ by $D_{k}[g\to Q\bar{Q}]$.
If $H$ is a ${}^{3}S_{1}$ quarkonium state, such as the $J/\psi$, then, in LO
in $v$, we must consider the LDME
$\langle 0|\mathcal{O}_{0}^{H}({}^{3}S_{1}^{[1]})|0\rangle=\langle
0|\chi^{\dagger}\sigma^{i}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi|0\rangle.$ (13)
In relative order $v^{2}$, we must consider the LDME
$\langle 0|\mathcal{O}_{2}^{H}({}^{3}S_{1}^{[1]})|0\rangle=\frac{1}{2}\langle
0|\chi^{\dagger}\sigma^{i}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{2}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi+\textrm{H.~{}c.}|0\rangle.$ (14)
In relative order $v^{3}$, we must consider the LDME
$\langle 0|{\cal O}_{0}^{H}({}^{1}S_{0}^{[8]})|0\rangle=\langle
0|\chi^{\dagger}T^{a}\psi\;{\cal P}_{H}\;\psi^{\dagger}T^{a}\chi|0\rangle.$
(15)
In relative order $v^{4}$, we must consider the LDMEs
$\displaystyle\langle 0|{\cal O}_{4,1}^{H}({}^{3}S_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\chi^{\dagger}\sigma^{i}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{2}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{2}\chi|0\rangle,$
(16a) $\displaystyle\langle
0|\mathcal{O}_{4,2}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle
0|\chi^{\dagger}\sigma^{i}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{4}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi+\textrm{H.~{}c.}|0\rangle,$ (16b)
$\displaystyle\langle 0|\mathcal{O}_{4,3}^{H}({}^{3}S_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\langle
0|\chi^{\dagger}\sigma^{i}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}(\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}}\\!\cdot
g_{s}\bm{E}+g_{s}\bm{E}\cdot\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})\chi$
(16c)
$\displaystyle\qquad-\chi^{\dagger}\sigma^{i}(\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}}\\!\cdot
g_{s}\bm{E}+g_{s}\bm{E}\cdot\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi|0\rangle,$ $\displaystyle\langle 0|{\cal
O}_{0}^{H}({}^{3}S_{1}^{[8]})|0\rangle$ $\displaystyle=$ $\displaystyle\langle
0|\chi^{\dagger}\sigma^{i}T^{a}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}T^{a}\chi|0\rangle,$ (16d)
$\displaystyle\langle 0|\mathcal{O}^{H}_{0}({}^{3}P_{0}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{d-1}\langle
0|\chi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}}\cdot\bm{\sigma})\psi\;{\cal
P}_{H}\;\psi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}}\cdot\bm{\sigma})\chi|0\rangle,\phantom{xxx}$
(16e) $\displaystyle\langle 0|\mathcal{O}^{H}_{0}({}^{3}P_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\chi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\phantom{}\\!\\!^{[i}\sigma^{j]})\psi\;{\cal
P}_{H}\;\psi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\phantom{}\\!\\!^{[i}\sigma^{j]})\chi|0\rangle,\phantom{xxx}$
(16f) $\displaystyle\langle 0|\mathcal{O}^{H}_{0}({}^{3}P_{2}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\chi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\\!\\!\phantom{}^{(i}\sigma^{j)})\psi\;{\cal
P}_{H}\;\psi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\\!\\!\phantom{}^{(i}\sigma^{j)})\chi|0\rangle.$
(16g)
Here, the symmetric traceless product is defined by
$A^{(i}B^{j)}=\frac{1}{2}(A^{i}B^{j}+A^{j}B^{i})-\frac{1}{d-1}\,\delta^{ij}A^{k}B^{k},$
(17)
and the antisymmetric product is defined by
$A^{[i}B^{j]}=\frac{1}{2}(A^{i}B^{j}-A^{j}B^{i}).$ (18)
For purposes of our calculation, it is also useful to define
$\displaystyle\langle 0|{\cal O}_{0}^{H}({}^{3}P^{[8]})|0\rangle$
$\displaystyle=$ $\displaystyle\sum_{J=0,\,1,\,2}\langle 0|{\cal
O}_{0}^{H}({}^{3}P_{J}^{[8]})|0\rangle$ (19) $\displaystyle=$
$\displaystyle\langle
0|\chi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{r}\sigma^{n}T^{a}\psi\;{\cal
P}_{H}\;\psi^{\dagger}(-\tfrac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{r}\sigma^{n}T^{a}\chi|0\rangle.$
It was shown in Ref. Bodwin:2002hg that, by making use of the NRQCD equations
of motion, one can express the LDME $\langle
0|\mathcal{O}_{4,3}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ in terms of the LDMEs
$\langle 0|\mathcal{O}_{4,1}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ and $\langle
0|\mathcal{O}_{4,2}^{H}({}^{3}S_{1}^{[1]})|0\rangle$. (Equivalently, one can
eliminate the LDME $\langle
0|\mathcal{O}_{4,3}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ by making use of a field
redefinition Brambilla:2008zg .) Hence, we need not consider $\langle
0|\mathcal{O}_{4,3}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ in our analysis.
In the LDMEs $\langle 0|\mathcal{O}_{4,1}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ and
$\langle 0|\mathcal{O}_{4,2}^{H}({}^{3}S_{1}^{[1]})|0\rangle$, one can replace
$|H+X\rangle\langle H+X|$ in the projector ${\cal P}_{H(M,\bm{0})}$ with
$|H\rangle\langle H|$ (vacuum-saturation approximation), making an error of
relative order $v^{4}$. If one takes this approximation and evaluates the
LDMEs in dimensional regularization in a potential model, then they are equal
Bodwin:2006dn . Since the static potential model is valid up to corrections of
order $v^{2}$, we have
$\langle 0|\mathcal{O}_{4,1}^{H}({}^{3}S_{1}^{[1]})|0\rangle=\langle
0|\mathcal{O}_{4,2}^{H}({}^{3}S_{1}^{[1]})|0\rangle+{\cal O}(v^{2}).$ (20)
Hence, up to corrections of relative order $v^{2}$, only the sum of short-
distance coefficients
$d_{4,1}({}^{3}S_{1}^{[1]})(z)+d_{4,2}({}^{3}S_{1}^{[1]})(z)$ appears in the
fragmentation function.
### 3.1 NRQCD factorization formulas for ${g\to J/\psi}$ through order
$v^{4}$
In summary, we have the following NRQCD factorization formulas for gluon
fragmentation into $J/\psi$ through relative order $v^{4}$.
In relative order $v^{0}$ we have
$D_{0}[g\to J/\psi]=d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle.$ (21)
The short-distance coefficient $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ was
calculated in Refs. Braaten:1993rw ; Braaten:1995cj .
In relative order $v^{2}$ we have
$D_{2}[g\to J/\psi]=d_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]\langle
0|\mathcal{O}_{2}^{J/\psi}({}^{3}S^{[1]})|0\rangle.$ (22)
The short-distance coefficient $d_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ was
calculated in Ref. Bodwin:2003wh .
In relative order $v^{3}$ we have
$D_{3}[g\to J/\psi]=d_{0}[g\to Q\bar{Q}({}^{1}S_{0}^{[8]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{1}S_{0}^{[8]})|0\rangle.$ (23)
The short-distance coefficient $d_{0}[g\to Q\bar{Q}({}^{1}S_{0}^{[1]})]$ was
calculated in Ref. Braaten:1996rp and differs from the short-distance
coefficient $d_{0}[g\to Q\bar{Q}({}^{1}S_{0}^{[8]})]$ in Eq. (23) only by a
color factor, which we provide in Sec. 7.
In relative order $v^{4}$ we have
$\displaystyle D_{4}[g\to J/\psi]$ $\displaystyle=$
$\displaystyle\big{\\{}\,d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]\,\big{\\}}\,\langle
0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$ (24) $\displaystyle+$
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}P^{[8]})|0\rangle$ $\displaystyle+$
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[8]})|0\rangle.$
The short-distance coefficient $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]$ was
calculated at LO in $\alpha_{s}$ in Refs. Braaten:1996rp ; Bodwin:2003wh and
at NLO in $\alpha_{s}$ in Refs. BL:gfrag-NLO ; Lee:2005jw . We verify the LO
calculations in Refs. Braaten:1996rp ; Bodwin:2003wh in the present paper,
giving our result in Sec. 7. We compute $d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]$
in this paper, giving the result in Sec. 7. The short-distance coefficient
$d_{0}[g\to Q\bar{Q}({}^{3}P^{[1]})]$ was calculated in Refs. Braaten:1996rp
and differs from the short-distance coefficient $d_{0}[g\to
Q\bar{Q}({}^{3}P^{[8]})]$ in Eq. (24) only by a color factor. The computation
of the combination of short-distance coefficients $d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]+d_{4,2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ is the
main goal of this paper. The result of that computation is given in Sec. 7.
## 4 Kinematics
In the calculations to follow, in both full QCD and NRQCD, we employ the
following kinematics.
We take the $Q$ and the $\bar{Q}$ to be free (on-shell) states with momenta
$\displaystyle p$ $\displaystyle=$ $\displaystyle\tfrac{1}{2}P+q,$ (25a)
$\displaystyle\bar{p}$ $\displaystyle=$ $\displaystyle\tfrac{1}{2}P-q,$ (25b)
respectively. The heavy quark has three-momentum $\bm{q}$ in the $Q\bar{Q}$
rest frame, and, so, the invariant mass of the $Q\bar{Q}$ state is
$P^{2}=M^{2}=4E^{2},$ (26)
where
$E=\sqrt{m^{2}+\bm{q}^{2}}.$ (27)
We work in the frame in which the transverse momentum of the $Q\bar{Q}$ pair
vanishes. In this frame, the initial-state gluon, the final-state $Q\bar{Q}$
pair and the final-state gluons, respectively, have the momenta
$\displaystyle k$ $\displaystyle=$
$\displaystyle\left(k^{+},k^{-}=\frac{k^{2}+(P_{\perp}/z)^{2}}{2k^{+}},-\frac{\bm{P}_{\perp}}{z}\right),$
(28a) $\displaystyle P$ $\displaystyle=$
$\displaystyle\left(zk^{+},\frac{M^{2}}{2zk^{+}},\bm{0}_{\perp}\right),$ (28b)
$\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\left(z_{1}k^{+},\frac{k_{1\perp}^{2}}{2z_{1}k^{+}},\bm{k}_{1\perp}\right),$
(28c) $\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\left(z_{2}k^{+},\frac{k_{2\perp}^{2}}{2z_{2}k^{+}},\bm{k}_{2\perp}\right),$
(28d)
where we have introduced the longitudinal momentum fractions
$\displaystyle z$ $\displaystyle=$ $\displaystyle\frac{P^{+}}{k^{+}},$ (29a)
$\displaystyle z_{1}$ $\displaystyle=$ $\displaystyle\frac{k_{1}^{+}}{k^{+}},$
(29b) $\displaystyle z_{2}$ $\displaystyle=$
$\displaystyle\frac{k_{2}^{+}}{k^{+}}.$ (29c)
Because of the conservation of four-momentum, $k=P+k_{1}+k_{2}$, the momenta
$k$, $k_{1}$ and $k_{2}$ depend implicitly on $P$, and, therefore, on $q$. We
can make the dependence on $q$ explicit by writing quantities in terms of
dimensionless momenta
$\displaystyle\bar{P}$ $\displaystyle=$ $\displaystyle\frac{P}{\sqrt{P^{2}}},$
(30a) $\displaystyle\bar{k}$ $\displaystyle=$
$\displaystyle\frac{k}{\sqrt{P^{2}}},$ (30b) $\displaystyle\bar{k}_{1}$
$\displaystyle=$ $\displaystyle\frac{k_{1}}{\sqrt{P^{2}}},$ (30c)
$\displaystyle\bar{k}_{2}$ $\displaystyle=$
$\displaystyle\frac{k_{2}}{\sqrt{P^{2}}}.$ (30d)
It is also useful to express the Lorentz invariants in terms of the following
dimensionless variables:
$\displaystyle e_{1}$ $\displaystyle=$ $\displaystyle\bar{k}_{1}\cdot\bar{P},$
(31a) $\displaystyle e_{2}$ $\displaystyle=$
$\displaystyle\bar{k}_{2}\cdot\bar{P},$ (31b) $\displaystyle x$
$\displaystyle=$
$\displaystyle\bar{k}_{1}\cdot\bar{k}_{2}=e_{1}e_{2}(1-\hat{\bm{k}}_{1}\cdot\hat{\bm{k}}_{2}),$
(31c)
where $\hat{\bm{k}}_{i}$ is the unit vector that is parallel to the three-
vector $\bm{k}_{i}$ in the $Q\bar{Q}$ rest frame.
The phase space in Eq. (10) can be expressed in terms of the dimensionless
variables as
$\displaystyle d\Phi_{0}$ $\displaystyle=$ $\displaystyle\frac{4\pi
M}{k^{+}}\,\delta(1-z),$ (32a) $\displaystyle d\Phi_{1}$ $\displaystyle=$
$\displaystyle\frac{4\pi
M^{d-1}}{k^{+}}\,\theta(z_{1})\,\delta(1-z-z_{1})\,\frac{dz_{1}}{4\pi
z_{1}}\frac{d^{d-2}\bar{\bm{k}}_{1\perp}}{(2\pi)^{d-2}},$ (32b) $\displaystyle
d\Phi_{2}$ $\displaystyle=$ $\displaystyle\frac{4\pi
M^{2d-3}}{Sk^{+}}\,\theta(z_{1})\,\theta(z_{2})\,\delta(1-z-z_{1}-z_{2})\,\frac{dz_{1}}{4\pi
z_{1}}\frac{dz_{2}}{4\pi
z_{2}}\frac{d^{d-2}\bar{\bm{k}}_{1\perp}}{(2\pi)^{d-2}}\frac{d^{d-2}\bar{\bm{k}}_{2\perp}}{(2\pi)^{d-2}}.\phantom{xxx}$
(32c)
We have not replaced the overall factor $1/k^{+}$ in Eq. (32) with
$1/(\bar{k}^{+}\sqrt{P^{2}})$, because it ultimately will be cancelled by the
factor $k^{+}$ in $C_{\rm frag}$ in Eq. (8). The ranges of the variables $z$,
$z_{1}$ and $z_{2}$ are completely determined by the $\delta$ and $\theta$
functions. When we expand the fragmentation function in powers of $q$, it is
convenient to make use of the phase space at LO in $q$,
$d\tilde{\Phi}_{n}=d\Phi_{n}\big{|}_{\bm{q}\to\bm{0}},$ (33)
where $\bm{q}\to\bm{0}$ means that, in the phase space in Eq. (32), we replace
$M$ with $2m$. Then $d\Phi_{n}$ can be expressed in terms of
$d\tilde{\Phi}_{n}$ as follows:
$\displaystyle d\Phi_{0}$ $\displaystyle=$
$\displaystyle\frac{E}{m}d\tilde{\Phi}_{0},$ (34a) $\displaystyle d\Phi_{1}$
$\displaystyle=$
$\displaystyle\left(\frac{E}{m}\right)^{3-2\epsilon}d\tilde{\Phi}_{1},$ (34b)
$\displaystyle d\Phi_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{E}{m}\right)^{5-4\epsilon}d\tilde{\Phi}_{2}.$ (34c)
We express our results for the fragmentation contributions in terms of
integrals over the phase spaces $d\tilde{\Phi}_{n}$:
$\tilde{D}[g\to Q\bar{Q}]=D[g\to Q\bar{Q}]\big{|}_{d\Phi_{n}\to
d\tilde{\Phi}_{n}}.$ (35)
The factors of $E/m$ in Eq. (34) are then an additional source of relativistic
corrections.
## 5 Full-QCD calculations
In this section, we compute the relevant fragmentation functions for free
$Q\bar{Q}$ states in full QCD. We have carried out the calculations by writing
independent codes using reduce REDUCE and using the feyncalc package
Mertig:an in mathematica MATH . At each stage of the calculations we have
checked that the independent codes give identical results.
The computations are carried out in $d=4-2\epsilon$ dimensions with
dimensional-regulariza-tion scale $\mu$. We use the modified-minimal-
subtraction ($\overline{\rm MS}$) scheme throughout. Then, in $d=4-2\epsilon$
dimensions, there is a factor $[\mu^{2}\exp({{\gamma}}_{\rm
E})/(4\pi)]^{\epsilon}$ that is associated with each factor of the strong
coupling $\alpha_{s}$, where ${{\gamma}}_{\rm E}$ is the Euler-Mascheroni
constant.
In computing the $Q\bar{Q}$ fragmentation functions, it is convenient to make
use of projection operators for the spin and color states of the $Q\bar{Q}$
pair. The projection operators for a $Q\bar{Q}$ pair in the color-singlet and
color-octet configurations are
$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{c}}}\mathbbm{1},$ (36a)
$\displaystyle\Lambda^{a}_{8}$ $\displaystyle=$ $\displaystyle\sqrt{2}T^{a},$
(36b)
where $\mathbbm{1}$ and $T^{a}$ are the identity matrix and the generator of
the fundamental (triplet) representation of SU(3), $a$ is the adjoint-
representation color index ($a=1,\ldots,N_{c}^{2}-1$), and $N_{c}=3$. Spin-
projection operators at LO in $v$ were first given in Refs. Barbieri:1975am ;
Barbieri:1976fp ; Chang:1979nn ; Guberina:1980dc ; Berger:1980ni . Projectors
accurate to all orders in $v$ were given in Ref. Bodwin:2002hg . For the spin-
triplet state, the projection operator, correct to all orders in $v$, is
$\Lambda(P,q,\epsilon_{S}^{*})=N(\not\\!{\overline{p}}-m)\not\\!{\epsilon}^{*}_{S}\,\frac{\not\\!{P}+2E}{4E}(\not\\!{p}+m),$
(37)
where $\epsilon_{S}$ is the spin polarization of the $Q\bar{Q}$ pair, and
$N=[2\sqrt{2}E(E+m)]^{-1}$. Note that we use nonrelativistic normalization for
the heavy-quark spinors.
The use of the spin projection (37) in $d$ dimensions requires some
justification. It accounts for only the $d-1$ vector polarization states,
which, in the $Q\bar{Q}$ rest frame, correspond to the $d-1$ Pauli matrices
$\sigma_{i}$. In general, in $d$ dimensions, one must consider states that
correspond to products of the $\sigma_{i}$ that are linearly independent of
the $\sigma_{i}$ Braaten:1996rp . These additional states vanish as $\epsilon$
goes to zero. Hence, they can contribute only in conjunction with a pole in
$\epsilon$. The poles in $\epsilon$ in our calculation correspond to soft
divergences. The divergent parts of soft interactions arise from the
convection current on fermions lines, and, hence, do not change the fermion
spin. Therefore, the additional states that correspond to products of the
$\sigma_{i}$ never mix in our calculation with the vector states that
correspond to the $\sigma_{i}$. Consequently, we need consider only the $d-1$
vector states in our calculation.222Some elements of this argument were
presented in Ref. Petrelli:1997ge .
The spin-triplet, color-singlet part of an amplitude $\mathcal{C}$ is
$\mathcal{M}=\textrm{Tr}[\,\mathcal{C}(\Lambda\otimes\Lambda_{1})\,],$ (38)
and the spin-triplet, color-octet part of an amplitude $\mathcal{C}$ is
$\mathcal{M}^{a}=\textrm{Tr}[\,\mathcal{C}(\Lambda\otimes\Lambda_{8}^{a})\,],$
(39)
where the traces are over the Dirac and color indices. The amplitude
$\mathcal{C}$ includes the propagator of the initial gluon, as well as the
associated polarization factor in Eq. (9). In our calculation, the amplitudes
$\mathcal{C}$, $\mathcal{M}$, and $\mathcal{M}^{a}$ are all expressed in terms
of the dimensionless variables in Eq. (30) or invariants that are formed from
them, and so the dependence on $q$ is explicit.
The $S$-wave part of $\mathcal{M}$ (with color index suppressed in the color-
octet case) can be written as an expansion in powers of
$v^{2}=\bm{q}^{2}/m^{2}$:
$\mathcal{M}_{S}=\mathcal{M}_{S0}+\mathcal{M}_{S2}+\mathcal{M}_{S4}+O(\bm{q}^{6}/m^{6}),$
(40)
where
$\displaystyle\mathcal{M}_{S0}$ $\displaystyle=$
$\displaystyle\left(\mathcal{M}\right)_{\bm{q}\to\bm{0}}\;,$ (41a)
$\displaystyle\mathcal{M}_{S2}$ $\displaystyle=$
$\displaystyle\frac{\bm{q}^{2}}{2!(d-1)}\,I^{\alpha\beta}\left(\frac{\partial^{2}\mathcal{M}}{\partial
q^{\alpha}\partial q^{\beta}}\right)_{\bm{q}\to\bm{0}}\;,$ (41b)
$\displaystyle\mathcal{M}_{S4}$ $\displaystyle=$
$\displaystyle\frac{\bm{q}^{4}}{4!(d-1)(d+1)}\,I^{\alpha\beta\gamma\delta}\left(\frac{\partial^{4}\mathcal{M}}{\partial
q^{\alpha}\partial q^{\beta}\partial q^{\gamma}\partial
q^{\delta}}\right)_{\bm{q}\to\bm{0}}\;,$ (41c)
and
$\displaystyle I^{\alpha\beta}$ $\displaystyle=$
$\displaystyle-g^{\alpha\beta}+P^{\alpha}P^{\beta}/(4E^{2}),$ (42a)
$\displaystyle I^{\alpha\beta\gamma\delta}$ $\displaystyle=$ $\displaystyle
I^{\alpha\beta}I^{\gamma\delta}+I^{\alpha\gamma}I^{\beta\delta}+I^{\alpha\delta}I^{\beta\gamma}.$
(42b)
In order to project out the $P$-wave part of the amplitude ${\cal M}$, we
multiply ${\cal M}$ by the $P$-wave orbital-angular-momentum state
$-\sqrt{d-1}\,\epsilon_{L}^{*}\cdot\hat{q}$ and average over the direction of
$\bm{q}$.333 In some calculations in NRQCD, the $P$-wave orbital-angular-
momentum state is normalized as $-\epsilon_{L}^{*}\cdot\hat{q}$. Here,
$\epsilon_{L}$ is the polarization vector for the orbital-angular-momentum
state, and $\hat{q}=(0,\hat{\bm{q}})$ in the rest frame of the $Q\bar{Q}$
pair. Then, the $P$-wave part of the amplitude is
$\mathcal{M}_{P}=\mathcal{M}_{P1}+O(\bm{q}^{3}/m^{3}),$ (43)
where
$\mathcal{M}_{P1}=-\frac{|\bm{q}|}{\sqrt{d-1}}\,\epsilon^{*}_{L\alpha}\,I^{\alpha\beta}\left(\frac{\partial\mathcal{M}}{\partial
q^{\beta}}\right)_{\bm{q}\to\bm{0}}\;.$ (44)
We define squared amplitudes for the color-singlet and color-octet states as
$\displaystyle\mathcal{A}({}^{2s+1}l_{j}^{[1]})$ $\displaystyle=$
$\displaystyle C_{\rm frag}|{\cal M}({}^{2s+1}l_{j}^{[1]})|^{2},$ (45a)
$\displaystyle\mathcal{A}({}^{2s+1}l_{j}^{[8]})$ $\displaystyle=$
$\displaystyle C_{\rm frag}\sum_{a}|{\cal M}^{a}({}^{2s+1}l_{j}^{[8]})|^{2},$
(45b)
where $C_{\rm frag}$ is given in Eq. (8), and it is implicit that there are
sums over the spin and orbital-angular-momentum polarizations of the
$Q\bar{Q}$ states and sums over the polarizations of the initial and final
gluons. Note that
$\sum_{\lambda}\epsilon_{S}^{\alpha*}(\lambda)\epsilon_{S}^{\beta}(\lambda)=\sum_{\lambda}\epsilon_{L}^{\alpha*}(\lambda)\epsilon_{L}^{\beta}(\lambda)=I^{\alpha\beta}.$
(46)
We denote the order-$v^{k}$ contribution to $\mathcal{A}$ by
$\mathcal{A}_{k}$.
### 5.1 $D_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]$
Figure 2: Feynman diagram for the fragmentation process $g\to
Q\bar{Q}({}^{3}S_{1}^{[8]})$. The dashed line represents the final-state cut.
The momenta for $Q$ and $\bar{Q}$ on the left side of the cut are
$p=\tfrac{1}{2}P+q$ and $\bar{p}=\tfrac{1}{2}P-q$, respectively. The momenta
on the right side of the cut are $\tfrac{1}{2}P+q^{\prime}$ and
$\tfrac{1}{2}P-q^{\prime}$, respectively. Here, $|\bm{q}^{\prime}|=|\bm{q}|$
in the rest frame of the $Q\bar{Q}$ pair, but we distinguish the directions of
$\bm{q}$ and $\bm{q}^{\prime}$ in order to be able to project out orbital-
angular-momentum states in the amplitude and its complex conjugate.
The diagram for gluon fragmentation into a ${}^{3}S_{1}$ color-octet
$Q\bar{Q}$ pair at order $v^{0}$ and $\alpha_{s}^{1}$ is shown in Fig. 2. A
straightforward computation yields
$\mathcal{A}_{0}({}^{3}S_{1}^{[8]})=\frac{\alpha_{s}k^{+}}{8m^{4}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}.$ (47)
Carrying out the trivial integration over the phase space $d\tilde{\Phi}_{0}$
in Eqs. (32a) and (34a), we obtain
$D_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[8]})]=\frac{\pi\alpha_{s}}{m^{3}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\delta(1-z).$ (48)
### 5.2 $D_{2}[g\to Q\bar{Q}({}^{3}P^{[8]})]$
Figure 3: One of the four Feynman diagrams for the fragmentation process
$g\to Q\bar{Q}({}^{3}P^{[8]})$. Three additional diagrams can be obtained by
permuting the gluon-fermion vertices on the left and right sides of the cut.
The diagrams for gluon fragmentation into a ${}^{3}P^{[8]}$ color-octet
$Q\bar{Q}$ pair at LO in $\alpha_{s}$ and $v$ are shown in Fig. 3. We obtain
$\mathcal{A}_{1}({}^{3}P^{[8]})=\frac{k^{+}\pi\alpha_{s}^{2}(N_{c}^{2}-4)\bm{q}^{2}z^{1-2\epsilon}}{8N_{c}(d-1)(d-2)m^{8}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{2\epsilon}\sum_{n=0}^{3}\frac{\rho_{n}(z)}{e_{1}^{n}(1+2e_{1})^{2}},$
(49)
where $\rho_{n}(z)$ are given by
$\displaystyle\rho_{0}(z)$ $\displaystyle=$
$\displaystyle(3-4\epsilon)(2-2\epsilon-4z+4z^{2}),$ (50a)
$\displaystyle\rho_{1}(z)$ $\displaystyle=$ $\displaystyle
2\big{[}5-10\epsilon+4\epsilon^{2}-z(5-12\epsilon)+2z^{2}(1-4\epsilon)\big{]},$
(50b) $\displaystyle\rho_{2}(z)$ $\displaystyle=$ $\displaystyle
3-12\epsilon+4\epsilon^{2}+2z(3+4\epsilon)-z^{2}(5+4\epsilon),$ (50c)
$\displaystyle\rho_{3}(z)$ $\displaystyle=$ $\displaystyle-2(1-z)^{2}.$ (50d)
We carry out the integration over the phase space $d\tilde{\Phi}_{1}$ in Eqs.
(32b) and (34b) by making use of the methods that are described in Appendix A.
Then, we obtain
$\displaystyle D_{2}[g\to Q\bar{Q}({}^{3}P^{[8]})]$ $\displaystyle=$
$\displaystyle\frac{8\alpha_{s}^{2}\,\bm{q}^{2}}{(d-1)m^{5}}\,\frac{N_{c}^{2}-4}{4N_{c}}(1-\epsilon)\Gamma(1+\epsilon)\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\left(\frac{\mu^{2}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}$ (51a)
$\displaystyle\times\left[-\frac{1}{2\epsilon_{\rm
IR}}\,\delta(1-z)+f(z)\right],$ where the finite function $f(z)$ is defined by
$\displaystyle f(z)$ $\displaystyle=$
$\displaystyle\left[\frac{1}{(1-z)^{1+2\epsilon}}\right]_{+}+\frac{1}{4(1-\epsilon)^{2}}\Bigg{\\{}\frac{(1-z)^{-\epsilon}-(1-z)^{-2\epsilon}}{\epsilon}\,(13-7z)$
(51b)
$\displaystyle+(1-z)^{-2\epsilon}[10+3z-5z^{2}+2\epsilon(2-2z+z^{2})-4\epsilon^{2}]$
$\displaystyle+\frac{1}{2}(1-z)^{-\epsilon}[-28+15z+\epsilon(8-11z)+4\epsilon^{2}z]\Bigg{\\}}.$
Here, the distribution $[g(z)]_{+}$ is defined by
$\int_{0}^{1}dz\,h(z)[g(z)]_{+}\equiv\int_{0}^{1}dz\,[h(z)-h(1)]g(z).$ (52)
In extracting the pole in Eq. (51), we have made use of the identity,
$\frac{1}{(1-z)^{1+n\epsilon}}=-\frac{1}{n\epsilon}\,\delta(1-z)+\left[\frac{1}{(1-z)^{1+n\epsilon}}\right]_{+},$
(53)
which applies when the domain of integration is $0\leq z\leq 1$. The
expression in Eq. (51b) gives the exact $\epsilon$ dependence. We can expand
the plus function $[1/(1-z)^{1+n\epsilon}]_{+}$ as
$\left[\frac{1}{(1-z)^{1+n\epsilon}}\right]_{+}=\sum_{k=0}^{\infty}\frac{(-n\epsilon)^{k}}{k!}\left[\frac{\log^{k}(1-z)}{1-z}\right]_{+}.$
(54)
In the analysis of $f(z)$, we need to keep only terms through order
$\epsilon^{1}$. Then, we can simplify $f(z)$ as follows:
$\displaystyle f(z)\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left(\frac{1}{1-z}\right)_{+}\\!\\!-2\epsilon\left[\frac{\log(1-z)}{1-z}\right]_{+}\\!\\!+\frac{1}{8}\big{[}\\!-8+21z-10z^{2}+2(13-7z)\log(1-z)\big{]}$
$\displaystyle+$
$\displaystyle\frac{\epsilon}{8}\big{[}(23-16z)z+5(8-11z+4z^{2})\log(1-z)-3(13-7z)\log^{2}(1-z)\big{]}+O(\epsilon^{2}).$
### 5.3 $D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$
Figure 4: One of the 36 Feynman diagrams for the fragmentation process $g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})$. Thirty-five additional diagrams can be obtained
by permuting the gluon-fermion vertices on the left and right sides of the
cut.
The diagrams for gluon fragmentation into a ${}^{3}S_{1}$ color-singlet
$Q\bar{Q}$ pair at LO in $\alpha_{s}$ are shown in Fig. 4. Through relative
order $v^{4}$, the relevant squared amplitudes are
$\displaystyle\mathcal{A}_{0}({}^{3}S_{1}^{[1]})$ $\displaystyle=$
$\displaystyle C_{\rm frag}\,|{\cal M}_{0}({}^{3}S_{1}^{[1]})|^{2},$ (56a)
$\displaystyle\mathcal{A}_{2}({}^{3}S_{1}^{[1]})$ $\displaystyle=$
$\displaystyle 2\,C_{\rm frag}\,{\rm Re}[{\cal M}_{2}({}^{3}S_{1}^{[1]}){\cal
M}_{0}^{*}({}^{3}S_{1}^{[1]})],$ (56b)
$\displaystyle\mathcal{A}_{4}({}^{3}S_{1}^{[1]})$ $\displaystyle=$
$\displaystyle C_{\rm frag}\left\\{|{\cal M}_{2}({}^{3}S_{1}^{[1]})|^{2}+2{\rm
Re}[{\cal M}_{4}({}^{3}S_{1}^{[1]}){\cal
M}_{0}^{*}({}^{3}S_{1}^{[1]})]\right\\}.$ (56c)
The order-$v^{0}$ contribution $\mathcal{A}_{0}({}^{3}S_{1}^{[1]})$ and the
order-$v^{2}$ contribution $\mathcal{A}_{2}({}^{3}S_{1}^{[1]})$ have been
computed previously in Ref. Bodwin:2003wh . Here, we wish to compute the
order-$v^{4}$ contribution $\mathcal{A}_{4}({}^{3}S_{1}^{[1]})$.
The integration of $\mathcal{A}_{4}({}^{3}S_{1}^{[1]})$ over the phase space
contains soft divergences. These can arise when one or both of the final-state
gluons become soft. We identify the divergent part of the integrand in Eq.
(56c) that arises when both gluons become soft by making the substitutions
$k_{1}\to k_{1}\lambda$ and $k_{2}\to k_{2}\lambda$, multiplying by
$\lambda^{4}$, and taking the limit $\lambda\to 0$. The result is
$\mathcal{S}_{12}=\frac{C_{{}_{\\!\mathcal{S}}}(d-2)}{e_{1}^{4}e_{2}^{4}}\left[(d-2)e_{1}^{2}e_{2}^{2}-2e_{1}e_{2}\,x+x^{2}\right],$
(57)
where
$C_{{}_{\\!\mathcal{S}}}=\frac{k^{+}\pi^{2}\alpha_{s}^{3}z^{d-3}}{(d-1)^{2}(d-2)m^{8}}\,\frac{N_{c}^{2}-4}{4N_{c}^{2}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{3\epsilon}\frac{\bm{q}^{4}}{m^{4}}.$ (58)
We identify the divergent part of the integrand in Eq. (56c) that arises when
only $k_{1}$ ($k_{2}$) becomes soft by subtracting $\mathcal{S}_{12}$, making
the substitution $k_{1}\to k_{1}\lambda$ ($k_{2}\to k_{2}\lambda$),
multiplying by $\lambda^{2}$, and taking the limit $\lambda\to 0$. The result
is
$\displaystyle\mathcal{S}_{1}$ $\displaystyle=$
$\displaystyle\frac{C_{{}_{\\!\mathcal{S}}}}{e_{1}^{4}e_{2}^{4}(1+2e_{2})^{2}}\sum_{n=0}^{2}x^{n}\,h_{n}(e_{1},e_{2},z_{1},z_{2}),$
(59a) $\displaystyle\mathcal{S}_{2}$ $\displaystyle=$
$\displaystyle\frac{C_{{}_{\\!\mathcal{S}}}}{e_{2}^{4}e_{1}^{4}(1+2e_{1})^{2}}\sum_{n=0}^{2}x^{n}\,h_{n}(e_{2},e_{1},z_{2},z_{1}),$
(59b)
where the $h_{n}(e_{1},e_{2},z_{1},z_{2})$ are given by
$\displaystyle h_{0}(e_{1},e_{2},z_{1},z_{2})$ $\displaystyle=$ $\displaystyle
2e_{2}(1+2e_{2})\big{\\{}-2(3-2\epsilon)(1-\epsilon)e_{1}^{2}e_{2}^{2}+z_{1}e_{1}e_{2}^{2}+z_{2}e_{1}^{2}e_{2}$
$\displaystyle\times[1-2(1-2\epsilon)e_{2}]-z_{1}^{2}e_{2}^{2}(1+\epsilon
e_{2})+z_{1}z_{2}e_{1}e_{2}[1-2(1-\epsilon)e_{2}]$
$\displaystyle+z_{2}^{2}e_{1}^{2}[-1+(1-3\epsilon)e_{2}+2(1-2\epsilon)e_{2}^{2}\,]\,\big{\\}}+4(3-2\epsilon)(1-\epsilon)e_{1}^{2}e_{2}^{4},$
$\displaystyle h_{1}(e_{1},e_{2},z_{1},z_{2})$ $\displaystyle=$ $\displaystyle
2e_{2}(1+2e_{2})\big{\\{}z_{1}e_{2}(1+2\epsilon
e_{2})+z_{2}e_{1}[1+(3-2\epsilon)e_{2}]-z_{1}z_{2}(1+2\epsilon e_{2}$
$\displaystyle+2\epsilon
e_{2}^{2})-z_{2}^{2}e_{1}[3-2\epsilon+2(1-\epsilon)e_{2}]\big{\\}}+2e_{1}e_{2}^{2}[5-4\epsilon+6(1-\epsilon)e_{2}],$
$\displaystyle h_{2}(e_{1},e_{2},z_{1},z_{2})$ $\displaystyle=$
$\displaystyle-(1+2e_{2})\big{\\{}e_{2}(4-3\epsilon+2\epsilon
e_{2})+2z_{2}e_{2}(1-2\epsilon-2\epsilon e_{2})$ (60c)
$\displaystyle-2z_{2}^{2}(1+e_{2})(1-\epsilon-\epsilon
e_{2})\big{\\}}-(2-\epsilon)e_{2}.$
We carry out the integrations of $\mathcal{S}_{12}$, $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ over the phase space $d\tilde{\Phi}_{2}$ in Eq. (34c) [see
also Eq. (32c)] by making use of the methods that are described in Appendix A.
Then, we obtain
$\displaystyle I[\mathcal{S}_{12}]$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{8\epsilon_{\rm
IR}^{2}}\,\delta(1-z)-\frac{1}{2\epsilon_{\rm
IR}}\left[\frac{1}{(1-z)^{1+4\epsilon}}\right]_{+}+\frac{1-z^{1+2\epsilon}}{2\epsilon_{\rm
IR}(1-z)^{1+4\epsilon}}\right\\}$ (61)
$\displaystyle\times\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{N_{c}^{2}-4}{16N_{c}^{2}}\,\frac{\pi\alpha_{s}}{(d-1)m^{3}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{\bm{q}^{4}}{d-1}$
$\displaystyle\times\left(\frac{\mu^{2}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{2\epsilon}\frac{\Gamma^{2}(1+\epsilon)\Gamma^{2}(1-2\epsilon)}{\Gamma(1-4\epsilon)}(1-\epsilon)(6-2\epsilon-\epsilon^{2}-2\epsilon^{3}),$
where we have used the identity in Eq. (53). If we expand Eq. (61) in powers
of $\epsilon$, then we find that
$\displaystyle I[\mathcal{S}_{12}]$ $\displaystyle=$
$\displaystyle\bigg{\\{}\frac{1}{8\epsilon_{\rm
IR}^{2}}\,\delta(1-z)-\frac{1}{2\epsilon_{\rm
IR}}\left[\delta(1-z)\left(\frac{1}{3}-\log\frac{\mu}{2m}\right)+\left(\frac{1}{1-z}\right)_{+}\\!\\!-1\right]$
(62)
$\displaystyle\;+\delta(1-z)\bigg{(}\frac{1-3\pi^{2}}{48}-\frac{2}{3}\log\frac{\mu}{2m}+\log^{2}\frac{\mu}{2m}\bigg{)}+\left(\frac{1}{1-z}\right)_{+}\\!\\!\left(\frac{2}{3}-2\log\frac{\mu}{2m}\right)$
$\displaystyle\;+2\left[\frac{\log(1-z)}{1-z}\right]_{+}\\!\\!-2\left(\frac{1}{3}-\log\frac{\mu}{2m}\right)-\frac{z\log
z}{1-z}-2\log(1-z)\bigg{\\}}$
$\displaystyle\times\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{N_{c}^{2}-4}{16N_{c}^{2}}\,\frac{\pi\alpha_{s}}{(d-1)m^{3}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{6\bm{q}^{4}}{d-1}+O(\epsilon).$
We also find that
$\displaystyle I[\mathcal{S}_{1}]$ $\displaystyle=$ $\displaystyle
I[\mathcal{S}_{2}]=\left(-\frac{\tau_{1}}{2\epsilon_{\rm
IR}}+\tau_{0}\right)\left(\frac{\mu^{2}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{2\epsilon}\frac{z^{-2+2\epsilon}(1-z)^{-4\epsilon}\Gamma^{2}(1+\epsilon)}{48(1-\epsilon)}$
(63) $\displaystyle\times$ $\displaystyle\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{N_{c}^{2}-4}{16N_{c}^{2}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{\pi\alpha_{s}}{(d-1)m^{3}}\frac{\bm{q}^{4}}{d-1}+O(\epsilon),$
where
$\displaystyle\tau_{0}$ $\displaystyle=$ $\displaystyle
3\bigg{\\{}12(13-7z)z^{2}\,\text{Li}_{2}(1-z)+3(7z-13)z^{2}\log^{2}(1-z)$
(64a) $\displaystyle-2z\left[z^{3}-(43+7\pi^{2})z^{2}+6(2z+9)z^{2}\log
z+(48+13\pi^{2})z-42\right]$
$\displaystyle+3(8z^{4}-7z^{3}+34z^{2}-44z+28)\log(1-z)\bigg{\\}},$
$\displaystyle{}\tau_{1}$ $\displaystyle=$ $\displaystyle
18z^{2}\left[z(21-10z)+2(13-7z)\log(1-z)\right].$ (64b)
Here, $\text{Li}_{2}(z)$ is the Spence function, which is defined by
$\text{Li}_{2}(x)=-\int_{0}^{x}\,dt\,\frac{\log(1-t)}{t}=\sum_{k=1}^{\infty}\frac{x^{k}}{k^{2}}.$
(65)
The fragmentation-function contribution of
$\mathcal{A}_{4}({}^{3}S_{1}^{[1]})$ is then
$\tilde{D}_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]=\tilde{D}_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}+I[\mathcal{S}_{12}]+I[\mathcal{S}_{1}]+I[\mathcal{S}_{2}],$ (66)
where
$\tilde{D}_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}=\int
d\tilde{\Phi}_{2}[\mathcal{A}_{4}({}^{3}S_{1}^{[1]})-\mathcal{S}_{12}-\mathcal{S}_{1}-\mathcal{S}_{2}].$
(67)
Since $\tilde{D}_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}$ contains
no soft divergences, we compute it in $d=4$ dimensions, using numerical
integration over the phase space. This computation is described in Sec. 8.
As was mentioned previously, $D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}$ also contains contributions that arise from the difference between
the fully relativistic phase space $d\Phi_{2}$ and the order-$v^{0}$ phase
space $d\tilde{\Phi}_{2}$ [Eq. (34c)]. Hence, we write
$D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]=D_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}+I[\mathcal{S}_{12}]+I[\mathcal{S}_{1}]+I[\mathcal{S}_{2}],$ (68)
where
$\displaystyle D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}$
$\displaystyle=$ $\displaystyle\tilde{D}_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}+\frac{5\bm{q}^{2}}{2m^{2}}\tilde{D}_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]$ (69)
$\displaystyle+\frac{15\bm{q}^{4}}{8m^{4}}\tilde{D}_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})],$ $\displaystyle\tilde{D}_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]$ $\displaystyle=$
$\displaystyle\int\\!d\tilde{\Phi}_{2}\,\mathcal{A}_{0}({}^{3}S_{1}^{[1]}),$
(70a) $\displaystyle\tilde{D}_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$
$\displaystyle=$
$\displaystyle\int\\!d\tilde{\Phi}_{2}\,\mathcal{A}_{2}({}^{3}S_{1}^{[1]}),$
(70b)
and we have used the expansion
$\left(\frac{E}{m}\right)^{5}=1+\frac{5\bm{q}^{2}}{2m^{2}}+\frac{15\bm{q}^{4}}{8m^{4}}+O(\bm{q}^{6}/m^{6}).$
(71)
$\tilde{D}_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ and $\tilde{D}_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]$ are finite, and they have been evaluated in
$d=4$ dimensions in Ref. Bodwin:2003wh . We have checked those calculations,
computing the phase-space integrations in $\tilde{D}_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]$ and $\tilde{D}_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]$ numerically.
## 6 NRQCD LDMEs
In this section we compute the NRQCD LDMEs for free
$Q\bar{Q}({}^{3}S_{1}^{[1]})$ states that are relevant through relative order
$v^{4}$. These computations are carried out in each case at the leading
nontrivial order in $\alpha_{s}$ in $d=4-2\epsilon$ dimensions, with
dimensional-regularization scale $\mu$. We remind the reader that, because we
use the $\overline{\rm MS}$ scheme in computing the QCD corrections to NRQCD
LDMEs, there is a factor $[\mu^{2}\exp({{\gamma}}_{\rm E})/(4\pi)]^{\epsilon}$
that is associated with each factor of the strong coupling $\alpha_{s}$ in $d$
dimensions.
### 6.1 Order $\alpha_{s}^{0}$
The matrix elements of the $Q\bar{Q}$ NRQCD operators at order
$\alpha_{s}^{0}$ are normalized as
$\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{1}S_{0}^{[8]})}({}^{1}S_{0}^{[8]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle(N_{c}^{2}-1),$ (72a) $\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle 2(d-1)N_{c},$ (72b) $\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle(d-1)(N_{c}^{2}-1),$ (72c)
$\displaystyle\langle
0|\mathcal{O}_{2}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\bm{q}^{2}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)},$
(72d) $\displaystyle\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\bm{q}^{4}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)},$
(72e) $\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\bm{q}^{2}(d-1)(N_{c}^{2}-1),$ (72f)
where a sum over the final-state polarizations is implied. The superscript
$(k)$ indicates the order in $\alpha_{s}$.
### 6.2 Order $\alpha_{s}$
#### 6.2.1 ${}^{3}P^{[8]}\to{}^{3}S_{1}^{[1]}$
Figure 5: One of the four Feynman diagrams for the computation of the LDME
$\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}[Q\bar{Q}({}^{3}P^{[8]})]|0\rangle^{(1)}$.
The solid circles represent the $Q\bar{Q}$ operators in the LDME. As in the
full-QCD calculation, we take the free $Q$ and $\bar{Q}$ states to have
momenta $\tfrac{1}{2}P+q$ and $\tfrac{1}{2}P-q$ on the left side of the cut
and $\tfrac{1}{2}P+q^{\prime}$ and $\tfrac{1}{2}P-q^{\prime}$ on the right
side of the cut, where, $|\bm{q}^{\prime}|=|\bm{q}|$ in the rest frame of the
$Q\bar{Q}$ pair, but we distinguish the directions of $\bm{q}$ and
$\bm{q}^{\prime}$ in order to be able to project out orbital-angular-momentum
states in the amplitude and its complex conjugate. Three additional diagrams
can be obtained by permuting the gluon-fermion and operator vertices on the
left and right sides of the cut.
In order $\alpha_{s}$, the ${}^{3}P^{[8]}$ $Q\bar{Q}$ operator can couple to
the ${}^{3}S_{1}^{[1]}$ state through the diagrams that are shown in Fig.
5.444We suppress Wilson lines in diagrams involving NRQCD production operators
Nayak:2005rw ; Nayak:2005rt because the diagrams involving interactions with
Wilson lines vanish for the orders in $\alpha_{s}$ and the operator and final-
state quantum numbers that we consider.
The set of diagrams in Fig. 5 is gauge invariant. We find it convenient to
work in the $Q\bar{Q}$ center-of-momentum frame and to compute the diagrams in
the Coulomb gauge. Then, the real gluons in the final state must be
transverse. Through relative order $v^{2}$, a straightforward computation in
dimensional regularization gives
$\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}=\left(\mathcal{M}_{a}^{(1)}+\mathcal{M}_{b}^{(1)}+\mathcal{M}_{c}^{(1)}+\mathcal{M}_{d}^{(1)}\right)_{{}^{3}S_{1}^{[1]}},$
(73)
where
$\displaystyle\mathcal{M}_{a}^{(1)}$ $\displaystyle=$
$\displaystyle\pi\alpha_{s}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\xi^{\dagger}\sigma^{k}\eta\,\eta^{\dagger}\sigma^{k}\xi$
(74a)
$\displaystyle\times\int\frac{d^{d-1}\bm{k}}{2|\bm{k}|(2\pi)^{d-1}}\frac{(k+2q)^{l}(k+2q)^{i}(\delta^{ij}-\hat{\bm{k}}^{i}\hat{\bm{k}}^{j})(k+2q^{\prime})^{j}(k+2q^{\prime})^{l}}{(\bm{k}^{2}+2\bm{k}\cdot\bm{q}+2m|\bm{k}|)(\bm{k}^{2}+2\bm{k}\cdot\bm{q}^{\prime}+2m|\bm{k}|)},\phantom{xxx}$
$\displaystyle\mathcal{M}_{b}^{(1)}$ $\displaystyle=$
$\displaystyle\pi\alpha_{s}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\xi^{\dagger}\sigma^{k}\eta\,\eta^{\dagger}\sigma^{k}\xi$
(74b)
$\displaystyle\times\int\frac{d^{d-1}\bm{k}}{2|\bm{k}|(2\pi)^{d-1}}\frac{(-k+2q)^{l}(-k+2q)^{i}(\delta^{ij}-\hat{\bm{k}}^{i}\hat{\bm{k}}^{j})(k+2q^{\prime})^{j}(k+2q^{\prime})^{l}}{(\bm{k}^{2}-2\bm{k}\cdot\bm{q}+2m|\bm{k}|)(\bm{k}^{2}+2\bm{k}\cdot\bm{q}^{\prime}+2m|\bm{k}|)},\phantom{xxx}$
$\displaystyle\mathcal{M}_{c}^{(1)}$ $\displaystyle=$
$\displaystyle\pi\alpha_{s}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\xi^{\dagger}\sigma^{k}\eta\,\eta^{\dagger}\sigma^{k}\xi$
(74c)
$\displaystyle\times\int\frac{d^{d-1}\bm{k}}{2|\bm{k}|(2\pi)^{d-1}}\frac{(-k+2q)^{l}(-k+2q)^{i}(\delta^{ij}-\hat{\bm{k}}^{i}\hat{\bm{k}}^{j})(-k+2q^{\prime})^{j}(-k+2q^{\prime})^{l}}{(\bm{k}^{2}-2\bm{k}\cdot\bm{q}+2m|\bm{k}|)(\bm{k}^{2}-2\bm{k}\cdot\bm{q}^{\prime}+2m|\bm{k}|)},\phantom{xxx}$
$\displaystyle\mathcal{M}_{d}^{(1)}$ $\displaystyle=$
$\displaystyle\pi\alpha_{s}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\xi^{\dagger}\sigma^{k}\eta\,\eta^{\dagger}\sigma^{k}\xi$
(74d)
$\displaystyle\times\int\frac{d^{d-1}\bm{k}}{2|\bm{k}|(2\pi)^{d-1}}\frac{(k+2q)^{l}(k+2q)^{i}(\delta^{ij}-\hat{\bm{k}}^{i}\hat{\bm{k}}^{j})(-k+2q^{\prime})^{j}(-k+2q^{\prime})^{l}}{(\bm{k}^{2}+2\bm{k}\cdot\bm{q}+2m|\bm{k}|)(\bm{k}^{2}-2\bm{k}\cdot\bm{q}^{\prime}+2m|\bm{k}|)}.\phantom{xxx}$
In Eq. (74), $\xi$ and $\eta^{\dagger}$ are the Pauli spinors for the free $Q$
and $\bar{Q}$ states, respectively. The subscript ${}^{3}S_{1}^{[1]}$ in Eq.
(73) indicates that the bispinors $\xi^{\dagger}\eta$ and $\eta^{\dagger}\xi$
are in color-singlet, spin-triplet states and that we project onto $S$-wave
states by averaging over the directions of $\bm{q}$ and $\bm{q}^{\prime}$. A
sum over the polarizations of the spin-triplet $Q\bar{Q}$ pair is implicit.
We expand the integrands in Eq. (74) in powers of $1/m$. In dimensional
regularization, only the leading power contributes because the expressions for
higher powers in $1/m$ produce power-divergent, homogeneous integrals.555This
approach was first used in Appendix B of Ref. BBL . It has been discussed
subsequently in Refs. Braaten:1996rp ; Manohar:1997qy ; Beneke:1997av . In
Ref. Bodwin:1998mn , it was pointed out that this approach allocates
contributions that are infrared finite to the short-distance coefficients,
rather than to the LDMEs, and is, therefore, the NRQCD analogue of the
standard methods for computing dimensionally regulated short-distance
coefficients for hard-scattering processes in collinear factorization in QCD.
The result is
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$
$\displaystyle=$
$\displaystyle\frac{8\pi\alpha_{s}}{m^{2}}\,\frac{d-2}{d-1}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{N_{c}^{2}-1}{4N_{c}^{2}}\left(\xi^{\dagger}q^{\prime
i}q^{\prime
l}\sigma^{k}\eta\,\eta^{\dagger}q^{i}q^{l}\sigma^{k}\xi\right)_{{}^{3}S_{1}^{[1]}}$
(75) $\displaystyle\times\int\frac{d^{d-1}\bm{k}}{|\bm{k}|^{3}(2\pi)^{d-1}}.$
The remaining integration over the spatial components of $k$ is
straightforward and yields
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}=\frac{8\alpha_{s}c(\epsilon)}{3\pi
m^{2}}\left(\frac{1}{2\epsilon_{\rm UV}}-\frac{1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-1}{4N_{c}^{2}}\left(\xi^{\dagger}q^{\prime
i}q^{\prime
l}\sigma^{k}\eta\,\eta^{\dagger}q^{i}q^{l}\sigma^{k}\xi\right)_{{}^{3}S_{1}^{[1]}}.$
Here, we have separated the ultraviolet (UV) and infrared (IR) divergent
contributions of the scaleless integral. The quantity $c(\epsilon)$ is given
by
$c(\epsilon)=\frac{(\mu^{2}e^{{{\gamma}}_{\rm
E}})^{\epsilon}(1-\epsilon)\Gamma(\tfrac{1}{2})}{\left(1-\frac{2}{3}\epsilon\right)(1-2\epsilon)\Gamma(\tfrac{1}{2}-\epsilon)}.$
(77)
Note that
$c(0)=1.$ (78)
Now,
$\displaystyle\frac{c(\epsilon)}{\epsilon_{\rm UV}}$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{\rm UV}}+\frac{c(\epsilon)-1}{\epsilon},$
(79a) $\displaystyle\frac{c(\epsilon)}{\epsilon_{\rm IR}}$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{\rm IR}}+\frac{c(\epsilon)-1}{\epsilon},$
(79b)
where we have dropped the subscripts “UV” and “IR” in the second terms of the
above equations because those terms are finite. Hence, we have
$c(\epsilon)\left(\frac{1}{2\epsilon_{\rm UV}}-\frac{1}{2\epsilon_{\rm
IR}}\right)=\frac{1}{2\epsilon_{\rm UV}}-\frac{1}{2\epsilon_{\rm IR}}.$ (80)
Therefore,
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}=\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{1}{2\epsilon_{\rm UV}}-\frac{1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-1}{4N_{c}^{2}}\left(\xi^{\dagger}q^{\prime
i}q^{\prime
l}\sigma^{k}\eta\,\eta^{\dagger}q^{i}q^{l}\sigma^{k}\xi\right)_{{}^{3}S_{1}^{[1]}}.$
We renormalize $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$ in the
$\overline{\rm MS}$ scheme. In the $\overline{\rm MS}$ scheme, one constructs
the counterterm for a UV-divergent subdiagram by subtracting the poles in
$\epsilon_{\rm UV}$ that appear in that subdiagram.666In some versions of the
$\overline{\rm MS}$ scheme, one subtracts constants, as well as poles, in
constructing the counterterms. These constants are accounted for in method
that we use in this paper by the factors $(\mu^{2}e^{\gamma_{\rm
E}})^{2\epsilon}$ that are associated with $g_{s}^{2}$. In order to insure
that the counterterm contribution to an LDME removes precisely the
contribution that is proportional to the UV divergence in the divergent
subdiagram, one must compute all factors that are external to the divergent
subdiagram, such as the projections of external momenta onto particular
angular-momentum states, in $d=4-2\epsilon$ dimensions. Hence, we find that
the $\overline{\rm MS}$-counterterm contribution to $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$ is
$\displaystyle\delta\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$
$\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{-1}{2\epsilon_{\rm
UV}}\right)\frac{N_{c}^{2}-1}{4N_{c}^{2}}\left(\xi^{\dagger}q^{\prime
i}q^{\prime
l}\sigma^{k}\eta\,\eta^{\dagger}q^{i}q^{l}\sigma^{k}\xi\right)_{{}^{3}S_{1}^{[1]}}$
(82) $\displaystyle=$
$\displaystyle\left[Z^{(1)}({}^{3}P^{[8]}\to{}^{3}S_{1}^{[1]})-1\right]\left(\xi^{\dagger}q^{\prime
i}q^{\prime
l}\sigma^{k}\eta\,\eta^{\dagger}q^{i}q^{l}\sigma^{k}\xi\right)_{{}^{3}S_{1}^{[1]}}.\phantom{xxxxx}$
Then, we have
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}+\delta\langle
0|{\cal O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$
(83) $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{-1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-1}{4N_{c}^{2}}\left(\xi^{\dagger}q^{\prime
i}q^{\prime
l}\sigma^{k}\eta\,\eta^{\dagger}q^{i}q^{l}\sigma^{k}\xi\right)_{{}^{3}S_{1}^{[1]}}.\phantom{xxxxx}$
Now we extract the $S$-wave part by averaging over the angles of $\bm{q}$ and
$\bm{q}^{\prime}$:
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{-1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\frac{1}{d-1}\langle 0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}.$
#### 6.2.2 ${}^{3}S_{1}^{[8]}\to{}^{3}P^{[8]}$
In order $\alpha_{s}$, the ${}^{3}S_{1}^{[8]}$ $Q\bar{Q}$ operator can couple
to the $Q\bar{Q}({}^{3}P^{[8]})$ state. By carrying out a calculation that is
very similar to the one in the preceding section, we find that
$Z^{(1)}({}^{3}S_{1}^{[8]}\to{}^{3}P^{[8]})-1=\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{-1}{2\epsilon_{\rm UV}}\right)\frac{N_{c}^{2}-4}{4N_{c}}$
(85)
and that
$\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}=\frac{8\alpha_{s}}{3\pi m^{2}}\left(\frac{-1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-4}{4N_{c}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)}.\phantom{xxx}$
(86)
The equivalent result for NRQCD decay LDMEs was obtained in Ref.
Petrelli:1997ge .
### 6.3 Order $\alpha_{s}^{2}$
Figure 6: One of the 36 Feynman diagrams for computing the LDME $\langle
0|\mathcal{O}_{n}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}[Q\bar{Q}({}^{3}S_{1}^{[8]})]|0\rangle^{(2)}$.
The solid circles represent the $Q\bar{Q}$ operators in the LDME. Thirty-five
additional diagrams can be obtained by permuting the gluon-fermion and
operator vertices on the left and right sides of the cut.
In order $\alpha_{s}^{2}$, the ${}^{3}S_{1}^{[8]}$ $Q\bar{Q}$ operator can
couple to the $Q\bar{Q}({}^{3}S_{1}^{[1]})$ state through the diagrams that
are shown in Fig. 6. As in the order-$\alpha_{s}$ case, we expand the
integrands for these diagrams in powers of $1/m$ and, again, only the leading
power contributes in dimensional regularization. Now we combine contributions
that differ in the order of the gluon vertices on a quark or an antiquark line
by making use of the fact that the color factor for the color-singlet
contribution is symmetric under the interchange of the gluon color indices and
by making use of the identity
$\frac{1}{|\bm{k}|}\frac{1}{|\bm{k}|+|\bm{\ell}|}+\frac{1}{|\bm{\ell}|}\frac{1}{|\bm{k}|+|\bm{\ell}|}=\frac{1}{|\bm{k}|}\frac{1}{|\bm{\ell}|}.$
(87)
After we combine the contributions of all of the diagrams in this way, the
loop integrations decouple, and we have
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
$\displaystyle=$
$\displaystyle\frac{(4\pi\alpha_{s})^{2}}{m^{4}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{2\epsilon}\frac{(N_{c}^{2}-4)(N_{c}^{2}-1)}{8N_{c}^{3}}$ (88)
$\displaystyle\times\left(\xi^{\dagger}q^{\prime i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{j}q^{s}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}}$
$\displaystyle\times\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\frac{\delta^{ij}-\hat{\bm{k}}^{i}\hat{\bm{k}}^{j}}{|\bm{k}|^{3}}\int\frac{d^{d-1}\ell}{(2\pi)^{d-1}}\frac{\delta^{rs}-\hat{\bm{\ell}}^{r}\hat{\bm{\ell}}^{s}}{|\bm{\ell}|^{3}}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{1}{2\epsilon_{\rm UV}}-\frac{1}{2\epsilon_{\rm
IR}}\right)\right]^{2}\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}$
$\displaystyle\times\left(\xi^{\dagger}q^{\prime i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{i}q^{r}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}},$
where we have made use of Eq. (80).
Figure 7: One of the four Feynman diagrams for computing the one-loop
correction to the counterterm $Z({}^{3}S_{1}^{[8]}\to{}^{3}P^{[8]})-1$. The
symbols $\otimes$ represent the counterterm
$Z({}^{3}S_{1}^{[8]}\to{}^{3}P^{[8]})-1$. Three additional diagrams can be
obtained by permuting the gluon-fermion and counterterm vertices on the left
and right sides of the cut.
We carry out the renormalization of $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$ in the
$\overline{\rm MS}$ scheme. First, we add the contribution of the one-loop
diagrams involving the counterterm
$Z^{(1)}({}^{3}P^{[8]}\to{}^{3}S_{1}^{[1]})-1$, which are shown in Fig. 7. The
contribution of these diagrams is
$\delta_{1}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}=\left[Z^{(1)}({}^{3}S_{1}^{[8]}\to{}^{3}P^{[8]})-1\right]\langle
0|{\cal O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)},$
(89)
where $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$ is given
in Eq. (6.2.1) and $Z^{(1)}({}^{3}S_{1}^{[8]}\to{}^{3}P^{[8]})-1$ is given in
Eq. (85). Thus,
$\displaystyle\delta_{1}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
$\displaystyle=$ $\displaystyle-\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{1}{2\epsilon_{\rm UV}}-\frac{1}{2\epsilon_{\rm
IR}}\right)\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}$ (90)
$\displaystyle\times\frac{8\alpha_{s}}{3\pi m^{2}}\,\frac{1}{2\epsilon_{\rm
UV}}\left(\xi^{\dagger}q^{\prime i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{i}q^{r}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}}.$
Adding this counterterm contribution to Eq. (88), we obtain
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}+\delta_{1}\langle
0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
$\displaystyle=\;\frac{1}{8}\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\biggl{(}\frac{-1}{\epsilon_{\rm
UV}^{2}}+\frac{1}{\epsilon_{\rm
IR}^{2}}\biggr{)}\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}\left(\xi^{\dagger}q^{\prime
i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{i}q^{r}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}}.\phantom{xxx}$
(91)
We see that the counterterm contribution removes the cross term between the
pole in $\epsilon_{\rm UV}$ and the pole in $\epsilon_{\rm IR}$. Hence, the
remaining overall UV divergence is not coupled to infrared contributions, as
is required for the consistency of the renormalization program. There are no
single poles in $\epsilon_{\rm UV}$. The remaining double pole in
$\epsilon_{\rm UV}$ is removed by adding the $\overline{\rm MS}$-counterterm
contribution
$\displaystyle\delta_{2}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
$\displaystyle=$ $\displaystyle\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{1}{8\epsilon_{\rm
UV}^{2}}\,\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}$
$\displaystyle\times\left(\xi^{\dagger}q^{\prime i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{i}q^{r}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}}$
$\displaystyle=$
$\displaystyle\left[Z^{(2)}({}^{3}S_{1}^{[8]}\to{}^{3}S_{1}^{[1]})-1\right]\left(\xi^{\dagger}q^{\prime
i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{i}q^{r}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}}.$
Adding this counterterm contribution to Eq. (91), we obtain
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$ $\displaystyle=$ $\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}+\delta_{1}\langle
0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$ (93)
$\displaystyle+\delta_{2}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
$\displaystyle=$ $\displaystyle\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{1}{8\epsilon_{\rm
IR}^{2}}\,\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}$
$\displaystyle\times\left(\xi^{\dagger}q^{\prime i}q^{\prime
r}\sigma^{n}\eta\,\eta^{\dagger}q^{i}q^{r}\sigma^{n}\xi\right)_{{}^{3}S_{1}^{[1]}}.$
We evaluate the $S$-wave part by carrying out the average over the directions
of $\bm{q}$ and $\bm{q}^{\prime}$, with the result
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$ $\displaystyle=$ $\displaystyle\frac{1}{8\epsilon_{\rm
IR}^{2}}\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}$ (94)
$\displaystyle\times\frac{1}{d-1}\langle 0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}.$
### 6.4 Renormalization-group evolution
In the renormalized LDMEs, we can identify the dimensional-regularization
scale $\mu$ with the NRQCD factorization scale $\mu_{\Lambda}$. We now work
out the renormalization-group evolution with respect to $\mu_{\Lambda}$ of the
LDMEs $\langle 0|{\cal O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}$
$({}^{3}P^{[8]})|0\rangle_{\overline{\rm MS}}^{(1)}$, $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$, and $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$. First, we take $d/d\log{\mu_{\Lambda}}$ of Eq.
(LABEL:3pj3s1msbar):
$\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{d\alpha_{s}}{d\log\mu_{\Lambda}}\frac{d}{d\alpha_{s}}\langle
0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\,\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\frac{1}{d-1}\langle 0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)},$
where we have used
$\frac{d\alpha_{s}}{d\log\mu_{\Lambda}}=-2\epsilon\alpha_{s}+O(\alpha_{s}^{2}).$
(96)
The result in Eq. (LABEL:3pj3s1-evo) agrees with the corresponding results of
Refs. Gremm:1997dq ; z-g-he .
Similarly, taking $d/d\log{\mu_{\Lambda}}$ of Eq. (86), we obtain
$\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\frac{N_{c}^{2}-4}{4N_{c}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)}.\phantom{xxx}$
(97)
Taking $d/d\log{\mu_{\Lambda}}$ of Eq. (94), we also obtain
$\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$ $\displaystyle=$ $\displaystyle\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\bigg{(}\frac{-1}{2\epsilon_{\rm
IR}}\bigg{)}\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}$ (98)
$\displaystyle\times\frac{1}{d-1}\langle 0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}.$
Substituting Eq. (LABEL:3pj3s1msbar) into the right side of Eq. (98) we find
that
$\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\,\frac{N_{c}^{2}-4}{4N_{c}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}.$ (99)
Equations (97) and (99) agree with the result in Eq. (B19b) of Ref. BBL at
the leading nontrivial order in $v$ and with the corresponding result in Ref
z-g-he , but disagree with the corresponding result in Ref. Gremm:1997dq .
## 7 NRQCD matching and short-distance coefficients
In the discussion to follow, and in remainder of this paper, the short-
distance coefficients $d_{n}$ that appear are always in the
$\overline{\textrm{MS}}$ scheme. For brevity, we do not indicate the scheme
explicitly.
In relative order $v^{4}$, the NRQCD factorization equation that relates full
QCD and NRQCD for the fragmentation of a gluon into a ${}^{3}S_{1}^{[1]}$
$Q\bar{Q}$ pair is
$\displaystyle D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ $\displaystyle=$
$\displaystyle\\{d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}\\}$ (100) $\displaystyle\times\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
$\displaystyle+$ $\displaystyle d_{0}[g\to
Q\bar{Q}({}^{3}P^{[8]})]^{(2)}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$
$\displaystyle+$ $\displaystyle d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)},$
where we have shown the contributions at the leading nontrivial order in
$\alpha_{s}$, namely, $\alpha_{s}^{3}$, and the LDME $\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
means either $\langle
0|\mathcal{O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
or $\langle 0|\mathcal{O}_{4,2}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}$
$({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$, since they are equal at the present
order of interest in $v$, as we have explained in Sec. 3. We wish to use this
matching equation to determine the sum of short-distance coefficients
$d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$. We have already computed the $Q\bar{Q}$
LDMEs on the right side of Eq. (100). In order to fix the short-distance
coefficients in the second and third terms on the right side of Eq. (100), we
make use of a matching equation at relative order $v^{2}$, which is
$\displaystyle D_{2}[g\to Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$ $\displaystyle=$
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]^{(2)}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)}$
(101) $\displaystyle+$ $\displaystyle d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle^{(1)},$
and a matching equation at relative order $v^{0}$, which is
$\displaystyle D_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}$ $\displaystyle=$
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle^{(0)}.$
(102)
First, we solve Eq. (102) for $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}$,
making use of Eq. (48) for $D_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}$ and
Eq. (72c) for $\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle^{(0)}$.
The result is
$d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}=\frac{\pi\alpha_{s}}{(d-1)(N_{c}^{2}-1)m^{3}}\left(\frac{\mu^{2}_{\Lambda}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\delta(1-z).$ (103)
The short-distance coefficient in Eq. (103) agrees with that in Refs.
BL:gfrag-NLO ; Lee:2005jw .
Next, we determine the short-distance coefficient $d_{0}[g\to
Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$ by making use of the order-$v^{2}$ matching
equation (101). We substitute $D_{2}[g\to Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$ in
Eq. (51), $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}$ in Eq. (103),
$\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)}$ in
Eq. (72f), and $\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle^{(1)}$
in Eq. (86) into Eq. (101). Then, solving for $d_{0}[g\to
Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$, we obtain
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$ $\displaystyle=$
$\displaystyle\frac{8\alpha_{s}^{2}}{3(d-1)(N_{c}^{2}-1)m^{5}}\left(\frac{N_{c}^{2}-4}{4N_{c}}\right)\left(\frac{\mu^{2}_{\Lambda}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}$ (104)
$\displaystyle\times\bigg{\\{}-\frac{\delta(1-z)}{2\epsilon_{\rm
IR}}\left[\frac{(1-\epsilon)\Gamma(1+\epsilon)}{1-\tfrac{2}{3}\epsilon}\left(\frac{\mu^{2}_{\Lambda}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}-1\right]$
$\displaystyle+\frac{(1-\epsilon)\Gamma(1+\epsilon)}{1-\tfrac{2}{3}\epsilon}\left(\frac{\mu^{2}_{\Lambda}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}f(z)\bigg{\\}},$
where $f(z)$ is given in Eq. (51b). In Ref. Braaten:1996rp , the color-singlet
short-distance coefficients $d_{0}[g\to Q\bar{Q}({}^{3}P_{J}^{[1]})]^{(2)}$
were computed. Summing the results for $d_{0}[g\to
Q\bar{Q}({}^{3}P_{J}^{[1]})]^{(2)}$ in Ref. Braaten:1996rp over $J=0$, 1 and
2 and multiplying by
$[(N_{c}^{2}-1)/(4N_{c}^{2})]^{-1}[(N_{c}^{2}-4)/(4N_{c})]$ in order to obtain
the corresponding short-distance coefficient for the color-octet channel, we
find agreement with our result in Eq. (104).
The expression in Eq. (104) gives the exact $\epsilon$ dependence. Expanding
this expression to order $\epsilon^{1}$, using the expression for $f(z)$ in
Eq. (5.2), we find that
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$
$\displaystyle\quad=\frac{8\alpha_{s}^{2}}{3(d-1)(N_{c}^{2}-1)m^{5}}\,\frac{N_{c}^{2}-4}{4N_{c}}\left(\frac{\mu^{2}_{\Lambda}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}$
$\displaystyle\qquad\times\Bigg{\\{}\frac{1}{6}\delta(1-z)\bigg{[}1-6\log\frac{\mu_{\Lambda}}{2m}+\epsilon\left(\frac{2}{3}-\frac{\pi^{2}}{4}+2\log\frac{\mu_{\Lambda}}{2m}-6\log^{2}\frac{\mu_{\Lambda}}{2m}\right)\bigg{]}$
$\displaystyle\qquad\phantom{xx}+\left(\frac{1}{1-z}\right)_{+}\left[1-\epsilon\left(\frac{1}{3}-2\log\frac{\mu_{\Lambda}}{2m}\right)\right]-2\epsilon\left[\frac{\log(1-z)}{1-z}\right]_{+}$
$\displaystyle\qquad\phantom{xx}+\frac{13-7z}{4}\left[1-\epsilon\,\Big{(}\,\frac{1}{3}+\frac{3}{2}\log(1-z)-2\log\frac{\mu_{\Lambda}}{2m}\,\Big{)}\right]\log(1-z)$
$\displaystyle\qquad\phantom{xx}-\frac{1}{8}(1-2z)(8-5z)\left(1+2\epsilon\log\frac{\mu_{\Lambda}}{2m}\right)$
$\displaystyle\qquad\phantom{xx}+\frac{\epsilon}{24}\Big{[}8+48z-38z^{2}+15(8-11z+4z^{2})\log(1-z)\Big{]}\Bigg{\\}}+O(\epsilon^{2}).$
(105)
Finally, we determine the sum of short-distance coefficients $d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ by making use of the matching equation at
relative order $v^{4}$ [Eq. (100)]. We substitute $D_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ in Eq. (68), $d_{0}[g\to
Q\bar{Q}({}^{3}P^{[8]})]^{(2)}$ in Eq. (104), $d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[8]})]^{(1)}$ in Eq. (103), $\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
in Eq. (72e), $\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle^{(1)}$
in Eq. (LABEL:3pj3s1msbar), and $\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
in Eq. (94) into Eq. (100). Solving for $d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$, we obtain
$\displaystyle d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ $\displaystyle\qquad=\,\,d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}\,\,+\,\,\frac{2\alpha_{s}^{3}(N_{c}^{2}-4)}{3\pi(d-1)^{3}N_{c}^{3}m^{7}}\,\,\bigg{\\{}\,\,\delta(1-z)\,\bigg{(}\frac{1}{24}-\frac{\pi^{2}}{6}$
$\displaystyle\phantom{xxxxx}-\frac{1}{3}\log\frac{\mu_{\Lambda}}{2m}+\log^{2}\frac{\mu_{\Lambda}}{2m}\bigg{)}+\left(\frac{1}{1-z}\right)_{\\!\\!+}\\!\\!\bigg{(}\frac{1}{3}-2\log\frac{\mu_{\Lambda}}{2m}\bigg{)}+2\left[\frac{\log(1-z)}{1-z}\right]_{\\!+}\\!$
$\displaystyle\phantom{xxxxx}-\frac{104-29z-10z^{2}}{24}+\frac{7[z+(1+z)\log(1-z)]}{2z^{2}}+\frac{(1-2z)(8-5z)}{4}\log\frac{\mu_{\Lambda}}{2m}$
$\displaystyle\phantom{xxxxx}+\frac{1+z}{4}\left(31-6z-\frac{36}{z}\right)\log(1-z)-\frac{z}{4}\left(\\!39-6z+\frac{8}{1-z}\\!\right)\log
z$
$\displaystyle\phantom{xxxxx}+\frac{13-7z}{2}\bigg{[}\,\bigg{(}\\!\log\frac{1-z}{z^{2}}-\log\frac{\mu_{\Lambda}}{2m}\bigg{)}\log(1-z)-\textrm{Li}_{2}(z)\,\bigg{]}\,\,\bigg{\\}}+O(\epsilon),$
(106)
where we have used the identity
$\textrm{Li}_{2}(1-z)=\pi^{2}/6-\textrm{Li}_{2}(z)-\log z\,\log(1-z)$.
$d_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}$ is defined by
$d_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}=\frac{D_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}}{\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}}=\frac{D_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}}{2(d-1)N_{c}\bm{q}^{4}},$ (107)
where $D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}$ and $\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
are given in Eqs. (69) and (72e), respectively. The result in Eq. (7) is new.
As expected, both the double and single poles in $\epsilon_{\rm IR}$ have
cancelled in Eq. (7). These cancellations rely nontrivially on the correctness
of the infrared subtractions and on the NRQCD operator renormalizations in our
calculation.
Finally, we note that the color-singlet short-distance coefficient $d_{0}[g\to
Q\bar{Q}({}^{1}S_{0}^{[1]})]^{(2)}$ was computed in Ref. Braaten:1996rp . We
can obtain the corresponding color-octet short-distance coefficient by
multiplying by the ratio of the color-octet and color-singlet color factors,
namely, $[(N_{c}^{2}-1)/(4N_{c}^{2})]^{-1}_{N_{c}=3}[(N_{c}^{2}-4)/(4N_{c})]$
. Then, we find that
$d_{0}[g\to
Q\bar{Q}({}^{1}S_{0}^{[8]})]^{(2)}=\frac{\alpha_{s}^{2}}{8m^{3}}\frac{N_{c}^{2}-4}{4N_{c}}\left[3z-2z^{2}+2(1-z)\log(1-z)\right].$
(108)
## 8 Numerical results
In this section we describe the numerical results that derive from our
calculations.
We have evaluated $d_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$ in
Eq. (107) by carrying out the integrations over the phase space
$d\tilde{\Phi}_{2}$ in Eqs. (67), (70a), and (70b) numerically at $100$ points
in each of the ranges $z=0$ to $z=10^{-2}$, $z=10^{-2}$ to $z=1-10^{-2}$, and
$z=1-10^{-2}$ to $z=1$. We have then used the parametrization in Eq. (B) to
obtain a best fit to the numerical results, which leads to the parameters that
are given in Table 3. Details of this procedure are given in Appendix B. We
have also applied this procedure to $d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$ and $d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$. In Fig. 8, we plot the results of the
numerical calculations of the short-distance coefficients.
Figure 8: The color-singlet short-distance coefficients $d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, $d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$ and $d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$, which are defined in Eqs. (21),
(22) and (7), respectively, as functions of $z$. The scaling factors are
$(\mathcal{N}_{0},\mathcal{N}_{2},\mathcal{N}_{4})=(10^{-3}\times\alpha_{s}^{3}/m^{3},10^{-2}\times\alpha_{s}^{3}/m^{5},10^{-2}\times\alpha_{s}^{3}/m^{7})$.
We would like to obtain estimates of the relative sizes of the contributions
of the various fragmentation functions to the cross section for $J/\psi$
production in hadron-hadron collisions. The contribution of a fragmentation
process to the $J/\psi$ production cross section, differential in the $J/\psi$
transverse momentum ${{p}}_{T}$, is
$\frac{d\sigma_{J/\psi}^{\rm
frag}}{d{{p}}_{T}}=\int_{0}^{1}\\!dz\,\frac{d\sigma_{g}}{d{{p}}_{T}}({{p}}_{T}/z)\,D(z),$
(109)
where $\frac{d\sigma_{g}}{d{{p}}_{T}}({{p}}_{T})$ is the cross section to
produce a gluon with transverse momentum ${{p}}_{T}$. Let us assume that
$\frac{d\sigma_{g}}{d{{p}}_{T}}({{p}}_{T})\propto 1/{{p}}_{T}^{\kappa}$, where
$\kappa$ is a fixed power. Then,
$\frac{d\sigma_{J/\psi}^{\rm frag}}{d{{p}}_{T}}\propto I_{\kappa}(D),$ (110)
where
$I_{\kappa}(D)=\int_{0}^{1}\\!dz\,z^{\kappa}\,D(z).$ (111)
Hence, we can obtain a rough estimate of the relative contribution of a
fragmentation process to the cross section by computing $I_{\kappa}(D)$.777The
precise calculation of a fragmentation contribution to a cross section would
require that one compute the convolution of the two-to-two partonic cross
sections that produce a final-state gluon with the fragmentation function and
with the appropriate parton distributions. Such a calculation is beyond the
scope of the present paper.
We can estimate $\kappa$ by taking advantage of the fact that $J/\psi$
production in the ${}^{3}S_{1}^{[8]}$ channel is dominated at LO in
$\alpha_{s}$ at large $p_{{}_{T}}$ by gluon fragmentation into a $J/\psi$ with
longitudinal-momentum fraction $z=1$ (Ref. Cho:1995vh ). Consequently, the
$p_{{}_{T}}$ dependence in this channel at high $p_{{}_{T}}$ is approximately
that of $d\sigma_{g}/dp_{{}_{T}}$. Furthermore, since the NLO $k$ factor for
this channel is essentially independent of $p_{{}_{T}}$ and amounts to a
correction of only about $14\%$ at the Tevatron Gong:2008ft , we do not expect
the NLO corrections to change the $p_{{}_{T}}$ dependence of this channel
significantly. Therefore, we estimate $\kappa$ by making use of the result for
the ${}^{3}S_{1}^{[8]}$ contribution to $d\sigma/dp_{{}_{T}}$ at NLO in
$\alpha_{s}$ that appears in Fig. 1(c) of Ref. Butenschoen:2010rq .
Specifically, we compute $(d/d\log p_{{}_{T}})\log[d\sigma/dp_{{}_{T}}]$ and
find that $\kappa\approx 5.2$ at $p_{{}_{T}}=20$ GeV.
We have computed $I_{\kappa}(d)$ for $\kappa=0$ and $5.2$, taking $d$ to be
the order-$v^{0}$, order-$v^{2}$ and order-$v^{4}$ fragmentation functions for
the fragmentation process $g\to Q\bar{Q}({}^{3}S_{1}^{[1]})$. The results are
shown in Table 1.
Table 1: Results of computing $I_{\kappa}(d)$ for the order-$v^{0}$, order-$v^{2}$, and order-$v^{4}$ fragmentation functions for the fragmentation process $g\to Q\bar{Q}({}^{3}S_{1}^{[1]})$. The first two rows are the normalization integral $I_{0}(d)$ for the short-distance coefficients $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ [Eq. (21)], $d_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ [Eq. (22)] and $d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ [Eq. (7)]. The third and fourth rows are $I_{0}(d)$ times the ratio of LDMEs $R_{n}\equiv\langle 0|\mathcal{O}_{n}({}^{3}S_{1}^{[1]})|0\rangle/\langle 0|\mathcal{O}_{0}({}^{3}S_{1}^{[1]})|0\rangle$, as is described in the text. The fifth and sixth rows are the integral $I_{5.2}(d)$. The seventh and eighth rows are $I_{5.2}(d)$ times $R_{n}$. In the case of $d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$, we have set $\mu_{\Lambda}=m$ and $2m$. The quantities in the table are divided by the factor $\mathcal{N}=10^{-4}\times\alpha_{s}^{3}/m^{3}$. x | $I_{\kappa}(d_{n})\,\backslash\,d_{n}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ | XXX | $d_{0}$ | XXX | $d_{2}$ | XX | $d_{4,1}+d_{4,2}$ x
---|---|---|---|---|---|---|---
| $\phantom{{}_{.2}}I_{0}(d_{n})|_{\mu_{\Lambda}=m}$ | | 8.29 | | 20.3$/m^{2}$ | | $-37.4\phantom{9}/m^{4}$
| $\phantom{{}_{.2}}I_{0}(d_{n})|_{\mu_{\Lambda}=2m}$ | | 8.29 | | 20.3$/m^{2}$ | | $-6.54/m^{4}$
| $\phantom{{}_{.2}}I_{0}(d_{n})|_{\mu_{\Lambda}=m\phantom{2}}\times R_{n}$ | | 8.29 | | 3.95 | | $-1.41$
| $\phantom{{}_{.2}}I_{0}(d_{n})|_{\mu_{\Lambda}=2m}\times R_{n}$ | | 8.29 | | 3.95 | | $-0.247$
| $I_{5.2}(d_{n})|_{\mu_{\Lambda}=m}$ | | 0.743 | | 11.2$/m^{2}$ | | $35.7\phantom{9}/m^{4}$
| $I_{5.2}(d_{n})|_{\mu_{\Lambda}=2m}$ | | 0.743 | | 11.2$/m^{2}$ | | $84.5\phantom{9}/m^{4}$
| $I_{5.2}(d_{n})|_{\mu_{\Lambda}=m\phantom{2}}\times R_{n}$ | | 0.743 | | 2.18 | | -391.35
| $I_{5.2}(d_{n})|_{\mu_{\Lambda}=2m}\times R_{n}$ | | 0.743 | | 2.18 | | -393.19
In the first and second rows of Table 1, we give the normalization integrals
$I_{0}(d)$ for the short-distance coefficients. In the third and fourth rows,
we give $I_{0}(d)$ multiplied by the relative value of corresponding LDME. In
the fifth and sixth rows we give $I_{5.2}(d)$. In the seventh and eighth rows
we give $I_{5.2}(d)$ multiplied by the relative value of corresponding LDME.
We take the relative values of the LDMEs to be given by the generalized Gremm-
Kapustin relation Bodwin:2006dn :
$\langle 0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle:\langle
0|\mathcal{O}_{2}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle:\langle
0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle=1:m^{2}\langle
v^{2}\rangle:m^{4}\langle v^{2}\rangle^{2}.$ (112)
We use the value of $m^{2}\langle v^{2}\rangle$ from Table I of Ref.
Bodwin:2007fz :
$m^{2}\langle
v^{2}\rangle=0.437\,\textrm{GeV}^{2}\,\,\,\textrm{at}\,\,\,m=1.5\,\textrm{GeV}.$
(113)
Examining the relative values in the fifth or sixth rows of Table 1, we see
that, if we exclude the factors from the NRQCD LDMEs, then the order-$v^{4}$
contribution is enhanced considerably relative to the order-$v^{0}$ and
order-$v^{2}$ contributions. We attribute this enhancement to the strong
peaking of $d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ near $z=1$, which arises primarily from
the $\delta$ functions and $+$ distributions that it contains. We expect such
strong peaking to occur whenever the full-QCD process contains soft
divergences that correspond to divergences in the $Q\bar{Q}$ LDMEs.
From the seventh and eighth rows of Table 1, we see that the order-$v^{4}$
fragmentation contribution to the cross section is enhanced by approximately a
factor of 2 for $\mu_{\Lambda}=m$ and by approximately a factor of 4 for
$\mu_{\Lambda}=2m$, relative to the order-$v^{0}$ contribution. However, the
order-$v^{0}$ contribution to the $J/\psi$ cross section at ${{p}}_{T}=10$ GeV
lies about a factor of 30 below the measured cross section. Thus, while the
enhancement of the order-$v^{4}$ fragmentation contribution to the cross
section is substantial, it is not sufficient to make the order-$v^{4}$
fragmentation contribution to the cross section important for the
phenomenology of $J/\psi$ production at the current level of precision.
We can also compare the relative contributions to $d\sigma_{J/\psi}^{\rm
frag}$ of each of the three $Q\bar{Q}$ channels in that appear in $D[g\to
J/\psi]$ in order $v^{4}$ [Eq. (24)]. For each channel, we compute
$I_{\kappa}(5.2)$. Then, we multiply by the following LDMEs:
$\displaystyle\langle 0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle m^{4}\langle v^{2}\rangle^{2}\times
1.32~{}{\rm GeV}^{3},$ (114a) $\displaystyle\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}P^{[8]})|0\rangle$ $\displaystyle=$
$\displaystyle-0.109~{}{\rm GeV}^{5},$ (114b) $\displaystyle\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[8]})|0\rangle$ $\displaystyle=$
$\displaystyle 3.12\times 10^{-3}~{}{\rm GeV}^{3},$ (114c)
$\displaystyle\langle 0|\mathcal{O}_{0}^{J/\psi}({}^{1}S_{0}^{[8]})|0\rangle$
$\displaystyle=$ $\displaystyle 4.50\times 10^{-2}~{}{\rm GeV}^{3},$ (114d)
where $m^{2}\langle v^{2}\rangle$ is given in Eq. (113), and the numerical
values on the right sides of Eq. (114) come from the fit at NLO in
$\alpha_{s}$ to the Tevatron and HERA data in Ref. Butenschoen:2010rq . The
results of this computation are shown in Table 2.
Table 2: Relative contributions to $d\sigma_{J/\psi}^{\rm frag}$ in order $v^{4}$. The first and second rows give $I_{0}(d)$ for the short-distance coefficients in Eq. (24). The second and third rows give $I_{0}(d)$ times the LDMEs in Eq. (114). The fourth and fifth rows give $I_{5.2}(d)$. The fifth and sixth rows give $I_{5.2}(d)$ times the LDMEs in Eq. (114). We take $m=m_{c}=1.5\,\textrm{GeV}$. For compatibility with Ref. Butenschoen:2010rq , we take $\alpha_{s}=\alpha_{s}({{m}}_{T})$, where ${{m}}_{T}=\sqrt{{{p}}_{T}^{2}+4m_{c}^{2}}$. We choose the point ${{p}}_{T}=20$ GeV, which implies that $\alpha_{s}({{m}}_{T})=0.154$. $I_{\kappa}(d)\,\backslash\,$channel | $\mathcal{O}_{0}^{J/\psi}({}^{1}S_{0}^{[8]})$ | | $\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[8]})$ | $\mathcal{O}_{0}^{J/\psi}({}^{3}P^{[8]})$ | $\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})$
---|---|---|---|---|---
$\phantom{{}_{.2}}I_{0}(d)|_{\mu_{\Lambda}=m\phantom{2}}\times\,10^{6}\,\textrm{GeV}^{3}$ | $122.$ | | $5970$ | $-171.\,\textrm{GeV}^{-2}$ | $-0.799\,\textrm{GeV}^{-4}$
$\phantom{{}_{.2}}I_{0}(d)|_{\mu_{\Lambda}=2m}\times\,10^{6}\,\textrm{GeV}^{3}$ | $122.$ | | $5970$ | $-271.\,\textrm{GeV}^{-2}$ | $-0.140\,\textrm{GeV}^{-4}$
$\phantom{{}_{.2}}I_{0}(d)|_{\mu_{\Lambda}=m\phantom{2}}\times\,10^{6}\,\textrm{LDME}$x | $\phantom{12}5.49$ | | $\phantom{59}18.6$ | $\phantom{-1}18.6$ | $-0.201$
$\phantom{{}_{.2}}I_{0}(d)|_{\mu_{\Lambda}=2m}\times\,10^{6}\,\textrm{LDME}$ | $\phantom{12}5.49$ | | $\phantom{59}18.6$ | $\phantom{-1}29.6$ | $-0.0353$
$I_{5.2}(d)|_{\mu_{\Lambda}=m\phantom{2}}\times\,10^{6}\,\textrm{GeV}^{3}$ | $\phantom{1}36.7$ | | $5970$ | $-300.\,\textrm{GeV}^{-2}$ | $\phantom{-}0.763\,\textrm{GeV}^{-4}$
$I_{5.2}(d)|_{\mu_{\Lambda}=2m}\times\,10^{6}\,\textrm{GeV}^{3}$ | $\phantom{1}36.7$ | | $5970$ | $-400.\,\textrm{GeV}^{-2}$ | $\phantom{-}1.81\phantom{2}\,\textrm{GeV}^{-4}$
$I_{5.2}(d)|_{\mu_{\Lambda}=m\phantom{2}}\times\,10^{6}\,\textrm{LDME}$ | $\phantom{12}1.65$ | | $\phantom{59}18.6$ | $\phantom{-1}32.7$ | $\phantom{-}0.192$
$I_{5.2}(d)|_{\mu_{\Lambda}=2m}\times\,10^{6}\,\textrm{LDME}$ | $\phantom{12}1.65$ | | $\phantom{59}18.6$ | $\phantom{-1}43.6$ | $\phantom{-}0.455$
We see from the second row of Table 2 that the $Q\bar{Q}({}^{3}S_{1}^{[1]})$
channel makes a small contribution to $D_{4}[g\to J/\psi]$ at ${p}_{T}=20$
GeV, confirming our previous conclusion that this channel is not important
phenomenologically at the current level of precision. We also see that the
$Q\bar{Q}({}^{3}S_{1}^{[8]})$ and $Q\bar{Q}({}^{3}P^{[8]})$ channels give
comparable contributions to $D_{4}[g\to J/\psi]$ at ${p}_{T}=20$ GeV. As we
have mentioned, at high $p_{{}_{T}}$ at LO in $\alpha_{s}$, the fragmentation
contribution gives the bulk of the contribution in the
$Q\bar{Q}({}^{3}S_{1}^{[8]})$ channel (Ref. Cho:1995vh ). As we have also
mentioned, the correction to the production rate in the
$Q\bar{Q}({}^{3}S_{1}^{[8]})$ channel at NLO in $\alpha_{s}$ is only about
$14\%$ at the Tevatron Gong:2008ft . Hence, we can estimate the fragmentation
contribution to $J/\psi$ production at the Tevatron in the
$Q\bar{Q}({}^{3}P^{[1]})$ channel by multiplying the ratio of the
$Q\bar{Q}({}^{3}P^{[8]})$ and $Q\bar{Q}({}^{3}S_{1}^{[8]})$ entries in the
seventh row of Table 2 by the value of $d\sigma/d{p}_{T}\times
B(J/\psi\to\mu\mu)$ at $20$ GeV from Fig. 1(c) of Ref. Butenschoen:2010rq and
by dividing by $1.14$ to account for the NLO correction in the
$Q\bar{Q}({}^{3}S_{1}^{[8]})$ channel. Our estimate is that the fragmentation
contribution to $d\sigma/d{p}_{T}\times B(J/\psi\to\mu\mu)$ at ${p}_{T}=20$
GeV from the $Q\bar{Q}({}^{3}P^{[8]})$ channel is about $6\times 10^{-3}$
nb/GeV. We see that this is comparable to (about a factor of $2$ larger than)
the total NLO contribution in the $Q\bar{Q}({}^{3}P^{[8]})$ channel in Fig.
1(c) of Ref. Butenschoen:2010rq . A more precise calculation of the
fragmentation contribution in the $Q\bar{Q}({}^{3}P^{[8]})$ channel will be
necessary in order to determine whether it is the dominant contribution in
that channel at NLO in $\alpha_{s}$ at high $p_{{}_{T}}$.
## 9 Summary
We have calculated NRQCD short-distance coefficients for gluon fragmentation
into a ${}^{3}S_{1}$ heavy-quarkonium state through relative order $v^{4}$.
Our principal new result is the expression for the sum of the relative-
order-$v^{4}$ short-distance coefficients $d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ for gluon fragmentation through the
${}^{3}S_{1}$ color-singlet channel. This expression is given in Eq. (7) and
in the parametrization of $d_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}$ in Eq. (B). As a byproduct of this calculation, we have computed the
short-distance coefficient for gluon fragmentation through the ${}^{3}S_{1}$
color-octet channel [Eq. (103)], finding agreement with the results in Refs.
BL:gfrag-NLO ; Lee:2005jw , and the sum of short-distance coefficients for
gluon fragmentation through the ${}^{3}P_{J}$ color-octet channel [Eq. (104)],
finding agreement with the result in Ref. Braaten:1996rp for the
${}^{3}P_{J}$ color-singlet short-distance coefficients. We have also computed
the short-distance coefficients for gluon fragmentation through the
${}^{3}S_{1}$ color-singlet channel at leading order in $v$ and at relative-
order $v^{2}$ and find agreement with the results in Refs. Braaten:1993rw ;
Braaten:1995cj and Ref. Bodwin:2003wh , respectively.
The analysis in this paper involves, for the first time in an NRQCD
factorization calculation, both single and double soft divergences in the
full-QCD process. These soft divergences manifest themselves as single and
double poles in $\epsilon=(4-d)/2$ in dimensional regularization. We have
dealt with the soft divergences by devising subtractions that remove both the
single and double poles in $\epsilon$. We have calculated the phase-space
integrals for the subtraction contributions analytically in $d=4-2\epsilon$
dimensions and have calculated the phase-space integrals for the finite
remainder contributions numerically in $d=4$ dimensions. We have also
calculated the perturbative NRQCD LDMEs that appear in the NRQCD matching
equations for the short-distance coefficients. These perturbative LDMEs
involve both one-loop and two-loop renormalizations of the NRQCD operators.
Our results for the renormalization-group evolution of the renormalized LDMEs
confirm some results in Refs. BBL ; Gremm:1997dq ; z-g-he , but disagree with
one of the results in Ref. Gremm:1997dq .
In Tables 1 and 2, we have given estimates of the relative sizes of the
contributions of the various channels to gluon fragmentation into $J/\psi$
through relative order $v^{4}$. As we have mentioned, the contribution of the
${}^{3}S_{1}$ color-octet channel is believed to be important
phenomenologically at large $p_{{}_{T}}$. Hence, one might expect the
order-$v^{4}$ contribution in the ${}^{3}S_{1}$ color-singlet channel to be
important as well, since the color-singlet channel mixes with the color-octet
channel through single and double logarithms of the NRQCD factorization scale
at order $v^{4}$. Indeed, we find that the contribution to the cross section
at $p_{{}_{T}}=20$ GeV of the ${}^{3}S_{1}$ color-singlet channel in order
$v^{4}$ is about a factor of $2$ larger than the contribution of the
${}^{3}S_{1}$ color-singlet channel at the leading order in $v$ when the
factorization scale is taken to be $\mu_{\Lambda}=m_{c}$ and is about a factor
of $4$ larger than the contribution of the ${}^{3}S_{1}$ color-singlet channel
at the leading order in $v$ when the factorization scale is taken to be
$\mu_{\Lambda}=2m_{c}$. This is a large enhancement, since one would nominally
expect the order-$v^{4}$ contribution to be about $\langle
v^{2}\rangle^{2}\approx 0.04$ times the order-$v^{0}$ contribution. In spite
of this large enhancement of the fragmentation contribution in order $v^{4}$,
the corresponding contribution to the $J/\psi$ production cross section at the
Tevatron or the LHC is not important at the current level of precision of the
phenomenology.
We attribute the large enhancement of the order-$v^{4}$ contribution to the
${}^{3}S_{1}$ color-singlet channel to the peaking of the sum of short-
distance coefficients near $z=1$. This peaking arises from terms that are
proportional to $\delta(1-z)$, $[1/(1-z)]_{+}$, and $[\log(1-z)/(1-z)]_{+}$.
These terms are remnants of the soft divergences that appear in the full-QCD
expression for the fragmentation process and that, ultimately, cancel in the
NRQCD matching equations for the short-distance coefficients. We expect such a
peaking, and the corresponding enhancement, to be present whenever soft
divergences appear in a full-QCD process and are cancelled in the NRQCD
matching equations.888There is also some peaking/enhancement in the
order-$v^{2}$ contribution to the ${}^{3}S_{1}$ color-singlet channel. The
order-$v^{2}$ contribution contains terms that are the product of an
order-$v^{0}$ amplitude with an order-$v^{2}$ amplitude. The order-$v^{2}$
amplitude is sufficiently singular that its square would produce a soft
divergence. However, in combination with the order-$v^{0}$ amplitude, it
produces only a peaking at $z=1$.
The signature of this peaking/enhancement in Table 2 is that the magnitude of
the $I_{5.2}$ entry is comparable to or greater than the magnitude of the
$I_{0}$ entry. We see this signature of peaking/enhancement in the
${}^{3}S_{1}$ and ${}^{3}P_{J}$ color-octet channels, as well as in the
${}^{3}S_{1}$ color-singlet channel in order $v^{4}$, but not in the
${}^{1}S_{0}$ color-octet channel. The ${}^{3}P_{J}$ color-octet channel
contains soft divergences in the full-QCD process, but the ${}^{1}S_{0}$
color-octet channel does not. (The fragmentation contribution in the
${}^{3}S_{1}$ color-octet channel is proportional to $\delta(1-z)$ at leading
order in $\alpha_{s}$.)
The fragmentation contribution to the ${}^{3}S_{1}$ color-octet channel
dominates the contribution of that channel to the $J/\psi$ production cross
section at hadron-hadron colliders at large $p_{{}_{T}}$, both at LO in
$\alpha_{s}$ (Ref. Cho:1995vh ) and at NLO in $\alpha_{s}$ (Ref. Gong:2008ft
). We estimate that the fragmentation contribution to the ${}^{3}P_{J}$ color-
octet channel gives a substantial part of the total contribution of that
channel at NLO in $\alpha_{s}$ to the $J/\psi$ production cross section at
hadron-hadron colliders at large $p_{{}_{T}}$. A more precise calculation of
the ${}^{3}P_{J}$ color-octet fragmentation contribution will be required in
order to determine if it is the dominant contribution at large $p_{{}_{T}}$ in
that channel.
###### Acknowledgements.
We thank Zhi-Guo He for providing us with his unpublished results for the
renormalization-group evolution of NRQCD LDMEs. We also thank Andrea Petrelli
for the use of some of his mathematica code. The work of G.T.B. in the High
Energy Physics Division at Argonne National Laboratory was supported by the U.
S. Department of Energy, Division of High Energy Physics, under Contract No.
DE-AC02-06CH11357. The work of U.R.K. and J.L. was supported by the MEST of
Korea under the NRF Grants No. 2011-0027559 and No. 2011-0003023,
respectively.
## Appendix A Analytic calculation of phase-space integrals
In this appendix, we give some of the details of the analytic integrations
over the final-state phase space for $S_{12}$, $S_{1}$ and $S_{2}$.
We first carry out the average over the angles of the transverse components of
the final-state gluon momenta in the phase-space (32c). Under this angular
averaging we have
$\displaystyle\langle x\rangle_{\perp}$ $\displaystyle=$
$\displaystyle\frac{z_{1}z_{2}}{z}\left(\frac{e_{1}}{z_{1}}+\frac{e_{2}}{z_{2}}-\frac{1}{z}\right),$
(115a) $\displaystyle\langle x^{2}\rangle_{\perp}$ $\displaystyle=$
$\displaystyle\frac{z_{1}^{2}z_{2}^{2}}{z^{2}}\left(\frac{e_{1}}{z_{1}}+\frac{e_{2}}{z_{2}}-\frac{1}{z}\right)^{2}+\frac{z_{1}z_{2}}{(d-2)z^{2}}\left(2e_{1}-\frac{z_{1}}{z}\right)\left(2e_{2}-\frac{z_{2}}{z}\right).$
(115b)
After this angular averaging, the squared amplitude is independent of the
directions of $\bar{\bm{k}}_{1\perp}$ and $\bar{\bm{k}}_{2\perp}$.
In evaluating the integrals over $|\bar{\bm{k}}_{1\perp}|$ and
$|\bar{\bm{k}}_{2\perp}|$, it is convenient to define dimensionless variables
$u_{1}$ and $u_{2}$:
$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{z}{z_{1}}\right)^{2}\bar{k}^{2}_{1\perp},$ (116a)
$\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{z}{z_{2}}\right)^{2}\bar{k}^{2}_{2\perp}.$ (116b)
Then $e_{1}$, $e_{2}$ and $x$ can be expressed as
$\displaystyle e_{1}$ $\displaystyle=$
$\displaystyle\frac{z_{1}}{2z}(1+u_{1}),$ (117a) $\displaystyle e_{2}$
$\displaystyle=$ $\displaystyle\frac{z_{2}}{2z}(1+u_{2}).$ (117b)
The integrals of $S_{12}$, $S_{1}$ and $S_{2}$ over the magnitudes of the
transverse components of the final-state gluon momenta can be expressed as
linear combinations of the following elementary integrals:
$\displaystyle J_{mn}^{i}$ $\displaystyle=$
$\displaystyle\int\frac{d^{d-2}\bm{\bar{k}}_{i\perp}}{(2\pi)^{d-2}}\frac{1}{e_{i}^{m}(1+2e_{i})^{n}}$
(118) $\displaystyle=$
$\displaystyle\frac{2^{m}}{(4\pi)^{1-\epsilon}\Gamma(1-\epsilon)}\left(\frac{z_{i}}{z}\right)^{2-m-n-2\epsilon}\int_{0}^{\infty}\frac{du_{i}}{u_{i}^{\epsilon}(1+u_{i})^{m}\left(1+\frac{z}{z_{i}}+u_{i}\right)^{n}},\phantom{xxxx}$
where $m$ takes on integer values equal to or greater than $-1$, and $n$ takes
on non-negative integer values. These integrals can be evaluated
straightforwardly to obtain
$\displaystyle J^{i}_{m0}$ $\displaystyle=$ $\displaystyle
2^{m}\left(\frac{z_{i}}{z}\right)^{2-m-2\epsilon}\frac{\Gamma(m-1+\epsilon)}{(4\pi)^{1-\epsilon}\Gamma(m)},$
(119a) $\displaystyle J^{i}_{0n}$ $\displaystyle=$
$\displaystyle\left(\frac{z_{i}}{z}\right)^{1-\epsilon}\left(1+\frac{z_{i}}{z}\right)^{1-n-\epsilon}\frac{\Gamma(n-1+\epsilon)}{(4\pi)^{1-\epsilon}\Gamma(n)},$
(119b) $\displaystyle J^{i}_{-1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(J^{i}_{01}-J^{i}_{02}\right),$ (119c)
$\displaystyle J_{mn}^{i}$ $\displaystyle=$
$\displaystyle(-1)^{m+n}\left(\frac{z_{i}}{z}\right)^{1-\epsilon}\frac{\Gamma(\epsilon)}{(4\pi)^{1-\epsilon}\Gamma(m)\Gamma(n)}$
(119d) $\displaystyle\times\bigg{(}\frac{\partial}{\partial
a}\bigg{)}^{m-1}\bigg{(}\frac{\partial}{\partial
b}\bigg{)}^{n-1}\left\\{\frac{2}{b-2a}\bigg{[}\left(2a+\frac{z_{i}}{z}\right)^{-\epsilon}-\left(b+\frac{z_{i}}{z}\right)^{-\epsilon}\bigg{]}\right\\}\Bigg{|}_{a=0,\,b=1}.\phantom{xxxx}$
After we have carried out the integrations over the transverse components of
the final-state gluon momenta, the remaining integrals over $z_{1}$ and
$z_{2}$ for $S_{12}$ are simple. The integrations over $z_{1}$ and $z_{2}$ for
$S_{1}$ and $S_{2}$ can be expressed as linear combinations of the integrals
$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\int_{z_{1}z_{2}}\frac{{z_{2}}^{n-\epsilon}}{{z_{1}}^{1+2\epsilon}(z+z_{2})^{1+\epsilon}},$
(120a) $\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle\int_{z_{1}z_{2}}\frac{{z_{2}}^{n-\epsilon}}{{z_{1}}^{1+2\epsilon}(z+z_{2})^{\epsilon}},$
(120b) $\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\int_{z_{1}z_{2}}\frac{{z_{2}}^{n-2\epsilon}}{{z_{1}}^{1+2\epsilon}},$
(120c) $\displaystyle D_{n}$ $\displaystyle=$
$\displaystyle\int_{z_{1}z_{2}}\frac{{z_{2}}^{n-2\epsilon}}{{z_{1}}^{1+2\epsilon}(z+z_{2})},$
(120d)
where
$\int_{{z_{1}}{z_{2}}}\\!\\!\\!\\!F({z_{1}},{z_{2}})\,\,=\int_{0}^{1}\\!d{z_{1}}\int_{0}^{1}\\!d{z_{2}}\,\,F({z_{1}},{z_{2}})\,\delta(1-z-{z_{1}}-{z_{2}}).$
(121)
By making use of the identity $z_{2}/(z+z_{2})=1-z/(z+z_{2})$, we derive the
relations
$\displaystyle A_{n}$ $\displaystyle=$ $\displaystyle B_{n-1}-zA_{n-1},$
(122a) $\displaystyle D_{n}$ $\displaystyle=$ $\displaystyle
C_{n-1}-zD_{n-1}.$ (122b)
Applying the recursion relations (122) repeatedly, we can reduce all of the
integrals to the forms $A_{0}$, $B_{n}$ and $C_{n}$, with $1\leq n\leq 4$. It
turns out that the coefficient of $D_{0}$ vanishes. The coefficient of $A_{0}$
is of order $\epsilon^{0}$, while the coefficients of the $B_{n}$ are of order
$\epsilon^{-1}$ or $\epsilon^{0}$. Therefore, we evaluate $A_{0}$ only through
order $\epsilon^{0}$, and we evaluate the $B_{n}$ only through order
$\epsilon$. The expressions for $A_{0}$, $B_{n}$ and $C_{n}$ can be obtained
conveniently by making use of the identity
$\frac{1}{z_{1}^{1+2\epsilon}}=-\frac{(1-z)^{-2\epsilon}}{2\epsilon}\,\delta(z_{1})+\left[\frac{1}{z_{1}^{1+2\epsilon}}\right]_{1-z},$
(123)
which applies when the domain of integration is $0\leq z_{1}\leq 1-z$. The
distribution in the second term of Eq. (123) is defined by
$\int_{0}^{1-z}\\!\\!dz_{1}\,f(z_{1})\left[\frac{1}{z_{1}^{1+2\epsilon}}\right]_{1-z}=\int_{0}^{1-z}\\!\\!dz_{1}\,\frac{f(z_{1})-f(0)}{z_{1}^{1+2\epsilon}}.$
(124)
A straightforward evaluation of the integrals then gives
$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\epsilon(1-z)^{3\epsilon}}-\log z+O(\epsilon),$
(125a) $\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle-\frac{(1-z)^{n-3\epsilon}}{2\epsilon}+(1-z)^{n-3\epsilon}\,\big{[}X_{n}+Y_{n}+Z_{n}+O(\epsilon^{2})\big{]},$
(125b) $\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle(1-z)^{n-4\epsilon}\left\\{-\frac{1}{2\epsilon}+\frac{1}{2\epsilon}\left[1-\frac{\Gamma(n+1-2\epsilon)\Gamma(1-2\epsilon)}{\Gamma(n+1-4\epsilon)}\right]\right\\},$
(125c)
where
$\displaystyle X_{n}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\\!dt\,\,\frac{(1-t)^{n}-1}{t}=-\sum_{k=1}^{n}\frac{1}{k},$
(126a) $\displaystyle Y_{n}$ $\displaystyle=$
$\displaystyle-\epsilon\int_{0}^{1}\\!dt\,\,\frac{(1-t)^{n}}{t}\big{\\{}\log(1-t)+\log[1-(1-z)t]\,\big{\\}},$
(126b) $\displaystyle Z_{n}$ $\displaystyle=$
$\displaystyle-2\epsilon\int_{0}^{1}\\!dt\,\,\frac{(1-t)^{n}-1}{t}\log
t=-2\epsilon\sum_{k=1}^{n}\frac{1}{k}\sum_{\ell=1}^{k}\frac{1}{\ell}.$ (126c)
In Eq. (125c), the first term in the braces is the pole contribution, and the
remainder is finite. The results for $Y_{n}$ for $0\leq n\leq 4$ are
$\displaystyle Y_{0}$ $\displaystyle=$
$\displaystyle\epsilon\Bigg{[}\frac{\pi^{2}}{6}+{\rm Li}_{2}(1-z)\Bigg{]},$
(127a) $\displaystyle Y_{1}$ $\displaystyle=$
$\displaystyle\epsilon\Bigg{[}\frac{\pi^{2}}{6}+{\rm
Li}_{2}(1-z)-2-\frac{z}{1-z}\log z\Bigg{]},$ (127b) $\displaystyle Y_{2}$
$\displaystyle=$ $\displaystyle\epsilon\Bigg{[}\frac{\pi^{2}}{6}+{\rm
Li}_{2}(1-z)-\frac{5-6z}{2(1-z)}-\frac{z(2-3z)\log z}{2(1-z)^{2}}\Bigg{]},$
(127c) $\displaystyle Y_{3}$ $\displaystyle=$
$\displaystyle\epsilon\Bigg{[}\frac{\pi^{2}}{6}+{\rm
Li}_{2}(1-z)-\frac{49-110z+67z^{2}}{18(1-z)^{2}}-\frac{z(6-15z+11z^{2})\log
z}{6(1-z)^{3}}\Bigg{]},\phantom{xxx}$ (127d) $\displaystyle Y_{4}$
$\displaystyle=$ $\displaystyle\epsilon\Bigg{[}\frac{\pi^{2}}{6}+{\rm
Li}_{2}(1-z)-\frac{205-669z+756z^{2}-310z^{3}}{72(1-z)^{3}}$ (127e)
$\displaystyle\quad-\frac{z(12-42z+52z^{2}-25z^{3})\log
z}{12(1-z)^{4}}\Bigg{]}.\phantom{xxxxxx}$
## Appendix B Parametrizations of the short-distance coefficients
In this appendix, we give parametrizations for the short-distance coefficients
$d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, $d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, and $d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$, which are defined in Eqs. (21),
(22), and (107), respectively.
We observe that $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$,
$d_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, and $d_{4}[g\to Q\bar{Q}$
$({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$ are continuous functions of $z$ over
the whole range $0\leq z\leq 1$ and that they are not analytic at the
endpoints $z=0$ and $1$. One can prove that $d_{n}(0)=0$ for $n=0$, 2, and 4.
In addition, $d_{n}(1)=\alpha_{s}^{3}b_{n}/m^{3+n}$ is finite and calculable
analytically. The values of $b_{n}$ for $n=0$, 2, and 4 are
$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle 0,$ (128a)
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\frac{22\pi^{2}-15}{4374\pi},$ (128b) $\displaystyle b_{4}$
$\displaystyle=$ $\displaystyle-\frac{2922\pi^{2}-2485}{229635\pi}.$ (128c)
Using this information, we parametrize $d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, $d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, and $d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$ as follows:
$\displaystyle d_{n}^{\rm fit}(z)=\frac{\alpha_{s}^{3}}{m^{3+n}}$
$\displaystyle\Bigg{[}$ $\displaystyle
b_{n}z+\log(1-z)\sum_{k=1}^{n_{a}}\alpha_{nk}(1-z)^{k}+\log^{2}(1-z)\sum_{k=1}^{n_{b}}\beta_{nk}(1-z)^{k}$
$\displaystyle+\log
z\sum_{k=1}^{n_{c}}\mu_{nk}z^{k}+\log^{2}z\sum_{k=1}^{n_{d}}\nu_{nk}z^{k}+\sum_{k_{1}=1}^{n_{1}}\sum_{k_{2}=1}^{n_{2}}\omega_{nk_{1}k_{2}}z^{k_{1}}(1-z)^{k_{2}}\Bigg{]}.$
In order to fix the parameters in Eq. (B), we have computed $d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, $d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, and $d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$ by numerical integration for
$100$ points in each of the ranges $z=0$ to $z=10^{-2}$, $z=10^{-2}$ to
$z=1-10^{-2}$, and $z=1-10^{-2}$ to $z=1$. We have carried out the
integrations in two ways: (1) using the change of variables that was proposed
in Refs. Braaten:1993rw ; Braaten:1995cj and is described in Appendix B of
Ref. Bodwin:2003wh and (2) using the change of variables that is given in Eq.
(116). At each value of $z$, the numerical results from the two methods of
integration agree to better than $\Delta^{\rm int}_{n}=r_{n}d_{n}(z)$, where
$(r_{0},r_{2},r_{4})=(7.8\times 10^{-4},1.7\times 10^{-3},1.8\times 10^{-3})$.
Table 3: Results of fitting the parametrization in Eq. (B) to numerical values for the color-singlet short-distance coefficients $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, $d_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, and $d_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$, which are defined in Eqs. (21), (22), and (107), respectively. The quantity $\Delta_{\rm max}^{\rm fit}$ is described in the text. Parameters | $d_{0}(z)$ | | $d_{2}(z)$ | | $d_{4}(z)^{\rm finite}$
---|---|---|---|---|---
$b_{\phantom{11}}$ | $\phantom{-}0\phantom{11111111111}$ | | $\phantom{-}1.4710\times 10^{-2}$ | | $-3.6531\times 10^{-2}$
$\alpha_{1\phantom{1}}$ | $-4.9866\times 10^{-3}$ | | $-1.8127\times 10^{-2}$ | | $\phantom{-}5.2157\times 10^{-2}$
$\alpha_{2\phantom{1}}$ | $\phantom{-}9.8448\times 10^{-3}$ | | | | $\phantom{-}2.4588\times 10^{-3}$
$\alpha_{3\phantom{1}}$ | $\phantom{-}1.9512\times 10^{-2}$ | | | |
$\beta_{1\phantom{1}}$ | $-5.1697\times 10^{-6}$ | | $-1.2282\times 10^{-2}$ | | $\phantom{-}3.6020\times 10^{-2}$
$\beta_{2\phantom{1}}$ | $\phantom{-}1.0462\times 10^{-2}$ | | | |
$\mu_{1\phantom{1}}$ | $-1.8921\times 10^{-3}$ | | $\phantom{-}4.1069\times 10^{-3}$ | | $-7.3565\times 10^{-3}$
$\mu_{2\phantom{1}}$ | | | $\phantom{-}1.8341\times 10^{-2}$ | | $-2.5387\times 10^{-2}$
$\mu_{3\phantom{1}}$ | | | | | $-9.3477\times 10^{-3}$
$\mu_{4\phantom{1}}$ | | | | | $-3.6926\times 10^{-3}$
$\nu_{1\phantom{1}}$ | $\phantom{-}1.2154\times 10^{-3}$ | | $-1.3653\times 10^{-3}$ | | $\phantom{-}1.3839\times 10^{-3}$
$\nu_{2\phantom{1}}$ | $\phantom{-}1.3039\times 10^{-3}$ | | | |
$\nu_{3\phantom{1}}$ | $-2.7246\times 10^{-3}$ | | | |
$\nu_{4\phantom{1}}$ | $-1.4814\times 10^{-3}$ | | | |
$\omega_{11}$ | $-1.6910\times 10^{-2}$ | | $-2.1832\times 10^{-2}$ | | $\phantom{-}7.7264\times 10^{-2}$
$\omega_{12}$ | $\phantom{-}3.8110\times 10^{-2}$ | | $\phantom{-}4.1531\times 10^{-3}$ | |
$\omega_{13}$ | | | $\phantom{-}8.4949\times 10^{-4}$ | |
$\omega_{14}$ | | | $-3.8207\times 10^{-3}$ | |
$\Delta_{\rm max}^{\rm fit}\times m^{3+n}/\alpha_{s}^{3}$ | $3.26\times 10^{-8}\phantom{1}$ | | $1.12\times 10^{-6}\phantom{1}$ | | $2.05\times 10^{-6}\phantom{1}$
We fit the parametrizations to the numerical integration results, using
$\chi^{2}=\sum_{i}[d(z_{i})-d^{\rm fit}(z_{i})]^{2}/[\sigma(z_{i})]^{2}$ as
the criterion for goodness of fit, where $\sigma(z_{i})$ is the error in the
numerical integration that is given by the VEGAS integration program
Lepage:1977sw . In cases in which it is possible to reduce the number of
parameters in the fit without significantly affecting $\chi^{2}$, we have done
so.
In Table 3, we show the results of this fitting procedure, along with the
value for each fit of $\Delta_{\rm max}^{\rm fit}=\textrm{Max}|d(z_{i})-d^{\rm
fit}(z_{i})|$. In these fits, $\chi^{2}$ per degree of freedom is much larger
than one because the error in the fit is generally considerably larger than
the errors in the numerical integrations that are given by VEGAS. A
conservative estimate of the overall error in a parametrization at each value
of $z$ is $\sqrt{[\Delta_{\rm max}^{\rm fit}]^{2}+[\Delta_{n}^{\rm
int}]^{2}}$.
## References
* (1) N. Brambilla, S. Eidelman, B.K. Heltsley, R. Vogt, G.T. Bodwin, E. Eichten, A.D. Frawley and A.B. Meyer et al., Heavy quarkonium: progress, puzzles, and opportunities, Eur. Phys. J. C 71 (2011) 1534 [arXiv:1010.5827] [InSPIRE].
* (2) G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D 51 (1995) 1125 [ Erratum ibid. D 55 (1997) 5853] [hep-ph/9407339] [InSPIRE].
* (3) G.C. Nayak, J.-W. Qiu and G.F. Sterman, Fragmentation, factorization and infrared poles in heavy quarkonium production, Phys. Lett. B 613 (2005) 45 [hep-ph/0501235] [InSPIRE].
* (4) G.C. Nayak, J.-W. Qiu and G.F. Sterman, Fragmentation, nonrelativistic QCD, and NNLO factorization analysis in heavy quarkonium production, Phys. Rev. D 72 (2005) 114012 [hep-ph/0509021] [InSPIRE].
* (5) M. Butenschoen and B.A. Kniehl, Complete Next-to-Leading-Order Corrections to $J/\psi$ Photoproduction in Nonrelativistic Quantum Chromodynamics, Phys. Rev. Lett. 104 (2010) 072001 [arXiv:0909.2798] [InSPIRE].
* (6) Y.-Q. Ma, K. Wang and K.-T. Chao, $J/\psi$ ($\psi^{\prime}$) Production at the Tevatron and LHC at $\mathcal{O}(\alpha_{s}^{4}v^{4})$ in Nonrelativistic QCD, Phys. Rev. Lett. 106 (2011) 042002 [arXiv:1009.3655] [InSPIRE].
* (7) M. Butenschoen and B.A. Kniehl, Reconciling $J/\psi$ Production at HERA, RHIC, Tevatron, and LHC with nonrelativistic QCD Factorization at Next-to-Leading Order, Phys. Rev. Lett. 106 (2011) 022003 [arXiv:1009.5662] [InSPIRE].
* (8) M. Butenschoen and B.A. Kniehl, $J/\psi$ production with NRQCD: HERA, Tevatron, RHIC and LHC, AIP. Conf. Proc. 1343 (2011) 409 [arXiv:1011.5619] [InSPIRE].
* (9) Y.-Q. Ma, K. Wang and K.-T. Chao, Complete next-to-leading order calculation of the $J/\psi$ and $\psi^{\prime}$ production at hadron colliders, Phys. Rev. D 84 (2011) 114001 [arXiv:1012.1030] [InSPIRE].
* (10) M. Butenschoen and B.A. Kniehl, World data of $J/\psi$ production consolidate nonrelativistic QCD factorization at next-to-leading order, Phys. Rev. D 84 (2011) 051501 [arXiv:1105.0820] [InSPIRE].
* (11) M. Butenschoen and B.A. Kniehl, Probing Nonrelativistic QCD Factorization in Polarized $J/\psi$ Photoproduction at Next-to-Leading Order, Phys. Rev. Lett. 107 (2011) 232001 [arXiv:1109.1476] [InSPIRE].
* (12) M. Butenschoen and B.A. Kniehl, $J/\psi$ Polarization at the Tevatron and the LHC: Nonrelativistic-QCD Factorization at the Crossroads, Phys. Rev. Lett. 108 (2012) 172002 [arXiv:1201.1872] [InSPIRE].
* (13) K.-T. Chao, Y.-Q. Ma, H.-S. Shao, K. Wang and Y.-J. Zhang, $J/\psi$ Polarization at Hadron Colliders in Nonrelativistic QCD, Phys. Rev. Lett. 108 (2012) 242004 [arXiv:1201.2675] [InSPIRE].
* (14) M. Butenschoen and B.A. Kniehl, $J/\psi$ production in NRQCD: A global analysis of yield and polarization, Nucl. Phys. Proc. Suppl. 222 (2012) 151 [arXiv:1201.3862] [InSPIRE].
* (15) B. Gong, L. -P. Wan, J. -X. Wang and H. -F. Zhang, Polarization for Prompt $J/\psi$ $(\psi^{\prime})$ production at the Tevatron and LHC, [arXiv:1205.6682] [InSPIRE]
* (16) P. Artoisenet, J.M. Campbell, J.P. Lansberg, F. Maltoni and F. Tramontano, $\Upsilon$ Production at Fermilab Tevatron and LHC Energies, Phys. Rev. Lett. 101 (2008) 152001 [arXiv:0806.3282] [InSPIRE].
* (17) Z.-B. Kang, J.-W. Qiu and G. Sterman, Factorization and Quarkonium Production, Nucl. Phys. Proc. Suppl. 214 (2012) 39 [InSPIRE].
* (18) Z.-B. Kang, J.-W. Qiu and G. Sterman, Heavy Quarkonium Production and Polarization, Phys. Rev. Lett. 108 (2012) 102002 [arXiv:1109.1520] [InSPIRE].
* (19) E. Braaten and T.C. Yuan, Gluon fragmentation into heavy quarkonium, Phys. Rev. Lett. 71 (1993) 1673 [hep-ph/9303205] [InSPIRE].
* (20) E. Braaten and T.C. Yuan, Gluon fragmentation into spin-triplet $S$-wave quarkonium, Phys. Rev. D 52 (1995) 6627 [hep-ph/9507398] [InSPIRE].
* (21) G.T. Bodwin and J. Lee, Relativistic corrections to gluon fragmentation into spin-triplet $S$-wave quarkonium, Phys. Rev. D 69 (2004) 054003 [hep-ph/0308016] [InSPIRE].
* (22) J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [InSPIRE].
* (23) M. Gremm and A. Kapustin, Annihilation of $S$-wave quarkonia and the measurement of $\alpha_{s}$, Phys. Lett. B 407 (1997) 323 [hep-ph/9701353] [InSPIRE].
* (24) L.D. Landau, The moment of a 2-photon system, Dokl. Akad. Nawk., USSR 60 (1948) 207.
* (25) C.-N. Yang, Selection Rules for the Dematerialization of a Particle into Two Photons, Phys. Rev. 77 (1950) 242 [InSPIRE].
* (26) G.T. Bodwin and A. Petrelli, Order $v^{4}$ corrections to $S$-wave quarkonium decay, Phys. Rev. D 66 (2002) 094011 [hep-ph/0205210] [InSPIRE].
* (27) N. Brambilla, E. Mereghetti and A. Vairo, Hadronic quarkonium decays at order $v^{7}$, Phys. Rev. D 79 (2009) 074002 [Erratum ibid. D 83 (2011) 079904] [arXiv:0810.2259] [InSPIRE].
* (28) G.T. Bodwin, D. Kang and J. Lee, Potential-model calculation of an order-$v^{2}$ nonrelativistic QCD matrix element, Phys. Rev. D 74 (2006) 014014 [hep-ph/0603186] [InSPIRE].
* (29) E. Braaten and Y.-Q. Chen, Dimensional regularization in quarkonium calculations, Phys. Rev. D 55 (1997) 2693 [hep-ph/9610401] [InSPIRE].
* (30) E. Braaten and J. Lee, Next-to-leading order calculation of the color-octet ${}^{3}S_{1}$ gluon fragmentation function for heavy quarkonium, Nucl. Phys. B 586 (2000) 427 [hep-ph/0004228] [InSPIRE].
* (31) J. Lee, Next-to-leading order calculation of a fragmentation function in a light-cone gauge, Phys. Rev. D 71 (2005) 094007 [hep-ph/0504285] [InSPIRE].
* (32) A.C. Hearn, REDUCE User’s Manual v. 3.7 (The RAND Corporation, Santa Monica, 1999) [E-mail: [email protected]].
* (33) R. Mertig, M. Böhm and A. Denner, Feyn Calc – Computer-algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [InSPIRE].
* (34) S. Wolfram, Mathematica, third edition (Wolfram Media, Champaign, Cambridge University Press, New York, 1996).
* (35) R. Barbieri, R. Gatto and R. Kogerler, Calculation of the annihilation rate Of $P$ wave quark-antiquark bound states, Phys. Lett. B 60 (1976) 183 [InSPIRE].
* (36) R. Barbieri, R. Gatto and E. Remiddi, Singular binding dependence in the hadronic widths of $1^{++}$ and $1^{+-}$ heavy quark antiquark bound states, Phys. Lett. B 61 (1976) 465 [InSPIRE].
* (37) C.-H. Chang, Hadronic production of $J/\psi$ associated with a gluon, Nucl. Phys. B 172 (1980) 425 [InSPIRE].
* (38) B. Guberina, J.H. Kühn, R.D. Peccei and R. Rückl, Rare decays of the $Z^{0}$, Nucl. Phys. B 174 (1980) 317 [InSPIRE].
* (39) E.L. Berger and D. Jones, Inelastic photoproduction of $J/\psi$ And $\Upsilon$ by gluons, Phys. Rev. D 23 (1981) 1521 [InSPIRE].
* (40) A. Petrelli, M. Cacciari, M. Greco, F. Maltoni and M.L. Mangano, NLO production and decay of quarkonium, Nucl. Phys. B 514 (1998) 245 [hep-ph/9707223] [InSPIRE].
* (41) A.V. Manohar, Heavy quark effective theory and nonrelativistic QCD Lagrangian to order $\alpha_{s}/m^{3}$, Phys. Rev. D 56 (1997) 230 [hep-ph/9701294] [InSPIRE].
* (42) M. Beneke, Non-relativistic effective theory for Quarkonium Production in Hadron Collisions, In “Stanford 1996, The strong interaction, from hadrons to partons” 549-574 [hep-ph/9703429] [InSPIRE].
* (43) G.T. Bodwin and Y.-Q. Chen, Renormalon ambiguities in NRQCD operator matrix elements, Phys. Rev. D 60 (1999) 054008 [hep-ph/9807492] [InSPIRE].
* (44) Zhi-Guo He, private communication.
* (45) P. Cho and A.K. Leibovich, Color-octet quarkonia production, Phys. Rev. D 53 (1996) 150 [hep-ph/9505329] [InSPIRE].
* (46) B. Gong, X.Q. Li and J.-X. Wang, QCD corrections to $J/\psi$ production via color octet states at Tevatron and LHC, Phys. Lett. B 673 (2009) 197 [Erratum ibid. B 693 (2010) 612] [arXiv:0805.4751] [InSPIRE].
* (47) G. T. Bodwin, H. S. Chung, D. Kang, J. Lee and C. Yu, Improved determination of color-singlet nonrelativistic QCD matrix elements for $S$-wave charmonium, Phys. Rev. D 77 (2008) 094017 [arXiv:0710.0994] [InSPIRE].
* (48) G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, J. Comput. Phys. 27 (1978) 192 [InSPIRE].
|
arxiv-papers
| 2012-08-27T06:47:34 |
2024-09-04T02:49:34.566381
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Geoffrey T. Bodwin (Argonne), U-Rae Kim, Jungil Lee (Korea U.)",
"submitter": "Jungil Lee",
"url": "https://arxiv.org/abs/1208.5301"
}
|
1208.5420
|
FTUV-2012-08-28 UAB-FT-719 IFIC-12-61
# MC generator TAUOLA: implementation of Resonance Chiral Theory for two and
three meson modes. Comparison with experiment
O. Shekhovtsova I. M. Nugent T. Przedzinski P. Roig Z. Was
###### Abstract
We present a partial upgrade of the Monte Carlo event generator TAUOLA with
the two and three hadron decay modes using the theoretical models based on
Resonance Chiral Theory. These modes account for 88% of total hadronic width
of the tau meson. First results of the model parameters have been obtained
using BaBar data for 3$\pi$ mode.
###### Keywords:
Tau physics, Monte Carlo event generator, chiral models
###### :
13.35.Dx; 12.39.Fe; 29.85.-c; 89.20.Ff
## 1 Introduction
Tau lepton is the only lepton that decays into hadrons. From the perspective
of low energies, as explored at B-factories, BaBar babar and Belle belle
experiments, hadronic decay modes of tau lepton provide an excellent
laboratory for modeling hadronic interactions at the energy scale of about 1-2
GeV, where neither perturbative QCD methods nor chiral lagrangians are
expected to work with a good precision Braaten:1990ef . From the perspective
of high energy experiments such as those at LHC, good understanding of tau
leptons properties contributes important ingredients of new physics
signatures. With the discovery of a new particle around the mass of 125-126GeV
higgs:2012 , tau decays are an important ingredient for determining if this is
the Standard Model Higgs. This is especially pertinent since CMS has reported
a deficit in the number of fermion decays from the new particle relative to
the Standard Model Higgs Prediction.
The first version of the program TAUOLA Jadach:1993hs was written in the
90’s. TAUOLA is Monte Carlo event generator which simulates tau decays for
both the leptonic and hadronic decays modes. The hadronic currents implemented
in TAUOLA are based on resonance dominance model (RDM) Kuhn 111Based on
Jadach:1993hs Cleo and Aleph collaborations developed their own versions, non
published, which differ from one another by the intermediate resonance states
because of different detector sensitivity.. In the framework of RDM the
hadronic current for a three-pseudoscalar decay mode is a sum of weighted
products of the Breit-Wigner functions. It is demonstrated in GomezDumm:2003ku
that this approach is able to reproduce only LO $\chi$PT result. Moreover, the
model was not sufficient to describe the Cleo $KK\pi$ data Coan:2004ep . This
resulted in the Cleo collaboration reshaping the model by introducing two ad-
hoc parameters that spoilt the QCD normalization for Weiss-Zumino
contribution. The parameters are obtained fitting to data. However, before
making conclusion that the Wess-Zumino anomaly normalization is spoilt it
should be checked whether on oversimplified theoretical approximation, like
RDM, was applied.
The alternative approach based on the Resonance Chiral Theory (R$\chi$T) rcht
was proposed in GomezDumm:2003ku . The computations done within R$\chi$T are
able to reproduce the low-energy limit of $\chi$PT at least up to NLO and
demonstrate the right falloff in the high energy region Roig:talk . The
hadronic currents for the two and three meson modes ($\pi\pi$, $K\pi$, $KK$,
$\pi\pi\pi$ and $KK\pi$) calculated in the framework of R$\chi$T were
implemented in TAUOLA Shekhovtsova:2012ra . The tar-ball version of the code
is available at web-page web:RChL .
## 2 Results for two and three meson final states
It is of utmost importance to implement the theory in a form that is as useful
for general applications as possible. Therefore the corresponding hadronic
currents ($J^{\mu}$) in the upgraded version of TAUOLA have been written in
the most general form compatible with Lorentz invariance. For $\tau$ decay
channels with two mesons ($h_{1}(p_{1})$ and $h_{2}(p_{2})$)
$J^{\mu}=N\bigl{[}(p_{1}-p_{2}-\frac{\Delta_{12}}{s}(p_{1}+p_{2}))^{\mu}F^{V}(s)+\frac{\Delta_{12}}{s}((p_{1}+p_{2})^{\mu}F^{S}(s)\bigr{]},$
(1)
where $\Delta_{12}=m_{1}^{2}-m_{2}^{2}$, $s=(p_{1}+p_{2})^{2}$. For the final
state of three pseudoscalars, with momenta $p_{1}$, $p_{2}$ and $p_{3}$,
Lorentz invariance determines the decomposition
$\displaystyle J^{\mu}$ $\displaystyle=N$
$\displaystyle\bigl{\\{}T^{\mu}_{\nu}\bigl{[}c_{1}(p_{2}-p_{3})^{\nu}F_{1}+c_{2}(p_{3}-p_{1})^{\nu}F_{2}+c_{3}(p_{1}-p_{2})^{\nu}F_{3}\bigr{]}$
(2) $\displaystyle+c_{4}q^{\mu}F_{4}-{i\over
4\pi^{2}F^{2}}c_{5}\epsilon^{\mu}_{.\
\nu\rho\sigma}p_{1}^{\nu}p_{2}^{\rho}p_{3}^{\sigma}F_{5}\bigr{\\}},$
where: $T_{\mu\nu}=g_{\mu\nu}-q_{\mu}q_{\nu}/q^{2}$ denotes the transverse
projector, $q^{\mu}=(p_{1}+p_{2}+p_{3})^{\mu}$ is the momentum of the hadronic
system, $F$ is the pion decay constant in the chiral limit.
The decomposition (1) and (2) is the most general one, model-dependence is
included in the hadronic form factors ($F_{V}$, $F_{S}$ as well as $F_{i}$, i
= 1…5). The hadronic form factors calculated within R$\chi$T can be written as
$F_{I}=F_{I}^{\chi}+F_{I}^{R}+F_{I}^{RR}$ (3)
where $F_{I}^{\chi}$ is the chiral contribution, $F_{I}^{R}$ is the one
resonance contribution and $F_{I}^{RR}$ is the double-resonance part. The
explict form of the functions $F_{i}$ for 3$\pi$ and $KK\pi$ modes can be
found in Shekhovtsova:2012ra , Section 2, as well as in GomezDumm:2003ku ;
Dumm . The theoretical results for the hadronic currents were obtained in the
isospin limit ($m_{\pi}=(2m_{\pi^{+}}+m_{\pi^{0}})/3$,
$m_{K}=(m_{K^{+}}+m_{K^{0}})/2$), except for the two pion and two kaons modes.
In the phase space generation, the differences between neutral and charged
pion and kaon masses is taken into account, i.e. physical values are chosen.
This has to be done to obtain proper kinematic configurations. The model
parameters, more specifically the masses of the resonances and the coupling
constants, were fitted to Aleph data, requiring correct high-energy behaviour
of the related form factors, see Appendix C in Shekhovtsova:2012ra .
To check stability of multidimensional integration in TAUOLA the MC results
were compared with the semi-analytical ones (Gauss integration of analytical
results was used). The difference between MC prediction and semi-analytical
results for the partial decay width is less than $0.02\%$ for all channels.
Both differential spectra and numerical tests are collected at web:RChL .
In Table 1, the partial decay width values from PDG Nakamura:2010zzi are
compared with our results obtained with isospin-averaged pseudoscalar masses
and with the physical ones. Comparison of the last two columns illustrates
numerical effect of physical masses. The difference between the MC result and
PDG one is $2\%-24\%$ depending on the channel. As expected, that agreement is
not good because only minimal attempts on adjusting to the model parameters
have been applied for the comparison with BaBar and Belle data. The next
section presents the first step toward this direction.
Channel | Width, [GeV]
---|---
| PDG | Equal masses | Phase space
| | | with masses
$\pi^{-}\pi^{0}\;\;\;\;$ | ($5.778\pm 0.35\%)\cdot 10^{-13}$ | ($5.2283\pm 0.005\%)\cdot 10^{-13}$ | $(5.2441\pm 0.005\%)\cdot 10^{-13}$
$K^{-}\pi^{0}\;\;\;\;$ | ($9.72\;\pm 3.5\%\;)\cdot 10^{-15}$ | ($8.3981\pm 0.005\%)\cdot 10^{-15}$ | $(8.5810\pm 0.005\%)\cdot 10^{-15}$
$\pi^{-}\bar{K}^{0}\;\;\;\;$ | ($1.9\;\;\;\pm 5\%\;\;\;)\cdot 10^{-14}$ | ($1.6798\pm 0.006\%)\cdot 10^{-14}$ | $(1.6512\pm 0.006\%)\cdot 10^{-14}$
$K^{-}K^{0}\;\;\;\;$ | ($3.60\;\pm 10\%\;\;)\cdot 10^{-15}$ | ($2.6502\pm 0.007\%)\cdot 10^{-15}$ | $(2.6502\pm 0.008\%)\cdot 10^{-15}$
$\pi^{-}\pi^{-}\pi^{+}$ | ($2.11\;\pm 0.8\%\;\;)\cdot 10^{-13}$ | ($2.1013\pm 0.016\%)\cdot 10^{-13}$ | $(2.0800\pm 0.017\%)\cdot 10^{-13}$
$\pi^{0}\pi^{0}\pi^{-}$ | ($2.10\;\pm 1.2\%\;\;)\cdot 10^{-13}$ | ($2.1013\pm 0.016\%)\cdot 10^{-13}$ | $(2.1256\pm 0.017\%)\cdot 10^{-13}$
$K^{-}\pi^{-}K^{+}$ | ($3.17\;\pm 4\%\;\;\;)\cdot 10^{-15}$ | ($3.7379\pm 0.024\%)\cdot 10^{-15}$ | $(3.8460\pm 0.024\%)\cdot 10^{-15}$
$K^{0}\pi^{-}\bar{K^{0}}$ | ($3.9\;\;\pm 24\%\;\;)\cdot 10^{-15}$ | ($3.7385\pm 0.024\%)\cdot 10^{-15}$ | $(3.5917\pm 0.024\%)\cdot 10^{-15}$
$K^{-}\pi^{0}K^{0}$ | ($3.60\;\pm 12.6\%\;\;)\cdot 10^{-15}$ | ($2.7367\pm 0.025\%)\cdot 10^{-15}$ | $(2.7711\pm 0.024\%)\cdot 10^{-15}$
Table 1: The $\tau$ decay partial widths. The PDG value Nakamura:2010zzi (2nd
collumn) is compared with numerical results from TAUOLA with the R$\chi$T
currents: in the isospin limit for pseudoscalar masses (3rd collumn), using
physical masses (4th collumn).
## 3 Fit for three pion mode to BaBar data
The main problem with upgrading the MC simulation of $\tau$ decays is a lack
of the published spectrum. Currently, only the two pion modes belle and three
pseudoscalar modes Nugent:2009zz are published.
The results from a preliminary fit to the three pion mode
($\pi^{+}\pi^{-}\pi^{-}$) Nugent:2009zz can be seen in Fig. 1 and Table 2.
Both the three particle and the $\pi^{+}\pi^{-}$ invariant mass distributions
have been considered. Disagreement about 12% level is visible in the low
energy region of two particle invariant mass distribution whereas for the
three invariant mass spectrum the difference between MC and data is less than
7%. However, the R$\chi$T parametrization is in better agreement with BaBar
data than CLEO one, see Fig. 1.
| $M_{\rho^{\prime}}$ | $\Gamma_{\rho^{\prime}}$ | $M_{a_{1}}$ | $F$ | $F_{V}$ | $F_{A}$ | $\beta_{\rho^{\prime}}$
---|---|---|---|---|---|---|---
Min. | 1.44 | 0.32 | 1.00 | 0.0920 | 0.12 | 0.1 | -0.36
Max | 1.48 | 0.39 | 1.24 | 0.0924 | 0.24 | 0.2 | -0.18
Default | 1.453 | 0.4 | 1.12 | 0.0924 | 0.18 | 0.149 | -0.25
Fit 222 | 1.4302 | 0.376061 | 1.21706 | 0.092318 | 0.121938 | 0.11291 | -0.208811
Table 2: Numerical values of the R$\chi$T parameters fitted to BaBar data for
three pion mode
The fit was done taking into account only $P$-wave contribution of two pion
system. As suggesting in Shibata:2002uv the discrepance in the low mass
region could be described using a contribution from the scalar particle,
$S$-wave contribution. We expect that inclusion of $S$-wave contribution
Roig:talk will improve the value of $\chi^{2}$. The values in the fifth row
of Table 2 are only the preliminary results. They will not necessarily
correspond to the minimum of $\chi^{2}/ndf$ of the final fit. Work is in
progress.
---
Figure 1: Invariant $\pi^{+}\pi^{-}\pi^{-}$ (up) and $\pi^{+}\pi^{-}$ (down)
mass distributions: the plots on the left-hand side correspond to the
differential decay distribution, the ones on the right-hand side to plot
ratios between MC and BaBar data. Lighter grey histograms are for R$\chi$T
parametrization, darker grey is for CLEO one.
## 4 Conclusion
The theoretical results for the hadronic currents of two and three
pseudoscalar modes, namely, $\pi\pi$, $K\pi$, $KK$, $\pi\pi\pi$ and $KK\pi$,
in the framework of R$\chi$T have been implemented in TAUOLA. These modes,
together with the one-meson decay modes, represent more than 88% of the
hadronic width of tau. R$\chi$T is a more controlled QCD-based model than the
usual used Breit-Wigner parametrizations. However, before making conclusion
about validity of the model the theoretical results have to be confronted with
the experimental data, which requires fit of the model parameters. Now that
the technical work on current installation is complete, the work on fits is in
progress in collaboration with theoreticians and experimentalists.
At LHC tau decays are only used for identification and are not used to study
their dynamic. However, the dynamics of tau decays are important for both
modeling the decays and -therefore the reconstruction and identification- and
for measuring the polarization of tau decays. Therefore, an upgrade to the
TAUOLA based on the BaBar and Belle results on tau decays is urgently needed
for LHC.
This research was supported by a Marie Curie Intra European Fellowship within
the 7th European Community Framework Programme (O.S.) and by Alexander von
Humboldt Foundation (I.N.), by the Spanish Consolider Ingenio 2010 Programme
CPAN (CSD2007-00042) and by MEC (Spain) under Grants FPA2007-60323,
FPA2011-23778 (O.S. and P.R.) and FPA2011-25948 (P.R.) and in part by the
funds of Polish National Science Centre under decision DEC-2011/03/B/ST2/00107
(T.P., Z.W.).
## References
* (1) B. Aubert et al., Phys.Rev.Lett. 100 (2008) 011801; J.P. Lees et al., arXiv: 1109.1527 [hep-ex].
* (2) M. Fujikawa et al., Phys. Rev. D78 (2008) 072006; H. Hayashii, M. Fujikawa, Nucl.Phys.Proc.Suppl. 198 (2010) 157.
* (3) E. Braaten and C.-S. Li, Phys.Rev. D42 (1990) 3888.
* (4) G. Aad et al., arXiv: 1207.7214 [hep-ex]; S. Chatrchyan, arXiv:1207.7235 [hep-ex].
* (5) S. Jadach, Z. Wa̧s, R. Decker and J. H. Kühn, Comput. Phys. Commun. 76 (1993) 361.
* (6) J.H. Kühn, E. Mirkes, Z. Phys. C56 (1992) 661; J.H. Kühn, A. Santamaría, Z. Phys. C48 (1990) 445.
* (7) D. G. Dumm, A. Pich and J. Portolés, Phys. Rev. D69 (2004) 073002.
* (8) T.E. Coan et al., Phys.Rev.Lett. 92 (2004) 232001.
* (9) G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael, Phys. Lett. B223 (1989) 425; Nucl. Phys. B321 (1989) 311.
* (10) P. Roig, I. Nugent, T. Przedzinski, O. Shekhovtsova, Z. Was, in these Proceedings.
* (11) O. Shekhovtsova, T. Przedzinski, P. Roig and Z. Was, arXiv: 1203.3955 [hep-ph].
* (12) T. Przedzinski, O. Shekhovtsova and Z. Was, http://annapurna.ifj.edu.pl/$\sim$wasm/RChL/RChL.htm.
* (13) D.G. Dumm, P. Roig, A. Pich, J. Portolés, Phys. Lett. B685 (2010) 158; Phys. Rev. D81 (2010) 034031.
* (14) K. Nakamura, J. Phys. G37 (2010) 075021.
* (15) I.M. Nugent, SLAC-R-936.
* (16) E.I. Shibata et al., arXiv: hep-ex/0210039.
|
arxiv-papers
| 2012-08-27T15:22:40 |
2024-09-04T02:49:34.580667
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "O. Shekhovtsova, I. M. Nugent, P. Roig, T. Przedzinski, Z. Was",
"submitter": "Olga Shekhovtsova",
"url": "https://arxiv.org/abs/1208.5420"
}
|
1208.5450
|
# Majorana fermions in one-dimensional spin-orbit coupled Fermi gases
Ran Wei Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China Laboratory of Atomic and Solid State Physics,
Cornell University, Ithaca, NY, 14850 Erich J. Mueller Laboratory of Atomic
and Solid State Physics, Cornell University, Ithaca, NY, 14850
###### Abstract
We theoretically study trapped one-dimensional Fermi gases in the presence of
spin-orbit coupling induced by Raman lasers. The gas changes from a
conventional (non-topological) superfluid to a topological superfluid as one
increases the intensity of the Raman lasers above a critical chemical-
potential dependent value. Solving the Bogoliubov-de Gennes equations self-
consistently, we calculate the density of states in real and momentum space at
finite temperatures. We study Majorana fermions (MFs) which appear at the
boundaries between topologically trivial and topologically non-trivial
regions. We linearize the trap near the location of a MF, finding an analytic
expression for the localized MF wavefunction and the gap between the MF state
and other edge states.
###### pacs:
67.85.Lm, 03.75.Ss, 05.30.Fk, 03.65.Vf
## I introduction
Majorana fermions (MFs), exotic excitations which are their own antiparticles,
have attracted a great deal of attention recently Mf . Condensed matter
systems with MFs possess degeneracies that are intrinsically nonlocal, and can
be manipulated geometrically. They can, in principle, be used to make a robust
quantum computer Sarma2008 . Condensed matter theorists have proposed various
ways to explore MFs during the past several years Read2000 ; Kitaev2001 ;
SSthoery ; Fuliang2008 ; Oreg2010 ; braiding . Four experimental groups have
recently reported evidence of MFs in semiconducting wires on superconducting
substrates SSexperiment . In those experiments, spin-orbit (SO) coupling was
important. Here we study MFs in a related cold atom system.
Two groups Zhang2012 ; Zwierlein2012 have successfully generated SO coupled
Fermi gases based on a Raman technique pioneered by Spielman’s group at NIST
Spielman2011 . Several theoretical groups have proposals for creating and
probing MFs in these SO coupled Fermi gases CAtheory ; Zoller2011 ; Huihu2012a
; Huihu2012b . We build upon the studies of Jiang et al. Zoller2011 and Liu
et al. Huihu2012a , which find MFs in a 1D geometry.
We study a 1D (pseudo) spin-$1/2$ Fermi gas with point interactions. In the
presence of Raman lasers, the energy spectrum has two helical bands. We study
this two-band model in a harmonic trap. Solving the Bogoliubov-de Gennes (BdG)
equations self-consistently, we calculate the density of states (DOS) in real
and momentum space at finite temperatures. We linearize the trap near the
location of a MF, finding an analytic expression for the localized MF
wavefunction and the gap between the MF state and other edge states.
Our numerical calculations extend the similar studies of Ref. Huihu2012a . We
explore a larger range of temperatures, and delve deeper into the physics near
the MFs. We also investigate a truncated one-band model.
One concern with mean-field calculations such as ours, is that they are unable
to capture the large phase fluctuations found in 1D. As shown by Ref.
multiwires , the MF physics is robust against these fluctuations. Moreover, an
actual experiment would be performed on a bundle of weakly coupled tubes
Hulet2010 . This latter setting also avoids issues of number conservation
multiwires . Our 1D model faithfully describes the properties of a single tube
within such a bundle when the tunneling is weak.
This paper is organized as follows. In Sec. II, we discuss the homogeneous
gas: We start with the two-band model, and in Sec. II(A) show how it relates
to a one-band model with $p$-wave interactions. In Sec. II(B), we describe the
band structure and topology of the two-band model. In Sec. III, we calculate
the properties of trapped gases: In Sec. III(A), we write the BdG equations
and self-consistently calculate the order parameter and density. In Sec.
III(B), we visualize the MFs by calculating the DOS in real space and momentum
space. In Sec. III(C), we introduce MF operators and construct the localized
MF states. In Sec. III(D), we linearize the trap near the location of a MF,
finding an analytic expression for the localized MF wavefunction and the gap
between the MF state and other edge states. Finally we conclude in Sec. IV.
## II homogeneous gas
We start from the Hamiltonian of the 1D (pseudo) spin-$1/2$ Fermi gases with
chemical potential $\mu$,
$\displaystyle
H=\int\bigg{(}\Psi^{\dagger}(x)\big{(}H_{0}(x)-\mu\big{)}\Psi(x)\bigg{)}dx+H_{I},$
(1)
where
$\Psi(x)=\big{(}\psi_{\uparrow}(x),\psi_{\downarrow}(x)\big{)}^{\intercal}$
annihilates the spin-up and spin-down states. In an experiment,
$\psi_{\uparrow}$ and $\psi_{\downarrow}$ correspond to two different
hyperfine states of a fermionic atom such as ${}^{40}K$. The single-particle
Hamiltonian
$H_{0}(x)=-\frac{\hbar^{2}}{2m}\partial_{x}^{2}+\frac{i\hbar^{2}k_{L}}{m}\partial_{x}\sigma_{z}+\frac{\hbar\Omega}{2}\sigma_{x}+E_{r}$
can be engineered by Raman lasers Spielman2011 , whose intensity is
characterized by the Rabi frequency $\Omega$. The recoil momentum of the Raman
lasers is $\hbar k_{L}$, $E_{r}=\frac{\hbar^{2}k_{L}^{2}}{2m}$ is the recoil
energy, and $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of
Pauli matrices. For ultra-cold fermions, the interaction may be modeled by
$H_{I}=g_{1D}\int\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\psi_{\downarrow}\psi_{\uparrow}dx$,
with coupling constant $g_{1D}$. This coefficient can be related to the three-
dimensional scattering length and the geometry of the confinement Olshanii1998
. In a typical experiment, $|g_{1D}|\sim 70a_{0}E_{r}$ Hulet2010 , where
$a_{0}$ is the Bohr radius. We restrict ourselves to attractive interactions,
$g_{1D}<0$. We note that if we rotate our spin basis
($\sigma_{x}\rightarrow\sigma_{z},\sigma_{z}\rightarrow\sigma_{y}$) and
identify $Z=\frac{\hbar\Omega}{2}$ as a Zeeman field, and
$\alpha=\frac{\hbar^{2}k_{L}}{2m}$ as the SO coupling strength, we recover the
Hamiltonian of a semiconducting wire. Note $H_{I}$ is very different for a
wire SSthoery . In the following sections we explore the physics of Eq. (1).
### II.1 One-band model
To get insight into Eq. (1), we first consider an approximation where we
truncate to a single band. We emphasize however that in all other sections, we
work with the full two-band Hamiltonian.
The physics of the single particle Hamiltonian is most transparent in momentum
space,
$H=\sum_{k}\Psi^{\dagger}_{k}\left(\frac{\hbar^{2}k^{2}}{2m}+E_{r}-\frac{\hbar^{2}kk_{L}}{m}\sigma_{z}+\frac{\hbar\Omega}{2}\sigma_{x}\right)\Psi_{k}$,
where $\Psi_{k}=\left(\psi_{k\uparrow},\psi_{k\downarrow}\right)^{\intercal}$.
This Hamiltonian is readily diagonalized by
$\displaystyle\left(\begin{array}[]{c}d_{k}\\\ c_{k}\\\ \end{array}\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}{\rm
cos}\frac{\theta_{k}}{2}&{\rm sin}\frac{\theta_{k}}{2}\\\ -{\rm
sin}\frac{\theta_{k}}{2}&{\rm cos}\frac{\theta_{k}}{2}\\\
\end{array}\right)\left(\begin{array}[]{c}\psi_{k\uparrow}\\\
\psi_{k\downarrow}\\\ \end{array}\right)$ (8)
with ${\rm tan}\theta_{k}=-\frac{m\Omega}{2k\hbar k_{L}}$, yielding
$H=\sum_{k}\left(E_{-}c_{k}^{\dagger}c_{k}+E_{+}d_{k}^{\dagger}d_{k}\right)$.
The energy spectrum has two helical bands, illustrated in Fig. 1.
Figure 1: (color online) Band structure of a 1D (pseudo) spin-$1/2$ gas. The
red (dashed) curves are the bare bands in the absence of SO coupling. The blue
(thick) curves are the upper band $E_{+}$ and lower band $E_{-}$ in the
presence of SO coupling, with the coupling strength $\hbar\Omega/E_{r}=1$.
If the effective chemical potential $\tilde{\mu}=\mu-
E_{r}\ll\frac{\hbar\Omega}{2}$, only the lower band $E_{-}$ is filled with
fermions. Projecting the interactions into this band, we find
$\displaystyle
H_{I}^{1B}=\tilde{g}_{1D}\sum_{kqq^{\prime}}\left(V_{kq}c^{\dagger}_{\frac{k}{2}+q}c^{\dagger}_{\frac{k}{2}-q}\right)\left(V_{kq^{\prime}}c_{\frac{k}{2}-q^{\prime}}c_{\frac{k}{2}+q^{\prime}}\right),$
(9)
where $\tilde{g}_{1D}=g_{1D}/L_{1D}$, with $L_{1D}$ the length of the gas. The
fermionic anti-commutation relation,
$c^{\dagger}_{\frac{k}{2}+q}c^{\dagger}_{\frac{k}{2}-q}=-c^{\dagger}_{\frac{k}{2}-q}c^{\dagger}_{\frac{k}{2}+q}$,
implies that the interaction coefficient $V_{kq}$ is odd with respective to
$q$, $V_{kq}=\frac{1}{2}{\rm sin}\frac{\theta_{k/2+q}-\theta_{k/2-q}}{2}$. At
zero center of mass momentum, $V_{q}\equiv
V_{k=0,q}=\frac{q}{2\sqrt{q^{2}+\hbar^{2}k_{L}^{2}\Omega^{2}/16E_{r}^{2}}}$.
In Fig. 2, we plot $V_{kq}$ as a function of $q$. The dependence on $k$ is
weak for $k\lesssim k_{L}$.
The interaction in Eq. (9) is separable. Given that $\tilde{g}_{1D}<0$, this
interaction can lead to pairing with zero center of mass and an order
parameter $\Delta_{q}=\tilde{g}_{1D}V_{q}\sum_{q^{\prime}}\langle
V_{q^{\prime}}c_{-q^{\prime}}c_{q^{\prime}}\rangle$, where
$\langle...\rangle\equiv\frac{{\rm Tr}(e^{-H/k_{b}T}...)}{{\rm
Tr}(e^{-H/k_{b}T})}$ is the thermal average, $k_{b}$ is the Boltzman constant
and $T$ is the temperature. The mean-field interaction becomes
$H_{I}^{1B}=\sum_{q}\left(\Delta_{q}c_{q}^{\dagger}c^{\dagger}_{-q}+\Delta_{q}^{*}c_{-q}c_{q}\right)-\tilde{g}_{1D}|\sum_{q}V_{q}\langle
c_{-q}c_{q}\rangle|^{2}$. By virtue of the symmetry of $V_{q}$, the order
parameter has a $p$-wave symmetry $\Delta_{-q}=-\Delta_{q}$. As is well
established, such a $p$-wave superfluid may possess Majorana edge modes
Kitaev2001 . We will discuss these Majorana modes at length in the two-band
model.
Figure 2: (color online) Interaction coefficient $V_{kq}$ versus dimensionless
momentum $q/k_{L}$ for $\hbar\Omega/E_{r}=2$. The blue (thick), green (dashed)
and red (dotted) curves correspond to $k=0,0.5k_{L}$ and $k_{L}$ respectively.
### II.2 Two-band model
While the one-band model connects the SO coupled gases and $p$-wave
superconductors, we will focus on the richer two-band model in the remainder
of the manuscript. Within the mean-field approach, the interaction term is
bilinear
$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle
g_{1D}\int\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\psi_{\downarrow}\psi_{\uparrow}dx$
(10) $\displaystyle\approx$
$\displaystyle\int\left(\Delta(x)\left(\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}+\psi_{\downarrow}\psi_{\uparrow}\right)-\frac{\Delta(x)^{2}}{g_{1D}}\right)dx,$
(11)
where the order parameter
$\Delta(x)=g_{1D}\langle\psi_{\downarrow}\psi_{\uparrow}\rangle$ is assumed to
be real. Defining the operator
$\tilde{\Psi}^{\dagger}(x)=\left(\psi_{\uparrow}^{\dagger}(x),\psi_{\downarrow}^{\dagger}(x),\psi_{\downarrow}(x),\psi_{\uparrow}(x)\right)$,
the Hamiltonian can be written as,
$\displaystyle
H=\int\left(\frac{1}{2}\tilde{\Psi}^{\dagger}(x)\mathcal{H}\tilde{\Psi}(x)-\frac{\Delta(x)^{2}}{g_{1D}}\right)dx+\frac{1}{2}\left(T_{-}+T_{+}\right),$
(12)
where
$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\left(-\frac{\hbar^{2}}{2m}\partial_{x}^{2}-\tilde{\mu}\right)\tau_{z}+\frac{i\hbar^{2}k_{L}}{m}\partial_{x}\tau_{z}\sigma_{z}$
(13) $\displaystyle+$
$\displaystyle\frac{\hbar\Omega}{2}\tau_{z}\sigma_{x}+\Delta(x)\tau_{x}\sigma_{z},$
$\displaystyle T_{\pm}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\left(-\frac{\hbar^{2}}{2m}\partial_{x}^{2}-\tilde{\mu}\pm\frac{i\hbar^{2}k_{L}}{m}\partial_{x}\right).$
(14)
The Pauli matrices $\bm{\sigma},\bm{\tau}$ operate in the spin subspace and
particle-hole subspace respectively,
$\displaystyle\sigma_{x}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}&1&&\\\ 1&&&\\\ &&&1\\\ &&1&\\\
\end{array}\right),\tau_{x}=\left(\begin{array}[]{cccc}&&1&\\\ &&&1\\\ 1&&&\\\
&1&&\\\ \end{array}\right)$ (23) $\displaystyle\sigma_{z}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&&&\\\ &-1&&\\\ &&1&\\\ &&&-1\\\
\end{array}\right),\tau_{z}=\left(\begin{array}[]{cccc}1&&&\\\ &1&&\\\
&&-1&\\\ &&&-1\\\ \end{array}\right).$ (32)
The elementary excitations can be found by solving the BdG equations
$\mathcal{H}W=EW$. When $\Delta(x)=\Delta$ is spatially homogeneous, one can
write the BdG equations in momentum space as $\mathcal{H}_{k}W(k)=E(k)W(k)$,
where $\mathcal{H}_{k}$ is the $4\times 4$ matrix produced by replacing
$-i\partial_{x}\rightarrow k$ in Eq. (13). The excitation spectrum $E(k)$ is
most simply calculated by squaring $\mathcal{H}_{k}$ twice, and extracting the
characteristic polynomial Oreg2010 . This procedure yields
$\displaystyle E_{\pm}^{2}(k)$ $\displaystyle=$
$\displaystyle\epsilon_{0}^{2}+2E_{r}\hbar^{2}k^{2}/m+\hbar^{2}\Omega^{2}/4+\Delta^{2}$
(33) $\displaystyle\pm$
$\displaystyle\hbar\sqrt{8E_{r}\epsilon_{0}^{2}k^{2}/m+\Omega^{2}\epsilon_{0}^{2}+\Omega^{2}\Delta^{2}},$
where $\epsilon_{0}=\frac{\hbar^{2}k^{2}}{2m}-\tilde{\mu}$. The four bands
$E_{+}(k),E_{-}(k),-E_{-}(k),-E_{+}(k)$, as shown in Fig. 3, correspond to the
four eigenvectors $W^{p+}(k)$, $W^{p-}(k)$, $W^{h-}(k)$, $W^{h+}(k)$. The
Hamiltonian $\mathcal{H}$ has the intrinsic symmetry,
$\\{\mathcal{H},\tau_{y}\\}=1$. Given two eigenvectors $W^{p\pm}$ with
eigenvalues $E^{\pm}$, one can always construct the other two
$W^{h\pm}=i\tau_{y}W^{p\pm}$ with eigenvalues $-E^{\pm}$. We therefore denote,
$\displaystyle W^{p+}(k)$ $\displaystyle=$
$\displaystyle\bigl{(}u^{+}_{k\uparrow},u^{+}_{k\downarrow},-v^{+}_{k\downarrow},-v^{+}_{k\uparrow}\bigr{)}^{\intercal}$
(34) $\displaystyle W^{p-}(k)$ $\displaystyle=$
$\displaystyle\bigl{(}u^{-}_{k\uparrow},u^{-}_{k\downarrow},-v^{-}_{k\downarrow},-v^{-}_{k\uparrow}\bigr{)}^{\intercal}$
(35) $\displaystyle W^{h-}(k)$ $\displaystyle=$
$\displaystyle\bigl{(}v^{-}_{k\downarrow},v^{-}_{k\uparrow},u^{-}_{k\uparrow},u^{-}_{k\downarrow}\bigr{)}^{\intercal}$
(36) $\displaystyle W^{h+}(k)$ $\displaystyle=$
$\displaystyle\bigl{(}v^{+}_{k\downarrow},v^{+}_{k\uparrow},u^{+}_{k\uparrow},u^{+}_{k\downarrow}\bigr{)}^{\intercal}.$
(37)
The unitary condition on the 4 by 4 matrix
$\big{(}W^{p+}(k),W^{p-}(k),W^{h-}(k),W^{h+}(k)\big{)}$ also leads to the
equalities $u^{\pm}_{k\downarrow}=(u^{\pm}_{-k\uparrow})^{*}$ and
$v^{\pm}_{k\downarrow}=-(v^{\pm}_{-k\uparrow})^{*}$.
Figure 3: (color online) Band structure of homogeneous gas. From the top to
the bottom, the four bands are $E_{+},E_{-},-E_{-},-E_{+}$ respectively. The
parameters are $\mu=E_{r},\Delta=0.5E_{r}$, and (a)$\Omega=0$,
(b)$\Omega=0.5E_{r}$, (c)$\Omega=E_{r}$, (d)$\Omega=1.5E_{r}$.
In Fig. 3, the spectrum is shown for a range of parameters. One important
feature is the $k=0$ gap $E_{0}\equiv 2E_{-}(k=0)\equiv 2|G|$, where
$G\equiv\sqrt{\tilde{\mu}^{2}+\Delta^{2}}-\hbar\Omega/2$. When $\Omega=0$, the
two positive energy bands touch at $k=0$, and $E_{0}>0$. The gas is in the
same universality class as a conventional $s$-wave superconductor. Increasing
the Raman laser strength such that
$0<\hbar\Omega<2\sqrt{\Delta^{2}+\tilde{\mu}^{2}}$ separates the two bands and
reduce $E_{0}$. At $\hbar\Omega=2\sqrt{\Delta^{2}+\tilde{\mu}^{2}}$, $E_{0}$
is zero, and there is a topological transition. Once
$\hbar\Omega>2\sqrt{\Delta^{2}+\tilde{\mu}^{2}}$, $E_{0}$ is again positive,
but the gas is no longer a conventional superfluid, instead it has a non-
trivial topological invariant.
The relevant topological invariant is a Berry phase. Eqs. (34-37) can be
thought of as maps from the real line ($-\infty\leq k\leq\infty$) to the space
of unit vectors in SU$(4)$. One can generate a closed path by taking
$\displaystyle\mathcal{C}:$ $\displaystyle W^{p+}(-\infty)$
$\displaystyle\rightarrow W^{p+}(\infty)=W^{p-}(-\infty)\rightarrow$ (38)
$\displaystyle W^{p-}(+\infty)$ $\displaystyle=W^{p+}(-\infty).$
The equalities follow from noting that up to phases
$\displaystyle W^{p+}(-\infty)$ $\displaystyle=$ $\displaystyle
W^{p-}(\infty)=\bigl{(}0,1,0,0\bigr{)}^{\intercal}$ (39) $\displaystyle
W^{p-}(-\infty)$ $\displaystyle=$ $\displaystyle
W^{p+}(\infty)=\bigl{(}1,0,0,0\bigr{)}^{\intercal}.$ (40)
Given this closed path, one can define the Berry phase
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
i\oint_{\mathcal{C}}\bm{W}^{*}\cdot\partial_{k}\bm{W}dk$ $\displaystyle=$
$\displaystyle
i\int_{-\infty}^{\infty}\big{(}W^{p+}(k)\big{)}^{*}\cdot\partial_{k}W^{p+}(k)dk$
$\displaystyle+$ $\displaystyle
i\int_{-\infty}^{\infty}\big{(}W^{p-}(k)\big{)}^{*}\cdot\partial_{k}W^{p-}(k)dk.$
(42)
In the case of a gauge which is not smooth, one would instead use
$\displaystyle e^{i\gamma}$ $\displaystyle=$ $\displaystyle\lim_{\delta
k\rightarrow
0}\bigg{(}\prod_{k=-\infty}^{\infty}\big{(}W^{p+}(k)\big{)}^{*}\cdot
W^{p+}(k-\delta k)$ $\displaystyle\times$
$\displaystyle\prod_{k=-\infty}^{\infty}\big{(}W^{p-}(k)\big{)}^{*}\cdot
W^{p-}(k-\delta k)\bigg{)}$ $\displaystyle\times$
$\displaystyle\big{(}W^{p+}(-\infty)\big{)}^{*}\cdot
W^{p-}(\infty)\big{(}W^{p-}(-\infty)\big{)}^{*}\cdot W^{p+}(\infty).$
Since $H$ has real valued matrix elements, $e^{i\gamma}=\pm 1$. The Berry
phase $\gamma$ is only well defined if the spectrum has no degeneracies. We
restrict $0\leq\gamma<2\pi$. For $G>0$, we find $\gamma=\pi$. For $G<0$,
$\gamma=0$. Somewhat counter-intuitively the $\gamma=0$ sector corresponds to
the “topologically non-trivial” state analogous to a 1D spinless $p$-wave
superconductor.
## III traps
In this section we will solve the BdG equations for a trapped gas. The
qualitative features of our results can be anticipated by treating the system
as locally homogeneous: the properties at position $x$ will be reminiscent of
those corresponding to a homogeneous gas with chemical potential
$\tilde{\mu}(x)=\tilde{\mu}-V(x)$. Within this local density approximation
(LDA), one can define a function
$G(x)=\sqrt{\tilde{\mu}(x)+\Delta(x)^{2}}-\hbar\Omega/2$, where $G(x)=0$
corresponds to the boundaries between topologically distinct regions. One
expects there will be Majorana excitations at the boundaries. We will use
numerical solution of the BdG equation to explore this physics beyond the LDA.
Further, in Sec. III(D) we will linearize the BdG equations about the points
$G(x)=0$, and analytically investigate these Majorana modes, without making a
LDA.
### III.1 Order parameter and density
In the presence of a trap, we write the BdG equations in real space,
$\displaystyle\mathcal{H}_{trap}W_{n}(x)$ $\displaystyle=$ $\displaystyle
E_{n}W_{n}(x),$ (44)
where
$\displaystyle\mathcal{H}_{trap}$ $\displaystyle=$
$\displaystyle\left(-\frac{\hbar^{2}}{2m}\partial_{x}^{2}-\tilde{\mu}+V(x)\right)\tau_{z}$
(45) $\displaystyle+$
$\displaystyle\frac{i\hbar^{2}k_{L}}{m}\partial_{x}\tau_{z}\sigma_{z}+\frac{\hbar\Omega}{2}\tau_{z}\sigma_{x}+\Delta(x)\tau_{x}\sigma_{z}.$
The eigenvectors $W_{n}(x)$ come in pairs, $W^{p}_{n}(x)$ and $W^{h}_{n}(x)$,
which correspond to eigenvalues $E_{n}\geq 0$ and $-E_{n}$,
$\displaystyle W_{n}^{p}(x)$ $\displaystyle=$
$\displaystyle\bigl{(}u_{n\uparrow}(x),u_{n\downarrow}(x),v_{n\downarrow}(x),v_{n\uparrow}(x)\bigr{)}^{\intercal}$
(46) $\displaystyle W_{n}^{h}(x)$ $\displaystyle=$
$\displaystyle\bigl{(}v^{*}_{n\uparrow}(x),v^{*}_{n\downarrow}(x),u^{*}_{n\downarrow}(x),u^{*}_{n\uparrow}(x)\bigr{)}^{\intercal}.$
(47)
To make contact with our previous discussion, we note that in the spatially
homogeneous case, $n$ can be represented by a momentum $k_{n}$ and a sign
$\varepsilon_{n}=\pm$, so that
$W_{n}^{p}(x)=e^{ik_{n}x}\big{(}u_{k_{n}\uparrow}^{\varepsilon_{n}},u_{k_{n}\downarrow}^{\varepsilon_{n}},v_{k_{n}\downarrow}^{\varepsilon_{n}},v_{k_{n}\uparrow}^{\varepsilon_{n}}\big{)}^{\intercal}$
and
$W_{n}^{h}(x)=e^{ik_{n}x}\big{(}(v_{k_{n}\uparrow}^{\varepsilon_{n}})^{*},(v_{k_{n}\downarrow}^{\varepsilon_{n}})^{*},(u_{k_{n}\downarrow}^{\varepsilon_{n}})^{*},(u_{k_{n}\uparrow}^{\varepsilon_{n}})^{*}\big{)}^{\intercal}$.
One then recovers Eqs. (34)-(37).
Fixing $\\{\tilde{\mu},\Omega,g_{1D}\\}$, we solve Eqs. (44) iteratively. We
discretize space into $n_{\rm grid}$ equally spaced points, and use a finite
difference method with a pseudo-spectral scheme to represent
$\mathcal{H}_{trap}$ as a $4n_{\rm grid}$ by $4n_{\rm grid}$ matrix. In the
$j$th iteration, we numerically diagonalize the matrix
$\mathcal{H}_{trap}^{(j)}$ with the order parameter $\Delta^{(j)}(x)$. We
start with a constant $\Delta^{(0)}(x)=\Delta_{0}$. We extract the
eigenvectors $W_{n}^{(j)}(x)$ and calculate the order parameter
$\Delta^{(j+1)}(x)=g_{1D}\sum_{n}\bigl{(}u_{n\downarrow}^{(j)}v^{*(j)}_{n\uparrow}\langle\xi_{n}\xi_{n}^{\dagger}\rangle+v_{n\downarrow}^{*(j)}u_{n\uparrow}^{(j)}\langle\xi_{n}^{\dagger}\xi_{n}\rangle\bigr{)}$,
where $\xi_{n}$ is the annihilation operator of the Bogoliubov particle. At
temperature $T$,
$\langle\xi_{n}^{\dagger}\xi_{n}\rangle=1/(e^{E_{n}/k_{b}T}+1)$. Then we
diagonalize $\mathcal{H}_{trap}^{(j+1)}$ and repeat the procedure. We stop
iterating when $|\Delta^{(j+1)}(x)-\Delta^{(j)}(x)|$ falls below a threshold.
The final convergent order parameter $\Delta^{(N)}(x)$ is largely independent
of $n_{\rm grid}$ and $\Delta^{(0)}(x)$ when $n_{\rm grid}\geq 1200$. In the
Appendix we explore the convergence with the real space grid size $n_{\rm
grid}$.
Figure 4: (color online) Profiles of order parameter
$\Delta(x)=g_{1D}\sum_{n}\bigl{(}u_{n\downarrow}(x)v^{*}_{n\uparrow}(x)\langle\xi_{n}\xi_{n}^{\dagger}\rangle+v_{n\downarrow}^{*}(x)u_{n\uparrow}(x)\langle\xi_{n}^{\dagger}\xi_{n}\rangle\bigr{)}$
(dashed curves) and density
$n(x)=\sum_{n\sigma}\bigl{(}|v_{n\sigma}(x)|^{2}\langle\xi_{n}\xi_{n}^{\dagger}\rangle+|u_{n\sigma}(x)|^{2}\langle\xi_{n}^{\dagger}\xi_{n}\rangle\bigr{)}$
(solid curves) at temperatures $T=0,0.1E_{r},0.2E_{r},0.3E_{r}$. Other
parameters are
$g_{1D}=-0.03E_{r}L,\hbar\Omega=2E_{r},\lambda=4,k_{L}L=100,(a)\mu=E_{r},(b)\mu=2.5E_{r}$.
The order parameters and density profiles for a gas in a harmonic trap
$V(x)=\lambda(x/L)^{2}E_{r}$ are shown in Fig. 4, where the dimensionless
parameter $\lambda=4$ characterizes the stiffness of the trap, and $2L$ is the
simulation length with $k_{L}L=100$. We choose the Rabi frequency to be
$\hbar\Omega=2E_{r}$, and take $g_{1D}=-0.03E_{r}L$, corresponding to the
dimensionless interaction strength $\beta=m|g_{1D}|/\hbar^{2}n_{0}\sim 2$,
where $n_{0}$ is the central density at zero temperature. For comparisons,
experiments on 1D Fermi gases at Rice have $\beta\sim 3$ Hulet2010 . If
$E_{r}/\hbar=50\rm{kHz}$ (a typical experimental value), then these parameters
correspond to a trap with small oscillation frequency $\omega=2\rm{kHz}$. The
order parameter has qualitatively different behavior if the center of the trap
has one or two bands occupied. For relatively small chemical potential
$E_{r}-\sqrt{(\hbar\Omega/2)^{2}-\Delta(x)^{2}}\lesssim\mu\lesssim
E_{r}+\sqrt{(\hbar\Omega/2)^{2}-\Delta(x)^{2}}$, the center of the trap will
be topologically non-trivial while the wings will be trivial. This regime is
illustrated in Fig. 4(a). The order parameter grows near the edge of the
cloud. This is a feature of 1D where, due to the divergence of the low energy
density of state, the interactions are stronger for lower density Petrov2004 .
For $\mu\gtrsim E_{r}+\sqrt{(\hbar\Omega/2)^{2}-\Delta(x)^{2}}$, the center
will be topologically trivial, but there will be a band of the non-trivial
state further out. Here the order parameter profile is quite rich, with a
central plateau, surrounded by two valleys and two peaks. The central plateau
roughly corresponds to where both bands are occupied. The order parameter is
sensitive to temperature. The bulk $\Delta$ is significantly suppressed and
vanishes for $T\gtrsim 0.2E_{r}$. The density has no notable structure and is
nearly independent of temperature for $T\lesssim 0.3E_{r}$.
### III.2 Density of states (DOS)
Figure 5: (color online) Density of states (DOS) in real space (left panel)
and momentum space (right panel) at $T=0,0.1E_{r},0.2E_{r},0.3E_{r}$ from the
top to the bottom, with order parameters identical to those in Fig. 4(a). The
grey (dashed) curves in the left panels is plotted with
$G(x)=\sqrt{\tilde{\mu}(x)^{2}+\Delta(x)^{2}}-\hbar\Omega/2$, where the zero
points of $G(x)$ pinpoint the position of MFs. The brighter color corresponds
to the higher spectral weight.
The elementary excitations are encoded in the single particle Green function
$G_{\sigma\sigma^{\prime}}(x,t,x^{\prime},t^{\prime})=\frac{1}{i}\langle\hat{T}\psi_{\sigma}(x,t)\psi_{\sigma^{\prime}}(x^{\prime},t^{\prime})\rangle$
and the associated spectral density
$A_{\sigma\sigma^{\prime}}(x,x^{\prime},E)=2{\rm Im}\int
e^{iEt}G_{\sigma\sigma^{\prime}}(x,t,x^{\prime},0)$, where $\hat{T}$ is the
time-ordering operator. A local tunneling experiment can measure the density
of states (DOS)
$\rho(E,x)=A_{\uparrow\uparrow}(x,x,E)+A_{\downarrow\downarrow}(x,x,E)$. This
quantity gives the number of single particle states with energy $E$ at
position $x$. It can be understood as an application of Fermi’s Golden rule to
the response to a tunneling probe. Within our mean-field theory
$\displaystyle\rho(E,x)$ $\displaystyle=$
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}\big{(}\rho^{h}_{\sigma}(-E,x)+\rho^{p}_{\sigma}(E,x)\big{)},$
(48)
where
$\displaystyle\rho^{h}_{\sigma}(E,x)$ $\displaystyle=$
$\displaystyle\sum_{n}\big{|}v_{n\sigma}(x)\big{|}^{2}\delta(E_{n}-E)$ (49)
$\displaystyle\rho^{p}_{\sigma}(E,x)$ $\displaystyle=$
$\displaystyle\sum_{n}\big{|}u_{n\sigma}(x)\big{|}^{2}\delta(E_{n}-E).$ (50)
We can similarly introduce a momentum resolved DOS $\rho(E,k)=\int
e^{ik(x-x^{\prime})}\big{(}A_{\uparrow\uparrow}(x,x^{\prime},E)+A_{\downarrow\downarrow}(x,x^{\prime},E)\big{)}dxdx^{\prime}$,
which can be measured with momentum resolved radio-frequency spectroscopy
Zwierlein2012 . In the present case
$\displaystyle\rho(E,k)$ $\displaystyle=$
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}\big{(}\rho_{\sigma}^{h}(-E,k)+\rho_{\sigma}^{p}(E,k)\big{)},$
(51)
where
$\displaystyle\rho_{\sigma}^{h}(E,k)$ $\displaystyle=$
$\displaystyle\sum_{n}\bigg{|}\int
v_{n\sigma}(x)e^{ikx}dx\bigg{|}^{2}\delta(E_{n}-E)$ (52)
$\displaystyle\rho_{\sigma}^{p}(E,x)$ $\displaystyle=$
$\displaystyle\sum_{n}\bigg{|}\int
u_{n\sigma}(x)e^{ikx}dx\bigg{|}^{2}\delta(E_{n}-E).$ (53)
Figure 6: (color online) Density of states (DOS) in real space (left panel)
and momentum space (right panel) at $T=0$ and $0.3E_{r}$, with parameters
identical to those in Fig. 4(b). The brighter color corresponds to the higher
spectral weight.
In Figs. 5 and 6 we plot the DOS for the trapped gas with the order parameters
calculated in Sec. III.1. We also show a dashed curve corresponding to
$G(x)=\sqrt{\tilde{\mu}(x)^{2}+\Delta(x)^{2}}-\hbar\Omega/2$. The point where
$G(x)=0$ represents the boundary between topologically distinct regions
defined in Sec. II.2. For the parameters in Fig. 5, $G(x)=0$ at two locations,
and we find that the BdG equations have two zero-energy modes, localized near
these points. As will be discussed later, these modes may be interpreted as
MFs. They are clearly spectrally separated from all other states. Fig. 6 shows
the case where $G(x)=0$ at four locations, representing four MFs. The right
panels of Figs. 5 and 6 show the momentum space DOS. The MF modes sit in a
large gap at $k=0$.
As we have shown in Sec. III.1, the order parameter decreases with
temperature. In real space, the bulk $\Delta$ becomes very small at
$T=0.2E_{r}$, while $\Delta$ at the edges remains large: the MFs at the edges
are very clear for $T\lesssim 0.2E_{r}$. At $T=0.3E_{r}$, the order parameter
is nearly zero and the gas becomes normal.
The evolution of the momentum space DOS parallels the real space DOS. As
temperature is increased from $T=0$, the gaps at large $k$ shrink. The gap at
$k=0$ remains robust until $T=0.3E_{r}$.
Finally for comparison, we plot the DOS within a LDA. As illustrated in Fig.
7, the LDA prediction for the DOS is qualitatively similar to the BdG result.
The main difference is that the LDA misses physics related to quantization. In
particular, the zero energy modes are not spectrally isolated in the LDA. They
are, however, still located at roughly the same place in space.
Figure 7: (color online) Density of states (DOS) at zero temperature under the
local density approximation (LDA). The parameters are identical to those in
Fig. 4, except $\Delta(x)$ is calculated within the LDA. The brighter color
corresponds to the higher spectral weight.
### III.3 Majorana fermions (MFs)
Here we explore the structure of the zero-energy states seen in Fig. 5. From
our numerical solutions to the BdG equations, we have two wavefunctions
$\displaystyle W_{0}^{p}(x)$ $\displaystyle=$ $\displaystyle
e^{i\varphi_{1}^{\prime}}\bigl{(}u_{0\uparrow}(x),u_{0\downarrow}(x),v_{0\downarrow}(x),v_{0\uparrow}(x)\bigr{)}^{\intercal}$
(54) $\displaystyle W_{0}^{h}(x)$ $\displaystyle=$ $\displaystyle
e^{i\varphi_{2}^{\prime}}\bigl{(}v^{*}_{0\uparrow}(x),v^{*}_{0\downarrow}(x),u^{*}_{0\downarrow}(x),u^{*}_{0\uparrow}(x)\bigr{)}^{\intercal},$
(55)
which define operators
$\displaystyle\xi_{0}$ $\displaystyle=$
$\displaystyle\int\left(\big{(}W_{0}^{p}(x)\big{)}^{\dagger}\cdot\tilde{\Psi}(x)\right)dx$
(56) $\displaystyle\xi_{0}^{\dagger}$ $\displaystyle=$ $\displaystyle
e^{i(\varphi_{2}^{\prime}-\varphi_{1}^{\prime})}\int\left(\big{(}W_{0}^{h}(x)\big{)}^{\dagger}\cdot\tilde{\Psi}(x)\right)dx,$
(57)
and obey
$\mathcal{H}_{trap}W_{0}^{p}(x)\approx\mathcal{H}_{trap}W_{0}^{h}(x)\approx
0$. The phases $\varphi_{1}^{\prime}$ and $\varphi_{2}^{\prime}$ are not
unique, and the factor in Eq. (54) must be introduced to make
$\xi_{0}^{\dagger}$ conjugate to $\xi_{0}$. By construction these are
fermionic operators $\\{\xi_{0},\xi_{0}^{\dagger}\\}=1$.
As zero-energy solutions to the BdG equations, both $\xi_{0}$ and
$\xi_{0}^{\dagger}$ commute with $H$. Hence the ground state is degenerate:
$\xi_{0}|GS_{1}\rangle=0$ and
$|GS_{2}\rangle=\xi_{0}^{\dagger}|GS_{1}\rangle$. These two degenerate states
can be used as a qubit for quantum information processing Sarma2008 .
The operator $\xi_{0}^{\dagger}$ which couples $|GS_{1}\rangle$ to
$|GS_{2}\rangle$ is intrinsically nonlocal, with weight at two spatially
separated points. One can however introduce operators
$\displaystyle\chi_{0}=\frac{1}{\sqrt{2}}e^{i\varphi}\left(\xi_{0}+e^{-2i\varphi}\xi_{0}^{\dagger}\right)=\int
f_{0}^{\dagger}(x)\cdot\tilde{\Psi}(x)dx$ (58)
$\displaystyle\bar{\chi}_{0}=\pm\frac{1}{\sqrt{2}i}e^{i\varphi}\left(\xi_{0}-e^{-2i\varphi}\xi_{0}^{\dagger}\right)=\int\bar{f}_{0}^{\dagger}(x)\cdot\tilde{\Psi}(x)dx$
(59)
where
$\displaystyle f_{0}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}e^{i\varphi_{1}}\big{(}W_{0}^{p}(x)+e^{-i\varphi_{2}}W_{0}^{h}(x)\big{)}$
(60) $\displaystyle\bar{f}_{0}(x)$ $\displaystyle=$
$\displaystyle\pm\frac{1}{\sqrt{2}i}e^{i\varphi_{1}}\big{(}W_{0}^{p}(x)-e^{-i\varphi_{2}}W_{0}^{h}(x)\big{)},$
(61)
with arbitrary phases $\varphi_{1}=\varphi-\varphi_{1}^{\prime}$ and
$\varphi_{2}=2\varphi-\varphi_{1}^{\prime}+\varphi_{2}^{\prime}$. By choosing
the appropriate $\varphi_{2}$, these operators can be made local. In
particular if $G(x)=0$ at $x=x_{1},x_{2}$, then $f_{0}(x)$ can be chosen to be
nonzero only near $x_{1}$, and $\bar{f}_{0}(x)$ only near $x_{2}$.
The operators $\chi_{0}$ and $\bar{\chi}_{0}$ obey the Majorana algebra:
$\chi_{0}^{\dagger}=\chi_{0}$, $\bar{\chi}_{0}^{\dagger}=\bar{\chi}_{0}$,
$\\{\chi_{0},\bar{\chi}_{0}\\}=0$,
$\\{\chi_{0},\chi_{0}\\}=\\{\bar{\chi}_{0},\bar{\chi}_{0}\\}=1$. They commute
with the Hamiltonian.
Note, as we will use in the next subsection, $f_{0}(x)\equiv
e^{i\varphi_{f}}\big{(}u_{f\uparrow}(x),u_{f\downarrow}(x),v_{f\downarrow}(x),v_{f\uparrow}(x)\big{)}$
obeys the BdG equations, but the resulting Bogoliubov transformation is not
unitary as it changes the commutation relations. Since
$\chi_{0}=\chi_{0}^{\dagger}$, we have
$u_{f\sigma}(x)=e^{-2i\varphi_{f}}v_{f\sigma}^{*}(x)$. For smaller systems,
coupling between these modes push them away from $E_{0}=0$.
### III.4 Eigen-energies of excited states near a MF
As seen in Figs. 5-6, the MFs are localized in real space and momentum space.
Thus we can calculate their properties by linearizing the trap around their
locations in position space, and linearizing momentum around $k=0$. As
previously discussed, the locations of the MFs can be found via the LDA. There
are generally four MFs, localized at $x_{m}=\pm L\sqrt{R_{\pm m}/\lambda
E_{r}}$, where $R_{\pm
m}\equiv\tilde{\mu}\pm\sqrt{\hbar^{2}\Omega^{2}/4-\Delta_{m}^{2}}$, with
$\Delta_{m}\equiv\Delta(x=x_{m})$. We restrict ourselves to the location of
one MF, $x_{m}=L\sqrt{R_{+m}/\lambda E_{r}}$. We write the linearized BdG
Hamiltonian as the sum of two terms
$\mathcal{H}_{lin}=\mathcal{H}_{0}+\mathcal{H}_{i}$,
$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$
$\displaystyle\frac{\hbar\Omega}{2}\tau_{z}\sigma_{x}+\Delta_{m}\tau_{x}+\sqrt{\hbar^{2}\Omega^{2}/4-\Delta_{m}^{2}}\tau_{x}\sigma_{z}$
(62) $\displaystyle\mathcal{H}_{i}$ $\displaystyle=$
$\displaystyle\tilde{\lambda}(x-x_{m})\tau_{z}-\kappa\tau_{z}\sigma_{z},$ (63)
where $\tilde{\lambda}=2\lambda x_{m}E_{r}/L^{2}$ and
$\kappa=\hbar^{2}k_{L}k/m$. The “interaction” term $\mathcal{H}_{i}$ can be
treated as a perturbation, and it vanishes as $x\rightarrow x_{m},k\rightarrow
0$. In the absence of perturbations, $\mathcal{H}_{lin}=\mathcal{H}_{0}$ has
two degenerate zero-energy states
$\displaystyle\mathcal{D}_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2}({\rm sin}\phi,-{\rm cos}\phi,-{\rm
cos}\phi,{\rm sin}\phi)$ (64) $\displaystyle\mathcal{D}_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2}(-{\rm cos}\phi,{\rm sin}\phi,-{\rm
sin}\phi,{\rm cos}\phi),$ (65)
where ${\rm sin}\phi=\sqrt{(\hbar\Omega/2+\Delta_{m})/\hbar\Omega}$. Following
the standard approach to first-order degenerate perturbation theory, we
diagonalize the Hamiltonian projected into the subspace
$\\{\mathcal{D}_{1},\mathcal{D}_{2}\\}$,
$\displaystyle\mathcal{\bar{H}}_{lin}=\left(\begin{array}[]{c}\mathcal{D}_{1}\\\
\mathcal{D}_{2}\end{array}\right)H_{lin}\big{(}\mathcal{D}_{1},\mathcal{D}_{2}\big{)}=K\bar{\sigma}_{z}+X\bar{\sigma}_{x},$
(68)
where $K=-2\hbar\Delta_{m}k_{L}k/m\Omega$ and $X=-4\lambda
E_{r}x_{m}(x-x_{m})R_{+m}/\hbar\Omega L^{2}$. The Pauli matrices
$\bar{\bm{\sigma}}$ operate in the subspace
$\\{\mathcal{D}_{1},\mathcal{D}_{2}\\}$. Noting that $\big{[}X,K\big{]}=iC$
with
$C=16\sqrt{\lambda}E_{r}^{3/2}R_{+m}^{3/2}\Delta_{m}/\hbar^{2}\Omega^{2}k_{L}L$,
one can define the operators
$a=\frac{K-iX}{\sqrt{2C}},a^{\dagger}=\frac{K+iX}{\sqrt{2C}}$ that satisfy
$\big{[}a,a^{\dagger}\big{]}=1$. The eigen-equations of
$\mathcal{\bar{H}}_{lin}$ then become
$\displaystyle\frac{\sqrt{2C}}{2}\left(\begin{array}[]{cc}-(a^{\dagger}+a)&i(a^{\dagger}-a)\\\
i(a^{\dagger}-a)&a^{\dagger}+a\end{array}\right)\left(\begin{array}[]{c}\bar{u}_{n}\\\
\bar{v}_{n}\end{array}\right)=\bar{E}_{n}\left(\begin{array}[]{c}\bar{u}_{n}\\\
\bar{v}_{n}\end{array}\right)$ (75)
where $\bar{u}_{n}=\mathcal{D}_{1}\cdot W_{n},\bar{v}_{n}=\mathcal{D}_{2}\cdot
W_{n}$. Combining $\bar{u}_{n},\bar{v}_{n}$ gives the equations
$\displaystyle-\sqrt{2C}\left(\begin{array}[]{cc}0&a^{\dagger}\\\
a&0\end{array}\right)\left(\begin{array}[]{c}\bar{u}_{n}+i\bar{v}_{n}\\\
\bar{u}_{n}-i\bar{v}_{n}\end{array}\right)=\bar{E}_{n}\left(\begin{array}[]{c}\bar{u}_{n}+i\bar{v}_{n}\\\
\bar{u}_{n}-i\bar{v}_{n}\end{array}\right).$ (82)
Squaring Eq. (82) yields harmonic oscillator Hamiltonian, and allows one to
read off
$\displaystyle\bar{E}_{n}=\pm\sqrt{2C}\sqrt{n}\quad(n=0,1,2,...).$ (83)
Not only is there a zero-energy mode $\bar{E}_{0}=0$ (the Majorana mode), but
there is a ladder of localized fermionic modes, whose energy spacing is
proportional to $\lambda^{1/4}$, and whose wavefunction components are excited
harmonic oscillator states. For a homogeneous gas where $\lambda=0$, the
energy spacing becomes zero.
Figure 8: (color online) The gap between the MF state and excited states as a
function of trap stiffness $\lambda^{1/4}$: the trapping potential is
$V(x)=\lambda(x/L)^{2}E_{r}$. The black (thick) curve is plotted based on the
analytic Eq. (83). The red (dot-dashed), green (dashed), blue (dotted) curves
are the energy levels $E_{1}/E_{r},E_{2}/\sqrt{2}E_{r},E_{3}/\sqrt{3}E_{r}$
respectively. They are numerically calculated from Eq. (44) with the
parameters identical to the thick curve.
In Fig. 8, we plot $\bar{E}_{n}/\sqrt{n}E_{r}$ as a function of
$\lambda^{1/4}$ (black thick curve) based on Eq. (83), and compare to the
numerical results calculated from Eq. (44). The dot-dashed (red), dashed
(green), and dotted (blue) curves show the energy levels of the first three
excited states. We see the analytic results agree well with the numerics for
small $\lambda$. For larger $\lambda$, the corrections to Eq. (68) are
important, and the discrepancy between the analytic and numerical results
becomes notable, especially for larger $n$.
At $n=0$ ($\bar{E}_{0}=0$), the zero-energy mode has wavefunction
$\displaystyle\bar{u}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sigma\sqrt{\pi}}e^{-(x-x_{m})^{2}/2\sigma^{2}}$ (84)
$\displaystyle\bar{v}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2i\sigma\sqrt{\pi}}e^{-(x-x_{m})^{2}/2\sigma^{2}},$
(85)
where the width
$\sigma=\sqrt{\Delta_{m}LE_{r}^{1/2}/R_{+m}^{3/2}k_{L}\lambda^{1/2}}$ is
proportional to $\lambda^{-1/4}$.
## IV Summary
We have investigated a (pseudo) spin-$1/2$ spin-orbit (SO) coupled Fermi gas
in a one-dimensional geometry. We first relate this system to a one-band model
with $p$-wave interactions. We then described the band structure and
calculated the Berry phase $\gamma$ of the full two-band model. We found
$\gamma$ distinguishes two topologically distinct sectors, with $\gamma=\pi$
corresponding to a conventional superconductor. By self-consistently solving
the Bogoliubov-de Gennes equations and calculating both the position resolved
and momentum resolved density of states, we visualized the Majorana fermion
(MF) states in real and momentum space at finite temperatures. These spectra
can be probed using the position resolved or momentum resolved radio-frequency
spectroscopy Ketterle2007 ; Zwierlein2012 . We introduced MF operators and
constructed the localized MF states. We further linearized the trap near the
location of a MF, finding an analytic expression for the localized MF
wavefunction and the gap between the MF state and other edge states.
This physics can be experimentally studied in a bundle of weakly coupled tubes
containing fermionic atoms Hulet2010 . By applying appropriate Raman lasers to
these quasi-1D tubes Zhang2012 ; Zwierlein2012 , one can produce an array of
quasi-1D SO coupled Fermi clouds. Our calculations show that the MFs can be
observed in such settings.
There are, however, significant experimental challenges. Most notably, the
Raman induced SO coupling relies on the ability of optical photons to flip the
atomic hyperfine spin. As Spielman argues Spielman2009 , if the Raman lasers
are detuned by a frequency $\Delta$ from an excited state multiplet (and
$\hbar\Delta$ is large compared to the fine structure splitting $A_{f}$), then
the coupling strength $\Omega$ scales as $1/\Delta^{2}$. (This is contrasted
with typical AC stark shifts, which instead scale as $1/\Delta$. The extra
suppression is due to quantum interference between the amplitudes arising from
different intermediate states.) The rate of inelastic light scattering
$\Gamma_{i}$ also scales as $1/\Delta^{2}$. The ratio
$\upsilon=\Gamma_{i}/\Omega$ is therefore roughly independent of detuning. In
terms of microscopic parameters, $\upsilon\propto\hbar/A_{f}\tau$, where
$\tau$ is the lifetime of the excited states. For ${}^{6}Li$,
$\hbar/A_{f}\tau\sim 5.8\times 10^{-4}$, for ${}^{40}K$, $\hbar/A_{f}\tau\sim
3.5\times 10^{-6}$ and for ${}^{87}Rb$, $\hbar/A_{f}\tau\sim 8.3\times
10^{-7}$. One sees ${}^{40}K$ has a much longer lifetime than ${}^{6}Li$ in a
SO coupled Fermi experiment. The situation is even less favorable at the
typical magnetic field $\sim 830G$ Ketterle2005 where one encounters Feshbach
resonances in ${}^{6}Li$. The large magnetic field decouples the electron spin
and the nuclear spin, and the relevant hyperfine states effectively only
differ by their nuclear spin. As a result, the Raman laser couplings vanish
between these states. However for ${}^{40}K$, the typical resonance field
$\sim 200G$ Jin2004 is much smaller, and the relevant hyperfine states have
larger Raman couplings. We therefore expect ${}^{40}K$ is a promising
candidate for producing an interacting SO coupled Fermi gas.
## V Acknowledgement
The work in Sec. II.1 is largely derived from notes by Bhuvanesh Sundar. The
authors thank Randall Hulet for discussions of experimental difficulties. R.
W. is supported by CSC, the NNSFC, the NNSFC of Anhui (under Grant No.
090416224), the CAS, and the National Fundamental Research Program (under
Grant No. 2011CB921304). This material is based upon work supported by the
National Science Foundation under Grant No. PHY-1068165 and a grant from the
Army Research Office with funding from the DARPA OLE program.
## References
* (1) E. Majorana, Nuovo Cimento 14, 171 (1937); F. Wilczek, Nat. Phys. 5, 614 (2009).
* (2) C. Nayak, S. H. Simon, A Stern, M. Freedman and S. DasSarma, Rev. Mod. Phys. 80, 1083 (2008).
* (3) N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
* (4) A. Kitaev, Phys. Usp. 44, 131 (2001).
* (5) S. DasSarma, C. Nayak, and S. Tewari, ibid. 73, 220502(R) (2006); J. Alicea, ibid. 81, 125318 (2010); L. Fidkowski, R. M. Lutchyn, C. Nayak, and M. P. A. Fisher, ibid. 84, 195436 (2011); T. Mizushima, M. Ichioka, and K. Machida, ibid. 101, 150409 (2008); J. D. Sau, R. M. Lutchyn, S. Tewari, and S. DasSarma, ibid. 104, 040502 (2010); R. M. Lutchyn, J. D. Sau, and S. D. Sarma, ibid. 105, 077001 (2010).
* (6) L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
* (7) Y. Oreg, G. Refael, and F. vonOppen, Phys. Rev. Lett. 105, 177002 (2010).
* (8) J. Alicea, Y. Oreg, G. Refael, F. vonOppen and M. P. A. Fisher, Nat. Phys. 7, 412 (2011).
* (9) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers and L. P. Kouwenhoven, Science 336, 1003 (2012); J. R. Williams, A. J. Bestwick, P. Gallagher, S. S. Hong, Y. Cui, A. S. Bleich, J. G. Analytis, I. R. Fisher and D. G.-Gordon, arXiv:1202.2323 (2012); M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff and H. Q. Xu, arXiv:1204.4130 (2012); A. Das. Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Arxiv:1205.7073 (2012).
* (10) Y.-J. Lin, K. Jiménez-García and I. B. Spielman, Nature (London) 471, 83 (2011).
* (11) P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Arxiv:1204.1887 (2012).
* (12) L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Arxiv:1205.3483 (2012).
* (13) C. Zhang, S. Tewari, R. M. Lutchyn, and S. DasSarma, Phys. Rev. Lett. 101, 160401 (2008); M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett. 103, 020401 (2009); S.-L. Zhu, L.-B. Shao, Z. D. Wang, and L.-M. Duan, Phys. Rev. Lett. 106, 100404 (2011); M. Gong, G. Chen, S. Jia, C. Zhang, arXiv:1201.2238 (2012).
* (14) L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. Rev. Lett. 106, 220402 (2011).
* (15) X.-J. Liu and H. Hu, Phys. Rev. A 85, 033622 (2012).
* (16) X.-J. Liu, L. Jiang, H. Pu, and H. Hu, Phys. Rev. A 85, 021603(R) (2012).
* (17) Y. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur and E. J. Mueller, Nature (London) 467, 567 (2010).
* (18) Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 99, 090403 (2007).
* (19) L. Fidkowski, R. M. Lutchyn, C. Nayak, and M. P. A. Fisher, Phys. Rev. B 84, 195436 (2011); Jay D. Sau, B. I. Halperin, K. Flensberg, and S. DasSarma, Phys. Rev. B 84, 144509 (2011); M. Cheng and H.-H. Tu, Phys. Rev. B 84, 094503 (2011).
* (20) M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
* (21) D.S. Petrov, D.M. Gangardt and G.V. Shlyapnikov, J. Phys. IV France 116, 3 (2004).
* (22) I. B. Spielman, Phys. Rev. A 79, 063613 (2009).
* (23) C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A. Simoni, E. Tiesinga, C. J. Williams, and P. S. Julienne, Phys. Rev. A 71, 045601 (2005).
* (24) C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004).
## VI Appendix
In this Appendix, we explore the convergence of our self-consistent
calculations with the grid spacing $\delta x=2L/n_{\rm grid}$. We show how the
energy and the order parameter for a zero-temperature homogeneous gas in a box
of size $2L$ with periodic boundary conditions depends on $n_{\rm grid}$.
Within our mean-field theory, the energy of this homogeneous gas is
$\displaystyle
E_{g}=\sum_{k}\bigg{(}\epsilon_{0}(k)-\frac{1}{2}\big{(}E_{+}(k)+E_{-}(k)\big{)}\bigg{)}-\frac{|\Delta|^{2}}{\tilde{g}_{1D}},$
(86)
where
$\tilde{g}_{1D}=g_{1D}/2L,\epsilon_{0}(k)=\frac{\hbar^{2}k^{2}}{2m}-\tilde{\mu}$,
and $E_{\pm}(k)$ is the excitation spectrum given in Eq. (33). The summation
index $k$ is discretized as $k=-\frac{n_{\rm grid}}{2L}\pi,-\left(\frac{n_{\rm
grid}-2}{2L}\right)\pi,...,\left(\frac{n_{\rm
grid}-4}{2L}\right)\pi,\left(\frac{n_{\rm grid}-2}{2L}\right)\pi$, and Eq.
(86) can be calculated numerically.
Figure 9: (color online) Ground state energy $E_{g}/E_{r}$ versus order
parameter $\Delta/E_{r}$. The blue (dashed), black (thick), red (dotted),
green (dot-dashed) curves correspond to $n_{\rm grid}=400,600,800,1000$
respectively. Other parameters are
$\tilde{g}_{1D}=-0.02E_{r},\hbar\Omega=2E_{r},\lambda=0,k_{L}L=100,\mu=E_{r}$.
Fig. 9 shows $E_{g}$ as a function of $\Delta$ for $n_{\rm
grid}=400,600,800,1000$. We find non-trivial behavior at intermediate $n_{\rm
grid}$. In particular, for these parameters and $n_{\rm grid}=600$, the energy
has two local minima, and the gap equations has four solutions, corresponding
$\Delta=0$ and other three stationary points. Such behavior is an artifact of
the discretization, as it goes away for $n_{\rm grid}\gtrsim 800$. It does,
however, indicate that in the presence of an appropriate tuned optical
lattice, there will be metastable superfluid states.
In Fig. 10, we show how the order parameter $\Delta$ depends on $n_{\rm
grid}$. We calculate $\Delta$ by minimizing $E_{g}$,
$\displaystyle\frac{\partial E_{g}}{\partial|\Delta|}\bigg{|}_{|\Delta|>0}=0.$
(87)
We see $\Delta$ converges to a finite value as $n_{\rm
grid}\rightarrow\infty$. For the simulation size $n_{\rm grid}=1200$ used in
the main text, the finite grid error is $\frac{|\Delta(n_{\rm
grid}=\infty)-\Delta(n_{\rm grid}=1200)|}{\Delta(n_{\rm grid}=\infty)}\leq
12\%$.
Figure 10: (color online) Order parameter $\Delta/E_{r}$ versus $10^{3}/n_{\rm
grid}$. The red dots are calculated from Eq. (87). The blue (thick) curve is
an extrapolation. The parameters here are identical to those in Fig. 4(a)
except for $\lambda=0$.
|
arxiv-papers
| 2012-08-27T17:18:27 |
2024-09-04T02:49:34.586684
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ran Wei, Erich J. Mueller",
"submitter": "Ran Wei",
"url": "https://arxiv.org/abs/1208.5450"
}
|
1208.5540
|
# Regularization of point vortices for the Euler equation in dimension two,
part II
Daomin Cao Institute of Applied Mathematics, Chinese Academy of Science,
Beijing 100190, P.R. China [email protected] , Zhongyuan Liu Institute of
Applied Mathematics, Chinese Academy of Science, Beijing 100190, P.R. China
[email protected] and Juncheng Wei Department of Mathematics, The Chinese
University of Hong Kong, Shatin, N.T., Hong Kong [email protected]
###### Abstract.
In this paper, we continue to construct stationary classical solutions of the
incompressible Euler equation approximating singular stationary solutions of
this equation. This procedure now is carried out by constructing solutions to
the following elliptic problem
$\begin{cases}-\varepsilon^{2}\Delta
u=\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\big{(}u-q-\frac{\kappa_{i}^{+}}{2\pi}\ln\frac{1}{\varepsilon}\big{)}_{+}^{p}-\sum_{j=1}^{n}\chi_{\Omega_{j}^{-}}\big{(}q-\frac{\kappa_{j}^{-}}{2\pi}\ln\frac{1}{\varepsilon}-u\big{)}_{+}^{p},\quad&x\in\Omega,\\\
u=0,\quad&x\in\partial\Omega,\end{cases}$
where $p>1$, $\Omega\subset\mathbb{R}^{2}$ is a simply connected bounded
domain, $\Omega_{i}^{+}$ and $\Omega_{j}^{-}$ are mutually disjoint subdomains
of $\Omega$, $q$ is a harmonic function.
We showed that if $\Omega$ is a simply-connected smooth domain, then for any
given $C^{1}$-stable critical point of Kirchhoff-Routh function
$\mathcal{W}(x_{1}^{+},\cdots,x_{m}^{+},x_{1}^{-},\cdots,x_{n}^{-})$ with
$\kappa^{+}_{i}>0\,(i=1,\cdots,m)$ and $\kappa^{-}_{j}>0\,(j=1,\cdots,n)$,
then there is a stationary classical solution approximating stationary $m+n$
points vortex solution of incompressible Euler equations with total vorticity
$\sum_{i=1}^{m}\kappa_{i}-\sum_{j=1}^{n}\kappa_{j}^{-}$.
_AMS 2000 Subject Classifications: Primary $35\mathrm{J}60$; Secondary
$35\mathrm{JB}05$; $35\mathrm{J}40$
Keywords: The Euler equation; Multiple non-vanishing vortices; Free boundary
problem. _
## 1\. Introduction and main results
The incompressible Euler equations
$\begin{cases}\mathbf{v}_{t}+(\mathbf{v}\cdot\nabla)\mathbf{v}=-\nabla P,\\\
\nabla\cdot\mathbf{v}=0,\end{cases}$ (1.1)
describe the evolution of the velocity $\mathbf{v}$ and the pressure $P$ in an
incompressible flow. In $\mathbb{R}^{2}$, the vorticity of the flow is defined
by $\omega=\nabla\times\mathbf{v}:=\partial_{1}v_{2}-\partial_{2}v_{1}$, which
satisfies the equation
$\omega_{t}+\mathbf{v}\cdot\nabla\omega=0.$
Suppose that $\omega$ is known, then the velocity $\mathbf{v}$ can be
recovered by Biot-Savart law as following:
$\mathbf{v}=\omega\,*\frac{1}{2\pi}\frac{-x^{\bot}}{|x|^{2}},$
where $x^{\bot}=(x_{2},\,-x_{1})$ if $x=(x_{1},\,x_{2})$. One special singular
solutions of Euler equations is given by
$\omega=\sum^{m}_{i=1}\kappa_{i}\delta_{x_{i}(t)}$, which is related
$\mathbf{v}=-\sum^{m}_{i=1}\frac{\kappa_{i}}{2\pi}\frac{(x-x_{i}(t))^{\bot}}{|x-x_{i}(t)|^{2}}.$
and the positions of the vortices $x_{i}:\mathbb{R}\rightarrow\mathbb{R}^{2}$
satisfy the following Kirchhoff law:
$\kappa_{i}\,\frac{dx_{i}}{dt}=(\nabla_{x_{i}}\mathcal{W})^{\bot}$
where $\mathcal{W}$ is the so called Kirchhoff-Routh function defined by
$\mathcal{W}(x_{1},\cdots,x_{m})=\frac{1}{2}\sum_{i\neq
j}^{m}\frac{\kappa_{i}\kappa_{j}}{2\pi}\log\frac{1}{|x_{i}-x_{j}|}.$
In simply-connected bounded domain $\Omega\subset\mathbb{R}^{2}$, similar
singular solutions also exist. Suppose that the normal component of
$\mathbf{v}$ vanishes on $\partial\Omega$, then the Kirchhoff-Routh function
is
$\mathcal{W}(x_{1},\cdots,x_{m})=\frac{1}{2}\sum_{i\neq
j}^{m}{\kappa_{i}\kappa_{j}}G(x_{i},\,x_{j})+\frac{1}{2}\sum_{i=1}^{m}{\kappa_{i}^{2}}H(x_{i},\,x_{i}),$
(1.2)
where $G$ is the Green function of $-\Delta$ on $\Omega$ with 0 Dirichlet
boundary condition and $H$ is its regular part (the Robin function).
Let $v_{n}$ be the outward component of the velocity $\mathbf{v}$ on the
boundary $\partial\Omega$, then we see that $\int_{\partial\Omega}v_{n}=0$ due
to the fact that $\nabla\cdot\mathbf{v}=0$. Suppose that $\mathbf{v}_{0}$ is
the unique harmonic field whose normal component on the boundary
$\partial\Omega$ is $v_{n}$, then $\mathbf{v}_{0}$ satisfies
$\begin{cases}\nabla\cdot\mathbf{v}_{0}=0,\,\,\text{in}\,\Omega,\\\
\nabla\times\mathbf{v}_{0}=0,\,\,\text{in}\,\Omega,\\\
n\cdot\mathbf{v}_{0}=v_{n},\,\,\text{on}\,\partial\Omega.\end{cases}$ (1.3)
If $\Omega$ is simply-connected, then $\mathbf{v}_{0}$ can be written
$\mathbf{v}_{0}=(\nabla\psi_{0})^{\bot}$, where the stream function $\psi_{0}$
is determined up to a constant by
$\begin{cases}-\Delta\psi_{0}=0,\,\,\text{in}\,\Omega,\\\
-\displaystyle\frac{\partial\psi_{0}}{\partial\tau}=v_{n},\,\,\text{on}\,\partial\Omega,\end{cases}$
(1.4)
where $\frac{\partial\psi_{0}}{\partial\tau}$ denotes the tangential
derivative on $\partial\Omega$. The Kirchhoff-Routh function associated to the
vortex dynamics becomes(see Lin [24])
$\mathcal{W}(x_{1},\cdots,x_{m})=\frac{1}{2}\sum_{i\neq
j}^{m}\kappa_{i}\kappa_{j}G(x_{i},x_{j})+\frac{1}{2}\sum^{m}_{i=1}\kappa^{2}_{i}H(x_{i},x_{i})+\sum^{m}_{i=1}\kappa_{i}\psi_{0}(x_{i}).$
(1.5)
For $m$ clockwise vortices motion (corresponding to $\kappa^{+}_{i}>0$) and
$n$ anti-clockwise vortices motion (corresponding to $-\kappa^{-}_{j}<0$), the
Kirchhoff-Routh function associated to the vortex dynamics becomes
$\begin{split}\mathcal{W}(x^{+}_{1},\cdots,x^{+}_{m},x^{-}_{1},\cdots,x^{-}_{n})=&\frac{1}{2}\sum_{i,k=1,i\neq
k}^{m}\kappa_{i}^{+}\kappa_{k}^{+}G(x_{i}^{+},x_{k}^{+})+\frac{1}{2}\sum^{n}_{j,l=1,j\neq
l}\kappa^{-}_{j}\kappa^{-}_{l}G(x_{j}^{-},x_{l}^{-})\\\
&+\frac{1}{2}\sum_{i=1}^{m}(\kappa_{i}^{+})^{2}H(x_{i}^{+},x_{i}^{+})+\frac{1}{2}\sum_{j=1}^{n}(\kappa_{j}^{-})^{2}H(x_{j}^{-},x_{j}^{-})\\\
&-\sum_{i=1}^{m}\sum_{j=1}^{n}\kappa_{i}^{+}\kappa_{j}^{-}G(x_{i}^{+},x_{j}^{-})+\sum^{m}_{i=1}\kappa_{i}^{+}\psi_{0}(x_{i}^{+})-\sum^{n}_{j=1}\kappa_{j}^{-}\psi_{0}(x_{j}^{-}).\end{split}$
(1.6)
It is known that critical points of the Kirchhoff-Routh function $\mathcal{W}$
give rise to stationary vortex points solutions of the Euler equations. As for
the existence of critical points of $\mathcal{W}$ given by (1.2), we refer to
[5].
Roughly speaking, there are two methods to construct stationary solutions of
the Euler equation, which are the vorticity method and the stream-function
method. The vorticity method was first established by Arnold and Khesin [3]
and further developed by Burton [7] and Turkington [32].
The stream-function method consists in observing that if $\psi$ satisfies
$-\Delta\psi=f(\psi)$ for some function $f\in C^{1}(\mathbb{R})$, then
$\mathbf{v}=(\nabla\psi)^{\bot}$ and $P=F(\psi)-\frac{1}{2}|\nabla\psi|^{2}$
is a stationary solution to the Euler equations, where
$(\nabla\psi)^{\bot}:=(\frac{\partial\psi}{\partial
x_{2}},-\frac{\partial\psi}{\partial x_{1}}),F(t)=\int_{0}^{t}f(s)ds$.
Moreover, the velocity $\mathbf{v}$ is irrotational on the set where
$f(\psi)=0$.
Set $q=-\psi_{0}$ and $u=\psi-\psi_{0}$, then $u$ satisfies the following
boundary value problem
$\begin{cases}-\Delta u=f(u-q),\quad&x\in\Omega,\\\
u=0,\quad&x\in\partial\Omega.\end{cases}$ (1.7)
In addition, if we suppose that $\inf_{\Omega}q>0$ and $f(t)=0,~{}t\leq 0$,
the vorticity set $\\{x:f(\psi)>0\\}$ is bounded away from the boundary.
The motivation to study (1.7) is to justify the weak formulation for point
vortex solutions of the incompressible Euler equations by approximating these
solutions with classical solutions.
Marchioro and Pulvirenti [26] have approximated these solutions on finite time
intervals by considering regularized initial data for the vorticity. On the
other hand, the stationary point vortex solutions can also be approximated by
stationary classical solutions. See e.g. [1, 2, 4, 6, 19, 28, 29, 31, 32, 33,
34] and the references therein.
In [18] Elcrat and Miller, by a rearrangements of functions, have studied
steady, inviscid flows in two dimensions which have concentrated regions of
vorticity. In particular, they studied such flows which ”desingularize” a
configuration of point vortices in stable equilibrium with an irrotational
flow, which generalized their earlier work for one vortex [16][17] which in
turn were based on results of Turkington [32]. As pointed by Elcrat and
Miller, an essential hypothesis in their existence proof was that the
vorticity was in a neighborhood of a stable point vortex configuration.
Saffman and Sheffield [30] have found an example of a steady flow in
aerodynamics with a single point vortex which is stable for a certain range of
the parameters. This has been generalized in [16], where some examples
computationally of stable configurations of two point vortices were briefly
discussed. Further examples of multiple point vortex configurations are given
in [27], where a theorem on the existence of such configurations is also
given.
It is worth pointing out that except [18] the above approximations can just
give explanation for the formulation to single point vortex solutions. D.
Smets and J. Van Schaftingen [31] investigated the following problem
$\begin{cases}-\varepsilon^{2}\Delta
u=\left(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\varepsilon}\right)_{+}^{p},&\text{in}\;\Omega,\\\
u=0,&\text{on}\;\partial\Omega,\end{cases}$ (1.8)
and gave the exact asymptotic behavior and expansion of the least energy
solution by estimating the upper bounds on the energy. The solutions for (1.8)
in [31] were obtained by finding a minimizer of the corresponding functional
in a suitable function space, which can only give approximation to a single
point non-vanishing vortex. In [13], we have shown that multi-point vortex
solutions can be approximated by stationary classical solutions.
Concerning regularization of pairs of vortices, D. Smets and J. Van
Schaftingen [31] also studied the following problem
$\begin{cases}-\varepsilon^{2}\Delta
u=\left(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\varepsilon}\right)_{+}^{p}-(q-\frac{\kappa}{2\pi}\ln\frac{1}{\varepsilon}-u)_{+}^{p},&\text{in}\;\Omega,\\\
u=0,&\text{on}\;\partial\Omega,\end{cases}$ (1.9)
and obtained the exact asymptotic behavior and expansion of the least energy
solution by similar methods for (1.8). This method is hard to obtain multiple
non-vanishing pairs of vortices solutions.
In this paper, we approximate stationary vortex solutions of Euler equations
(1.1) with multiple non-vanishing pairs of vortices solutions by stationary
classical solutions. Our main result concerning (1.1) is the following:
###### Theorem 1.1.
Suppose that $\Omega\subset\mathbb{R}^{2}$ is a bounded simply-connected
smooth domain. Let $v_{n}:\partial\Omega\rightarrow\mathbb{R}$ be such that
$v_{n}\in L^{s}(\partial\Omega)$ for some $s>1$ satisfying
$\int_{\partial\Omega}v_{n}=0$. Let
$\kappa^{+}_{i}>0,\kappa^{-}_{j}>0,~{}i=1,\cdots,m,~{}j=1,\dots,n$. Then, for
any given $C^{1}$-stable critical point
$(x_{1,\ast}^{+},\cdots,x_{m,\ast}^{+},x_{1,\ast}^{-},\cdots,x_{n,\ast}^{-})$
of Kirchhoff-Routh function $\mathcal{W}$ defined by (1.6), there exists
$\varepsilon_{0}>0$, such that for each $\varepsilon\in(0,\varepsilon_{0})$,
problem (1.1) has a stationary solution $\mathbf{v}_{\varepsilon}$ with
outward boundary flux given by $v_{n}$, such that its vorticities
$\omega^{\pm}_{\varepsilon}$ satisfying
$supp(\omega^{+}_{\varepsilon})\subset\cup_{i=1}^{m}B(x_{i,\,\varepsilon},C\varepsilon)~{}~{}\text{for}~{}~{}x^{+}_{i,\,\varepsilon}\in\Omega,~{}~{}i=1,\cdots,m,$
$supp(\omega^{-}_{\varepsilon})\subset\cup_{j=1}^{n}B(x^{-}_{j,\,\varepsilon},C\varepsilon)~{}~{}\text{for}~{}~{}x^{-}_{j,\,\varepsilon}\in\Omega,~{}~{}j=1,\cdots,n$
and as $\varepsilon\rightarrow 0$
$\int_{\Omega}\omega_{\varepsilon}\rightarrow\sum_{i=1}^{m}\kappa_{i}^{+}-\sum_{j=1}^{n}\kappa^{-}_{j},$
$(x^{+}_{1,\,\varepsilon},\cdots,x^{+}_{m,\,\varepsilon},x_{1,\,\varepsilon}^{-},\cdots,x^{-}_{n,\varepsilon})\rightarrow(x_{1,\ast}^{+},\cdots,x_{m,\ast}^{+},x_{1,\ast}^{-},\cdots,x_{n,\ast}^{-}).$
###### Remark 1.2.
The simplest case, corresponding to pairs of vortices $(m=n=1)$ was studied by
Smets and Van Schaftingen [31] by minimizing the corresponding energy
functional. In their paper as $\varepsilon\to 0$,
$\mathcal{W}(x^{+}_{1,\,\varepsilon},x^{-}_{1,\varepsilon})\rightarrow\sup\limits_{x_{1}^{+},x^{-}_{1}\in\Omega,\,x_{1}^{+}\neq
x^{-}_{1}}\mathcal{W}(x_{1}^{+},x_{1}^{-})$. Even in the case $m=n=1$, our
result extends theirs to general critical points (with additional assumption
that the critical point is non-degenerate or stable in the sense of $C^{1}$).
The method used in [31] can not be applied to deal with general critical point
cases. The method used here is constructive and is completely different from
theirs.
###### Remark 1.3.
In this case that $m=n=1$ suppose that $(x^{+}_{1,\ast},\,x^{-}_{1,\ast})$ is
a strict local maximum(or minimum) point of Kirchhoff-Routh function
$\mathcal{W}(x^{+},\,x^{-})$ defined by (1.6), statement of Theorem 1.1 still
holds which can be proved similarly (see Remark 1.5). Thus we can obtain
corresponding existence result in [31].
Theorem 1.1 is proved via considering the following problem
$\begin{cases}-\varepsilon^{2}\Delta
u=\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\big{(}u-q-\frac{\kappa_{i}^{+}}{2\pi}\ln\frac{1}{\varepsilon}\big{)}_{+}^{p}-\sum_{j=1}^{n}\chi_{\Omega_{j}^{-}}\big{(}q-\frac{\kappa_{j}^{-}}{2\pi}\ln\frac{1}{\varepsilon}-u\big{)}_{+}^{p},\quad&x\in\Omega,\\\
u=0,\quad&x\in\partial\Omega,\end{cases}$ (1.10)
where $p>1$, $q\in C^{2}(\Omega)$, $\Omega\subset\mathbb{R}^{2}$ is a bounded
domain, $\Omega_{i}^{+}(i=1\cdots,m)$ and $\Omega_{j}^{-}(\,j=1\cdots,n)$ are
mutually disjoint subdomains of $\Omega$ such that
$x^{+}_{i,*}\in\Omega^{+}_{i},$ and $x^{-}_{j,*}\in\Omega^{-}_{j}$.
###### Theorem 1.4.
Suppose $q\in C^{2}(\Omega)$. Then for any given
$\kappa^{+}_{i}>0,\kappa^{-}_{j}>0,~{}i=1,\cdots,m,~{}j=1,\dots,n$ and for any
given $C^{1}$-stable critical point
$(x_{1,\ast}^{+},\cdots,x_{m,\ast}^{+},x_{1,\ast}^{-},\cdots,x_{n,\ast}^{-})$
of Kirchhoff-Routh function $\mathcal{W}$ defined by (1.6), there exists
$\varepsilon_{0}>0$, such that for each $\varepsilon\in(0,\varepsilon_{0})$,
(1.10) has a solution $u_{\varepsilon}$, such that the set
$\Omega_{\varepsilon,i}^{+}=\\{x:u_{\varepsilon}(x)-\frac{\kappa_{i}^{+}}{2\pi}\,\ln\frac{1}{\varepsilon}-q(x)>0\\}\subset\subset\Omega_{i}^{+},\,i=1,\cdots,m$,
$\Omega_{\varepsilon,j}^{-}=\\{x:u_{\varepsilon}(x)-\frac{\kappa_{j}^{-}}{2\pi}\,\ln\frac{1}{\varepsilon}-q(x)>0\\}\subset\subset\Omega_{j}^{-},\,j=1,\cdots,n$
and as $\varepsilon\to 0$, each $\Omega^{\pm}_{\varepsilon,\,i}$ shrinks to
$x_{i,\ast}^{\pm}\in\Omega$.
###### Remark 1.5.
For the case $m=n=1$, suppose that $(x^{+}_{1,\ast},\,x^{-}_{1,\ast})$ is a
strict local maximum(or minimum) point of Kirchhoff-Routh function
$\mathcal{W}(x)$ defined by (1.6), then statement of Theorem 1.4 still holds.
This conclusion can be proved by making corresponding modification of the
proof of Theorem 1.4 in obtaining critical point of $K(z)$ defined by
(4.1)(see Propositions 2.3,2.5 and 2.6 in [12] for detailed arguments).
As in [13], we prove Theorem 1.4 by considering an equivalent problem of
(1.10) instead. Let $w=\frac{2\pi}{|\ln\varepsilon|}u$ and
$\delta=\varepsilon(\frac{2\pi}{|\ln\varepsilon|})^{\frac{p-1}{2}}$, then
(1.10) becomes
$\begin{cases}-\delta^{2}\Delta
w=\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\left(w-\kappa_{i}^{+}-\frac{2\pi}{|\ln\varepsilon|}q(x)\right)_{+}^{p}-\sum_{j=1}^{m}\chi_{\Omega_{j}^{-}}\left(\frac{2\pi}{|\ln\varepsilon|}q(x)-\kappa_{j}^{-}-w\right)_{+}^{p},&\text{in}\;\Omega,\\\
w=0,&\text{on}\;\partial\Omega.\end{cases}$ (1.11)
We will use a reduction argument to prove Theorem 1.4. To this end, we need to
construct an approximate solution for (1.11). For the problem studied in this
paper, the corresponding “limit” problem in $\mathbb{R}^{2}$ has no bounded
nontrivial solution. So, we will follow the method in [14, 15] to construct an
approximate solution. Since there are two parameters $\delta,~{}\varepsilon$
in problem (1.11) and two terms in nonlinearity, which causes some difficulty,
we must take this influence into careful consideration and give delicate
estimates in order to perform the reduction argument. For example we need to
consider
$(s^{+}_{1,\delta},\cdots,s^{+}_{m,\delta},s^{-}_{1,\delta},\cdots,s^{-}_{n,\delta})$
and
$(a^{+}_{1,\delta},\cdots,a^{+}_{m,\delta},a_{1,\delta}^{-},\cdots,a_{n,\delta}^{-})$
together in Lemma 2.1.
As a final remark, we point out that problem (1.11) can be considered as a
free boundary problem. Similar problems have been studied extensively. The
reader can refer to [11, 13, 14, 15, 20, 23] for more results on this kind of
problems.
This paper is organized as follows. In section 2, we construct the approximate
solution for (1.11). We will carry out a reduction argument in section 3 and
the main results will be proved in section 4. We put some basic estimates used
in sections 3 and 4 in the appendix.
## 2\. Approximate solutions
In the section, we will construct approximate solutions for (1.11).
Let $R>0$ be a large constant, such that for any $x\in\Omega$,
$\Omega\subset\subset B_{R}(x)$. Consider the following problem:
$\begin{cases}-\delta^{2}\Delta w=(w-a)_{+}^{p},&\text{in}\;B_{R}(0),\\\
w=0,&\text{on}\;\partial B_{R}(0),\end{cases}$ (2.1)
where $a>0$ is a constant. Then, (2.1) has a unique solution $W_{\delta,a}$,
which can be written as
$W_{\delta,a}(x)=\begin{cases}a+\delta^{2/(p-1)}s_{\delta}^{-2/(p-1)}\phi\bigl{(}\frac{|x|}{s_{\delta}}\bigr{)},&|x|\leq
s_{\delta},\\\
a\ln\frac{|x|}{R}/\ln\frac{s_{\delta}}{R},&s_{\delta}\leq|x|\leq
R,\end{cases}$ (2.2)
where $\phi(x)=\phi(|x|)$ is the unique solution of
$-\Delta\phi=\phi^{p},\quad\phi>0,~{}~{}\phi\in
H_{0}^{1}\bigl{(}B_{1}(0)\bigr{)}$
and $s_{\delta}\in(0,R)$ satisfies
$\delta^{2/(p-1)}s_{\delta}^{-2/(p-1)}\phi^{\prime}(1)=\frac{a}{\ln(s_{\delta}/R)},$
which implies
$\frac{s_{\delta}}{\delta|\ln\delta|^{(p-1)/2}}\rightarrow\left(\frac{|\phi^{\prime}(1)|}{a}\right)^{(p-1)/2}>0,\quad\text{as}~{}~{}\delta\rightarrow
0.$
Moreover, by Pohozaev identity, we can get that
$\int_{B_{1}(0)}\phi^{p+1}=\frac{\pi(p+1)}{2}|\phi^{\prime}(1)|^{2}~{}~{}\text{and}~{}~{}\int_{B_{1}(0)}\phi^{p}=2\pi|\phi^{\prime}(1)|.$
For any $z\in\Omega$, define $W_{\delta,z,a}(x)=W_{\delta,a}(x-z)$. Because
$W_{\delta,z,a}$ does not vanish on $\partial\Omega$, we need to make a
projection. Let $PW_{\delta,z,a}$ be the solution of
$\begin{cases}-\delta^{2}\Delta w=(W_{\delta,z,a}-a)_{+}^{p},&\text{in
}\;\Omega,\\\ w=0,&\text{on}\;\partial\Omega.\end{cases}$
Then
$PW_{\delta,z,a}=W_{\delta,z,a}-\frac{a}{\ln\frac{R}{s_{\delta}}}g(x,z),$
(2.3)
where $g(x,z)$ satisfies
$\begin{cases}-\Delta g=0,&\text{in }\;\Omega,\\\
g=\ln\frac{R}{|x-z|},&\text{on}\;\partial\Omega.\end{cases}$
It is easy to see that
$g(x,z)=\ln R+2\pi h(x,z),$
where $h(x,z)=-H(x,z)$.
Let $Z=(Z_{m}^{+},Z_{n}^{-})$, where $Z_{m}^{+}=(z_{1}^{+},\cdots,z_{m}^{+})$,
$Z_{n}^{+}=(z_{1}^{-},\cdots,z_{n}^{-})$. We will construct solutions for
(1.11) of the form
$\sum_{i=1}^{m}PW_{\delta,z_{i}^{+},a_{\delta,i}^{+}}-\sum_{j=1}^{n}PW_{\delta,z_{j}^{-},a_{\delta,j}^{-}}+\omega_{\delta},$
where $z_{i}^{+},z_{j}^{-}\in\Omega$, $a^{+}_{\delta,i}>0,a^{-}_{\delta,j}>0$
for $i=1,\cdots,m$, $j=1,\cdots,n$, $\omega_{\delta}$ is a perturbation term.
To make $\omega_{\delta}$ as small as possible, we need to choose
$a^{+}_{\delta,i},\,a^{-}_{\delta,j}$ properly.
In this paper, we always assume that $z_{i}^{+},z_{j}^{-}\in\Omega$ satisfies
$\begin{split}&d(z_{i}^{+},\partial\Omega)\geq\varrho,~{}d(z_{j}^{-},\partial\Omega)\geq\varrho,\quad|z_{i}^{+}-z_{k}^{+}|\geq\varrho^{\bar{L}},\quad
i,k=1,\cdots,m,\;i\neq k\\\
&~{}~{}|z_{j}^{-}-z_{l}^{-}|\geq\varrho^{\bar{L}},\quad|z_{i}^{+}-z_{j}^{-}|\geq\varrho^{\bar{L}},\quad
j,l=1,\cdots,n,\;j\neq l,\end{split}$ (2.4)
where $\varrho>0$ is a fixed small constant and $\bar{L}>0$ is a fixed large
constant.
###### Lemma 2.1.
For $\delta>0$ small, there exist
$(s_{\delta,1}^{+}(Z),\cdots,s_{\delta,m}^{+}(Z),s_{\delta,1}^{-}(Z),\cdots,s^{-}_{\delta,n}(Z))$
and
$(a_{\delta,1}^{+}(Z),\cdots,a_{\delta,m}^{+}(Z),a_{\delta,1}^{-}(Z),\cdots,a^{-}_{\delta,n}(Z))$
satisfying the following system
$\delta^{2/(p-1)}(s_{i}^{+})^{-2/(p-1)}\phi^{\prime}(1)=\frac{a_{i}^{+}}{\ln(s_{i}^{+}/R)},\qquad
i=1,\cdots,m$ (2.5)
$\delta^{2/(p-1)}(s_{j}^{-})^{-2/(p-1)}\phi^{\prime}(1)=\frac{a_{j}^{-}}{\ln(s_{j}^{-}/R)},\qquad
j=1,\cdots,n$ (2.6)
and
$a_{i}^{+}=\kappa_{i}^{+}+\frac{2\pi
q(z_{i}^{+})}{|\ln\varepsilon|}+\frac{g(z_{i}^{+},z_{i}^{+})}{\ln\frac{R}{s_{i}^{+}}}a_{i}^{+}-\sum_{\alpha\neq
i}^{m}\frac{\bar{G}(z_{i}^{+},z_{\alpha}^{+})}{\ln\frac{R}{s_{\alpha}^{+}}}a_{\alpha}^{+}+\sum_{l=1}^{n}\frac{\bar{G}(z_{i}^{+},z_{l}^{-})}{\ln\frac{R}{s_{l}^{-}}}a_{l}^{-},\quad
i=1,\cdots,m,$ (2.7) $a_{j}^{-}=\kappa_{j}^{-}-\frac{2\pi
q(z_{j}^{-})}{|\ln\varepsilon|}+\frac{g(z_{i}^{-},z_{i}^{-})}{\ln\frac{R}{s_{j}^{-}}}a_{j}^{-}-\sum_{\beta\neq
j}^{n}\frac{\bar{G}(z_{\beta}^{-},z_{j}^{-})}{\ln\frac{R}{s_{\beta}^{+}}}a_{\beta}^{+}+\sum_{k=1}^{m}\frac{\bar{G}(z_{j}^{-},z_{k}^{+})}{\ln\frac{R}{s_{k}^{-}}}a_{k}^{-},\quad
j=1,\cdots,n,$ (2.8)
where $\bar{G}(x,y)=\ln\frac{R}{|x-y|}-g(x,y)$ for $x\neq y$.
Since the proof is exactly the same as in Lemma 2.1 in [13], we omit it here
therefore.
To simplify our notations, for given $Z=(Z_{m}^{+},Z_{n}^{-})$, in this paper,
we will use $a_{\delta,i}^{\pm}$, $s_{\delta,i}^{\pm}$ instead of
$a_{\delta,i}^{\pm}(Z)$, $s_{\delta,i}^{\pm}(Z)$. From now on we will always
choose
$(a_{\delta,1}^{+},\cdots,a_{\delta,m}^{+},a_{\delta,1}^{-},\cdots,a_{\delta,n}^{-})$
and
$(s_{\delta,1}^{+},\cdots,s_{\delta,m}^{+},s_{\delta,1}^{-},\cdots,s_{\delta,n}^{-})$
such that (2.5)–(2.8) hold. For
$(a_{\delta,1}^{+},\cdots,a_{\delta,m}^{+},a_{\delta,1}^{-},\cdots,a_{\delta,n}^{-})$
and
$(s_{\delta,1}^{+},\cdots,s_{\delta,m}^{+},s_{\delta,1}^{-},\cdots,s_{\delta,n}^{-})$
chosen in such a way let us define
$P_{\delta,Z,i}^{+}=PW_{\delta,z^{+}_{i},\,a_{\delta,i}^{+}},~{}~{}P_{\delta,Z,j}^{-}=PW_{\delta,z^{-}_{j},\,a_{\delta,j}^{-}}.$
(2.9)
###### Remark 2.2.
As in [13], we have the following asymptotic expansions:
$\frac{1}{\ln\frac{R}{s^{+}_{\delta,i}}}=\frac{1}{\ln\frac{R}{\varepsilon}}+0\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right),\,i=1,\cdots,m,$
(2.10)
$a_{\delta,i}^{+}=1+\frac{2\pi
q(z_{i}^{+})}{\kappa|\ln\varepsilon|}+\frac{g(z_{i}^{+},z_{i}^{+})}{\ln\frac{R}{\varepsilon}}-\sum_{\alpha\neq
i}^{m}\frac{\bar{G}(z_{i}^{+},z_{\alpha}^{+})}{\ln\frac{R}{\varepsilon}}+\sum_{l=1}^{n}\frac{\bar{G}(z_{i}^{+},z_{l}^{-})}{\ln\frac{R}{\varepsilon}}+0\Bigl{(}\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\Bigr{)},\,i=1,\cdots,m,$
(2.11)
$\frac{\partial a_{\delta,i}^{+}}{\partial
z_{k,h}^{\pm}}=0\left(\frac{1}{|\ln\varepsilon|}\right),\quad~{}~{}\frac{\partial
s_{\delta,i}^{+}}{\partial
z_{k,h}^{\pm}}=0\left(\frac{\varepsilon}{|\ln\varepsilon|}\right),\,i=1,\cdots,m,\,h=1,2.$
(2.12)
Moreover, $a_{\delta,j}^{-}$ and $s_{\delta,j}^{-}$ have similar expansions.
To simplify notations, set
$P^{+}_{\delta,Z}=\sum_{\alpha=1}^{m}P^{+}_{\delta,Z,\alpha},~{}~{}P^{-}_{\delta,Z}=\sum_{\beta=1}^{n}P^{-}_{\delta,Z,\beta}.$
Then, we find that for $x\in B_{Ls_{\delta,i}^{+}}(z_{i}^{+})$, where $L>0$ is
any fixed constant,
$\begin{split}&P_{\delta,Z,i}^{+}(x)-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}=W_{\delta,z^{+}_{i},\,a_{\delta,i}^{+}}(x)-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}g(x,z_{i}^{+})-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\\\
=&W_{\delta,z^{+}_{i},\,a_{\delta,i}^{+}}(x)-\kappa_{i}^{+}-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}g(z_{i}^{+},z_{i}^{+})-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\Bigl{(}\left\langle
Dg(z_{i}^{+},z_{i}^{+}),x-z_{i}^{+}\right\rangle+O(|x-z_{i}^{+}|^{2})\Bigr{)}\\\
\quad&-\frac{2\pi
q(z_{i}^{+})}{|\ln\varepsilon|}-\frac{2\pi}{|\ln\varepsilon|}\left(\left\langle
Dq(z_{i}^{+}),x-z_{i}^{+}\right\rangle+O(|x-z_{i}^{+}|^{2})\right)\\\
=&W_{\delta,z^{+}_{i},\,a^{+}_{\delta,i}}(x)-\kappa_{i}^{+}-\frac{2\pi
q(z^{+}_{i})}{|\ln\varepsilon|}-\frac{2\pi}{|\ln\varepsilon|}\left\langle
Dq(z_{i}^{+}),x-z^{+}_{i}\right\rangle\\\
\quad&-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}g(z_{i}^{+},z_{i}^{+})-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\left\langle
Dg(z_{i}^{+},z_{i}^{+}),x-z_{i}^{+}\right\rangle+O\left(\frac{(s^{+}_{\delta,i})^{2}}{|\ln\varepsilon|}\right),\end{split}$
and for $k\neq i$ and $x\in B_{Ls^{+}_{\delta,i}}(z_{i}^{+})$, by (2.2)
$\begin{split}&P^{+}_{\delta,Z,k}(x)=W_{\delta,z_{k}^{+},a^{+}_{\delta,k}}(x)-\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}g(x,z_{k}^{+})=\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}\bar{G}(x,z_{k}^{+})\\\
=&\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}\bar{G}(z_{i}^{+},z_{k}^{+})+\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}\left\langle
D\bar{G}(z_{i}^{+},z_{k}^{+}),x-z_{i}^{+}\right\rangle+O\Bigl{(}\frac{(s^{+}_{\delta,i})^{2}}{|\ln\varepsilon|}\Bigr{)}\end{split}$
and
$P^{-}_{\delta,Z,j}(x)=\frac{a^{-}_{\delta,j}}{\ln\frac{R}{s^{-}_{\delta,j}}}\bar{G}(z_{i}^{+},z_{j}^{-})+\frac{a^{-}_{\delta,j}}{\ln\frac{R}{s^{-}_{\delta,j}}}\left\langle
D\bar{G}(z_{i}^{+},z_{j}^{-}),x-z_{i}^{+}\right\rangle+O\Bigl{(}\frac{(s^{+}_{\delta,i})^{2}}{|\ln\varepsilon|}\Bigr{)}.$
So, by using (2.7), we obtain
$\begin{split}&P^{+}_{\delta,Z}(x)-P^{-}_{\delta,Z}(x)-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\\\
=&W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}(x)-a^{+}_{\delta,i}-\frac{2\pi}{|\ln\varepsilon|}\left\langle
Dq(z_{i}^{+}),x-z^{+}_{i}\right\rangle-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\left\langle
Dg(z_{i}^{+},z_{i}^{+}),x-z^{+}_{i}\right\rangle\\\ &+\sum_{k\neq
i}^{m}\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}\left\langle
D\bar{G}(z_{i}^{+},z_{k}^{+}),x-z_{i}^{+}\right\rangle-\sum_{l=1}^{n}\frac{a^{-}_{\delta,l}}{\ln\frac{R}{s^{-}_{\delta,l}}}\left\langle
D\bar{G}(z_{i}^{+},z_{l}^{-}),x-z_{i}^{+}\right\rangle\\\
&+O\left(\frac{(s^{+}_{\delta,i})^{2}}{|\ln\varepsilon|}\right),\quad x\in
B_{Ls^{+}_{\delta,i}}(z^{+}_{i}).\end{split}$ (2.13)
Similarly, we have
$\begin{split}&P^{-}_{\delta,Z}(x)-P^{+}_{\delta,Z}(x)-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\\\
=&W_{\delta,z_{j}^{-},a^{-}_{\delta,j}}(x)-a^{-}_{\delta,j}+\frac{2\pi}{|\ln\varepsilon|}\left\langle
Dq(z_{j}^{-}),x-z^{-}_{j}\right\rangle-\frac{a^{-}_{\delta,j}}{\ln\frac{R}{s^{-}_{\delta,j}}}\left\langle
Dg(z_{j}^{-},z_{j}^{-}),x-z^{-}_{j}\right\rangle\\\ &+\sum_{l\neq
j}^{n}\frac{a^{-}_{\delta,l}}{\ln\frac{R}{s^{-}_{\delta,l}}}\left\langle
D\bar{G}(z_{j}^{-},z_{l}^{-}),x-z_{j}^{-}\right\rangle-\sum_{k=1}^{m}\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}\left\langle
D\bar{G}(z_{j}^{-},z_{k}^{+}),x-z_{j}^{-}\right\rangle\\\
&+O\left(\frac{(s^{-}_{\delta,j})^{2}}{|\ln\varepsilon|}\right),\quad x\in
B_{Ls^{-}_{\delta,j}}(z_{j}^{-}).\end{split}$ (2.14)
We end this section by giving the following formula which can be obtained by
direct computation and will be used in the next two sections.
$\begin{array}[]{ll}\displaystyle\frac{\partial
W_{\delta,z_{i}^{\pm},a^{\pm}_{\delta,i}}(x)}{\partial z^{\pm}_{i,h}}&\\\
=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{\delta}\Bigl{(}\frac{a^{\pm}_{\delta,i}}{|\phi^{\prime}(1)||\ln\frac{R}{s^{\pm}_{\delta,i}}|}\Bigr{)}^{(p+1)/2}\phi^{\prime}\bigl{(}\frac{|x-z^{\pm}_{i}|}{s^{\pm}_{\delta,i}}\bigr{)}\frac{z^{\pm}_{i,h}-x_{h}}{|x-z_{i}^{\pm}|}+O\left(\frac{1}{|\ln\varepsilon|}\right),~{}~{}x\in
B_{s^{\pm}_{\delta,i}}(z^{\pm}_{i}),\\\ \\\
\displaystyle-\frac{a^{\pm}_{\delta,i}}{\ln\frac{R}{s^{\pm}_{\delta,i}}}\frac{z^{\pm}_{i,h}-x_{h}}{|x-z^{\pm}_{i}|^{2}}+O\left(\frac{1}{|\ln\varepsilon|}\right),\qquad\qquad\qquad\qquad\qquad\quad
x\in\Omega\setminus B_{s^{\pm}_{\delta,i}}(z^{\pm}_{i}).\end{array}\right.\\\
\end{array}$ (2.15)
## 3\. the reduction
Let
$w(x)=\begin{cases}\phi(|x|),&|x|\leq 1,\\\
\phi^{\prime}(1)\ln|x|,&|x|>1.\end{cases}$
Then $w\in C^{1}(\mathbb{R}^{2})$. Since $\phi^{\prime}(1)<0$ and $\ln|x|$ is
harmonic for $|x|>1$, we see that $w$ satisfies
$-\Delta w=w_{+}^{p},\quad\text{in}\;\mathbb{R}^{2}.$ (3.1)
Moreover, since $w_{+}$ is Lip-continuous, by the Schauder estimate, $w\in
C^{2,\alpha}$ for any $\alpha\in(0,1)$.
Consider the following problem:
$-\Delta v-pw_{+}^{p-1}v=0,\quad v\in L^{\infty}(\mathbb{R}^{2}),$ (3.2)
It is easy to see that $\frac{\partial w}{\partial x_{i}}$, $i=1,2,$ is a
solution of (3.2). Moreover, from Dancer and Yan [15], we know that $w$ is
also non-degenerate, in the sense that the kernel of the operator $Lv:=-\Delta
v-pw_{+}^{p-1}v,~{}~{}v\in D^{1,2}(\mathbb{R}^{2})$ is spanned by
$\bigl{\\{}\frac{\partial w}{\partial x_{1}},\frac{\partial w}{\partial
x_{2}}\bigr{\\}}$.
Let $P_{\delta,Z,i}^{+},~{}P_{\delta,Z,j}^{-}$ be the functions defined in
(2.9). Set
$\begin{split}F_{\delta,Z}=\Bigg{\\{}u:u\in
L^{p}(\Omega),&\int_{\Omega}\frac{\partial P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}u=0,\int_{\Omega}\frac{\partial P^{-}_{\delta,Z,j}}{\partial
z^{-}_{j,h}}u=0,\\\
&i=1,\cdots,m,\;\;j=1,\cdots,n,\;\;h=1,2\Bigg{\\}},\end{split}$
and
$\begin{split}E_{\delta,Z}=\Bigg{\\{}u:\;u\in W^{2,p}(\Omega)\cap
H_{0}^{1}(\Omega),&\int_{\Omega}\Delta\left(\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\right)u=0,\;\int_{\Omega}\Delta\left(\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,h}}\right)u=0,\\\
&i=1,\cdots,m,\;\;j=1,\cdots,n,\;\;h=1,2\Bigg{\\}}.\end{split}$
For any $u\in L^{p}(\Omega)$, define $Q_{\delta}u$ as follows:
$Q_{\delta}u=u-\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h}\left(-\delta^{2}\Delta\Bigl{(}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\Bigr{)}\right)-\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h}}\left(-\delta^{2}\Delta\Bigl{(}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\Bigr{)}\right),$
where the constants $b^{+}_{i,h}$, $b^{-}_{j,\bar{h}}$ satisfy
$\begin{split}&\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h}\left(-\delta^{2}\int_{\Omega}\Delta\Bigl{(}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\Bigr{)}\frac{\partial
P^{+}_{\delta,Z,k}}{\partial z^{+}_{k,\hat{h}}}\right)\\\
&+\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h}}\left(-\delta^{2}\int_{\Omega}\Delta\Bigl{(}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\Bigr{)}\frac{\partial
P^{+}_{\delta,Z,k}}{\partial
z^{+}_{k,\hat{h}}}\right)=\int_{\Omega}u\frac{\partial
P^{+}_{\delta,Z,k}}{\partial z^{+}_{k,\hat{h}}},\end{split}$ (3.3)
and
$\begin{split}&\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h}\left(-\delta^{2}\int_{\Omega}\Delta\Bigl{(}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\Bigr{)}\frac{\partial
P^{-}_{\delta,Z,l}}{\partial z^{-}_{l,\tilde{h}}}\right)\\\
&+\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h}}\left(-\delta^{2}\int_{\Omega}\Delta\Bigl{(}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\Bigr{)}\frac{\partial
P^{-}_{\delta,Z,l}}{\partial
z^{-}_{l,\tilde{h}}}\right)=\int_{\Omega}u\frac{\partial
P^{-}_{\delta,Z,l}}{\partial z^{-}_{l,\tilde{h}}}.\end{split}$ (3.4)
Since $\int_{\Omega}\frac{\partial P^{+}_{\delta,Z,k}}{\partial
z^{+}_{k,\hat{h}}}Q_{\delta}u=0$, $\int_{\Omega}\frac{\partial
P^{-}_{\delta,Z,l}}{\partial z^{-}_{l,\tilde{h}}}Q_{\delta}u=0$, the operator
$Q_{\delta}$ can be regarded as a projection from $L^{p}(\Omega)$ to
$F_{\delta,Z}$. In order to show that we can solve (3.3) and (3.4) to obtain
$b^{+}_{i,h}$ and $b^{-}_{j,\bar{h}}$, we just need the following estimate (
by (2.12) and (2.15)):
$\begin{split}&-\delta^{2}\int_{\Omega}\Delta\Bigl{(}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\Bigr{)}\frac{\partial
P^{+}_{\delta,Z,k}}{\partial z^{+}_{k,\hat{h}}}\\\
=&p\int_{\Omega}\bigl{(}W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a_{\delta,i}^{+}\bigr{)}_{+}^{p-1}\left(\frac{\partial
W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}}{\partial z^{+}_{i,h}}-\frac{\partial
a^{+}_{\delta,i}}{\partial z^{+}_{i,h}}\right)\frac{\partial
P^{+}_{\delta,Z,k}}{\partial z^{+}_{k,\hat{h}}}\\\
=&\delta_{ikh\hat{h}}\frac{c}{|\ln\varepsilon|^{p+1}}+0\left(\frac{\varepsilon}{|\ln\varepsilon|^{p+1}}\right),\end{split}$
(3.5)
where $c>0$ is a constant, $\delta_{ikh\hat{h}}=1$, if $i=k$ and $h=\hat{h}$;
otherwise, $\delta_{ijh\hat{h}}=0$.
Similarly,
$\begin{split}-\delta^{2}\int_{\Omega}\Delta\Bigl{(}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\Bigr{)}\frac{\partial
P^{-}_{\delta,Z,l}}{\partial
z^{-}_{l,\tilde{h}}}=\delta_{jl\bar{h}\tilde{h}}\frac{c}{|\ln\varepsilon|^{p+1}}+0\left(\frac{\varepsilon}{|\ln\varepsilon|^{p+1}}\right),\end{split}$
(3.6)
where $c>0$ is a constant, $\delta_{jl\bar{h}\tilde{h}}=1$, if $j=l$ and
$\bar{h}=\tilde{h}$; otherwise, $\delta_{jl\bar{h}\tilde{h}}=0$.
Set
$\begin{split}L_{\delta}u=-\delta^{2}\Delta
u&-\sum_{i=1}^{m}p\chi_{\Omega_{i}^{+}}\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p-1}u\\\
&-\sum_{j=1}^{m}p\chi_{\Omega_{j}^{-}}\left(P_{\delta,Z}^{-}-P_{\delta,Z}^{+}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p-1}u,\end{split}$
and
$B_{\delta,Z}=\Bigl{(}\cup_{i=1}^{m}B_{Ls_{\delta,i}^{+}}(z_{i}^{+})\Bigr{)}\bigcup\Bigl{(}\cup_{j=1}^{n}B_{Ls_{\delta,j}^{-}}(z_{j}^{-})\Bigr{)}.$
We have the following lemma.
###### Lemma 3.1.
There are constants $\rho_{0}>0$ and $\delta_{0}>0$, such that for any
$\delta\in(0,\delta_{0}]$, $Z$ satisfying (2.4), $u\in E_{\delta,Z}$ with
$Q_{\delta}L_{\delta}u=0$ in $\Omega\setminus B_{\delta,Z}$ for some $L>0$
large, then
$\|Q_{\delta}L_{\delta}u\|_{L^{p}(\Omega)}\geq\frac{\rho_{0}\delta^{\frac{2}{p}}}{|\ln\delta|^{\frac{(p-1)^{2}}{p}}}\|u\|_{L^{\infty}(\Omega)}.$
###### Proof.
Set $s_{N,j}^{\pm}=s_{\delta_{N},j}^{\pm}$. In the sequel, we will use
$\|\cdot\|_{p},\|\cdot\|_{\infty}$ to denote $\|\cdot\|_{L^{p}(\Omega)}$ and
$\|\cdot\|_{L^{\infty}(\Omega)}$ respectively.
We argue by contradiction. Suppose that there are $\delta_{N}\to 0$, $Z_{N}$
satisfying (2.4) and $u_{N}\in E_{\delta_{N},Z_{N}}$ with
$Q_{\delta_{N}}L_{\delta_{N}}u_{N}=0$ in $\Omega\setminus
B_{\delta_{N},Z_{N}}$ and $\|u_{N}\|_{\infty}=1$ such that
$\|Q_{\delta_{N}}L_{\delta_{N}}u_{N}\|_{p}\leq\frac{1}{N}\frac{\delta_{N}^{\frac{2}{p}}}{|\ln\delta_{N}|^{\frac{(p-1)^{2}}{p}}}.$
First, we estimate $b^{+}_{i,h,N}$ and $b^{-}_{j,\bar{h},N}$ in the following
formula:
$\begin{split}Q_{\delta_{N}}L_{\delta_{N}}u_{N}=L_{\delta_{N}}u_{N}&-\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h,N}\left(-\delta_{N}^{2}\Delta\frac{\partial
P^{+}_{\delta_{N},Z_{N},i}}{\partial z^{+}_{i,h}}\right)\\\
&-\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h},N}\left(-\delta_{N}^{2}\Delta\frac{\partial
P^{-}_{\delta_{N},Z_{N},j}}{\partial z^{-}_{j,\bar{h}}}\right).\end{split}$
(3.7)
For each fixed $k$, multiplying (3.7) by $\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}$, noting that
$\int_{\Omega}\bigl{(}Q_{\delta_{N}}L_{\delta_{N}}u_{N}\bigr{)}\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}=0,$
we obtain
$\begin{split}&\int_{\Omega}u_{N}\,L_{\delta_{N}}\left(\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial
z^{+}_{k,\hat{h}}}\right)=\int_{\Omega}\bigl{(}L_{\delta_{N}}u_{N}\bigr{)}\,\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}\\\
&=\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h,N}\int_{\Omega}\left(-\delta_{N}^{2}\Delta\frac{\partial
P^{+}_{\delta_{N},Z_{N},i}}{\partial z^{+}_{i,h}}\right)\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}\\\
&\quad+\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h},N}\int_{\Omega}\left(-\delta_{N}^{2}\Delta\frac{\partial
P^{-}_{\delta_{N},Z_{N},j}}{\partial z^{-}_{j,\bar{h}}}\right)\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}.\end{split}$
Using (2.13), (2.14) and Lemma A.1, we obtain
$\begin{split}&\int_{\Omega}u_{N}\,L_{\delta_{N}}\left(\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}\right)\\\
&=\int_{\Omega}\Bigg{[}-\delta_{N}^{2}\Delta\left(\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial
z^{+}_{k,\hat{h}}}\right)-\sum_{i=1}^{m}p\chi_{\Omega_{i}^{+}}\left(P^{+}_{\delta_{N},Z_{N}}-P_{\delta_{N},Z_{N}}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon_{N}|}\right)_{+}^{p-1}\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}\\\
&\quad-\sum_{j=1}^{n}p\chi_{\Omega_{j}^{-}}\left(P^{-}_{\delta_{N},Z_{N}}-P^{+}_{\delta_{N},Z_{N}}-1+\frac{2\pi
q(x)}{\kappa|\ln\varepsilon_{N}|}\right)_{+}^{p-1}\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}\Bigg{]}u_{N}\\\
&=p\int_{\Omega}\left(W_{\delta_{N},z^{+}_{k,N},a^{+}_{\delta_{N},k}}-a^{+}_{\delta_{N},k}\right)_{+}^{p-1}\left(\frac{\partial
W_{\delta_{N},z^{+}_{k,N},a^{+}_{\delta_{N},k}}}{\partial
z^{+}_{k,\hat{h}}}-\frac{\partial a^{+}_{\delta_{N},k}}{\partial
z^{+}_{k,\hat{h}}}\right)u_{N}\\\
&\quad-p\sum_{\alpha=1}^{m}\int_{\Omega_{\alpha}^{+}}\left(W_{\delta_{N},z^{+}_{\alpha,N},a^{+}_{\delta_{N},\alpha}}-a^{+}_{\delta_{N},\alpha}+O\left(\frac{s^{+}_{N,\alpha}}{|\ln\varepsilon_{N}|}\right)\right)_{+}^{p-1}\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}u_{N}\\\
&\quad-p\sum_{\beta=1}^{n}\int_{\Omega_{\beta}^{-}}\left(W_{\delta_{N},z^{-}_{\beta,N},a^{-}_{\delta_{N},\beta}}-a^{-}_{\delta_{N},\beta}+O\left(\frac{s^{-}_{N,\beta}}{|\ln\varepsilon_{N}|}\right)\right)_{+}^{p-1}\frac{\partial
P^{+}_{\delta_{N},Z_{N},k}}{\partial z^{+}_{k,\hat{h}}}u_{N}\\\
&=0\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p}}\right).\end{split}$
Using (3.5) and (3.6), we find that
$b^{+}_{i,h,N}=0\left(\varepsilon_{N}^{2}|\ln\varepsilon_{N}|\right).$
Similarly,
$b^{-}_{i,h,N}=0\left(\varepsilon_{N}^{2}|\ln\varepsilon_{N}|\right).$
Therefore,
$\begin{split}&\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h,N}\left(-\delta_{N}^{2}\Delta\frac{\partial
P^{+}_{\delta_{N},Z_{N},i}}{\partial
z^{+}_{i,h}}\right)+\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h},N}\left(-\delta_{N}^{2}\Delta\frac{\partial
P^{-}_{\delta_{N},Z_{N},j}}{\partial z^{-}_{j,\bar{h}}}\right)\\\
&=p\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h,N}\left(W_{\delta_{N},z^{+}_{i,N},a^{+}_{\delta_{N},i}}-a^{+}_{\delta_{N},i}\right)_{+}^{p-1}\left(\frac{\partial
W_{\delta_{N},z^{+}_{i,N},a^{+}_{\delta_{N},i}}}{\partial
z^{+}_{i,h}}-\frac{\partial a^{+}_{\delta_{N},i}}{\partial
z^{+}_{i,h}}\right)\\\
&\quad+p\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h},N}\left(W_{\delta_{N},z^{-}_{j,N},a^{-}_{\delta_{N},j}}-a^{-}_{\delta_{N},j}\right)_{+}^{p-1}\left(\frac{\partial
W_{\delta_{N},z^{-}_{j,N},a^{-}_{\delta_{N},j}}}{\partial
z^{-}_{j,\bar{h}}}-\frac{\partial a^{-}_{\delta_{N},j}}{\partial
z^{-}_{j,\bar{h}}}\right)\\\
&=0\left(\sum_{i=1}^{m}\sum_{h=1}^{2}\frac{\varepsilon_{N}^{\frac{2}{p}-1}|b^{+}_{i,h,N}|}{|\ln\varepsilon_{N}|^{p}}\right)+0\left(\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}\frac{\varepsilon_{N}^{\frac{2}{p}-1}|b^{-}_{j,\bar{h},N}|}{|\ln\varepsilon_{N}|^{p}}\right)\\\
&=0\left(\frac{\varepsilon_{N}^{\frac{2}{p}+1}}{|\ln\varepsilon_{N}|^{p-1}}\right)\quad\text{in}~{}~{}L^{p}(\Omega).\end{split}$
Thus, we obtain
$L_{\delta_{N}}u_{N}=Q_{\delta_{N}}L_{\delta_{N}}u_{N}+O\left(\frac{\varepsilon_{N}^{\frac{2}{p}+1}}{|\ln\varepsilon_{N}|^{p-1}}\right)=O\left(\frac{1}{N}\frac{\delta_{N}^{\frac{2}{p}}}{|\ln\delta_{N}|^{\frac{(p-1)^{2}}{p}}}\right).$
For any fixed $i,j$, define
$\tilde{u}^{+}_{i,N}(y)=u_{N}(s^{+}_{N,i}y+z^{+}_{i,N}),\quad\tilde{u}^{-}_{j,N}(y)=u_{N}(s^{-}_{N,j}y+z^{-}_{j,N}).$
Let
$\begin{split}\tilde{L}_{N}^{\pm}u=&-\Delta
u-\sum_{k=1}^{m}p\frac{(s_{N,i}^{\pm})^{2}}{\delta_{N}^{2}}\chi_{\Omega_{k}^{+}}\left(P^{+}_{\delta_{N},Z_{N}}(s^{\pm}_{N,i}y+z^{\pm}_{i,N})-P^{-}_{\delta_{N},Z_{N}}(s^{\pm}_{N,i}y+z^{\pm}_{i,N})-\kappa_{k}^{+}-\frac{2\pi
q}{|\ln\varepsilon_{N}|}\right)_{+}^{p-1}u\\\
&-\sum_{l=1}^{n}p\frac{(s_{N,i}^{\pm})^{2}}{\delta_{N}^{2}}\chi_{\Omega_{l}^{-}}\left(P^{-}_{\delta_{N},Z_{N}}(s^{\pm}_{N,i}y+z^{\pm}_{i,N})-P^{+}_{\delta_{N},Z_{N}}(s^{\pm}_{N,i}y+z^{\pm}_{i,N})-\kappa_{l}^{-}+\frac{2\pi
q}{|\ln\varepsilon_{N}|}\right)_{+}^{p-1}u.\end{split}$
Then
$(s_{N,i}^{\pm})^{\frac{2}{p}}\times\frac{\delta_{N}^{2}}{(s^{\pm}_{N,i})^{2}}\|\tilde{L}_{N}^{\pm}\tilde{u}^{\pm}_{i,N}\|_{p}=\|L_{\delta_{N}}u_{N}\|_{p}.$
Noting that
$\left(\frac{\delta_{N}}{s^{\pm}_{N,i}}\right)^{2}=O\left(\frac{1}{|\ln\delta_{N}|^{p-1}}\right),$
we find that
$L_{\delta_{N}}u_{N}=o\left(\frac{\delta_{N}^{\frac{2}{p}}}{|\ln\delta_{N}|^{\frac{(p-1)^{2}}{p}}}\right).$
As a result,
$\tilde{L}_{N}^{\pm}\tilde{u}_{i,N}^{\pm}=o(1),\quad\text{in}\;L^{p}(\Omega_{N}^{\pm}),$
where
$\Omega_{N}^{\pm}=\bigl{\\{}y:s^{\pm}_{N,i}y+z^{\pm}_{i,N}\in\Omega\bigr{\\}}$.
Since $\|\tilde{u}_{i,N}^{\pm}\|_{\infty}=1$, by the regularity theory of
elliptic equations, we may assume that
$\tilde{u}_{i,N}^{\pm}\to
u_{i}^{\pm},\quad\text{in}\;C_{loc}^{1}(\mathbb{R}^{2}).$
It is easy to see that
$\begin{split}&\sum_{k=1}^{m}\frac{(s^{+}_{N,i})^{2}}{\delta_{N}^{2}}\chi_{\Omega_{k}^{+}}\left(P^{+}_{\delta_{N},Z_{N}}(s^{+}_{N,i}y+z^{+}_{i,N})-P^{-}_{\delta_{N},Z_{N}}(s_{N,i}^{+}y+z_{i,N}^{+})-\kappa_{k}^{+}-\frac{2\pi
q}{|\ln\varepsilon_{N}|}\right)_{+}^{p-1}\\\
&=\frac{(s_{N,i}^{+})^{2}}{\delta_{N}^{2}}\left(W_{\delta_{N},z^{+}_{i,N},a^{+}_{\delta_{N},i}}-a^{+}_{\delta_{N},i}+O\left(\frac{s^{+}_{N,i}}{|\ln\varepsilon_{N}|}\right)\right)_{+}^{p-1}+o(1)\\\
&\rightarrow w_{+}^{p-1}.\end{split}$
Similarly,
$\begin{split}&\sum_{l=1}^{n}\frac{(s^{-}_{N,j})^{2}}{\delta_{N}^{2}}\chi_{\Omega_{l}^{-}}\left(P^{-}_{\delta_{N},Z_{N}}(s_{N,j}^{-}y+z_{j,N}^{-})-P^{+}_{\delta_{N},Z_{N}}(s^{-}_{N,j}y+z^{-}_{j,N})-\kappa_{l}^{-}+\frac{2\pi
q}{|\ln\varepsilon_{N}|}\right)_{+}^{p-1}\\\ &\rightarrow
w_{+}^{p-1}.\end{split}$
Then, by Lemma A.1, we find that $u_{i}^{\pm}$ satisfies
$-\Delta u-pw_{+}^{p-1}u=0.$
Now from the Proposition 3.1 in [15], we have
$u_{i}^{\pm}=c_{1}^{\pm}\frac{\partial w}{\partial
x_{1}}+c_{2}^{\pm}\frac{\partial w}{\partial x_{2}}.$ (3.8)
Since
$\int_{\Omega}\Delta\bigl{(}\frac{\partial
P^{\pm}_{\delta_{N},Z_{N},i}}{\partial z^{\pm}_{i,h}}\bigr{)}u_{N}=0,$
we find that
$\int_{\mathbb{R}^{2}}\phi_{+}^{p-1}\frac{\partial\phi}{\partial
z_{h}}u_{i}^{\pm}=0,$
which, together with (3.8), gives $u_{i}^{\pm}\neq 0$. Thus,
$\tilde{u}_{i,N}^{\pm}\to 0,\quad\text{in}\;C^{1}(B_{L}(0)),$
for any $L>0$, which implies that $u_{N}=o(1)$ on $\partial
B_{Ls^{\pm}_{N,i}}(z^{\pm}_{i,N})$.
By assumption,
$Q_{\delta_{N}}L_{\delta_{N}}u_{N}=0,\quad\text{in}\;\Omega\setminus
B_{\delta_{N},Z_{N}}.$
On the other hand, by Lemma A.1, for $i=1,\cdots,m$, $j=1,\cdots,n$, we have
$\left(P^{+}_{\delta_{N},Z_{N}}-P_{\delta_{N},Z_{N}}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon_{N}|}\right)_{+}=0,\quad x\in\Omega_{i}^{+}\setminus
B_{Ls^{+}_{N,i}}(z^{+}_{i,N}),$
$\left(P^{-}_{\delta_{N},Z_{N}}-P_{\delta_{N},Z_{N}}^{+}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon_{N}|}\right)_{+}=0,\quad x\in\Omega_{j}^{-}\setminus
B_{Ls^{-}_{N,j}}(z^{-}_{j,N}).$
Thus, we find that
$-\Delta u_{N}=0,\quad\text{in}~{}\Omega\setminus B_{\delta_{N},Z_{N}}.$
However, $u_{N}=0$ on $\partial\Omega$ and $u_{N}=o(1)$ on $\partial
B_{\delta_{N},Z_{N}}$. So we have
$u_{N}=o(1).$
This is a contradiction.
∎
From Lemma 3.1, using Fredholm alternative, we can prove, as in [13], the
following result:
###### Proposition 3.2.
$Q_{\delta}L_{\delta}$ is one to one and onto from $E_{\delta,Z}$ to
$F_{\delta,Z}$.
Now consider the equation
$Q_{\delta}L_{\delta}\omega=Q_{\delta}l_{\delta}^{+}-Q_{\delta}l_{\delta}^{-}+Q_{\delta}R^{+}_{\delta}(\omega)-Q_{\delta}R^{-}_{\delta}(\omega),$
(3.9)
where
$l_{\delta}^{+}=\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\left(P_{\delta,Z}^{+}-P^{-}_{\delta,Z}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\sum_{i=1}^{m}\left(W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p},$
(3.10)
$l_{\delta}^{-}=\sum_{j=1}^{n}\chi_{\Omega_{j}^{-}}\left(P_{\delta,Z}^{-}-P^{+}_{\delta,Z}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\sum_{j=1}^{n}\left(W_{\delta,z_{j}^{-},a^{-}_{\delta,j}}-a^{-}_{\delta,j}\right)_{+}^{p},$
(3.11)
and
$\begin{split}R^{+}_{\delta}(\omega)=&\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\Bigg{[}\left(P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\left(P^{+}_{\delta,Z}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\\\
&-p\left(P^{+}_{\delta,Z}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p-1}\omega\Bigg{]},\end{split}$ (3.12)
$\begin{split}R^{-}_{\delta}(\omega)=&\sum_{j=1}^{n}\chi_{\Omega_{j}^{-}}\Bigg{[}\left(P^{-}_{\delta,Z}-P_{\delta,Z}^{+}-\omega-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\left(P^{-}_{\delta,Z}-P_{\delta,Z}^{+}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\\\
&+p\left(P^{-}_{\delta,Z}-P_{\delta,Z}^{+}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p-1}\omega\Bigg{]}.\end{split}$ (3.13)
Using Proposition 3.2, we can rewrite (3.9) as
$\omega=G_{\delta}\omega=:(Q_{\delta}L_{\delta})^{-1}Q_{\delta}\bigl{(}l_{\delta}^{+}-l_{\delta}^{-}+R^{+}_{\delta}(\omega)-R^{-}_{\delta}(\omega)\bigr{)}.$
(3.14)
The next Proposition enables us to reduce the problem of finding a solution
for (1.11) to a finite dimensional problem.
###### Proposition 3.3.
There is an $\delta_{0}>0$, such that for any $\delta\in(0,\delta_{0}]$ and
$Z$ satisfying (2.4), (3.9) has a unique solution $\omega_{\delta}\in
E_{\delta,Z}$, with
$\|\omega_{\delta}\|_{\infty}=0\Bigl{(}\delta|\ln\delta|^{\frac{p-1}{2}}\Bigr{)}.$
###### Proof.
It follows from Lemma A.1 that if $L$ is large enough, $\delta$ is small then
$\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}=0,\quad x\in\Omega_{i}^{+}\setminus
B_{Ls^{+}_{\delta,i}}(z^{+}_{i}),\,i=1,\cdots,m$
$\left(P^{-}_{\delta,Z}-P_{\delta,Z}^{+}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}=0,\quad x\in\Omega_{j}^{-}\setminus
B_{Ls^{-}_{\delta,j}}(z^{-}_{j}),\,j=1,\cdots,n.$
Let
$M=E_{\delta,Z}\cap\Bigl{\\{}\|\omega\|_{\infty}\leq\delta|\ln\delta|^{\frac{p-1}{2}}\Big{\\}}.$
Then $M$ is complete under $L^{\infty}$ norm and $G_{\delta}$ is a map from
$E_{\delta,Z}$ to $E_{\delta,Z}$. We will show that $G_{\delta}$ is a
contraction map from $M$ to $M$.
Step 1. $G_{\delta}$ is a map from $M$ to $M$.
For any $\omega\in M$, similar to Lemma A.1, it is easy to prove that for
large $L>0$, $\delta$ small
$\begin{split}\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}=0,\quad x\in\Omega_{i}^{+}\setminus
B_{Ls^{+}_{\delta,i}}(z^{+}_{i}),\\\
\left(P^{-}_{\delta,Z}-P^{+}_{\delta,Z}-\omega-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}=0,\quad x\in\Omega_{j}^{-}\setminus
B_{Ls^{-}_{\delta,j}}(z^{-}_{j}).\end{split}$ (3.15)
Note also that for any $u\in L^{\infty}(\Omega)$,
$Q_{\delta}u=u\quad\text{in}\;\Omega\setminus B_{\delta,Z}.$
Therefore, using Lemma A.1, (3.10)–(3.13), we find that for any $\omega\in M$,
$\begin{split}&Q_{\delta}(l_{\delta}^{+}-l_{\delta}^{-})+Q_{\delta}(R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega))\\\
=&l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)\\\
=&0,\quad\text{in}\;\Omega\setminus B_{Z,\delta}.\end{split}$
So, we can apply Lemma 3.1 to obtain
$\begin{split}&\|(Q_{\delta}L_{\delta})^{-1}\bigl{(}Q_{\delta}(l_{\delta}^{+}-l_{\delta}^{-})+Q_{\delta}(R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega))\bigr{)}\|_{\infty}\\\
\leq&\frac{C|\ln\delta|^{\frac{(p-1)^{2}}{p}}}{\delta^{\frac{2}{p}}}\|Q_{\delta}(l_{\delta}^{+}-l_{\delta}^{-})+Q_{\delta}(R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega))\|_{p}.\end{split}$
Thus, for any $\omega\in M$, we have
$\begin{array}[]{ll}\|G_{\delta}(\omega)\|_{\infty}=\|(Q_{\delta}L_{\delta})^{-1}Q_{\delta}\bigl{(}l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)\bigr{)}\|_{\infty}&\\\
\qquad\quad\quad\quad\leq\frac{C|\ln\delta|^{\frac{(p-1)^{2}}{p}}}{\delta^{\frac{2}{p}}}\|Q_{\delta}\bigl{(}l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)\bigr{)}\|_{p}.\end{array}$
(3.16)
It follows from (3.3)–(3.6) that the constant $b_{k,\hat{h}}^{\pm}$,
corresponding to $u\in L^{\infty}(\Omega)$, satisfies
$|b_{k,\hat{h}}^{\pm}|\leq
C|\ln\delta|^{p+1}\Bigg{(}\sum_{i,\,h}\int_{\Omega}\Bigl{|}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\Bigr{|}|u|+\sum_{j,\,\bar{h}}\int_{\Omega}\Bigl{|}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\Bigr{|}|u|\Bigg{)}.$
Since
$l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)=0,\quad\text{in}\;\Omega\setminus
B_{\delta,Z},$
we find that the constant $b_{k,\hat{h}}^{\pm}$, corresponding to
$l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)$
satisfies
$\begin{split}|b_{k,\hat{h}}^{\pm}|\leq&C|\ln\delta|^{p+1}\sum_{i,\,h}\left(\sum_{\alpha=1}^{m}\int_{B_{Ls^{+}_{\delta,\alpha}}(z_{\alpha}^{+})}\Bigl{|}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\Bigr{|}|l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)|\right)\\\
&+C|\ln\delta|^{p+1}\sum_{j,\,\bar{h}}\left(\sum_{\beta=1}^{n}\int_{B_{Ls^{-}_{\delta,\beta}}(z_{\beta}^{-})}\Bigl{|}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial
z^{-}_{j,\bar{h}}}\Bigr{|}|l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)|\right)\\\
\leq&C\varepsilon^{1-\frac{2}{p}}|\ln\varepsilon|^{p}\|l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)\|_{p}.\end{split}$
As a result,
$\begin{split}&\|Q_{\delta}(l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega))\|_{p}\\\
\leq&\|l_{\delta}^{+}-l_{\delta}^{-}+R_{\delta}^{+}(\omega)-R_{\delta}^{-}(\omega)\|_{p}+C\sum_{i,\,h}|b_{i,h}^{+}|\left\|-\delta^{2}\Delta\Bigl{(}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\Bigr{)}\right\|_{p}\\\
&+C\sum_{j,\,\bar{h}}|b_{j,\bar{h}}^{-}|\left\|-\delta^{2}\Delta\Bigl{(}\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\Bigr{)}\right\|_{p}\\\
\leq&C\bigl{(}\|l^{+}_{\delta}\|_{p}+\|l^{-}_{\delta}\|_{p}+\|R^{+}_{\delta}(\omega)\|_{p}+\|R^{-}_{\delta}(\omega)\|_{p}\bigr{)}.\end{split}$
On the other hand, from Lemma A.1 and (2.13), we can deduce
$\begin{split}\|l_{\delta}^{+}\|_{p}=&\left\|\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\left(P_{\delta,Z}^{+}-P^{-}_{\delta,Z}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\sum_{i=1}^{m}\left(W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p}\right\|_{p}\\\
\leq&\sum_{i=1}^{m}\frac{Cs^{+}_{\delta,i}}{|\ln\varepsilon|}\Big{\|}\bigl{(}W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}_{+}^{p-1}\Big{\|}_{p}\\\
=&O\left(\frac{\delta^{1+\frac{2}{p}}}{|\ln\delta|^{\frac{p-1}{2}+\frac{1}{p}}}\right).\end{split}$
For the estimate of $\|R^{+}_{\delta}(\omega)\|_{p}$, we have
$\begin{split}\|R^{+}_{\delta}(\omega)\|_{\infty}=&\bigg{\|}\sum_{i=1}^{n}\chi_{\Omega_{i}^{+}}\bigg{[}\Big{(}P_{\delta,Z}^{+}-P_{\delta,Z}^{-}+\omega-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\Big{)}_{+}^{p}-\Big{(}P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\Big{)}_{+}^{p}\\\
&-p\Big{(}P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\Big{)}_{+}^{p-1}\omega\bigg{]}\bigg{\|}_{p}\\\
\leq&C\|\omega\|_{\infty}^{2}\left\|\sum_{i=1}^{n}\chi_{\Omega_{i}^{+}}\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p-2}\right\|_{p}\\\
=&O\left(\frac{\delta^{\frac{2}{p}}\|\omega\|_{\infty}^{2}}{|\ln\delta|^{p-3+\frac{1}{p}}}\right).\end{split}$
(3.17)
Similarly, we have
$\|l_{\delta}^{-}\|_{p}=O\left(\frac{\delta^{1+\frac{2}{p}}}{|\ln\delta|^{\frac{p-1}{2}+\frac{1}{p}}}\right),\quad\|R^{-}_{\delta}(\omega)\|_{p}=O\left(\frac{\delta^{\frac{2}{p}}\|\omega\|_{\infty}^{2}}{|\ln\delta|^{p-3+\frac{1}{p}}}\right).$
Thus, we obtain
$\begin{split}\|G_{\delta}(\omega)\|_{\infty}\leq&\frac{C|\ln\delta|^{\frac{(p-1)^{2}}{p}}}{\delta^{\frac{2}{p}}}\Bigl{(}\|l^{+}_{\delta}\|_{p}+\|l^{-}_{\delta}\|_{p}+\|R^{+}_{\delta}(\omega)\|_{p}+\|R^{-}_{\delta}(\omega)\|_{p}\Bigr{)}\\\
\leq&C|\ln\delta|^{\frac{(p-1)^{2}}{p}}\left(\frac{\delta}{|\ln\delta|^{\frac{p-1}{2}+\frac{1}{p}}}+\frac{\|\omega\|_{\infty}^{2}}{|\ln\delta|^{p-3+\frac{1}{p}}}\right)\\\
\leq&\delta|\ln\delta|^{\frac{p-1}{2}}\end{split}$ (3.18)
Thus, $G_{\delta}$ is a map from $M$ to $M$.
Step 2. $G_{\delta}$ is a contraction map.
In fact, for any $\omega_{i}\in M$, $i=1,2$, we have
$G_{\delta}\omega_{1}-G_{\delta}\omega_{2}=(Q_{\delta}L_{\delta})^{-1}Q_{\delta}\bigl{[}R^{+}_{\delta}(\omega_{1})-R^{+}_{\delta}(\omega_{2})-(R^{-}_{\delta}(\omega_{1})-R^{-}_{\delta}(\omega_{2}))\bigr{]}.$
Noting that
$R^{+}_{\delta}(\omega_{1})=R^{+}_{\delta}(\omega_{2})=0,\quad\text{in}\;\Omega\setminus\cup_{i=1}^{m}B_{Ls^{+}_{\delta,i}}(z_{i}^{+}),$
and
$R^{-}_{\delta}(\omega_{1})=R^{-}_{\delta}(\omega_{2})=0,\quad\text{in}\;\Omega\setminus\cup_{j=1}^{n}B_{Ls^{-}_{\delta,j}}(z_{j}^{-}),$
we can deduce as in Step 1 that
$\begin{split}\|G_{\delta}\omega_{1}-G_{\delta}\omega_{2}\|_{\infty}\leq&\frac{C|\ln\delta|^{\frac{(p-1)^{2}}{p}}}{\delta^{\frac{2}{p}}}(\|R^{+}_{\delta}(\omega_{1})-R^{+}_{\delta}(\omega_{2})\|_{p}+\|R^{-}_{\delta}(\omega_{1})-R^{-}_{\delta}(\omega_{2})\|_{p})\\\
\leq&C|\ln\delta|^{p-1}\left(\frac{\|\omega_{1}\|_{\infty}}{|\ln\delta|^{p-2}}+\frac{\|\omega_{2}\|_{\infty}}{|\ln\delta|^{p-2}}\right)\|\omega_{1}-\omega_{2}\|_{\infty}\\\
\leq&C\delta|\ln\delta|^{\frac{p+1}{2}}\|\omega_{1}-\omega_{2}\|_{\infty}\leq\frac{1}{2}\|\omega_{1}-\omega_{2}\|_{\infty}.\end{split}$
Combining Step 1 and Step 2, we have proved that $G_{\delta}$ is a contraction
map from $M$ to $M$. By the contraction mapping theorem, there is an unique
$\omega_{\delta}\in M$, such that $\omega_{\delta}=G_{\delta}\omega_{\delta}$.
Moreover, it follows from (3.18) that
$\|\omega_{\delta}\|_{\infty}\leq\delta|\ln\delta|^{\frac{p-1}{2}}.$
∎
## 4\. Proof of The main results
In this section, we will choose $Z$, such that
$P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega_{\delta}$, where $\omega_{\delta}$
is the map obtained in Proposition 3.3, is a solution of (1.11).
Define
$\begin{split}I(u)=\frac{\delta^{2}}{2}\int_{\Omega}|Du|^{2}&-\sum_{i=1}^{m}\frac{1}{p+1}\int_{\Omega}\chi_{\Omega_{i}^{+}}\left(u-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p+1}\\\
&-\sum_{j=1}^{n}\frac{1}{p+1}\int_{\Omega}\chi_{\Omega_{j}^{-}}\left(\frac{2\pi
q(x)}{|\ln\varepsilon|}-\kappa_{j}^{-}-u\right)_{+}^{p+1}\end{split}$
and
$K(Z)=I\left(P_{\delta,Z}^{+}-P^{-}_{\delta,Z}+\omega_{\delta}\right).$ (4.1)
It is well known that if $Z$ is a critical point of $K(Z)$, then
$P_{\delta,Z}^{+}-P^{-}_{\delta,Z}+\omega_{\delta}$ is a solution of (1.11).
In the following, we will prove that $K(Z)$ has a critical point.
###### Lemma 4.1.
We have
$K(Z)=I\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\right)+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p}}\right).$
###### Proof.
Recall that $P^{+}_{\delta,Z}=\sum_{i=1}^{m}P^{+}_{\delta,Z,i},\quad
P^{-}_{\delta,Z}=\sum_{j=1}^{n}P^{-}_{\delta,Z,j}.$ We have
$\begin{split}K(Z)&=I\bigl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\bigr{)}+\delta^{2}\int_{\Omega}D\bigl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\bigr{)}D\omega_{\delta}+\frac{\delta^{2}}{2}\int_{\Omega}|D\omega_{\delta}|^{2}\\\
&\quad\quad-\sum_{i=1}^{m}\frac{1}{p+1}\int_{\Omega}\chi_{\Omega_{i}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega_{\delta}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}\\\
&\quad\quad\quad-\biggl{(}P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}\Biggr{]}\\\
&\quad\quad-\sum_{j=1}^{n}\frac{1}{p+1}\int_{\Omega}\chi_{\Omega_{j}^{-}}\Biggl{[}\biggl{(}P^{-}_{\delta,Z}-P^{+}_{\delta,Z}-\omega_{\delta}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}\\\
&\quad\quad\quad-\biggl{(}P_{\delta,Z}^{-}-P_{\delta,Z}^{+}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}\Biggr{]}.\end{split}$
Using Proposition 3.3 and (3.15), we find
$\begin{split}&\int_{\Omega_{i}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega_{\delta}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}-\biggl{(}P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}\Biggr{]}\\\
=&\int_{B_{Ls^{+}_{\delta,i}}(z^{+}_{i})}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega_{\delta}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}-\biggl{(}P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p+1}\Biggr{]}\\\
=&O\left(\frac{(s^{+}_{\delta,i})^{2}\|\omega_{\delta}\|_{\infty}}{|\ln\varepsilon|^{p}}\right)\\\
=&O\Bigl{(}\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p}}\Bigr{)}.\end{split}$
On the other hand,
$\begin{split}&\delta^{2}\int_{\Omega}DP^{+}_{\delta,Z}D\omega_{\delta}=\sum_{i=1}^{m}\int_{\Omega}\left(W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p}\omega_{\delta}\\\
=&\sum_{i=1}^{m}\int_{B_{s^{+}_{\delta,k}}(z^{+}_{k})}(W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i})_{+}^{p}\omega_{\delta}\\\
=&O\Bigl{(}\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p}}\Bigr{)}.\end{split}$
Next, we estimate $\delta^{2}\int_{\Omega}|D\omega_{\delta}|^{2}$. Note that
$\begin{split}-\delta^{2}\Delta\omega_{\delta}=&\sum_{i=1}^{m}\chi_{\Omega_{i}^{+}}\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega_{\delta}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\sum_{i=1}^{m}\left(W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p}\\\
&-\sum_{j=1}^{n}\chi_{\Omega_{j}^{-}}\left(P^{-}_{\delta,Z}-P^{+}_{\delta,Z}-\omega_{\delta}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}+\sum_{j=1}^{n}\left(W_{\delta,z_{j}^{-},a^{-}_{\delta,j}}-a^{-}_{\delta,j}\right)_{+}^{p}\\\
&+\sum_{i=1}^{m}\sum_{h=1}^{2}b_{i,h}^{+}\left(-\delta^{2}\Delta\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\right)+\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h}}\left(-\delta^{2}\Delta\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\right).\end{split}$
Hence, by (2.13)–(2.14), we have
$\begin{split}\delta^{2}\int_{\Omega}|D\omega_{\delta}|^{2}=&\sum_{i=1}^{m}\int_{\Omega_{i}^{+}}\Biggl{[}\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}+\omega_{\delta}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\left(W_{\delta,z_{i}^{+},a_{\delta,i}^{+}}-a^{+}_{\delta,i}\right)_{+}^{p}\Biggr{]}\omega_{\delta}\\\
&-\sum_{j=1}^{n}\int_{\Omega_{j}^{-}}\Biggl{[}\left(P^{-}_{\delta,Z}-P^{+}_{\delta,Z}-\omega_{\delta}-\kappa_{j}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}-\left(W_{\delta,z_{j}^{-},a_{\delta,j}^{-}}-a^{-}_{\delta,j}\right)_{+}^{p}\Biggr{]}\omega_{\delta}\\\
&+\sum_{i=1}^{m}\sum_{h=1}^{2}b^{+}_{i,h}\int_{\Omega}\left(-\delta^{2}\Delta\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\right)\omega_{\delta}+\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}b^{-}_{j,\bar{h}}\int_{\Omega}\left(-\delta^{2}\Delta\frac{\partial
P^{-}_{\delta,Z,j}}{\partial z^{-}_{j,\bar{h}}}\right)\omega_{\delta}\\\
=&p\sum_{i=1}^{m}\int_{\Omega_{i}^{+}}\left(W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p-1}\left(\frac{s^{+}_{\delta,i}}{|\ln\varepsilon|}+\omega_{\delta}\right)\omega_{\delta}+0\left(\sum_{i=1}^{m}\sum_{h=1}^{2}\frac{\varepsilon|b^{+}_{i,h}|\|\omega_{\delta}\|_{\infty}}{|\ln\varepsilon|^{p}}\right)\\\
&-p\sum_{j=1}^{n}\int_{\Omega_{j}^{-}}\left(W_{\delta,z^{-}_{j},a^{-}_{\delta,j}}-a^{-}_{\delta,j}\right)_{+}^{p-1}\left(\frac{s^{-}_{\delta,j}}{|\ln\varepsilon|}+\omega_{\delta}\right)\omega_{\delta}+0\left(\sum_{j=1}^{n}\sum_{\bar{h}=1}^{2}\frac{\varepsilon|b^{-}_{j,\bar{h}}|\|\omega_{\delta}\|_{\infty}}{|\ln\varepsilon|^{p}}\right)\\\
=&O\left(\frac{\varepsilon^{4}}{|\ln\varepsilon|^{p-1}}\right).\end{split}$
Other terms can be estimated as above. So our assertion follows.
∎
###### Lemma 4.2.
We have
$\frac{\partial K(Z)}{\partial z^{+}_{i,h}}=\frac{\partial}{\partial
z^{+}_{i,h}}I\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\right)+O\Bigl{(}\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p-1}}\Bigr{)},\,\,i=1,\cdots,m,$
$\frac{\partial K(Z)}{\partial z^{-}_{j,\bar{h}}}=\frac{\partial}{\partial
z^{-}_{j,\bar{h}}}I\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\right)+O\Bigl{(}\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p-1}}\Bigr{)},\,\,j=1,\cdots,n.$
###### Proof.
We only give the proof of the first estimate.
First, we have
$\begin{split}&\frac{\partial K(Z)}{\partial z^{+}_{i,h}}=\left\langle
I^{\prime}\Bigl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}\Bigr{)},\frac{\partial
P^{+}_{\delta,Z}}{\partial z^{+}_{i,h}}-\frac{\partial
P^{-}_{\delta,Z}}{\partial
z^{+}_{i,h}}+\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}\right\rangle\\\ =&\frac{\partial}{\partial
z^{+}_{i,h}}I\Bigl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}\Bigr{)}+\left\langle
I^{\prime}\big{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}\big{)},\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}\right\rangle\\\
&-\sum_{k=1}^{m}\int_{\Omega_{k}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}-\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}\Biggr{]}\\\
&\qquad\times\left(\frac{\partial P^{+}_{\delta,Z}}{\partial
z^{+}_{i,h}}-\frac{\partial P^{-}_{\delta,Z}}{\partial z^{+}_{i,h}}\right)\\\
&-\sum_{l=1}^{n}\int_{\Omega_{l}^{-}}\Biggl{[}\biggl{(}P^{-}_{\delta,Z}-P_{\delta,Z}^{+}-\omega_{\delta}-\kappa_{l}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}-\biggl{(}P^{-}_{\delta,Z}-P^{+}_{\delta,Z}-\kappa_{l}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}\Biggr{]}\\\
&\qquad\times\left(\frac{\partial P^{-}_{\delta,Z}}{\partial
z^{+}_{i,h}}-\frac{\partial P^{+}_{\delta,Z}}{\partial
z^{+}_{i,h}}\right).\end{split}$
Since $\omega_{\delta}\in E_{\delta,Z}$, we have
$\int_{\Omega}\left(W_{\delta,z_{k}^{\pm},a_{\delta,k}^{\pm}}-a_{\delta,k}^{\pm}\right)_{+}^{p-1}\left(\frac{\partial
W_{\delta,z_{k}^{\pm},a^{\pm}_{\delta,k}}}{\partial
z^{\pm}_{k,h}}-\frac{\partial a_{\delta,k}^{\pm}}{\partial
z_{k,h}^{\pm}}\right)\omega_{\delta}=0.$
Differentiating the above relation with respect to $z^{+}_{i,h}$, we can
deduce
$\begin{split}&\Bigg{\langle}I^{\prime}\bigl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}\bigr{)},\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}\Bigg{\rangle}\\\
=&\sum_{\alpha=1}^{m}\sum_{\hat{h}=1}^{2}b^{+}_{\alpha,\hat{h}}\int_{\Omega}\left(-\delta^{2}\Delta\frac{\partial
P^{+}_{\delta,Z,\alpha}}{\partial
z^{+}_{\alpha,\hat{h}}}\right)\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}+\sum_{\beta=1}^{n}\sum_{\tilde{h}=1}^{2}b^{-}_{\beta,\tilde{h}}\int_{\Omega}\left(-\delta^{2}\Delta\frac{\partial
P^{-}_{\delta,Z,\beta}}{\partial
z^{-}_{\beta,\tilde{h}}}\right)\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}\\\
=&\sum_{\alpha=1}^{m}\sum_{\hat{h}=1}^{2}pb_{\alpha,\hat{h}}^{+}\int_{\Omega}\left(W_{\delta,z_{\alpha}^{+},a^{+}_{\delta,\alpha}}-a_{\delta,\alpha}^{+}\right)_{+}^{p-1}\left(\frac{\partial
W_{\delta,z_{\alpha}^{+},a^{+}_{\delta,\alpha}}}{\partial
z^{+}_{\alpha,\hat{h}}}-\frac{\partial a^{+}_{\delta,\alpha}}{\partial
z^{+}_{\alpha,\hat{h}}}\right)\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}\\\
&+\sum_{\beta=1}^{n}\sum_{\tilde{h}=1}^{2}pb_{\beta,\tilde{h}}^{-}\int_{\Omega}\left(W_{\delta,z_{\beta}^{-},a^{-}_{\delta,\beta}}-a_{\delta,\beta}^{-}\right)_{+}^{p-1}\left(\frac{\partial
W_{\delta,z_{\beta}^{-},a^{-}_{\delta,\beta}}}{\partial
z^{-}_{\beta,\tilde{h}}}-\frac{\partial a^{-}_{\delta,\beta}}{\partial
z^{-}_{\beta,\tilde{h}}}\right)\frac{\partial\omega_{\delta}}{\partial
z^{+}_{i,h}}\\\
=&O\left(\sum_{\alpha=1}^{m}\sum_{\hat{h}=1}^{2}\frac{\varepsilon|b^{+}_{\alpha,\hat{h}}|}{|\ln\varepsilon|^{p}}+\sum_{\beta=1}^{n}\sum_{\tilde{h}=1}^{2}\frac{\varepsilon|b^{-}_{\beta,\tilde{h}}|}{|\ln\varepsilon|^{p}}\right)=O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p-1}}\right).\end{split}$
On the other hand, using (3.17) (for the definition of
$R^{+}_{\delta}(\omega)$, see (3.12)), we obtain
$\begin{split}&\sum_{k=1}^{m}\int_{\Omega_{k}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}-\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}\Biggr{]}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\\\
=&\sum_{k=1}^{m}\int_{\Omega_{k}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}-\biggl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}\\\
&-p\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p-1}\omega_{\delta}\Biggr{]}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\\\
&+\sum_{k=1}^{m}p\int_{\Omega_{k}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p-1}-\bigl{(}W_{\delta,z^{+}_{k},a^{+}_{\delta,k}}-a^{+}_{\delta,k}\bigr{)}_{+}^{p-1}\Biggr{]}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\omega_{\delta}\\\
&+O\left(\frac{(s^{+}_{\delta,k})^{2}\|\omega_{\delta}\|_{\infty}}{|\ln\varepsilon|^{p}}\right)\\\
=&\int_{\Omega}R^{+}_{\delta}(\omega_{\delta})\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}+\sum_{k=1}^{m}p\int_{\Omega_{k}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p-1}\\\
&-\bigl{(}W_{\delta,z_{k}^{+},a^{+}_{\delta,k}}-a^{+}_{\delta,k}\bigr{)}_{+}^{p-1}\Biggr{]}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial
z^{+}_{i,h}}\omega_{\delta}+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p}}\right)\\\
=&O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p-1}}\right).\end{split}$
In addition, we have
$\begin{split}&\int_{\Omega_{l}^{+}}\Biggl{[}\biggl{(}P^{+}_{\delta,Z}-P_{\delta,Z}^{-}+\omega_{\delta}-\kappa_{l}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}-\biggl{(}P^{+}_{\delta,Z}-P^{-}_{\delta,Z}-\kappa_{l}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p}\Biggr{]}\frac{\partial
P^{-}_{\delta,Z,i}}{\partial z^{-}_{i,h}}\\\
&=p\int_{\Omega_{l}^{+}}\biggl{(}P_{\delta,Z}^{+}-P^{-}_{\delta,Z}-\kappa_{l}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\biggr{)}_{+}^{p-1}\frac{\partial
P^{-}_{\delta,Z,i}}{\partial z^{-}_{i,h}}\omega_{\delta}\\\
&=O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p}}\right).\end{split}$
Other teams can be estimated as above. Thus, the estimate follows. ∎
###### Proof of Theorem 1.4.
Recall that $Z=(Z_{m}^{+},Z_{n}^{-}).$ Set
$\begin{split}\Phi(Z_{m}^{+},Z_{n}^{-})=&\sum_{i=1}^{m}4\pi^{2}\kappa_{i}^{+}q(z_{i}^{+})-\sum_{j=1}^{n}4\pi^{2}\kappa_{j}^{-}q(z_{j}^{-})+\sum_{i=1}^{m}\pi(\kappa_{i}^{+})^{2}g(z_{i}^{+},z_{i}^{+})\\\
&+\sum_{j=1}^{n}\pi(\kappa_{j}^{-})^{2}g(z_{j}^{-},z_{j}^{-})-\sum_{i\neq
k}\pi\kappa_{i}^{+}\kappa_{k}^{+}\bar{G}(z_{i}^{+},z_{k}^{+})-\sum_{j\neq
l}\pi\kappa_{j}^{-}\kappa_{l}^{-}\bar{G}(z_{l}^{-},z_{j}^{-})\\\
&+\sum_{i=1}^{m}\sum_{j=1}^{n}2\pi\kappa_{i}^{+}\kappa_{j}^{-}\bar{G}(z_{i}^{+},z_{j}^{-}).\end{split}$
Note that the Kirchhoff–Routh function associated to the vortex dynamics now
is
$\begin{split}\mathcal{W}(Z_{m}^{+},Z_{n}^{-})=&\frac{1}{2}\sum_{i,k=1,i\neq
k}^{m}\kappa_{i}^{+}\kappa_{k}^{+}G(z_{i}^{+},z_{k}^{+})+\frac{1}{2}\sum^{n}_{j,l=1,j\neq
l}\kappa^{-}_{j}\kappa^{-}_{l}G(z_{j}^{-},z_{l}^{-})\\\
&+\frac{1}{2}\sum_{i=1}^{m}(\kappa_{i}^{+})^{2}H(z_{i}^{+},z_{i}^{+})+\frac{1}{2}\sum_{j=1}^{n}(\kappa_{j}^{-})^{2}H(z_{j}^{-},z_{j}^{-})\\\
&-\sum_{i=1}^{m}\sum_{j=1}^{n}\kappa_{i}^{+}\kappa_{j}^{-}G(z_{i}^{+},z_{j}^{-})+\sum^{m}_{i=1}\kappa_{i}^{+}\psi_{0}(z_{i}^{+})-\sum^{n}_{j=1}\kappa_{j}^{-}\psi_{0}(z_{j}^{-}).\end{split}$
Recall that $h(z_{i},z_{j})=-H(z_{i},z_{j})$, it is easy to check that
$\Phi(Z_{m}^{+},Z_{n}^{-})=-4\pi^{2}\mathcal{W}(Z_{m}^{+},Z_{n}^{-})+\pi\ln
R\left(\sum_{i=1}^{m}(\kappa_{i}^{+})^{2}+\sum_{j=1}^{n}(\kappa_{j}^{-})^{2}\right).$
Hence, $\Phi(Z_{m}^{+},Z_{N}^{-})$ and $\mathcal{W}(Z_{m}^{+},Z_{N}^{-})$
possess the same critical points.
By Lemma 4.1, 4.2 and Proposition A.2, A.3, we have
$K(Z)=\frac{C\delta^{2}}{\ln\frac{R}{\varepsilon}}+\frac{\pi(p-1)\delta^{2}}{4(\ln\frac{R}{\varepsilon})^{2}}\left(\sum_{i=1}^{m}(\kappa_{i}^{+})^{2}+\sum_{j=1}^{n}(\kappa_{j}^{-})^{2}\right)+\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\Phi(Z)+0\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{3}}\right)$
and
$\frac{\partial K(Z)}{\partial
z^{\pm}_{i,h}}=\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\frac{\partial\Phi(Z)}{\partial
z^{\pm}_{i,h}}+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{3}}\right).$
Thus, the existence of a $C^{1}$-stable critical point of Kirchhoff-Routh
function $\mathcal{W}(Z)$ implies that $K(Z)$ has a critical point.
Thus we get a solution $w_{\delta}$ for (1.11). Let
$u_{\varepsilon}=\frac{|\ln\varepsilon|}{2\pi}w_{\delta},\,\delta=\varepsilon\left(\frac{|\ln\varepsilon|}{2\pi}\right)^{\frac{1-p}{2}}$,
it is not difficult to check that $u_{\varepsilon}$ has all the properties
listed in Theorem 1.4 and thus the proof of Theorem 1.4 is complete. ∎
Now we are in the position to prove Theorem 1.1.
###### Proof of Theorem 1.1.
By Theorem 1.4, we obtain that $u_{\varepsilon}$ is a solution to (1.10).
Set
$\begin{split}&\mathbf{v}_{\varepsilon}=(\nabla(u_{\varepsilon}-q))^{\bot},\qquad\omega_{\varepsilon}=\nabla\times\mathbf{v}_{\varepsilon},\\\
P_{\varepsilon}=&\sum_{i=1}^{m}\frac{1}{p+1}\chi_{\Omega_{i}^{+}}\left(u_{\varepsilon}-q-\frac{\kappa_{i}^{+}|\ln\varepsilon|}{2\pi}\right)_{+}^{p+1}\\\
&+\sum_{j=1}^{n}\frac{1}{p+1}\chi_{\Omega_{j}^{-}}\left(q-\frac{\kappa_{j}^{-}|\ln\varepsilon|}{2\pi}-u_{\varepsilon}\right)_{+}^{p+1}-\frac{1}{2}|\nabla(u_{\varepsilon}-q)|^{2}.\end{split}$
Then $(\mathbf{v}_{\varepsilon},P_{\varepsilon})$ forms a stationary solution
for problem (1.1).
We now just need to verify as $\varepsilon\rightarrow 0$
$\int_{\Omega}\omega_{\varepsilon}\rightarrow\sum_{j=1}^{m}\kappa_{j}^{+}-\sum_{j=1}^{n}\kappa_{j}^{-}.$
By direct calculations, we find that
$\begin{split}\int_{\Omega}\omega_{\varepsilon}&=\sum_{i=1}^{m}\frac{1}{\varepsilon^{2}}\int_{\Omega}\chi_{\Omega_{i}^{+}}\left(u_{\varepsilon}-q-\frac{\kappa_{i}^{+}|\ln\varepsilon|}{2\pi}\right)_{+}^{p}-\sum_{j=1}^{n}\frac{1}{\varepsilon^{2}}\int_{\Omega}\chi_{\Omega_{j}^{-}}\left(q-\frac{\kappa_{j}^{-}|\ln\varepsilon|}{2\pi}-u_{\varepsilon}\right)_{+}^{p}\\\
&=\sum_{i=1}^{m}\frac{|\ln\varepsilon|^{p}}{(2\pi)^{p}\varepsilon^{2}}\int_{\Omega_{i}^{+}}\left(w_{\delta}-\kappa_{i}^{+}-\frac{2\pi
q}{|\ln\varepsilon|}\right)^{p}_{+}-\sum_{j=1}^{n}\frac{|\ln\varepsilon|^{p}}{(2\pi)^{p}\varepsilon^{2}}\int_{\Omega_{j}^{-}}\left(\frac{2\pi
q}{|\ln\varepsilon|}-\kappa_{j}^{-}-w_{\delta}\right)^{p}_{+}\\\
&=\frac{|\ln\varepsilon|^{p}}{(2\pi)^{p}\varepsilon^{2}}\sum_{i=1}^{m}\int_{B_{Ls^{+}_{\delta,i}(z^{+}_{i})}}\left(W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}+O\Big{(}\frac{s^{+}_{\delta,i}}{|\ln\varepsilon|}\Big{)}\right)^{p}_{+}\\\
&\quad-\frac{|\ln\varepsilon|^{p}}{(2\pi)^{p}\varepsilon^{2}}\sum_{j=1}^{n}\int_{B_{Ls^{-}_{\delta,j}(z^{-}_{j})}}\left(W_{\delta,z^{-}_{j},a^{-}_{\delta,j}}-a^{-}_{\delta,j}+O\Big{(}\frac{s^{-}_{\delta,j}}{|\ln\varepsilon|}\Big{)}\right)^{p}_{+}\\\
&=\sum_{i=1}^{m}\frac{(s^{+}_{\delta,i})^{2}|\ln\varepsilon|^{p}}{(2\pi)^{p}\varepsilon^{2}}\left(\frac{\delta}{s^{+}_{\delta,i}}\right)^{\frac{2p}{p-1}}\int_{B_{1}(0)}\phi^{p}\\\
&\quad-\sum_{j=1}^{n}\frac{(s^{-}_{\delta,j})^{2}|\ln\varepsilon|^{p}}{(2\pi)^{p}\varepsilon^{2}}\left(\frac{\delta}{s^{-}_{\delta,j}}\right)^{\frac{2p}{p-1}}\int_{B_{1}(0)}\phi^{p}+o(1)\\\
&=\sum_{i=1}^{m}\frac{a^{+}_{\delta,i}|\ln\varepsilon|}{\ln\frac{R}{s^{+}_{\delta,i}}}-\sum_{j=1}^{n}\frac{a^{-}_{\delta,j}|\ln\varepsilon|}{\ln\frac{R}{s^{-}_{\delta,j}}}+o(1)\\\
&\rightarrow\sum_{j=1}^{m}\kappa_{j}^{+}-\sum_{j=1}^{n}\kappa_{j}^{-},\quad\text{as}~{}~{}\varepsilon\rightarrow
0.\end{split}$
Therefore, the result follows. ∎
###### Remark 4.3.
To regularize pairs of vortices with equi-strength $\kappa$, we do not need
$\chi_{\Omega_{i}^{+}}$ and $\chi_{\Omega_{j}^{-}}$, that is, we only need to
consider the following problem
$\begin{cases}-\varepsilon^{2}\Delta
u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\varepsilon})_{+}^{p}-(q-\frac{\kappa}{2\pi}\ln\frac{1}{\varepsilon}-u)_{+}^{p},\quad&x\in\Omega,\\\
u=0,\quad&x\in\partial\Omega.\end{cases}$
Acknowledgements: D. Cao and Z. Liu were supported by the National Center for
Mathematics and Interdisciplinary Sciences, CAS. D. Cao and J. Wei were also
supported by CAS Croucher Joint Laboratories Funding Scheme.
## Appendix A Energy expansion
In this section we will give precise expansions of
$I\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\right)$ and
$\frac{\partial}{\partial
z^{\pm}_{i,h}}I\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}\right)$, which have
been used in section 4.
We always assume that $z_{i}^{+},z_{j}^{-}\in\Omega$ satisfies
$\begin{split}&d(z_{i}^{+},\partial\Omega)\geq\varrho,~{}d(z_{j}^{-},\partial\Omega)\geq\varrho,\quad|z_{i}^{+}-z_{k}^{+}|\geq\varrho^{\bar{L}},\quad
i,k=1,\cdots,m,\;i\neq k\\\
&~{}~{}|z_{j}^{-}-z_{l}^{-}|\geq\varrho^{\bar{L}},\quad|z_{i}^{+}-z_{j}^{-}|\geq\varrho^{\bar{L}},\quad
j,l=1,\cdots,n,\;j\neq l,\end{split}$
where $\varrho>0$ is a fixed small constant and $\bar{L}>0$ is a fixed large
constant.
###### Lemma A.1.
For $x\in\Omega_{i}^{+},\,i=1,2,\cdots,m$ and
$x\in\Omega_{j}^{-},\,j=1,2,\cdots,m$, we have
$P^{+}_{\delta,Z}(x)-P^{-}_{\delta,Z}(x)>\kappa_{i}^{+}+\frac{2\pi
q(x)}{|\ln\varepsilon|},\quad x\in
B_{s^{+}_{\delta,i}(1-Ts^{+}_{\delta,i})}(z^{+}_{i}),$
$P^{-}_{\delta,Z}(x)-P^{+}_{\delta,Z}(x)>\kappa_{j}^{-}-\frac{2\pi
q(x)}{|\ln\varepsilon|},\quad x\in
B_{s^{-}_{\delta,j}(1-Ts^{-}_{\delta,j})}(z_{j}^{-}),$
where $T>0$ is a large constant; while
$P_{\delta,Z}^{+}(x)-P_{\delta,Z}^{-}(x)<\kappa_{i}^{+}+\frac{2\pi
q(x)}{|\ln\varepsilon|},\quad x\in\Omega_{i}^{+}\setminus
B_{s^{+}_{\delta,i}(1+(s^{+}_{\delta,i})^{\sigma})}(z^{+}_{i}),$
$P_{\delta,Z}^{-}(x)-P_{\delta,Z}^{+}(x)<\kappa_{j}^{-}-\frac{2\pi
q(x)}{|\ln\varepsilon|},\quad x\in\Omega_{j}^{-}\setminus
B_{s^{-}_{\delta,j}(1+(s^{-}_{\delta,j})^{\sigma})}(z^{-}_{j}),$
where $\sigma>0$ is a small constant.
###### Proof.
The proof is exactly same as Lemma A.1 in [13]. For reader’s convenience, we
give the proof for $P_{\delta,Z}^{+}-P_{\delta,Z}^{-}$ here.
Suppose that $x\in B_{s^{+}_{\delta,i}(1-Ts^{+}_{\delta,i})}(z_{i}^{+})$. It
follows from (2.13) and $\phi_{1}^{\prime}(s)<0$ that
$\begin{split}&P^{+}_{\delta,Z}(x)-P_{\delta,Z}^{-}(x)-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}=W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}(x)-a^{+}_{\delta,i}+O\left(\frac{s^{+}_{\delta,i}}{|\ln\varepsilon|}\right)\\\
=&\frac{a^{+}_{\delta,i}}{|\phi^{\prime}(1)||\ln\frac{R}{s^{+}_{\delta,i}}|}\phi\Bigl{(}\frac{|x-z^{+}_{i}|}{s^{+}_{\delta,i}}\Bigr{)}+O\Bigl{(}\frac{\varepsilon}{|\ln\varepsilon|}\Bigr{)}>0,\end{split}$
if $T>0$ is large. On the other hand, if $x\in\Omega_{i}^{+}\setminus
B_{(s^{+}_{\delta,i})^{\tilde{\sigma}}}(z_{i}^{+})$, where
$\tilde{\sigma}>\sigma>0$ is a fixed small constant, then
$\begin{split}&P^{+}_{\delta,Z}(x)-P^{-}_{\delta,Z}(x)-\kappa_{i}^{+}-\frac{2\pi
q(x)}{\kappa|\ln\varepsilon|}\\\
\leq&\sum_{i=1}^{m}a^{+}_{\delta,i}\ln\frac{R}{|x-z^{+}_{i}|}/\ln\frac{R}{s^{+}_{\delta,i}}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{\kappa|\ln\varepsilon|}+o(1)\\\
\leq&C\tilde{\sigma}-\kappa_{i}^{+}+o(1)<0.\end{split}$
Finally, if $x\in B_{(s^{+}_{\delta,i})^{\tilde{\sigma}}}(z^{+}_{i})\setminus
B_{s^{+}_{\delta,i}(1+T(s^{+}_{\delta,i})^{\tilde{\sigma}})}(z^{+}_{i})$ for
some $i$, then
$\begin{split}&P_{\delta,Z}^{+}(x)-P_{\delta,Z}^{-}(x)-\kappa_{i}^{+}-\frac{2\pi
q(x)}{\kappa|\ln\varepsilon|}=W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}(x)-a^{+}_{\delta,i}+O\left(\frac{(s^{+}_{\delta,i})^{\tilde{\sigma}}}{\ln\frac{R}{s^{+}_{\delta,i}}}\right)\\\
=&a^{+}_{\delta,i}\frac{\ln\frac{R}{|x-z_{i}^{+}|}}{\ln\frac{R}{s^{+}_{\delta,i}}}-a^{+}_{\delta,i}+O\left(\frac{(s^{+}_{\delta,i})^{\tilde{\sigma}}}{\ln\frac{R}{s^{+}_{\delta,i}}}\right)\\\
\leq&-a^{+}_{\delta,i}\frac{\ln(1+T(s^{+}_{\delta,i})^{\tilde{\sigma}})}{\ln\frac{R}{s^{+}_{\delta,i}}}+O\left(\frac{(s^{+}_{\delta,i})^{\tilde{\sigma}}}{\ln\frac{R}{s^{+}_{\delta,i}}}\right)<0,\end{split}$
if $T>0$ is large. Note that by the choice of $\tilde{\sigma}$,
$B_{s^{+}_{\delta,i}(1+(s^{+}_{\delta,i})^{\sigma})}(z_{i}^{+})\supset
B_{s^{+}_{\delta,i}(1+T(s^{+}_{\delta,i})^{\tilde{\sigma}})}(z^{+}_{i})$ for
small $\delta$. We therefore derive our conclusion. ∎
###### Proposition A.2.
We have
$\begin{split}I\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}\right)=&\frac{C\delta^{2}}{\ln\frac{R}{\varepsilon}}+\frac{\pi(p-1)\delta^{2}}{4(\ln\frac{R}{\varepsilon})^{2}}\left(\sum_{i=1}^{m}(\kappa_{i}^{+})^{2}+\sum_{j=1}^{n}(\kappa_{j}^{-})^{2}\right)+\sum_{i=1}^{m}\frac{4\pi^{2}\delta^{2}\kappa_{i}^{+}q(z_{i}^{+})}{|\ln\varepsilon||\ln\frac{R}{\varepsilon}|}\\\
&-\sum_{j=1}^{n}\frac{4\pi^{2}\delta^{2}\kappa_{j}^{-}q(z_{j}^{-})}{|\ln\varepsilon||\ln\frac{R}{\varepsilon}|}+\sum_{i=1}^{m}\frac{\pi\delta^{2}(\kappa_{i}^{+})^{2}g(z_{i}^{+},z^{+}_{i})}{(\ln\frac{R}{\varepsilon})^{2}}+\sum_{j=1}^{n}\frac{\pi\delta^{2}(\kappa_{j}^{-})^{2}g(z_{j}^{-},z^{-}_{j})}{(\ln\frac{R}{\varepsilon})^{2}}\\\
&-\sum_{k\neq
i}^{m}\frac{\pi\delta^{2}\kappa_{i}^{+}\kappa_{k}^{+}\bar{G}(z_{k}^{+},z^{+}_{i})}{{(\ln\frac{R}{\varepsilon})^{2}}}-\sum_{l\neq
j}^{n}\frac{\pi\delta^{2}\kappa_{j}^{-}\kappa_{l}^{-}\bar{G}(z_{l}^{-},z^{-}_{j})}{{(\ln\frac{R}{\varepsilon})^{2}}}\\\
&+\sum_{i=1}^{m}\sum_{j=1}^{n}\frac{2\pi\delta^{2}\kappa_{i}^{+}\kappa_{j}^{-}\bar{G}(z_{i}^{+},z_{j}^{-})}{(\ln\frac{R}{\varepsilon})^{2}}+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{3}}\right).\end{split}$
where $C$ is a positive constant.
###### Proof.
Taking advantage of (2.3), we have
$\begin{split}&\delta^{2}\int_{\Omega}\big{|}D(P_{\delta,Z}^{+}-P_{\delta,Z}^{-})\big{|}^{2}=\sum_{k=1}^{m}\sum_{i=1}^{m}\int_{\Omega}\bigl{(}W_{\delta,z^{+}_{k},a^{+}_{\delta,k}}-a^{+}_{\delta,k}\bigr{)}_{+}^{p}P_{\delta,Z,i}^{+}\\\
&+\sum_{l=1}^{n}\sum_{j=1}^{n}\int_{\Omega}\bigl{(}W_{\delta,z^{-}_{l},a^{-}_{\delta,l}}-a^{-}_{\delta,l}\bigr{)}_{+}^{p}P_{\delta,Z,j}^{-}-2\sum_{j=1}^{n}\sum_{i=1}^{m}\int_{\Omega}\bigl{(}W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}_{+}^{p}P_{\delta,Z,j}^{-}.\end{split}$
First, we estimate
$\begin{split}&\int_{B_{s^{+}_{\delta,i}}(z^{+}_{i})}\left(W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p}\left(W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}g(x,z^{+}_{i})\right)\\\
=&\int_{B_{s^{+}_{\delta,i}}(z^{+}_{i})}\bigl{(}W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}^{p+1}+a^{+}_{\delta,i}\int_{B_{s^{+}_{\delta,i}}(z^{+}_{i})}\bigl{(}W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}^{p}\\\
&-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\int_{B_{s^{+}_{\delta,i}}(z_{i})}\bigl{(}W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}^{p}g(x,z_{i}^{+})\\\
=&\Bigl{(}\frac{\delta}{s^{+}_{\delta,i}}\Bigr{)}^{\frac{2(p+1)}{p-1}}(s^{+}_{\delta,i})^{2}\int_{B_{1}(0)}\phi^{p+1}+a^{+}_{\delta,i}\Bigl{(}\frac{\delta}{s^{+}_{\delta,i}}\Bigr{)}^{\frac{2p}{p-1}}(s^{+}_{\delta,i})^{2}\int_{B_{1}(0)}\phi^{p}\\\
&-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\Bigl{(}\frac{\delta}{s^{+}_{\delta,i}}\Bigr{)}^{\frac{2p}{p-1}}g(z^{+}_{i},z^{+}_{i})(s^{+}_{\delta,i})^{2}\int_{B_{1}(0)}\phi^{p}+O\left(\frac{(s^{+}_{\delta,i})^{3}}{|\ln\varepsilon|^{p+1}}\right)\\\
=&\frac{\pi(p+1)}{2}\frac{\delta^{2}(a^{+}_{\delta,i})^{2}}{(\ln\frac{R}{s^{+}_{\delta,i}})^{2}}+\frac{2\pi\delta^{2}(a^{+}_{\delta,i})^{2}}{\ln\frac{R}{s^{+}_{\delta,i}}}-\frac{2\pi\delta^{2}(a^{+}_{\delta,i})^{2}}{(\ln\frac{R}{s^{+}_{\delta,i}})^{2}}g(z^{+}_{i},z^{+}_{i})+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
Next, for $k\neq i$,
$\begin{split}&\int_{B_{s^{+}_{\delta,k}}(z^{+}_{k})}\bigl{(}W_{\delta,z_{k}^{+},a^{+}_{\delta,k}}-a_{\delta,k}^{+}\bigr{)}_{+}^{p}\left(W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}g(x,z^{+}_{i})\right)\\\
=&\Bigl{(}\frac{\delta}{s_{\delta,k}^{+}}\Bigr{)}^{\frac{2p}{p-1}}\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\int_{B_{s^{+}_{\delta,k}}(z^{+}_{k})}\phi^{p}\Bigl{(}\frac{|x-z_{k}^{+}|}{s_{\delta,k}^{+}}\Bigr{)}\bar{G}(x,z_{i}^{+})\\\
=&\Bigl{(}\frac{\delta}{s^{+}_{\delta,k}}\Bigr{)}^{\frac{2p}{p-1}}\frac{a^{+}_{\delta,i}(s^{+}_{\delta,k})^{2}}{\ln\frac{R}{s^{+}_{\delta,i}}}\bar{G}(z_{k}^{+},z^{+}_{i})\int_{B_{1}(0)}\phi^{p}+O\left(\frac{(s^{+}_{\delta,k})^{3}}{|\ln\varepsilon|^{p+1}}\right)\\\
=&\frac{2\pi\delta^{2}a_{\delta,i}^{+}a_{\delta,k}^{+}}{|\ln\frac{R}{s^{+}_{\delta,i}}||\ln\frac{R}{s^{+}_{\delta,k}}|}\bar{G}(z_{i}^{+},z_{k}^{+})+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
Moreover, we have
$\begin{split}&\int_{B_{s^{+}_{\delta,i}}(z^{+}_{i})}\bigl{(}W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a_{\delta,i}^{+}\bigr{)}_{+}^{p}\left(W_{\delta,z_{j}^{-},a^{-}_{\delta,j}}-\frac{a^{-}_{\delta,j}}{\ln\frac{R}{s^{-}_{\delta,j}}}g(x,z^{-}_{j})\right)\\\
=&\Bigl{(}\frac{\delta}{s_{\delta,i}^{+}}\Bigr{)}^{\frac{2p}{p-1}}\frac{a^{-}_{\delta,j}}{\ln\frac{R}{s^{-}_{\delta,j}}}\int_{B_{s^{+}_{\delta,i}}(z^{+}_{i})}\phi^{p}\Bigl{(}\frac{|x-z_{i}^{+}|}{s_{\delta,i}^{+}}\Bigr{)}\bar{G}(x,z_{j}^{-})\\\
=&\Bigl{(}\frac{\delta}{s^{+}_{\delta,i}}\Bigr{)}^{\frac{2p}{p-1}}\frac{a^{-}_{\delta,j}(s^{+}_{\delta,i})^{2}}{\ln\frac{R}{s^{-}_{\delta,j}}}\bar{G}(z_{j}^{-},z^{+}_{i})\int_{B_{1}(0)}\phi^{p}+O\left(\frac{(s^{+}_{\delta,i})^{3}}{|\ln\varepsilon|^{p+1}}\right)\\\
=&\frac{2\pi\delta^{2}a_{\delta,i}^{+}a_{\delta,j}^{-}}{|\ln\frac{R}{s^{+}_{\delta,i}}||\ln\frac{R}{s^{-}_{\delta,j}}|}\bar{G}(z_{i}^{+},z_{j}^{-})+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
By Lemma A.1 and (2.13),
$\begin{split}&\sum_{k=1}^{m}\int_{\Omega}\chi_{\Omega_{k}^{+}}\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p+1}\\\
=&\sum_{k=1}^{m}\int_{B_{Ls^{+}_{\delta,k}}(z_{k}^{+})}\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p+1}\\\
=&\sum_{k=1}^{m}\int_{B_{Ls^{+}_{\delta,k}}(z_{k}^{+})}\left(W_{\delta,z_{k}^{+},a_{\delta,k}^{+}}-a^{+}_{\delta,k}+O\bigg{(}\frac{s_{\delta,k}^{+}}{|\ln\varepsilon|}\bigg{)}\right)_{+}^{p+1}\\\
=&\sum_{k=1}^{m}\left(\frac{\delta}{s_{\delta,k}^{+}}\right)^{\frac{2(p+1)}{p-1}}\int_{B_{s^{+}_{\delta,k}}(z^{+}_{k})}\phi^{p+1}\Bigl{(}\frac{|x-z_{k}^{+}|}{s^{+}_{\delta,k}}\Bigr{)}+O\left(\frac{(s_{\delta,k}^{+})^{3}}{|\ln\varepsilon|^{p+1}}\right)\\\
=&\sum_{k=1}^{m}\left(\frac{\delta}{s^{+}_{\delta,k}}\right)^{\frac{2(p+1)}{p-1}}(s_{\delta,k}^{+})^{2}\int_{B_{1}(0)}\phi^{p+1}+O\left(\frac{(s^{+}_{\delta,k})^{3}}{|\ln\varepsilon|^{p+1}}\right)\\\
=&\sum_{k=1}^{m}\frac{\pi(p+1)}{2}\frac{\delta^{2}(a^{+}_{\delta,k})^{2}}{(\ln\frac{R}{s^{+}_{\delta,k}})^{2}}+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
Other terms can be estimated as above. So, we have proved
$\begin{split}I\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}\right)=&\sum_{i=1}^{m}\left[\frac{\pi(p+1)}{4}\frac{\delta^{2}(a^{+}_{\delta,i})^{2}}{|\ln\frac{R}{s^{+}_{\delta,i}}|^{2}}+\frac{\pi\delta^{2}(a^{+}_{\delta,i})^{2}}{|\ln\frac{R}{s^{+}_{\delta,i}}|}-\frac{\pi
g(z_{i}^{+},z_{i}^{+})\delta^{2}(a^{+}_{\delta,i})^{2}}{|\ln\frac{R}{s^{+}_{\delta,i}}|^{2}}\right]\\\
&+\sum_{j=1}^{n}\left[\frac{\pi(p+1)}{4}\frac{\delta^{2}(a^{-}_{\delta,j})^{2}}{|\ln\frac{R}{s^{-}_{\delta,j}}|^{2}}+\frac{\pi\delta^{2}(a^{-}_{\delta,j})^{2}}{|\ln\frac{R}{s^{-}_{\delta,j}}|}-\frac{\pi
g(z_{j}^{-},z_{j}^{-})\delta^{2}(a^{-}_{\delta,j})^{2}}{|\ln\frac{R}{s^{-}_{\delta,j}}|^{2}}\right]\\\
&+\sum_{k\neq
i}^{m}\frac{\pi\bar{G}(z_{k}^{+},z^{+}_{i})\delta^{2}a^{+}_{\delta,i}a^{+}_{\delta,k}}{|\ln\frac{R}{s^{+}_{\delta,i}}||\ln\frac{R}{s^{+}_{\delta,k}}|}+\sum_{l\neq
j}^{n}\frac{\pi\bar{G}(z_{l}^{-},z^{-}_{j})\delta^{2}a^{-}_{\delta,l}a^{-}_{\delta,j}}{|\ln\frac{R}{s^{-}_{\delta,l}}||\ln\frac{R}{s^{-}_{\delta,j}}|}\\\
&-\sum_{i=1}^{m}\sum_{j=1}^{n}\frac{2\pi\bar{G}(z_{i}^{+},z^{-}_{j})\delta^{2}a^{+}_{\delta,i}a^{-}_{\delta,j}}{|\ln\frac{R}{s^{+}_{\delta,i}}||\ln\frac{R}{s^{-}_{\delta,j}}|}-\frac{\pi\delta^{2}}{2}\left(\sum_{i=1}^{m}\frac{(a_{\delta,i}^{+})^{2}}{|\ln\frac{R}{s_{\delta,i}^{+}}|^{2}}\right)\\\
&-\frac{\pi\delta^{2}}{2}\left(\sum_{j=1}^{n}\frac{(a_{\delta,j}^{-})^{2}}{|\ln\frac{R}{s^{-}_{\delta,j}}|^{2}}\right)+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
Thus, the result follows from Remark 2.2.
∎
###### Proposition A.3.
We have
$\begin{split}\frac{\partial}{\partial
z^{+}_{i,h}}&I\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\right)=\frac{4\pi^{2}\delta^{2}\kappa_{i}^{+}}{|\ln\varepsilon||\ln\frac{R}{\varepsilon}|}\frac{\partial
q(z_{i}^{+})}{\partial
z^{+}_{i,h}}+\frac{2\pi\delta^{2}(\kappa_{i}^{+})^{2}}{(\ln\frac{R}{\varepsilon})^{2}}\frac{\partial
g(z_{i}^{+},z_{i}^{+})}{\partial z^{+}_{i,h}}\\\ &-\sum_{k\neq
i}^{m}\frac{2\pi\delta^{2}\kappa_{i}^{+}\kappa_{k}^{+}}{(\ln\frac{R}{\varepsilon})^{2}}\frac{\partial\bar{G}(z_{k}^{+},z_{i}^{+})}{\partial
z^{+}_{i,h}}+\sum_{l=1}^{n}\frac{2\pi\delta^{2}\kappa_{i}^{+}\kappa_{l}^{-}}{(\ln\frac{R}{\varepsilon})^{2}}\frac{\partial\bar{G}(z_{i}^{+},z_{l}^{-})}{\partial
z^{+}_{i,h}}+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{3}}\right),\end{split}$
$\begin{split}\frac{\partial}{\partial
z^{-}_{j,\bar{h}}}&I\left(P^{+}_{\delta,Z}-P^{-}_{\delta,Z}\right)=-\frac{4\pi^{2}\delta^{2}\kappa_{j}^{-}}{|\ln\varepsilon||\ln\frac{R}{\varepsilon}|}\frac{\partial
q(z_{j}^{-})}{\partial
z^{-}_{j,\bar{h}}}+\frac{2\pi\delta^{2}(\kappa_{j}^{-})^{2}}{(\ln\frac{R}{\varepsilon})^{2}}\frac{\partial
g(z_{j}^{-},z_{j}^{-})}{\partial z^{-}_{j,\bar{h}}}\\\ &-\sum_{l\neq
j}^{n}\frac{2\pi\delta^{2}\kappa_{j}^{-}\kappa_{l}^{-}}{(\ln\frac{R}{\varepsilon})^{2}}\frac{\partial\bar{G}(z_{l}^{-},z_{j}^{-})}{\partial
z^{-}_{j,\bar{h}}}+\sum_{k=1}^{m}\frac{2\pi\delta^{2}\kappa_{j}^{-}\kappa_{k}^{+}}{(\ln\frac{R}{\varepsilon})^{2}}\frac{\partial\bar{G}(z_{j}^{-},z_{k}^{+})}{\partial
z^{-}_{j,\bar{h}}}+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{3}}\right).\end{split}$
###### Proof.
Direct computation yields that
$\begin{split}\frac{\partial}{\partial
z^{+}_{i,h}}&I\left(P_{\delta,Z}^{+}-P^{-}_{\delta,Z}\right)\\\
=&\sum_{k=1}^{m}\int_{B_{Ls^{+}_{\delta,k}}(z_{k}^{+})}\left[\left(W_{\delta,z_{k}^{+},a_{\delta,k}^{+}}-a_{\delta,k}^{+}\right)_{+}^{p}-\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P_{\delta,Z}^{+}}{\partial z^{+}_{i,h}}\\\
&+\sum_{l=1}^{n}\int_{B_{Ls^{-}_{\delta,l}}(z^{-}_{l})}\left[\left(W_{\delta,z_{l}^{-},a_{\delta,l}^{-}}-a^{-}_{\delta,l}\right)_{+}^{p}-\left(P_{\delta,Z}^{-}-P_{\delta,Z}^{+}-\kappa_{l}^{-}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P_{\delta,Z}^{-}}{\partial z^{+}_{i,h}}\\\
&-\sum_{k=1}^{m}\int_{B_{Ls^{+}_{\delta,k}}(z_{k}^{+})}\left[\left(W_{\delta,z_{k}^{+},a_{\delta,k}^{+}}-a_{\delta,k}^{+}\right)_{+}^{p}-\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P_{\delta,Z}^{-}}{\partial z^{+}_{i,h}}\\\
&-\sum_{l=1}^{n}\int_{B_{Ls^{-}_{\delta,l}}(z^{-}_{l})}\left[\left(W_{\delta,z_{l}^{-},a_{\delta,l}^{-}}-a^{-}_{\delta,l}\right)_{+}^{p}-\left(P_{\delta,Z}^{-}-P_{\delta,Z}^{+}-\kappa_{l}^{+}+\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P_{\delta,Z}^{+}}{\partial z^{+}_{i,h}}.\end{split}$
For $k\neq i$, from (2.13), we have
$\begin{split}&\int_{B_{Ls_{\delta,k}^{+}}(z_{k}^{+})}\left[\bigl{(}W_{\delta,z_{k}^{+},a_{\delta,k}^{+}}-a_{\delta,k}^{+}\bigr{)}_{+}^{p}-\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{k}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\\\
=&\int_{B_{Ls^{+}_{\delta,k}}(z^{+}_{k})}\left[\left(W_{\delta,z_{k}^{+},a^{+}_{\delta,k}}-a_{\delta,k}^{+}\right)^{p-1}\frac{s_{\delta,k}^{+}}{|\ln\varepsilon|}\right]\frac{C}{\ln\frac{R}{s^{+}_{\delta,i}}}\\\
=&O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
Using (2.13), Lemma A.1 and Remark 2.2, we find that
$\begin{split}&\int_{B_{Ls^{+}_{\delta,i}}(z_{i}^{+})}\left[\left(W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\right)_{+}^{p}-\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P_{\delta,Z,i}^{+}}{\partial z^{+}_{i,h}}\\\
=&\int_{B_{s^{+}_{\delta,i}(1+(s^{+}_{\delta,i})^{\sigma})}(z_{i})}\left[\bigl{(}W_{\delta,z^{+}_{i},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}_{+}^{p}-\left(P_{\delta,Z}^{+}-P_{\delta,Z}^{-}-\kappa_{i}^{+}-\frac{2\pi
q(x)}{|\ln\varepsilon|}\right)_{+}^{p}\right]\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\\\
=&p\int_{B_{s^{+}_{\delta,i}}(z^{+}_{i})}\bigl{(}W_{\delta,z_{i}^{+},a^{+}_{\delta,i}}-a^{+}_{\delta,i}\bigr{)}_{+}^{p-1}\bigg{[}\frac{2\pi}{|\ln\varepsilon|}\bigl{\langle}Dq(z^{+}_{i}),x-z^{+}_{i}\bigr{\rangle}+\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\bigl{\langle}Dg(z^{+}_{i},z^{+}_{i}),x-z^{+}_{i}\bigr{\rangle}\\\
\quad&-\sum_{k\neq
i}^{m}\frac{a_{\delta,k}^{+}}{\ln\frac{R}{s^{+}_{\delta,k}}}\bigl{\langle}D\bar{G}(z_{i}^{+},z_{k}^{+}),x-z_{i}^{+}\bigr{\rangle}+\sum_{l=1}^{n}\frac{a_{\delta,l}^{-}}{\ln\frac{R}{s^{-}_{\delta,l}}}\bigl{\langle}D\bar{G}(z_{i}^{+},z_{l}^{-}),x-z_{i}^{+}\bigr{\rangle}\bigg{]}\frac{\partial
P^{+}_{\delta,Z,i}}{\partial z^{+}_{i,h}}\\\
&\,+O\Bigl{(}\frac{\varepsilon^{2+\sigma}}{|\ln\varepsilon|^{p+1}}\Bigr{)}\\\
=&-\frac{p\delta^{2}a^{+}_{\delta,i}}{|\phi^{\prime}(1)||\ln\frac{R}{s^{+}_{\delta,i}}|}\bigg{(}\frac{2\pi}{|\ln\varepsilon|}\frac{\partial
q(z_{i}^{+})}{\partial
z^{+}_{i,h}}+\frac{a^{+}_{\delta,i}}{\ln\frac{R}{s^{+}_{\delta,i}}}\frac{\partial
g(z_{i}^{+},z_{i}^{+})}{\partial z^{+}_{i,h}}-\sum_{k\neq
i}^{m}\frac{a^{+}_{\delta,k}}{\ln\frac{R}{s^{+}_{\delta,k}}}\frac{\partial\bar{G}(z_{i}^{+},z_{k}^{+})}{\partial
z^{+}_{i,h}}\\\
&+\sum_{l=1}^{n}\frac{a^{-}_{\delta,l}}{\ln\frac{R}{s^{-}_{\delta,l}}}\frac{\partial\bar{G}(z_{i}^{+},z_{l}^{-})}{\partial
z^{+}_{i,h}}\bigg{)}\int_{B_{1}(0)}\phi^{p-1}(|x|)\phi^{\prime}(|x|)\frac{x_{h}^{2}}{|x|}+0\Bigl{(}\frac{\varepsilon^{2+\sigma}}{|\ln\varepsilon|^{p+1}}\Bigr{)}\\\
=&\frac{4\pi^{2}\delta^{2}a^{+}_{\delta,i}}{|\ln\varepsilon||\ln\frac{R}{s^{+}_{\delta,i}}|}\frac{\partial
q(z_{i}^{+})}{\partial
z^{+}_{i,h}}+\frac{2\pi\delta^{2}(a^{+}_{\delta,i})^{2}}{(\ln\frac{R}{s^{+}_{\delta,i}})^{2}}\frac{\partial
g(z_{i}^{+},z^{+}_{i})}{\partial z^{+}_{i,h}}-\sum_{k\neq
i}^{m}\frac{2\pi\delta^{2}a^{+}_{\delta,i}a^{+}_{\delta,k}}{|\ln\frac{R}{s^{+}_{\delta,k}}||\ln\frac{R}{s^{+}_{\delta,i}}|}\frac{\partial\bar{G}(z^{+}_{i},z_{k}^{+})}{\partial
z^{+}_{i,h}}\\\
&+\sum_{l=1}^{n}\frac{2\pi\delta^{2}a^{+}_{\delta,i}a^{-}_{\delta,l}}{|\ln\frac{R}{s^{-}_{\delta,l}}||\ln\frac{R}{s^{+}_{\delta,i}}|}\frac{\partial\bar{G}(z^{+}_{i},z_{l}^{-})}{\partial
z^{+}_{i,h}}+O\Bigl{(}\frac{\varepsilon^{2+\sigma}}{|\ln\varepsilon|^{p+1}}\Bigr{)},\end{split}$
since
$\int_{B_{1}(0)}\phi^{p-1}(|x|)\phi^{\prime}(|x|)\frac{x_{h}^{2}}{|x|}=-\frac{2\pi}{p}|\phi^{\prime}(1)|.$
Other terms can be estimated as above. Thus, the result follows.
∎
## References
* [1] A. Ambrosetti and M. Struwe, Existence of steady vortex rings in an ideal fluid, Arch. Rational Mech. Anal., 108(1989), 97–109.
* [2] A. Ambrosetti and J. Yang, Asymptotic behaviour in planar vortex theory, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1(1990), 285–291.
* [3] V.I. Arnold and B.A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, Vol. 125. Springer, New York, 1998.
* [4] T.V. Badiani, Existence of steady symmetric vortex pairs on a planar domain with an obstacle, Math. Proc. Cambridge Philos. Soc., 123(1998), 365–384.
* [5] T.Bartsch, A.Pistoia and T.Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Comm.Math.Phys., 297(2010), 653–686.
* [6] M.S. Berger and L.E. Fraenkel, Nonlinear desingularization in certain free-boundary problems, Comm. Math. Phys., 77(1980), 149–172.
* [7] G.R. Burton, Vortex rings in a cylinder and rearrangements, J. Diff. Equat., 70(1987), 333–348.
* [8] G. R. Burton, Steady symmetric vortex pairs and rearrangements, Proc. Royal Soc. Edinburgh, 108A (1988), 269–290.
* [9] G. R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincare. Analyse Nonlineare, 6(1989), 295–319.
* [10] G. R. Burton, Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex, Acta Math., 163(1989), 291–309.
* [11] L. Caffarelli and A. Friedman, Asymptotic estimates for the plasma problem, Duke Math. J., 47(1980), 705–742.
* [12] D.Cao and T.Küpper, On the existence of multi-peaked solutions to a semilinear Neumann problem, Duke Math. J., 97(1999), 261–300.
* [13] D. Cao, Z. Liu and J. Wei, Regularization of point vortices for the Euler equation in dimension two, arXiv:12083200v2, 32 pages.
* [14] D. Cao, S. Peng and S. Yan, Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225(2010), 2741–2785.
* [15] E.N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. London Math. Soc., 78(2008), 639–662.
* [16] A. R. Elcrat and K. G. Miller, Steady vortex flows with circulation past asymmetric obstacles, Comm. Part. Diff. Equat., 12 (1987), 1095–1115.
* [17] A. R. Elcrat and K. G. Miller, Rearrangements in steady vortex jows with circulation, Proc. Amer. Math. Soc., 111 (1991), 1051–1055.
* [18] A.R.Elcrat and K.G.Miller, Rearrangements in steady multiple vortex flows,Comm.Part.Diff.Equat., 20(1995), 1481–1490.
* [19] L.E. Fraenkel and M.S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math., 132(1974), 13–51.
* [20] M. Flucher and J. Wei, Asymptotic shape and location of small cores in elliptic free-boundary problems, Math. Z., 228(1998), 638–703.
* [21] W. Kulpa, The Poincaré-Miranda theorem, The Amer. Math. Monthly, 104(1997), 545–550.
* [22] G. Li, S. Yan and J. Yang, An elliptic problem related to planar vortex pairs, SIAM J. Math. Anal., 36(2005), 1444–1460.
* [23] Y. Li and S. Peng, Multiple solutions for an elliptic problem related to vortex pairs, J. Diff. Equat., 250(2011), 3448–3472.
* [24] C.C.Lin, On the motion of vortices in two dimension – I. Existence of the Kirchhoff-Routh function, Proc. Natl. Acad. Sci. USA, 27(1941), 570–575.
* [25] C. Miranda, Un’osservazione su un teorema di Brouwer., Boll. Un. Mat. Ital., 3(1940), 5–7.
* [26] C. Marchioro and M. Pulvirenti, Euler evolution for singular initial data and vortex theory, Comm. Math. Phys., 91(1983), 563–572.
* [27] K. G. Miller, Stationary corner vortex configurations, Z. Angew. Math. Phys., 47(1996),39-56.
* [28] W.-M. Ni, On the existence of global vortex rings, J. Anal. Math., 37(1980), 208–247.
* [29] J. Norbury, Steady planar vortex pairs in an ideal fluid, Comm. Pure Appl. Math., 28(1975), 679–700.
* [30] P. G. Saffman and J. Sheffield, Flow over a wing with an attached free vortex, Studies in Applied Math., 57 (1977), 107–117.
* [31] D. Smets and J. Van Schaftingen, Desingulariation of vortices for the Euler equation, Arch. Rational Mech. Anal., 198(2010), 869–925.
* [32] B. Turkington, On steady vortex flow in two dimensions. I, II, Comm. Partial Diff. Equat., 8(1983), 999–1030, 1031–1071.
* [33] J. Yang, Existence and asymptotic behavior in planar vortex theory, Math. Models Methods Appl. Sci., 1(1991), 461–475.
* [34] J. Yang, Global vortex rings and asymptotic behaviour, Nonlinear Anal., 25(1995), 531–546.
|
arxiv-papers
| 2012-08-28T02:37:26 |
2024-09-04T02:49:34.594761
|
{
"license": "Public Domain",
"authors": "Daomin Cao, Zhongyuan Liu and Juncheng Wei",
"submitter": "Daomin Cao",
"url": "https://arxiv.org/abs/1208.5540"
}
|
1208.5601
|
# Charmonia production in ALICE
Christophe Suire for the ALICE Collaboration Institut de Physique Nucléaire
d’Orsay, CNRS-IN2P3, Université Paris-Sud, France
###### Abstract
Quarkonia states are expected to provide essential information on the
properties of the high-density strongly-interacting system formed in the early
stages of high-energy heavy-ion collisions. ALICE is the LHC experiment
dedicated to the study of nucleus-nucleus collisions and can study charmonia
at forward rapidity ($2.5<y<4$) via the $\mu^{+}\mu^{-}$ decay channel and at
mid rapidity ($|y|<0.9$) via the $e^{+}e^{-}$ decay channel. In both cases
charmonia are measured down to zero transverse momentum. The inclusive
$J/\psi$ production as a function of transverse momentum and rapidity in pp
collisions at $\sqrt{s}=$ 2.76 and 7 TeV are presented. For pp collisions at
$\sqrt{s}=$ 7 TeV, the inclusive $J/\psi$ production as a function of the
charged particle multiplicity, the inclusive $J/\psi$ polarization at forward
rapidity and the $J/\psi$ prompt to non-prompt fraction are discussed.
Finally, the analysis of the inclusive $J/\psi$ production in the Pb-Pb data
collected fall 2011 at a center of mass energy of $\sqrt{s_{NN}}=2.76$ TeV is
presented. Results on the nuclear modification factor are then shown as a
function of centrality, transverse momentum and rapidity and compared to model
predictions. First results on inclusive $J/\psi$ elliptic flow are given.
###### keywords:
Hadron-induced high- and super-high-energy interactions, Relativistic heavy-
ion collisions, Quark-Gluon plasma, $J/\psi$ production and suppression
mechanisms.
## 1 Charmonia in heavy-ion collisions
Charmonia suppression via color-screening of the heavy-quark potential was
originally proposed as a probe of the QCD matter formed in relativistic heavy-
ion collisions in 1986 [1]. $J/\psi$ production was extensively studied at the
Super Proton Synchrotron (SPS) and at the Relativistic Heavy Ion Collider
(RHIC). Indeed, $J/\psi$ suppression in most central heavy-ion collisions was
observed over a large energy range ($\approx$ 20 to 200 GeV/$c$). The LHC has
opened a new energy regime for the study of quarkonium in heavy-ion
collisions. In a Pb-Pb collision at $\sqrt{s_{NN}}=2.76$ TeV, an average of
one $J/\psi$ particle is expected to be produced in every central Pb-Pb
collision, together with about 100 c$\bar{\rm{c}}$ pairs. Several models [2,
3, 4, 5] have included, already at RHIC energy, a $J/\psi$ regeneration
component from deconfined charm quarks in the medium which counteracts the
J/$\psi$ suppression in a QGP. At LHC, this regeneration component may become
important, even dominant.
The in-medium modification of the $J/\psi$ production can be quantified with
the nuclear modification factor $R_{\mathrm{AA}}$, defined as the $J/\psi$
yield measured in nucleus-nucleus collisions divided by the yield measured in
pp collisions and the number of binary nucleon-nucleon collisions occurring in
the nucleus-nucleus collision. To interpret the $R_{\mathrm{AA}}$, one must
keep in mind the following points. First, prompt $J/\psi$ production in
hadronic interactions consists of the sum of direct $J/\psi$ ($\approx$ 65%)
and excited $\rm{c\overline{c}}$ states such as $\chi_{\rm{c}}$ and
$\psi\rm{(2S)}$ ($\approx$ 35%). Since the $\chi_{\rm{c}}$ and $\psi\rm{(2S)}$
have lower dissociation temperatures than the $J/\psi$, a $J/\psi$
$R_{\mathrm{AA}}$ measurement around 0.65 is compatible within errors with the
suppression of excited states only. In addition to these prompt $J/\psi$ one
should also take into account that a non-prompt component from beauty hadron
decays is present at LHC energy. The $R_{\mathrm{AA}}$ includes cold nuclear
matter (CNM) effects, dominated by nuclear absorption and (anti-) shadowing.
These CNM effects can be responsible for $J/\psi$ suppression independently
from the creation of a deconfined medium. To quantify CNM effects, proton-
nucleus collisions are needed. At SPS energy, the observed $J/\psi$
suppression would be compatible with the dissociation of excited states, once
the CNM effects are taken into account. At RHIC, a $J/\psi$ $R_{\mathrm{AA}}$
of $\approx$ 0.25 in most central Au-Au collisions was measured by the PHENIX
experiment [6] with a strong centrality dependence. After estimating the
correction due to the CNM effects, the suppression of direct $J/\psi$ is at
least $\approx$ 40% or more. At the LHC, $J/\psi$ are abundantly produced and
detailed studies of its production are possible in both elementary and heavy-
ion collisions, such as azimuthal asymmetry, polarization,
$R_{\mathrm{AA}}$dependence on rapidity and on transverse momentum, etc. Such
studies may give us some answers about the balance between suppression and
recombination mechanisms of $J/\psi$.
The asymmetry the azimuthal distribution of $J/\psi$ in the plane
perpendicular to the beam direction, the so-called elliptic flow or v2is
indeed a very interesting experimental observable. When heavy-ions collide at
finite impact parameter (non-central collisions), the geometrical overlap
region and therefore the initial matter distribution is anisotropic and is
converted into a momentum anisotropy of the produced particles. The possible
onset of $J/\psi$ production via recombination mechanisms should be, according
to models, accompanied with a non-zero or possibly large v2 value [7] at low
$p_{{\mathrm{T}}}$. Indeed, if charm quarks reach some level of thermalization
in the medium, they may acquire an elliptic flow that can be further
transferred to the $J/\psi$ assuming the $J/\psi$ is formed via recombination.
In the following section, the ALICE experiment and the data samples will be
described. Then, $J/\psi$ production results in pp collisions at $\sqrt{s}=$
2.76 TeV and 7 TeV will be presented. The $J/\psi$ production in Pb-Pb
collisions at$\sqrt{s_{\mathrm{NN}}}=$ 2.76 will be studied through the
$R_{\mathrm{AA}}$ dependence on centrality, $p_{{\mathrm{T}}}$ and $y$.
Finally, $J/\psi$ elliptic flow measurement will be shown.
## 2 Experimental apparatus and data sample
ALICE is a general purpose heavy-ion experiment and is described in [8]. It
consists of a central part covering the pseudo-rapidity $|\eta|<0.9$ and a
muon spectrometer covering $-4<\eta<-2.5$. At forward (mid) rapidity, $J/\psi$
production is measured in the dimuon (dielectron) decay channel; in both cases
the $p_{{\mathrm{T}}}$ coverage extends down to zero. Only detectors that are
relevant to the analysis will be presented.
At mid rapidity, the $J/\psi$ $\longrightarrow e^{+}e^{-}$ analysis makes use
of the high precision tracking and particle identification of the Inner
Tracking System (ITS) and the Time Projection Chamber (TPC). The ITS consists
of six cylindrical layers of silicon detectors; at a radius 3.9 and 7.6 cm,
the first two layers are equipped with silicon pixels (SPD), then two layers
of silicon drifts at radius 15 and 23.9 cm and finally, two layers of silicon
strips at radius 38 and 43 cm. Its main tasks are the primary and secondary
vertex reconstruction; the resolution on the primary vertex ranges from 100
$\mu$m (pp collisions) to 10 $\mu$m (central Pb-Pb collisions). The SPD has
triggering capabilities and can provide a signal at level 0. The large
cylindrical TPC has full azimuthal coverage and extends from z = -2.50 m to z
= 2.50 m 111The z axis is defined here as the beam line axis in the counter
clockwise direction and its origin is at the center of the ALICE detector..
The TPC radial coverage ranges from r = 85 cm to r = 247 cm. This large drift
detector is the main track reconstruction device at central rapidity since it
can provide up to 159 space points per track. Particle identification (PID) is
achieved via the measurement of the specific energy loss (dE/dx) of particles
in the detector gas (Ne/CO2/N2). The excellent dE/dx resolution of $\approx$
5% allows to identify electrons by using inclusion cut around the Bethe-Bloch
fit for electrons and exclusion cuts for protons and pions. ALICE has further
capabilities to improve electron identification and triggering thanks to the
Time-Of-Flight (TOF), the Transition Radiation Detector (TRD) and the
Electromagnetic Calorimeter (EMCAL) detectors. However, these detectors have
not been used in the analysis presented here.
At forward rapidity ($2.5<$ $y$ $<4$) the production of quarkonium states is
measured in the muon spectrometer 222In the ALICE reference frame, the muon
spectrometer covers a negative $\eta$ range and consequently a negative $y$
range. We have chosen to present our results with a positive $y$ notation..
The spectrometer consists of a ten interaction length thick absorber ( -0.9 m
$<$ z $<$ -5.0 m) filtering the muons in front of five tracking stations (MCH)
made of two planes of cathode pad chambers each. The third station is located
inside a dipole magnet with a 3 Tm field integral. The MCH chambers are
positioned between z=-5.2 and z=-14.4 m. The tracking apparatus is completed
by a triggering system (MTR) made of two stations, located at z=-16.1 and
z=-17.1 m, each equipped of two planes of resistive plate chambers. The MTR
chambers are downstream of a 1.2 m thick iron wall, which absorbs secondary
hadrons escaping from the front absorber and low momentum muons coming mainly
from $\pi$ and K decays. Throughout its full length, a conical absorber made
of tungsten, lead and steel protects the muon spectrometer against secondary
particles produced by the interaction of large-$\eta$ primaries in the beam
pipe. The forward VZERO detectors, two arrays of 32 scintillator tiles
covering the range $2.8\leq\eta\leq 5.1$ (VZERO-A) and $-3.7\leq\eta\leq-1.7$
(VZERO-C), are positioned at z=340 and z=-90 cm. And finally, the zero degree
calorimeters (ZDC) placed 116 m down and up-stream ALICE can detect spectator
neutrons and protons.
In proton-proton collisions, the minimum bias (MB) trigger uses information of
the SPD and VZERO detectors. It is defined as the logical OR of the three
following conditions: (i) a signal in two readout chips in the outer layer of
the SPD, (ii) a signal in VZERO-A, (iii) a signal in VZERO-C. This MB trigger
requires the coincidence of the crossing of two proton bunches at the
experiment interaction point (IP). ALICE MB trigger selects about 86% of the
proton-proton inelastic cross section. Specific cross sections were measured
during van der Meer scan in pp collisions at 7 and 2.76 TeV and allowed to
determine the absolute normalization of the inclusive $J/\psi$ cross section.
The muon minimum bias trigger ($\mu$-MB) requires, in addition to the MB
conditions given above, a signal in the MTR system. The MTR can reconstruct a
trigger track, determine its $p_{{\mathrm{T}}}$ and select different
thresholds ($p_{{\mathrm{T}}}$ $\approx$ 0.5, 1 and 4 GeV/$c$). The $\mu$-MB
trigger helps to take advantage of the full luminosity delivered by the LHC in
the muon spectrometer. The MB trigger used in Pb-Pb collisions collected in
2010 requires the logical AND of the conditions (i),(ii) and (iii) given
above. The centrality of the collision is determined from the amplitude of the
VZERO signal fitted with a geometrical-Glauber model [9]. In 2011, the MB
conditions were reduced to the AND of conditions (ii) and (iii) but additional
requirements were added to select rare events. In particular, event
multiplicity and dimuon triggers were added and ZDC were used for rejecting
electromagnetic Pb-Pb interactions and satellite Pb-Pb collisions. Once the
centrality selection cut has been applied, triggers are fully efficient with
negligible contamination.
## 3 $J/\psi$ production in pp collisions
The $J/\psi$ production in pp collisions is extensively studied in ALICE and
only a selection of the available results will be presented in this section.
Further details on the related analysis can be found in [10].
The inclusive $J/\psi$ production was measured in pp collisions at $\sqrt{s}=$
7 TeV in the dimuon and dielectron channels in the rapidity ranges $2.5<y<4$
and $|y|<0.9$ down to $p_{{\mathrm{T}}}$ = 0. The analysis was made with an
integrated luminosity $\mathcal{L}_{\rm int}\approx 16\;(6)\;\mathrm{nb}^{-1}$
in the dimuon (dielectron) channel. The measured cross section values are
$\sigma_{J/\psi}(|y|<0.9)=10.7\pm 1.0(\mathrm{stat.})\pm
1.6(\mathrm{syst.})^{+1.6}_{-2.3}(\mathrm{syst.pol.})\;\mu\mathrm{b}$ and
$\sigma_{J/\psi}(2.5<y<4.)=6.31\pm 0.25(\mathrm{stat.})\pm
0.76(\mathrm{syst.})^{+0.95}_{-1.96}(\mathrm{syst.pol.})\;\mu\mathrm{b}$. At
forward rapidity differential cross section
d${}^{2}\sigma$/d$p_{{\mathrm{T}}}$d$y$ measurement from ALICE fully overlaps
with LHCb and a good agreement is found. At mid rapidity, the situation is
different since ATLAS and CMS cannot measure $J/\psi$ with $p_{{\mathrm{T}}}$
$\lesssim$ 6 GeV/$c$. Thus combining ALICE and CMS/ATLAS data offers a rather
complete inclusive $J/\psi$ production measurement over a large rapidity
range. These comparisons are available in [11].
The same analysis was carried out with pp collisions at $\sqrt{s}=$ 2.76 TeV
collected in March 2011. Since the center of mass energy per nucleon-nucleon
collisions is identical to the one of the Pb-Pb collisions, this analysis
provides an essential reference data to measure the $J/\psi$ nuclear
modification factor. The integrated luminosity for the analysis is
$\mathcal{L}_{\rm int}\approx 20\;(1)\;\mathrm{nb}^{-1}$ in the dimuon
(dielectron) channel. The integrated cross sections are
$\sigma_{J/\psi}(|y|<0.9)=6.71\pm 1.24(\mathrm{stat.})\pm
1.22(\mathrm{syst.})^{+1.01}_{-1.41}(\mathrm{syst.pol.})\;\mu\mathrm{b}$ and
$\sigma_{J/\psi}(2.5<y<4.)=3.34\pm 0.13(\mathrm{stat.})\pm
0.28(\mathrm{syst.})^{+0.53}_{-1.07}(\mathrm{syst.pol.})\;\mu\mathrm{b}$. Note
that the uncertainties quoted here on the pp measurement are one of the main
source of uncertainty of the nuclear modification factor discussed in the next
section. The differential cross sections
d${}^{2}\sigma$/d$p_{{\mathrm{T}}}$d$y$ have been extracted down to
$p_{{\mathrm{T}}}$ = 0 at both rapidities. These results are compared to a
theoretical model, NRQCD calculation that includes Color Singlet and Color
Octet terms at NLO, which describes reasonably well the measurement at
$\sqrt{s}=$ 2.76 TeV and also the one at $\sqrt{s}=$ 7 TeV [12].
In the previous results, one could remark that the $J/\psi$ cross section has
a large uncertainty related to the unknown polarization. ALICE has studied
$J/\psi$ polarization in pp collisions $\sqrt{s}=$ 7 TeV in the dimuon
channel. Measurements of the polar and azimuthal angle distributions of the
decay muons allowed us to extract the $J/\psi$ polarization for 2 $<$
$p_{{\mathrm{T}}}$ $<$ 8 GeV/$c$ and 2.5 $<$ $y$ $<$ 4\. The parameters
describing the $J/\psi$ polarizations are consistent with zero in the
kinematic range under study [13]. This measurement is, at the present date,
the only $J/\psi$ polarization measurement at the LHC. It is crucial in the
near future to extend the polarization measurement down to zero
$p_{{\mathrm{T}}}$ and to high $p_{{\mathrm{T}}}$ in order to provide more
stringent tests to theoretical calculations. In addition, since the pp cross
section enters the nuclear modification factor calculation, the polarization,
if different from zero, may have a strong impact at low transverse momentum.
Such a measurement needs a large statistic and strengthens the requirement to
collect a large amount of data at the same center of mass energy as the Pb-Pb
collisions.
An interesting feature of the $J/\psi$ production in pp collisions at
$\sqrt{s}=$ 7 TeV arises from its dependence on the charged particle
multiplicity. The d$N_{\rm{ch}}$/d$\eta$ is calculated from the number of
tracks reconstructed in $|\eta|<1$ using pairs of hits (tracklets) in the SPD.
These measurements were performed at both rapidities in the dimuon and
dielectron channels. Expressed in terms of the relative $J/\psi$ yield
$\frac{\mathrm{d}N_{J/\psi}/\mathrm{d}\eta}{\langle\mathrm{d}N_{J/\psi}/\mathrm{d}\eta\rangle}$
as a function of the relative charged multiplicity
$\frac{\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta}{\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle}$,
a linear increase is clearly seen at both rapidities. For
$\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta/\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle=4$
($\approx 24/6$), the relative $J/\psi$ yield is enhanced by a factor of about
5 at forward rapidity and about 8 at mid rapidity. This trend is not
reproduced by PYTHIA 6.4.25 in the Perugia 2011 tune which exhibits an
opposite tendency, i.e. a decrease of the $J/\psi$ multiplicity with respect
to the event multiplicity [14]. One could infer that the $J/\psi$ production
is strongly connected with the underlying hadronic activity. Whether this
hadronic activity comes from multiple parton interactions remains an open
question. Further investigations are needed to better understand this
measurement that challenges our understanding of the $J/\psi$ production in pp
collisions. In particular, the event multiplicity dependence should be
completed by the $p_{{\mathrm{T}}}$ dependence and extended to the open charm
cross section (e.g. D mesons).
All the results presented up to now refer to inclusive $J/\psi$ production
which sums three distinct contributions: the prompt $J/\psi$ produced directly
in the pp collisions, the prompt $J/\psi$ produced indirectly via the decay of
heavier charmonia states and the non-prompt $J/\psi$ produced in the decay of
beauty hadrons. At central rapidity ($|y|<0.9$), the measurement of the non-
prompt $J/\psi$ was achieved in pp collisions at $\sqrt{s}=$ 7 TeV for $1.3<$
$p_{{\mathrm{T}}}$ $<10$ GeV/$c$. This measurement is only accessible in ALICE
since the other experiments cannot detect $J/\psi$ at mid rapidity below a
$p_{{\mathrm{T}}}$ of 6.5 GeV/$c$ where most of the cross section lies. The
integrated luminosity for the analysis is $\mathcal{L}_{\rm
int}=5.6\mathrm{nb}^{-1}$. This measurement relies on the discrimination of
$J/\psi$ produced detached from the primary vertex of the pp collisions thanks
to the good spatial resolution of the ITS. By fitting simultaneously the
invariant mass spectra and the pseudo-proper decay length of the reconstructed
$J/\psi$, one can can measure the relative abundances of prompt and non-prompt
$J/\psi$, and the background. The fraction of $J/\psi$ from beauty hadrons
($f_{\mathrm{B}}$) in the measured kinematic range is about 15% with a strong
$p_{{\mathrm{T}}}$ dependence. Then, $f_{\mathrm{B}}$ is combined with the the
inclusive $J/\psi$ cross section measured in [11] to extract the prompt
$J/\psi$ cross section
$\sigma_{J/\psi}^{\mathrm{prompt}}(|y|<0.9,p_{\mathrm{T}}>1.3{\mathrm{~{}GeV}/c})=7.2\pm
0.7(\mathrm{stat.})\pm
1.0(\mathrm{syst.})^{+1.3}_{-1.2}(\mathrm{syst.pol.})\;\mu\mathrm{b}$.
Comparisons with models lead to a good description of the prompt $J/\psi$
dependence with $p_{{\mathrm{T}}}$ and of the total beauty cross section [15].
## 4 Nuclear modification factor in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV
Inclusive $J/\psi$ production was studied in Pb-Pb collisions at 2.76 TeV at
mid and forward rapidity in the dielectron and dimuon decay channels using
respectively an integrated luminosity $\mathcal{L}_{\rm int}\approx$ 2.1 and
$\mathcal{L}_{\rm int}\approx 70$ $\mu$b-1. A crucial feature of the ALICE
detector is to measure, in both channels, the $J/\psi$ production down to
$p_{{\mathrm{T}}}$ = 0 GeV/c. The large data sample analyzed in the dimuon
channel allowed us to perform differential analysis of the nuclear
modification factor ($R_{\mathrm{AA}}$) as function of centrality,
$p_{{\mathrm{T}}}$, and $y$. In the dielectron channel, only the centrality
dependence in 3 centrality classes (0–10%, 10–40% and 40–80%) could be
achieved. One should note that the acceptance times efficiency factor in the
dielectron (dimuon) channel is quite high $\approx 8\%$ ($14\%$) and weakly
depends on the collision centrality with a maximum relative loss of 12% (8%)
from peripheral to most central collisions. Details on both analysis are given
in [16].
Figure 1: (Color online) Inclusive $J/\psi$ $R_{\rm{AA}}$ measured in Pb-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV at forward and mid rapidity in
ALICE is shown on the left side. On the right side the $J/\psi$
$R_{\mathrm{AA}}$ at high-$p_{{\mathrm{T}}}$ from ALICE at forward $y$ is
compared to one measured by CMS at central $y$.
On the left side of Fig. 1, the inclusive $J/\psi$ $R_{\mathrm{AA}}$ is shown
as a function of the number of nucleons participating in the collision [9] at
mid and forward rapidity. At forward rapidity, a clear suppression is seen for
$N_{\mathrm{part}}$ $>70$ with almost no centrality dependence. These results
show a good agreement with the ones published in [17] based on
$\mathcal{L}_{\rm int}\approx 2.9$ $\mu$b-1 collected in 2010. At mid
rapidity, a similar pattern could be possible but the coarser centrality
classes and larger uncertainties prevent to draw any firm conclusion. The
centrality integrated $J/\psi$ $R_{\mathrm{AA}}$ at forward and mid rapidity
are $R^{0\%-90\%}_{\rm{AA}}=0.497\pm 0.006\rm{(stat.)}\pm 0.078\rm{(syst.)}$
and $R^{0\%-80\%}_{\rm{AA}}=0.66\pm 0.10\rm{(stat.)}\pm 0.24\rm{(syst.)}$. In
both cases, the systematic uncertainty is dominated by the pp reference. On
the right side of Fig. 1, the centrality dependence of $J/\psi$
$R_{\mathrm{AA}}$ at high-$p_{{\mathrm{T}}}$ is compared between ALICE and CMS
[18]. A larger suppression, $R_{\mathrm{AA}}$ $\approx 0.25-0.30$, is measured
in the most central collisions with a clear centrality dependence. One could
see here an indication that the $J/\psi$ $R_{\mathrm{AA}}$ is
$p_{{\mathrm{T}}}$ dependent at forward rapidity and possibly at mid rapidity;
selecting high-$p_{{\mathrm{T}}}$ $J/\psi$ drives down the $R_{\mathrm{AA}}$.
The $p_{{\mathrm{T}}}$ dependence of the $J/\psi$ $R_{\mathrm{AA}}$ is
confirmed and can be better observed in Fig. 2 (left side). The inclusive
$J/\psi$ $R_{\mathrm{AA}}$ is shown as a function of $p_{{\mathrm{T}}}$ for
the 0%–90% most central Pb-Pb collisions and exhibits a decrease from 0.6 to
0.35 approximately. At high-$p_{{\mathrm{T}}}$ a rather direct comparison with
CMS results [18] is possible; the only difference is that the CMS measurement
covers a more central rapidity range ($1.6<|y|<2.4$). A reasonable agreement
between the two measurements is found for high-$p_{{\mathrm{T}}}$ $J/\psi$
$R_{\mathrm{AA}}$. For $p_{{\mathrm{T}}}$ smaller than 4 GeV/$c$, the
difference with PHENIX measurement [19] is striking. The PHENIX result concern
the 0%–20% most central Au-Au collisions whereas the ALICE result is for a
much wider centrality range (0%–90%). However, the bulk of the $J/\psi$
production ($\approx 60\%$) occurring in 0%–20% most central collisions, the
comparison remains meaningful. In addition, work is ongoing to extract the
$R_{\mathrm{AA}}$ versus $p_{{\mathrm{T}}}$ in smaller centrality classes.
The $J/\psi$ $R_{\mathrm{AA}}$ dependence on rapidity has been measured over a
wide range thanks to the combination of our measurement in the central barrel
and in the muon spectrometer, and is displayed on the right side of Fig. 2. At
forward rapidity, the $J/\psi$ $R_{\mathrm{AA}}$ decreases by $\approx 40\%$
from $y=2.5$ to $y=4$. The measurement at mid rapidity, because of its large
uncertainties, does not allow to draw a clear conclusion but hints towards a
rather flat behavior between $y$ = 2.5 and $y$ = 0. On the same figure, an
estimate of the $J/\psi$ $R_{\mathrm{AA}}$ due only to shadowing effects is
given for two models. Indeed at LHC energies, modification of the gluon
distribution function is dominated by shadowing effects [20]. The first model
is a Next to Leading Order calculation within the Color Evaporation Model [21]
with the EPS09 nuclear PDF (nPDF). The second model is a Leading Order
calculation within the CS Model [22] with the nDSg nPDF. In the first model,
the upper and lower limits correspond to the uncertainty of the EPS09 nPDF,
and in the second model the band covers the uncertainty in the factorization
scale of the for nDSg PDF.
Figure 2: (Color online) ALICE $p_{{\mathrm{T}}}$ dependence of the inclusive
$J/\psi$ $R_{\mathrm{AA}}$ measured in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV is compared to PHENIX and CMS measurements
on the left side. On the right side, the rapidity dependence of the $J/\psi$
$R_{\mathrm{AA}}$ measured by ALICE is compared with model predictions [21,
22] that implement only shadowing effects.
One could not exclude that shadowing effects are responsible for a large part
of the $J/\psi$ suppression observed in $R_{\mathrm{AA}}$ from $y$ = 0 to $y$
$\approx$ 3, this would imply that the expected color screening $J/\psi$
suppression observed at lower energy (RHIC) or higher $p_{{\mathrm{T}}}$ (CMS)
is either small or compensated by recombination mechanisms. In the rapidity
range from 3 to 4, our results show that the suppression goes beyond the
shadowing-only prediction given by models with our current knowledge of nPDF.
The influence of the contribution of beauty hadron feed-down to the inclusive
$J/\psi$ yield in our $y$ and $p_{{\mathrm{T}}}$ range was estimated. Non-
prompt $J/\psi$ are indeed different since their suppression or production is
insensitive to color screening or recombination phenomena that are expected to
occur in the hot and dense medium created in the Pb-Pb collisions. The beauty
hadron decay mostly occurs outside the fireball, and a measurement of the non-
prompt $J/\psi$ $R_{\mathrm{AA}}$ is connected to the beauty quark in-medium
energy loss. At forward rapidity, the non-prompt $J/\psi$ was measured by the
LHCb collaboration to be about 10% in pp collisions at $\sqrt{s}=7$ TeV [23]
in our $p_{{\mathrm{T}}}$ range. Assuming the scaling of beauty production
with the number of binary nucleon-nucleon collisions and neglecting the
shadowing effects, the prompt $J/\psi$ $R_{\mathrm{AA}}$ would be, and this is
an upper limit, 11% smaller than our inclusive measurement. To estimate the
influence of non-prompt $J/\psi$ as a function of $p_{{\mathrm{T}}}$ and $y$
on our inclusive $R_{\mathrm{AA}}$ measurement, we have extrapolated the LHCb
measurement at $\sqrt{s}=7$ TeV down $\sqrt{s}=2.76$ TeV using an center of
mass energy dependence extracted from CDF and CMS data. Assuming a range of
energy loss for the beauty quarks from $R_{\mathrm{AA}}$(b) = 0.2 to
$R_{\mathrm{AA}}$(b)= 1, we have found that the $J/\psi$ from beauty hadrons
have a negligible influence on our measurement.
In Fig. 3, our $J/\psi$ $R_{\mathrm{AA}}$ measurement is compared with
theoretical models that all include a $J/\psi$ regeneration component from
deconfined charm quarks in the medium.
Figure 3: (Color online) Centrality and $p_{{\mathrm{T}}}$ dependence of the
inclusive $J/\psi$ $R_{\mathrm{AA}}$ measured in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV at forward rapidity compared to the
predictions by Statistical Hadronization Model [24], Transport Model I [25]
and II [26], see text for details.
The Statistical Hadronization Model [2, 24] assumes deconfinement and a
thermal equilibration of the bulk of the c$\bar{\rm{c}}$ pairs. Then,
charmonium production occurs only at phase boundary by statistical
hadronization of charm quarks. The prediction is given for two values of ${\rm
d}\sigma_{\rm{c}\bar{\rm{c}}}/{\rm d}y$ since no measurements are available at
this rapidity for Pb-Pb collisions. The two transport model results [25, 26]
presented in the same figure differ mostly in the rate equation controlling
the $J/\psi$ dissociation and regeneration. Both are shown as a band that
connects the results obtained with (lower limit) and without (higher limit)
shadowing and can be interpreted as the uncertainty of the prediction. The
model from Zhao & al. implements a simple shadowing estimate leading to a 30%
suppression in most central Pb-Pb collisions. The charm cross-section
d$\sigma_{c\bar{c}}/$d$y$ at $y=3.25$ is $\approx 0.5$ mb and the $J/\psi$
from beauty hadrons is estimated at 10% and no quenching is assumed. The model
from Liu & al. takes the shadowing from EKS98 and uses a smaller charm cross-
section d$\sigma_{c\bar{c}}/$d$y\approx 0.38$ mb. The $J/\psi$ from beauty
hadrons is estimated at 10% and b quenching is fixed at $R_{\mathrm{AA}}$(b)=
0.4 for all the $p_{{\mathrm{T}}}$ range. In both transport models, the amount
of regenerated $J/\psi$ in the most central collisions contributes to about
50% of the production yield, the rest being from initial production. We can
see on the left side of Fig. 3 that all three models reproduce correctly the
centrality dependence of the forward $J/\psi$ $R_{\mathrm{AA}}$ for
$N_{\mathrm{part}}$ $>70$. A similar observation can be made for the mid
rapidity results [16]. The $p_{{\mathrm{T}}}$ dependence of the forward
$J/\psi$ $R_{\mathrm{AA}}$ is also successfully reproduced by the transport
models, as shown on Fig. 3 right side. In addition, both models predict that a
large fraction of $J/\psi$ from regeneration have a $p_{{\mathrm{T}}}$ below
$\approx$ 3.5 GeV/$c$.
## 5 Elliptic flow in Pb-Pb collisions $\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV
The elliptic flow of inclusive $J/\psi$ has been measured as a function of the
transverse momentum of the $J/\psi$. For this measurement, the reaction plane
has been determined with the VZERO-A detector. The large rapidity gap between
the $J/\psi$ acceptance and the VZEROA detector minimizes the influence of
non-flow effects. One the left side of Fig. 4, an example of the $v_{2}$
signal extraction is given; one can clearly see the cosine shape of the
measured $J/\psi$ signal in the 6 $\Delta\varphi$ bins, where $\Delta\varphi$
is the difference between the azimuthal angle of the $J/\psi$ and the angle of
the reaction plane. Further analysis details can be found in [27]. Figure 4
shows, on the right side, the first measurement of $J/\psi$ elliptic flow at
the LHC. The $J/\psi$ $v_{2}$ is given as a function of $p_{{\mathrm{T}}}$ in
the 20%–60% centrality range. A non-null $J/\psi$ $v_{2}$ seems to be present
at intermediate $p_{{\mathrm{T}}}$ and would tend to vanish at low and high
$p_{{\mathrm{T}}}$. Uncertainties are still too large to draw definitive
conclusions, nevertheless we have a non-zero $v_{2}$ signal for $J/\psi$ with
$p_{{\mathrm{T}}}$ between 2 to 4 GeV/$c$ with 2.2 $\sigma$ significance. At
lower energy, the $J/\psi$ elliptic flow was measured by STAR and appear to be
consistent with zero at $p_{{\mathrm{T}}}$ $<$ 10 GeV/$c$ in 20%–60%
centrality range, whereas charged hadrons and $\phi$ exhibit a rather strong
flow in this same kinematic domain [28]. Model prediction (private
communication) for ALICE was provided by the authors of [26] and is shown on
Fig. 4. The full line assumes a thermalization of the beauty quarks in the
medium, for which there is no evidence so far, and should be considered as an
upper limit. The dashed line, a more realistic prediction in which beauty
quarks are not thermalized, shows indeed a non-zero $J/\psi$ $v_{2}$, which
matches qualitatively our data. It is important to add that this model
reproduces successfully the ALICE $J/\psi$ $R_{\mathrm{AA}}$measurement.
Figure 4: (Color online) Example of the $J/\psi$ $v_{2}$ signal extraction in
$\Delta\varphi$ bins (left). Inclusive $J/\psi$ $v_{2}$ measured in the
20%–60% centrality range for Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$
TeV (right side) compared to the STAR measurement and to the prediction from a
parton transport model.
## 6 Conclusion
Quarkonia production in ALICE in pp and Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 2.76 and 7 TeV has been presented. In pp
collisions, the $p_{{\mathrm{T}}}$, $y$ and multiplicity dependence of
$J/\psi$ production, $J/\psi$ polarization and non-prompt $J/\psi$ have been
studied. Results have shown good agreement or complementarity with other LHC
results. All these measurements provide stringent constraints to model
predictions. In Pb-Pb collisions, the $J/\psi$ nuclear modification factor was
studied as a function of centrality, $p_{{\mathrm{T}}}$ and $y$. The $J/\psi$
$R_{\mathrm{AA}}$ dependence on the number of participant nucleons is flat and
centrality integrated values are large at mid and forward rapidity ($\approx
0.7-0.5$). This result is clearly different from the ones seen at lower
energies (e.g. RHIC and SPS). The rapidity dependence of the $J/\psi$
$R_{\mathrm{AA}}$ shows that suppression at large rapidity ( 2.5 $<$ $y$ $<$ 4
) is beyond the one that could be expected from shadowing only predictions.
The $J/\psi$ $R_{\mathrm{AA}}$ is large at low $p_{{\mathrm{T}}}$ and then
decreases with increasing $p_{{\mathrm{T}}}$. The trends observed in the data
as a function of centrality and $p_{{\mathrm{T}}}$ can be reproduced by models
based on deconfinement followed by charm recombination. In these models,
$J/\psi$ from recombination mostly occur at low $p_{{\mathrm{T}}}$ and account
for half of the produced $J/\psi$ in the most central collisions. Finally, we
have presented the $J/\psi$ elliptic flow in semi-central Pb-Pb collisions.
For the first time, a non-zero $J/\psi$ $v_{2}$ is observed in the
intermediate $p_{{\mathrm{T}}}$ range. We have now accumulated hints that the
$J/\psi$ production in Pb-Pb collisions at LHC energy may be governed, for an
important part, by charm quarks recombination processes. In order to confirm
this observation, the shadowing must be measured and constrained since it
remains unknown at LHC energies and this will be addressed by a pPb run
scheduled at the beginning of 2013. One should add here that the uncertainties
in $J/\psi$ $R_{\mathrm{AA}}$ results depend directly on the pp reference
data; thus it is crucial to collect a large amount of pp collisions at the
same collision energy that of Pb-Pb sample, in order to have precise
measurements of $J/\psi$ and charm differential cross section and $J/\psi$
polarization.
## References
* [1] T. Matsui, H. Satz, $J/\psi$ Suppression by Quark-Gluon Plasma Formation, Phys. Lett. B178 (1986) 416.
* [2] P. Braun-Munzinger, J. Stachel, (Non)thermal aspects of charmonium production and a new look at $J/\psi$ suppression, Phys. Lett. B490 (2000) 196–202.
* [3] R. L. Thews, M. Schroedter, J. Rafelski, Enhanced $J/\psi$ production in deconfined quark matter, Phys. Rev. C63 (2001) 054905.
* [4] A. Andronic, P. Braun-Munzinger, K. Redlich, J. Stachel, Evidence for charmonium generation at the phase boundary in ultra-relativistic nuclear collisions, Phys. Lett. B652 (2007) 259–261.
* [5] X. Zhao, R. Rapp, Transverse Momentum Spectra of $J/\psi$ in Heavy-Ion Collisions, Phys. Lett. B664 (2008) 253–257.
* [6] A. Adare, et al., $J/\psi$ production vs centrality, transverse momentum, and rapidity in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$=200 GeV, Phys. Rev. Lett. 98 (2007) 232301.
* [7] Y. Liu, N. Xu, P. Zhuang, J/psi elliptic flow in relativistic heavy ion collisions, Nucl.Phys. A834 (2010) 317C–319C.
* [8] K. Aamodt, et al., The ALICE experiment at the CERN LHC, JINST 3 (2008) S08002.
* [9] K. Aamodt, et al., Centrality Dependence of the Charged-Particle Multiplicity Density at Midrapidity in Pb-Pb Collisions at $\sqrt{{s}_{\mathrm{NN}}}=2.76$ TeV, Phys. Rev. Lett. 106 (2011) 032301.
* [10] C.Geuna, Open Heavy-Flavour and $J/\psi$ production in pp collisions measured with the ALICE experiment at LHC, these proceedings.
* [11] K. Aamodt, et al., Rapidity and transverse momentum dependence of inclusive $J/\psi$ production in pp collisions at $\sqrt{s}=7$ TeV, Phys.Lett. B704 (2011) 442–455. arXiv:1105.0380.
* [12] B. Abelev, et al., Inclusive J$/\psi$ production in pp collisions at $\sqrt{s}$ = 2.76 TeV, arXiv:1203.3641.
* [13] B. Abelev, et al., $J/\psi$ polarization in pp collisions at $\sqrt{s}=7$ TeV, Phys.Rev.Lett. 108 (2012) 082001.
* [14] B. Abelev, et al., $J/\psi$ Production as a Function of Charged Particle Multiplicity in pp Collisions at $\sqrt{s}=7$ TeV, Phys.Lett. B712 (2012) 165–175. arXiv:1202.2816.
* [15] B. Abelev, et al., Measurement of prompt and non-prompt $J/\psi$ production cross sections at mid-rapidity in pp collisions at $\surd s=7$ TeVarXiv:1205.5880.
* [16] J.Wiechula, Nuclear modification of $J/\psi$ production in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, these proceedings.
* [17] B. Abelev, et al., $J/\psi$ Suppression at Forward Rapidity in Pb-Pb Collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, Phys. Rev. Lett. 109 (2012) 072301\.
* [18] S. Chatrchyan, et al., Suppression of non-prompt J/$\psi$, prompt J/$\psi$, and $\Upsilon$(1S) in PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV , JHEP 1205 (2012) 063.
* [19] A. Adare, et al., $J/\psi$ suppression at forward rapidity in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV, Phys.Rev. C84 (2011) 054912.
* [20] C. Lourenço, R. Vogt, H. K. Wöhri, Energy dependence of $J/\psi$ absorption in proton-nucleus collisions, JHEP 02 (2009) 014.
* [21] R. Vogt, Cold Nuclear Matter Effects on $J/\psi$ and $\Upsilon$ Production at the LHC, Phys. Rev. C81 (2010) 044903, and Priv. Comm.
* [22] E. Ferreiro, F. Fleuret, J. Lansberg, N. Matagne, A. Rakotozafindrabe, Cold Nuclear Matter Effects on extrinsic $J/\psi$ production at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV at the LHC, Nucl.Phys. A855 (2011) 327–330.
* [23] R. Aaij, et al., Measurement of $J/\psi$ production in pp collisions at $\sqrt{s}$=7 TeV, Eur.Phys.J. C71 (2011) 1645.
* [24] A. Andronic, P. Braun-Munzinger, K. Redlich, J. Stachel, The thermal model on the verge of the ultimate test: particle production in Pb-Pb collisions at the LHC, J.Phys.G G38 (2011) 124081.
* [25] X. Zhao, R. Rapp, Medium Modifications and Production of Charmonia at LHC, Nucl.Phys. A859 (2011) 114–125.
* [26] Y.-P. Liu, Z. Qu, N. Xu, P.-F. Zhuang, $J/\psi$ Transverse Momentum Distribution in High Energy Nuclear Collisions at RHIC, Phys.Lett. B678 (2009) 72–76, and Priv. Comm.
* [27] L.Massacrier, $J/\psi$ elliptic flow measurement in Pb-Pb collisions at forward rapidity in the ALICE experiment, these proceedings.
* [28] Z. Tang, $J/\psi$ production and correlation in p+p and Au+Au collisions at STAR, J.Phys.G G38 (2011) 124107. arXiv:1107.0532.
|
arxiv-papers
| 2012-08-28T09:44:53 |
2024-09-04T02:49:34.604489
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christophe Suire (for the ALICE Collaboration)",
"submitter": "Christophe Suire",
"url": "https://arxiv.org/abs/1208.5601"
}
|
1208.5692
|
# Thermopower of an SU(4) Kondo resonance under an SU(2) symmetry-breaking
field
P. Roura-Bas Dpto de Física, Centro Atómico Constituyentes, Comisión Nacional
de Energía Atómica, Buenos Aires, Argentina L. Tosi Centro Atómico Bariloche
and Instituto Balseiro, Comisión Nacional de Energía Atómica, 8400 Bariloche,
Argentina A. A. Aligia Centro Atómico Bariloche and Instituto Balseiro,
Comisión Nacional de Energía Atómica, 8400 Bariloche, Argentina P. S.
Cornaglia Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional
de Energía Atómica, 8400 Bariloche, Argentina
###### Abstract
We calculate the thermopower of a quantum dot described by two doublets
hybridized with two degenerate bands of two conducting leads, conserving
orbital (band) and spin quantum numbers, as a function of the temperature $T$
and a splitting $\delta$ of the quantum dot levels which breaks the SU(4)
symmetry. The splitting can be regarded as a Zeeman (spin) or valley (orbital)
splitting. We use the non-crossing approximation (NCA), the slave bosons in
the mean-field approximation (SBMFA) and also the numerical renormalization
group (NRG) for large $\delta$. The model describes transport through clean C
nanotubes and in Si fin-type field effect transistors, under an applied
magnetic field. The thermopower as a function of temperature $S(T)$ displays
two dips that correspond to the energy scales given by the Kondo temperature
$T_{K}$ and $\delta$ and one peak when $k_{B}T$ reaches the charge-transfer
energy. These features are much more pronounced than the corresponding ones in
the conductance, indicating that the thermopower is a more sensitive probe of
the electronic structure at intermediate or high energies. At low temperatures
($T\ll T_{K}$) $T_{K}S(T)/T$ is a constant that increases strongly near the
degeneracy point $\delta=0$. We find that the SBMFA fails to provide an
accurate description of the thermopower for large $\delta$. Instead, a
combination of Fermi liquid relations with the quantum-dot occupations
calculated within the NCA gives reliable results for $T\ll T_{K}$.
###### pacs:
72.20.Pa, 72.15.Qm,73.23.Hk, 73.63.Nm
## I Introduction
Materials with potentially useful thermoelectric properties and in particular
large thermopower, are currently a subject of intense research due to their
potential applications, for example in refrigeration or conversion of waste
heat into electricity.tritt ; sales ; tera ; bent ; kana ; klie ; maso In
addition, the thermopower is a useful tool to obtain additional insight on
fundamental problems, like the Kondo effect.bic ; velmo ; cost ; vel The
focus of research on thermal and thermoelectric properties has moved in the
last years to nanostructures such as quantum dots,vel ; harm ; sche ; boese ;
dong ; kim carbon nanotubes,hone ; yu molecules red nanowires, and spin
systems.sogo ; sogo2 ; roz ; arra As we will show, the thermopower can be a
more useful tool than the conductance, to study some features of the
electronic structure of Kondo systems at finite energies.
Electronic and thermal transport through single level quantum dots is well
described by the SU(2) Anderson model, and the Seebeck coefficient
(thermopower) has been calculated using this model.vel ; boese ; dong ; kim
In particular, Costi and Zlatić have made a comprehensive study of the
transport properties using the numerical renormalization group (NRG).vel A
particularly interesting multilevel system is the SU(4) Anderson model, which
describes quantum dots in carbon nanotubes,lim ; ander ; lipi ; buss ; fcm ;
grove and silicon nanowires.tetta In contrast to the SU(2) case, the
electron spectral density near the Fermi level of the SU(4) Anderson model in
the strong coupling (Kondo) limit for nearly one electron at the dot is highly
asymmetric,lim ; fcm leading to a high derivative at the Fermi energy, which
(neglecting phonon-drag effects) in turn is proportional to the Seebeck
coefficient $S$ at low temperatures.vel
The interest in Si nanowires increased due to the possibility of reducing the
thermal conductance $\kappa$, leading to a large figure of merit $\sigma
S^{2}T/\kappa$, where $\sigma$ is the conductivity.he Recently, the
conductance of Si fin-type field effect transistors has been measured under an
applied magnetic field $B$, which leads to a crossover from an SU(4) to an
orbital SU(2) Kondo effect.tetta The results were interpreted using the non-
crossing approximation (NCA). In C nanotubes, even for $B=0$, there might be a
symmetry reduction to SU(2) due to disorder-induced intervalley mixing.grove
In this work, we calculate the thermopower of the Anderson impurity model for
two doublets, each one either spin or orbital degenerate, and infinite on-site
Coulomb repulsion in the Kondo limit, as a function of the level splitting
$\delta$, which corresponds to the Zeeman splitting for two orbitally
degenerate levels in presence of an applied magnetic field, as in Si fin-type
field effect transistors.tetta The symmetry of the model is SU(4) for
$\delta=0$ and SU(2) for $\delta\neq 0$. We use complementary theoretical
approaches: the NCA,bic ; velmo ; fcm slave bosons in the mean-field
approximation (SBMFA),col ; hew ; lady ; soc and Fermi liquid theory for low
temperatures. We also use NRG to test the other approaches in the SU(2) limit.
The SBMFA satisfies Fermi liquid relationships at $T=0$ and is expected to
capture the low-energy physics in the Kondo limit. In turn, while NCA has an
error of about 15% in the value of the spectral density at very low
temperature according to the Friedel sum rule,fcm it describes better the
whole behavior, as shown for example in a previous comparison of NCA and NRG
results in the SU(2) case.compa In addition, if the differential conductance
$G$ is normalized at $T=0$, the leading behavior of $G$ for small voltage and
temperature roura agrees with alternative Fermi liquid approaches,ogu ; rpt ;
sela ; scali and the temperature dependence of the conductance practically
coincides with the NRG result over several decades of temperature.roura
Calculations of the thermopower for more than one spin-degenerate level within
the NCA, compared well with experiments in some Ce compounds.bic ; velmo NRG
is a very accurate technique at low temperatures.bulla However, for two
bands, the Hilbert space is increased 16 times in each iteration, instead of 4
times for one band, making the technique much more demanding if the same
degree of accuracy is wished. In addition, the SU(4) symmetry cannot be used
to reduce the size of the matrices, since this symmetry is broken by $\delta$.
Therefore we use here NRG only in the limit $\delta\rightarrow+\infty$, in
which only one doublet and the band that mixes with it play a significant
role.
The paper is organized as follows. In Section II we describe the model used.
The approximations and the equation for the Seebeck coefficient are described
in Section III. The numerical results are presented in Section IV. Section V
contains a summary and a short discussion.
## II Model
We start with a generalization of the Anderson model for infinite on-site
Coulomb repulsion, which contains a singlet $|s\rangle$ with $\mathcal{N}$
(even) particles and two spin doublets $|i\sigma\rangle$ ($i=1,2$ is the
valley index; $\sigma=\uparrow$ or $\downarrow$) with $\mathcal{N}+1$ (or
$\mathcal{N}-1$) particles representing the four spin and valley degenerate
states of a quantum dot created for example by depleting the density at two
points of a C nanotube or in a Si nanowire. The dot is connected to two
conducting leads which are also spin and valley degenerate. The SU(4) symmetry
is then broken by applying a magnetic field or breaking the valley degeneracy
in a simple way. In both cases, interchanging spin ($\sigma$) and valley ($i$)
indices if necessary, the Hamiltonian can be written in the form
$\displaystyle H$ $\displaystyle=$ $\displaystyle E_{s}|s\rangle\langle
s|+\sum_{i\sigma}E_{i}|i\sigma\rangle\langle i\sigma|+\sum_{\nu
k\sigma}\epsilon_{\nu k}c_{\nu ki\sigma}^{\dagger}c_{\nu ki\sigma}$ (1)
$\displaystyle+\sum_{i\nu k\sigma}(V_{\nu}|i\sigma\rangle\langle s|c_{\nu
ki\sigma}+\mathrm{H.c}.),$
where $c_{\nu ki\sigma}^{\dagger}$ create conduction states at the left
($\nu=L$) or right ($\nu=R$) lead, and $V_{\nu}$ is the hopping between the
lead $\nu$ and both doublets, assumed independent of $k$ for simplicity. The
doublets are split by an energy $\delta=E_{2}-E_{1}$. This corresponds to the
Zeeman splitting when the SU(4) symmetry of the model for $\delta=0$ is broken
by an applied magnetic field.
While there are four spin degenerate bands of mobile electrons, depending on
valley index $i$ or position with respect to the quantum dot (left or right),
for each energy $\epsilon_{Lk}=\epsilon_{Rk^{\prime}}$ for which there are
states at the left and the right, only the linear combination
$V_{Lk}c_{Lki\sigma}^{\dagger}+V_{Rk^{\prime}}c_{Rk^{\prime}i\sigma}^{\dagger}$
hybridizes with the state $|i\sigma\rangle$. Thus, the model is effectively a
two-band model.
We note that in the case of intervalley mixing induced by disorder in C
nanotubes, the hopping elements for small magnetic field depend on the valley
and lead indices.grove In this case, the formalism used in this work is not
applicable. It seems that a full non-equilibrium formalism is needed to treat
the most general case, as that developed for the conductance in Ref. benzene,
and sketched in Ref. desint, . The model is not applicable either for the case
of magnetic impurities in C nanotubes, for which symmetry-breaking geometrical
effects play an important role.baru
## III The formalism
### III.1 Equations for the thermopower
In the limit of vanishing applied bias voltage and temperature difference
between the leads, the electronic part of the transport coefficients can be
evaluated in terms of the total spectral density at the dot
$\rho_{d}(\omega)=\sum\limits_{i\sigma}\rho_{i\sigma}(\omega)$. This is
possible due to the fact that the couplings between the quantum dot and the
right or left leads are proportional.meir If this were not the case (as in
nanotubes affected by disorder grove or for some molecules benzene ) a
different formalism would be needed.benzene ; desint
The Seebeck coefficient is simply given by vel
$S=\frac{-I_{1}(T)}{eTI_{0}(T)},$ (2)
where $e$ is the absolute value of the electronic charge and
$I_{n}=\int\omega^{n}\rho_{d}(\omega)\left(-\frac{\partial
f}{\partial\omega}\right)d\omega,$ (3)
where $f(\omega)$ is the Fermi function, and the zero of energy is taken at
the Fermi energy $\epsilon_{F}=0$
### III.2 Approximations for the spectral density
To calculate the total spectral density $\rho_{d}(\omega)$ that enters Eqs.
(3) we have used mainly the NCA. At $T=0$ have used the SBMFA and also a
combination of Fermi liquid relationships and quantum-dot occupations obtained
using NCA. In the limit $\delta\rightarrow+\infty$, we have also used NRG to
shed light on the virtues and shortcomings of the other approximations.
In the NCA and SBMFA, an auxiliary boson $b$, and four auxiliary fermions
$f_{i\sigma}$ are introduced, so that the localized states are represented as
$|s\rangle=b^{{\dagger}}|0\rangle\text{,
}|i\sigma\rangle=f_{i\sigma}^{{\dagger}}|0\rangle,$ (4)
where $|0\rangle$ is the vacuum. These pseudoparticles should satisfy the
constraint
$b^{{\dagger}}b+\sum_{i\sigma}f_{i\sigma}^{{\dagger}}f_{i\sigma}=1.$ (5)
The NCA solves a system of self-consistent equations to obtain the Green
functions of the auxiliary particles, which is equivalent to sum an infinite
series of diagrams (all those without crossings) in the corresponding
perturbation series in the hopping, and afterwards a projection on the
physical subspace of the constraint is made. The formalism of the NCA for this
problem or similar ones is described in previous papers.bic ; velmo ; lim ;
benzene In particular, the more general case of complex hoppings is treated
in Ref. benzene, . Therefore, we do not give more details here. The
application of NRG in the case of only one doublet
($\delta\rightarrow+\infty$) has also been explained before.vel
In the SBMFA, the boson operators are replaced by a number $b_{0}=\langle
b\rangle$, and the energy is minimized with respect to $b_{0}$ and a Lagrange
multiplier $\lambda$ that enforces Eq. (5). Assuming a constant density of
unperturbed conduction electrons $\rho$ extending from $-D$ to $D$ and filled
to the Fermi level $\epsilon_{F}=0$, a simple generalization of the case of
one doublet,hew ; lady leads to the following change of energy after
introduction of the impurity
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sum\limits_{i}\left[\tilde{\Delta}\ln\left(\frac{\tilde{E}_{i}^{2}+\tilde{\Delta}^{2}}{D^{2}}\right)+2\tilde{E}_{i}\arctan\left(\frac{\tilde{\Delta}}{\tilde{E}_{i}}\right)\right]$
(6) $\displaystyle-\frac{4\tilde{\Delta}}{\pi}+\lambda(b_{0}^{2}-1),$
where $\tilde{E}_{i}=E_{i}+\lambda$, $\tilde{\Delta}=b_{0}^{2}\Delta$, and
$\Delta=\pi\rho(V_{L}^{2}+V_{R}^{2})$ is the total half resonant level width
(adding the contributions from both leads). Above $D\gg\tilde{\Delta}$ was
assumed. Minimizing $E$ with respect to $\lambda$ one obtains an equation that
allows to relate $\tilde{E}_{1}$ (or $\lambda$) with half the quasiparticle
level width $\tilde{\Delta}$ (which is of the order of the Kondo temperature).
After some algebra we obtain
$\displaystyle\tilde{E}_{1}$ $\displaystyle=$
$\displaystyle\left[\frac{\delta^{2}}{4}+(1+\beta^{2})\tilde{\Delta}^{2}\right]^{1/2}+\beta\tilde{\Delta}-\frac{\delta}{2},$
$\displaystyle\beta^{-1}$ $\displaystyle=$
$\displaystyle\tan\left[\frac{\pi}{2}\left(1-\frac{\tilde{\Delta}}{\Delta}\right)\right].$
(7)
Minimizing $E$ with respect to $\tilde{\Delta}$ one obtains
$\frac{1}{\pi}\ln\frac{(\tilde{E}_{1}^{2}+\tilde{\Delta}^{2})((\tilde{E}_{1}+\delta)^{2}+\tilde{\Delta}^{2})}{D^{4}}+\frac{\tilde{E}_{1}-E_{1}}{\Delta}=0,$
(8)
and replacing $\tilde{E}_{1}$ from Eq. (7) in Eq. (8) an equation for the
single unknown $\tilde{\Delta}$ is obtained, which we solve numerically.
The occupation and the spectral density near the Fermi energy for each doublet
are
$\displaystyle n_{i\sigma}$ $\displaystyle=$
$\displaystyle\langle|i\sigma\rangle\langle
i\sigma|\rangle=\frac{1}{\pi}\arctan\left(\frac{\tilde{\Delta}}{\tilde{E}_{i}}\right),$
$\displaystyle\rho_{i\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\frac{b_{0}^{2}\tilde{\Delta}/\pi}{(\omega-\tilde{E}_{i})^{2}+\tilde{\Delta}^{2}}.$
(9)
### III.3 Fermi liquid theory
For an SU(N) model which channel index $j=1$ to $N$, and a simple symmetry
breaking perturbation (like a generalized magnetic field) that does not mix
channels, so that spin and channel are conserved, the Friedel sum rule relates
the spectral density at the Fermi level $\epsilon_{F}$ with the number of
displaced electrons for each channel.yoshi2 The latter coincides with the
occupation for a constant unperturbed density of conduction electrons with a
wide band $D\gg\tilde{\Delta}$ as we assume, where $\tilde{\Delta}$ (of the
order of the Kondo temperature $T_{K}$) is the resonant level width of the
quasiparticles [as in the SBMFA, Eq. (9)]
Then one has fcm ; yoshi2
$\rho_{j}(\epsilon_{F})=\frac{1}{\pi\Delta}\sin^{2}(\pi n_{j}).$ (10)
In addition, the derivatives at $\epsilon_{F}$ are also known (using for
example renormalized perturbation theory) scali ; yoshi ; kirchner ,
$\frac{\partial\rho_{j}(\omega)}{\partial\omega}|_{\epsilon_{F}}=\sin(2\pi
n_{j})\frac{\rho_{j}(\epsilon_{F})}{\tilde{\Delta}}.$ (11)
From Eqs. (9), it is apparent that the SBMFA (in which we have chosen
$\epsilon_{F}=0$) satisfies these relationships.
Using Eqs. (10) and (11) and a Sommerfeld expansion in Eqs (3), the Seebeck
coefficient Eq. (2) for $T\rightarrow 0$ becomes
$S=-\frac{2\pi^{2}k_{B}T}{3\tilde{\Delta}}\frac{\sum_{i\sigma}\sin^{3}(\pi
n_{i\sigma})\cos(\pi n_{i\sigma})}{\sum_{i\sigma}\sin^{2}(\pi n_{i\sigma})}.$
(12)
## IV Numerical results
Without loss of generality, we take $\epsilon_{F}=E_{s}=0$, where
$\epsilon_{F}$ is the Fermi level of the leads. For the numerical
calculations, we assume a constant density of states per spin of the leads
$\rho=1/(2D)$ between $-D$ and $D$. We take the unit of energy as the total
level width of both doublets $\Gamma=2\Delta=2\pi\rho(V_{L}^{2}+V_{R}^{2})$.
The energy of both doublets is denoted as $E_{1}=E_{d}$, $E_{2}=E_{d}+\delta$.
### IV.1 The limit of one doublet
For $\delta\rightarrow+\infty$, the model is equivalent to the limit of
infinite on-site repulsion $U$ of the simplest Anderson model, studied in
detail before using the NRG.vel Here we compare results obtained with NRG
with those of NCA and SBMFA to see the limitations of the different methods,
which will be useful for the analysis of the general case.
Figure 1: (Color online) Thermopower as a function of temperature for
$E_{d}=-4\Gamma$ and $\delta\rightarrow+\infty$. Squares: NRG, dots: NCA,
dashed line: Lorentzian spectral density (see text).
In Fig. 1 we display the thermopower as a function of temperature in the Kondo
regime. There is a dip near the Kondo temperature $T_{K}$ due to the fact that
the Kondo peak in the spectral density is slightly above the Fermi level
$\epsilon_{F}$, and a peak near the charge-transfer energy
$\epsilon_{F}-E_{d}$ due to the corresponding peak below $\epsilon_{F}$. We
mean “near” as an estimation of the order of magnitude. For example, the
maximum of the curve within NCA is at $T=1.10\Gamma$, while
$\epsilon_{F}-E_{d}=4\Gamma$ in the figure.
The spectral density $\rho_{d}(\omega)$ of the Anderson model for only one
doublet ($\delta\rightarrow+\infty$) is well known. In particular, using a
local-moment approach, Logan and coworkers have shown that in the Kondo
regime, the charge-transfer peaks (which carry most of the spectral weight in
this regime) have a Lorentzian shape of a width two times larger than that of
the non-interacting case.logan We have included in Fig. 1 the thermopower
that results replacing $\rho_{d}(\omega)$ in Eqs. (2) and (3), by a simple
Lorentzian of total width $2\Gamma$ centered at $E_{d}$. The agreement with
the NCA result at high temperatures is remarkable. Instead, the charge-
transfer peak in the thermopower within NRG has a small shift to higher
temperatures and an intensity about 10 % lower. We have verified that this
also happens for other values of $E_{d}$. This is probably related with
resolution problems of the NRG at large energies, as discussed below.
At low temperatures, the NRG results are more reliable than those of NCA and
SBMFA. We have extracted the linear term in $T$ in the Seebeck coefficient,
plotting $S/T$ vs $T$ for small $T$ and looking at the extrapolation to $T=0$.
The results are shown in Table I. They have an uncertainty of the order of 10
%. We have also calculated the conductance $G(T)$ (not shown) and the total
occupancy $n$ adding both spins. Using the results for $G(T)$, we estimated
the Kondo temperature from the requirement that $G(T_{K}^{G})=G(0)/2$. Using
$n_{1\uparrow}=n_{1\downarrow}=n/2$, $n_{2\uparrow}=n_{2\downarrow}=0$, and
the Fermi liquid expression Eq. (12), $S/T$ is calculated in an independent
way. Taking into account the exponential variation of $T_{K}$ with $E_{d}$ one
can see a semiquantitative agreement between both results in Table 1. The
remaining quantitative discrepancy can be ascribed to the difference between
$T_{K}^{G}$ and $\tilde{\Delta}$ as a measure of the Kondo temperature. In
fact from renormalized perturbation theory one obtains
$T_{K}^{G}/\tilde{\Delta}=0.746$ in the extreme Kondo limit while this ratio
increases beyond 1 when valence fluctuations are allowed,rpt and the Wilson
ratio decreases.ogu ; rpt ; sela ; scali
Table 1: Kondo temperature from $G(T)$ ($T_{K}^{G}$), total occupation of the lowest doublet ($n$), linear coefficient of $S(T)$ from NRG ($ST_{K}^{G}/T$) and from a Fermi liquid theory ($S\tilde{\Delta}/T$) for different values of the charge-transfer energy $\epsilon_{F}-E_{d}$. $E_{d}/\Gamma$ | $T_{K}^{G}/\Gamma$ | $n$ | $ST_{K}^{G}/T$ | $S\tilde{\Delta}/T$
---|---|---|---|---
-1 | $2.22\times 10^{-2}$ | 0.732 | -3.48 | -2.450
-2 | $6.15\times 10^{-4}$ | 0.897 | -0.93 | -1.047
-3 | $2.22\times 10^{-5}$ | 0.941 | -0.51 | -0.611
-4 | $8.38\times 10^{-7}$ | 0.958 | -0.38 | -0.437
-5 | $3.16\times 10^{-8}$ | 0.967 | -0.27 | -0.339
-6 | $1.18\times 10^{-9}$ | 0.973 | -0.22 | -0.317
The NCA has the drawback that it does not fulfill Fermi liquid relationships.
For example, the spectral density at the Fermi energy $\rho_{d}(0)$ at
temperatures well below $T_{K}$ differs by about 10 or 20% in the Kondo regime
from the value predicted by the Friedel sum rule.fcm A detailed comparison of
$\rho_{d}(\omega)$ between NRG and NCA in the Kondo regime is given by Fig. 10
of Ref. compa, . One can see that in addition to the larger value of
$\rho_{d}(0)$, the spectral density is more asymmetric for the NCA. This is
probably the main reason of the factor near five between the magnitude of the
dip in $S(T)$ near $T_{K}$ calculated with NCA with respect to the NRG result.
Part of the discrepancy is probably also due to lack of resolution at finite
energies within the NRG. For example in models of two quantum dots, the split
Kondo peaks in the spectral density are considerably broadened, losing
intensity (Fig. 11 of Ref. vau, ). Another example is the plateau in the
conductance $G(T)$ observed at intermediate temperatures $T$ in transport
through C60 molecules for gate voltages for which triplet states are
important,roch ; serge which was missed in early NRG studies, but captured by
the NCA.st1 ; st2 More recent NRG calculations using tricks to improve the
resolution,freyn have confirmed this plateau.serge
In any case, the one-level SU(2) limit is the worst case for the NCA, because
the real spectral density tends to be symmetric, while the NCA improves with
increasing N for SU(N) models.bic
The absolute value of $S$ is exaggerated within NCA for $T\rightarrow 0$.
However, as shown in Fig. 2, there is a good agreement between the occupancies
calculated with NRG and NCA. This suggests that using the occupancies
calculated with NCA and Eq. (12) a semiquantitatively correct result for the
linear part of $S(T)$ as $T\rightarrow 0$ can be obtained. Instead, while the
SBMFA satisfies Fermi liquid relationships, the occupancies are not accurate
for $\delta\rightarrow+\infty$. Even perturbation theory up to second order in
$V_{\nu}$ neglecting spin flip leads to a better result for $n$, although (in
contrast to SBMFA) this approach is unable to predict the right magnitude of
$T_{K}$.
Figure 2: (Color online) Total occupancy as a function of $E_{d}$ for
$\delta\rightarrow+\infty$ and different approximations.
### IV.2 Temperature dependence in the general case
Figure 3: (Color online) Thermopower as a function of temperature for
$E_{d}=-4\Gamma$ and different values of the level splitting.
In Fig. 3 we represent the NCA results for the Seebeck coefficient $S(T)$ as a
function of temperature for different values of $\delta$, which represents the
splitting due to a Zeeman term for example. The spectral density
$\rho_{d}(\omega)$ for finite $\delta$ has been studied before.lim ; fcm At
$\delta=0$, $\rho_{d}(\omega)$ has a peak just above the Fermi energy
$\epsilon_{F}$ with width of the order of $2T^{4}_{K}$, where $T^{4}_{K}$ is
the Kondo temperature of the SU(4) limit $\delta=0$. For the parameters of
Fig. 3, $T^{4}_{K}$ is of the order of $0.01\Gamma$.fcm As $\delta$ increases
above $T^{4}_{K}$, the Kondo peak splits and another peak at an energy of the
order of $\delta$ above $\epsilon_{F}$ appears. This peak above $\epsilon_{F}$
originates a dip in $S(T)$ for $T\sim\delta$. Note that peaks in
$\rho_{d}(\omega)$ above $\epsilon_{F}$ lead to a positive contribution to
$I_{1}$ [see Eq. (3)], which in turn lead to a negative contribution to $S(T)$
at the corresponding temperature [see Eq. (2)] Most of the spectral weight of
the spectral density lies in the charge-transfer peak at energy $E_{d}$, which
lies below the Fermi energy. Thus, when the temperature reaches values of the
order of the charge-transfer energy $\epsilon_{F}-E_{d}$, the thermopower
becomes positive [$I_{0}>0$, $I_{1}<0$ in Eq. (2)].
As discussed above, for $\delta\rightarrow+\infty$ (the case of one SU(2)
doublet), $S(T)$ shows one dip at $T_{K}\sim 10^{-6}\Gamma$ and a peak near
the charge-transfer energy. For $\delta=0$, these qualitative features remain,
but the Kondo temperature is four orders of magnitude larger, and the dip near
$T^{4}_{K}$ is more pronounced, due to the larger asymmetry of the peak in the
spectral density with respect to $\epsilon_{F}$,fcm leading to a larger
integral $I_{1}$ [see Eqs. (2) and (3)]. Since $T_{K}$ changes with $\delta$,
as discussed below, the dip at $T_{K}$ displaces towards larger temperatures
as $\delta$ decreases. Instead, the charge-transfer peak remains approximately
at the same temperature and decreases in magnitude due mainly to additional
broadening of the corresponding peak in the spectral density $\rho(\omega)$ as
the SU(4) limit is approached, and also due to some transfer of the spectral
weight to the Kondo peak. While this transfer is not large, the Kondo peak has
a larger weight in the integrals $I_{n}$ due to the factor of the derivative
of the Fermi function [see Eqs. (2) and (3)].
For $\delta>T^{4}_{K}$, for example $\delta=0.02\Gamma$ in Fig. 3, an
additional dip develops due to the peak near $\delta$ in $\rho_{d}(\omega)$.
As $\delta$ increases further, the dip moves to higher temperatures, as it is
apparent for $\delta=0.05$ and 0.1 in the figure. The relative minimum of
$S(T)$ near the dip lies at temperatures of the order of $\delta$ but smaller,
probably because of the large intensity of the peak near the charge-transfer
energy, which pushes this minimum to lower temperatures. For $\delta=1$ this
dip turns to a shoulder at the left of the charge-transfer peak and for larger
$\delta$, the dip and the peak cross, interchanging the order of temperatures
for which they appear.
Figure 4: (Color online) Thermopower as a function of $T/T_{K}$ for
$E_{d}=-4\Gamma$ and different values of the level splitting.
In Fig. 4 we show the thermopower as a function of $T/T_{K}(\delta)$, where
$T_{K}(\delta)$ is the Kondo temperature for each value of $\delta$. Here we
determined $T_{K}(\delta)$ from the requirement that the conductance
$G(T_{K}(\delta))=e^{2}/h$ for symmetric leads ($V_{L}=V_{R}$). To a good
degree of accuracy, it is given by the expression
$T_{K}(\delta)=\left\\{(D+\delta)D\exp\left[\frac{\pi
E_{1}}{2\Delta}\right]+\frac{\delta^{2}}{4}\right\\}^{1/2}-\frac{\delta}{2},$
(13)
obtained from a simple variational wave function.desint ; fcm With increasing
$\delta$, $T_{K}$ stays roughly constant until $\delta>T^{4}_{K}=T_{K}(0)$ and
then it decreases strongly. From Fig. 4 it is apparent that the dip at smaller
temperatures remains at $T\sim T_{K}(\delta)$ for all values of $\delta$. One
can also see that the magnitude of the dip and (the absolute value of the
thermopower for $T\sim T_{K}$) increases as $\delta$ decreases, being maximum
at the SU(4) point $\delta=0$. Note that while the magnitude of this dip is
exaggerated by the NCA for $\delta\rightarrow+\infty$, as explained in the
previous section, we believe that this is not the case for $\delta=0$, because
the spectral density is naturally asymmetric in this case, and since the NCA
is a 1/N expansion, its accuracy improves with N in SU(N) models.bic
Furthermore NCA calculations of the thermopower of Ce compounds, in which
orbital degeneracy is important, compared well with experiment.bic ; velmo
Figure 5: (Color online) Conductance as a function of $T/T_{K}$ for
$E_{d}=-4\Gamma$ and different values of the level splitting.
In Fig. 5 we show the conductance $G(T)$ calculated with the NCA for different
values of $\delta$. $G(0)$ is slightly larger for $\delta\rightarrow+\infty$
and lower for $\delta=0$ with respect to the correct values due to the failure
of the NCA spectral density to satisfy Friedel sum rule.fcm In spite of this
shortcoming, the overall shape and temperature dependence of $G$ in these
limits agree with those obtained using NRG.vel ; ander
The point that we want to stress here is that although the finite energy
features for $T\sim\delta$ and $\epsilon_{F}-E_{d}$ are present not only in
$S(T)$ but also in $G(T)$, they are much weaker in $G(T)$. Thus, the Seebeck
coefficient might be the appropriate tool to study the electronic structure of
the system at intermediate energies.
### IV.3 Dependence on splitting for $T\rightarrow 0$
Figure 6: (Color online) Coefficient of the linear dependence of the
thermopower as a function of the level splitting, for $E_{d}=-4\Gamma$.
In this subsection, we present results for the term linear in temperature $T$
of the thermopower, as $T\rightarrow 0$, using two techniques: SBMFA and the
Fermi liquid expression Eq. (12) with occupations calculated with the NCA. In
Fig. 6 we represent $-S/T$ as a function of $\delta$. In the SU(4) limit
$\delta=0$, both techniques agree and indicate a very large absolute value of
$S\tilde{\Delta}/T$. In fact, in the extreme Kondo limit of an SU(N) model,
one has $n_{j}=1/$N and from Eqs. (2), (3), (10), (11) and a Sommerfeld
expansion one obtains
$S=-\frac{\pi^{2}T}{3\tilde{\Delta}}\sin(2\pi/N),$ (14)
and $|S\tilde{\Delta}/T|$ reaches its maximum value $\pi^{2}/3=3.29$ for N=4.
The value in Fig. 6 is slightly smaller due to some degree of intermediate
valence.
As $\delta$ increases, there is little variation until the Kondo temperature
of the SU(4) limit $T_{K}^{4}$ ($\sim 0.01\Gamma$ in the figure) is reached.
For larger $\delta$, $S\tilde{\Delta}/T$ falls due to the change of
occupations (see Fig. 7): the lower doublet becomes more populated, while the
occupation of the higher one decreases, keeping a total occupation slightly
below 1. While the trend of the curve is the same for both approaches, NCA and
SBMFA, the absolute value of the thermopower decreases too much within the
SBMFA for $\delta>10T_{K}^{4}$. This is due to the fact discussed in Section
IV.1, that the occupation predicted by the SBMFA of the lower lying doublet is
too large for $\delta\rightarrow+\infty$. While this shortcoming affects the
conductance or thermodynamic properties in a few %, the effect of this
increase is more dramatic in the thermopower.
Instead, since the NCA occupations agree with NRG in the limit
$\delta\rightarrow+\infty$, the approach that combines NCA occupations and
Fermi liquid relationships is reliable in this limit. The agreement with SBMFA
and general expectations for the SU(4) model in the Kondo limit, indicates
that this approach is also reliable for $\delta=0$.
Figure 7: (Color online) Occupation of the two doublets
$n_{i}=\sum_{\sigma}n_{i\sigma}$ as a function of the level splitting, for
$E_{d}=-4\Gamma$.
## V Summary and discussion
We have calculated the thermopower of a model that describes electronic
transport through quantum dots in C nanotubes, in which the disorder does not
play an essential role, and Si fin-type field effect transistors in the Kondo
regime. This regime can be easily controlled by applying a gate voltage that
modifies the energy of the localized levels $E_{d}$. Without disorder and
applied magnetic field, the model has SU(4) symmetry, as explained in Section
I. In this case, the thermopower as a function of temperature $S(T)$ has a dip
(negative $S$) at temperatures near the Kondo temperature $T_{K}$, and a peak
(positive $S$) at temperatures near the charge-transfer energy. For SU(N)
symmetry, varying N with width of the Kondo resonance $\sim 2T_{K}$ fixed, N=4
is the most favorable case to have a large thermopower at temperatures lower
that the Kondo temperature.
When the SU(4) symmetry is broken, leading to an energy splitting $\delta$
between two SU(2) doublets, by a simple symmetry breaking field (like a
magnetic field), a new dip appears in $S(T)$ at temperatures of the order of
$\delta$. While this dip the peak for positive $S$ at the charge-transfer
energy are clearly displayed in $S(T)$, the corresponding features in the
conductance $G(T)$ at equilibrium are very weak. This suggest that the study
of the thermopower might be a more useful tool to study the electronic
structure at finite energies. Another alternative is to study the conductance
out of equilibrium from which peaks at a bias voltage of the order of
$\pm\delta/e$ are expected.tetta ; desint
As $\delta$ increases, the characteristic energy scale $T_{K}(\delta)$, which
determines among several physical scales, the width of the Kondo resonance and
the temperature at which $G(T)$ falls to half its value at $T=0$, decreases
following a simple expression Eq. (13). For $T\ll T_{K}(\delta)$, $S$ is
linear in $T$ with a coefficient that stays approximately constant for
$\delta<T_{K}(0)$. As $\delta$ increases further, $ST_{K}(\delta)/T$ for
$T\rightarrow 0$ decreases by an order of magnitude but $T_{K}(\delta)$
decreases by nearly three orders of magnitude, so that the linear coefficient
increases. For this calculation, an approach that combines Fermi liquid
relationships with occupation numbers calculated with NCA gives more reliable
results that the SBMFA.
Because of the nature of the underlying SU(4) symmetry, the couplings of the
four states involved to the conducting leads are proportional, and independent
of the state ($V_{L}$ to the left and $V_{R}$ to the right), This fact which
simplifies the calculations is no more true for weak magnetic field in C
nanotubes with disorder,grove and in general in multilevel quantum dots.izum2
In these cases, as well as in situations with total or partial destructive
interference, such as transport through molecules,desint ; benzene ; mole the
present formalism does not apply and a full non-equilibrium formalism seems
necessary.benzene In addition here, the total occupation of the dot is below
1, in contrast to models with multilevel systems in which even occupation is
allowed.serge
## Acknowledgments
We thank CONICET from Argentina for financial support. This work was partially
supported by PIP 11220080101821 of CONICET and PICT R1776 of the ANPCyT,
Argentina. P. R. is sponsored by Escuela de Ciencia y Tecnología, Universidad
Nacional de San Martín.
## References
* (1) T. M. Tritt, Annu. Rev. Mater. Res. 41, 433 (2011).
* (2) B. C. Sales, D. Mandrus, and R. K. Williams, Science 272, 1325 (1996).
* (3) I. Terasaki, Y. Sasago, and K. Uchinokura, Phys. Rev. B 56, R12685 (1997).
* (4) A. Bentien, S. Johnsen, G. K. H. Madsen, B. B. Iversen, and F. Steglich, EPL 80, 17008 (2007).
* (5) M. G. Kanatzidis, Chem. Mater. 22, 648 (2010).
* (6) R. F. Klie, Q. Qiao, T. Paulauskas, A. Gulec, A. Rebola, S. Öğüt, M. P. Prange, J. C. Idrobo, S. T. Pantelides, S. Kolesnik, B. Dabrowski, M. Ozdemir, C. Boyraz, D. Mazumdar, and A. Gupta, Phys. Rev. Lett. 108, 196601 (2012).
* (7) G. D. Mahan, and J. O. Sofo, Proc. Natl. Acad. Sci. USA, 93, 7436 (1996).
* (8) N. E. Bickers, D. L. Cox, and J. W. Wilkins, Phys. Rev. B 36, 2036 (1987).
* (9) V. Zlatić and R. Monnier, Phys. Rev. B 71, 165109 (2005).
* (10) T. A. Costi, A. C. Hewson, and V. Zlatić, J. Phys.: Condens. Matter 6, 2519 (1994); T. A. Costi and A. C. Hewson, ibid. 5, L361 (1993).
* (11) T. A. Costi and V. Zlatić, Phys. Rev. B 81, 235127 (2010).
* (12) T. C. Harman, P. J. Taylor, M. P. Walsh, and B. E. Laforge, Science 297, 2229 (2002).
* (13) R. Scheibner, H. Buhmann, D. Reuter, M. N. Kiselev, and L. W. Molenkamp, Phys. Rev. Lett. 95, 176602 (2005).
* (14) D. Boese and R. Fazio, Europhys. Lett. 56, 576 (2001).
* (15) B. Dong and X. L. Lei, J. Phys.: Condens. Matter 14, 11747 (2002).
* (16) T.-S. Kim and S. Hershfield, Phys. Rev. Lett. 88, 136601 (2002).
* (17) J. Hone, I. Ellwood, M. Muno, A. Mizel, M. L. Cohen, A. Zettl, A. G. Rinzler, and R. E. Smalley, Phys. Rev. Lett. 80, 1042 (1998).
* (18) C. H.Yu; L. Shi, Z. Yao, D. Y. Li and A. Majumdar, A. Nano Lett. 5, 1842 (2005).
* (19) P. Reddy, S-Y. Jang, R. A. Segalman, and A. Majumdar, Science 315, 1568 (2007).
* (20) A. V. Sologubenko, E. Felder, K. Giannò, H. R. Ott, A. Vietkine, and A. Revcolevschi, Phys. Rev. B 62, R6108 (2000).
* (21) A. V. Sologubenko, H. R. Ott, G. Dhalenne, and A. Revcolevschi, Europhys. Lett. 62, 540 (2003).
* (22) A. V. Rozhkov and A. L. Chernyshev, Phys. Rev. Lett. 94, 087201 (2005).
* (23) L. Arrachea, G. S. Lozano, and A. A. Aligia, Phys. Rev. B 80, 014425 (2009).
* (24) J. S. Lim, M.-S. Choi, M. Y. Choi, R. López, and R. Aguado, Phys. Rev. B 74, 205119 (2006).
* (25) F. B. Anders, D. E. Logan, M. R. Galpin, and G. Finkelstein, Phys. Rev. Lett. 100, 086809 (2008).
* (26) S. Lipinski and D. Krychowski, Phys. Rev. B 81, 115327 (2010).
* (27) C. A. Büsser, E. Vernek, P. Orellana, G. A. Lara, E. H. Kim, A. E. Feiguin, E. V. Anda, and G. B. Martins, Phys. Rev. B 83, 125404 (2011).
* (28) L. Tosi, P. Roura-Bas, and A. A. Aligia, Physica B 407, 3259 (2012).
* (29) K. Grove-Rasmussen, S. Grap, J. Paaske, K. Flensberg, S. Andergassen, V. Meden, H. I. Jorgensen, K. Muraki, and T. Fujisawa, Phys. Rev. Lett. 108, 176802 (2012).
* (30) G. C. Tettamanzi, J. Verduijn, G. P. Lansbergen, M. Blaauboer, M. J. Calderón, R. Aguado, and S. Rogge, Phys. Rev. Lett. 108, 046803 (2012).
* (31) Y. He and G. Galli, Phys. Rev. Lett. 108, 215901 (2012).
* (32) P. Coleman, Phys. Rev. B 29, 3035 (1984).
* (33) A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge, England, 1993 .
* (34) A. A. Aligia and L. A. Salguero, Phys. Rev. B 70, 075307 (2004); Phys. Rev. B 71, 169903(E).
* (35) A. M. Lobos and A. A. Aligia, Phys. Rev. Lett. 100, 016803 (2008); Physica B 404, 3306 (2009).
* (36) T. A. Costi, J. Kroha and P. Wölfle, Phys. Rev. B 53, 1850 (1996).
* (37) P. Roura-Bas, Phys. Rev. B 81, 155327 (2010).
* (38) A. Oguri, J. Phys. Soc. Jpn. 74, 110 (2005).
* (39) J. Rincón, A. A. Aligia, and K. Hallberg, Phys. Rev. B 79, 121301(R) (2009); Phys. Rev. B 80, 079902(E) (2009); Phys. Rev. B 81, 039901(E) (2010).
* (40) E. Sela and J. Malecki, Phys. Rev. B 80, 233103 (2009).
* (41) A. A. Aligia, J. Phys. Condens. Matter 24, 015306 (2012); references therein.
* (42) R. Bulla, A.C. Hewson, and Th. Pruschke, J. Phys. Cond. Matt. 10, 8365 (1998).
* (43) L. Tosi, P. Roura-Bas, and A. A. Aligia, J. Phys. Condens. Matter 24, 365301 (2012).
* (44) P. Roura-Bas, L. Tosi, A. A. Aligia, and K. Hallberg, Phys. Rev. B 84, 073406 (2011).
* (45) P. P. Baruselli, A. Smogunov, M. Fabrizio, and E. Tosatti, Phys. Rev. Lett. 108, 206807 (2012).
* (46) Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).
* (47) A Yoshimori and A Zawadowski, J. Phys. C 15, 5241 (1982).
* (48) A. Yoshimori, Prog. Theor. Phys. 55, 67 (1976).
* (49) S. Kirchner, J. Kroha, and P. Wölfle, Phys. Rev. B 70, 165102 (2004).
* (50) D. E. Logan, M. P. Eastwood and M. A. Tusch, J. Phys.: Condens. Matter 10, 2673 (1998).
* (51) L. Vaugier, A.A. Aligia and A.M. Lobos, Phys. Rev. B 76, 165112 (2007).
* (52) N. Roch, S. Florens, V. Bouchiat, W. Wernsdorfer, and F. Balestro, Nature 453, 633 (2008).
* (53) S. Florens, A, Freyn, N. Roch, W. Wernsdorfer, F. Balestro, P. Roura-Bas and A. A. Aligia, J. Phys. Condens. Matter 23, 243202 (2011); references therein.
* (54) P. Roura-Bas and A. A. Aligia, Phys. Rev. B 80, 035308 (2009).
* (55) P. Roura-Bas and A. A. Aligia, J. Phys. Cond. Matt. 22, 025602 (2010).
* (56) A. Freyn and S. Florens, Phys. Rev. Lett. 107, 017201 (2011).
* (57) W. Izumida, O. Sakai, and Y. Shimizu, J. Phys. Soc. Jpn. 67, 2444 (1998).
* (58) J. Rincón, K. Hallberg, A. A. Aligia, and S. Ramasesha, Phys. Rev. Lett. 103, 266807 (2009); references therin.
|
arxiv-papers
| 2012-08-28T15:46:26 |
2024-09-04T02:49:34.610900
|
{
"license": "Public Domain",
"authors": "P. Roura-Bas, L. Tosi, A. A. Aligia and P. S. Cornaglia",
"submitter": "Pablo Roura-Bas Dr.",
"url": "https://arxiv.org/abs/1208.5692"
}
|
1208.5776
|
††thanks: [email protected]
# Hot electron bolometer heterodyne receiver with a 4.7-THz quantum cascade
laser as a local oscillator
J. L. Kloosterman [email protected] Department of Electrical and
Computer Engineering, 1230 E. Speedway Blvd., University of Arizona, Tucson,
AZ 85721 USA D. J. Hayton SRON Netherlands Institute for Space Research,
Groningen/Utrecht, Netherlands Y. Ren Kavli Institute of NanoScience, Delft
University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Purple
Mountain Observatory (PMO), Chinese Academy of Sciences, 2 West Beijing Road,
Nanjing, JiangSu 210008, China, and Graduate School, Chinese Academy of
Sciences, 19A Yu Quan Road, Beijing 100049, China T. Y. Kao Department of
Electrical Engineering and Computer Science, Massachusetts Institute of
Technology (MIT), Cambridge, Massachusetts 02139, USA J. N. Hovenier Kavli
Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628
CJ Delft, The Netherlands J. R. Gao SRON Netherlands Institute for Space
Research, Groningen/Utrecht, Netherlands Kavli Institute of NanoScience,
Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
T. M. Klapwijk Kavli Institute of NanoScience, Delft University of
Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Q. Hu Department of
Electrical Engineering and Computer Science, Massachusetts Institute of
Technology (MIT), Cambridge, Massachusetts 02139, USA C. K. Walker Steward
Observatory, 933 N Cherry Ave., Rm N204, University of Arizona, Tucson, AZ
85721 USA J. L. Reno Sandia National Laboratories, Albuquerque, NM
87185-0601, USA
###### Abstract
We report on a heterodyne receiver designed to observe the astrophysically
important neutral atomic oxygen [OI] line at 4.7448 THz. The local oscillator
is a third-order distributed feedback Quantum Cascade Laser operating in
continuous wave mode at 4.741 THz. A quasi-optical, superconducting NbN hot
electron bolometer is used as the mixer. We recorded a double sideband
receiver noise temperature (T${}^{DSB}_{rec}$) of 815 K, which is $\sim$7
times the quantum noise limit ($\frac{\rm{h}\nu}{2\rm{k_{B}}}$) and an Allan
variance time of 15 s at an effective noise fluctuation bandwidth of 18 MHz.
Heterodyne performance was confirmed by measuring a methanol line spectrum.
Astronomers have long been interested in the fine structure line of [OI] at
4.7448 THz. [OI] probes the star formation process and is the most important
cooling line in the interstellar medium (ISM)(Tielens and Hollenbach, 1985)
for gas clouds with densities of n $>10^{4}$ cm-3. Large scale surveys with
extremely high spectral resolution and sensitivity are required to disentangle
large scale kinematics and energetics within these clouds. Such high spectral
resolution observations require the development of super-THz ($>$ 3 THz)
heterodyne receivers. Due to strong water absorption in the atmosphere, it is
not possible to observe the [OI] line from ground-based telescopes. Therefore,
an astronomical [OI] receiver requires a compact local oscillator (LO) that
can be readily integrated into space-based or suborbital observatories.
There are several candidate THz technologies for use in LO systems. These
technologies include Schottky diode based multiplier chains, gas lasers, and
quantum cascade lasers (QCLs). QCL’s are currently the only technological
approach that leads to devices small and powerful enough (Köhler _et al._ ,
2002) to be used in a variety of space-based, super-THz applications.
Furthermore, THz QCLs operating in CW mode have yielded line widths of
$\sim$100 Hz (Vitiello _et al._ , 2012), excellent power stability (Gao _et
al._ , 2005), and output powers over 100 mW (Williams _et al._ , 2005),
making them well-suited for high resolution spectroscopy. Much progress has
been made toward overcoming the challenges associated with using a QCL as an
LO. Frequency stabilization without the need of another THz source has been
achieved using an absorption line within a methanol gas cell (Richter _et
al._ , 2010; Ren _et al._ , 2012). Diverging far-field beam patterns and mode
selectivity have been improved by using a 3rd-order distributed feedback (DFB)
grating (Amanti _et al._ , 2010; Kao, Hu, and Reno, 2012).
In this letter we report on a full demonstration of a heterodyne receiver
using a THz quantum cascade laser as local oscillator. In contrast to previous
publications (Gao _et al._ , 2005; Ren _et al._ , 2011), significant
progress has been made on DFB QCLs by changing the tapered corrugations. At
4.7 THz, this QCL is the highest frequency ever reported using the 3rd-order
DFB structure. Furthermore, by introducing an array of 21 DFB lasers with a
linear frequency coverage and a 7.5 GHz frequency spacing, we can target a
specific LO frequency. An unprecedented high sensitivity for a heterodyne
receiver was measured at 4.741 THz along with a 15 s Allan variance time, the
first time such stability has been reported with this combination. Lastly, a
theoretical model for methanol molecular lines has been verified at 4.7 THz,
which was not possible until now.
Figure 1: (Color online) CW spectra of a 4.7 THz QCL (at 77 K) measured at
different bias voltages. (a) Frequencies of an array with 11 devices in pulse
mode (at 10 K) demonstrating the frequency selectivity of a 21-element third-
order DFB array. (b) Beam pattern of the QCL. (c) Scanning Electron Microscope
(SEM) image for a taper-horn third-order DFB laser. The contact pad connects
to the side of the last period of the DFB grating.
The THz QCL active region is based on a four-well resonant-phonon depopulation
design in a metal–-metal waveguide. Cavity structure with a lateral corrugated
third-order DFB grating similar to those demonstrated in Amanti et al. (Amanti
_et al._ , 2010) were used to provide frequency-selectivity and also to the
improve far-field beam pattern. We improve upon this design by changing the
shape of the corrugated gratings from a traditional square tooth to a tapered
shape as shown in Fig. 1c. According to an electromagnetic finite-element
(FEM) simulation, the taper-horn shape increases the radiation loss from the
unwanted upper band-edge mode while marginally reducing the radiation loss for
the desired third-order DFB mode, hence improving mode selectivity in order to
ensure a robust single-mode operation. Effectively, this approach leverages a
trade-off between the output power efficiency and mode discrimination. With
this improved frequency selectivity, we realize a linear frequency coverage of
440 GHz, from 4.61 to 5.05 THz as shown in Figure 1a with robust single-mode
operation on the same gain medium. These grating periods range from 28.5 to 25
$\mu$m, which cover $\sim$80% of the gain spectrum.
The third-order DFB QCL arrays were fabricated using standard metal–-metal
waveguide fabrication techniques, contact lithography, and inductively-
coupled–-reactive ion etching (ICP-RIE) to define the laser mesas with the
Ti/Au top contact acting as the self-aligned etch mask. A 300 nm SiO2
electrical insulation layer was used for the isolation of the contact pads.
Each array consists of 21 DFB lasers arranged in a similar manner as in Lee et
al. (Wei Min Lee _et al._ , 2012) with a $\sim$7.5 GHz frequency spacing. The
position where the contact pads connect to the DFB laser was chosen to
minimize unwanted perturbation to the grating boundary condition.
The device used in the heterodyne experiment has a width of 17 $\mu$m and 27
grating periods with an overall device length of $\sim$0.76 mm. The measured
CW output power is 0.25 mW with $\sim$0.7 W DC of power dissipation at 10 K
and a main beam divergence of $\sim$12∘, as shown in Figure 1b. CW lasing at
4.7471 THz is realized at 77 K with a 12.4 V bias voltage, which is within 3
GHz of the target [OI] line (see Fig. 1a). For the heterodyne measurement
described below, the device is operated at $\sim$10 K.
HEBs are the preferred mixer for frequencies above 1.5 THz and have been used
up to 1.9 THz in the Herschel Space Telescope (Cherednichenko _et al._ ,
2008) and the Stratospheric THz Observatory (Walker _et al._ , 2010), and up
to 2.5 THz in the Stratospheric Observatory For Infrared Astronomy (Pütz _et
al._ , 2012; Heyminck _et al._ , 2012). We use a NbN HEB mixer, which was
developed by SRON and TU Delft. Nb contact pads connect a 2 $\times$ 0.2
$\mu$m2 superconducting bridge to a tight winding spiral antenna, which is
suitable for super-THz frequencies. With the application of both electrical
bias and optical pumping from an LO source, a temperature distribution of hot
electrons is maintained producing a resistive state in the center of the
bridge. Incoming signals modulate temperature causing a modulation in the
resistance to create heterodyne mixing (Barends _et al._ , 2005).
Figure 2: (Color online) Laboratory setup for heterodyne QCL-HEB measurements.
The test setup for measuring T${}^{DSB}_{rec}$ is shown in Fig. 2. The QCL was
mounted in a liquid helium cryostat and operated at $\sim$10 K. The beam was
focused by an ultra-high molecular weight polyethylene (UHMW-PE) lens, which
is $\sim$80% transmissive at 4.7-THz. A voice coil attenuator together with a
proportional - integral - derivative (PID) feedback loop is used to stabilize
the power output of the QCL during T${}^{DSB}_{rec}$ and Allan variance
measurements (Hayton _et al._ , 2012) (where noted). In this case the HEB DC
current is used as a power reference signal. The beam entered a blackbody
hot/cold vacuum setup attached to the HEB cryostat via a second UHMW-PE window
and then was reflected by a 3 $\mu$m mylar beam splitter. This cryostat was
cooled to 4.2 K. The HEB was mounted to the back side of a 10 mm Si lens with
an anti-reflection coating designed for 4.25 THz. The first stage low noise
amplifier (LNA) was attached to the cold plate and operated at 4.2 K. The LNA
noise temperature was 3 K with a gain of 42 dB measured at 15 K. Outside the
dewar, room temperature amplifiers and an 80 MHz wide band pass filter (BPF)
centered at 1.5 GHz were used to further condition the intermediate frequency
(IF) signal before the total power was read using an Agilent E4418B power
meter.
The total optical losses in the setup are $\sim$20 dB or about 99% of the QCL
emission, including the mylar beam splitter efficiency and atmospheric
absorption in the optical path from the LO to the HEB mixer (opt, ). Based on
the QCL output power of 250 $\mu$W, a maximum of 2.5 $\mu$W can couple into
the detector. By using the isothermal method based on IV curves(Ekstrom _et
al._ , 1995), the maximum LO power recorded by the detector is $\sim$290 nW.
Thus, with a lens, 10% to 15% of the available power was coupled into the HEB.
Because of the beam pattern of the third-order DFB structure, this is
considerably improved over the 1.4% coupling efficiency of previous generation
QCLs(Gao _et al._ , 2005).
Receiver sensitivity was measured using the Y-factor method. Eq. 1 is used to
convert a Y-factor to a T${}^{DSB}_{rec}$. The Callen-Welton temperatures at
4.7 THz are T${}_{\rm{eff,hot}}=309$ K and T${}_{\rm{eff,cold}}=126$ K (Callen
and Welton, 1951).
$\rm{T_{N,rec}=\frac{T_{eff,hot}-YT_{eff,cold}}{Y-1}}$ (1)
The Y-factor was measured using three different methods. In Fig. 3a the
measured IF power was swept as a function of bias voltage with a fixed LO
power. We corrected for direct detection by adjusting the LO power so that the
IV curves were on top of one another. The best T${}^{DSB}_{rec}$ was found to
be 825 K at a bias of 0.7 mV and 30 $\mu$A. Recently it has become possible to
accurately sweep LO power by attenuating the LO signal with a stabilized voice
coil attenuator (Hayton _et al._ , 2012) and plot the resulting HEB bias
current as a function of output power (see Fig. 3b). This method reduces
direct detection and results in an average T${}^{DSB}_{rec}$ of 810 K around a
bias of 0.65 mV and 29 $\mu$A. This current corresponds to 220 nW of LO power.
The third method, not shown, chops between hot and cold loads with a
stabilized current. It also produced a T${}^{DSB}_{rec}$ of 810 K at the same
operating point. Thus we obtain a T${}^{DSB}_{rec}$ of 815 K by averaging the
three methods. This T${}^{DSB}_{rec}$ is $\sim$7 times the quantum noise limit
$\left(\frac{h\nu}{2k_{B}}\right)$.
Figure 3: (Color Online) (a) IF power measurements as functions of bias
voltage with the calculated T${}^{DSB}_{rec}$ plotted on the right hand side.
(b) IF power measurements as functions of stabilized bias current with the
calculated T${}^{DSB}_{rec}$ plotted on the right hand side. (c)
T${}^{DSB}_{rec}$ for 4.25, 4.74, and 5.25 THz using a 3 $\mu$m beam splitter.
A gas laser was used as an LO at 4.25 ad 5.25 THz and a QCL was used as an LO
at 4.74 THz. Ten times the quantum noise limit is also shown with the dashed
line.
To demonstrate the QCL adds no additional noise to the receiver system,
T${}^{DSB}_{rec}$ measurements were taken with a gas laser at 4.25 and 5.25
THz. We recorded 750 K at 4.25 THz and 950 K at 5.25 THz with the same HEB
receiver. Fig. 3c shows that all three T${}^{DSB}_{rec}$ scale linearly with
frequency. This suggests that the QCL is a clean LO source. These measurements
improve upon the previously published T${}^{DSB}_{rec}$ of 860 K at 4.25 THz
and 1150 K at 5.25 THz(Zhang _et al._ , 2010). We attribute most of this
($\gtrsim$12%) improvement to a new IF mixer circuit.
HEB receivers have been plagued by stability issues, which can now largely be
attributed to instability in the received LO power at the detector (Hayton
_et al._ , 2012). Allan variance measurements are important in determining the
optimum integration time on a source between instrument calibrations. For this
purpose the noise temperature setup of Fig. 2 was modified to include a two-
way power splitter at the end of the IF chain. Each of the output channels was
then sent through a band pass filter, one centered at 1.25 GHz and the other
at 1.75 GHz. This enabled measurements of the Allan variance in the
spectroscopic configuration (the spectral difference between the two
channels), which yields greater Allan variance times because it effectively
filters out longer period baseline variations.
Figure 4: (Color online) Measurements in air for (a) non-stabilized
spectroscopic and (b) stabilized spectroscopic Allan variance curves. In the
inset is an Allan variance curve for a non-stabilized spectroscopic
measurement taken with the air purged by nitrogen gas.
The results from our receiver are shown in Fig. 4. We found that the non-
stabilized Allan variance time was $\sim$1 s and the stabilized Allan variance
time was $\sim$ 15 s with an 18 MHz noise fluctuation bandwidth. The resulting
Allan variance time from a shorted IF chain was sufficiently long enough to
eliminate the IF chain as a source of instability. Next, we purged the air
between the QCL window and vacuum setup with nitrogen gas. This improves the
non-stabilized, spectroscopic Allan time to $\sim$7 s as shown in the inset of
Fig 4, suggesting that atmospheric turbulence at 4.7 THz may be a large
contributor to the instability in the system.
Figure 5: (Color online) A DSB spectrum of methanol with an LO frequency of
4.74093 THz compared with the predicted spectrum from the JPL catalog.
In order to demonstrate the functionality of the receiver for heterodyne
spectroscopy, the receiver was used to measure a spectrum of methanol gas
(CH3OH). A methanol gas cell was attached to an external input port on the
hot/cold vacuum setup so that there was no air in the signal path. The QCL was
operated at a bias voltage of 11.8 V. The results, averaged over 18 s of
integration time, are shown in Fig. 5 along with a simulation (Ren _et al._ ,
2010) at 0.25 mbar that predicts line widths based on the frequencies and line
strengths from the JPL spectral catalog (Pickett _et al._ , 1998; Xu _et
al._ , 2008). The lines from 1500-1700 MHz are attenuated because the FFTS
upper band (1500-3000 MHz) high pass filter has a cut-off frequency of 1700
MHz. The best-fit frequency for the QCL is 4.740493 THz, which is close to the
HEB bandwidth ($\sim$4 GHz) for the [OI] line. The verification of the JPL
spectral catalog is also important for the frequency locking of the QCL Ren
_et al._ (2012).
In conclusion, we have demonstrated a 4.7-THz HEB-QCL receiver with a measured
sensitivity of 815 K and spectroscopic Allan time of 15 s. This
T${}^{DSB}_{rec}$ is 85 times lower than a previous Schottky receiver (Boreiko
and Betz, 1996). Heterodyne performance was verified by observing a methanol
spectrum. The performance of this receiver indicates THz receiver technology
has reached a level of maturity that will permit large-scale [OI] surveys of
the interstellar medium to take place, such as those planned by the Gal/Xgal
Ultra-Long Duration Spectroscopic Stratospheric Terahertz Observatory
(GUSSTO).
We acknowledge G. Goltsman’s group at MSPU for the provision of NbN films. We
would like to thank John C. Pearson for his help in understanding methanol
lines in the JPL catalog near 4.7 THz. The work of the University of Arizona
was supported by NASA grant NN612PK37C. The work in the Netherlands is
supported by NWO, KNAW, and NATO SFP. The work at MIT is supported by NASA and
NSF. The work at Sandia was performed, in part, at the Center for Integrated
Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences
user facility. Sandia National Laboratories is a multiprogram laboratory
managed and operated by Sandia Corporation, a wholly owned subsidiary of
Lockheed Martin Corporation, for the U.S. Department of Energy‚ National
Nuclear Security Administration under contract DE-AC04-94AL85000.
## References
* Tielens and Hollenbach (1985) A. G. G. M. Tielens and D. Hollenbach, Astrophys. J. 291, 722 (1985).
* Köhler _et al._ (2002) R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, Nature 417, 156 (2002).
* Vitiello _et al._ (2012) M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, Nat. Photonics 6, 525 (2012).
* Gao _et al._ (2005) J. R. Gao, J. N. Hovenier, Z. Q. Yang, J. J. A. Baselmans, A. Baryshev, M. Hajenius, T. M. Klapwijk, A. J. L. Adam, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 86, 244104 (2005).
* Williams _et al._ (2005) B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, Opt. Express 13, 3331 (2005).
* Richter _et al._ (2010) H. Richter, S. G. Pavlov, A. D. Semenov, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and H.-W. Hübers, Appl. Phys. Lett. 96, 071112 (2010).
* Ren _et al._ (2012) Y. Ren, J. N. Hovenier, M. Cui, D. J. Hayton, J. R. Gao, T. M. Klapwijk, S. C. Shi, T.-Y. Kao, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 100, 041111 (2012).
* Amanti _et al._ (2010) M. I. Amanti, G. Scalari, F. Castellano, M. Beck, and J. Faist, Opt. Express 18, 6390 (2010).
* Kao, Hu, and Reno (2012) T.-Y. Kao, Q. Hu, and J. L. Reno, Opt. Lett. 37, 2070 (2012).
* Ren _et al._ (2011) Y. Ren, J. N. Hovenier, R. Higgins, J. R. Gao, T. M. Klapwijk, S. C. Shi, B. Klein, T.-Y. Kao, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 98, 231109 (2011).
* Wei Min Lee _et al._ (2012) A. Wei Min Lee, T.-Y. Kao, D. Burghoff, Q. Hu, and J. L. Reno, Opt. Lett. 37, 217 (2012).
* Cherednichenko _et al._ (2008) S. Cherednichenko, V. Drakinskiy, T. Berg, P. Khosropanah, and E. Kollberg, Rev. Sci. Instrum. 79, 034501 (2008).
* Walker _et al._ (2010) C. Walker, C. Kulesa, P. Bernasconi, H. Eaton, N. Rolander, C. Groppi, J. Kloosterman, T. Cottam, D. Lesser, C. Martin, A. Stark, D. Neufeld, C. Lisse, D. Hollenbach, J. Kawamura, P. Goldsmith, W. Langer, H. Yorke, J. Sterne, A. Skalare, I. Mehdi, S. Weinreb, J. Kooi, J. Stutzki, U. Graf, M. Brasse, C. Honingh, R. Simon, M. Akyilmaz, P. Puetz, and M. Wolfire, in _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7733 (2010).
* Pütz _et al._ (2012) P. Pütz, C. E. Honingh, K. Jacobs, M. Justen, M. Schultz, and J. Stutzki, Astron. Astrophys. 542, L2 (2012), arXiv:1204.2381 [astro-ph.IM] .
* Heyminck _et al._ (2012) S. Heyminck, U. U. Graf, R. Güsten, J. Stutzki, H. W. Hübers, and P. Hartogh, Astron. Astrophys. 542, L1 (2012), arXiv:1203.2845 [astro-ph.IM] .
* Barends _et al._ (2005) R. Barends, M. Hajenius, J. R. Gao, and T. M. Klapwijk, Appl. Phys. Lett. 87, 263506 (2005).
* Hayton _et al._ (2012) D. J. Hayton, J. R. Gao, J. W. Kooi, Y. Ren, W. Zhang, and G. de Lange, Appl. Phys. Lett. 100, 081102 (2012).
* (18) The optical losses are due primarily to the UHMW-PE lens and cryostat windows ($\sim$3 dB), air ($\sim$3.5 dB), mylar beam splitter ($\sim$9 dB), QMC IR filter ($\sim$0.8 dB), coated Si lens ($\sim$1 dB), and spiral antenna ($\sim$3 dB).
* Ekstrom _et al._ (1995) H. Ekstrom, B. Karasik, E. Kollberg, and K. Yngvesson, IEEE Trans. Microwave Theory Tech. 43, 938 (1995).
* Callen and Welton (1951) H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
* Zhang _et al._ (2010) W. Zhang, P. Khosropanah, J. R. Gao, T. Bansal, T. M. Klapwijk, W. Miao, and S. C. Shi, J. Appl. Phys. 108, 093102 (2010).
* Ren _et al._ (2010) Y. Ren, J. N. Hovenier, R. Higgins, J. R. Gao, T. M. Klapwijk, S. C. Shi, A. Bell, B. Klein, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 97, 161105 (2010).
* Pickett _et al._ (1998) H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Müller, J. Quant. Spectros. Radiat. Transfer 60, 883 (1998).
* Xu _et al._ (2008) L.-H. Xu, J. Fisher, R. M. Lees, H. Y. Shi, J. T. Hougen, J. C. Pearson, B. J. Drouin, G. A. Blake, and R. Braakman, J. Mol. Spectrosc. 251, 305 (2008).
* Boreiko and Betz (1996) R. T. Boreiko and A. L. Betz, Astrophys. J. Lett. 464, L83 (1996).
|
arxiv-papers
| 2012-08-28T20:03:18 |
2024-09-04T02:49:34.617817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jenna L. Kloosterman, Darren J. Hayton, Yuan Ren, Tsung-Yu Kao, Neils\n Hovenier, Jian-Rong Gao, Teun M. Klapwijk, Qing Hu, Christopher K. Walker,\n John L. Reno",
"submitter": "Jenna Kloosterman",
"url": "https://arxiv.org/abs/1208.5776"
}
|
1208.5810
|
# Bounding the pseudogap with a line of phase transitions in YBa2Cu3O6+δ.
Arkady Shekhter Pulsed Field Facility, NHMFL, Los Alamos National Laboratory,
Los Alamos, NM 87545 B. J. Ramshaw Pulsed Field Facility, NHMFL, Los Alamos
National Laboratory, Los Alamos, NM 87545 Ruixing Liang Department of
Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada,
V6T 1Z1 Canadian Institute for Advanced Research, Toronto, Canada, M5G 1Z8
W. N. Hardy Department of Physics and Astronomy, University of British
Columbia, Vancouver, BC, Canada, V6T 1Z1 Canadian Institute for Advanced
Research, Toronto, Canada, M5G 1Z8 D. A. Bonn Department of Physics and
Astronomy, University of British Columbia, Vancouver, BC, Canada, V6T 1Z1
Canadian Institute for Advanced Research, Toronto, Canada, M5G 1Z8 Fedor F.
Balakirev Pulsed Field Facility, NHMFL, Los Alamos National Laboratory, Los
Alamos, NM 87545 Ross D. McDonald Pulsed Field Facility, NHMFL, Los Alamos
National Laboratory, Los Alamos, NM 87545 Jon B. Betts Pulsed Field
Facility, NHMFL, Los Alamos National Laboratory, Los Alamos, NM 87545 Scott
C. Riggs Stanford Institute of Materials and Energy Sciences, Stanford
University, Stanford, CA 94305, USA Departments of Physics and Applied
Physics, and Geballe Laboratory for Advanced Materials, Stanford University,
Stanford, CA 94305, USA Albert Migliori Pulsed Field Facility, NHMFL, Los
Alamos National Laboratory, Los Alamos, NM 87545
Close to optimal doping, the copper oxide superconductors show ’strange metal’
behaviorTransport ; TransportHussey2009 , suggestive of strong fluctuations
associated with a quantum critical pointOrensteinMillis2000 ; vanderMarel2003
; VarmaReports ; MarginalFL1989 . Such a critical point requires a line of
classical phase transitions terminating at zero temperature near optimal
doping inside the superconducting ’dome’. The underdoped region of the
temperature-doping phase diagram from which superconductivity emerges is
referred to as the ’pseudogap’Timusk ; NeutronsYBCO1 ; NeutronsYBCO2 ;
NeutronsHg1201 ; Kaminski ; VarmaPG ; AjiVarma because evidence exists for
partial gapping of the conduction electrons, but so far there is no compelling
thermodynamic evidence as to whether the pseudogap is a distinct phase or a
continuous evolution of physical properties on cooling. Here we report that
the pseudogap in YBa2Cu3O6+δ is a distinct phase, bounded by a line of phase
transitions. The doping dependence of this line is such that it terminates at
zero temperature inside the superconducting dome. From this we conclude that
quantum criticality drives the strange metallic behavior and therefore
superconductivity in the cuprates.
Resonant ultrasound spectroscopy (RUS) measures the frequencies $f_{n}$ and
widths $\Gamma_{n}$ of the vibrational normal modes of a crystal acting as a
free mechanical resonator. The frequencies of the normal modes are determined
by density and geometry of the crystal as well as its elastic properties. The
elastic component of the temperature evolution of these frequencies, $\Delta
f_{n}(T)$, depends on a linear combination of all elastic moduli and reflects
changes in the thermodynamic state of the system such as those associated with
a phase transition. The width of a resonance $\Gamma_{n}(T)$ is proportional
to the energy dissipation caused by time-dependent (dynamic) fluctuations in
the system. Measuring many resonances provides access to elastic properties
and fluctuations with different symmetries.RUS ; Migliori-RSI ; Migliori-YBCO
; Birss Recent advances in the quality of single crystal YBa2Cu3O6+δ (YBCO)
have pushed the boundary of possible measurements, as evidenced by the
observation of quantum oscillationsDoironLeyraud . Advances in resonant
ultrasound enable determination of the thermodynamics of these sub-millimeter
crystals to part-per million accuracy.
The narrow temperature range over which the resonances evolve across the
superconducting transition illustrates the quality of the crystals and the
accuracy of the measurementBishop1987 (Figure 1). For the underdoped crystal,
YBa2Cu3O6.60, we observe a sharp ($0.5$K wide) discontinuity in the resonance
frequency, $\Delta{f}/f\approx 10^{-4}$, at the superconducting transition
(Figure 1). A sharper discontinuity is observed in the overdoped crystal,
YBa2Cu3O6.98, a possible consequence of the reduction in oxygen disorder near
optimal doping. The step discontinuity in resonance frequency and the
accompanying discontinuous change (break) in slope are thermodynamic
signatures of the superconducting transition (SI).
RUS measurements across the temperature range encompassing the pseudogap in
the two YBCO crystals are shown in Figure 2. The temperature dependence of the
resonance frequencies in underdoped YBa2Cu3O6.60 reveals a break in slope at
the pseudogap boundary $T^{*}=245K$—in itself a standard thermodynamic marker
for a phase transition (Figure 2(a,c)). It differs from the signature of the
superconducting transition in that there is no resolvable discontinuity in the
frequency itself. This temperature is the same as the onset temperature of
magnetic order observed by neutron scattering measurements of YBCO specimens
of similar composition (Figure 3).NeutronsYBCO1 ; NeutronsYBCO2 In the
overdoped crystal, YBa2Cu3O6.98, the break in slope of the temperature
dependence is observed at $T^{*}=68$K, Figure 2(b). To emphasize the break in
slope in these data, we use the redundant information contained in all
observed resonances to extract the different contributions to their
temperature dependences (Figure 4(c)). This process reduces the temperature
dependence of all fifteen normal modes measured to three dominant components
(see SI). The blue and red curves in Figure 4(c) capture the effects of
superconductivity and of fluctuations in the vicinity of the pseudogap,
respectively. The green curve, which has a break in slope at $T^{*}=68K$,
corresponds to the thermodynamic effects at the pseudogap, revealing that the
pseudogap occurs via a phase transition.
The ‘strange metal’ behavior that cuprates exhibit universally at higher
temperature breaks down in the pseudogap region of the temperature-doping
phase diagram,Timusk ; TransportHussey2009 ; NMR ; ARPES-AD ; ARPES-AK ;
Nernst ; LeridonMonod where measurements indicate the presence of magnetic
orderKaminski ; NeutronsYBCO1 ; NeutronsYBCO2 ; NeutronsHg1201 (Figure 3).
The break in slope that we observe in both underdoped and overdoped YBCO
establishes the pseudogap as a thermodynamic phase that moves to lower
temperature with increased doping. Observation of the pseudogap boundary below
the superconducting transition temperature in overdoped YBCO indicates that
the superconducting dome surrounds the zero-temperature end point of the
pseudogap phase boundary.
At both dopings the pseudogap is accompanied by a strong (up to hundred-fold
in the overdoped crystal) increase in the width of the resonances at
temperatures above the pseudogap phase boundary (Figure 2(c,d)). The width of
the resonances are determined by the ultrasonic energy absorption
(attenuation)Bhatia , revealing strong fluctuations in the dynamics of the
metallic state as it approaches $T^{*}$. From the width of the resonances we
estimate the thermodynamic effects accompanying the pseudogap phase transition
to be $\Gamma/f\sim 5\times 10^{-3}$: about $50$ times larger than the
relative modulus shift across the superconducting phase transition for both
dopings. Energy absorption is highest when the measurement frequency matches
the characteristic relaxation time of the system: $2\pi f\tau(T)=1$. $\tau$
diverges as the phase transition temperature is approached (critical slowing
down)LandauKhalatnikov , therefore the maximum in ultrasonic energy absorption
is closer to the pseudogap phase boundary for resonances of lower frequency.
For the underdoped crystal the width of the maximum and the contribution of
the large phonon background at $245K$ obscures this effect. The overdoped
crystal, with its narrower maxima and smoother background exhibits this effect
clearly: $1/{\tau(T)}$ extrapolated from resonances at different frequencies
vanishes at the pseudogap phase boundary (Figure 4(a,b)). Causality requires
that the maxima in energy absorption are accompanied by elastic stiffening
over the same temperature range. This stiffening is observed in addition to
the distinct break in slope at $T^{*}$ (Figure 2(b)).
The potential for RUS to determine the broken symmetry in the pseudogap phase
was limited in this study by the precision with which crystal shape could be
controlled, an issue that may be resolvable as sample preparation techniques
improve. The pseudogap phase transition is located by our RUS measurements
with $\pm 3$K uncertainty, improving on the $\pm 30$K uncertainty in onset of
neutron spin-flip scattering. This clearly separates the onset of magnetic
orderKaminski ; NeutronsYBCO1 ; NeutronsYBCO2 ; NeutronsHg1201 at $T^{*}$
from the onset $T_{K}$ of the Kerr rotation signalKerr and charge orderXray
at lower temperature (Figure 3). In our measurements we observe an increase in
energy absorption over a broad region near $T_{K}$ (Figure 2(c)), however we
do not observe an accompanying thermodynamic signature there. Our observed
evolution of the pseudogap phase boundary from underdoped to overdoped
establishes the presence of a quantum critical point inside the
superconducting dome, suggesting a quantum-critical origin for both the
strange metallic behavior and the mechanism of superconducting pairing.
## References
* (1) Ando, Y., Komiya, S., Segawa, K., Ono, S., & Kurita, Y., Electronic Phase Diagram of High-T${}_{\textrm{c}}$ Cuprate Superconductors from a Mapping of the In-Plane Resistivity Curvature, _Phys. Rev. Lett._ 93, 267001 (2004).
* (2) Hussey, N.E., Phenomenology of the normal state in-plane transport properties of high-T${}_{\textrm{c}}$ cuprates, _J. Phys.: Condens. Matter_ 20, 123201 (2008)
* (3) van der Marel, D., Molegraaf, H.J.A., Zaanen, J., Nussinov, Z., Carbone, F., Damascelli, A., Eisaki, H., Greven, M., Kes, P.H., & Li, M., Quantum critical behaviour in a high-T${}_{\textrm{c}}$ superconductor, _Nature_ 425, 271-274 (2003).
* (4) Orenstein, J., & Millis, A.J., Advances in the Physics of High-Temperature Superconductivity, _Nature_ 288, 468-474 (2000).
* (5) Varma, C.M., Littlewood, P.B., Schmitt-Rink, S., Abrahams, E., and Ruckenstein, A., Phenomenology of the normal state of Cu-O high-temperature superconductors, _Phys. Rev. Lett._ 63, 1996-1999 (1989).
* (6) Varma, C.M., Nussinov, Z., van Saarloos, W., Singular or non-Fermi liquids, _Physics Reports_ 361 267–417 (2002).
* (7) Timusk, T., & Statt, B., The pseudogap in high-temperature superconductors: an experimental survey, _Rep. Prog. Phys._ 62, 61-122 (1999).
* (8) Fauqué, B., Sidis, Y., Hinkov, V., Pailhès, S., Lin, C.T., Chaud, X., & Bourges, P., Magnetic Order in the Pseudogap Phase of High-$T_{c}$ Superconductors, _Phys. Rev. Lett._ 96, 197001 (2006).
* (9) Mook, H.A., Sidis Y., Fauqué, B., Balédent, V., & Bourges, P., Observation of magnetic order in a superconducting YBa2Cu3O6.6 single crystal using polarized neutron scattering, _Phys. Rev. B_ 78, 020506 (2008).
* (10) Li, Y., Baledent, V., Barisic, N., Bourges, P., Cho, Y., Fauqué, B., Sidis, Y., Yu, G., Zhao, X., & Greven, M., Unusual magnetic order in the pseudogap region of the superconductor HgBa2CuO4+δ, _Nature_ 455, 372-375 (2008).
* (11) Kaminski, A., Rosenkranz, S., Fretwell, H.M., Campuzano, J.C., Li, Z., Raffy, H., Cullen, W.G., You, H., Olson, C.G., Varma, C.M., & Höchst, H., Spontaneous breaking of time-reversal symmetry in the pseudogap state of a high-T${}_{\textrm{c}}$ superconductor, _Nature_ 416, 610-613 (2002).
* (12) Varma, C.M., Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals, _Phys. Rev. B_ 55, 14554-14580 (1997).
* (13) Aji, V., & Varma, C.M., Quantum criticality in dissipative quantum two-dimensional XY and Ashkin-Teller models: Application to the cuprates, _Phys. Rev. B_ 79, 184501 (2009).
* (14) Migliori, A., Sarrao, J.M., _Resonant Ultrasound Spectroscopy_ , Wiley-Interscience (1997).
* (15) Migliori, A., Maynard, J.D., Implementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic moduli of small solid specimens, _Review of Scientific Instruments_ 76, 121301-121308 (2005).
* (16) Birss, R.R., _Symmetry and Magnetism_ , (Wiley-Interscience Inc., New York, 1964).
* (17) Lei, M., Sarrao, J.L., Visscher, W.M., Bell, T.M., Thompson, J.D., Migliori, A., Welp, U.W., & Veal, B.W., Elastic constants of a monocrystal of superconducting YBa2Cu3O7-δ, _Phys. Rev. B_ 47, 6154-6156 (1993).
* (18) Doiron-Leyraud, N., Proust, C., LeBoeuf, D., Levallois, J., Bonnemaison, J.B., Liang, R., Bonn, D.A., Hardy, W.N., Taillefer, L., Quantum oscillations and the Fermi surface in an underdoped high-$T_{c}$ superconductor, _Nature_ 447, 565-568 (2007).
* (19) Bishop, D.J., Ramirez, A.P., Gammel, P.L., Batlogg, B., Rietman, E. A., Cava, R.J., & Millis, A.J., Bulk-modulus anomalies at the superconducting transition of single-phase YBa2Cu3O7, _Phys. Rev. B_ 36, 2408–2410 (1987).
* (20) Walstedt, R.E., Warren, Jr. W.W., Bell, R.F., Cava, R.J., Espinosa, G.P., Schneemeyer, L.F., & Waszczak, J.V., 63Cu NMR shift and linewidth anomalies in the Tc=60K phase of YBaCuO, _Phys. Rev. B_ 41, 9574-9577 (1990).
* (21) Vishik, I.M., Lee, W.S., He, R.H., Hashimoto, M., Hussain, Z., Devereaux, T.P., & Shen, Z.X., ARPES studies of cuprate Fermiology: superconductivity, pseudogap and quasiparticle dynamics, _New J. Phys._ 12, 105008 (2010).
* (22) Daou, R., Chang, J., LeBoeuf, D., Cyr-Choinière, O., Laliberte, F., Doiron-Leyraud, N., Ramshaw, B.J., Liang, R., Bonn, D.A., Hardy, W.N., & Taillefer, L., Broken rotational symmetry in the pseudogap phase of a high-T${}_{\textrm{c}}$ superconductor, _Nature_ 463, 519-522 (2010).
* (23) Kondo, T., Hamaya, Y., Palczewski, A.D., Takeuchi, T., Wen, J.S., Xu, Z.J., Gu, G., Schmalian, J., & Kaminski, A., Disentangling Cooper-pair formation above the transition temperature from the pseudogap state in the cuprates, _Nature Physics_ 7, 21-25 (2011).
* (24) Leridon, B., Monod, P., & Colson, D., Thermodynamic signature of a phase transition in the pseudogap phase of YBa2Cu3Ox high-T${}_{\textrm{c}}$ superconductor, _Europhys. Lett._ 87, 17011 (2009).
* (25) Bhatia, A.B., _Ultrasonic Absorption_ , (Oxford, Clarendon Press 1967).
* (26) Landau, L.D., & Khalatnikov, I.M., _Dokl. Akad. Nauk SSSR_ 96, 469 (1954) [English translation: in _Collected papers of Landau, L.D._ , edited by ter Haar, D., (Pergamon, London 1965)].
* (27) Xia, J., Schemm, E., Deutscher, G., Kivelson, S.A., Bonn, D.A., Hardy, W.N., Liang, R., Siemons, W., Koster, G., Fejer, M.M., & Kapitulnik, A., Polar Kerr-Effect Measurements of the High-Temperature YBa2Cu3O6+x Superconductor: Evidence for Broken Symmetry near the Pseudogap Temperature, _Phys. Rev. Lett._ 100, 127002 (2008).
* (28) Chang, J., Blackburn, E., Holmes, A.T., Christensen, N.B., Larsen, J., Mesot, J., Liang, R., Bonn, D.A., Hardy, W.N., Watenphul, A., Zimmermann, M., Forgan, E.M., & Hayden, S.M., Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67, _Nature Physics_ 8, 871-876 (2012).
* (29) Liang, R., Hardy, W.N., & Bonn, D.A., Evaluation of CuO2 plane hole doping in YBa2Cu3O6+x single crystals, _Phys. Rev. B_ 73, 180505 (2006).
* (30) Varshni, Y., Temperature Dependence of the Elastic Constants, _Phys. Rev. B_ 2, 3952-3958 (1970).
Acknowledgements We thank Elihu Abrahams, James Analytis, Philippe Bourges,
Alexander Finkel’stein, Martin Greven, Neil Harrison, Kim Modic, Chandra
Varma, Inna Vishik, and Guichuan Yu for critical reading of the manuscript and
informative discussions. Work at Los Alamos National Laboratory (LANL) was
supported by NSF-DMR-0654118, DOE, and the State of Florida. LANL is operated
by LANS LLC. Work at the University of British Columbia was supported by the
Canadian Institute for Advanced Research and the Natural Science and
Engineering Research Council. This work was supported in part by the NSF under
Grant No. PHYS-1066293 and the hospitality of Aspen Center for Physics.
Figure 1: Figure 1. The temperature evolution of resonances in underdoped and
overdoped YBCO crystals. Superconductivity. a A typical resonance frequency
scan (normalized at room temperature) from room temperature to 10K for
underdoped YBa2Cu3O6.60 in blue with $T_{c}=61.6K$, and overdoped YBa2Cu3O6.98
in red with $T_{c}=88K$. The scan for overdoped crystal is offset vertically
for clarity. The smooth increase in frequency, which saturates at low
temperature, is driven by the anharmonicity of the lattice and is typical of
most solids.Varshni . b,c Superconducting transition in the underdoped (b) and
overdoped (c) crystals. Measurements were made at approximately $70mK$ steps.
The elastic moduli drop discontinuously at the transition. The discontinuity
is approximately one part in $10^{-4}$ in the underdoped crystal, and five
parts in $10^{-4}$ in the overdoped. The form of the smooth monotonic
background subtracted to obtain (b) and (c)was chosen only to emphasize the
discontinuityBishop1987 . d,e Resonance width for underdoped (d) and overdoped
(e) YBCO. In the underdoped crystal no feature at the superconducting
transition can be resolved. A broad maximum in resonance width well below
$T_{c}$ in the overdoped crystal is an effect of the pseudogap (see text).
Figure 2: Figure 2. The temperature evolution of resonances across the
pseudogap phase boundary. At both dopings a discontinuous change in slope of
the temperature dependence of the frequency reveals a phase transition:
underdoped (a) at $T^{*}=245K$, and overdoped (b) at $T^{*}=68K$. At both
dopings the resonance width has a broad maximum above $T^{*}$ (underdoped (c)
and overdoped (d)). The break in slope is 5K wide in the underdoped crystal,
3K wide in overdoped. The increase in scatter of points near the break in
slope in panel (b) is a result of strong increase in resonance width at this
temperature, panel (d).
Figure 3: Figure 3. The phase diagram of YBa2Cu3O6+δ. The pseudogap boundary
in YBCO cuprates is indicated by a thick grey line (guide to the eye), as
determined by neutron diffraction measurementsNeutronsYBCO1 ; NeutronsYBCO2
(blue squares) and resonant ultrasound (red circles). The superconducting
transition temperature is indicated by black circlesLiangHardyBonn . The
temperature of the onset of Kerr rotationKerr where recent X-ray measurements
detect an onset of charge order Xray is shown in purple diamonds. Error bars
represent the uncertainty in the determination of the onset temperature. The
thin grey line is a guide for the eye.
Figure 4: Figure 4. The pseudogap boundary inside the superconducting dome. a
Evolution of resonance width with temperature across $T^{*}$ for several
resonances. To illustrate the evolution of the maximum in resonance width with
resonance frequency each curve is offset vertically by an amount proportional
to resonance frequency. b Evolution of the temperature of the resonance width
maxima in (a) with resonance frequency ( $2\pi{f}\tau=1$). The characteristic
time $\tau$ increases as the pseudogap temperature is approached (critical
slowing down) c. Three different components of the temperature dependence of
all resonance modes in the overdoped crystal: blue is dominated by
superconductivity, red by fluctuations, and green by the pseudogap. The smooth
anharmonic background, which dominates Figure 1(a), is not shown. Each curve
is scaled vertically for clarity.
Supplementary information.
## Appendix A Resonant ultrasound spectroscopy measurement system.
The resonant ultrasound spectroscopy (RUS) measurement system comprises a
piezoelectric driver and receiver, each in sufficiently weak point contact
that the measured crystal acts as a free mechanical resonator. The essence of
the measurement is the stress-stress response of the crystal: the driving
transducer generates a stress at the point of contact on the crystal at a
frequency $\omega$, and the receiving transducer generates a voltage
proportional to the stress at different point on the crystal. The in-phase
(real) and quadrature (imaginary) components of the voltage on the receiving
transducer are recorded as a complex number $V(\omega)$. The measurement
proceeds by sequentially changing the frequency of the driving transducer in a
range that encompasses the lowest mechanical resonances of the crystal
(typically about 20 resonances in the range 0.1 MHz to a few MHz for 1mm sized
crystals).RUS-book Generation of the driving signal and the phase-sensitive
receiver logic are implemented on a custom-built electronic board (similar to
a heterodyne lock-in amplifier).Migliori-RSI The shifts of the resonant
frequencies with temperature, proportional to the shifts in the elastic moduli
of the crystal, provide symmetry-specific information about the changes in the
thermodynamic state of the system. The widths of the resonances are
proportional to ultrasonic energy dissipation accompanying each resonance.
High quality crystals are required so that the Q factor of the mechanical
resonances are sufficiently high for this measurement. Detwinned crystals are
necessary for this measurement because the motion of un-pinned twin boundaries
in ultra-high purity crystals can be a large source of ultrasonic energy
dissipation. This broadens the resonances to the point where they can not be
resolved. Ultra-high-quality detwinned YBCO crystals are available in typical
dimensions of 1x1x0.2mm, and sub-milligram mass. The small mass, small size,
and plate geometry of the crystals require lower excitation power, better
vibration isolation, and smaller transducer contact force than what is
commercially available or previously used for RUS on larger crystals.alpha-Pu-
reference . Measuring a relatively large number of modes during a single
temperature sweep that takes of order one week requires intelligent data
acquisition; linear frequency sweeps across the whole range would take of
order one year. These points define the experimental challenges.
Standard lithium niobate compressional mode transducers (disk-shaped, 1.5mm
diameter, 200$\mu{m}$ thick) are used for both driver and receiver. To prevent
vibrational crosstalk, the transducers must be acoustically isolated from one
another and from the environment. Balsa-wood conveniently provides such
isolation over a temperature range from liquid helium to room temperature. To
obtain the sharpest resonances, the crystal was mounted on its corners,
providing a single point of contact to each transducer and consistent coupling
(Figure 5). The ultrasonic power is adjusted as the temperature is swept to
avoid non-linear effects and heating, while still maintaining the signal-to-
noise ratio. By employing a non-uniform frequency scan, in which the frequency
steps are small in the immediate vicinity of the resonance and large away from
the resonance, we speed up (compared to a linear frequency sweep) the
measurement across multiple resonances by about a factor of hundred while
maintaining resonance frequency resolution exceeding one part per million.
### A.1 Determining the resonance frequency and width using in-phase and
quadrature components of the measured signal.
The resonance measured by the pickup transducer has a Lorentzian shape,
$V(\omega)=z_{\infty}+Ae^{i\phi}/(\omega-\omega_{0}+i\Gamma/2)$, where
$\Gamma$ is the resonance width and $z_{\infty}$ is the background in the
vicinity of the resonance. The resonance frequency and the width can be
determined by finding the frequency of the maximum amplitude of the signal and
the width at half maximum. However, in measurements with marginal signal-to-
noise, or in this case very weak coupling to maintain free mechanical
vibration, this procedure does not produce reliable frequencies and widths.
Our approach is to use both the amplitude and the phase information of the
signal $V(\omega)$, and then fit the data to a circle in the complex plane
(Figure 6). This first requires finding the centre of the circle, $z_{c}$, and
then the resonant frequency and the width are fit via regression of all
available data points in the vicinity of the resonance (as described in the
caption of Figure 6.) The center of the circle is found by using a variant of
a Hough transform: circular waves emanating from each data point in the
complex plane will interfere constructively at the center of the circle, and
destructively elsewhere. This procedure is robust against noise and distortion
of the circle caused by transients and non-linear effects.
### A.2 Understanding the effects of the superconducting and pseudogap
physics on the observed frequency shifts.
The temperature dependence of the resonance frequencies is determined by a
superposition of the effects of several physical processes, each with a
distinct temperature dependence of its own, for example superconductivity and
the pseudogap. The problem of extracting the dependence of these two
components from about 15 measured frequencies is that of solving 15 linear
equations for only 2 or 3 unknown temperature dependent contributions. This
overdetermination makes it possible to extract the evolution of the effects of
pseudogap and superconductivity, as shown in Figure 4(d) in the main text. The
problem is to find the $D$ functions $\phi_{i}(T)$ that capture the
temperature dependence of $N>D$ frequencies with minimal error,
$\Delta{f_{n}(T)}=\sum\limits_{i=1..D}\beta^{n}_{i}\phi_{i}(T)$. Here the
temperature $T$ is a vector index. This defines the functions $\phi_{i}(T)$ as
the first $D$ eigenvectors of the matrix
$M_{T,T^{\prime}}=\sum\limits_{n=1..N}\Delta{f_{n}(T)}\Delta{f_{n}(T^{\prime})}$
with the largest eigenvalues. The expansion coefficients $\beta_{i}^{n}$ can
subsequently found by expanding vectors $\Delta{f_{n}(T)}$ in the basis of
${\phi_{i}(T)}$. The benefit of this approach is that you do not need to
identify the mode corresponding to each resonance frequency, however, as a
result this does not reveal the symmetry of the separate physical processes.
## Appendix B Background on the thermodynamic and transport phenomena
associated with the elastic response.
To examine the ultrasonic signature across a phase transition (discontinuities
in the resonance frequency and the attenuation response), we present the
elastic strain and stress in terms of the irreducible representations of the
underlying lattice. For simplicity, we illustrate this point for a tetragonal
system, the weak orthorhombicity of YBa2Cu3O6+δ can then be introduced as a
small perturbation.
The six components of the strain tensor in the tetragonal lattice fall into
five irreducible representations of the lattice point group: two compressional
strain components, $\varepsilon_{xx}+\varepsilon_{yy}$ and $\varepsilon_{zz}$,
are crystallographic scalars belonging to the $A_{1g}$ representation.Birss ;
four shear strain components, $\varepsilon_{xx}-\varepsilon_{yy}$,
$\epsilon_{xy}$, and a pair $(\varepsilon_{xz},\varepsilon_{yz})$, fall into
three non-trivial irreducible representations, $B_{1g}$,$B_{2g}$ and $E_{g}$,
respectively. We define the index $m=A_{1g},B_{1g},B_{2g},{E_{g}}$, and use it
to label the components of strain, stress, and the elastic moduli. As an
example, in the presence of a (scalar) order parameter $\eta$, the free energy
can be written as $d\mathbb{F}(T,\eta,\varepsilon_{m})=-SdT+\phi
d\eta+\sigma_{m}d\varepsilon_{m}$, where $\phi$ is the thermodynamic conjugate
to the order parameter $\eta$—a restoring force equal to zero in thermodynamic
equilibrium—and $\sigma_{m}$ is an elastic stress.
Elastic deformations perturb the local thermodynamic equilibrium of the
crystal, resulting in a coupling to thermodynamic variables such as
temperature or order parameters. Crystallographic scalar components of strain
can couple linearly to temperature and to order parameters. In this notation
the elastic moduli are
$\lambda_{mn}=\left(\nicefrac{{\partial{\sigma_{m}}}}{{\partial{\varepsilon_{n}}}}\right)_{T}$.
In a tetragonal crystal, the linear couplings of elastic strain to temperature
$\delta{T}$, and to a scalar order parameter $\delta\eta$, are controlled by
the thermodynamic coefficients
$\beta_{m}=-\left(\nicefrac{{\partial{\sigma_{m}}}}{{\partial{T}}}\right)_{\varepsilon_{m}}$
and
$Z_{m}=\left(\nicefrac{{\partial{\sigma_{m}}}}{{\partial{\eta}}}\right)_{T,\varepsilon}$
via $\Delta\mathbb{F}=-\delta{T}\beta_{m}\epsilon_{m}+\delta\eta
Z_{m}\epsilon_{m}$, where $Z_{m}$ and $\beta_{m}$ are only non-zero in the
scalar crystallographic symmetry channels. We note that while shear strains
are non-scalar in a tetragonal crystal, they can be scalar in a lower symmetry
environment. For example, the weak orthorhombicity of the YBCO crystal
structure (characterized by a small $B_{1g}$ distortion) introduces a small
linear coupling of $B_{1g}$ shear strain to heat and scalar order parameters
via a scalar $\zeta_{o}(\epsilon_{xx}-\epsilon_{yy})$, where $\zeta_{o}$ is
the magnitude of distortion. In addition, for non-scalar order parameters, the
shear strain can couple linearly in some situations. For example, a polar
magnetic vector AjiVarma ; ShekhterVarma , $\bm{\eta}=(\eta_{x},\eta_{y})$ can
couple linearly to shear elastic strain in the $B_{1g}$ and $B_{2g}$ symmetry
channels via order parameter bilinears $B_{1g}=\eta_{x}^{2}-\eta_{y}^{2}$ and
$B_{2g}=\eta_{x}\eta_{y}$.
When elastic strain is coupled to the dynamics of the order parameter or to
heat flow, the elastic response acquires a frequency (and momentum)
dispersion. This can be described by introducing dynamic elastic response
functions: $\lambda_{mn}(\omega)=\lambda_{mn}+R_{mn}A(\omega)$ where $R_{mn}$
is the difference between the fast and slow values of the elastic moduli, and
$A(\omega)$ is proportional to the full dynamic correlation function of the
physical process in question, such as heat flow or order parameter
fluctuation. $A(\omega)$ is normalized such that
$A(\omega\rightarrow\infty)=1,\quad A(\omega\rightarrow 0)=0$, and must be
analytic in the upper half of the complex plane of $\omega$. For example, in
the case of coupling elastic strain to heat, $R_{mn}$ is equal to the
difference between the adiabatic (fast) and isothermal (slow) elastic
moduli:Bhatia
$R_{mn}=\left(\lambda_{mn}\right)_{S}-\left(\lambda_{mn}\right)_{T}=T\beta_{m}\beta_{n}/C_{\varepsilon}$,
where $C_{\varepsilon}$ is the heat capacity at constant strain, and
$\beta_{m}$ is the inverse of the thermal expansion coefficient (defined in
the previous section). When elastic strain couples to the dynamics of an order
parameter,
$R_{mn}=\left(\lambda_{mn}\right)_{\eta}-\left(\lambda_{mn}\right)_{\phi}={Z_{m}Z_{n}}/{Y}$
where $Y=\left(\nicefrac{{\partial{\phi}}}{{\partial{\eta}}}\right)$ is the
order parameter stiffness, and $Z_{m}$ is the coupling coefficient between
order parameter and stress as defined above. In a tetragonal crystal, $R_{mn}$
is only non-zero for compressional strains.
In a tetragonal crystal the change in the compressional elastic moduli across
a second order phase transition is equal to
$\left(\lambda_{mn}\right)_{\eta}-\left(\lambda_{mn}\right)_{\phi}={Z_{m}Z_{n}}/{Y}$.
Near the phase transition $Z\propto T_{c}\eta_{0}$, and $Y\propto
T_{F}\eta_{0}^{2}$, which gives an estimate for the jump in elastic moduli (on
warming) across the transition, $\delta\lambda\propto T_{c}^{2}/T_{F}$. Thus
the relative change in frequency $\delta{f}/f_{0}$, which is proportional to
the relative change in elastic moduli, is estimated as
$\delta\lambda/\lambda\propto(T_{c}/T_{F})^{2}$, where we assume that
compressional elastic moduli in a metal are within a factor of order unity
equal to the Fermi energy per unit cell volume. This estimate compares well
with the observed jump across the superconducting transition in YBCO, equal to
$\Delta{f}/f\sim(T_{c}/T_{F})^{2}\sim 10^{-4}$, where $T_{F}$ is the Fermi
temperature which is 5000K.
Near a phase transition the order parameter can have relaxational
dynamicsLandauKhalatnikov ; Kinetics ; the rate of relaxation of a perturbed
order parameter is proportional to the order parameters restoring force
$\phi$: ${d\delta\eta}/{dt}=-\gamma\phi=-\gamma(\delta F/\delta\eta)$. In this
conventional case, the function $A(\omega)$ has the form
$A(\omega)=\nicefrac{{-i\omega}}{{-i\omega+\frac{1}{\tau}}}$ where
$\tau=1/(\gamma Y)$. The frequency dispersion of the elastic response hence
shifts the resonant frequencies and gives them a width proportional to real
and imaginary parts of $A(\omega)$,
$\displaystyle\frac{\Gamma}{\omega_{0}}=$ $\displaystyle-
R_{s}{\mathrm{Im}}A(\omega_{0})$
$\displaystyle\frac{\delta\omega}{\omega_{0}}=$ $\displaystyle\quad
R_{s}{\mathrm{Re}}A(\omega_{0})\,.$ (1)
Here $R_{s}$ is of order of $R_{mn}/\lambda_{mn}$ and depends on the geometry
of a normal mode associated with the resonance.Bhatia For example, for
magnetic phase transitions in solids $R_{s}$ is few percent.Bhatia At $T^{*}$
the specific form of $A(w)$ is not as simple as in our example, but Eq. B
still applies.
The width $\Gamma$ in Eq. B is the width of a resonance measured in the RUS
experiment, and is proportional to the energy dissipation that accompanies the
vibrational mode associated with that resonance.LandauVol5 Attenuation
typically reaches a maximum when the timescale of the fluctuations match the
frequency of the measured resonance. As the system approaches a phase
transition, the relaxation time $\tau$ becomes very long (critical slowing
down) and the condition $\omega\tau=1$ can be satisfied for ultrasonic
frequencies.
## References
* (1) Migliori, A., & Sarrao, J. M., _Resonant Ultrasound Spectroscopy_ , (Wiley-Interscience 1997).
* (2) Migliori, A., Pantea, C., Ledbetter, H., Stroe, I., Betts, J. B., Mitchell, J. N., Ramos, M., Freibert, F., Dooley, D., Harrington, S., & Mielke, C. H., _J. Acoust. Soc. Am._ 122, 4 (2007).
* (3) Landau L. D., & Lifshitz, E. M., _Course of Theoretical Physics, Vol. 5 (Statistical Physics)_ (Pergamon, Oxford 1980).
* (4) Landau, L. D., & Lifshitz, E. M., _Course of Theoretical Physics, Vol. 10_ , Lifshitz, E. M., & Pitaevskii, L. P., _Physical Kinetics_ (Pergamon, Oxford 1980).
* (5) Shekhter, A., & Varma, C. M., _Phys. Rev. B_ 80, 214501 (2009).
Figure 5: (a) Geometry of the crystal-transducer assembly. The crystals are
approximately $200\mu{m}$ thick, and one mm square. Mounting the crystal on
its corners ensures weak coupling to the crystal, allowing a free mechanical
resonator conditions. (d) The RUS probe. Balsa-wood provides vibrational
isolation over a broad temperate range, preventing acoustic crosstalk between
the driving and pick-up transducers.
Figure 6: Frequency scan across the resonances and accurate determination of
resonance frequency and width. (a) A non-uniform scan in a broad frequency
range encompassing two resonances: in-phase (red) and quadrature (blue)
components of measured signal. (b) The same scan in the complex plane of
voltage, i.e., quadrature and in-phase components on the receiving transducer
are on the vertical and horizontal axes respectively. This panel illustrates a
scan which is uniform in the complex plane of response voltage $V$: each data
point is acquired at an equal time intervals (1ms), the frequency steps are
adjusted in such a way that the complex response advances at a constant speed
in the complex plane. (c) Scan across a single resonance illustrating the
nonuniform-in-frequency scan in the close vicinity of the resonance. (d) All
measured resonances have a Lorentzian shape,
$z(\omega)=z_{\infty}+Ae^{i\phi}/(\omega-\omega_{0}+i\Gamma/2)$. This panel
illustrate basic geometrical facts that are necessary to accurately determine
frequency and width of the resonance. A point
$z(\omega)=z_{\infty}+Ae^{i\phi}/(\omega-\omega_{0}+i\Gamma/2)$, in the
complex $V$ plane traces a circle as we scan across a resonance, centered at
$z_{c}$. The tails of the Lorentzian map to $z_{\infty}$, and the centre of
the resonance at frequency $\omega_{0}$ maps to $z_{0}$. To determine the
resonance frequency and width, we use the identity
$\theta=\arctan\big{(}(\omega-\omega_{0})/(\Gamma/2)\big{)}$, where $\theta$
is the angle shown in the figure. A complementary strategy is to use the fact
that the distance $\ell$ between the point $z^{\prime}$, defined via
$z^{\prime}(\omega)-z_{\infty}=|z_{0}-z_{\infty}|^{2}/[z(\omega)-z_{\infty}]^{*}$,
and the point $z_{0}$, is a linear function of frequency:
$\ell=|z_{0}-z_{\infty}|(\omega-\omega_{0})/(\Gamma/2)$. This reduces the
determination of the resonance frequency and the width to a linear regression
of all available data points.
|
arxiv-papers
| 2012-08-28T23:35:38 |
2024-09-04T02:49:34.623255
|
{
"license": "Public Domain",
"authors": "Arkady Shekhter, B. J. Ramshaw, Ruixing Liang, W. N. Hardy, D. A.\n Bonn, Fedor F. Balakirev, Ross D. McDonald, Jon B. Betts, Scott C. Riggs,\n Albert Migliori",
"submitter": "Arkady Shekhter",
"url": "https://arxiv.org/abs/1208.5810"
}
|
1208.5904
|
# A burst with double radio spectrum observed up to 212 GHz
C.G. Giménez de Castro1,2G.D. Cristiani2,3P.J.A. Simões6C.H. Mandrini2,3E.
Correia1,4P. Kaufmann1,5 1 CRAAM, Universidade Presbiteriana Mackenzie,
01302-907, São Paulo, Brasil; email: [email protected];
[email protected]; [email protected]
2 Instituto de Astronomía y Física del Espacio, CONICET-UBA, CC. 67 Suc. 28,
1428, Buenos Aires, Argentina; email: [email protected];
[email protected]
3 Facultad de Ciencias Exactas y Naturales, FCEN-UBA, Buenos Aires, Argentina
4 Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil
5 Centro de Componentes Semicondutores, Universidade Estadual de Campinas,
Campinas, Brasil
6 School of Physics & Astronomy, University of Glasgow, Glasgow, Scotland
###### Abstract
We study a solar flare that occurred on September 10, 2002, in active region
NOAA 10105 starting around 14:52 UT and lasting approximately 5 minutes in the
radio range. The event was classified as M2.9 in X-rays and 1N in H$\alpha$.
Solar Submillimeter Telescope observations, in addition to microwave data give
us a good spectral coverage between 1.415 and 212 GHz. We combine these data
with ultraviolet images, hard and soft X-rays observations and full-disk
magnetograms. Images obtained from Ramaty High Energy Solar Spectroscopic
Imaging data are used to identify the locations of X-ray sources at different
energies and to determine the X-ray spectrum, while ultra violet images allow
us to characterize the coronal flaring region. The magnetic field evolution of
the active region is analyzed using Michelson Doppler Imager magnetograms. The
burst is detected at all available radio-frequencies. X-ray images (between 12
keV and 300 keV) reveal two compact sources and 212 GHz data, used to estimate
the radio source position, show a single compact source displaced by 25′′ from
one of the hard X-ray footpoints. We model the radio spectra using two
homogeneous sources, and combine this analysis with that of hard X-rays to
understand the dynamics of the particles. Relativistic particles, observed at
radio wavelengths above 50 GHz, have an electron index evolving with the
typical soft–hard–soft behaviour.
###### keywords:
Radio Bursts, Association with Flares; Radio Bursts, Microwave; X-Ray Bursts,
Association with Flares; Flares, Relation to Magnetic Field; Chromosphere,
Active
## 1 Introduction
High frequency radio observations, above 50 GHz, bring information about
relativistic particles (see e.g. Ramatyetal:1994 and Trottetetal:1998).
Moreover, the efficiency of synchrotron emission, responsible for the radio
radiation, increases as the electron energy increases, contrary to the
bremsstrahlung mechanism which is the origin of the Hard X-ray (HXR) emission
[White et al. (2011)]. This makes observations at high frequencies very
attractive for the analysis of high energy particles. For typical magnetic
fields on the Chromosphere and mildly relativistic electrons, gyrosynchrotron
theory expects a peak frequency at approximately 10 GHz. Therefore the caveat
of submillimeter observations is that flare emission becomes weaker as the
observing frequency increases. At the same time, at high frequencies, earth
atmosphere becomes brighter and absorbs much of the incoming radiation.
Notwithstanding some X-class flares have shown a second spectrum besides the
microwaves spectrum, with an optically thick emission at submillimeter
frequencies, sometimes described as an upturn (see e.g. Kaufmannetal:2004,
Silvaetal:2007, Luthietal:2004b). Nonetheless, Cristianietal:2008 found, in a
medium size flare, a second radio component peaking around 200 GHz. We call
these cases double radio spectrum events.
Although different mechanisms were proposed to explain the double radio
spectrum events [Kaufmann and Raulin (2006), Fleishman and Kontar (2010)], the
conservative approach of two distinct synchrotron sources can fit reasonably
well to the observations [Silva et al. (2007), Trottet et al. (2008)]. We
note, however, that observations at higher frequencies are needed to
completely determine the radiation mechanism of those events that only show
the optically thick emission of the second component, like in
Kaufmannetal:2004 and, because of their strong fluxes, have much stringent
requirements.
Therefore, the double radio spectrum bursts may represent a kind of events
whose low frequency component is the classical gyrosynchrotron from mildly
relativistic particles peaking around 10 GHz, and the high frequency component
is also synchrotron emission with peak frequency around or above 50 GHz,
depending on the flare characteristics (in some cases above 400 GHz).
In this work we present a detailed analysis of a double radio spectrum burst
occurred during a GOES M class event on September 10, 2002 and observed in
radio from 1.415 to 212 GHz. We’ll show that the low frequency component is
well correlated with the HXR observed with RHESSI up to 300 keV, hence we can
study the dynamics of the mildly relativistic electrons inside the coronal
loop. On the other hand, the high frequency component, which is also well
represented by an electron synchrotron source, should be produced by a
different particle population and, likely, in a different place.
We first present the data analyzed, explain the reduction methods and give the
clues that justify the interpretations in Section sec:observations. The
spectral analysis, both at radio and Hard X-rays is the kernel of our work,
thus it deserves the entire Section sec:analysis. We divide the interpretation
of the event in two wide energy bands for the mildly relativistic and the
relativistic particles in Section sec:discusion. The consequences of our
interpretations are presented as our final remarks in Section sec:fin.
## 2 Observations
sec:observations
### 2.1 The data
sec:data
The impulsive phase of the solar burst SOL2002-09-10T14:53 started at 14:52:30
UT, in active region (AR) NOAA 10105 (S10E43) and lasted a few minutes. The
flare is classified as GOES M2.9 in soft X-Rays and 1N in H$\alpha$. The burst
was observed by various radiotelescopes around the world: (1) the United
States Air Force (USAF) Radio Solar Telescope Network (RSTN, Guidiceetal:1981)
at 1.415, 2.695, 4.995, 8.8 and 15.4 GHz with 1 s time resolution; (2) the
solar polarimeter at 7 GHz of the Itapetinga Observatory with 12 ms time
resolution [Kaufmann (1971), Correia, Kaufmann, and Melnikov (1999)]; (3) the
solar patrol radiotelescopes of the Bern University at 11.8, 19.6, 35 and 50
GHz with 100 ms time resolution (unfortunately, at that time, the channel at
8.4 GHz was not working); (4) the null interferometer at 89.4 GHz of the
University of Bern with 15 ms time resolution [Lüthi, Magun, and Miller
(2004)] and (5) the Solar Submillimeter Telescope (SST) at 212 and 405 GHz and
40 ms time resolution [Kaufmann et al. (2008)]. All telescopes, except the
SST, have beams greater than the solar angular size. The null interferometer
can be used to remove the Quiet Sun contribution. The SST beam sizes are 4′
and 2′ at 212 and 405 GHz respectively, and form a focal array that may locate
the centroid position of the emitting source and correct the flux density for
mispointing. Below we comment more on this. Since there are no overlapping
frequencies between the different instruments, we have to rely on the own
telescope calibration procedures, which were successfully verified along their
long operation history. All of them, except the SST, claim to determine the
flux density with an accuracy of the order of 10%. When applying the multi-
beam method [Giménez de Castro et al. (1999)] to SST data, the accuracy is of
the order of 20%. We have used these figures in the present work.
HXR data for this event were obtained with the Reuven Ramaty High Energy Solar
Spectroscopic Imager (RHESSI, Linetal:2002). RHESSI provides imaging and
spectral observations with high spatial (0.5′′) and spectral (1 keV)
resolution in the 3 keV–17 MeV energy range. We also used complementary data:
extreme–ultraviolet (EUV) images from the Extreme ultraviolet Imaging
Telescope (SOHO/EIT, Delaboudiniereetal:1995), and full disk magnetograms from
the Michelson Doppler Imager (SOHO/MDI, Scherreretal:1995). During September
10, 2002, EIT was working in CME watch mode. Observations were taken at half
spatial resolution and with a temporal cadence of 12 minutes, which is too low
compared to the flare duration to detect any kind of flare temporal evolution.
Early during the day, and up to $\sim$14:48 UT, data were taken at 195 Å. At
around the flare time, from 15:00 UT, and later, images were obtained at 304
Å. There is only one image at this wavelength displaying the flare
brightening. The Transition Region and Coronal Explorer (TRACE) was not
observing this AR.
### 2.2 The photospheric evolution around the flare time
sec:photosphere
The active region, where the flare occurred, arrived at the eastern solar limb
on September 7, 2002, as an already mature region. It appeared formed by an
unusually strong leading negative polarity and a weaker dispersed positive
following polarity. AR 10105 is the recurrence of AR 10069 seen on the disk
during the previous solar rotation.
Figure 1.: Magnetic field evolution seen in MDI line of sight magnetograms
during September 9 and 10, 2002. All images have been rotated to solar Sun
center and saturated above (below) 300 G (-300 G). Notice that the field
intensity in the negative spot is such that saturation is very clear in MDI
data in Figure fig:MDI-HXR-sst. The arrows in panel (a) point to a chain of
small bipoles located in the moat region surrounding the spot. The green box
in panel (c) represents the field of view of the Figure fig:MDI-HXR-
sst.fig:field-evolution
Adjacent to AR 10105, on its west, there is a dispersed bipolar facular region
and a new AR to its SE (AR 10108, see Figure fig:field-evolutiona). Vigorous
moving magnetic feature (MMF, see HarveyHarvey:1973) activity is seen (compare
Figure fig:field-evolutiona to Figure fig:field-evolutionb and see the movie
mag-field-evol.mpg that accompanies this paper) around the big leading spot
and a moat boundary is visible in Figure fig:field-evolutiona mainly in a E-SE
segment, where magnetic field aggregates at what appears the confluence of
three supergranular cells. In particular, we have indicated within arrows in
Figure fig:field-evolutiona a chain of small bipoles. Though this image is
affected by projection effects, these bipolar regions persist in MDI
magnetograms. Figure fig:field-evolutionb shows that the eastern negative
polarity of the eastermost bipole rotates counter-clockwise around the
positive one. This motion should increase the magnetic field shear in the
region and, at the same time, favour the interaction with nearby bipoles in
the MMFs to its north. Figure fig:field-evolutionc (see a zoom of this image
in Figure fig:MDI-HXR-sst) shows the magnetic field at 14:27 UT on September
10, this is the closest in time magnetogram to the flare occurrence at 14:52
UT. In this magnetogram the negative polarity to the east (which is along the
moat boundary) has decreased in size as flux cancellation proceeds with the
positive bipole polarity. Along this period, flux cancellation also proceeds
between this positive polarity and the nearby negative polarity of the bipole
to the west. Finally, this small bipole is hardly observable by the end of the
day (see Figure fig:field-evolutiond). The magnetic field evolution just
discussed lets us infer the origin of the flare eruption and the location of
the pre-reconnected set of loops and the reconnected loops of which only one
is visible in the XR and EUV data (see Figure fig:MDI-HXR-sst and Section
fig:MDI-HXR-sst).
### 2.3 Time profiles in radio and X-rays
sec:profiles
Figure 2.: Radio data at selected frequencies. The vertical bar on the 2.695
GHz plot indicates the short pulse. We include also 75–300 keV HXR from RHESSI
and 1-8 Å SXR from GOES.fig:profiles
The time profiles are shown in Fig fig:profiles. The flare appears as a single
impulsive peak in GOES data with a start at around 14:50 UT, peak time at
14:56 UT and a total duration of about 20 minutes. The soft-X rays (SXR)
emission has a maximum flux $2.9\times 10^{-5}\ \mathrm{Wm}^{-2}$ (M2.9). We
have derived a temperature $T_{goes}=14.4$ MK and emission measure
$EM_{goes}=3.4\times 10^{48}\ \mathrm{cm}^{-3}$ using the Chianti 6.0.1
Coronal Model, during the HXR impulsive peak interval (14:52:50 - 14:53:00).
The impulsive phase is clearly visible at all radio frequencies covering
almost two decades from 1 to 200 GHz, starting at approximately 14:52:30 UT. A
short pulse is observed at the beginning of the event between 14:52:50 and
14:53 UT (indicated by a vertical bar on the 2.695 GHz panel) with a flux in
excess of the main emission well defined between 1.415 and 4.995 GHz, above
this range it is hardly distinguishable. The short duration and the narrow
spectrum of this pulse reminds us of a similar one that also occurred during
the rising phase of an event (SOL2002-08-30T13:28, GimenezdeCastroetal:2006).
At 212 GHz, the flux density time profile is composed of a single peak, with
maximum at 14:53:20 UT with a total duration of around 2 minutes. There is no
clear evidence of an extended phase like in Luthietal:2004a and
Trottetetal:2011. Besides, the HXR emission above 50 keV ends around the peak
time of the radio emission, although the start time of both are similar, below
30 keV the emission extends longer. We note that this is one of the weakest
GOES events that has a clear submillimeter counterpart. Indeed, as a
comparison, the flare SOL2002-08-30T13:28 (X1.5) had a 212 GHz peak density
flux of 150 s.f.u. [Giménez de Castro et al. (2009)]
We observe a frequency dependent time delay in the radio data, which in the
four second integrated HXR data is not evident. Normalized time profiles
integrated in one second bins are shown in Figure fig:delays (top) at each
frequency with different colours. This figure gives the impression that each
different frequency starts at slightly different times. To further investigate
this, we analyzed the rising times at different frequencies. Since the start
time is subjected to big uncertainties, we compared the time elapsed between a
reference time and the time at which the emission reaches a certain normalized
flux density. Moreover, we find more accurate to measure delays during the
rising phase than during the, almost flat, maximum. We used two different
normalized levels, 10% and 50%. For each frequency $\nu$ we found
$t_{10}(\nu)$ and $t_{50}(\nu)$, the times when the flux density reaches the
10% and 50% relative level respectively. The reference is the 75–300 keV
emission, with $t_{10}^{*}$ and $t_{50}^{*}$ defined when the HXR flux is 10%
and 50% of the peak flux respectively. Delays are then defined as
$\Delta_{10}(\nu)=t_{10}(\nu)-t_{10}^{*}$ and
$\Delta_{50}(\nu)=t_{50}(\nu)-t_{50}^{*}$. To some extent $t_{10}^{*}$ and
$t_{50}^{*}$ are arbitrarily defined, but in doing so we simultaneously have
an indication of the relation between radio and HXR. Delays are determined
with 0.5 s accuracy, derived from the worst time resolution we have in the
radio data. The result is shown in Figure fig:delays (bottom). Above 10 GHz
there is a continuous shift, which is, within the uncertainties, independent
of the level (10% or 50%). This is qualitatively observed in the time profiles
(Fig. fig:delays, top). Previous works have shown that
millimeter/submillimeter emission is delayed from microwaves (see e.g.
Limetal:1992,Trottetetal:2002, Luthietal:2004a), but this is the first time to
our knowledge that a spectrum of the delay is presented. Below 7 GHz the
presence of the short pulse distorts this trend.
Figure 3.: Top: normalized flux densities from 1.415 to 212 GHz. Normalization
fluxes are taken in the interval 14:5310 – 14:5330 UT. The dashed curve is the
normalized 75–300 keV flux. Dotted horizontal lines, represent 10% and 50%
levels. Bottom: Measured delays between radio and HXR. The horizontal gray
band represents the uncertainty at 75–300 keV rising levels.fig:delays
### 2.4 X-Ray imaging and radio-source positions
sec:images
We produced X-ray images using RHESSI data. These images are constructed with
the PIXON algorithm [Hurford et al. (2002)] with an accumulation time of four
seconds (from 14:52:52 UT to 14:52:56 UT) for lower energy bands (below 80
keV), and twelve seconds (from 14:52:52 UT to 14:53:04 UT) for higher energy
bands (above 100 keV). We used collimators 1–6 and a pixel size of 0.5′′. We
considered the following energy bands: 12–25 keV, 40–80 keV, 100–250 keV,
150–300 keV and 250–300 keV. The images show two sources clearly defined (see
Fig. fig:MDI-HXR-sst). There is a loop or arcade connecting the two footpoints
visible in the 12-25 keV energy band image (see Fig. fig:MDI-HXR-sst).
Figure 4.: MDI magnetogram closest in time to the event. The black lines
denote magnetic polarity inversion lines. Blue contours are RHESSI 12–25 keV
for the interval 14:52:52 to 14:52:56 UT, magenta lines are RHESSI 40–80 keV
contour levels at 20%, 50% and 80% of the image maximum for the same time
interval. A contour corresponding to 60% of the maximum intensity of the
closest in time EIT image in 304 Å is also shown (red dashed contour). Green
points indicate the centroid positions of the submillimeter emission from
14:52:40 to 14:54:40 UT (impulsive peak in radio) averaged every 0.4 s. The
yellow dot–dashed circle denotes the absolute uncertainty in the determination
of the radio–source positions ($\sim$30′′), mainly limited by pointing
accuracy.fig:MDI-HXR-sst
Fig. fig:MDI-HXR-sst shows the magnetogram closest in time ($\sim$14:27 UT) to
the flare overlaid by RHESSI contours in the 12–25 (blue contours) and 40–80
(magenta contours) keV energy bands. Also included is a 60% of the maximum
intensity contour of the closest in time 304 Å EIT image (red lines). Black
thin contours correspond to the magnetic polarity inversion line, i.e. they
separate positive from negative line of sight magnetic field. It is evident
that both RHESSI footpoints overlay opposite sign polarities, with one of them
located on the positive polarity, the evolution of which we discussed in
Section sec:photosphere, and the other one on the negative polarity, north of
it. Whithin the uncertainties of the reconstruction method, we did not observe
any displacement in the HXR sources.
We determined the centroid of the sources emitting at 212 GHz every 40 ms
during the impulsive phase of the event, assuming that the source size is
small compared to the SST beam sizes (For a review of the multi–beam method
see GimenezdeCastroetal:1999). At the same time, we corrected the flux density
for mispointing. We note that since beam sizes are of the order of arcminutes,
when they are not aligned with the emitting source the flux obtained from a
single beam may be wrong.
In Figure fig:MDI-HXR-sst we have superimposed the positions of the 212 GHz
burst emission centroids averaged every 0.4 s. They seem to be separated by
25′′ from one of the HXR footpoints. The dot–dashed yellow circle represents
the absolute uncertainty in the determination of the radio source position
($\sim$30”); this uncertainty is mainly due to the radiotelescope pointing
accuracy. Because of this large absolute uncertainty, it is not possible to
determine where the submillimeter source is located inside the loop, at the
footpoints or at the loop top. The two X-ray sources in the flare seem to be
associated with the bright ultraviolet enhancement seen by EIT encircling
them. This brightening would correspond to an upper chromospheric loop also
traced by the lowest energy RHESSI isocontours. Timing analysis does not
reveal a displacement of the 212 GHz emission centroids in a privileged
direction during the impulsive phase of the event. We can infer the
compactness of the submillimeter source or sources because of the low level of
spread, lower than 10′′.
## 3 Spectral Analysis
sec:analysis
Figure 5.: Radio spectra ad selected times. The data, integrated in 1 s bins,
are represented with their error bars. Dashed curves correspond to the low
frequency fitting and dot-dashed curves to the high frequencies fitting.
Continuous lines represent the total solution (low frequency + high
frequency). fig:rspec-fit
Figure fig:rspec-fit shows integrated radio spectra at selected one second
time intervals. The existence of composed spectra is evident after 14:52:50
UT. We distinguish two components that we call the low frequency and high
frequency components. The low frequency one has a peak frequency around 10
GHz, while the high frequency is maximum at around 35 GHz. Because only the
high frequency component shows clearly its optically thin part, we have used
frequencies 50, 89.4 and 212 GHz to compute the spectral index $\alpha_{hf}$
along the event. Figure fig:delta bottom shows $\alpha_{hf}$ in function of
time together with error bars. During the maximum of the event the spectral
index remains stable, but at the beginning and the end a softening is
observed, i.e. the index has an SHS behaviour. To get insight into the
characteristics of electron populations that produced this emission, we have
fitted the data to two homogeneous gyrosynchrotron sources using the
traditional Ramaty:1969 procedure. The suprathermal electron distributions are
represented by a power law with electron indices $\delta_{l\\!f}$ and
$\delta_{h\\!f}$ for the low frequency and high frequency sources
respectively. At each time interval, a model was fitted to the low frequency
and another model to the high frequency, the sum of both was compared to the
data until the best solution was obtained. In general we fixed the source
size, the magnetic intensity, medium density and the electron energy cutoffs
(see Table tbl:parametros). We allowed changes only in the electron indices
$\delta_{l\\!f}$, $\delta_{h\\!f}$ and the total number of accelerated
electrons. The fittings are shown in the Figure fig:rspec-fit. It is evident
that below 9 GHz the fittings are rather poor, which is an indication that the
source is not totally homogeneous [Klein, Trottet, and Magun (1986)],
nonetheless we may be confident on the electron index and number, which depend
on the optically thin part, and on the magnetic field which is defined by the
peak frequency. Moreover, the good agreement with similar parameters derived
from HXR is another indication of the goodness of the fittings (see below).
| Low Frequency | High Frequency |
---|---|---|---
Parameter | Component | Component | Unit
Mag. Field | 380 | 2000 | G
Diameter | 18 | 5 | arc sec
Height | $10^{9}$ | $10^{8}$ | cm
Low En. Cutoff | 20 | 20 | keV
High En. Cutoff | 10 | 10 | MeV
Maximum Total Number of $e^{-}$ | $4.7\times 10^{37}$ | $7.6\times 10^{34}$ |
Table 1.: Fixed parameters used to fit the spectral data and maximum total
electron number derived from the computations.tbl:parametros
Figure 6.: HXR spectra at selected interval times. Data are integrated in 4 s
bins and represented with error bars. Blue curves represent the thermal
solution, green curves are the broken power law solution and red curves are
the total solution (Thernal+broken power law). Magenta curves represent the
expected emission from the relativistic electrons used to fit the radio high
frequency component. Dashed curves are the background.fig:xspec-fit
HXR spectra were taken from 14:52:00UT to 14:54:00UT in 4 second intervals,
using front detectors 1, 3, 4, 5, 6, 8 and 9, for energies between 3 and 290
keV. We excluded the time bin when there was a change in attenuator state,
from 0 to 1. Due to the high flare activity during this RHESSI observation
interval, we selected the background emission from the subsequent night period
(15:18:48 – 15:22:12 UT). Figure fig:xspec-fit shows the photon flux spectra
at selected time intervals along with the fitting used to calibrate the data.
Using standard OSPEX111see ’OSPEX, Reference Guide’, Kim Tolbert, at
http://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex˙explanation.htm
procedures we applied a model of a thermal source, including continuum and
lines, and a double power law. We found significant counts up to 300 keV,
during peak time (14:52:59 UT). The thermal component is well represented by
an isothermal source with a mean temperature $T_{hsi}=20.7$ MK and emission
meassure $EM_{hsi}=1.2\times 10^{48}\ \mbox{cm}^{-3}$ during the HXR impulsive
peak interval (14:52:50 - 14:53:00). The break energy remains always below 80
keV, and the electron index below it is always harder than the index above.
The later was used to compare with radio data because electrons with energies
$<100$ keV affect the gyroemission only in the optically thick part of the
spectrum [White et al. (2011)] which we did not try to fit as noted before. We
also have to take into account that HXR emission depends on the electron flux,
therefore for non relativistic particles we should add 0.5 to $\delta_{X}$ to
compare with radio [Holman et al. (2011)], and, since the HXR spectra are
computed below 300 keV, we should compare $\delta_{X}$ with $\delta_{l\\!f}$,
because relativistic electrons are needed to produce gyrosynchrotron emission
for frequencies above 50 GHz. In the bottom panel of Figure fig:delta we can
see the evolution of electron indices. We observe that $\delta_{X}$ (dot-
dashed curve) remains stable at the beginning and increases at the end. The
low frequency index $\delta_{l\\!f}$ remains stable until 14:53:25 UT and is
comparable to $\delta_{X}$, then a sudden change takes place making it harder.
On the other hand, the high frequency index (continuous line) $\delta_{h\\!f}$
is much harder than $\delta_{l\\!f}$ and $\delta_{X}$ as it was observed in
previous works (see e.g. GimenezdeCastroetal:2009, Trottetetal:1998) and it
evolves in the same way as the spectral index $\alpha_{hf}$.
Figure 7.: Top panel: time profiles at 15.8 GHz (dark gray), 75–300 keV (light
gray) and 212 GHz (continuous line). The bottom panel shows the time evolution
of the electron indices together with the values and error bars of the high
frequency optically thin spectral index $\alpha_{h\\!f}$. fig:delta
## 4 Discussion
sec:discusion
The optically thin emission above 50 GHz is produced by relativistic
particles, while the optically thick emission at microwaves and the HXR
observed by RHESSI are emitted by mildly relativistic particles. We roughly
divide the analysis in these two energy bands.
### 4.1 Dynamics of mildly relativistic electrons
sec:mild-relativistic
In order to understand the dynamics of the mildly relativistic electrons we
compare the HXR emission, which it was observed up to 300 keV with the low
frequency radio data. The comparison of the temporal evolution of both sets of
data (see Figures fig:profiles and fig:delta) supports the existence of
trapped electrons because: 1) the duration of the impulsive phase in HXR is
shorter than in radio and 2) the peak time in HXR occurs before the radio
peak, even at low frequencies. Therefore, the HXR time profile is not
necessarily the representation of the injected electrons, since there are
transport effects along the loop, or at least, HXR may represent the injected
electrons that precipitate directly, without being subject to trapping.
We can use the spectral analysis to derive the rate of the injected electrons
in the emitting area in function of time. To do so, we write a simplified
continuity equation that depends only on time, since we are not interested on
how the electron distribution changes in terms of energy, pitch angle, or
depth. In this simplified model, we are interested only in the total
instantaneous number of electrons, $N(t)$, inside the magnetic loop,
incremented by a source, $Q(t)$, from the acceleration site and decremented by
the precipitated electrons, $P(t)$. Therefore, the continuity equation should
be
$\frac{dN(t)}{dt}=Q(t)-P(t)\ .\ilabel{eq:cont}$ (1)
Integrating the above equation in the interval $(t_{i},t_{i+1})$ (with
$i=0,1,2\dots$) and solving for $Q(t_{i})$ yields
$Q(t_{i})\Delta t_{i+1}=N(t_{i+1})-N(t_{i})+P(t_{i})\Delta t_{i+1}\
,\ilabel{eq:injection}$ (2)
with $\Delta t_{i+1}=t_{i+1}-t_{i}$. Since we are comparing $<300$ keV
emission with gyrosynchrotron, we can identify the instantaneous number of
electrons inside the loop $N(t)$ with the trapped particles emitting the lower
frequency component. On the other hand, the particles leaving the volume
$P(t)$ produce the HXR emission observed by RHESSI. In our picture, the low
frequency component is produced all along a loop with a length of $10^{9}$ cm,
while the HXR emission is produced in a narrow slab (see e.g. Holmanetal:2011)
with a very small surface (see Figure fig:MDI-HXR-sst); therefore, we can
neglect the gyroemission produced within this small volume. Furthermore, no
change would be appreciated if one includes the particles responsible for the
high frequency component since they are two orders of magnitude less than
those that produce the low frequency component. (See Table tbl:parametros)
We divided the event in 10 second intervals and assumed that within these
intervals the conditions do not change. The computation of Equation
eq:injection is straightforward and the result is presented in Figure
fig:injection. Since $>100$ keV data have good S/N ratio only between 14:52:50
UT and 14:53:30 UT, (see Figure fig:xspec-fit) the analysis is restricted to
this interval, although is clear from the time profiles that there are
emitting particles before and after. We observe a continuous injection with
two peaks, one at the beginning and the second during the decay of the
impulsive phase. To verify our results, we sum the precipitated electrons,
$\sum_{i}P(t_{i})\Delta t=3.4\times 10^{37}$, and we compare this number with
the maximum number of electrons existing instantaneously inside the loop,
$\max[N(t_{i})]=4.7\times 10^{37}$. The difference between these two numbers
maybe due to the fact that we are limited to the time interval in which the
HXR data are statistically meaningful; therefore, we cannot track the
precipitation until the end of the gyrosynchrotron emission.
Figure 8.: Top: time profiles at 15.4 (blue) and 75–300 keV (red). Bottom: the
precipitated electrons $P(t)\Delta t$ (green); the instantaneous number of
electrons, $N(t)$ (blue), and the injected electrons $Q(t)\Delta t$
(red).fig:injection
We observe that changes in $\delta_{l\\!f}$ (Fig. fig:delta) can be related to
the injections occurred at around 14:52:51 UT and 14:53:15 UT (Figure
fig:injection). The electron index $\delta_{X}$ lies around 5, as
$\delta_{l\\!f}$ for the same period, although, it increases slowly until
14:53:20 UT when it suddenly softens by around 0.5. The difference in time
evolution of $\delta_{X}$ and $\delta_{l\\!f}$ is an indication that the
softening of the former is a consequence of the trapping. Since
$\delta_{l\\!f}$ has two constant values, we conclude that it is not affected
by the medium. In the above analysis we rely upon the parameters derived from
the radio and HXR data fittings. Although we do not claim that the obtained
solutions are unique, the fact that two independent fittings give very
comparable results, give us confidence on them.
The progressive delay of the radio emission observed in Figure fig:delays must
be interpreted differently depending on the frequency range. Between 1 and 5
GHz, the short pulse dominates the emission during the rising phase;
therefore, its contribution should be removed to asses the delay of the main
emission. This may lead to ambiguous results, hence, we preferred not to
analyze delays in this range. In the range between 9 and 30 GHz lies the peak
frequency, i.e. the optical opacity $\tau$ is approximately 1, hence, the
gyrosynchrotron self absorption is critical. The peak frequency ($\tau\simeq
1$) of the low frequency component shifts from approximately 7 GHz to 15 GHz
from 14:52:40 UT to the peak time around 14:5259 UT (Fig. fig:rspec-fit).
Since there is no change in the magnetic field, this can be interpreted by the
accumulation of accelerated electrons inside the loop due to the trapping that
increases its density, which is the dominant factor of the self-absorption
mechanism. The shift makes the low frequencies more absorbed and thus
increases the relative importance of higher frequencies. Therefore, even with
a rather constant $\delta_{l\\!f}$ (Fig. fig:delta) the progressive delay
should be observed. In our fittings at 14:52:40 UT we have $\tau(\nu=7\
\mbox{GHz})=0.7$ and $\tau(\nu=10\ \mbox{GHz})=0.15$. Later on, during peak
time $\tau(\nu=7\ \mbox{GHz})=60$ and $\tau(\nu=10\ \mbox{GHz})=3.7$. We note
that the emission is proportional to $1-e^{-\tau}$, hence, we have a relative
amplification of $(1-e^{-60})/(1-e^{-0.7})\simeq 1.4$ at 7 GHz, while it is
$(1-e^{-3.7})/(1-e^{-0.15})\simeq 7.5$ at 10 GHz. The relative amplification
changes the rate at which the emission rises and, therefore, the time when the
signal reaches a certain level with respect to its maximum.
### 4.2 Relativistic Electrons
sec:relativistic
The centroid position of the emitting source at 212 GHz remains quite stable
during the flare which may imply that the source is compact. Moreover, it is
placed 25′′ far from one of the HXR footpoints (Figure fig:MDI-HXR-sst). A
similar result was obtained by Trottetetal:2008 for the SOL2003-10-28T11:10
flare; during the impulsive phase (interval B in their work) the centroid
positions of the 210 GHz emission lie at approximately 10′′ from the center of
one of the HXR footpoints (250–300 keV), but are coincident with the location
of precipitating high energy protons with energies above 30 MeV seen in
$\gamma$-ray imaging of the 2.2 MeV line emission. Since for our work we do
not have $\gamma$-ray imaging to compare with, we should be cautious because
the uncertainty in position is of the order of the position shift.
Since we observe the optically thin part of the high frequency spectra, the
obtained $\delta_{h\\!f}$ is not affected by the medium and, within data
uncertainties, it must be correct. On the other hand, we assumed a standard
viewing angle $\theta=45^{\circ}$ which gives us a mean value of the magnetic
field $B$ and total number of electrons $N$. Increasing (reducing) $\theta$
results in smaller (larger) $B$ and $N$. Although we cannot rightly evaluate
$\theta$ with our data, we do not expect an extreme value for it since the AR
is located not far from Sun center (E43). Furthermore we did consider an
isotropic electron distribution. The total number of accelerated electrons is
$8\times 10^{34}$ during peak time, and they should not produce enough
bremsstrahlung flux to be detected by RHESSI detectors. We confirmed this by
computing its HXR emission using the bremthick222Developed by G. Holman, last
revision May 2002. Obtained from the RHESSI site:
http://hesperia.usfc.nasa.gov/hessi/modelware.htm program (magenta curves in
Figure fig:xspec-fit) which is orders of magnitude smaller than the mildly
relativistic electron emission and remains below (except during one time
interval) the background.
The spectral index of the high frequency component $\alpha_{hf}$ shows a SHS
behaviour, but in this case we cannot conclude whether its origin comes from
the acceleration mechanism or from the interaction with the medium as before.
We tend to think that the former should be the cause, since these are
relativistic particles and their interaction with the medium should be less
effective. The emission must come from a compact region with a strong magnetic
field and the electron index $\delta_{h\\!f}$ should be harder than
$\delta_{X}$ and $\delta_{l\\!f}$, as is the case. These arguments support the
evidence of the existence of a separated source where relativistic electrons
are the responsible for the emission.
The progressive delay of the radio emission above 50 GHz (Figure fig:delays)
can be interpreted considering the initial hardening of the spectral index
$\alpha_{hf}$. If it is due to the acceleration mechanism that accelerates
first the low energy particles and later the high energy particles, then, the
progressive delay is a consequence.
## 5 Final Remarks
sec:fin
From our simple continuity model, we inferred the evolution of injected number
of electrons, appearing as a continuous injection with two distinctive pulses
separated by approximately 30 seconds. From the timing of those pulses, the
first one produces the HXR and initiates the radio low frequency emission. The
second pulse, while slightly stronger than the first, builds up into the radio
emission but do not contributes to generate more HXR. This description
suggests that the two injection might have different initial pitch angle
distributions, because our fittings do not show magnetic field changes during
the burst. The first injection might be formed by an electron beam aligned
with the magnetic field direction, a fraction of the population is trapped by
magnetic mirroring, while the other fraction enter the loss cone and
precipitates, producing HXR. The second pulse should then be formed by a beam
with a wider pitch angle distribution, or even isotropic, keeping most of the
electrons trapped, producing radio emission, with little precipitation (no
significant increase in HXR).
From our observations we conclude that the radio spectra cannot be explained
by an homogeneous gyrosynchrotron source, therefore we adopted a model based
on two homogeneous sources since it is the simplest hypothesis that fits
reasonably the data. Although we don’t have images to support this assumption,
it is plausible that it is the case. As shown by the magnetic field evolution
discussed in Section 2.2, we can conclude that the flare originates by the
interaction of magnetic loops anchored in the MMF polarities. Magnetic energy
is probably increased in the configuration by shearing motions, in particular,
the rotation of the negative polarity around the positive one. After magnetic
reconnection occurs, two sets of reconnected loops should be present. In the
example we have analyzed, we observe a set of loops in SXR (and also in EUV)
with two HXR footpoints. Considering our description in Section
sec:photosphere, we speculate that the second set of reconnected loops (not
visible) could be anchored in the higher magnetic field positive polarity
located north of the northern HXR footpoint and the negative polarity that
rotates around the positive bipole polarity. Such a set of reconnected loops
would have a larger volume than the ones that are observed; therefore,
considering that the same amount of energy is injected in both sets, the
emission in the second one could be less intense and, then, not visible. This
second magnetic structure is also suggested by the 25′′ displacement of the
emitting source at 212 GHz respect to one of the HXR footpoints (Figure
fig:MDI-HXR-sst). While it has been shown that the reconnection of many
different magnetic loops is more efficient to accelerate high energy electrons
(see Trottetetal:2008 and references therein), we might conclude that this
complex mechanism operates even in medium size events as the one we analyzed
in this work.
Another possible scenario is a loop structure where the low frequency source
represents the coronal part of the loop (with a lower effective magnetic field
strength), and the high frequency source represents the low coronal or
chromospheric footpoints of the loop (with a higher effective magnetic field
strength). Melnikovetal:2011, using simulations of electron dynamics and
gyrosynchrotron emission in a loop structure, demonstrated that two spectral
components can be produced from one single loop, where mildly-relativistic
electrons produce microwave emission in the loop, while relativistic electrons
produce higher frequency (reaching sub-THz frequencies) emission from the
footpoints.
#### Acknowledgements
CGGC is grateful to FAPESP (Proc. 2009/18386-7). CHM acknowledge financial
support from the Argentinean grants UBACyT 20020100100733, PIP 2009-100766
(CONICET), and PICT 2007-1790 (ANPCyT). PJAS is grateful to FAPESP (Proc.
2008/09339-2) and to the European Commission (project HESPE
FP7-2010-SPACE-1-263086). GDC and CHM are members of the Carrera del
Investigador Científico (CONICET), CGGC is level 2 fellow of CNPq and
Investigador Correspondiente (CONICET).
## References
* Correia, Kaufmann, and Melnikov (1999) Correia, E., Kaufmann, P., Melnikov, V.: 1999, Itapetinga 7 GHz polarimeter. In: Bastian, T., Gopalswamy, N., Shibasaki, K. (eds.) Proc. of the Nobeyama Symposium, 263 – +.
* Cristiani et al. (2008) Cristiani, G., Giménez de Castro, C.G., Mandrini, C.H., Machado, M.E., Silva, I.D.B.E., Kaufmann, P., Rovira, M.G.: 2008, A solar burst with a spectral component observed only above 100 GHz during an M class flare. A&A 492, 215 – 222. doi:10.1051/0004-6361:200810367.
* Delaboudiniere et al. (1995) Delaboudiniere, J.-P., Artzner, G.E., Brunaud, J., Gabriel, A.H., Hochedez, J.F., Millier, F., Song, X.Y., Au, B., Dere, K.P., Howard, R.A., Kreplin, R., Michels, D.J., Moses, J.D., Defise, J.M., Jamar, C., Rochus, P., Chauvineau, J.P., Marioge, J.P., Catura, R.C., Lemen, J.R., Shing, L., Stern, R.A., Gurman, J.B., Neupert, W.M., Maucherat, A., Clette, F., Cugnon, P., van Dessel, E.L.: 1995, EIT: Extreme-Ultraviolet Imaging Telescope for the SOHO Mission. Sol. Phys. 162, 291 – 312.
* Fleishman and Kontar (2010) Fleishman, G.D., Kontar, E.P.: 2010, Sub-Thz Radiation Mechanisms in Solar Flares. ApJ 709, L127 – L132. doi:10.1088/2041-8205/709/2/L127.
* Giménez de Castro et al. (1999) Giménez de Castro, C.G., Raulin, J.-P., Makhmutov, V.S., Kaufmann, P., Costa, J.E.R.: 1999, Instantaneous positions of microwave solar bursts: Properties and validity of the multiple beam observations. A&AS 140, 373 – 382.
* Giménez de Castro et al. (2006) Giménez de Castro, C.G., Costa, J.E.R., Silva, A.V.R., Simões, P.J.A., Correia, E., Magun, A.: 2006, A very narrow gyrosynchrotron spectrum during a solar flare. A&A 457, 693 – 697. doi:10.1051/0004-6361:20054438.
* Giménez de Castro et al. (2009) Giménez de Castro, C.G., Trottet, G., Silva-Valio, A., Krucker, S., Costa, J.E.R., Kaufmann, P., Correia, E., Levato, H.: 2009, Submillimeter and X-ray observations of an X class flare. A&A 507, 433 – 439. doi:10.1051/0004-6361/200912028.
* Guidice et al. (1981) Guidice, D.A., Cliver, E.W., Barron, W.R., Kahler, S.: 1981, The Air Force RSTN System. In: Bulletin of the American Astronomical Society, Bulletin of the American Astronomical Society 13, 553 – +.
* Harvey and Harvey (1973) Harvey, K., Harvey, J.: 1973, Observations of Moving Magnetic Features near Sunspots. Sol. Phys. 28, 61 – 71. doi:10.1007/BF00152912.
* Holman et al. (2011) Holman, G.D., Aschwanden, M.J., Aurass, H., Battaglia, M., Grigis, P.C., Kontar, E.P., Liu, W., Saint-Hilaire, P., Zharkova, V.V.: 2011, Implications of X-ray Observations for Electron Acceleration and Propagation in Solar Flares. Space Sci. Rev., 260 – +. doi:10.1007/s11214-010-9680-9.
* Hurford et al. (2002) Hurford, G.J., Schmahl, E.J., Schwartz, R.A., Conway, A.J., Aschwanden, M.J., Csillaghy, A., Dennis, B.R., Johns-Krull, C., Krucker, S., Lin, R.P., McTiernan, J., Metcalf, T.R., Sato, J., Smith, D.M.: 2002, The RHESSI Imaging Concept. Sol. Phys. 210, 61 – 86. doi:10.1023/A:1022436213688.
* Kaufmann (1971) Kaufmann, P.: 1971, The New Itapetinga Radio Observatory, from Mackenzaie University, São Paulo, Brazil. Sol. Phys. 18, 336\.
* Kaufmann and Raulin (2006) Kaufmann, P., Raulin, J.-P.: 2006, Can microbunch instability on solar flare accelerated electron beams account for bright broadband coherent synchrotron microwaves? Physics of Plasmas 13, 701 – 704. doi:10.1063/1.2244526.
* Kaufmann et al. (2004) Kaufmann, P., Raulin, J.-P., de Castro, C.G.G., Levato, H., Gary, D.E., Costa, J.E.R., Marun, A., Pereyra, P., Silva, A.V.R., Correia, E.: 2004, A New Solar Burst Spectral Component Emitting Only in the Terahertz Range. ApJ 603, L121 – L124.
* Kaufmann et al. (2008) Kaufmann, P., Levato, H., Cassiano, M.M., Correia, E., Costa, J.E.R., Giménez de Castro, C.G., Godoy, R., Kingsley, R.K., Kingsley, J.S., Kudaka, A.S., Marcon, R., Martin, R., Marun, A., Melo, A.M., Pereyra, P., Raulin, J.-P., Rose, T., Silva Valio, A., Walber, A., Wallace, P., Yakubovich, A., Zakia, M.B.: 2008, New telescopes for ground-based solar observations at submillimeter and mid-infrared. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 7012. doi:10.1117/12.788889.
* Klein, Trottet, and Magun (1986) Klein, K.-L., Trottet, G., Magun, A.: 1986, Microwave diagnostics of energetic electrons in flares. Sol. Phys. 104, 243 – 252.
* Lüthi, Lüdi, and Magun (2004) Lüthi, T., Lüdi, A., Magun, A.: 2004, Determination of the location and effective angular size of solar flares with a 210 GHz multibeam radiometer. A&A 420, 361 – 370.
* Lüthi, Magun, and Miller (2004) Lüthi, T., Magun, A., Miller, M.: 2004, First observation of a solar X-class flare in the submillimeter range with KOSMA. A&A 415, 1123 – 1132.
* Lim et al. (1992) Lim, J., White, S.M., Kundu, M.R., Gary, D.E.: 1992, The high-frequency characteristics of solar radio bursts. Sol. Phys. 140, 343 – 368. doi:10.1007/BF00146317.
* Lin and et al. (2002) Lin, R.P., et al.: 2002, The Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI). Sol. Phys. 210, 3 – 32.
* Melnikov, Costa, and Simões (2012) Melnikov, V., Costa, J.E.R., Simões, P.J.A.: 2012, Sub-THz flare emission: an evidence for relativistic electron beams in the dense chromosphere. Sol. Phys.. In preparation.
* Ramaty (1969) Ramaty, R.: 1969, Gyrosynchrotron Emission and Absorption in a Magnetoactive Plasma. ApJ 158, 753 – +.
* Ramaty et al. (1994) Ramaty, R., Schwartz, R.A., Enome, S., Nakajima, H.: 1994, Gamma-ray and millimeter-wave emissions from the 1991 June X-class solar flares. ApJ 436, 941 – 949.
* Scherrer et al. (1995) Scherrer, P.H., Bogart, R.S., Bush, R.I., Hoeksema, J.T., Kosovichev, A.G., Schou, J., Rosenberg, W., Springer, L., Tarbell, T.D., Title, A., Wolfson, C.J., Zayer, I., MDI Engineering Team: 1995, The Solar Oscillations Investigation - Michelson Doppler Imager. Sol. Phys. 162, 129 – 188.
* Silva et al. (2007) Silva, A.V.R., Share, G.H., Murphy, R.J., Costa, J.E.R., de Castro, C.G.G., Raulin, J.-P., Kaufmann, P.: 2007, Evidence that Synchrotron Emission from Nonthermal Electrons Produces the Increasing Submillimeter Spectral Component in Solar Flares. Sol. Phys. 245, 311 – 326. doi:10.1007/s11207-007-9044-0.
* Trottet et al. (1998) Trottet, G., Vilmer, N., Barat, C., Benz, A., Magun, A., Kuznetsov, A., Sunyaev, R., Terekhov, O.: 1998, A multiwavelength analysis of an electron-dominated gamma-ray event associated with a disk solar flare. A&A 334, 1099 – 1111.
* Trottet et al. (2002) Trottet, G., Raulin, J.-P., Kaufmann, P., Siarkowski, M., Klein, K.-L., Gary, D.E.: 2002, First detection of the impulsive and extended phases of a solar radio burst above 200 GHz. A&A 381, 694 – 702.
* Trottet et al. (2008) Trottet, G., Krucker, S., Lüthi, T., Magun, A.: 2008, Radio Submillimeter and $\gamma$-Ray Observations of the 2003 October 28 Solar Flare. ApJ 678, 509 – 514. doi:10.1086/528787.
* Trottet et al. (2011) Trottet, G., Raulin, J.-P., Giménez de Castro, C.G., Lüthi, T., Caspi, A., Mandrini, C., Luoni, M.L., Kaufmann, P.: 2011, Origin of the submillimeter radio emission during the time-extended phase of a solar flare. Sol. Phys.. submitted.
* White et al. (2011) White, S.M., Benz, A.O., Christe, S., Fárník, F., Kundu, M.R., Mann, G., Ning, Z., Raulin, J.-P., Silva-Válio, A.V.R., Saint-Hilaire, P., Vilmer, N., Warmuth, A.: 2011, The Relationship Between Solar Radio and Hard X-ray Emission. Space Sci. Rev., 263\. doi:10.1007/s11214-010-9708-1.
|
arxiv-papers
| 2012-08-29T13:00:00 |
2024-09-04T02:49:34.630239
|
{
"license": "Public Domain",
"authors": "C. G. Gim\\'enez de Castro, G. D. Cristiani, P. J. A. Sim\\~oes, C. H.\n Mandrini, E. Correia and P. Kaufmann",
"submitter": "Carlos Guillermo Gim\\'enez de Castro",
"url": "https://arxiv.org/abs/1208.5904"
}
|
1208.6105
|
# Eccentricity and elliptic flow at fixed centrality in Au+Au collisions at
$\sqrt{s_{\rm NN}}$=200GeV in AMPT model ††thanks: Supported by ′the
Fundametal Research Funds for Central Universities′ (GUGL 100237) and National
Natural Science Foundation of China (10835005).
WANG Mei-Juan1;1) CHEN Gang1 WU Yuan-Fang2 [email protected] 1 Physics
Department, China University of Geoscience, Wuhan 430074, China
2 Institute of Particle Physics, Huazhong Normal University, Wuhan 430079,
China
###### Abstract
In this paper, elliptic flow is studied at fixed centrality in Au+Au collision
at $\sqrt{s_{\rm NN}}$=200GeV in the AMPT model. It is observed that with the
participant increasing, elliptic flow has an increase or a decrease at
different fixed impact parameter, but it does not have a trivial fluctuation.
It is analyzed that the initial space anisotropy dominates the participant
dependence of elliptic flow in near-central collisions(b=5fm) and mid-central
collisions(b=8fm), while the interaction between particles can mainly answer
for the behavior of elliptic flow with participant in peripheral
collisions(b=12fm). To distinguish the pure geometrical effect, elliptic flow
scaled by initial eccentricity is studied. It is found that the ratio
$v_{2}/\epsilon$ increases with participant and reaches a saturation when the
participant is large enough, indicating that the collision system may reach
the local equilibrium.
###### keywords:
elliptic flow, initial eccentricity, at fixed centrality, local equilibrium.
###### pacs:
2
5.40.Cm, 28.75.Gz, 21.60.-n
## 1 Introduction
The discovery of a large azimuthal anisotropic flow of hadrons at RHIC
provides a conclusive evidence for the created dense partonic matter in
ultrarelativistic nucleus-nucleus collisions [2] [3] [4]. The strong
interaction medium in the collision zone can be expected to achieve a local
equilibrium and exhibit an approximately hydrodynamics flow [5] [6] [7].
Moreover, the study of anisotropic flow has a potential to offer insights into
the equation of state of the produced matter [8] [9].
The momentum anisotropy of final particles is generated due to the transverse
density gradient based on an initial geometry of an $"$almond-shaped$"$
overlap region produced in non-central collisions. The pressure gradient
converts the initial coordinate space asymmetry into the momentum anisotropy
of final particles, such as elliptic flow. The magnitude of the elliptic flow
depends on both initial spatial asymmetry in non-central collisions and the
subsequent interaction between the particles. Therefore, the study of elliptic
flow is very crucial to understand the properties of the dense matter formed
during the initial stage of heavy ion collisions [10] and parton dynamics [11]
at the relativistic heavy ion energies.
In this paper, we focus on the study of elliptic flow at fixed centrality
where the impact parameter b is a constant. The elliptic flow at fixed
centrality in Au+Au collisions at $\sqrt{s_{\rm NN}}$=200GeV is presented in
detail, such as the variation with the number of participating nucleons and
the behavior after scaled by the initial eccentricity. Here, the model we used
is the AMPT with string melting [12].
The layout of this paper is in the following. Section 2 briefly introduces the
AMPT model, a transport model based on parton level. In Section 3, we mainly
study the elliptic flow and initial eccentricity at fixed centrality in Au+Au
collisions, the dependence of participating nucleons, the ratio
$v_{2}/\epsilon$, and so on. Finally, a conclusion is given in Section 4.
## 2 A brief introduction to AMPT
The AMPT model [12] is based on parton level transport dynamics. There are two
versions of AMPT model, one is the default AMPT, and the other is AMPT with
string melting. It turns out that the default AMPT (v1.11) is able to give a
reasonable description on hadron rapidity distributions and transverse
momentum spectra observed in heavy ion collisions at both SPS and RHIC.
However, it fails to reproduce the experimental data about elliptic flow and
two-pion correlation function. On the other hand, the AMPT model with string
melting (v2.11) can well describe the elliptic flow and two-pion correlation
function [13] [14] but agrees bad with the hadron rapidity and transverse
momentum spectra. According to our demand, the AMPT with string melting is
chosen.
The AMPT model with string melting contains four main components: the initial
conditions, partonic interactions, conversion from the partonic to the
hadronic matter and hadronic interactions. The initial conditions are obtained
from the HIJING model [15]. The time evolution of partons is then modeled by
the ZPC parton cascade model [16]. After partons stop interacting, a simple
quark coalescence model is used to combine the two nearest quarks into a meson
and three nearest quarks (antiquarks) into a baryon (antibaryon). After
hadronization, scatterings among the resulting hadrons are described by a
relativistic transport (ART) model [17] which includes baryon-baryon, baryon-
meson and meson-meson elastic and inelastic scatterings.
In the following we will utilize the AMPT with string melting to generate
Au+Au collision events at $\sqrt{s_{\rm NN}}=200$ GeV. The parton cross
section is taken to be 10 mb.
## 3 The elliptic flow, initial eccentricity and their ratio
$v_{2}/\epsilon$
In Section 3.1, we will firstly study the elliptic flow in Au+Au collisions
from minibias events, and it is observed that the strongest collective
behavior appears in the mid-central collisions. In Section 3.2, we present the
participant dependence of elliptic flow and initial eccentricity at fixed
centrality. Finally, the participant dependence of the ratio $v_{2}/\epsilon$
is shown in Section 3.3.
### 3.1 The elliptic flow from minibias events
When two nuclei collide at nonzero impact parameter, their overlap in the
transverse plane has a short axis, parallel to the impact parameter, and a
long axis perpendicular to it. This initial space anisotropy is converted by
the pressure gradient into a momentum asymmetry, so more particles are emitted
along the short axis [5]. This magnitude of this effect is characterized by
the elliptic flow, defined as
$\displaystyle v_{2}=\langle\cos 2(\varphi-\Phi_{R})\rangle,$ (1)
where $\varphi$ is the azimuthal angle of an outgoing particle, $\Phi_{R}$ is
the azimuthal angle of the impact parameter, and angular brackets denote an
average over many particles and many events.
In this section, we study the elliptic flow signal as a function of the impact
parameter b(Fig1.(a)), the initial participants(Fig1.(b))and the final charged
particles(Fig1.(c)) in Au + Au collisions at $\sqrt{s_{\rm NN}}$=200GeV from
the AMPT model with string melting. Here, all the charged particles with
rapidity in the region y$\in$(-5,+5) are included.
From the figure we can see that the elliptic flow first increases, reaches its
maximum value and then decreases, indicating the strongest collective behavior
is produced in mid-central Au+Au collisions. The elliptic flow is built and
formed from the anisotropic geometrical effect and the hadron and parton
interaction between particles during the system expansion. In peripheral
collisions, we can obtain the strongest initial anisotropy, while the
interaction between produced particles is weak when the number of participants
is so small. In the near-central collisions, most nucleons can take part in
the collision, while the initial anisotropy is not large enough. It is
reasonable that that the strongest elliptic flow appears in the mid-central
collision where the interaction between particles is strong enough to convert
the initial space anisotropy into final momentum anisotropy completely.
The impact parameter (a), the participant (b) and the final charged particles
(c) dependence of elliptic flow in Au + Au collisions at 200 GeV with AMPT
model.
### 3.2 The participant dependence of the elliptic flow and initial
eccentricity at fixed centrality
In this section, we focus on the study of the elliptic flow at fixed
centrality where the impact parameter b is a constant. Here, we choose the
fixed impact parameter b =5fm, 8fm and 12fm, which responds to the case in the
near-central, the mid-central and the peripheral collisions respectively. In
general, the geometric overlap region of Au+Au collisions is fixed, elliptic
flow is expected to remain roughly unchanged or at most have a trivial
fluctuation.
The participant dependence of elliptic flow in Au+Au collisions at fixed
centrality at$\sqrt{s_{\rm NN}}$=200GeV is shown in the upper panel of Fig. 2.
From the figure, we can see that even if the impact parameter b is fixed, the
fluctuation of participating nucleons is so large that it can’t be ignored.
With the participant increasing, the elliptic flow has a slow increase in the
peripheral collisions for b =12fm (left), while decreases monotonously in the
mid-central and near-central collisions, e.g., b=8fm (middle) and b=5fm
(right). As a result, it is argued that the participant dependence of elliptic
flow at fixed centrality has some physics origin, but not the only statistical
fluctuation effect.
The participant dependence of elliptic flow(the upper panel) and eccentricity
(the lower panel) for 200 GeV Au+Au collisions in AMPT with string melting at
fixed impact parameter b=12fm (left), b=8fm (middle) and b=5fm (right).
It is widely believed that the anisotropic collective flow is mainly driven by
the initial eccentricity of the matter created in nuclear overlap zone. In
this point, we will study simply the participant dependence of the
corresponding initial eccentricity $\epsilon$ at fixed centrality. The Monte
Carlo Glauber model [18] is used to estimate the initial eccentricity from the
distribution of participant nucleons in the transverse plane. The participant
eccentricity is defined by [19]
$\displaystyle{\rm\varepsilon_{\rm
part}=\frac{\sqrt{(\sigma_{y}^{2}-\sigma_{x}^{2})^{2}+4\sigma_{xy}^{2}}}{\sigma_{x}^{2}+\sigma_{y}^{2}},}$
(2)
where $\sigma_{x}^{2}=\langle x^{2}\rangle-\langle x\rangle^{2}$ and
$\sigma_{y}^{2}=\langle y^{2}\rangle-\langle y\rangle^{2}$ are the variances
of the nucleon distribution in the x- and the y-direction, and
$\sigma_{xy}=\langle xy\rangle-\langle x\rangle\langle y\rangle$ is the
covariance of the position of participant nucleons .
The participant dependence of initial eccentricity at fixed centrality in
Au+Au collisions at $\sqrt{s_{\rm NN}}$=200 GeV is presented in the lower
panel of Fig. 2. From the figure, we can see the initial eccentricity
$\epsilon$ for the cases of b=12fm (left), 8fm (middle) and 5fm (right), all
have a decrease with the participant increasing.
Comparing the results of the upper and down panel in Fig. 2, it is obvious
that the elliptic flow and the initial eccentricity have a consistent decrease
with the increasing participant both for b=8fm (middle) and b=5fm (right),
while they have the contrast case for b=12fm (left). As we know, the elliptic
flow is formed and built from the initial anisotropy of the overlap region and
the subsequent interaction between the produced particles. On this basis, we
argue that the initial space anisotropy dominates the participant dependence
of elliptic flow in near-central (b=5fm) and mid-central collisions (b=8fm).
The subsequent interaction between the produced particles, which becomes
stronger with the participant increasing, can mainly answer for the
participant dependence of elliptic flow in peripheral collisions(b=12fm), such
as a slow increase.
### 3.3 The participant dependence of $v_{2}/\epsilon$ at fixed centrality
The participant dependence of the ratio $v_{2}/\epsilon$ in Au + Au collisions
at 200GeV at fixed impact parameter b=12fm (left), b=8fm (middle) and b=5fm
(right) in the AMPT model with string melting.
In order to distinguish the collision dynamics from the purely geometrical
effects, it has been suggested that the measured $v_{2}$ should be scaled by
the initial eccentricity of the nuclear overlap. If the produced matter
equilibrates, it behaves as an ideal fluid. Hydrodynamics predicts that
$v_{2}$ scaled by eccentricity $\epsilon$ has a saturation when the collision
system achieves the local equilibrium [20] [21]. However, if equilibration is
not incomplete, then the eccentricity scaling is indeed broken, e.g., in the
peripheral Au+Au collisions [22] [23].
The participant dependence of $v_{2}/\epsilon$ at fixed centrality in Au+Au
collisions at $\sqrt{s_{\rm NN}}$=200 GeV is shown in Fig. 3\. From the
figure, we can see that the ratio $v_{2}/\epsilon$ keeps increasing both for
b=12fm (left) and b=8fm (middle), while it becomes saturated in the near-
central collisions for b=5fm (right). It is reasonable that more participants
lead to stronger interaction between particles, hence a larger ratio
$v_{2}/\epsilon$ can be obtained in more central collisions. As expected in an
equilibrium scenario, the ratio $v_{2}/\epsilon$ in the near-central
collisions shows little sensitivity to the participants. This indicates that
the system created in 200GeV Au+Au near-central collision by the AMPT model
including both parton cascade and hadron scattering, may reach local
thermalization when the interaction between particles is strong enough.
## 4 Conclusion
To summarize, we have studied the elliptic flow and initial eccentricity and
their ratio at fixed centrality in Au+Au collisions at $\sqrt{s_{\rm NN}}=200$
GeV . It is observed that they have a monotonous behavior(an increase or a
decrease) with the participant increasing for different fixed impact parameter
b, but they do not have a trivial fluctuation. It is argued that the initial
eccentricity dominates the participant dependence of elliptic flow in the mid-
central collisions (b=8fm) and near-central collisions (b=5fm), while the
interaction between particles dominates the behavior of elliptic flow with the
participants in peripheral collisions (b=12fm). Moreover, it is found that the
elliptic flow scaled with initial eccentricity keeps increasing with the
participants. When the number of participants is large enough, the ratio
$v_{2}/\epsilon$ has a saturation, indicating that the collision system may
reach local equilibrium.
## References
* [1]
* [2] Marcus Bleicher and Horst Stöcker, Phys. Lett. B, 2002, 526, 309-314.
* [3] K. Adcox et al.(PHENIX Collaboration), Nucl. Phys. A, 2005, 757, 184-283.
* [4] John Adams et al.(STAR Collaboration), Nucl. Phys. A, 2005, 757, 102-183.
* [5] J. Y. Ollitrault, Phys. Rev . D, 1992, 46, 229\.
* [6] U. Heinz and P. Kolb, Nucl. Phys. A, 2002, 702, 269.
* [7] E. Shuryak, Prog. Part. Nucl. Phys., 2009, 62, 48.
* [8] J. Brachmann, S. Soff, A. Dumitru, H. Stöcker, J. A. Maruhn, W. Greiner, D. H. Rischke, Phys. Rev. C, 2000, 61, 024909\.
* [9] J. H. Chen et al, Phys. Rev. C, 2006, 74, 064902; T. Z. Yan et al, Phys. Lett. B, 2006, 638, 50.
* [10] P. F Kolb, P. Huovinen, U. Heinz and H. Heiselberg, Phys. Lett. B, 2001, 500, 232.
* [11] B. Zhang, M. Gyulassy and Che-Ming Ko, Phys. Lett. B, 1999, 455, 45.
* [12] Zi-Wei Lin, Che Ming Ko, Bao-An Li and Bin Zhang and Subrata Pal, Phys. Rev. C, 2005, 72, 064901.
* [13] Zi-Wei Lin and C. M. Ko, Phys. Rev. C, 2002, 65, 034904.
* [14] Zi-Wei Lin, C.M. Ko and Subrata Pal, Phys. Rev. Lett, 2002, 89, 152301.
* [15] X. N. Wang, Phys. Rev. D, 1991, 43, 104; M. Gyulassy and X. N. Wang, Comput. Phys. Commun., 1994, 83, 307.
* [16] B. Zhang, Comput. Phys. Commun., 1998, 109, 193.
* [17] B. A. Li and C. M. Ko, Phys. Rev. C, 1995, 52, 2037.
* [18] Michael L. Miller, Klaus Reygers, Stephen J. Sanders, Peter Steinberg, Ann. Rev. Nucl. Part. Sci. , 2007, 57, 205-243.
* [19] B. Alver et al. (PHOBOS), Phys. Rev. Lett, 2007, 98, 242302.
* [20] J. Y. Ollireault, Phys. Rev. D, 1992, 46, 229.
* [21] H. Sorge, Phys. Rev. Lett, 1999, 82, 2048.
* [22] Hans-Joachim Drescher, Adrian Dumitru, Clement Gombeaud, and Jeans-Yves Ollitrault, Phys. Rev. C, 2005, 76, 024905\.
* [23] B. l. Abelev, et al.(STAR), Phys. Rev. C, 2008, 77, 054901.
|
arxiv-papers
| 2012-08-30T08:12:26 |
2024-09-04T02:49:34.640658
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wang Meijuan, Chen Gang, Wu Yuanfang",
"submitter": "Meijuan Wang",
"url": "https://arxiv.org/abs/1208.6105"
}
|
1208.6171
|
# Measurement of heavy-flavour decay muon production at forward rapidity in pp
and Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with the ALICE experiment
D. Stocco [email protected] for the ALICE Collaboration
SUBATECH (Ecole des Mines, CNRS-IN2P3, Université de Nantes), Nantes, France
###### Abstract
The ALICE experiment measured the heavy-flavour production in the semi-muonic
decay channel at forward rapidities ($2.5<y<4$) in pp and Pb–Pb collisions at
$\sqrt{s_{{NN}}}~{}=~{}2.76$ TeV. We report on the first results on the
$p_{\rm t}$-differential cross-sections in pp collisions as well as on the
nuclear modification factors as a function of the transverse momentum and
centrality.
###### keywords:
heavy-ion collisions , heavy-flavour hadrons , semi-muonic decay
###### PACS:
25.75.Dw , 13.20.Fc , 13.20.He
## 1 Introduction
The main goal of the ALICE experiment [1] is the study of the properties of
the state of strongly-interacting matter at very high energy density created
in ultra-relativistic heavy-ion collisions at the LHC. Heavy-flavour quarks
(charm and beauty) have an important role in the investigation: being produced
in the early stage of the collision, they are sensitive probes of the Quark-
Gluon Plasma and allow us to study the parton-medium interaction. The study of
the spectra of heavy-flavoured hadrons provides information on the mechanisms
of in-medium energy-loss and hadronization of heavy quarks. The nuclear
modification factor is a sensitive observable for this analysis. It is defined
as the ratio between the transverse momentum ($p_{\rm t}$) spectra measured in
ion–ion (A–A) collisions (corrected for the detector acceptance and response)
and the $p_{\rm t}$-differential cross-section measured in proton–proton (pp)
collisions, rescaled by the nuclear overlap function estimated through the
Glauber model [2]:
$R_{AA}(p_{\rm t})=\frac{1}{\langle
T_{AA}\rangle}\frac{\mathrm{d}N_{AA}/\mathrm{d}p_{\rm
t}}{\mathrm{d}\sigma_{pp}/\mathrm{d}p_{\rm t}}$ (1)
The ALICE experiment has measured the nuclear modification factor of charmed
mesons [3] and of heavy flavours in the semi-electronic [4] and semi-muonic
decay channels. The latter will be detailed in the following.
## 2 Analysis
The ALICE experiment is equipped with several detectors for tracking, particle
identification, triggering and centrality estimation [1]. The most relevant
detectors for the current analysis are the Silicon Pixel Detector (SPD) which
covers the range $|\eta|<2$ and is used both for triggering and to measure the
interaction vertex position; the VZERO, consisting of an array of scintillator
hodoscopes covering the ranges $2.8<\eta<5.1$ and $-3.7<\eta<-1.7$, which is
used for triggering and centrality estimation through a Glauber model fit of
the signal amplitudes [5]; and the Muon Spectrometer ($-4<\eta<-2.5$),
consisting of a passive front absorber, five tracking stations (the central
one placed inside a 3 T$\cdot$m integrated dipole magnetic field) and two
trigger stations placed downstream of an iron filter, which is used to track
and identify muons with momenta higher than 4 GeV/$c$.
The analysis was performed using data from pp and Pb–Pb collisions at
$\sqrt{s_{NN}}$ = 2.76 TeV, collected in spring 2011 and fall 2010,
respectively. The data sample in Pb–Pb collisions consists of Minimum Bias
(MB) events obtained requiring a signal in the SPD or in either of the two
VZERO arrays in coincidence with the beam-beam counters. In pp collisions, an
additional muon trigger is used which requires, on top of the MB conditions,
the detection of a muon with a transverse momentum above 0.5 GeV/$c$ by the
muon trigger chambers.
Muons are identified by requiring that a track reconstructed in the tracking
chambers matches a corresponding track segment in the trigger chambers. This
condition rejects most of the reconstructed hadrons, which are absorbed in the
iron wall. Geometrical cuts on the track pseudo-rapidity are then applied to
remove the contamination of particles leaking into the spectrometer from
outside the front absorber acceptance. Moreover, the correlation between the
track momentum and the distance of closest approach to the interaction point
(DCA111The DCA is defined as the distance between the interaction point and
the extrapolation of the track to the plane orthogonal to the beam-line and
containing the interaction point itself.) is used to remove fake tracks and
tracks from beam-gas interactions. The main background contribution after
selection cuts consists of muons from the decay in flight of light hadrons
produced in the collision. In the pp analysis, such contribution was estimated
through Monte Carlo simulations, using the Phojet and Pythia event generators
as input, and then subtracted from the measured inclusive spectrum (see [6]
and references therein for details). This approach, however, could not be used
in the Pb–Pb analysis, due to the presence of unknown nuclear effects, in
particular medium-induced parton energy loss at forward rapidity. Hence, a
data driven method was developed, based on the pion and kaon distributions
measured in pp and Pb–Pb collisions in the ALICE central barrel. Such
distributions are extrapolated to forward rapidities and used to generate the
corresponding decay muons through simulations of the decay kinematics and the
front absorber (see [7] for further details). The background contribution
decreases with increasing transverse momentum: the systematic uncertainty on
its subtraction can therefore be limited by restricting to $p_{\rm t}>2$ (4)
GeV/$c$ in pp (Pb–Pb) collisions.
The $p_{\rm t}$-differential cross-section of muons from heavy-flavour decays,
measured in pp collisions at $\sqrt{s}$ = 2.76 TeV with an integrated
luminosity of $\mathcal{L}_{int}=19$ nb-1, is shown in Figure 1. The vertical
bars are the statistical uncertainties. The boxes are the uncorrelated
systematic uncertainties, accounting for detection efficiencies, alignment and
background subtraction. The results are compared to Fixed-Order Next-to-
Leading Log (FONLL) calculations [8, 9], which show a good agreement with data
within errors.
Figure 1: Transverse momentum differential cross-section of muons from heavy-
flavour decay in pp collisions at $\sqrt{s}$ = 2.76 TeV (red open circles).
The vertical bars (boxes) are the statistical (systematic) uncertainties. The
data points are compared to the sum (grey band) of the contribution of muons
from charm (blue dashes) and beauty (magenta dash-dotted) decays, estimated
with FONLL calculations [8, 9]. The ratio between data and FONLL is shown in
the bottom panel.
Figure 2 shows the nuclear modification factor as a function of the transverse
momentum for muons from heavy-flavour decay in the 0–10% (left panel) and
40–80% (right panel) most central collisions. The vertical bars (boxes) are
the statistical (uncorrelated systematic) uncertainties. The correlated
uncertainties on $\langle T_{AA}\rangle$ and on the cross-section
normalization of the pp reference are shown as a filled box at $R_{AA}=1$. The
nuclear modification factor is independent of $p_{\rm t}$ within
uncertainties, and exhibits a reduction of a factor 3–4 in the most central
collisions. It is worth noting that the in-medium energy loss is not the only
mechanism that could lead to a reduction of the $R_{AA}$. In particular, the
nuclear modification of the parton distributions in nuclei could lead to a
variation of the initial hard-scattering probability, and a consequent
variation of the heavy-flavour yield. In the kinematic range relevant for
heavy-flavour production the main effect is the nuclear shadowing, which
reduces the parton distribution functions for partons carrying a fraction of
the nucleon momentum smaller than $10^{-2}$. This effect was estimated by
using perturbative calculations by Mangano, Nason and Ridolfi [10] and the
EPS09NLO [11] parameterization of the shadowing. The result is shown in Figure
2 (grey dot-dot-dot-dashed curve): the initial state effect is expected to be
small in the $p_{\rm t}$ region studied, thus suggesting that the strong
reduction is a final-state effect. The $R_{AA}$ in the most central collisions
(left panel) is compared to models implementing collisional (BAMPS) [12],
radiative (BDMPS-APW) [13] and radiative with in-medium hadronization (Vitev
et al.) [14] in-medium energy loss: a good agreement with data is found for
the last two models, while the BAMPS tends to underestimate the heavy-flavour
muons $R_{AA}$. It is worth noting that the comparison of these models with
the $R_{AA}$ of D mesons measured at mid-rapidity with ALICE [15] leads to
similar observations.
Figure 2: Nuclear modification factor as a function of transverse momentum for
muons from heavy-flavour decay (red circles) in the 0-10% central (left) and
40-80% peripheral (right) collisions. The vertical bars (boxes) are the
statistical (uncorrelated systematic) uncertainties. The grey box at 1 is the
correlated error on the centrality estimation and the pp cross-section
normalization. The expected contribution of shadowing, estimated using
perturbative calculations by Mangano, Nason and Ridolfi [10] and the EPS09NLO
[11] parameterization, is also shown (dot-dot-dot-dashed curve). The
$R_{AA}(p_{\rm t})$ in the most central collisions is compared with models
implementing collisional (BAMPS) [12] radiative (BDMPS-ASW) [13] and radiative
with in-medium hadronization (Vitev et al.) [14] energy-loss.
Finally, the centrality dependence of the nuclear modification factor is shown
in Figure 3. The vertical bars (boxes) are the statistical (uncorrelated
systematic) uncertainties, while the grey filled boxes are the systematic
uncertainties on the nuclear overlap function and on the normalization of the
pp reference. The result refers to muons from heavy-flavour decay with a
transverse momentum higher than 6 GeV/$c$. The high $p_{\rm t}$ cut allows the
selection of a region dominated by beauty decay according to the FONLL
predictions (see also Figure 1). This nuclear modification factor is similar
to the one of D mesons measured at mid-rapidity in $6<p_{\rm t}<12$ GeV/$c$
with ALICE [15] and of non-prompt J/$\psi$ measured in $|y|<2.4$ and
$6.5<p_{\rm t}<30$ GeV/$c$ by the CMS Collaboration [16].
Figure 3: Nuclear modification factor as a function of centrality for muons
from heavy-flavour decay with $p_{\rm t}>6$ GeV/$c$. The vertical bars are the
statistical uncertainties. The empty (filled) boxes are the uncorrelated
(correlated) systematic uncertainties.
## 3 Conclusion
The ALICE experiment has measured the heavy-flavour production in the semi-
muonic decay channels at forward rapidities ($2.5<y<4$) in pp and Pb–Pb
collisions at 2.76 TeV center of mass energy. The measured $p_{\rm
t}$-differential cross section is well described by the FONLL perturbative QCD
calculations.
The resulting nuclear modification factor as a function of $p_{\rm t}$ (for
$p_{\rm t}>4$ GeV/$c$) was also shown, for the 0-10% most central and the
40-80% most peripheral collisions. A reduction of a factor of 3–4 is observed
in the most central collisions, independent of $p_{\rm t}$. It is worth noting
that the initial state effects are expected to be small in this transverse
momentum region. A similar reduction is observed when measuring the nuclear
modification factor as a function of centrality for muons from heavy-flavour
decay with $p_{\rm t}>6$ GeV/$c$, a region where the beauty decay contribution
is expected to be dominant according to FONLL calculations.
## References
* [1] K. Aamodt, et al., JINST 3 (2008) S08002.
* [2] M. L. Miller, K. Reygers, S. J. Sanders, P. Steinberg, Ann.Rev.Nucl.Part.Sci. 57 (2007) 205–243. arXiv:nucl-ex/0701025.
* [3] Z. Conesa del Valle, These proceedings.
* [4] M.-J. Kweon, These proceedings.
* [5] K. Aamodt, et al., Phys.Rev.Lett. 106 (2011) 032301. arXiv:1012.1657.
* [6] B. Abelev, et al., Phys.Lett. B708 (2012) 265–275. arXiv:1201.3791.
* [7] B. Abelev, et al.Submitted to PRL. arXiv:1205.6443.
* [8] M. Cacciari, M. Greco, P. Nason, JHEP 9805 (1998) 007. arXiv:hep-ph/9803400.
* [9] M. Cacciari, S. Frixione, N. Houdeau, M. L. Mangano, P. Nason, G. Ridolfi, arXiv:1205.6344.
* [10] M. L. Mangano, P. Nason, G. Ridolfi, Nucl.Phys. B373 (1992) 295–345.
* [11] K. Eskola, H. Paukkunen, C. Salgado, JHEP 0904 (2009) 065. arXiv:0902.4154.
* [12] J. Uphoff, O. Fochler, Z. Xu, C. Greiner, arXiv:1205.4945.
* [13] N. Armesto, A. Dainese, C. A. Salgado, U. A. Wiedemann, Phys.Rev. D71 (2005) 054027\. arXiv:hep-ph/0501225.
* [14] R. Sharma, I. Vitev, B.-W. Zhang, Phys.Rev. C80 (2009) 054902. arXiv:0904.0032.
* [15] B. Abelev, et al.Submitted to JHEP. arXiv:1203.2160.
* [16] S. Chatrchyan, et al., JHEP 1205 (2012) 063. arXiv:1201.5069.
|
arxiv-papers
| 2012-08-30T13:34:32 |
2024-09-04T02:49:34.645743
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Diego Stocco (for the ALICE Collaboration)",
"submitter": "Diego Stocco",
"url": "https://arxiv.org/abs/1208.6171"
}
|
1208.6570
|
—
$J/\psi$ production and polarization
Maddalena Frosini
on behalf of LHCb Collaboration
Department of Physics
University of Florence, INFN of Florence, ITALY
> The study of the production of heavy quarkonium is crucial for a thorough
> understanding of Quantum Chromodynamics (QCD). This note reports the
> measurements of the $J/\psi$, $\chi_{c}$ and double charm production cross
> section, and discusses the prospects for the $J/\psi$ polarization at LHCb.
> PRESENTED AT
>
>
>
>
> CHARM 2012
> Honolulu, Hawaii, May 14–17, 2012
## 1 Introduction
Understanding the charmonium hadroproduction mechanism has been a long-term
program both experimentally and theoretically. At the LHC two different
components contribute: the “prompt” component, which includes “direct”
production in the $pp$ collisions and the “feed down” from higher charmonium
states, and the “delayed” component, coming from the $b$-hadron decays. In the
direct production the $c\overline{c}$ pairs are expected to be created mainly
through gluon-gluon fusion at the leading order (LO) and the bound states are
formed in the final color-singlet states. The most recent models allow the
formation of the $c\overline{c}$ pairs also through color-octet states, which
evolve toward the final state via exchange of soft gluons. Such evolution is
described in terms of a non-perturbative QCD (NRQCD) factorization approach.
The LHCb collaboration has given many contributions in understanding the
quarkonium production mechanism, some of them reported here. The measurement
of the $J/\psi$ production cross section together with the status of the
polarization analysis will be presented. Both measurements provide a critical
test for the color-singlet [1, 2] and color-octet models [3]. The study of
$\chi_{c}$, double $J/\psi$ and $J/\psi$ production associated with an open
charm hadron will also be presented. In particular the last two are rare
processes and they can be useful to investigate the contributions from other
mechanisms, such as the Double Parton Scattering (DPS) [7, 8, 9].
## 2 The LHCb detector
The LHCb detector [10] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the proton-
proton interaction region, a large-area silicon-strip detector located
upstream of a dipole magnet with a bending power of about 4 Tm, and three
stations of silicon-strip detectors and straw drift-tubes placed downstream.
The combined tracking system has a momentum resolution $\delta p/p$ that
varies from 0.4% at 5 GeV/$c^{2}$ to 0.6% at 100 GeV/$c^{2}$, and an impact
parameter resolution of 20 $\mu$m for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadron calorimeter. Muons are identified by a muon system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction.
## 3 $J/\psi$ cross section measurement
The cross section is measured selecting $J/\psi$ decaying to two muons: the
data sample corresponds to an integrated luminosity $\mathcal{L}=\left(5.2\pm
0.5\right)$ pb-1 of $pp$ collisions at $\sqrt{s}=7$ TeV recorded by the
experiment during September 2010. The double differential cross section in
$J/\psi$ $p_{T}$ and $y$ is defined as following:
$\frac{d^{2}\sigma}{dp_{T}dy}=\frac{N(J/\psi\rightarrow\mu^{+}\mu^{-})}{L\times\varepsilon_{tot}\times\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})\Delta
p_{T}\Delta y},$ (1)
where $N(J/\psi\rightarrow\mu^{+}\mu^{-})$ is the number of selected $J/\psi$
decaying in two muons, $\mathcal{L}$ is the integrated luminosity,
$\varepsilon_{tot}$ is the total efficiency (estimated from Monte Carlo
including the detector acceptance, the reconstruction and trigger efficiency),
$\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})$ is the branching ratio of the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, $\Delta p_{T}$ and $\Delta y$ are
respectively the $J/\psi$ transverse momentum and rapidity bin sizes. The
analysis selection requires at least one reconstructed primary vertex in each
event. The $J/\psi$ candidates are formed from pairs of opposite sign charged
tracks reconstructed in the tracking system and identified as muons. The two
muons must have a good quality of the track fit and originate from a common
vertex. To separate the prompt and the delayed component the $J/\psi$ pseudo
proper time is used, defined as
$t_{z}=\frac{(z_{J/\psi}-z_{PV})m_{J/\psi}}{p_{z}},$ where $z_{J/\psi}$ and
$z_{PV}$ are the $J/\psi$ decay vertex and the primary vertex positions along
the beam axis and $m_{J/\psi}$ and $p_{z}$ are respectively the mass and the
momentum component of the $J/\psi$ along the beam axis.
Figure 1: Double differential cross section of $J/\psi$ prompt component.
Figure 2: Prompt $J/\psi$ differential cross section compared with different
theoretical models. The top plots show the direct component only and the
bottom include the feed down.
Figure 3: Double differential cross section of the delayed $J/\psi$ component.
Figure 4: Differential cross section of delayed $J/\psi$ component compared
with FONLL computation.
Figures 2 and 2 show respectively the double differential prompt cross section
and the differential prompt cross section integrated over rapidity as a
function of $p_{T}$. Results are compared with the prediction of three
different theoretical models (Colour Singlet Model, Colour Octet Model and
Colour Evaporation Model). In Fig.4 the double differential cross section of
the delayed component is shown and in Fig.4 it is integrated over rapidity and
compared with the FONLL computation. The total integrated cross sections are
$\sigma_{prompt}=(10.52\pm 0.04(stat.)\pm
1.40(sys.)^{+1.64}_{-2.20}(pol.))\mu\mathrm{b},$ (2)
$\sigma_{from\>b}=\left[1.14\pm 0.01(stat.)\pm
0.16(sys.)\right]\mu\mathrm{b}.$ (3)
The first and the second uncertainties are the statistical and the systematic,
where the main sources of systematic uncertainty come from the luminosity
measurement, the tracking and trigger efficiency. The third uncertainty on the
prompt cross section is due to the unknown polarization of the $J/\psi$ and it
is estimated calculating the total efficiency in two possible extreme
scenarios of fully transverse and fully longitudinal polarization. The
deviation from the case of zero polarization is assigned as systematic
uncertainty to the measurement. From Eq.3 the $b\overline{b}$ cross section is
extrapolated to the full solid angle using the formula
$\sigma(pp\rightarrow
b\overline{b}X)=\alpha_{4\pi}\frac{\sigma_{fromb}}{2\mathcal{B}(b\rightarrow
J/\psi X)}=\left[288\pm 4(stat.)\pm 48(sys.)\right]\mu\mathrm{b}.$ (4)
All these results have been published in Ref.[11].
## 4 Outlook for polarization measurement
With the full 2011 data sample the prompt $J/\psi$ polarization will be
measured studying the full angular distribution of the two muons:
$\frac{dN}{d(\cos\theta)d\phi}\propto
1+\lambda_{\theta}\cos^{2}\theta+\lambda_{\phi}\sin^{2}\theta\cos
2\phi+\lambda_{\theta\phi}\sin 2\theta\cos\phi,$ (5)
where $\theta$ and $\phi$ are the polar and azimuthal angles in the helicity
frame (using the $J/\psi$ momentum as polarization axis). The measurement will
be performed in bins of $J/\psi$ transverse momentum and rapidity. The
statistical sensitivity should be under 0.15 for $\lambda_{\theta}$ and about
0.01 for $\lambda_{\phi}$ and $\lambda_{\theta\phi}$. The systematic
uncertainty is expected to be of the same order of magnitude.
## 5 $\chi_{c}$ production
The study of the $J/\psi$ production through the radiative decays of the
$\chi_{c}$ states provides a useful test of both the color-singlet and color-
octet model. Moreover it is fundamental for the $J/\psi$ polarization
measurement, as the directly produced $J/\psi$ and those coming from
$\chi_{c}$ decays can carry different polarization and this represents a
possible source of uncertainty for the polarization measurement of the prompt
component. The measurement of the fraction of $J/\psi$ coming from $\chi_{c}$
decays can quantify this uncertainty.
Figure 5: Invariant mass difference spectrum $\Delta
m=m(\chi_{c})-m(J/\psi)=m(e^{+}e^{-}\mu^{+}\mu^{-})-m(\mu^{+}\mu^{-})$ using
converted photons.
Figure 6: Relative cross section $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ in bins
of $J/\psi$ transverse momentum.
The relative cross section $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ is measured
using two different data sample acquired by the LHCb experiment during the
2010 and 2011, respectively of 37 pb-1 [12] and 370 pb-1 [13]. In both cases
the $\chi_{c}$ states are identified through their radiative decay
$\chi_{c}\rightarrow J/\psi\gamma$ with the $J/\psi$ decaying to two muons
$J/\psi\rightarrow\mu^{+}\mu^{-}$. For the first measurement, made with a
smaller data sample, the photons reconstructed in the calorimeter system have
been used. This allows to have a higher statistics but the poor resolution of
the calorimeter doesn’t permit to separate the two $\chi_{c1}$ and $\chi_{c2}$
states. In the second measurement the photons converted in the detector
material before the magnet have been used, $\gamma\rightarrow e^{+}e^{-}$. In
this way it is possible to take advantage of the good resolution of the
tracker, which allows to resolve the two states, as it is shown in Fig.6.
In both measurements the efficiency is determined from the Monte Carlo
simulation and the number of signal events is extracted with a fit to the
invariant mass difference spectra, in four bins of $J/\psi$ transverse
momentum.
The results for the relative cross section
$\sigma(\chi_{c2})/\sigma(\chi_{c1})$ are shown in Fig.6. In the plot the
inner error bars correspond to the statistical uncertainties and the outer
bars correspond to the sum of all the sources of systematic uncertainties. The
shaded area represents the maximum effect due to the unknown $\chi_{c}$
polarization. In green and black the results obtained reconstructing the
photons in the calorimeter (2010 statistics) and the converted photons (2011
statistics) are shown. The results from the CDF collaboration are shown in
magenta [14]. The blue and red shaded area correspond to the color-singlet and
NRQCD prediction respectively.
Figure 7: Relative cross section $\sigma(\chi_{c})/\sigma(J/\psi)$ in bins of
$J/\psi$ transverse momentum. The results (black points) are compared with the
CDF measurement [16] and with the color-singlet and color-octet prediction
(respectively the blue and red area.)
The $\chi_{c}$ to $J/\psi$ ratio has been measured with the 36 pb-1 data
sample acquired by the experiment in the 2010 [15]. The $\chi_{c}$ states are
reconstructed through their radiative decay $\chi_{c}\rightarrow J/\psi\gamma$
with the $J/\psi$ decaying into two muons $J/\psi\rightarrow\mu^{+}\mu^{-}$.
The results are shown in Fig.7, compared with the CDF measurement and with two
theoretical models, the color-singlet (blue) and NRQCD approach (red).
## 6 Double charm production
Recently, both the production of double quarkonium and also the associated
production of quarkonium together with open charm have been suggested as
probes of the production mechanism. In the $pp$ collisions also other
mechanisms, as the DPS [7, 8, 9], can be involved in the production and their
contribution can be estimated with respect to the Single Parton Scattering
(SPS) [4, 5, 6].
Such a study has been performed at LHCb through the measurement of the double
$J/\psi$, taken from Ref.[17], and $J/\psi$ production associated with an open
charm hadron (such as $D^{0}$, $D^{+}$, $D^{+}_{s}$ and $\Lambda_{c}^{+}$)
[18], with the 2010 and 2011 datasets (respectively 37 pb-1 and 355 pb-1).
Figure 8: Invariant mass distribution of the first muon pair in bins of the
second muon pair for the double $J/\psi$ production.
The double $J/\psi$ production cross section has been measured reconstructing
the two $J/\psi$ mesons in their decay to two muons. Both the $J/\psi$ mesons
have been required to have rapidity and transverse momentum lying respectively
in the ranges $2<y<4.5$ and $p_{T}<10$ GeV/$c$.
The signal yield is determined fitting the invariant mass distribution of the
first muon pair in bins of the second muon pair and correcting the number of
signal events by the total efficiency.
The total efficiency is factorized in three different terms
$\varepsilon_{J/\psi J/\psi}^{tot}=\varepsilon_{J/\psi
J/\psi}^{sel\&reco\&acc}\times\varepsilon_{J/\psi J/\psi}^{\mu
ID}\times\varepsilon_{J/\psi J/\psi}^{trg},$ (6)
where $\varepsilon_{J/\psi J/\psi}^{sel\&reco\&acc}$ is the acceptance,
selection and reconstruction efficiency, $\varepsilon_{J/\psi J/\psi}^{\mu
ID}$ is the efficiency of the muon identification and $\varepsilon_{J/\psi
J/\psi}^{trg}$ is the trigger efficiency. To take into account the distortion
due to the unknown $J/\psi$ polarization $\varepsilon_{J/\psi
J/\psi}^{sel\&reco\&acc}$ is a function of the $J/\psi$ $\cos\theta$ where
$\theta$ is the angle between the $\mu^{+}$ in the $J/\psi$ center of mass
frame and the Lorentz boost from the laboratory frame to the $J/\psi$ frame.
The corrected invariant mass distribution of the first muon pair in bins of
the second muon pair is shown in Fig.8, in three bins of $J/\psi$ transverse
momentum in a particular bin of rapidity.
The double $J/\psi$ cross section is estimated to be
$\sigma_{J/\psi J/\psi}=\frac{N^{corr}_{J/\psi
J/\psi}}{\mathcal{L}\,\mathcal{B}^{2}(J/\psi\rightarrow\mu^{+}\mu^{-})}=\left(5.1\pm
1.0(stat.)\pm 1.1(sys.)\right)\mathrm{nb},$ (7)
where $N^{corr}_{J/\psi J/\psi}$ is the efficiency corrected signal yield,
$\mathcal{L}=37$ pb-1 is the integrated luminosity and
$\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})$ is the branching ratio of the
$J/\psi$ decay into a muon pair.
The uncertainty are respectively statistical and systematic. The main
contributions to the systematic uncertainty come from the tracking and trigger
efficiency and from the unknown $J/\psi$ polarization.
The experimental result obtained by LHCb has been compared with the
theoretical contribution calculated in color-singlet model, from the SPS and
the DPS. The two contributions, estimated in the LHCb acceptance, are listed
together with the related uncertainties, in Tab.1 [17]. The sum of the two
contributions is in agreement with the experimental value reported in Eq.7,
although the uncertainties on the theoretical expectations are too large to
draw a definite conclusion on the production mechanism.
Model | Cross section (nb) | Uncertainty
---|---|---
Single Parton S. | 4.15 | 30%
Double Parton S. contribution | 2 | 50%
Table 1: SPS and DPS contribution to the double $J/\psi$ production cross
section estimated in the LHCb acceptance. The theoretical uncertainties are
also listed.
Figure 9: $J/\psi$ production cross section with associated open charm. The
experimental results (black points) are compared with the gluon-gluon fusion
expectation (yellow and green areas [4, 5, 6]).
Figure 10: Results for the prompt open charm cross sections and double open
charm cross section ratio compared with the theoretical expectation computed
with the DPS approach [7, 8, 9].
The production cross sections of a $J/\psi$ meson associated with an open
charm hadron, $D^{0}$, $D^{+}$, $D^{+}_{s}$ or $\Lambda_{c}^{+}$ have been
measured using 355 pb-1 out of the 2011 datasets. As control channels, the
$c\overline{c}$ events with two open charm hadrons reconstructed in the LHCb
acceptance have been also studied. The $J/\psi$, $D^{0}$, $D^{+}$, $D^{+}_{s}$
and $\Lambda_{c}^{+}$ hadrons have been reconstructed through the following
decays: $J/\psi\rightarrow\mu^{+}\mu^{-}$, $D^{0}\rightarrow\pi^{+}K^{-}$,
$D^{+}\rightarrow\pi^{+}\pi^{+}K^{-}$,
$D^{+}_{s}\rightarrow\pi^{+}K^{+}K^{-}$, $\Lambda^{+}_{c}\rightarrow
p\pi^{+}K^{-}$.
In Fig.10 (taken from Ref.[18]) the results for the production cross section
of the $J/\psi D$ processes are shown, compared with the theoretical
expectation estimated with gluon-gluon fusion model. In Fig.10 (from Ref.[18])
the ratios of the product of the prompt open charm cross sections and the
double open charm cross section show a good agreement with the theoretical
expectation from the DPS, assuming the effective cross section measured in
multi-jet events at Tevatron [19]. In the plots the inner error bars represent
the statistical uncertainties while the outer error bars are the sum in
quadrature of the statistical and systematic uncertainties. The main
contributions to the systematics are coming from the trigger efficiency, the
unknown $J/\psi$ polarization and the luminosity.
## 7 Conclusion
LHCb has provided many contributions in understanding the quarkonium
production mechanism in particular with the measurement of the $J/\psi$ and
$\chi_{c}$ production. Also the double $J/\psi$ and $J/\psi$ associated with
an open charm hadron production has been investigated, which provide a useful
test for the SPS and DPS approach. The measurement of the polarization of the
prompt $J/\psi$ component is ongoing and it will provide a critical test for
the validity of the color-singlet and color-octet models.
## References
* [1] Berger E.L. and Jones D. Phys. Rev. D 23 1521, 1981
* [2] Baier R. and Rückl Phys. Lett. B 102 364, 1981
* [3] Bodwin G.T., Braaten E. and Lepage G.P. Phys. Rev. D 51 1125, 1995
* [4] Phys. Rev. D57 (1998) 4385
* [5] Phys. Rev. D73 (2006) 074021
* [6] Eur. Phys. J. C61 (2009) 693
* [7] Phys. Rev. Lett. 107 (2011) 082002
* [8] Phys. Lett. B705 (2011) 116
* [9] arXiv:1106.2184
* [10] The LHCb Collaboration,
JINST 3 (2008) S08005
* [11] The LHCb Collaboration,
Eur. Phys. J. C71 1645, 2011
* [12] arXiv:1202.1080
* [13] LHCb-CONF-2011-062
* [14] Phys. Rev. Lett. 98 (2007) 232001
* [15] arXiv:1204.1462
* [16] Phys. Rev. Lett. 79 (1997), 578-583
* [17] The LHCb Collaboration,
Phys. Lett. B 707 (2012), pp. 52-59
* [18] arXiv:1205.0975
* [19] Phys. Rev. D 56 (1997), 3811-3832
|
arxiv-papers
| 2012-08-31T18:23:30 |
2024-09-04T02:49:34.659540
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maddalena Frosini (for the LHCb Collaboration)",
"submitter": "Maddalena Frosini",
"url": "https://arxiv.org/abs/1208.6570"
}
|
1209.0118
|
# New results on the spectroscopy of X, Y, Z states from LHC experiments
A. Augusto Alves Jr, on behalf of LHCb Collabration
Universita degli studi di Roma “La Sapienza” and INFN sezione di Roma
P.le Aldo Moro, 2 - 00185 Roma - Italy
Presented at 5th International Workshop on Charm Physics
Honolulu, Hawaii, May 17 2012
###### Abstract
The main results from LHC experiments on XYZ charmonium-like candidates are
summarized.
## 1 Introduction
According to our current understanding, the forces responsible to bind quarks
into hadrons are described by the non-Abelian field theory called Quantum
Chromodynamics (QCD). In QCD-motivated quark potential models, the quarkonia
states are described as a quark-antiquark pair bound by an interquark force
with a short- distance behavior that is approximately Coulombic, plus an
increasing confining potential that dominates at large separations. In one of
the simplest approaches, the energy levels can be determined by solving the
corresponding non-relativistic Schrodinger equation in order to obtain the
expected masses of the charmonium spectrum, characterized by the radial
quantum number n and the relative orbital angular momentum between the quark
and the antiquark, L. In particular, all predicted states lying under the
$\mathrm{D}\kern 1.99997pt\overline{\kern-1.99997pt\mathrm{D}}{}$ mass
threshold have been observed[1, 2, 3, 4].
On the other hand, the possible existence of more sophisticated states than
mesons and baryons, like the multiquark states, hybrid mesons and mesonic
molecules has been discussed since the early days of the quark model[2, 5, 6,
7, 8].
In the last decade, considerable experimental evidence has been collected
about the existence of new states, lying in the charmonium mass range, but not
fitting well the charmonium mass spectrum picture[9, 10, 11, 12, 13, 14, 15].
Most of the observations also suggested that these candidates are exotic.
These studies have been performed at Babar and Belle, two experiments which
took data at the $\mathrm{e}^{+}\mathrm{e}^{-}$ Beauty Factories at SLAC
(Stanford Linear Accelerator Center, USA) and KEK (High-Energy Accelerator
Research Organization, Japan), respectively. Confirmations have also come from
the CDF experiment, collecting data from $\mathrm{p}\overline{}\mathrm{p}$
interactions at Fermilab,USA.
In these notes the main results from LHC experiments on XYZ charmonium-like
candidates are summarized. In Section 1 the main features of the LHCb detector
are presented. Section 2 is dedicated to the discussion of the measurements of
the $\mathrm{X}(3872)$ mass and cross-section in the LHCb and CMS experiments.
In Section 4, the results of the search for the $\mathrm{X}(4140)$ and
$\mathrm{X}(4274)$ states in
$\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ decays at LHCb are presented. The conclusions are presented in
Section 5.
## 2 LHCb and CMS detectors
LHCb is an experiment dedicated to heavy flavour physics at the LHC[16]. Its
primary goal is to search for indirect evidence of new physics in CP violation
and rare decays of beauty and charm hadrons.
LHCb detector is a single-arm spectrometer (see figure 1) with a forward
angular coverage from approximately 10 mrad to 300 (250) mrad in the bending
(non-bending) plane, corresponding to a pseudorapidity range of 2 $<\eta<$ 5\.
In fact, the detector geometry is optimized to cover the region where the
$\mathrm{b}\overline{}\mathrm{b}$ cross-section peaks in such way that, even
if just covering about 4% of the solid angle, the LHCb detects about 40% of
heavy quark hadrons produced in the proton-proton colisions.
Figure 1: YZ view of the LHCb detector.
The spectrometer consists of a vertex locator, a warm dipole magnet, a
tracking system, two RICH detectors, a calorimeter system and a muon system.
The track momenta are measured to a precision of $\delta p/p$ between 0.35%
and 0.5%. The Ring Imaging Cherenkov Detector (RICH) system provides excellent
charged hadron identification in a momentum range 2-100 GeV/c. The calorimeter
system identifies high transverse energy hadron, electron and photon
candidates and provides information for the trigger. The muon system provides
information for the trigger and muon identification with an efficiency of
about 95% for a misidentification rate of about 1-2 % for momenta above 10
GeV/c.
The luminosity for the LHCb experiment can be tuned by changing the beam focus
at its interaction point independently from the other interaction points,
allowing LHCb to maintain the optimal luminosity in order not to saturate the
trigger or to damage the delicate sub-detectors parts. In fact, due this
capability, LHCb was able to keep its luminosity at the constant value of
$3.5\times 10^{32}\,{\rm cm^{-2}s^{-1}}$ during most of 2011 data taking.
The trigger chain is composed by a first level hardware trigger and two levels
of software triggers. LHCb uses hadrons, muons, electrons and photons
throughout the trigger chain, maximizing the trigger efficiency on all heavy
quark decays and making the experiment sensitive to many different final
states.
In 2010 and 2011, the detector recorded about $1.1\,{\rm fb^{-1}}$ integrated
luminosity in proton-proton collisions at $\sqrt{s}=7$ TeV, corresponding to
90% of the luminosity delivered by the Large Hadron Collider(LHC) to LHCb.
The CMS is a multi-purpose experiment at LHC, designed with the main goal of
search for new physics phenomena at large transverse momentum scales. CMS
cover a rapidity range up to $\arrowvert\eta\arrowvert<2.5$ and since -̱quark
production peaks at large rapidities, CMS is most able to search for
charmonium-like candidates produced primary in the proton-proton collisions.
For a complete description of the CMS detector see [17].
## 3 $\mathrm{X}(3872)$ mass and cross-section measurements at LHC and CMS
The $\mathrm{X}(3872)$ resonance was discovered in 2003 by the Belle
collaboration in the
$\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}\mathrm{X}(3872)$,
$\mathrm{X}(3872)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ decay chain [18]. Its existence was confirmed by the
CDF[19], D$\emptyset$ [20] and BaBar[21] collaborations.
The $\mathrm{X}(3872)$ mass is currently known with <
1.0${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ precision, the dipion mass
spectrum in the decay
$\mathrm{X}(3872)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ [22, 23] has been studied and the $\mathrm{X}(3872)$
quantum numbers have been constrained to be either $J^{PC}=2^{-+}$ or $1^{++}$
[24] and are still not established. However, despite the cumulated
experimental and theoretical effort, the nature of the $\mathrm{X}(3872)$
remains uncertain. Among the possible interpretations for this state currently
discussed in the literature, one can remark the mesonic molecule, the hybrid
meson and the tetraquark hypotesis. The conventional charmonium interpretation
is not excluded.
Figure 2: Invariant mass distribution of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$(black points with statistical error bars) and same-
sign ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{+}$(blue filled histogram) candidates. The solid red
curve is the result of the fit described in the text. The inset shows a zoom
of the $\mathrm{X}(3872)$ region.
In LHCb the analysis is performed on $34.7\mbox{\,pb}^{-1}$ dataset collected
in 2010 in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}$ =
7$\mathrm{\,Te\kern-1.00006ptV}$. The $\mathrm{X}(3872)$ signal has been
isolated applying tight cuts in order to reduce the combinatorial background,
generated when a correctly reconstructed
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson is combined
with a random $\uppi^{+}\uppi^{-}$ pair from the primary
$\mathrm{p}\mathrm{p}$ interaction. The selection cuts are optimized using
reconstructed
$\uppsi(2S)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ decays, as well as “same-sign pion” candidates
satisfying the same criteria as used for the $\mathrm{X}(3872)$ and
$\uppsi(2S)$ selection. A further background suppression is reached applying
the requirement Q < 300 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where
$Q=\mathrm{M}_{\upmu^{+}\upmu^{-}\uppi^{+}\uppi^{-}}-\mathrm{M}_{\upmu^{+}\upmu^{-}}-\mathrm{M}_{\uppi^{+}\uppi^{-}}$.
See [25] for a detailed discussion on the selection procedure.
The masses of the $\uppsi(2S)$ and $\mathrm{X}(3872)$ mesons are determined
from an extended unbinned maximum likelihood fit of the reconstructed
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\uppi^{+}\uppi^{-}$
mass in the interval
$3.60<\mathrm{M}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}}<3.95{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
$\uppsi(2S)$ and $\mathrm{X}(3872)$ signals are described with a non-
relativistic Breit-Wigner function convolved with a Gaussian resolution
function. The intrinsic width of the $\uppsi(2S)$ is fixed to the PDG value
and the $\mathrm{X}(3872)$ width is fixed to zero in the nominal fit. The
ratio of the mass resolutions for the $\mathrm{X}(3872)$ and the $\uppsi(2S)$
is fixed to the value $\sigma_{\mathrm{X}(3872)}/\sigma_{\uppsi(2S)}=1.31$.
The background shape is described by the functional form
$\mathrm{f}(\mathrm{M})\propto(\mathrm{M}-\mathrm{M}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}-2\mathrm{M}_{\uppi})^{\mathrm{c}_{0}}\mathrm{e}^{(-\mathrm{c}_{1}\mathrm{M}-\mathrm{c}_{2}\mathrm{M}^{2})}$.The
results of the fit are summarized in the table 1.
Fit parameter | $\uppsi(2S)$ | $\mathrm{X}(3872)$
---|---|---
Number of signal events | $3998\pm 83$ | $565\pm 62$
Mass [ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | $3686.10\pm 0.06$ | $3871.88\pm 0.48$
Mass resolution [ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | $2.54\pm 0.06$ | $3.33\pm 0.08$
S/B in $\pm 3\sigma$ window | 1.5 | 0.15
Number of background events | $73094\pm 282$ | –
Table 1: Fit results on LHCb $\mathrm{X}(3872)$ studies.
At LHCb, the same sample used to measure the $\mathrm{X}(3872)$ mass has been
used to perform $\mathrm{X}(3872)$ production studies. The product of the
inclusive production cross-section
$\upsigma(\mathrm{p}\mathrm{p}\rightarrow\mathrm{X}(3872)+\cdots)$ and the
branching fraction ${\cal
B}(\mathrm{X}(3872)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-})$ is determined according the expression
$\upsigma(\mathrm{p}\mathrm{p}\rightarrow\mathrm{X}(3872)+\cdots)\times{\cal
B}(\mathrm{X}(3872)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-})=\frac{\mathrm{N}^{corr}_{\mathrm{X}(3872)}}{\xi\times\mathcal{L}_{\rm
int}\times{\cal B}({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-})}$
where $\mathrm{N}_{\mathrm{X}(3872)}$ is the efficiency-corrected signal
yield, $\upxi$ is a correction factor to the simulation-derived efficiency
that accounts for known differences between data and simulation, ${\cal
B}({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-})=(5.93\pm 0.06)\times 10^{-2}$ is the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ branching fraction, and $\mathcal{L}_{\rm
int}$ is the integrated luminosity. See [25] for detailed discussion about the
calibration procedure and the treatment of the different sources of systematic
uncertainty. The studies are performed just considering candidates lying
inside the fiducial region for the measurement defined by
$2.5<y<4.5\text{ and }5<p_{T}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
where $y$ and $p_{T}$ are the rapidity and transverse momentum of the
$\mathrm{X}(3872)$. The X(3872) production cross section at LHCb is measured
to be
$\upsigma(\mathrm{p}\mathrm{p}\rightarrow\mathrm{X}(3872)+\cdots)\times{\cal
B}(\mathrm{X}(3872)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-})=4.7\pm 1.1\mathrm{(stat)}\pm
0.7\mathrm{(syst)}\rm\,nb$
The CMS Collabration also performed studies on the $\mathrm{X}(3872)$
production. CMS uses a dataset of 40$\mbox{\,pb}^{-1}$ collected in
$\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}$
=7$\mathrm{\,Te\kern-1.00006ptV}$ to measure the ratio of the branching
fractions of $\uppsi(2S)$ $\rightarrow$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ $\uppi^{+}\uppi^{-}$
and $\mathrm{X}(3872)$ $\rightarrow$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ $\uppi^{+}\uppi^{-}$
which is defined as
$R=\frac{\sigma({\rm
pp}\rightarrow\mathrm{X}(3872)+\cdots)\times\mathcal{B}(\mathrm{X}(3872)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-})}{\upsigma({\rm
pp}\rightarrow\mathrm{X}(3872)+\cdots)\times\mathcal{B}(\uppsi(2S)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-})}$
inside the fiducial region defined by
$\mbox{$p_{\rm T}$}>8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\text{ and }|y|<2.2$
.
The result of the CMS analysis is
$R=0.087\pm 0.017\mathrm{(stat)}\pm 0.009\mathrm{(syst)},$
where the first error refers to the statistical uncertainty and the second
error contains the sum of all systematic uncertainties, as described in [26],
added in quadrature. See [26] for a detailed discussion of the selection
procedure and uncertainties estimation.
## 4 Search for the X(4140) state in
$\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ decays
The CDF collaboration has reported a $3.8\sigma$ evidence for the
$\mathrm{X}(4140)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ state using data collected in proton-antiproton collisions at the
Tevatron ($\sqrt{s}$ = 1.96 TeV)[27]. In a preliminary update on the analysis
[28], the CDF collaboration reported $115\pm 12$
$\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ events and $19\pm 6$ $\mathrm{X}(4140)$ candidates with a
statistical significance of more than $5\sigma$. The mass and width were
determined to be $4143.4^{+2.9}_{-3.0}\pm 0.6$MeV/$c^{2}$ and
$15.3^{10.4}_{-6.1}\pm 2.5$ MeV/$c^{2}$, respectively. The relative branching
ratio was measured to be
$\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}\mathrm{X}(4140))\times\mathcal{B}(\mathrm{X}(4140)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)/\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)=0.149\pm 0.039\mathrm{(stat)}\pm 0.024\mathrm{(syst)}$.
Since a charmonium state at this mass is expected to have much larger width
because of open flavor decay channels, the decay rate of the
$\mathrm{X}(4140)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ mode, so near to kinematic threshold, should be small and
unobservable. Due to these issues, the CDF’s report rejuvenated the
discussions on exotic hadronic states. It was cogitated that the
$\mathrm{X}(4140)$ resonance could be a molecular state [29, 30, 31], a
tetraquark state [32, 33], a hybrid state [34, 35] or even a rescattering
effect [15, 16].
The CDF data also suggested the presence of a second state, referred here as
$\mathrm{X}(4274)$ with mass $4274.4^{+8.4}_{-6.4}\pm 1.9$ MeV/$c^{2}$ and
width $32.3^{+21.9}_{-15.3}\pm{7.6}$ MeV/$c^{2}$. The corresponding event
yield was $22\pm 8$ with $3.1\sigma$ significance. This observation has also
received attention in the literature [36, 37]. On the other hand, the Belle
experiment found no evidence for the $\mathrm{X}(4140)$ and $\mathrm{X}(4274)$
states[38, 39].
Figure 3: Distribution of the mass difference
$M({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)-M({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu})$. Fit
of the $\mathrm{X}(4140)$ signal on top of a smooth background is superimposed
(solid red line). The dashed blue (dotted blue) line on top illustrates the
expected $\mathrm{X}(4140)$ ($\mathrm{X}(4274)$) signal yield from the CDF
measurement. The top and bottom plots differ by the background function
(dashed black line) used in the fit: (a) a background efficiency-corrected
three-body phase-space; (b) background efficiency-corrected quadratic
function.
The LHCb analysis[40, 41] starts reconstructing a $\mathrm{B}^{+}$ candidate
as five-track $(\mu^{+}\mu^{-}\mathrm{K}^{+}\mathrm{K}^{-}\mathrm{K}^{+})$
vertex using well reconstructed and identified muons and kaons candidates. The
$\mathrm{B}^{+}$ candidates are required to have $p_{T}>4.0$ GeV/$c$ and a
decay time of at least 0.25 ps. The invariant mass of the
$(\mu^{+}\mu^{-}\mathrm{K}^{+}\mathrm{K}^{-}\mathrm{K}^{+})$ combination is
evaluated after the muon pair is constrained to the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass, and all final
state particles are constrained to a common vertex. Further background
suppression is provided using the likelihood ratio discriminator method.
The
$\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ invariant mass distribution, with at least one $\mathrm{K}^{+}$
$\mathrm{K}^{-}$ combination having an invariant mass within $\pm 15$
MeV/$c^{2}$ of the nominal $\phi$ mass was fitted by a Gaussian and a
quadratic function resulting in $346\pm 20$ $\mathrm{B}^{+}$ events with a
mass resolution of $5.2\pm 0.3$ MeV/$c^{2}$.
The $\mathrm{X}(4140)$ state was searched selecting events within $\pm 15$
MeV/$c^{2}$ of the $\phi$ mass. Figure 3 shows the mass difference
$M({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)-M({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu})$
distribution without ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
or $\phi$ mass constraints. No narrow structure is observed near the
threshold. The fit results are $N_{\mathrm{X}(4140)}^{(a)}=6.9\pm 4.9$ or
$N_{\mathrm{X}(4140)}^{(b)}=0.6\pm 7.1$, depending on the background shape
used.
The CDF’s fit model was used to quantify the compatibility of the two
measurements and considering the LHCb
$\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi$ yield, the efficiency ratio, and the CDF value for
$\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}\mathrm{X}(4140))/\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)$, one concludes that LHCb should have observed $35\pm 9\pm 6$
events, where the first uncertainty is statistical from the CDF data and the
second includes both the CDF and LHCb systematic uncertainties. The LHCb
results disagree with the CDF observation by $\sim 2.7\sigma$. In the case of
the $\mathrm{X}(4274)$ candidate, the same procedure predicts that LHCb should
have observed $53\pm 19$ $\mathrm{X}(4274)$ candidates. The final results are
the following upper limits at 90%CL
$\frac{\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}\mathrm{X}(4140))\times\mathcal{B}(\mathrm{X}(4140)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)}{\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)}<0.07,$
$\frac{\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}\mathrm{X}(4274))\times\mathcal{B}(\mathrm{X}(4274)\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)}{\mathcal{B}(\mathrm{B}^{+}\rightarrow\mathrm{K}^{+}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\phi)}<0.08.$
## 5 Conclusions
A selection of results on XYZ states spectroscopy at the LHC have been
summarized. Many new results are expected from the analysis of the 2011 and
2012 datasets and as well from the news studies currently on-going.
The LHC experiments are in a privileged position to explore the production
mechanisms and spectra of the XYZ states, delivering competitive results in
the heavy flavor sector.
## References
* [1] S. Godfrey and S. L. Olsen, Ann.Rev.Nucl.Part.Sci. 58, 51 (2008), 0801.3867.
* [2] E. Swanson, Int. J. Mod. Phys. A21, 733 (2006), hep-ph/0509327.
* [3] N. Drenska et al., Riv. Nuovo Cim. 033, 633 (2010), 1006.2741.
* [4] N. Brambilla et al., Eur.Phys.J. C71, 1534 (2011), 1010.5827.
* [5] F. E. Close, (2008), 0801.2646.
* [6] J. L. Rosner, J. Phys. Conf. Ser. 69, 012002 (2007), hep-ph/0612332.
* [7] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B634, 214 (2006), hep-ph/0512230.
* [8] L. Maiani, V. Riquer, F. Piccinini, and A. D. Polosa, Phys. Rev. D72, 031502 (2005), hep-ph/0507062.
* [9] S. K. Choi et al., BELLE, Phys. Rev. Lett. 100, 142001 (2008), 0708.1790.
* [10] B. Aubert et al., BABAR, Phys. Rev. Lett. 95, 142001 (2005), hep-ex/0506081.
* [11] Q. He et al., CLEO, Phys. Rev. D74, 091104 (2006), hep-ex/0611021.
* [12] C. Z. Yuan et al., Belle, Phys. Rev. Lett. 99, 182004 (2007), 0707.2541.
* [13] K. Abe et al., Belle, Phys. Rev. Lett. 98, 082001 (2007), hep-ex/0507019.
* [14] K. Abe et al., Belle, Phys. Rev. Lett. 94, 182002 (2005), hep-ex/0408126.
* [15] S. Uehara et al., Belle, Phys. Rev. Lett. 96, 082003 (2006), hep-ex/0512035.
* [16] LHCb Collaboration, JINST 3, S08005 (2008).
* [17] S. Chatrchyan et al., CMS Collaboration, Journal of Instrumentation 3, S08004 (2008).
* [18] S. Choi et al., Belle Collaboration, Phys.Rev.Lett. 91, 262001 (2003), hep-ex/0309032.
* [19] D. Acosta et al., CDF Collaboration, Phys.Rev.Lett. 93, 072001 (2004), hep-ex/0312021.
* [20] V. Abazov et al., D0 Collaboration, Phys.Rev.Lett. 93, 162002 (2004), hep-ex/0405004.
* [21] B. Aubert et al., BABAR Collaboration, Phys.Rev. D71, 071103 (2005), hep-ex/0406022.
* [22] S.-K. Choi et al., Phys.Rev. D84, 052004 (2011), 1107.0163.
* [23] A. Abulencia et al., CDF Collaboration, Phys.Rev.Lett. 96, 102002 (2006), hep-ex/0512074.
* [24] A. Abulencia et al., CDF Collaboration, Phys.Rev.Lett. 98, 132002 (2007), hep-ex/0612053.
* [25] LHCb Collaboration, hep-ex/1112.5310.
* [26] S. Chatrchyan et al., CMS Collaboration, CMS-PAS-BPH-10-018 (2011).
* [27] CDF Collaboration, Phys. Rev. Lett. 102, 242002 (2009).
* [28] CDF Collaboration, (2011), hep-ex/1101.6058.
* [29] Z.-G. Wang, Z.-C. Liu, and X.-H. Zhang, The European Physical Journal C - Particles and Fields 64, 373 (2009), 10.1140/epjc/s10052-009-1156-2.
* [30] R. M. Albuquerque, M. E. Bracco, and M. Nielsen, Physics Letters B 678, 186 (2009).
* [31] J.-R. Zhang and M.-Q. Huang, Journal of Physics G: Nuclear and Particle Physics 37, 025005 (2010).
* [32] F. Stancu, Journal of Physics G: Nuclear and Particle Physics 37, 075017 (2010).
* [33] N. V. Drenska, R. Faccini, and A. D. Polosa, Phys. Rev. D 79, 077502 (2009).
* [34] N. Mahajan, Physics Letters B 679, 228 (2009).
* [35] Z.-G. Wang, The European Physical Journal C - Particles and Fields 63, 115 (2009), 10.1140/epjc/s10052-009-1097-9.
* [36] X. Liu, Physics Letters B 680, 137 (2009).
* [37] S. I. Finazzo, M. Nielsen, and X. Liu, Physics Letters B 701, 101 (2011).
* [38] C. P. Shen et al., Belle Collaboration, Phys. Rev. Lett. 104, 112004 (2010).
* [39] J. Brodzicka, Heavy flavour spectroscopy, in Lepton Photon 2009 (LP09) Vol. DESY-PROC-2010-04, 2010.
* [40] LHCb Collaboration, hep-ex/1202.5087.
* [41] LHCb Collaboration, LHCb-CONF-2011-045,.
|
arxiv-papers
| 2012-09-01T18:36:24 |
2024-09-04T02:49:34.669657
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Augusto Alves Jr (on behalf of LHCb Collaboration)",
"submitter": "Antonio Augusto Alves Jr",
"url": "https://arxiv.org/abs/1209.0118"
}
|
1209.0140
|
# Iron based superconductors: magnetism, superconductivity and electronic
structure
(Review Article)
A. A. Kordyuk Institute of Metal Physics of National Academy of Sciences of
Ukraine, 03142 Kyiv, Ukraine
###### Abstract
Angle resolved photoemission spectroscopy (ARPES) reveals the features of the
electronic structure of quasi-two-dimensional crystals, which are crucial for
the formation of spin and charge ordering and determine the mechanisms of
electron-electron interaction, including the superconducting pairing. The
newly discovered iron based superconductors (FeSC) promise interesting physics
that stems, on one hand, from a coexistence of superconductivity and magnetism
and, on the other hand, from complex multi-band electronic structure. In this
review I want to give a simple introduction to the FeSC physics, and to
advocate an opinion that all the complexity of FeSC properties is encapsulated
in their electronic structure. For many compounds, this structure was
determined in numerous ARPES experiments and agrees reasonably well with the
results of band structure calculations. Nevertheless, the existing small
differences may help to understand the mechanisms of the magnetic ordering and
superconducting pairing in FeSC.
superconductivity, iron based superconductors, electronic band structure,
angle resolved photoemission spectroscopy
###### pacs:
74.20.-z, 74.25.Jb, 74.62. f, 74.70.Xa, 79.60. i
###### Contents
1. I Introduction
2. II Iron based superconductors
3. III Magnetism
1. III.1 Magnetic ordering
2. III.2 Spin-fluctuations
3. III.3 Pseudogap
4. IV Superconductivity
1. IV.1 Pairing models
2. IV.2 Superconducting gap
3. IV.3 Electronic structure and $T_{c}$
5. V Conclusions
## I Introduction
Four years ago, the discovery of LaO1-xFxFeAs KamiharaJACS2008 , a new
superconductor with transition temperature at 26 K, has marked the beginning
of a new era in superconducting research. The Copper Age has been replaced by
the Iron Age, i.e. all the researchers and fundings have switched from the
high-$T_{c}$ cuprates (HTSC or CuSC) to the iron based superconductors (FeSC),
as it is clear from a number of early reviews on the subject SadovskiiPU2008 ;
IvanovskiiPU2008 ; IzyumovPU2008 ; IshidaJPSJ2009 ; MazinPhC2009 ;
JohnstonAPh2010 ; PaglioneNPh2010 . Today, after four years of active
research, Ref. KamiharaJACS2008, has been cited more than 3000 times and the
investigation of FeSC is in the mainstream of the condensed matter physics
WenARoCMP2011 ; StewartRMP2011 ; HirschfeldRPP2011 ; ChubukovAR2012 .
There are several good reasons why the FeSC are so interesting. First, they
promise interesting physics that stems from a coexistence of superconductivity
and magnetism. Second, providing much larger variety of compounds for research
and having multi-band electronic structure, they give hopes to resolve finally
the mechanism of high temperature superconductivity and find the way of
increasing $T_{c}$. Lastly, the FeSC are quite promising for applications.
Having much higher $H_{c}$ than cuprates and high isotropic critical currents
PuttiSST2010 ; MollNM2010 ; GurevichRPP2011 , they are attractive for
electrical power and magnet applications, while the coexistence of magnetism
and superconductivity makes them interesting for spintronics PatelAPL2009 .
Figure 1: Crystal structures of some of iron based superconductors, after
PaglioneNPh2010 . Figure 2: (a) FeAs lattice indicating As above and below
the Fe plane. Dashed green and solid blue squares indicate 1- and 2-Fe unit
cells, respectively. (b) Schematic 2D Fermi surface in the 1-Fe BZ whose
boundaries are indicated by a green dashed square. (c) Fermi sheets in the
folded BZ whose boundaries are now shown by a solid blue square. After
HirschfeldRPP2011 .
To date, there is a number of useful and comprehensive reviews on the diverse
properties of FeSC SadovskiiPU2008 ; IvanovskiiPU2008 ; IzyumovPU2008 ;
IshidaJPSJ2009 ; JohnstonAPh2010 ; PaglioneNPh2010 ; WenARoCMP2011 and on the
pairing models WenARoCMP2011 ; HirschfeldRPP2011 ; ChubukovAR2012 . The scope
of this review is smaller but twofold. On one hand, I want to give a simple,
even oversimplified introduction to the FeSC physics. On the other hand, I
want to advocate an opinion that all the complexity of FeSC properties are
encapsulated in their complex but well defined and rather common multi-band
electronic structure. For many compounds, this structure has been determined
in numerous angle resolved photoemission experiments (ARPES), and one of the
scopes of this review is to show that while the overall agreement between the
measured and calculated band structures is very good, it is the observed small
differences KordyukFPS2011 that may help to understand the mechanisms of the
magnetic ordering and superconducting pairing in FeSC.
Figure 3: Examples of the FeSC phase diagrams: a schematic one ChubukovAR2012
, and the diagrams measured for (Ba1-xKx)Fe2As2 WenARoCMP2011 ,
Ba(Fe1-xCox)2As2 NandiPRL2010 , La(O1-xFx)FeAs LuetkensNM2009 , Fe1+ySexTe1-x
Katayama2010 , BaFe2(As1-xPx)2 JiangJPCM2009 .
## II Iron based superconductors
There are many families of FeSC with different structure and composition
already known SadovskiiPU2008 ; IvanovskiiPU2008 ; IzyumovPU2008 ;
IshidaJPSJ2009 ; JohnstonAPh2010 ; PaglioneNPh2010 ; WenARoCMP2011 but all
share a common iron-pnictogen (P, As) or iron-chalcogen planes (Se, Te), as
shown in Fig. 1 PaglioneNPh2010 . All the compounds share similar electronic
band structure in which the electronic states at the Fermi level are occupied
predominantly by the Fe $3d$ electrons. The structure itself is quite complex
and, in most cases, consist of five conduction bands that result in rather
complex Fermiology that changes rapidly with doping and, consequently, leads
to many unusual superconducting and normal state properties. Fig. 2 shows the
square FeAs lattice and the corresponding Fermi surface for a stichometric
parent compound. Fig. 3 provides examples of the FeSC phase diagrams with
distinct areas of the antiferromagnetically ordered spin density wave (marked
as AFM or SDW and bordered by the Néel temperature $T_{N}$) and
superconducting (SC, $T_{c}$) phases, reminding the extensively discussed
phase diagram of CuSC KordyukLTP2006 . Here I briefly review some of the most
interesting and most studied FeSC materials with the references to their
properties and experimental (ARPES) studies of their electronic structure,
which will be important for the following discussion.
1111. Starting from LaO1-xFxFeAs KamiharaJACS2008 , the 1111 family keeps the records of $T_{c}$: NdFeAsO1-y (54 K), SmFeAsO1-xFx (55 K), Gd0.8Th0.2FeAsO (56.3 K), but the material is hard to study. First, the available single crystals are too small—for all members of the family they grow as thin platelets up to $200\times 200\times 10$ $\mu\textrm{m}^{3}$ only MollNM2010 . Second, the termination of the crystal reveals a polar surface with distinct surface states that are markedly different from the bulk electronic structure EschrigPRB2010 and highly complicate the use of any surface sensitive experimental probe such as ARPES LiuPRB2010 .
122. The 122 family consists of a variety of different compounds with wide ranges of doping in both hole and electron sides WenARoCMP2011 that form a rich phase diagram (see Fig. 3) where the superconductivity and magnetism compete or coexist. The most studied compounds are the hole doped Ba1-xKxFe2As2 (BKFA) with $T_{c}^{\textrm{max}}=$ 38 K RotterPRL2008 and the electron doped Ba(Fe1-xCox)2As2 (BFCA), 22 K SefatPRL2008 ; NiPRB2008 . Both share the same parent compound, BaFe2As2 (BFA), which is a compensated metal, i.e. the total volume of its three hole Fermi surfaces (FS’s) is equal to the total volume of two electron FS’s BrouetPRB2009 ; KondoPRB2010 . BFA goes into magnetically ordered phase below 140 K RotterPRB2008 and never superconducts. An extremely overdoped BKFA is a stoichiometric KFe2As2 (KFA) SatoPRL2009 , which is non-magnetic, with $T_{c}=$ 3 K. There is also an interesting case of isovalent dopping, BaFe2(As1-xPx)2 (BFAP) ($T_{c}=$ 30 K) JiangJPCM2009 with similar phase diagram (see Fig. 3).
To this, one can add a number of similar compounds: Ba1-xNaxFe2As2 (BNFA)
($T_{c}^{\textrm{max}}=$ 34 K) CortesGil2010 ; Aswartham2012 , Ca1-xNaxFe2As2
($\sim$20 K) WuJPCM2008 , CaFe2As2 ($T_{N}=$ 170 K, $T_{c}>$ 10 K under
pressure) ParkJPCM2008 , EuFe2(As1-xPx)2 ($T_{c}=$ 26 K) RenPRL2009 , etc.
JohnstonAPh2010
As a consequence of good crystal quality and variety of compounds, the 122
family is the most studied by ARPES RichardRPP2011 (see LiuPRL2008 ;
DingEPL2008 ; ZabolotnyyN2009 ; EvtushinskyPRB2009 ; RichardPRL2009 ;
EvtushinskyNJP2009 ; EvtushinskyJPSJ2011 for BKFA, BrouetPRB2009 ;
ThirupathaiahPRB2010 ; LiuNP2010 ; LiuPRB2011 ; YiPNAS2011 ; HeumenPRL2011
for BFCA, YiPRB2009 ; FinkPRB2009 ; KondoPRB2010 ; RichardPRL2010 ; KimPRB2011
for BFA, SatoPRL2009 ; YoshidaJPCS2011 for KFA, LiuPRL2009 ; KondoPRB2010
for CaFe2As2, YoshidaPRL2011 ; ZhangNP2012 for BFAP, ThirupathaiahPRB2011
for EuFe2(As1-xPx)2). The ARPES spectra well represent the bulk electronic
structure of this family, at least, for the hole doped BKFA, BNFA, and BFAP,
where the superconducting gap is routinely observed DingEPL2008 ;
EvtushinskyPRB2009 ; Evtushinsky2011 and is in a good agreement with the bulk
probes EvtushinskyNJP2009 ; EvtushinskyJPSJ2011 . This poses the 122 family as
the main arena to study the rich physics of the iron-based superconductors.
Figure 4: Fermi surface (FS) maps measured by ARPES for LiFeAs
BorisenkoPRL2010 (left) and an optimally doped Ba1-xKxFe2As2 (BKFA)
ZabolotnyyN2009 (right).
111. Being highly reactive with air and, consequently, more challenging to study, the 111 family has gave many interesting results that keep growing. The main representative of the family, LiFeAs WangSSC2008 ; TappPRB2008 , is the most “arpesable” compound BorisenkoPRL2010 ; KordyukPRB2011 ; BorisenkoSym2012 . It grows in good quality single crystals MorozovCGD2010 that cleave between the two Li layers, thus revealing a non-polar surface with protected topmost FeAs layer; it is stoichiometric, i.e. impurity clean; it has the transition temperature about 18 K and one can measure the superconducting gap by ARPES and compare its value to bulk techniques; it is non-magnetic and, consequently, the observed band structure is free of SDW replicas; and, finally, its electronic bands are the most separated from each other that allows one to disentangle them most easily and analyze their fine structure KordyukPRB2011 . Fig. 4 shows the FS maps measured by ARPES for LiFeAs (left) and an optimally doped BKFA (right).
NaFeAs is another member of 111 family. It shows three successive phase
transitions at around 52, 41, and 23 K, which correspond to structural,
magnetic, and superconducting transitions, respectively ChenPRL2009 ;
LiPRB2009 . The compound is less reactive with the environment than LiFeAs but
the exposure to air strongly affects $T_{c}$ TanatarPRB2012 . Replacing Fe by
either Co or Ni suppresses the magnetism and enhances superconductivity
ParkerPRL2010 . For ARPES on NaFeAs, see HePRL2010 ; HeJPCS2011 .
11. The binary FeAs does not crystallize in the FeAs-layered structure (it adopts an orthorhombic structure consisting of distorted FeAs6 octahedra unlike the superconducting ferro-pnictides in which FeAs4 tetrahedra form square lattices of iron atoms SegawaJPSJ2009 ), but the FeSe does. So, the 11 family is presented by simplest ferro-chalcogenides FeSe and FeTe, and their ternary combination FeSexTe1-x SalesPRB2009 . The FeSe has been found to superconduct at approximately 8 K HsuPNAS2008 ), and up to 37 K under pressure ImaiPRL2009 . Fe1+ySexTe1-x shows maximum $T_{c}$ about 14 K at $x=0.5$ SalesPRB2009 ; Katayama2010 ; FedorchenkoLTP2011 . The crystals grow with excess ($y$) Fe atoms that present beyond those needed to fill the Fe square lattice layers and go into interstitial positions within the Te layers JohnstonAPh2010 . For ARPES on 11 family, see XiaPRL2009 ; TamaiPRL2010 ; NakayamaPRL2010 ; MiaoPRB2012 .
245 or x22. The attempts to intercalate FeSe, the simplest FeSC, resulted in
discovery of a new family AxFe2-ySe2 (A stands for alkali metal: K, Rb, Cs,
Tl) with $T_{c}$ up to 30 K and with exceptionally high Néel temperature
($>$500 K) and magnetic moment ($>$$3\mu_{B}$) GuoPRB2010 ; WangNC2011 ;
LiPRB2011 . This family called most often “245” because of its parent compound
A0.8Fe1.6Se2 $\equiv$ A2Fe4Se5. It is interesting that their resistivity shows
insulating behavior down to 100 K and superconductivity seems to occur from an
antiferromagnetic semiconductor YanSR2012 . This, however, is not consistent
with ARPES results which show the presence of a Fermi surface QianPRL2011 ;
MouPRL2011 ; ZhangNM2011 ; ChenPRX2011 . It is even more interesting that the
observed FS is completely electron-like that, seemingly, contradicts to the
most popular $s\pm$ scenario for superconducting pairing WenARoCMP2011 ;
HirschfeldRPP2011 ; ChubukovAR2012 . Recently, it has been shown
BorisenkoXXX2012 , that the puzzling behavior of these materials is the result
of separation into metallic and antiferromagnetic insulating phases, from
which only the former becomes superconducting, while the later has hardly any
relation to superconductivity. The superconducting phase has an electron doped
composition AxFe2Se2 (so, the family can be called “x22”). Similar conclusion
has been made based on neutron scattering experiments FriemelPRB2012 .
## III Magnetism
Naturally, the magnetic properties of FeSC are very rich and far from be
completely understood LumsdenJPCM2010 , but since the focus of this review is
in superconductivity, I will discuss only two issues: coexistence of static
magnetism and superconductivity and role of spin fluctuations.
### III.1 Magnetic ordering
Nearly perfect FS nesting in many parent compounds (which are compensated
metals) suggests us to expect some static density wave with the nesting vector
($\pi,\pi$), as a way to lower the kinetic energy of the electrons (Peierls
transition) GrunerRMP1988 . Therefore, the realization of the
antiferromagnetic spin density wave in those compounds is quite natural—the
most easy ordering for Fe lattice is the spin ordering DongEPL2008 . Indeed,
almost all parent compounds enters the antiferromagnetic SDW below Néel
temperature with exactly the same wavevector. Such a most common spin
configuration on Fe atoms is shown in Fig. 5 (left) Li2009FeTe . This said,
there are different opinions based on importance of interaction of the
localized spins YildirimPRL2008 ; MazinPhC2009 ; YinPRL2010 (see also
LumsdenJPCM2010 for review on this topic). From experiment, there are both
pro and con arguments on this problem. Pro: any time when the FS nesting is
good (BFA YiPRB2009 ; KondoPRB2010 ; RichardPRL2010 and other parent
compounds of 122 family YoshidaPRL2011 ; ZhangNP2012 ; ThirupathaiahPRB2011 ,
NaFeAs HePRL2010 ; HeJPCS2011 ), the SDW is present, and when nesting is poor
or absent (superconducting BKFA LiuPRL2008 ; DingEPL2008 , BFCA BrouetPRB2009
; ThirupathaiahPRB2010 , BFAP YoshidaPRL2011 , and stoichiometric LiFeAs
BorisenkoPRL2010 ), there is no magnetic ordering. Con: Fe1+yTe shows
different spin order, see Fig. 5 (right) Li2009FeTe , despite having very
similar FS topology (as it follows from calculations SubediPRB2008 and ARPES
study XiaPRL2009 ). So, one may conclude that the mechanism of the magnetic
ordering in FeAS is not yet clear, but for the scope of this review it is
important to know that this ordering is routinely observed in many compounds,
_always_ neighboring the superconducting phase and often coexisting with it.
Since static magnetism and superconductivity coexist on the phase diagrams for
a number of FeSC StewartRMP2011 , it is important to answer the questions: (1)
do they coexist microscopically and (2) do magnetism and superconductivity
evolve from the same conduction electrons? The latter is related to the
“itinerant _vs._ localized” problem and was briefly discussed above. The
problem of coexistence on the microscopic scale is related to sample
homogeneity and has been addressed in a number of publications (see
StewartRMP2011 for a short review). In particular, for BFCA crystals, the
homogeneity of superconducting state was demonstrated by magneto-optic imaging
ProzorovPhC2009 down to 2 $\mu$m and by NMR JulienEPL2009 down to the sub-
nanometer scale. Another 122 compound, BKFA, is known to be inhomogeneous
EvtushinskyPRB2009 , and some separation of the magnetic and superconducting
regions has been found on a nanometer scale ParkPRL2009 . An evidence for
homogeneity has been reported for one of 245 family, K0.8Fe1.6Se2,
ShermadiniPRL2011 , but not confirmed by magnetic measurements on similar
samples ShenEPL2011 . Clear phase separation in other, Rb based 245, has been
recently demostrated by ARPES BorisenkoXXX2012 and by inelastic neutron
scattering (INS) FriemelPRB2012 . Also, it has been shown, that in EuFe2As2
under pressure KuritaPRB2011 , similarly to the quaternary borocarbides
GuptaAP2006 , the antiferromagnetism is realized on the Eu sublattice,
affecting the superconductivity on the Fe sublattice. So, one may conclude
that while on some systems like 245 the magnetic and superconducting phases
are spatially separated, the question of coexistence in other FeSC systems
requires more careful study.
The neighboring is close and interesting issue. FeSC are perfect systems for
realization of the CDW (or SDW) induced superconductivity, the idea which had
been suggested long ago KopaevZETF1970 ; MattisPRL1970 ; RusinovJETP1973 and
widely discussed BalseiroPRB1979 ; KopaevPLA1987 ; GabovichLTP2000 ;
GabovichACMP2010 . For a slightly non-stoichiometric system, the band gap
cannot kill the FS completely since some extra carriers should form small FS
pockets and place the Van Hove singularity (vHs) close to the Fermi level.
This mechanism is supported empirically since there are many known systems
where superconductivity occurs at the edge of CDW or SDW phase FangPRB2005 ;
MorosanNP2006 ; Kato1988 . On the other hand, the related increase of the
density of states seems to be too small to explain the observed $T_{c}$’s
within the standard BCS model. I this sense, the conclusion of this review
about importance of the proximity of FS to Lifshitz transition for
superconductivity can help to understand the density wave induced
superconductivity, in general.
Figure 5: In-plane magnetic structure common for the 1111 and 122 parent
compounds (left) and for parent 11 compound (FeTe, right). The shaded areas
indicate the magnetic unit cells. After Li2009FeTe .
### III.2 Spin-fluctuations
If a magnetically mediated pairing mechanism takes place in FeSC, the spin-
fluctuation spectrum must contain the necessary spectral weight to facilitate
pairing LumsdenJPCM2010 . It is also expected that fingerprints of its
structure will be recognizable in one-particle spectral function, like in case
of cuprates (for example, see DahmNP2009 ; KordyukEPJ2010 ).
The spin dynamics in FeSC is revealed primarily by INS and, in some cases,
supplemented by NMR measurements (see LumsdenJPCM2010 for review).
First, the correlation between the spectral weight of the spin-fluctuations
and superconductivity is observed. In at least two cases (BFCA NingPRL2010 ;
MatanPRB2010 and LaFeAsO1-xFx WakimotoJPSJ2010 ), when antiferromagnetically
ordered parent compounds are overdoped by electron doping, the spin
fluctuations vanish together with the FS hole pocket BrouetPRB2009 and
superconductivity. This is compatible with the idea that the spin fluctuations
are completely defined by the electronic band structure and play important
role in superconductivity.
Second, the correlation between the normal state spin excitations and
electronic structure is found to be common for all FeSC LumsdenJPCM2010 . In
particurar, even in Fe1+ySexTe1-x IikuboJPSJ2009 , an interesting early
development in the study of the spin excitations was that, in contrast to the
parent FeTe, the spin fluctuations in superconducting samples were found at a
similar wavevector as found in the other Fe-based materials. Also, there is
another common feature, a quartet of low energy incommensurate inelastic peaks
characterized by the square lattice wavevectors ($\pi\pm\xi$ ,$\pi$) and
($\pi$, $\pi\pm\xi$), observed for BFCA LesterPRB2010 ; LiPRB2010 ,
Fe1+ySexTe1-x ArgyriouPRB2010 , and ${\text{CaFe}}_{2}{\text{As}}_{2}$
DialloPRB2010 , in analogy to CuSC VignolleNP2007 .
Third, the “resonance peak” in the spin-fluctuation spectrum has been observed
in many FeSC compounds in superconducting state, that is considered by many
authors as an evidence for a sign change of the superconducting order
parameter LumsdenJPCM2010 ; StewartRMP2011 .
The spin resonance, the resonance in the dynamic spin susceptibility, occurs
indeed because of its divergence through a sign change of the superconducting
order parameter on different parts of the Fermi surface MonthouxPRL1994 . In
cuprates, it was associated with the “resonance peak”, observed in INS
experiments, and considered as one of the arguments for $d$-wave symmetry of
the superconducting gap. In FeSC, the resonance peak was predicted to be the
most pronounced for the $s\pm$ gap KorshunovPRB2008 ; MaierPRB2008 and,
indeed, the peaks in INS spectra had been observed for a number of compounds:
BKFA ChristiansonN2008 , BFCA LumsdenPRL2009 ; InosovNP2010 , Fe1+ySexTe1-x
ArgyriouPRB2010 ; LeePRB2010 ,
${\mathrm{Rb}}_{2}{\mathrm{Fe}}_{4}{\mathrm{Se}}_{5}$ ParkPRL2011 ;
FriemelPRB2012 , etc. LumsdenJPCM2010 . However, one should realize, that the
peak in the dynamic susceptibility is not necessarily caused by the spin
resonance but can be due to a peak in the bare susceptibility (Lindhard
function), which, as a result of self-correlation of electronic Green’s
function, is expected to be peaked in energy at about $2\Delta$ and in
momentum at the FS nesting vectors InosovPRB2007 . In OnariPRB2010 , in
contrast to MaierPRB2008 , it has been shown that a prominent hump structure
appears just above the spectral gap by taking into account the quasiparticle
damping in SC state. The obtained hump structure looks similar to the
resonance peak in the $s\pm$-wave state, although the height and the weight of
the peak in the latter state is much larger. This shows that in order to
support the sign charge scenario, not only the presence of the peak in INS
spectra but also its spectral weight should be considered. The later is not
trivial task. In Ref. InosovNP2010, , for example, the INS measurements were
calibrated in the absolute scale and the spectral weight of the resonance in
BFCA has been found to be comparable to ones in cuprates.
In summary, the spin-fluctuation spectra in FeSC, looks, at first glance,
similar to the ones in CuSC in terms of appearance and correlation with
electronic structure, but its accurate interpretation requires more efforts.
As very last example for this, a combined analysis of neutron scattering and
photoemission measurements on superconducting FeSe0.5Te0.5 LeePRB2010 has
shown that while the spin resonance occurs at an incommensurate wave vector
compatible with nesting, neither spin-wave nor FS nesting models can describe
the magnetic dispersion. The authors propose that a coupling of spin and
orbital correlations is key to explaining this behavior.
Figure 6: Superconducting gap symmetry in LiFeAs. Experimental Fermi surface
(left). The experimental dispersions (center) measured along the cuts A and B.
A sketch of distribution of the superconducting gap magnitude over Fermi
surfaces (right). After HirschfeldRPP2011 .
### III.3 Pseudogap
Surprisingly, the pseudogap in FeSC is not a hot topic like in cuprates
TimuskRPP1999 . From a nearly perfect FS nesting one would expect the
pseudogap due to incommensurate ordering like in transition metal
dichalcogenides BorisenkoPRL2008 and, may be, in cuprates KordyukPRB2009 . If
the pseudogap in cuprates is due to superconducting fluctuations Huefner2008 ,
then it would be also natural to expect it in FeSC.
In NMR data, the decrease in $1/T_{1}T$ in some of 1111 compounds and BFCA
IshidaJPSJ2009 was associated with the pseudogap. The interplane resistivity
data for BFCA over a broad doping range also shows a clear correlation with
the NMR Knight shift, assigned to the formation of the pseudogap
TanatarPRB2010 . In SmFeAsO1-x, the pseudogap was determined from resistivity
measurements SolovevLTP2009 . The evidence for the superconducting pairs in
the normal state (up to temperature $T\approx 1.3T_{c}$) has been obtained
using point-contact spectroscopy on BFCA film.
An evidence for the pseudogap has been reported from photoemission experiments
on polycrystalline samples (e.g., see SatoJPSJ2008 ; HaiYun2008 ) and in some
ARPES experiments on single crystals XuNC2011 , but this is supported neither
by other numerous ARPES studies EvtushinskyPRB2009 ; RichardPRL2009 ;
Evtushinsky2011 ; ShimojimaSci2011 ; BorisenkoSym2012 ; ZhangNP2012 ;
Evtushinsky2012 nor by STM measurements YinPhC2009 ; MasseeEPL2010 . The
absence of the pseudogap in ARPES spectra may be just a consequence of low
spectral weight modulation by the magnetic ordering that may question its
importance for superconductivity, discussed in previous section.
## IV Superconductivity
In 1111 and 122 systems, superconductivity emerges upon electron or hole
doping, or can be induced by pressure SefatRPP2011 or by isovalent doping. In
111 systems, superconductivity emerges already at zero doping instead of
magnetic order (in LiFeAs) or together with it (in NaFeAs). There are several
important experimentally established tendencies, which are followed by many
representatives of iron-based family with highest $T_{c}$ Evtushinsky2012 :
large difference in superconducting gap magnitude on different FS pockets
DingEPL2008 ; EvtushinskyNJP2009 ; HardyEPL2010 ; PopovichPRL2010 ,
$\Delta/T_{c}$ value, that is similar to cuprates and much higher than
expected from BCS HardyEPL2010 ; EvtushinskyNJP2009 ; Evtushinsky2011 ,
correlation of $T_{\rm\text{c}}$ with anion height OkabePRB2010 . The
complexity of the electronic structure of FeSC was originally an obstacle on
the way to its understanding DingEPL2008 ; ZabolotnyyN2009 , but at closer
look, such a variety of electronic states turned out to be extremely useful
for uncovering the correlation between orbital character and pairing strength
Evtushinsky2012 and, more general, between electronic structure and
superconductivity KordyukFPS2011 .
In this section I briefly discuss the existent pairing models, the
experimental, mainly ARPES data on superconducting gap symmetry, and the
observed general correlation of the electronic band structure with $T_{c}$.
### IV.1 Pairing models
From similarity of the phase diagrams for FeSC and cuprates, it was proposed
that the pairing in FeSC is also mediated by spin-fluctuations that assumes
the sign change of the superconducting order parameter. Then, to adopt the FS
geometry of FeSC, the symmetry of the sign change should be different from
$d$-wave symmetry of cuprates and can be satisfied by an extended s-wave
pairing with a sign reversal of the order parameter between different Fermi
surface pockets MazinPRL2008 . Today, the most of researchers do believe that
the gap does have $s\pm$ symmetry, at least in weakly and optimally doped
FeSCs (see recent reviews HirschfeldRPP2011 ; ChubukovAR2012 ).
This said, numerous studies of superconductivity in FeSCs demonstrated that
the physics of the pairing could be more involved than it was originally
thought because of the multiorbital/multiband nature of low-energy electronic
excitations ChubukovAR2012 . It turns out that both the symmetry and the
structure of the pairing gap result from rather nontrivial interplay between
spin-fluctuation exchange, Coulomb repulsion, and the momentum structure of
the interactions. In particular, an $s\pm$-wave gap can be with or without
nodes, depending on the orbital content of low-energy excitations, and can
even evolve into a $d$-wave gap with hole or electron overdoping. In addition
to spin fluctuations, FeSCs also possess charge fluctuations that can be
strongly enhanced OnariPRL2009 ; YinPRL2010 due to proximity to a transition
into a state with an orbital order. This interaction can give rise to a
conventional $s$-wave pairing.
The experimental data on superconductivity show very rich behavior,
superconducting gap structures appear to vary substantially from family to
family, and even within families as a function of doping or pressure
HirschfeldRPP2011 . The variety of different pairing states raises the issue
of whether the physics of FeSCs is model dependent or is universal, governed
by a single underlying pairing mechanism ChubukovAR2012 .
In favor of $s\pm$ symmetry, there are natural expectation that spin-
fluctuations mediate pairing in FeSC, the observation of spin resonances by
INS, which implies the sign change of $\Delta$ as discussed above, and
numerous experimental evidences for the nodal gap NakaiPRB2010 ;
YamashitaPRB2011 ; ZhangNP2012 (see also references in HirschfeldRPP2011 ;
ChubukovAR2012 ). It was also argued VorontsovPRB2010 that the very presence
of the coexistence region between SC and stripe magnetism in FeSCs is a
fingerprint of an $s\pm$ gap, because for an $s^{++}$ gap a first-order
transition between a pure magnetic and a pure SC state is much more likely
ChubukovAR2012 .
On the other hand, several cons come from ARPES. There is an evidence for
strong electron-phonon coupling in LiFeAs KordyukPRB2011 ; BorisenkoSym2012 .
The accurately measured gap anisotropy is difficult to reconcile with the
existent $s\pm$ models but with $s^{++}$ models based on orbital fluctuations
assisted by phonons OnariPRL2009 ; KontaniPRL2010 ; YanagiPRB2010 . The
remnant superconductivity in KFe2As2, and, actually, for all overdoped BKFA
started from the optimally doped one ZabolotnyyN2009 , should have different
symmetry since only hole like FSs are present KordyukFPS2011 . The same is
applicable for AxFe2-ySe2 where only electron-like FSs are present QianPRL2011
; MouPRL2011 ; ZhangNM2011 ; ChenPRX2011 ; BorisenkoXXX2012 .
In ChubukovAR2012 it was suggested that in both AxFe2-ySe2 and KFe2As2 cases
the gap symmetry may be $d$-wave, though with different nodes. In Khodas2012
it is argued that $s\pm$ symmetry in AxFe2-ySe2 can be realized due to inter-
pocket pairing, i.e. $\Delta$ changes sign between electron pockets. Another
possibility HirschfeldRPP2011 for the order parameter to change sign in
AxFe2-ySe2, is taking into account the finite energy of the coupling boson
that should be higher than the binding energy of the top of the hole band in
$\Gamma$-point, but one can hardly describe rather high $T_{c}$ in 245 family
within such a mechanism.
### IV.2 Superconducting gap
The best FeSC for ARPES and, consequently, the systems on which the most
reliable data on superconducting gap can be obtained, are LiFeAs, BKFA (and
similar hole doped compounds), and BFAP.
Figure 7: Three-dimensional distribution of the superconducting gap and
orbital composition of the electronic states at the Fermi level of
Ba1-xKxFe2As2 (BKFA). (a) Distribution of the superconducting gap (plotted as
height) and distribution of the orbital composition for the states at the
Fermi level (shown in color: $d_{xz,yz}$ — red, $d_{xy}$ — green, $d_{xz,yz}$
with admixture of other orbitals — orange) as function of $k_{x}$ and $k_{y}$
at constant $k_{z}=0$; (b) the same, only for $k_{z}=\pi$; (c) same
distributions as function of in-plane momentum, directed along BZ diagonal,
and $k_{z}$. Note unambiguous correlation between the color and height, i.e.
there is strong correlation between the orbital composition and
superconducting gap magnitude. After Evtushinsky2012 .
LiFeAs allows the most careful determination of the gap value BorisenkoPRL2010
; BorisenkoSym2012 . Accurate measurements at 1 K have allowed to detect the
variations of $\Delta$ over the FS with relative precision of 0.3 meV and the
result is the following BorisenkoSym2012 (see Fig. 6): On the small hole-like
FS at $\Gamma$-point of $d_{xz/yz}$ origin, that, at some $k_{z}$, only
touches the Fermi level, the largest superconducting energy gap of the size of
6 meV opens and is in agreement with tunneling spectroscopy HankePRL2012 .
Along the large 2D hole-like FS of $d_{xy}$ character the gap varies around
3.4 meV roughly as 0.5 meV $\cos(4\phi)$, being minimal at the direction
towards the electron-like FS. The gap on the outer electron pocket is smaller
than on the inner one and both vary around 3.6 meV as 0.5 meV $\cos(4\phi)$,
having maximal values at the direction towards $\Gamma$-point. The detected
gap anisotropy is difficult to reconcile with coupling through spin
fluctuations and the sign change of the order parameter but fits better to the
model of orbital fluctuations assisted by phonons OnariPRL2009 ;
KontaniPRL2010 ; YanagiPRB2010 .
In BKFA, the superconducting gap was studied by means of various experimental
techniques EvtushinskyNJP2009 ; PopovichPRL2010 , and vast majority of the
results can be interpreted in terms of presence of comparable amount of
electronic states gapped with a large gap ($\Delta_{\rm large}=$10–11 meV) and
with a small gap ($\Delta_{\rm small}<4$ meV). The in-plane momentum
dependence of the superconducting gap, determined in early ARPES studies, is
the following: the large gap is located on all parts of the FS except for the
outer hole-like FS sheet around $\Gamma$-point DingEPL2008 ;
EvtushinskyPRB2009 . In Evtushinsky2012 , a clear correlation between the
orbital character of the electronic states and their propensity to
superconductivity is observed in hole-doped BaFe2As2: the magnitude of the
superconducting gap maximizes at 10.5 meV exclusively for iron $3d_{xz,yz}$
orbitals, while for others drops to 3.5 meV (see Fig. 7).
In BFAP, motivated by earlier reported evidences for the nodal gap from NMR
NakaiPRB2010 and angle-resolved thermal conductivity YamashitaPRB2011 , the
superconducting gap was measured by ARPES ZhangNP2012 as function of $k_{z}$,
the out-of-plane momentum. A “circular line node” on the largest hole FS
around the Z point at the Brillouin zone (BZ) boundary was found. This result
was considered as an evidence for $s\pm$ symmetry ZhangNP2012 . Alternatively,
taking into account the observed correlation of the gap value with orbital
character of the electronic states Evtushinsky2012 , the “circular line node”
can be explained as a location of extremely small gap due to lack of
$d_{xz/yz}$ character of given FS sheet at the BZ boundary.
Figure 8: Electronic band structure of LiFeAs (a-c), a representative 111
compound, and BaFe2As2 (BFA) / Ba0.6K0.4Fe2As2 (BKFA) (d-f), the
parent/optimally doped 122 compound: the electronic bands, calculated (a, d)
and derived from ARPES experiment (b, e), and the Fermi surfaces of LiFeAs (c)
and BKFA (f), as seen by ARPES. The bands and FS contours are colored by the
most pronounced orbital character: Fe 3$d_{xy}$, 3$d_{xz}$, and 3$d_{yz}$.
Figure 9: PhD. Phase diagram of the 122 family of ferro-pnictides complemented
by the 122(Se) family as a generalized band structure driven diagram for the
iron based superconductors. The insets show that the Fermi surfaces for every
compound close to $T_{c\mathrm{max}}$ are in the proximity of Lifshitz
topological transitions: the corresponding FS sheets are highlighted by color
(blue for hole- and red for electron-like).
### IV.3 Electronic structure and $T_{c}$
One can safely say that the visiting card of the iron based superconductors is
their complex electronic band structure that usually results in five Fermi
surface sheets (see Fig. 8): three around the center of the Fe2As2 BZ and two
around the corners. Band structure calculations predict rather similar
electronic structure for all FeSCs (see AndersenAdP2011 ; Sadovskii and
references therein). ARPES experiments show that it is indeed the case: one
can fit the calculated bands to the experiment if it is allowed to renormalize
them about 3 times and shift slightly with respect to each other YiPRB2009 ;
DingJoPCM2011 ; BorisenkoPRL2010 ; BorisenkoJPCS2011 . In this section, I
focus first on the most “arpesable” LiFeAs and BKFA compounds, to discuss
their electronic structure in details.
LiFeAs. Fig. 8(a) shows a fragment of the low energy electronic band structure
of LiFeAs calculated using the LMTO method in the atomic sphere approximation
AndersenPRB1975 . The same calculated bands but 3 times renormalized are
repeated in panel (b) by the dotted lines to compare with the dispersions
derived from the numerous ARPES spectra BorisenkoPRL2010 ; KordyukPRB2011
shown in the same panel by the thick solid lines. The experimental Fermi
surface is sketched in panel (c). The five bands of interest are colored in
accordance to the most pronounced orbital character: Fe 3$d_{xy}$, 3$d_{xz}$,
and 3$d_{yz}$ LeePRB2008 ; GraserNJoP2009 . Those characters have helped us to
identify uniquely the bands in the experimental spectra using differently
polarized photons BorisenkoPRL2010 .
Comparing the results of the experiment and renormalized calculations, one can
see that the strongest difference is observed around $\mathrm{\Gamma}$ point:
the experimental $d_{xy}$ band is shifted up about 40 meV (120 meV, in terms
of the bare band structure) while the $d_{xz}$/$d_{yz}$ bands are shifted
about 40 (120) meV downwards. Around the corners of the BZ (X point) the
changes are different, the up-shift of the $d_{xy}$ band in X point is about
60 meV while the $d_{xz}$/$d_{yz}$ bands are also shifted up slightly (about
10 meV). At the Fermi level, the largest hole-like FS sheet around
$\mathrm{\Gamma}$ point, formed by $d_{xy}$ band, is essentially larger in
experiment than in calculations. This is compensated by the shrunk
$d_{xz}$/$d_{yz}$ FSs where the larger one has become three-dimensional, i.e.
closed also in $k_{z}$ direction, and the smallest one has disappeared
completely. The electron-like FSs have changed only slightly, alternating its
character in $\mathrm{\Gamma}$X direction due to shift of the crossing of
$d_{xz}$ and $d_{xy}$ bands below the Fermi level, see Fig. 8(b). So, the
experimental electronic band structure of LiFeAs has the following very
important differences from the calculated one BorisenkoPRL2010 : (i) there is
no FS nesting, see 4 (left), and (ii) the vHs, the tops of the
$d_{xz}$/$d_{yz}$ bands at $\mathrm{\Gamma}$ point, stays in the vicinity of
the Fermi level, i.e. the system is very close to a Lifshitz transition
LifshitzZETF1960 . The latter makes the band structure of LiFeAs similar to
the structure of optimally doped Ba(Fe1-xCo${}_{x})$As2 (BFCA) LiuPRB2011 , as
discussed below.
BKFA. I start from the parent stoichiometric BaFe2As2, a representative
fragment of the calculated electronic band structure for which is shown in
Fig. 8(d). It is very similar to the band structure of LiFeAs with a small
complication at the bottom of the $d_{xy}$ bands in X point that is a
consequence of body-centered tetragonal stacking of FeAs layers instead of
simple tetragonal stacking in LiFeAs.
With the highest, in 122 family, transition temperature ($T_{c}$ = 38 K) and
the sharpest ARPES spectra, the hole doped BKFA and BNFA are the most
promising and the most popular objects for trying to understand the mechanism
of superconductivity in ferro-pnictides. This said, it is important to stress
that the FS of the optimally doped Ba0.6K0.4Fe2As2 and Ba0.6Na0.4Fe2As2 is
topologically different from the expected one: instead of two electron-like
pockets around the corners of the Fe2As2 BZ (X and Y points) there is a
propeller-like FS with the hole-like blades and a very small electron-like
center ZabolotnyyN2009 ; ZabolotnyyPhC2009 , as shown in Fig. 4 (right).
Curiously enough, despite the experimental reports of the propeller like FS,
the “parent” FS is still used in a number of theoretical models and as a basis
for interpretation of experimental results such as superconducting gap
symmetry.
Our first interpretation of the propeller-like FS, as an evidence for an
additional electronic ordering ZabolotnyyN2009 , was based on temperature
dependence of the photoemission intensity around X point and on the similarity
of its distribution to the parent BFA, but the interpretation based on a shift
of the electronic band structure YiPRB2009 was also discussed. Now, while it
seems that the electronic ordering plays a certain role in spectral weight
redistribution EvtushinskyJPSJ2011 , we have much more evidence for the
“structural” origin of the propellers: (1) The propeller-like FS, such as
shown Fig. LABEL:Fig_PropoMaps(a), is routinely observed for every optimally
doped BKFA or BNFA crystals we have studied. (2) In extremely overdoped KFA
SatoPRL2009 ; YoshidaJPCS2011 , where the magnetic ordering is not expected at
all, they naturally (according to rigid band approximation) evolve to larger
hole-like propellers. (3) The same propellers in the spectrum of the overdoped
($T_{c}=$ 10 K) BFCA at 90 meV below the Fermi level KordyukFPS2011 .
Fig. 8(e) shows the experimental bands (solid lines), derived from a number of
ARPES spectra KordyukFPS2011 , on top of the bands (thin dotted lines)
calculated for parent BFA, 3 times renormalized, and shifted by 30 meV, as
discussed above, to model the band structure expected for Ba0.6K0.4Fe2As2. One
can see that the difference between the experimental and “expected”
dispersions is even smaller than in case of LiFeAs and mainly appears near X
point as 40 meV shifts of the $d_{xz}$/$d_{yz}$ bands and one of $d_{xy}$
bands. These small shifts, however, result in the topological Lifshitz
transition of the FS and the question is how it is related to
superconductivity.
Naturally, one would like to examine whether one of the peaks in the
electronic density of states (DOS), related to the Lifshitz transitions, can
be responsible for the enhancement of superconductivity in BKFA. Comparing the
DOS calculated for the parent BFA and the model Fermi surfaces KordyukFPS2011
(see also SI ) one can see that the chemical potential, for which the FS would
be the most similar to the experimental FS of BKFA, drops in the region where
DOS of $d_{xz}$/$d_{yz}$ bands exhibits singularities. Strictly speaking, at
the energy of $-228$ meV DOS is not peaked but is increasing with lowering
energy, hinting that a simple correlation between DOS and $T_{c}$, as
suggested in Sadovskii , does not work for BKFA. From this procedure one can
also conclude that the extremely doped KFA should have much higher DOS than
any of BKFA, that clearly contradicts to the idea of simple relation between
DOS and $T_{c}$. On the other hand, the high $T_{c}$ superconductivity
scenario driven by interband pairing in a multiband system in the proximity of
a Lifshitz topological transition InnocentiPRB2010 ; InnocentiSUST2011 , looks
more promising alternative for BKFA. This said, it seems extremely challenging
task for chemists to go with overdoping still further in order to reach the
$d_{xz/yz}$ saddle points responsible for the largest DOS peak at $-282$ eV.
Interestingly, the same can be suggested for LiFeAs, where DOS SI shows a
much higher peak of the same $d_{xz/yz}$ origin.
Going back to the Lifshitz transitions in iron based superconductors, let us
overview their electronic band structures now accessible by ARPES. Recently,
the correlation of the Lifshitz transition with the onset of superconductivity
has been observed in BFCA LiuPRB2011 ; LiuNP2010 . The study has been mainly
concentrated on the outer hole-like FS formed by $d_{xy}$ orbitals,
nevertheless, it has been also found LiuPRB2011 that the tops of the
$d_{xz}$/$d_{yz}$ bands go to the Fermi level for the samples with the optimal
doping and $T_{c}=24$ K. Thus, the FS of optimally doped BFCA is similar to
the FS formed by $d_{xz}$/$d_{yz}$ bands of LiFeAs, i.e. for the case when the
$\mathrm{\Gamma}$-centered $d_{xz/yz}$ FS pocket is in the proximity of a
Lifshitz transition. One can add another 111 compound here, NaFeAs, that also
has the tops of $d_{xz}$/$d_{yz}$ bands very close to the Fermi level
HePRL2010 .
One more example to support this picture comes from 245 family (see Sadovskii
and references therein). The ARPES spectra from these compounds
BorisenkoXXX2012 are not very sharp yet, but one can confidently say that the
bottom of the electron pocket at the center of the BZ is very close to the
Fermi level, that allows us to place this family on the electron overdoped
side of the generalized phase diagram, as shown in Fig. 9. At the end we note
that in all known cases the bands those Lifshitz transitions do correlate with
$T_{c}$ have predominantly Fe 3$d_{xz/yz}$ orbital character.
## V Conclusions
While the mechanisms of superconductivity and magnetism in FeSC remain
unresolved issues, the experimental determination of electronic band structure
allows us to make useful conclusions. Now we can say that the electronic
structure of FeSC is either clear or can be easily clarified by experiment so
that one can easily fit the calculated bands to the experiment if it is
allowed to renormalize them about 3 times and shift slightly with respect to
each other. So, one can suggest the following algorithm:
$\displaystyle\mathrm{experiment}$ $\displaystyle=$
$\displaystyle(\mathrm{calculation}+\mathrm{shifts})\times\mathrm{renormalization}$
$\displaystyle\mathrm{calculation}$ $\displaystyle\Rightarrow$
$\displaystyle\mathrm{orbital~{}character}$ $\displaystyle\mathrm{shifts}$
$\displaystyle\Rightarrow$
$\displaystyle\mathrm{FS~{}topology}+\mathrm{nesting~{}conditions,}$
i.e. from comparison of the experiment and calculations one can get the
correct electronic structure with known orbital symmetry and estimate the
self-energy (renormalization). From the former one gets the Fermi surface
topology that is necessary for understanding superconductivity and FS geometry
(nesting conditions) that may or may not be important for understanding the
magnetism here. From renormalization one can get the information about
electronic interaction.
Considering all the electronic band structures of FeSCs that can be derived
from ARPES, it has been found that the Fermi surface of every optimally doped
compound (the compounds with highest $T_{c}$) has the Van Hove singularities
of the Fe 3$d_{xz/yz}$ bands in the vicinity to the Fermi level. This suggests
that the proximity to an electronic topological transition, known as Lifshitz
transition, for one of the multiple Fermi surfaces makes the superconductivity
dome at the phase diagram. Since the parent band structure is known, one can
consciously move the essential vHs to the Fermi level by charge doping, by
isovalent doping, or by pressure. Based on this empirical observation, one can
predict, in particular, that hole overdoping of KFe2As2 and LiFeAs compounds
is a possible way to increase the $T_{c}$.
To summarize, the iron based superconductors promise interesting physics and
applications. While the interplay of superconductivity and magnetism, as well
as their mechanisms remain the issues of active debates and studies, one thing
in FeSC puzzle is clear, namely that it is the complex multi-band electronic
structure of these compounds that determines their rich and puzzling
properties. What is important and fascinating is that this complexity seems to
play a positive role in the struggle for understanding the FeSC physics and
also for search of the materials with higher $Tc$’s. This is because the
multiple electronic bands and resulting complex Fermiology offer exceptionally
rich playground for establishing useful empirical correlations. This is also
because this electronic structure is well understood—the band structure
calculations well reproduce its complexity: all the bands and their symmetry.
The role of the experiment, in this case, is just to define exact position and
renormalization for each band. This piece of experimental knowledge, however,
appears to be vitally important for understanding of all electronic properties
of these new compounds.
###### Acknowledgements.
I am pleased to dedicate this review to 80th anniversary of Prof. V. V.
Eremenko. I acknowledge numerous discussions with members of the ARPES group
at IFW Dresden: S. V. Borisenko, D. V. Evtushinsky, and V. B. Zabolotnyy, as
well as with A. Bianconi, A. V. Boris, B. Büchner, A. V. Chubukov, A. M.
Gabovich, G. E. Grechnev, P. J. Hirschfeld, D. S. Inosov, T. K. Kim, Yu. V.
Kopaev, M. M. Korshunov, I. I. Mazin, I. V. Morozov, I. A. Nekrasov, S. G.
Ovchinnikov, E. A. Pashitskii, S. M. Ryabchenko, M. V. Sadovskii, S.
Thirupathaiah, M. A. Tanatar, and A. N. Yaresko. The project was supported by
the DFG priority program SPP1458, Grants No. KN393/4, BO1912/2-1.
## References
* (1) Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008).
* (2) M. V. Sadovskii, Physics-Uspekhi 51, 1201 (2008).
* (3) A. L. Ivanovskii, Physics-Uspekhi 51, 1229 (2008).
* (4) Y. A. Izyumov and E. Z. Kurmaev, Physics-Uspekhi 51, 1261 (2008).
* (5) K. Ishida, Y. Nakai, and H. Hosono, J. Phys. Soc. Jpn. 78, 062001 (2009).
* (6) I. Mazin and J. Schmalian, Physica C 469, 614 (2009).
* (7) D. C. Johnston, Advances in Physics 59, 803 (2010).
* (8) J. Paglione and R. L. Greene, Nat Phys 6, 645 (2010).
* (9) H.-H. Wen and S. Li, Annual Review of Condensed Matter Physics 2, 121 (2011).
* (10) G. R. Stewart, Rev. Mod. Phys. 83, 1589 (2011).
* (11) P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. Prog. Phys. 74, 124508 (2011).
* (12) A. Chubukov, Annual Review of Condensed Matter Physics 3, 57 (2012).
* (13) M. Putti et al., Supercond. Sci. Technol. 23, 034003 (2010).
* (14) P. J. W. Moll et al., Nat. Mater. 9, 628 (2010).
* (15) A. Gurevich, Rep. Prog. Phys. 74, 124501 (2011).
* (16) U. Patel et al., Appl. Phys. Lett. 94, 082508 (2009).
* (17) A. A. Kordyuk et al., arXiv:1111.0288 (2011).
* (18) A. A. Kordyuk and S. V. Borisenko, Low Temp. Phys. 32, 298 (2006).
* (19) S. Nandi et al., Phys. Rev. Lett. 104, 057006 (2010).
* (20) H. Luetkens et al., Nat. Mater. 8, 305 (2009).
* (21) N. Katayama et al., J. Phys. Soc. Jpn. 79, 113702 (2010).
* (22) S. Jiang et al., J. Phys.: Condens. Matter 21, 382203 (2009).
* (23) H. Eschrig, A. Lankau, and K. Koepernik, Phys. Rev. B 81, 155447 (2010).
* (24) C. Liu et al., Phys. Rev. B 82, 075135 (2010).
* (25) M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett. 101, 107006 (2008).
* (26) A. S. Sefat et al., Phys. Rev. Lett. 101, 117004 (2008).
* (27) N. Ni et al., Phys. Rev. B 78, 214515 (2008).
* (28) V. Brouet et al., Phys. Rev. B 80, 165115 (2009).
* (29) T. Kondo et al., Phys. Rev. B 81, 060507 (2010).
* (30) M. Rotter et al., Phys. Rev. B 78, 020503 (2008).
* (31) T. Sato et al., Phys. Rev. Lett. 103, 047002 (2009).
* (32) R. Cortes-Gil et al., Chem. Mater. 22, 4304 (2010).
* (33) S. Aswartham et al., arXiv:1203.0143 (2012).
* (34) G. Wu et al., J. Phys.: Condens. Matter 20, 422201 (2008).
* (35) T. Park et al., J. Phys.: Condens. Matter 20, 322204 (2008).
* (36) Z. Ren et al., Phys. Rev. Lett. 102, 137002 (2009).
* (37) P. Richard et al., Rep. Prog. Phys. 74, 124512 (2011).
* (38) C. Liu et al., Phys. Rev. Lett. 101, 177005 (2008).
* (39) H. Ding et al., EPL 83, 47001 (2008).
* (40) V. B. Zabolotnyy et al., Nature 457, 569 (2009).
* (41) D. V. Evtushinsky et al., Phys. Rev. B 79, 054517 (2009).
* (42) P. Richard et al., Phys. Rev. Lett. 102, 047003 (2009).
* (43) D. V. Evtushinsky et al., New J. Phys. 11, 055069 (2009).
* (44) D. V. Evtushinsky et al., J. Phys. Soc. Jpn. 80, 023710 (2011).
* (45) S. Thirupathaiah et al., Phys. Rev. B 81, 104512 (2010).
* (46) C. Liu et al., Nat Phys 6, 419 (2010).
* (47) C. Liu et al., Phys. Rev. B 84, 020509 (2011).
* (48) M. Yi et al., PNAS 108, 6878 (2011).
* (49) E. van Heumen et al., Phys. Rev. Lett. 106, 027002 (2011).
* (50) M. Yi et al., Phys. Rev. B 80, 024515 (2009).
* (51) J. Fink et al., Phys. Rev. B 79, 155118 (2009).
* (52) P. Richard et al., Phys. Rev. Lett. 104, 137001 (2010).
* (53) Y. Kim et al., Phys. Rev. B 83, 064509 (2011).
* (54) T. Yoshida et al., J. Phys. Chem. Solids 72, 465 (2011).
* (55) C. Liu et al., Phys. Rev. Lett. 102, 167004 (2009).
* (56) T. Yoshida et al., Phys. Rev. Lett. 106, 117001 (2011).
* (57) Y. Zhang et al., Nat. Phys. 8, 371 (2012).
* (58) S. Thirupathaiah et al., Phys. Rev. B 84, 014531 (2011).
* (59) D. V. Evtushinsky et al., arXiv:1106.4584v1 (2011).
* (60) X. Wang et al., Solid State Commun. 148, 538 (2008).
* (61) J. H. Tapp et al., Phys. Rev. B 78, 060505 (2008).
* (62) S. V. Borisenko et al., Phys. Rev. Lett. 105, 067002 (2010).
* (63) A. A. Kordyuk et al., Phys. Rev. B 83, 134513 (2011).
* (64) S. V. Borisenko et al., Symmetry 4, 251 (2012).
* (65) I. Morozov et al., Cryst. Growth Des. 10, 4428 (2010).
* (66) G. F. Chen et al., Phys. Rev. Lett. 102, 227004 (2009).
* (67) S. Li et al., Phys. Rev. B 80, 020504 (2009).
* (68) M. A. Tanatar et al., Phys. Rev. B 85, 014510 (2012).
* (69) D. R. Parker et al., Phys. Rev. Lett. 104, 057007 (2010).
* (70) C. He et al., Phys. Rev. Lett. 105, 117002 (2010).
* (71) C. He et al., J. Phys. Chem. Solids 72, 479 (2011).
* (72) K. Segawa and Y. Ando, J. Phys. Soc. Jpn. 78, 104720 (2009).
* (73) B. C. Sales et al., Phys. Rev. B 79, 094521 (2009).
* (74) F.-C. Hsu et al., PNAS 105, 14262 (2008).
* (75) T. Imai et al., Phys. Rev. Lett. 102, 177005 (2009).
* (76) A. V. Fedorchenko et al., Low Temp. Phys. 37, 83 (2011).
* (77) Y. Xia et al., Phys. Rev. Lett. 103, 037002 (2009).
* (78) A. Tamai et al., Phys. Rev. Lett. 104, 097002 (2010).
* (79) K. Nakayama et al., Phys. Rev. Lett. 105, 197001 (2010).
* (80) H. Miao et al., Phys. Rev. B 85, 094506 (2012).
* (81) J. Guo et al., Phys. Rev. B 82, 180520 (2010).
* (82) M. Wang et al., Nat. Commun. 2, 580 (2011).
* (83) C.-H. Li et al., Phys. Rev. B 83, 184521 (2011).
* (84) Y. J. Yan et al., Sci. Rep. 2, 212 (2012).
* (85) T. Qian et al., Phys. Rev. Lett. 106, 187001 (2011).
* (86) D. Mou et al., Phys. Rev. Lett. 106, 107001 (2011).
* (87) Y. Zhang et al., Nat. Mater. 10, 273 (2011).
* (88) F. Chen et al., Phys. Rev. X 1, 021020 (2011).
* (89) S. V. Borisenko et al., arXiv:1204.1316 (2012).
* (90) G. Friemel et al., Phys. Rev. B 85, 140511 (2012).
* (91) M. D. Lumsden and A. D. Christianson, Journal of Physics: Condensed Matter 22, 203203 (2010).
* (92) G. Grüner, Rev. Mod. Phys. 60, 1129 (1988).
* (93) J. Dong et al., EPL 83, 27006 (2008).
* (94) S. Li et al., Phys. Rev. B 79, 054503 (2009).
* (95) T. Yildirim, Phys. Rev. Lett. 101, 057010 (2008).
* (96) W.-G. Yin, C.-C. Lee, and W. Ku, Phys. Rev. Lett. 105, 107004 (2010).
* (97) A. Subedi et al., Phys. Rev. B 78, 134514 (2008).
* (98) R. Prozorov et al., Physica C 469, 667 (2009).
* (99) M.-H. Julien et al., EPL 87, 37001 (2009).
* (100) J. T. Park et al., Phys. Rev. Lett. 102, 117006 (2009).
* (101) Z. Shermadini et al., Phys. Rev. Lett. 106, 117602 (2011).
* (102) B. Shen et al., EPL 96, 37010 (2011).
* (103) N. Kurita et al., Phys. Rev. B 83, 214513 (2011).
* (104) L. C. Gupta, Advances in Physics 55, 691 (2006).
* (105) Y. V. Kopaev, Zh. Eksp. Teor. Fiz. 58, 1012 (1970) [Sov. Phys. JETP. 31544 (1970)].
* (106) D. C. Mattis and W. D. Langer, Phys. Rev. Lett. 25, 376 (1970).
* (107) A. I. Rusinov, D. C. Kat, and K. Y. V., Zh. Eksp. Teor. Fiz. 65, 1984 (1973) [Sov. Phys. JETP 38, 991 (1974)].
* (108) C. A. Balseiro and L. M. Falicov, Phys. Rev. B 20, 4457 (1979).
* (109) Y. Kopaev and A. Rusinov, Phys. Lett. A 121, 300 (1987).
* (110) A. M. Gabovich and A. I. Voitenko, Low Temp. Phys. 26, 305 (2000).
* (111) A. M. Gabovich et al., Advances in Condensed Matter Physics 2010 (2010).
* (112) L. Fang et al., Phys. Rev. B 72, 014534 (2005).
* (113) E. Morosan et al., Nat. Phys. 2, 544 (2006).
* (114) M. Kato and K. Machida, Phys. Rev. B 37, 1510 (1988).
* (115) T. Dahm et al., Nat. Phys. 5, 217 (2009).
* (116) A. Kordyuk et al., Eur. Phys. J. Special Topics 188, 153 (2010).
* (117) F. L. Ning et al., Phys. Rev. Lett. 104, 037001 (2010).
* (118) K. Matan et al., Phys. Rev. B 82, 054515 (2010).
* (119) S. Wakimoto et al., Journal of the Physical Society of Japan 79, 074715 (2010).
* (120) S. Iikubo et al., Journal of the Physical Society of Japan 78, 103704 (2009).
* (121) C. Lester et al., Phys. Rev. B 81, 064505 (2010).
* (122) H.-F. Li et al., Phys. Rev. B 82, 140503 (2010).
* (123) D. N. Argyriou et al., Phys. Rev. B 81, 220503 (2010).
* (124) S. O. Diallo et al., Phys. Rev. B 81, 214407 (2010).
* (125) B. Vignolle et al., Nat. Phys. 3, 163 (2007).
* (126) P. Monthoux and D. J. Scalapino, Phys. Rev. Lett. 72, 1874 (1994).
* (127) M. M. Korshunov and I. Eremin, Phys. Rev. B 78, 140509 (2008).
* (128) T. A. Maier and D. J. Scalapino, Phys. Rev. B 78, 020514 (2008).
* (129) A. D. Christianson et al., Nature 456, 930 (2008).
* (130) M. D. Lumsden et al., Phys. Rev. Lett. 102, 107005 (2009).
* (131) D. S. Inosov et al., Nat. Phys. 6, 178 (2010).
* (132) S.-H. Lee et al., Phys. Rev. B 81, 220502 (2010).
* (133) J. T. Park et al., Phys. Rev. Lett. 107, 177005 (2011).
* (134) D. S. Inosov et al., Phys. Rev. B 75, 172505 (2007).
* (135) S. Onari, H. Kontani, and M. Sato, Phys. Rev. B 81, 060504 (2010).
* (136) T. Timusk and B. Statt, Reports on Progress in Physics 62, 61 (1999).
* (137) S. V. Borisenko et al., Phys. Rev. Lett. 100, 196402 (2008).
* (138) A. A. Kordyuk et al., Phys. Rev. B 79, 020504 (2009).
* (139) S. Huefner et al., Rep. Prog. Phys. 71, 062501 (2008).
* (140) M. A. Tanatar et al., Phys. Rev. B 82, 134528 (2010).
* (141) A. L. Solov’ev et al., Low Temp. Phys. 35, 826 (2009).
* (142) T. Sato et al., J. Phys. Soc. Jpn. 77, 063708 (2008).
* (143) H.-Y. Liu et al., Chin. Phys. Lett. 25, 3761 (2008).
* (144) Y.-M. Xu et al., Nat. Commun. 2, 392 (2011).
* (145) T. Shimojima et al., Science 332, 564 (2011).
* (146) D. V. Evtushinsky et al., arXiv:1204.2432v1 (2012).
* (147) Y. Yin et al., Physica C 469, 535 (2009).
* (148) F. Massee et al., EPL 92, 57012 (2010).
* (149) A. S. Sefat, Rep. Prog. Phys. 74, 124502 (2011).
* (150) F. Hardy et al., EPL 91, 47008 (2010).
* (151) P. Popovich et al., Phys. Rev. Lett. 105, 027003 (2010).
* (152) H. Okabe et al., Phys. Rev. B 81, 205119 (2010).
* (153) I. I. Mazin et al., Phys. Rev. Lett. 101, 057003 (2008).
* (154) S. Onari and H. Kontani, Phys. Rev. Lett. 103, 177001 (2009).
* (155) Y. Nakai et al., Phys. Rev. B 81, 020503 (2010).
* (156) M. Yamashita et al., Phys. Rev. B 84, 060507 (2011).
* (157) A. B. Vorontsov, M. G. Vavilov, and A. V. Chubukov, Phys. Rev. B 81, 174538 (2010).
* (158) H. Kontani and S. Onari, Phys. Rev. Lett. 104, 157001 (2010).
* (159) Y. Yanagi, Y. Yamakawa, and Y. Ōno, Phys. Rev. B 81, 054518 (2010).
* (160) M. Khodas and A. V. Chubukov, Phys. Rev. Lett. (2012) (arXiv:1202.5563).
* (161) T. Hänke et al., Phys. Rev. Lett. 108, 127001 (2012).
* (162) O. K. Andersen and L. Boeri, Annalen der Physik 523, 8 (2011).
* (163) M. V. Sadovskii, E. Z. Kuchinskii, and I. A. Nekrasov, arXiv:1106.3707v1 (2011).
* (164) H. Ding et al., J. Phys.: Condens. Matter 23, 135701 (2011).
* (165) S. V. Borisenko et al., J. Phys. Chem. Solids 72, 562 (2011).
* (166) O. K. Andersen, Phys. Rev. B 12, 3060 (1975).
* (167) P. A. Lee and X.-G. Wen, Phys. Rev. B 78, 144517 (2008).
* (168) S. Graser et al., New Journal of Physics 11, 025016 (2009).
* (169) I. M. Lifshitz, Zh. Eksp. Teor. Fiz. 38, 1569 (1960) [Sov. Phys. JETP 11, 1130 (1960)].
* (170) V. B. Zabolotnyy et al., Physica C: Superconductivity 469, 448 (2009).
* (171) D. Innocenti et al., Phys. Rev. B 82, 184528 (2010).
* (172) D. Innocenti et al., Supercond. Sci. Technol. 24, 015012 (2011).
* (173) For details, see www.imp.kiev.ua/~kord/papers/FPS11.
|
arxiv-papers
| 2012-09-01T23:03:59 |
2024-09-04T02:49:34.678293
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. A. Kordyuk",
"submitter": "Alexander Kordyuk",
"url": "https://arxiv.org/abs/1209.0140"
}
|
1209.0157
|
# Global classical solutions to the two-dimensional compressible Navier-Stokes
equations in $\mathbb{R}^{2}$
Quansen Jiu,1,3 Yi Wang2,3 and Zhouping Xin3 The research is partially
supported by National Natural Sciences Foundation of China (No. 11171229) and
Project of Beijing Education Committee. E-mail: [email protected]
research is partially supported by National Natural Sciences Foundation of
China (No. 11171326) and by the National Center for Mathematics and
Interdisciplinary Sciences, CAS. E-mail: [email protected] research is
partially supported by Zheng Ge Ru Funds, Hong Kong RGC Earmarked Research
Grant CUHK4042/08P and CUHK4041/11P, and a grant from the Croucher Foundation.
Email: [email protected]
1 School of Mathematical Sciences, Capital Normal University, Beijing 100048,
P. R. China
2 Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, P. R. China
3The Institute of Mathematical Sciences, Chinese University of HongKong,
HongKong
Abstract: In this paper, we prove the global well-posedness of the classical
solution to the 2D Cauchy problem of the compressible Navier-Stokes equations
with arbitrarily large initial data when the shear viscosity $\mu$ is a
positive constant and the bulk viscosity $\lambda(\rho)=\rho^{\beta}$ with
$\beta>\frac{4}{3}$. Here the initial density keeps a non-vacuum states
$\bar{\rho}>0$ at far fields and our results generalize the ones by Vaigant-
Kazhikhov [41] for the periodic problem and by Jiu-Wang-Xin [26] and Huang-Li
[18] for the Cauchy problem with vacuum states $\bar{\rho}=0$ at far fields.
It shows that the solution will not develop the vacuum states in any finite
time provided the initial density is uniformly away from vacuum. And the
results also hold true when the initial data contains vacuum states in a
subset of $\mathbb{R}^{2}$ and the natural compatibility conditions are
satisfied. Some new weighted estimates are obtained to establish the upper
bound of the density.
Key Words: compressible Navier-Stokes equations, Cauchy problem, global well-
posedness, large data, vacuum
## 1 Introduction
We consider the following compressible and isentropic Navier-Stokes equations
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}\rho+{\rm div}(\rho
u)=0,\\\ \partial_{t}(\rho u)+{\rm div}(\rho u\otimes u)+\nabla
P(\rho)=\mu\Delta u+\nabla((\mu+\lambda(\rho)){\rm div}u),&\quad
x\in\mathbb{R}^{2},t>0,\end{array}\right.$ (1.3)
where $\rho(t,x)\geq 0$, $u(t,x)=(u_{1},u_{2})(t,x)$ represent the density and
the velocity of the fluid, respectively. And
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$ and $t\in[0,T]$ for any fixed $T>0$. Here
it is assumed that the shear viscosity $\mu>0$ is a positive constant and the
bulk viscosity
$\displaystyle\lambda(\rho)=\rho^{\beta}$ (1.4)
with $\beta>0$ in general such that the operator
$\mathcal{L}_{\rho}u\equiv\mu\Delta u+\nabla((\mu+\lambda(\rho)){\rm div}u)$
is strictly elliptic. The pressure function is given by
$P(\rho)=A\rho^{\gamma},$ where $\gamma>1$ denotes the adiabatic exponent and
$A>0$ is a constant which is normalized to be $1$ for simplicity. We impose
the initial values as
$(\rho,u)(t=0,x)=(\rho_{0},u_{0})(x)\rightarrow(\bar{\rho},0),\quad{\rm
as}\quad|x|\rightarrow+\infty,$ (1.5)
where $\bar{\rho}>0$ is a given positive constant.
In the case that both the shear and bulk viscosities are positive constants,
there are a large number of literatures on the well-posedness theories of the
compressible Navier-Stokes equations. In particular, the one-dimensional
theory is rather satisfactory, see [16, 32, 28, 29] and the references
therein. In multi-dimensional case, the local well-posedness theory of
classical solutions was established in the absence of vacuum (see [37], [20]
and [40]) and the global well-posedness theory of classical solutions was
obtained for initial data close to a non-vacuum steady state (see [35], [14],
[8], [3] and references therein). The local well-posedness of classical
solutions containing vacuum was studied by Cho-Kim [6] and Luo [34] and the
global well-posedness of classical solutions to the 3D isentropic compressible
Navier-Stokes equations with small energy was proved by Huang-Li-Xin [19]. For
the large initial data permitting vacuums, the global existence of weak
solutions was investigated in [31], [11], [22]. It should be noted that if the
initial data are arbitrarily large and the vacuums are permitted, the solution
will also contain possible vacuums and one could not expect the global well-
posedness in general, see [43] [39] and [44] for blow-up results of classical
solutions.
The case that both the shear and bulk viscosities depend on the density has
also received a lot attention recently, see [1, 2, 8, 10, 13, 21, 22, 23, 24,
27, 30, 33, 36, 45, 46, 47] and the references therein. When deriving by
Chapman-Enskog expansions from the Boltzmann equation, the viscosity of the
compressible Navier-Stokes equations depends on the temperature and thus on
the density for isentropic flows. Moreover, in geophysical flows, the viscous
Saint-Venant system for the shallow water corresponds exactly to a kind of
compressible Navier-Stokes equations with density-dependent viscosities.
However, except for the one-dimensional problems, few results are available
for the multi-dimensional problems and even the short time well-posedness of
classical solutions in the presence vacuum remains open.
The system (1.3) was first proposed and studied by Vaigant-Kazhikhov in [41].
For the periodic problem on the torus $\mathbb{T}^{2}$ and under assumptions
that the initial density is uniformly away from vacuum and $\beta>3$ in (1.4),
Vaigant-Kazhikhov established the well-posedness of the classical solution to
(1.3) in [41] and the global existence and large time behavior of weak
solutions was stuided by Perepelitsa in [38]. Recently, Jiu-Wang-Xin [25]
improved the result in [41] and obtained the global well-posedness of the
classical solution to the periodic problem with large initial data permitting
vacuum. Later on, Huang-Li relaxed the index $\beta$ to be $\beta>\frac{4}{3}$
and studied the large time behavior of the solutions in [17]. However, all the
above results are concerned with the 2D periodic problems. For the 2D Cauchy
problems with vacuum states at far fields, Jiu-Wang-Xin [26] and Huang-Li [18]
independently considered the global well-posedness of classical solution in
different weighted spaces.
In the present paper, we study the global well-posedness of the classical
solution to the Cauchy problem (1.3)-(1.5) with large data which keeps a non-
vacuum states $\bar{\rho}>0$ at far fields. In particular, our results show
that the solution will not develop the vacuum states in any finite time
provided the initial density is uniformly away from vacuum. The results of
this paper generalize the ones by Vaigant-Kazhikhov in [41] to the Cauchy
problem and the index $\beta$ is relaxed to be $\beta>\frac{4}{3}$. The
results also improve ones by Jiu-Wang-Xin [26] and Huang-Li [18] for the
Cauchy problem with vacuum states $\bar{\rho}=0$ at far fields. Moreover, the
results hold true if the initial data contains vacuum states in a subset of
$\mathbb{R}^{2}$ under appropriate compatibility conditions (see (1.11) in
Theorem 1.2).
To study the global well-posedness of the classical solution of the
compressible Navier-Stokes equations, it is crucial to obtain the uniformly
upper bound of the density. To do that, similar to [41], [25] and [26], we
first obtain any $L^{p}(2\leq p<\infty)$ estimates of the density
$\rho-\bar{\rho}$ and then obtain the estimates of the first order derivative
of the velocity. A new transport equation (3.36) is derived by means of the
effective viscous flux $F=(2\mu+\lambda(\rho)){\rm
div}u-(P(\rho)-P(\bar{\rho}))$ and two new functions $\xi$ and $\eta$
satisfying the elliptic problems
$-\Delta\xi={\rm div}(\rho u),\qquad-\Delta\eta={\rm div}[{\rm div}(\rho
u\otimes u)],$ (1.6)
respectively, which was introduced in [41]. Comparing with the periodic
problem and the Cauchy problem with vacuum at far fields, new difficulties
will be encountered. Since no integrability is expected for the density
itself, we will decompose the elliptic problem (1.6) into the following two
parts:
$-\Delta\xi_{1}={\rm div}(\sqrt{\rho}u(\sqrt{\rho}-\sqrt{\bar{\rho}})),$ (1.7)
$-\Delta\xi_{2}=\sqrt{\bar{\rho}}~{}{\rm div}(\sqrt{\rho}u).$ (1.8)
For the elliptic problem (1.7), one can make use of the similar properties as
the periodic case and the Cauchy problem with vanishing density at the far
fields thanks to the expected integrability of
$\sqrt{\rho}-\sqrt{\bar{\rho}}$. For the second elliptic problem (1.8), since
it is expected that $\rho\in L^{\infty}$ and $\sqrt{\rho}u\in
L^{\infty}([0,T];L^{2}(\mathbb{R}^{2}))$ by the elementary energy estimate, it
follows from (1.8) that $\nabla\xi_{2}\in D^{1}(\mathbb{R}^{2})$ which is a
homogeneous and critical Sobolev space. Therefore, the integrability of
$\xi_{2}$ can not be derived in a direct way. However, the integrability of
$\xi_{2}$ is crucial to obtain the $L^{p}(2\leq p<\infty)$ estimates of
$\rho-\bar{\rho}$ and the upper bound of the density $\rho$. In order to
circumvent this difficulty, some new weighted estimates are needed and the
integrability of the velocity and $\xi_{2}$ is proved by using Cafferelli-
Kohn-Nirenberg type inequality [4, 5]. It should be remarked that these
weighted estimates are motivated by our previous work [26] and in comparison
with the uniform constant in [26], the weight power $\alpha$ here depends on
the ratio $\frac{\lambda(\bar{\rho})}{\mu}$. At the same time, if
$\bar{\rho}=0$, then the weight $\alpha$ is exactly same as the one in our
previous work [26]. Moreover, when deriving the first-order derivative
estimates of the velocity, since $L^{p}$-integrability $(2\leq p<\infty)$ is
not available, it would be required to use the $L^{\infty}$-norm of the
density $\rho$ in a priori way which is motivated by the work [38]. In this
way, a $\log$-type inequality of the first-order derivative of the velocity
can be obtained (see Lemma 3.6). Finally, with help of a higher energy
estimate in Lemma 3.7, one can get a upper bound of the density under the
restriction $\beta>\frac{4}{3}$ (see [17, 18]).
Denote the potential energy by
$\Psi(\rho,\bar{\rho})=\frac{1}{\gamma-1}\big{[}\rho^{\gamma}-\bar{\rho}^{\gamma}-\gamma\bar{\rho}^{\gamma-1}(\rho-\bar{\rho})\big{]}.$
Our main results can be stated as follows.
###### Theorem 1.1
Let $\beta>\frac{4}{3}$ and $1<\gamma\leq 2\beta$. Suppose that the initial
values $(\rho_{0},u_{0})(x)$ satisfy
$\begin{array}[]{ll}\displaystyle 0<c\leq\rho_{0}\leq
C,\quad(\rho_{0}-\bar{\rho},P(\rho_{0})-P(\bar{\rho}))\in
W^{2,q}(\mathbb{R}^{2})\times W^{2,q}(\mathbb{R}^{2}),\quad u_{0}(x)\in
H^{2}(\mathbb{R}^{2}),\\\
\displaystyle\Psi(\rho_{0},\bar{\rho})(1+|x|^{\alpha})\in
L^{1}(\mathbb{R}^{2}),\quad\sqrt{\rho_{0}}u_{0}|x|^{\frac{\alpha}{2}}\in
L^{2}(\mathbb{R}^{2}),\end{array}$
where $q,c,C$ and $\alpha$ are positive constants satisfying $q>2$, $0<c<C$
and
$0<\alpha^{2}<\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}$
respectively. Then, for any $T>0$, there exists a unique global classical
solution $(\rho,u)(t,x)$ to the Cauchy problem (1.3)-(1.5) satisfying
$0<c_{1}\leq\rho\leq C_{1}$
for some positive constants $c_{1}$ and $C_{1}$. Moreover, one has
$\begin{array}[]{ll}\displaystyle(\rho-\bar{\rho},P(\rho)-P(\bar{\rho}))(t,x)\in
C([0,T];W^{2,q}(\mathbb{R}^{2})),\\\
\displaystyle\Psi(\rho,\bar{\rho})(1+|x|^{\alpha})\in
C([0,T];L^{1}(\mathbb{R}^{2})),\quad\displaystyle\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\in
C([0,T];L^{2}(\mathbb{R}^{2})),\\\ u\in C([0,T];H^{2}(\mathbb{R}^{2}))\cap
L^{2}(0,T;H^{3}(\mathbb{R}^{2})),~{}~{}\sqrt{t}u\in
L^{\infty}(0,T;H^{3}(\mathbb{R}^{2})),\\\ tu\in
L^{\infty}(0,T;W^{3,q}(\mathbb{R}^{2})),~{}~{}u_{t}\in
L^{2}(0,T;H^{1}(\mathbb{R}^{2})),\\\ \displaystyle\sqrt{t}u_{t}\in
L^{2}(0,T;H^{2}(\mathbb{R}^{2}))\cap
L^{\infty}(0,T;H^{1}(\mathbb{R}^{2})),~{}~{}tu_{t}\in
L^{\infty}(0,T;H^{2}(\mathbb{R}^{2})),\\\ \sqrt{t}\sqrt{\rho}u_{tt}\in
L^{2}(0,T;L^{2}(\mathbb{R}^{2})),~{}~{}t\sqrt{\rho}u_{tt}\in
L^{\infty}(0,T;L^{2}(\mathbb{R}^{2})),~{}~{}t\nabla u_{tt}\in
L^{2}(0,T;L^{2}(\mathbb{R}^{2})).\end{array}$ (1.9)
If the initial values contain vacuum states in a subset of $\mathbb{R}^{2}$,
then the following results can be obtained.
###### Theorem 1.2
Suppose that the initial values $(\rho_{0},u_{0})(x)$ satisfy
$\begin{array}[]{ll}\displaystyle\rho_{0}\geq
0,\quad(\rho_{0}-\bar{\rho},P(\rho_{0})-P(\bar{\rho}))\in
W^{2,q}(\mathbb{R}^{2})\times W^{2,q}(\mathbb{R}^{2}),\quad u_{0}(x)\in
H^{2}(\mathbb{R}^{2}),\\\
\displaystyle\Psi(\rho_{0},\bar{\rho})(1+|x|^{\alpha})\in
L^{1}(\mathbb{R}^{2}),\quad\sqrt{\rho_{0}}u_{0}|x|^{\frac{\alpha}{2}}\in
L^{2}(\mathbb{R}^{2}),\end{array}$ (1.10)
with $q,\alpha,\gamma$ and $\beta$ being the same as in Theorem 1.1. Suppose
that the compatibility conditions
$\mathcal{L}_{\rho_{0}}u_{0}-\nabla P(\rho_{0})=\sqrt{\rho}_{0}g(x)$ (1.11)
are satisfied for some $g\in L^{2}(\mathbb{R}^{2})$. Then, for any $T>0$,
there exists a unique global classical solution $(\rho,u)(t,x)$ to the Cauchy
problem (1.3)-(1.5) satisfying $0\leq\rho\leq C_{2}$ for some positive
constant $C_{2}$ and (1.9) in Theorem 1.1.
###### Remark 1.1
If $\lambda(\bar{\rho})<7\mu,$ one has
$\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}>1.$
Then one can choose a weight $\alpha$ satisfying
$1<\alpha^{2}<\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}$.
In this case, the condition $\gamma\leq 2\beta$ in Theorems 1.1 and 1.2 can be
removed and both theorems hold true for any $\gamma>1$ and $\beta>\frac{4}{3}$
(see [26] for more details).
###### Remark 1.2
If $\bar{\rho}=0,$ then
$\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}=4(\sqrt{2}-1)$.
This is exactly same as our previous work [26] for the Cauchy problem with the
vanishing density at the far fields.
The rest of the paper is organized as follows. In Section 2, we present some
elementary facts which will be used later. In Section 3, we derive a priori
estimates which are needed to extend the local solution to a global one. The
sketch of proof of our main results is given in Section 4.
_Notations._ Throughout this paper, positive generic constants are denoted by
$c$ and $C$, which are independent of $\delta$, $m$ and $t\in[0,T]$, without
confusion, and $C(\cdot)$ stands for some generic constant(s) depending only
on the quantity listed in the parenthesis. For functional spaces,
$L^{p}(\mathbb{R}^{2}),1\leq p\leq\infty$, denote the usual Lebesgue spaces on
$\mathbb{R}^{2}$ and $\|\cdot\|_{p}$ denotes its $L^{p}$ norm.
$W^{k,p}(\mathbb{R}^{2})$ denotes the standard $k^{th}$ order Sobolev space
and $H^{k}(\mathbb{R}^{2}):=W^{k,2}(\mathbb{R}^{2})$. For $1<p<\infty$, the
homogenous Sobolev space $D^{k,p}(\mathbb{R}^{2})$ is defined by
$D^{k,p}(\mathbb{R}^{2})=\\{u\in
L^{1}_{loc}(\mathbb{R}^{2})|\|\nabla^{k}u\|_{p}<+\infty\\}$ with
$\|u\|_{D^{k,p}}:=\|\nabla^{k}u\|_{p}$ and
$D^{k}(\mathbb{R}^{2}):=D^{k,2}(\mathbb{R}^{2})$.
## 2 Preliminaries
Motivated by [41], we introduce the following variables. First denote the
effective viscous flux by
$F=(2\mu+\lambda(\rho)){\rm div}u-(P(\rho)-P(\bar{\rho})),$ (2.1)
and the vorticity by
$\omega=\partial_{x_{1}}u_{2}-\partial_{x_{2}}u_{1}.$
Also, we define that
$H=\frac{1}{\rho}(\mu\omega_{x_{1}}+F_{x_{2}}),\qquad\quad
L=\frac{1}{\rho}(-\mu\omega_{x_{2}}+F_{x_{1}}).$
Then the momentum equation $\eqref{CNS}_{2}$ can be rewritten as
$\left\\{\begin{array}[]{ll}\dot{u}_{1}=u_{1t}+u\cdot\nabla
u_{1}=\frac{1}{\rho}(-\mu\omega_{x_{2}}+F_{x_{1}})=L,\\\
\dot{u}_{2}=u_{2t}+u\cdot\nabla
u_{2}=\frac{1}{\rho}(\mu\omega_{x_{1}}+F_{x_{2}})=H,\end{array}\right.$
that is,
$\dot{u}=(\dot{u}_{1},\dot{u}_{2})^{t}=(L,H)^{t}.$
Then the effective viscous flux $F$ and the vorticity $\omega$ solve the
following system:
$\left\\{\begin{array}[]{ll}\omega_{t}+u\cdot\nabla\omega+\omega{\rm
div}u=H_{x_{1}}-L_{x_{2}},\\\
(\frac{F+P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})_{t}+u\cdot\nabla(\frac{F+P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})+(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}=H_{x_{2}}+L_{x_{1}}.\end{array}\right.$
Due to the continuity equation $\eqref{CNS}_{1}$, it holds that
$\left\\{\begin{array}[]{ll}\omega_{t}+u\cdot\nabla\omega+\omega{\rm
div}u=H_{x_{1}}-L_{x_{2}},\\\ F_{t}+u\cdot\nabla
F-\rho(2\mu+\lambda(\rho))[F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})^{\prime}]{\rm
div}u\\\
\qquad+(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]=(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}}).\end{array}\right.$
(2.2)
Furthermore, the system for $(H,L)$ can be derived as
$\left\\{\begin{array}[]{ll}\rho H_{t}+\rho u\cdot\nabla H-\rho H{\rm
div}u+u_{x_{2}}\cdot\nabla F+\mu u_{x_{1}}\cdot\nabla\omega+\mu(\omega{\rm
div}u)_{x_{1}}\\\
\qquad-\big{\\{}\rho(2\mu+\lambda(\rho))[F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})^{\prime}]{\rm
div}u\big{\\}}_{x_{2}}\\\
\qquad+\big{\\{}(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]\big{\\}}_{x_{2}}\\\
\qquad=[(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})]_{x_{2}}+\mu(H_{x_{1}}-L_{x_{2}})_{x_{1}},\\\
\rho L_{t}+\rho u\cdot\nabla L-\rho L{\rm div}u+u_{x_{1}}\cdot\nabla F-\mu
u_{x_{2}}\cdot\nabla\omega-\mu(\omega{\rm div}u)_{x_{2}}\\\
\qquad-\big{\\{}\rho(2\mu+\lambda(\rho))[F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})^{\prime}]{\rm
div}u\big{\\}}_{x_{1}}\\\
\qquad+\big{\\{}(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]\big{\\}}_{x_{1}}\\\
\qquad=[(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})]_{x_{1}}-\mu(H_{x_{1}}-L_{x_{2}})_{x_{2}}.\end{array}\right.$
In the following, we will utilize the above systems in different steps. Note
that these systems are equivalent to each other for the smooth solution to the
original system (1.3).
Several elementary Lemmas are needed later. The first one is the various
Gagliardo-Nirenberg inequalities.
###### Lemma 2.1
* (1)
$\forall h\in W^{1,m}(\mathbb{R}^{2})\cap L^{r}(\mathbb{R}^{2})$, it holds
that
$\|h\|_{q}\leq C\|\nabla h\|_{m}^{\theta}\|h\|_{r}^{1-\theta},$
where
$\theta=(\frac{1}{r}-\frac{1}{q})(\frac{1}{r}-\frac{1}{m}+\frac{1}{2})^{-1}$,
and if $m<2,$ then $q$ is between $r$ and $\frac{2m}{2-m}$, that is,
$q\in[r,\frac{2m}{2-m}]$ if $r<\frac{2m}{2-m}$, $q\in[\frac{2m}{2-m},r]$ if
$r\geq\frac{2m}{2-m},$ if $m=2,$ then $q\in[r,+\infty)$, if $m>2$, then
$q\in[r,+\infty].$
* (2)
(Best constant for the Gagliardo-Nirenberg inequality)
$\forall h\in\mathbb{D}^{m}(\mathbb{R}^{2})\doteq\Big{\\{}h\in
L^{m+1}(\mathbb{R}^{2})\Big{|}\nabla h\in L^{2}(\mathbb{R}^{2}),h\in
L^{2m}(\mathbb{R}^{2})\Big{\\}}$ with $m>1$, it holds that
$\|h\|_{2m}\leq A_{m}\|\nabla h\|_{2}^{\theta}\|u\|_{m+1}^{1-\theta},$
where $\theta=\frac{1}{2}-\frac{1}{2m}$ and
$A_{m}=\Big{(}\frac{m+1}{2\pi}\Big{)}^{\frac{\theta}{2}}\Big{(}\frac{2}{m+1}\Big{)}^{\frac{1}{2m}}\leq
Cm^{\frac{1}{4}}$
with the positive constant $C$ independent of $m$, and $A_{m}$ is the optimal
constant.
* (3)
$\forall h\in W^{1,m}(\mathbb{R}^{2})$ with $1\leq m<2,$ then
$\|h\|_{\frac{2m}{2-m}}\leq C(2-m)^{-\frac{1}{2}}\|\nabla h\|_{m},$
where the positive constant $C$ is independent of $m.$
Proof: The proof of (1) can be found in [41] while the proof of (2) can be
found in [7]. The proof of (3) can be found in [12].
The following Lemma is the Caffarelli-Kokn-Nirenberg weighted inequalities,
which is crucial to the weighted estimates in the two-dimensional Cauchy
problem.
###### Lemma 2.2
* (1)
$\forall h\in C^{\infty}_{0}(\mathbb{R}^{2})$, it holds that
$\||x|^{\kappa}h\|_{r}\leq C\||x|^{\alpha}|\nabla
h|\|^{\theta}_{p}~{}\||x|^{\beta}h\|^{1-\theta}_{q}$
where $1\leq p,q<\infty,0<r<\infty,0\leq\theta\leq
1,\frac{1}{p}+\frac{\alpha}{2}>0,\frac{1}{q}+\frac{\beta}{2}>0,\frac{1}{r}+\frac{\kappa}{2}>0$
and satisfying
$\frac{1}{r}+\frac{\kappa}{2}=\theta(\frac{1}{p}+\frac{\alpha-1}{2})+(1-\theta)(\frac{1}{q}+\frac{\beta}{2}),$
and
$\kappa=\theta\sigma+(1-\theta)\beta,$
with $0\leq\alpha-\sigma$ if $\theta>0$ and $0\leq\alpha-\sigma\leq 1$ if
$\theta>0$ and $\frac{1}{p}+\frac{\alpha-1}{2}=\frac{1}{r}+\frac{\kappa}{2}.$
* (2)
(Best constant for Caffarelli-Kohn-Nirenberg inequality)
$\forall h\in C^{\infty}_{0}(\mathbb{R}^{2})$, it holds that
$\||x|^{b}h\|_{p}\leq C_{a,b}\||x|^{a}\nabla h\|_{2}$ (2.3)
where $a>0,a-1\leq b\leq a$ and $p=\frac{2}{a-b}$. If $b=a-1$, then $p=2$ and
the best constant in the inequality (2.3) is
$C_{a,b}=C_{a,a-1}=a.$
Proof: The proof of (1) can be found in [4] while the proof of (2) can be
found in [5].
The following lemma is known and the proof is referred to [26].
###### Lemma 2.3
* (1)
It holds that for $1<p<\infty$ and $u\in C_{0}^{\infty}(\mathbb{R}^{2})$,
$\|\nabla u\|_{p}\leq C(\|{\rm div}u\|_{p}+\|\omega\|_{p});$
* (2)
It holds that for $1<p<\infty$, $-2<\alpha<2(p-1)$ and $u\in
C_{0}^{\infty}(\mathbb{R}^{2})$,
$\||x|^{\frac{\alpha}{p}}|\nabla u|\|_{p}\leq C(\||x|^{\frac{\alpha}{p}}{\rm
div}u\|_{p}+\||x|^{\frac{\alpha}{p}}\omega\|_{p}).$
## 3 A priori estimates
In this section, we will obtain various a priori estimates and a upper bound
of the density.
Step 1. Elementary energy estimates:
###### Lemma 3.1
There exists a positive constant $C$ depending on $(\rho_{0},u_{0})$, such
that
$\sup_{t\in[0,T]}\big{(}\|\sqrt{\rho}u\|^{2}_{2}+\|\Psi(\rho,\bar{\rho})\|_{1}\big{)}+\int_{0}^{T}\big{(}\|\nabla
u\|_{2}^{2}+\|\omega\|_{2}^{2}+\|(2\mu+\lambda(\rho))^{\frac{1}{2}}{\rm
div}u\|_{2}^{2}\big{)}dt\leq C.$
Proof: Multiplying the equation $\eqref{CNS}_{2}$ by $u$, the continuity
equation $\eqref{CNS}_{1}$ by $\frac{\gamma}{\gamma-1}\rho^{\gamma-1}$, then
summing the resulting equations, and using the continuity equation
$\eqref{CNS}_{1}$, yield that
$\begin{array}[]{ll}\displaystyle\big{[}\rho\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})\big{]}_{t}+{\rm
div}\big{[}\rho
u\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})u+(P(\rho)-P(\bar{\rho}))u\big{]}\\\
\displaystyle={\rm
div}\big{[}\mu\nabla\frac{|u|^{2}}{2}+(\mu+\lambda(\rho))({\rm
div}u)u\big{]}-\mu|\nabla u|^{2}-(\mu+\lambda(\rho))({\rm
div}u)^{2}.\end{array}$ (3.1)
Therefore, integrating the above equality over $[0,t]\times\mathbb{R}^{2}$
with respect to $t$ and $x$ and noting that
$\int\big{[}\mu|\nabla u|^{2}+(\mu+\lambda(\rho))({\rm
div}u)^{2}\big{]}dx=\int\big{[}\mu\omega^{2}+(2\mu+\lambda(\rho))({\rm
div}u)^{2}\big{]}dx,$
complete the proof of Lemma 3.1. $\hfill\Box$
Step 2. Weighted energy estimates:
The following weighted energy estimates are fundamental and crucial in our
analysis.
###### Lemma 3.2
For $\alpha>0$ satisfying
$\alpha^{2}<\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}$
and $\gamma\leq 2\beta$, it holds that for sufficiently large $m>1$ and
$\forall t\in[0,T]$,
$\begin{array}[]{ll}\displaystyle\int_{\mathbb{R}^{2}}|x|^{\alpha}\big{[}\rho|u|^{2}+\Psi(\rho,\bar{\rho})\big{]}(t,x)dx+\int_{0}^{t}\big{[}\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2}(s)+\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}(s)+\||x|^{\frac{\alpha}{2}}\sqrt{\lambda(\rho)}{\rm
div}u\|_{2}^{2}(s)\big{]}ds\\\ \displaystyle\leq
C_{\alpha}\Big{[}1+\int_{0}^{t}(\|\rho-\bar{\rho}\|^{\beta}_{2m\beta+1}(s)+1)(\|\nabla
u\|_{2}^{2}(s)+1)ds\Big{]},\end{array}$ (3.2)
where the positive constant $C_{\alpha}$ may depend on $\alpha$ but is
independent of $m$.
Proof: Multiplying the equality (3.1) by $|x|^{\alpha}$ yields that
$\begin{array}[]{ll}\displaystyle\big{[}|x|^{\alpha}(\rho\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho}))\big{]}_{t}+\big{[}\mu|\nabla
u|^{2}+(\mu+\lambda(\rho))({\rm div}u)^{2}\big{]}|x|^{\alpha}\\\
\displaystyle=-{\rm div}\big{[}|x|^{\alpha}\big{(}\rho
u\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})u+(P(\rho)-P(\bar{\rho}))u\big{)}\big{]}+{\rm
div}\big{[}\big{(}\mu\nabla\frac{|u|^{2}}{2}+(\mu+\lambda(\rho))({\rm
div}u)u\big{)}|x|^{\alpha}\big{]}\\\ \displaystyle\quad+\big{[}\rho
u\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})u+(P(\rho)-P(\bar{\rho}))\big{]}u\cdot\nabla(|x|^{\alpha})-\big{[}\mu\nabla\frac{|u|^{2}}{2}+(\mu+\lambda(\rho))({\rm
div}u)u\big{]}\cdot\nabla(|x|^{\alpha}).\end{array}$ (3.3)
Integrating the above equation (3.3) with respect to $x$ over $\mathbb{R}^{2}$
yields that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\int|x|^{\alpha}\big{[}\rho\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})\big{]}(t,x)dx+\Big{[}\mu\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2}+\mu\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}+\||x|^{\frac{\alpha}{2}}\sqrt{\lambda(\rho)}{\rm
div}u\|_{2}^{2}\Big{]}(t)\\\ \displaystyle=\int\big{[}\rho
u\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})u+(P(\rho)-P(\bar{\rho}))\big{]}u\cdot\nabla(|x|^{\alpha})dx\\\
\displaystyle\qquad\qquad\qquad-\int\big{[}\mu\nabla\frac{|u|^{2}}{2}+(\mu+\lambda(\rho))({\rm
div}u)u\big{]}\cdot\nabla(|x|^{\alpha})dx.\end{array}$ (3.4)
Now we estimate the terms on the right hand side of (3.4). First, it holds
that
$\begin{array}[]{ll}\displaystyle|\int\rho\frac{|u|^{2}}{2}u\cdot\nabla(|x|^{\alpha})dx|\displaystyle=|\int\frac{|u|^{2}}{2}\big{(}(\sqrt{\rho}-\sqrt{\bar{\rho}})+\sqrt{\bar{\rho}}\big{)}\sqrt{\rho}u\cdot\nabla(|x|^{\alpha})dx|\\\
\displaystyle\quad\leq|\int\frac{|u|^{2}}{2}\big{(}\sqrt{\rho}-\sqrt{\bar{\rho}}\big{)}\sqrt{\rho}u\cdot\nabla(|x|^{\alpha})dx|+\sqrt{\bar{\rho}}~{}|\int\frac{|u|^{2}}{2}\sqrt{\rho}u\cdot\nabla(|x|^{\alpha})dx|:=I_{11}+I_{12}.\end{array}$
(3.5)
Then, it follows that
$\begin{array}[]{ll}\displaystyle
I_{11}=|\int\frac{|u|^{2}}{2}\big{(}\sqrt{\rho}-\sqrt{\bar{\rho}}\big{)}\big{(}{\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}+{\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\big{)}\sqrt{\rho}u\cdot\nabla(|x|^{\alpha})dx|\\\
\displaystyle\leq
C\|\sqrt{\rho}u\|_{2}\Big{[}\|(\sqrt{\rho}-\sqrt{\bar{\rho}}){\bf
1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{p_{1}}\||x|^{\alpha-1}|u|^{2}\|_{q_{1}}+\|(\sqrt{\rho}-\sqrt{\bar{\rho}}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|_{2\gamma}\||x|^{\alpha-1}|u|^{2}\|_{\frac{2\gamma}{\gamma-1}}\Big{]}\\\
\displaystyle\leq C\big{[}\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{p_{1}}\||x|^{\frac{\alpha-1}{2}}u\|^{2}_{2q_{1}}+\|\Psi(\rho,\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{1}{2\gamma}}_{1}\||x|^{\frac{\alpha-1}{2}}u\|^{2}_{\frac{4\gamma}{\gamma-1}}\big{]}\\\
\displaystyle\leq C\big{[}\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{p_{1}}\|\nabla
u\|_{2}^{2\theta_{1}}\||x|^{\frac{\alpha}{2}}\nabla
u\|^{2(1-\theta_{1})}_{2}+\|\nabla
u\|_{2}^{\frac{2}{\alpha\gamma}}\||x|^{\frac{\alpha}{2}}\nabla
u\|^{2(1-\frac{1}{\alpha\gamma})}_{2}\big{]}\\\
\displaystyle\leq\sigma\||x|^{\frac{\alpha}{2}}\nabla
u\|^{2}_{2}+C_{\sigma}\big{[}1+\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{1}{\theta_{1}}}_{p_{1}}\big{]}\|\nabla
u\|_{2}^{2},\end{array}$ (3.6)
where and in the sequel $\sigma>0$ is a small constant to be determined,
$C_{\sigma}$ is a positive constant depending on $\sigma$. By the
H${\rm\ddot{o}}$lder inequality and the Caffarelli-Kohn-Nirenberg inequality
in Lemma 2.2 (1), the positive constants $p_{1}>2,q_{1}>2,\theta_{1}\in(0,1]$
in the above inequality (3.8) satisfying
$\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{2},$
and
$\frac{1}{2q_{1}}+\frac{\frac{\alpha-1}{2}}{2}=\theta_{1}(\frac{1}{2}+\frac{0-1}{2})+(1-\theta_{1})(\frac{1}{2}+\frac{\frac{\alpha}{2}-1}{2})=\frac{\alpha}{4}(1-\theta_{1}).$
The combination of the above two equalities yields that
$p_{1}=\frac{2}{\alpha\theta_{1}},$ (3.7)
with $\alpha>0$, $\theta_{1}\in(0,1)$ and $p_{1}>2$. Therefore, it holds that
$\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{1}{\theta_{1}}}_{p_{1}}\leq C\|(\rho-\bar{\rho}){\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}\|^{\frac{1}{\theta_{1}}}_{2}\leq
C\|\Psi(\rho,\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{1}{2\theta_{1}}}_{1}\leq C,$
which together with (3.6) gives that
$\displaystyle I_{11}\leq\sigma\||x|^{\frac{\alpha}{2}}\nabla
u\|^{2}_{2}+C_{\sigma}\|\nabla u\|_{2}^{2}.$ (3.8)
Then, one can obtain
$\begin{array}[]{ll}\displaystyle\quad
I_{12}\leq\frac{\alpha}{2}\sqrt{\bar{\rho}}\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}\||x|^{\frac{\frac{\alpha}{2}-1}{2}}|u|\|^{2}_{4}\\\
\displaystyle\qquad\leq C\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}\|\nabla
u\|_{2}\||x|^{\frac{\alpha}{2}}\nabla u\|_{2}\\\
\displaystyle\qquad\leq\sigma\||x|^{\frac{\alpha}{2}}\nabla
u\|^{2}_{2}+C_{\sigma}\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{2}\|\nabla
u\|_{2}^{2}.\end{array}$ (3.9)
Then it holds that
$\begin{array}[]{ll}\displaystyle|\int\big{[}\Psi(\rho,\bar{\rho})+(P(\rho)-P(\bar{\rho}))\big{]}u\cdot\nabla(|x|^{\alpha})dx|\\\
\displaystyle\quad=|\int\big{[}\Psi(\rho,\bar{\rho})+(P(\rho)-P(\bar{\rho}))\big{]}\big{(}{\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}+{\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\big{)}u\cdot\nabla(|x|^{\alpha})dx|\\\
\displaystyle\quad\leq C|\int\big{[}|\rho-\bar{\rho}|{\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}+|\rho-\bar{\rho}|^{\gamma}{\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\big{]}u\cdot\nabla(|x|^{\alpha})dx|\\\
\displaystyle\quad\leq C\big{[}\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{2}\||x|^{\alpha-1}u\|_{2}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\gamma}_{\gamma
p_{2}}\||x|^{\alpha-1}u\|_{q_{2}}\big{]}\\\ \displaystyle\quad\leq
C\big{[}\|\Psi(\rho,\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{1}{2}}_{1}\|\nabla u\|_{2}^{\frac{1}{2}}\|\nabla
u|x|^{\frac{\alpha}{2}}\|_{2}^{\frac{1}{2}}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\gamma}_{\gamma p_{2}}\|\nabla
u\|_{2}^{\theta_{2}}\||x|^{\frac{\alpha}{2}}\nabla
u\|^{1-\theta_{2}}_{2}\big{]}\\\
\displaystyle\quad\leq\sigma\||x|^{\frac{\alpha}{2}}\nabla
u\|^{2}_{2}+C_{\sigma}\big{[}1+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{2\gamma}{1+\theta_{2}}}_{\gamma
p_{2}}\big{]}(\|\nabla u\|_{2}^{2}+1),\end{array}$ (3.10)
By the H${\rm\ddot{o}}$lder inequality and the Caffarelli-Kohn-Nirenberg
inequality in Lemma 2.2 (1), the positive constants
$p_{2}>1,q_{2}>1,\theta_{2}\in(0,1]$ in the above inequality (3.10) satisfying
$\frac{1}{p_{2}}+\frac{1}{q_{2}}=1,$
$\frac{1}{q_{2}}+\frac{\alpha-1}{2}=\theta_{2}(\frac{1}{2}+\frac{0-1}{2})+(1-\theta_{2})(\frac{1}{2}+\frac{\frac{\alpha}{2}-1}{2})=\frac{\alpha}{4}(1-\theta_{2}).$
The combination of the above three equalities yields that
$p_{2}=\frac{4}{2+\alpha(1+\theta_{2})},$ (3.11)
with the parameters $\alpha>0$, $\theta_{2}\in(0,1)$ and $p_{2}>1$. Note that
$p_{2}>1$ is equivalent to the condition that
$\frac{\alpha}{2}(1+\theta_{2})<1.$ (3.12)
Then one can compute that
$\begin{array}[]{ll}\displaystyle|-\int\mu\nabla\frac{|u|^{2}}{2}\cdot\nabla(|x|^{\alpha})dx|=\mu\alpha|\int
u\cdot\nabla u\cdot x|x|^{\alpha-2}dx|\\\
\displaystyle\quad\leq\mu\alpha\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}\||x|^{\frac{\alpha}{2}-1}u\|_{2}\leq\frac{\mu\alpha^{2}}{2}\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2},\end{array}$ (3.13)
where in the last inequality one has used the best constant $\frac{\alpha}{2}$
for the Caffarelli-Kohn-Nirenberg inequality in Lemma 2.2 (2). Similarly, it
holds that
$\begin{array}[]{ll}\displaystyle|-\int\mu({\rm
div}u)u\cdot\nabla(|x|^{\alpha})dx|=\mu\alpha|\int({\rm
div}u)|x|^{\alpha-2}u\cdot xdx|\\\
\displaystyle\quad\leq\mu\alpha\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}\||x|^{\frac{\alpha}{2}-1}u\|_{2}\leq\frac{\mu\alpha^{2}}{2}\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}\||x|^{\frac{\alpha}{2}}\nabla u\|_{2}.\end{array}$ (3.14)
Then it follows that
$\begin{array}[]{ll}\displaystyle|-\int\lambda(\rho)({\rm
div}u)u\cdot\nabla(|x|^{\alpha})dx|\displaystyle=\alpha|\int\sqrt{\lambda(\rho)}({\rm
div}u)\big{[}(\sqrt{\lambda(\rho)}-\sqrt{\lambda(\bar{\rho})})+\sqrt{\lambda(\bar{\rho})}\big{]}|x|^{\alpha-2}u\cdot
xdx|\\\ \displaystyle\leq\alpha|\int\sqrt{\lambda(\rho)}({\rm
div}u)\big{(}\sqrt{\lambda(\rho)}-\sqrt{\lambda(\bar{\rho})}\big{)}|x|^{\alpha-2}u\cdot
xdx|+\sqrt{\lambda(\bar{\rho})}\alpha|\int\sqrt{\lambda(\rho)}({\rm
div}u)|x|^{\alpha-2}u\cdot xdx|\\\ \displaystyle:=I_{21}+I_{22}.\end{array}$
(3.15)
It holds that
$\begin{array}[]{ll}\displaystyle
I_{21}\leq\alpha\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}\|\sqrt{\lambda(\rho)}-\sqrt{\lambda(\bar{\rho})}\|_{p_{3}}\||x|^{\frac{\alpha}{2}-1}u\|_{q_{3}}\\\
\displaystyle\leq C\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}\big{[}\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{p_{3}}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{2}}_{\frac{\beta
p_{3}}{2}}\big{]}\|\nabla u\|_{2}^{\theta_{3}}\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{1-\theta_{3}}\\\
\displaystyle\leq\sigma\big{[}\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}+\||x|^{\frac{\alpha}{2}}\nabla u\|_{2}^{2}\big{]}\\\
\displaystyle\qquad\qquad\qquad+C_{\sigma}\big{[}\|(\rho-\bar{\rho}){\bf
1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{2}{\theta_{3}}}_{p_{3}}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}}_{\frac{\beta
p_{3}}{2}}\big{]}\|\nabla u\|_{2}^{2}.\end{array}$ (3.16)
By the H${\rm\ddot{o}}$lder inequality and the Caffarelli-Kohn-Nirenberg
inequality in Lemma 2.2 (1), the positive constants
$p_{3}>2,q_{3}>2,\theta_{3}\in(0,1]$ in the above inequality (3.18) satisfying
$\frac{1}{p_{3}}+\frac{1}{q_{3}}=\frac{1}{2},$
$\frac{1}{q_{3}}+\frac{\frac{\alpha}{2}-1}{2}=\theta_{3}(\frac{1}{2}+\frac{0-1}{2})+(1-\theta_{3})(\frac{1}{2}+\frac{\frac{\alpha}{2}-1}{2})=\frac{\alpha}{4}(1-\theta_{3}).$
The combination of the above three equalities yields that
$p_{3}=\frac{4}{\alpha\theta_{3}}.$ (3.17)
with $\alpha>0$, $\theta_{3}\in(0,1)$ and $p_{3}>2$. Therefore, it holds that
$\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{2}{\theta_{3}}}_{p_{3}}\leq C\|(\rho-\bar{\rho}){\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}\|^{\frac{2}{\theta_{3}}}_{2}\leq
C\|\Psi(\rho,\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|^{\frac{1}{\theta_{3}}}_{1}\leq C,$
which together with (3.16) gives that
$I_{21}\leq\sigma\big{[}\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}+\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2}\big{]}+C_{\sigma}\big{[}1+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}}_{\frac{\beta
p_{3}}{2}}\big{]}\|\nabla u\|_{2}^{2}.$ (3.18)
Meanwhile, it holds that
$\begin{array}[]{ll}\displaystyle\quad
I_{22}\leq\sqrt{\lambda(\bar{\rho})}\alpha\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}\||x|^{\frac{\alpha}{2}-1}u\|_{2}\\\
\displaystyle\qquad\leq\frac{\alpha^{2}\sqrt{\lambda(\bar{\rho})}}{2}\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}\||x|^{\frac{\alpha}{2}}\nabla u\|_{2}.\end{array}$ (3.19)
Substituting (3.8) and (3.9) into (3.5), (3.18) and (3.19) into (3.15) and
then substituting the resulting (3.5), (3.15) and (3.10), (3.13) and (3.14)
into (3.4) yield that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\int|x|^{\alpha}\big{[}\rho\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})\big{]}(t,x)dx+J(t)\leq\sigma\big{[}\|\sqrt{\lambda(\rho)}|x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}+4\||x|^{\frac{\alpha}{2}}\nabla u\|_{2}^{2}\big{]}\\\
\displaystyle~{}+C_{\sigma}\big{[}1+\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{2}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{2\gamma}{1+\theta_{2}}}_{p_{2}\gamma}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}}_{\frac{\beta
p_{3}}{2}}\big{]}(\|\nabla u\|_{2}^{2}+1),\end{array}$ (3.20)
where $\theta_{i}\in(0,1]$, $p_{i}~{}(i=1,2,3)$ are given in (3.7), (3.11) and
(3.17), respectively, and $p_{1},p_{3}>2$ and $p_{2}>1$ and
$\begin{array}[]{ll}\displaystyle
J(t)=\mu(1-\frac{\alpha^{2}}{2})\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2}-\frac{\mu\alpha^{2}}{2}\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}+\mu\||x|^{\frac{\alpha}{2}}{\rm div}u\|_{2}^{2}\\\
\displaystyle\qquad+\||x|^{\frac{\alpha}{2}}\sqrt{\lambda(\rho)}{\rm
div}u\|_{2}^{2}(t)-\frac{\alpha^{2}\sqrt{\lambda(\bar{\rho})}}{2}\||x|^{\frac{\alpha}{2}}\sqrt{\lambda(\rho)}{\rm
div}u\|_{2}\||x|^{\frac{\alpha}{2}}\nabla u\|_{2}.\end{array}$ (3.21)
The corresponding matrix of the above quadratic term (3.21) is
$A=\left(\begin{array}[]{ccc}\displaystyle\mu(1-\frac{\alpha^{2}}{2})&\displaystyle-\frac{\mu\alpha^{2}}{4}&\displaystyle-\frac{\alpha^{2}\sqrt{\lambda(\bar{\rho})}}{4}\\\\[5.69054pt]
\displaystyle-\frac{\mu\alpha^{2}}{4}&\mu&0\\\\[5.69054pt]
\displaystyle-\frac{\alpha^{2}\sqrt{\lambda(\bar{\rho})}}{4}&0&1\end{array}\right).$
The matrix $A$ is positively definite if and only if all the principal minor
determinant of $A$ is positive, that is,
$\begin{array}[]{ll}\displaystyle\mu(1-\frac{\alpha^{2}}{2})>0,\qquad\left|\begin{array}[]{ccc}\displaystyle\mu(1-\frac{\alpha^{2}}{2})&\displaystyle-\frac{\mu\alpha^{2}}{4}\\\\[5.69054pt]
\displaystyle-\frac{\mu\alpha^{2}}{4}&\mu\\\ \end{array}\right|>0,~{}~{}{\rm
and}~{}~{}~{}\displaystyle\left|\begin{array}[]{ccc}\displaystyle\mu(1-\frac{\alpha^{2}}{2})&\displaystyle-\frac{\mu\alpha^{2}}{4}&\displaystyle-\frac{\alpha^{2}\sqrt{\lambda(\bar{\rho})}}{4}\\\\[5.69054pt]
\displaystyle-\frac{\mu\alpha^{2}}{4}&\mu&0\\\\[5.69054pt]
\displaystyle-\frac{\alpha^{2}\sqrt{\lambda(\bar{\rho})}}{4}&0&1\end{array}\right|>0.\end{array}$
Therefore, if the weight $\alpha$ satisfies
$0<\alpha^{2}<\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}},$
(3.22)
then the matrix $A$ is positively definite, and then there exists a positive
constant $C_{\alpha}$ such that
$J(t)\geq C^{-1}_{\alpha}\Big{[}\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2}(t)+\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}(t)+\||x|^{\frac{\alpha}{2}}\sqrt{\lambda(\rho)}{\rm
div}u\|_{2}^{2}(t)\Big{]}.$ (3.23)
Consequently, if the weight $\alpha$ satisfies (3.22), then substituting
(3.23) into (3.20) and choosing $\sigma$ suitably small yield that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\int|x|^{\alpha}\big{[}\rho\frac{|u|^{2}}{2}+\Psi(\rho,\bar{\rho})\big{]}(t,x)dx+\frac{\mu}{2}C^{-1}_{\alpha}\Big{[}\||x|^{\frac{\alpha}{2}}\nabla
u\|_{2}^{2}+\||x|^{\frac{\alpha}{2}}{\rm
div}u\|_{2}^{2}+\||x|^{\frac{\alpha}{2}}\sqrt{\lambda(\rho)}{\rm
div}u\|_{2}^{2}(t)\Big{]}\\\ \displaystyle\leq
C\big{[}1+\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{2}++\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{2\gamma}{1+\theta_{2}}}_{p_{2}\gamma}+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}}_{\frac{\beta
p_{3}}{2}}\big{]}(\|\nabla u\|_{2}^{2}+1).\end{array}$ (3.24)
Now choose $m>1$ sufficiently large such that
$2m\beta+1\geq\max\Big{\\{}p_{2}\gamma,\frac{\beta p_{3}}{2}\Big{\\}}.$
Then, it holds that
$\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{2\gamma}{1+\theta_{2}}}_{p_{2}\gamma}\leq\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{2a_{2}\gamma}{1+\theta_{2}}}_{\gamma}\|\rho-\bar{\rho}\|^{\frac{2(1-a_{2})\gamma}{1+\theta_{2}}}_{2m\beta+1}\leq
C\|\Psi(\rho,\bar{\rho})\|^{\frac{2a_{2}}{1+\theta_{2}}}_{1}\|\rho-\bar{\rho}\|^{\frac{2\gamma(1-a_{2})}{1+\theta_{2}}}_{2m\beta+1},$
(3.25)
with $a_{2}\in(0,1)$ satisfying
$\frac{a_{2}}{\gamma}+\frac{1-a_{2}}{2m\beta+1}=\frac{1}{p_{2}\gamma}=\frac{2+\alpha(1+\theta_{2})}{4\gamma},$
which implies that
$a_{2}=\frac{(2+\alpha(1+\theta_{2}))(2m\beta+1)-4\gamma}{4(2m\beta+1-\gamma)}\rightarrow\frac{2+\alpha(1+\theta_{2})}{4},~{}~{}{\rm
as}~{}m\rightarrow+\infty.$
The following restriction should be imposed to (3.25)
$\frac{2\gamma(1-a_{2})}{1+\theta_{2}}\leq\beta,$
which is satisfied provided
$(1+\theta_{2})(\frac{\beta}{\gamma}+\frac{\alpha}{2})>1$ (3.26)
and $m\gg 1.$ Since $\gamma\leq 2\beta$, then
$\frac{\beta}{\gamma}\geq\frac{1}{2},$ thus one can choose
$\theta_{2}\in(0,1)$ such that $(1+\theta_{2})\frac{\beta}{\gamma}\geq 1$ and
$\frac{\alpha}{2}(1+\theta_{2})<1$, and thus satisfies the restrictions (3.26)
and (3.12) if $m\gg 1$. If $\lambda(\bar{\rho})<7\mu,$ then
$\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}>1.$
Thus we can choose the weight $\alpha>0$ satisfying
$1<\alpha^{2}<\frac{4(\sqrt{2+\frac{\lambda(\bar{\rho})}{\mu}}-1)}{1+\frac{\lambda(\bar{\rho})}{\mu}}$.
In this case, one can choose $\theta_{2}\in(0,1)$ satisfying the restrictions
(3.12) and (3.26) for any fixed $\gamma,\beta>1$, that is, the condition
$\gamma\leq 2\beta$ in the Theorem 1.2 can be removed as in Remark 2. Then it
follows from (3.25) that
$\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|_{p_{2}\gamma}^{\frac{2\gamma}{1+\theta_{2}}}\leq
C(\|\rho-\bar{\rho}\|^{\beta}_{2m\beta+1}+1)$ (3.27)
with the positive constant $C$ independent of $m$.
Similarly, one has
$\begin{array}[]{ll}\displaystyle\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}}_{\frac{\beta
p_{3}}{2}}\leq\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}a_{3}}_{1}\|\rho-\bar{\rho}\|^{\frac{\beta}{\theta_{3}}(1-a_{3})}_{2m\beta+1}\\\
\displaystyle\leq\|\Psi(\rho,\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}a_{3}}_{1}\|\rho-\bar{\rho}\|^{\frac{\beta}{\theta_{3}}(1-a_{3})}_{2m\beta+1},\end{array}$
(3.28)
with with $a_{3}\in(0,1)$ satisfying
$\frac{a_{3}}{1}+\frac{1-a_{3}}{2m\beta+1}=\frac{2}{p_{3}\beta}=\frac{\alpha\theta_{3}}{2\beta},$
which implies that
$a_{3}=\frac{\alpha\theta_{3}(2m\beta+1)-2\beta}{4m\beta^{2}}\rightarrow\frac{\alpha\theta_{3}}{2\beta},~{}~{}{\rm
as}~{}m\rightarrow+\infty.$
The following restriction should be imposed to (3.28)
$\frac{1-a_{3}}{\theta_{3}}\leq 1,$
which is satisfied provided we choose $m\gg 1$ and $\theta_{3}\in(0,1)$ such
that
$\theta_{3}(1+\frac{\alpha}{2\beta})>1.$
Then it follows from (3.28) that
$\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{\beta}{\theta_{3}}}_{\frac{\beta
p_{3}}{2}}\leq C(\|\rho-\bar{\rho}\|^{\beta}_{2m\beta+1}+1)$ (3.29)
with the positive constant $C$ independent of $m$. Substituting (3.27) and
(3.29) into (3.24), then integrating the resulting inequality over $[0,t]$
with $t\in[0,T]$ and using Gronwall inequality yield the estimate (3.2) in
Lemma 3.2. $\hfill\Box$
Step 3. Density estimates:
Applying the operator $div$ to the momentum equation $\eqref{CNS}_{2}$, it
holds that
$[{\rm div}(\rho u)]_{t}+{\rm div}[{\rm div}(\rho u\otimes u)]=\Delta F.$
(3.30)
Consider the following three elliptic problems on the whole space
$\mathbb{R}^{2}$:
$-\Delta\xi_{1}={\rm div}(\sqrt{\rho}u(\sqrt{\rho}-\sqrt{\bar{\rho}})),$
(3.31) $-\Delta\xi_{2}=\sqrt{\bar{\rho}}~{}{\rm div}(\sqrt{\rho}u),$ (3.32)
$-\Delta\eta={\rm div}[{\rm div}(\rho u\otimes u)],$ (3.33)
all with the boundary conditions $\xi_{1},\xi_{2},\eta\rightarrow 0$ as
$|x|\rightarrow\infty$.
By the elliptic estimates and H${\rm\ddot{o}}$lder inequality, it holds that
###### Lemma 3.3
* (1)
$\|\nabla\xi_{1}\|_{2m}\leq
Cm\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|u\|_{2mk},$ for any $k>1,m\geq 1;$
* (2)
$\|\nabla\xi_{2}\|_{2m}\leq
Cm\Big{[}\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|u\|_{2mk}+\sqrt{\bar{\rho}}\|u\|_{2m}\Big{]},$
for any $k>1,m\geq 1;$
* (3)
$\|\nabla\xi_{2}|x|^{\frac{\alpha}{2}}\|_{2}\leq
C\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2},$ for $\alpha$ satisfying (3.22);
* (4)
$\|\eta\|_{2m}\leq
Cm\big{[}\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|u\|^{2}_{4mk}+\bar{\rho}\|u\|_{4m}^{2}\big{]},$
for any $k>1,m\geq 1;$
where $C$ are positive constants independent of $m,k$ and $r$.
Proof: By the elliptic estimates to the equations (3.31), (3.32),
respectively, and then using the H${\rm\ddot{o}}$lder inequality, one has for
any $k>1,m\geq 1$,
$\begin{array}[]{ll}\|\nabla\xi_{1}\|_{2m}\leq
Cm\|\sqrt{\rho}u(\sqrt{\rho}-\sqrt{\bar{\rho}})\|_{2m}=Cm\|u(\rho-\bar{\rho})\frac{\sqrt{\rho}}{\sqrt{\rho}+\sqrt{\bar{\rho}}}\|_{2m}\\\
\displaystyle\qquad\leq
Cm\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|\frac{\sqrt{\rho}}{\sqrt{\rho}+\sqrt{\bar{\rho}}}\|_{\infty}\|u\|_{2mk}\leq
Cm\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|u\|_{2mk},\end{array}$
and
$\begin{array}[]{ll}\|\nabla\xi_{2}\|_{2m}\leq Cm\|\sqrt{\rho}u\|_{2m}\leq
Cm\big{[}\|(\sqrt{\rho}-\sqrt{\bar{\rho}})u\|_{2m}+\sqrt{\bar{\rho}}\|u\|_{2m}\big{]}\\\
\displaystyle\qquad\leq
Cm\big{[}\|\sqrt{\rho}-\sqrt{\bar{\rho}}\|_{\frac{2mk}{k-1}}\|u\|_{2mk}+\sqrt{\bar{\rho}}\|u\|_{2m}\big{]}\leq
Cm\big{[}\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|u\|_{2mk}+\sqrt{\bar{\rho}}\|u\|_{2m}\big{]}.\end{array}$
Thus the proofs of (1) and (2) are completed.
By the similar proof as in Lemma 2.3 (2) in [26], the statements (3) can be
proved.
Now we prove (4). By the elliptic estimates to the equation (3.33) and then
using the H${\rm\ddot{o}}$lder inequality, one has for any $k>1,m\geq 1$,
$\begin{array}[]{ll}\|\eta\|_{2m}\leq
Cm\|\rho|u|^{2}\|_{2m}=Cm\big{[}\|(\rho-\bar{\rho})|u|^{2}\|_{2m}+\bar{\rho}\||u|^{2}\|_{2m}\big{]}\\\
\displaystyle\qquad\leq
Cm\big{[}\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}\|u\|^{2}_{4mk}+\bar{\rho}\|u\|_{4m}^{2}\big{]}.\end{array}$
Thus Lemma 3.3 is proved. $\hfill\Box$
Based on Lemmas 2.1-2.3 and Lemma 3.3, it holds that
###### Lemma 3.4
* (1)
$\|\xi_{1}\|_{2m}\leq Cm^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m},$ for any
$m\geq 2;$
* (2)
$\|\xi_{2}\|_{2m}\leq
Cm^{\frac{1}{2}}\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{\frac{2}{m\alpha}},$
for any $m+1\geq\frac{4}{\alpha}$ and $\alpha$ satisfying (3.22);
* (3)
$\|u\|_{2m}\leq Cm^{\frac{1}{2}}\big{[}\|\nabla u\|_{2}+1\big{]},$ for any
$m\geq 1;$
* (4)
$\|\nabla\xi_{1}\|_{2m}\leq
Cm^{\frac{3}{2}}k^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}(\|\nabla
u\|_{2}+1),$ for any $k>1,m\geq 1;$
* (5)
$\|\nabla\xi_{2}\|_{2m}\leq
Cm^{\frac{3}{2}}\big{[}k^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}+1\big{]}(\|\nabla
u\|_{2}+1),$ for any $k>1,m\geq 1;$
* (6)
$\|\eta\|_{2m}\leq
Cm^{2}\big{[}k\|\rho-\bar{\rho}\|_{\frac{2mk}{k-1}}+1\big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)},$ for any $k>1,m\geq 1;$
where $C$ are positive constants independent of $m,k$.
Proof: (1) By Lemma 2.2, it holds that
$\begin{array}[]{ll}\displaystyle\|\xi_{1}\|_{2m}\leq
Cm^{\frac{1}{2}}\|\nabla\xi_{1}\|_{\frac{2m}{m+1}}&\displaystyle\leq
Cm^{\frac{1}{2}}\|\sqrt{\rho}u\|_{2}\|\sqrt{\rho}-\sqrt{\bar{\rho}}\|_{2m}\leq
Cm^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m},\end{array}$
where in the last inequality one has used the elementary energy estimates.
(2). If $m+1>\frac{4}{\alpha}$, then by interpolation inequality, Caffarelli-
Kohn-Nirenberg inequality and Lemma 3.3 (2), it holds that
$\|\xi_{2}\|_{m+1}\leq\|\xi_{2}\|^{\theta}_{2m}\|\xi_{2}\|_{\frac{4}{\alpha}}^{1-\theta}\leq
C\|\xi_{2}\|^{\theta}_{2m}\||x|^{\frac{\alpha}{2}}\nabla\xi_{2}\|_{2}^{1-\theta}\leq
C\|\xi_{2}\|^{\theta}_{2m}\||x|^{\frac{\alpha}{2}}\sqrt{\rho}u\|_{2}^{1-\theta}$
(3.34)
where
$\theta=\frac{\frac{1}{m+1}-\frac{\alpha}{4}}{\frac{1}{2m}-\frac{\alpha}{4}}.$
Then it follows from Lemma 2.1 (2) and (3.34) that
$\begin{array}[]{ll}\displaystyle\|\xi_{2}\|_{2m}\leq
Cm^{\frac{1}{4}}\|\nabla\xi_{2}\|_{2}^{\frac{1}{2}-\frac{1}{2m}}\|\xi_{2}\|_{m+1}^{\frac{1}{2}+\frac{1}{2m}}\leq
Cm^{\frac{1}{4}}\|\sqrt{\rho}u\|_{2}^{\frac{1}{2}-\frac{1}{2m}}\|\xi_{2}\|_{2m}^{(\frac{1}{2}+\frac{1}{2m})\theta}\||x|^{\frac{\alpha}{2}}\sqrt{\rho}u\|_{2}^{(\frac{1}{2}+\frac{1}{2m})(1-\theta)}\\\
\displaystyle\qquad\quad\leq
Cm^{\frac{1}{4}}\|\xi_{2}\|_{2m}^{(\frac{1}{2}+\frac{1}{2m})\theta}\||x|^{\frac{\alpha}{2}}\sqrt{\rho}u\|_{2}^{(\frac{1}{2}+\frac{1}{2m})(1-\theta)},\end{array}$
which implies Lemma 3.4 (2) immediately.
Now we prove (3). First,
$\begin{array}[]{ll}\displaystyle\bar{\rho}\int|u|^{2}dx=\int(\bar{\rho}-\rho)|u|^{2}dx+\int\rho|u|^{2}dx\\\
\displaystyle=\int(\bar{\rho}-\rho)\big{(}{\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}+{\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\big{)}|u|^{2}dx+\int\rho|u|^{2}dx\\\
\displaystyle\leq\|(\bar{\rho}-\rho){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{2}\|u\|_{4}^{2}+\|(\bar{\rho}-\rho){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|_{\gamma}\|u\|_{\frac{2\gamma}{\gamma-1}}^{2}+C\\\
\displaystyle\leq\|\Psi(\rho,\bar{\rho})\|_{1}^{\frac{1}{2}}\|u\|_{2}\|\nabla
u\|_{2}+\|\Psi(\rho,\bar{\rho})\|_{1}^{\frac{1}{\gamma}}\|u\|_{2}^{2-\frac{2}{\gamma}}\|\nabla
u\|_{2}^{\frac{2}{\gamma}}+C\\\
\displaystyle\leq\sigma\|u\|_{2}^{2}+C_{\sigma}\|\nabla
u\|_{2}^{2}+C.\end{array}$
Choosing $\sigma=\frac{\bar{\rho}}{2}$ in the above inequality yields that
$\|u\|_{2}^{2}\leq C\big{(}\|\nabla u\|_{2}^{2}+1\big{)}.$ (3.35)
By Lemma 2.1 and the interpolation inequality, it holds that
$\|u\|_{2m}\leq Cm^{\frac{1}{4}}\|\nabla
u\|_{2}^{\frac{1}{2}-\frac{1}{2m}}\|u\|_{m+1}^{\frac{1}{2}+\frac{1}{2m}}\leq
Cm^{\frac{1}{4}}\|\nabla
u\|_{2}^{\frac{1}{2}-\frac{1}{2m}}\big{(}\|u\|^{\frac{1}{m+1}}_{2}\|u\|^{\frac{m}{m+1}}_{2m}\big{)}^{\frac{1}{2}+\frac{1}{2m}},$
thus one has
$\|u\|_{2m}\leq Cm^{\frac{1}{2}}\|\nabla
u\|_{2}^{1-\frac{1}{m}}\|u\|_{2}^{\frac{1}{m}}\leq Cm^{\frac{1}{2}}(\|\nabla
u\|_{2}+1),$
where in the last inequality we have used (3.35). The statement (3) is proved.
The assertions (3), (4) and (5) in Lemma 3.4 are the direct consequences of
Lemma 3.4 (2) and Lemma 3.3 (1), (2), (4), respectively. Thus the proof of
Lemma 3.4 is completed. $\hfill\Box$
Substituting (3.31), (3.32) and (3.33) into (3.30) yields that
$-\Delta\Big{(}\xi_{1t}+\xi_{2t}+\eta+F\Big{)}=0,$
which implies that
$\xi_{1t}+\xi_{2t}+\eta+F=0.$
It follows from the definition (2.1) of the effective viscous flux $F$ that
$\xi_{1t}+\xi_{2t}+(2\mu+\lambda(\rho)){\rm
div}u-(P(\rho)-P(\bar{\rho}))+\eta=0.$
Then the continuity equation $\eqref{CNS}_{1}$ yields that
$\xi_{1t}+\xi_{2t}-\frac{2\mu+\lambda(\rho)}{\rho}(\rho_{t}+u\cdot\nabla\rho)-(P(\rho)-P(\bar{\rho}))+\eta=0.$
Define
$\Lambda(\rho)=\int_{\bar{\rho}}^{\rho}\frac{2\mu+\lambda(s)}{s}ds=2\mu\ln\frac{\rho}{\bar{\rho}}+\frac{1}{\beta}(\rho^{\beta}-\bar{\rho}^{\beta}).$
Then we obtain a new transport equation
$(\Lambda(\rho)-\xi_{1}-\xi_{2})_{t}+u\cdot\nabla(\Lambda(\rho)-\xi_{1}-\xi_{2})+(P(\rho)-P(\bar{\rho}))+u\cdot\nabla(\xi_{1}+\xi_{2})-\eta=0,$
(3.36)
which is crucial in the following Lemma for the higher integrability of the
density function.
###### Lemma 3.5
For any $k\geq 2$ and $\beta>1$, it holds that
$\sup_{t\in[0,T]}\|(\rho-\bar{\rho})(t,\cdot)\|_{k}\leq
Ck^{\frac{2}{\beta-1}}.$ (3.37)
Proof: Multiplying the equation (3.36) by
$\rho[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}$ with $m$ being sufficiently
large integer, here and in what follows, the notation $(\cdots)_{+}$ denotes
the positive part of $(\cdots)$, one can get that
$\begin{array}[]{ll}\displaystyle\frac{1}{2m}\frac{d}{dt}\int\rho[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m}dx+\int\rho(P(\rho)-P(\bar{\rho})){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx\\\
\displaystyle=-\int\rho(P(\rho)-P(\bar{\rho})){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx\\\
\displaystyle\quad+\int\rho\eta[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx-\int\rho
u\cdot\nabla(\xi_{1}+\xi_{2})[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx.\end{array}$
(3.38)
Denote
$f(t)=\big{\\{}\int\rho[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m}dx\big{\\}}^{\frac{1}{2m}},\qquad
t\in[0,T].$
Now we estimate the three terms on the right hand side of (3.38). First, it
holds that
$\begin{array}[]{ll}\displaystyle|\int\rho(P(\rho)-P(\bar{\rho})){\bf
1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx|\\\
\displaystyle\leq f(t)^{2m-1}\Big{(}\int\rho|P(\rho)-P(\bar{\rho})|^{2m}{\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}dx\Big{)}^{\frac{1}{2m}}\\\
\displaystyle\leq Cf(t)^{2m-1}\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{2}\leq Cf(t)^{2m-1}.\end{array}$ (3.39)
Then, it follows that
$\begin{array}[]{ll}\displaystyle|\int\rho\eta[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx|\leq\int\rho^{\frac{1}{2m}}|\eta|\big{[}\rho(\Lambda(\rho)-\xi_{1}-\xi_{2})^{2m}_{+}\big{]}^{\frac{2m-1}{2m}}dx\\\
\displaystyle=\int\big{[}(\rho-\bar{\rho})+\bar{\rho}\big{]}^{\frac{1}{2m}}|\eta|\big{[}\rho(\Lambda(\rho)-\xi_{1}-\xi_{2})^{2m}_{+}\big{]}^{\frac{2m-1}{2m}}dx\\\
\displaystyle\leq
C\Big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{1}{2m}}\|\eta\|_{2m+\frac{1}{\beta}}+\|\eta\|_{2m}\Big{]}\|\rho(\Lambda(\rho)-\xi_{1}-\xi_{2})^{2m}_{+}\|_{1}^{\frac{2m-1}{2m}}\\\
\displaystyle\leq
C\Big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{1}{2m}}(m+\frac{1}{2\beta})^{2}\big{(}k_{1}\|\rho-\bar{\rho}\|_{\frac{2(m+\frac{1}{2\beta})k_{1}}{k_{1}-1}}+1\big{)}\\\
\displaystyle\qquad\qquad\qquad\qquad+m^{2}\big{(}k_{2}\|\rho-\bar{\rho}\|_{\frac{2mk_{2}}{k_{2}-1}}+1\big{)}\Big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}f(t)^{2m-1}\\\ \displaystyle\leq
Cm^{2}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}+1\big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}f(t)^{2m-1},\end{array}$ (3.40)
where in the last inequality we have taken $k_{1}=\frac{\beta}{\beta-1}$ and
$k_{2}=\frac{2m\beta+1}{2m(\beta-1)+1}.$
Next, for $\frac{1}{2m\beta+1}+\frac{1}{p_{1}}+\frac{1}{q_{1}}=1$ and
$\frac{1}{p_{2}}+\frac{1}{q_{2}}=1$ with $p_{i},q_{i}>1,(i=1,2)$, one has
$\begin{array}[]{ll}\displaystyle|-\int\rho
u\cdot\nabla(\xi_{1}+\xi_{2})[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx|\\\
\qquad\displaystyle\leq\int\big{[}(\rho-\bar{\rho})+\bar{\rho}\big{]}^{\frac{1}{2m}}|u||\nabla(\xi_{1}+\xi_{2})|\big{[}\rho(\Lambda(\rho)-\xi_{1}-\xi_{2})^{2m}_{+}\big{]}^{\frac{2m-1}{2m}}dx\\\
\qquad\displaystyle\leq
C\Big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{1}{2m}}\|u\|_{2mp_{1}}\|\nabla(\xi_{1}+\xi_{2})\|_{2mq_{1}}+\|u\|_{2mp_{2}}\|\nabla(\xi_{1}+\xi_{2})\|_{2mq_{2}}\Big{]}f(t)^{2m-1}\\\
\qquad\displaystyle\leq
C\Big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{1}{2m}}(mp_{1})^{\frac{1}{2}}(mq_{1})^{\frac{3}{2}}\big{(}k_{1}^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{\frac{2mq_{1}k_{1}}{k_{1}-1}}+1\big{)}\\\
\qquad\qquad\displaystyle+(mp_{2})^{\frac{1}{2}}(mq_{2})^{\frac{3}{2}}\big{(}k_{2}\|\rho-\bar{\rho}\|_{\frac{2mq_{2}k_{2}}{k_{2}-1}}+1\big{)}\Big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}f(t)^{2m-1}\\\ \qquad\displaystyle\leq
Cm^{2}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}+1\big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}f(t)^{2m-1},\end{array}$ (3.41)
where in the last inequality one has chosen
$p_{1}=\frac{(2m\beta+1)(\beta+1)}{2m\beta(\beta-1)},q_{1}=\frac{(\beta+1)(2m\beta+1)}{4m\beta},k_{1}=\frac{2\beta}{\beta-1}$
and
$p_{2}=\frac{2\beta}{\beta-1},q_{2}=\frac{2\beta}{\beta+1},k_{2}=\frac{(\beta+1)(2m\beta+1)}{2m\beta(\beta-1)+(\beta+1)}.$
Substituting (3.39), (3.40) and (3.41) into (3.38) yields that
$\begin{array}[]{ll}\displaystyle\frac{1}{2m}\frac{d}{dt}(f^{2m}(t))+\int\rho(P(\rho)-P(\bar{\rho})){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}[(\Lambda(\rho)-\xi_{1}-\xi_{2})_{+}]^{2m-1}dx\\\
\displaystyle\leq
Cm^{2}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}+1\big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}f(t)^{2m-1}.\end{array}$
Then it holds that
$\frac{d}{dt}f(t)\leq
Cm^{2}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}+1\big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}.$
Integrating the above inequality over $[0,t]$ gives that
$f(t)\leq
f(0)+Cm^{2}\int_{0}^{t}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}+1\big{]}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}d\tau.$ (3.42)
Now we calculate the quantity
$f(0)=\Big{(}\int\rho_{0}[(\Lambda(\rho_{0})-\xi_{10}-\xi_{20})_{+}]^{2m}dx\Big{)}^{\frac{1}{2m}}.$
By Lemma 3.3 (1), (2) and Lemma 3.4 (1), (2) with $t=0$, we can easily get
$\|\xi_{10}+\xi_{20}\|_{L^{\infty}}\leq C.$
Furthermore, by the definition of
$\Lambda(\rho_{0})=2\mu\ln\frac{\rho_{0}}{\bar{\rho}}+\frac{1}{\beta}((\rho_{0})^{\beta}-\bar{\rho}^{\beta})$,
we have
$\Lambda(\rho_{0})-\xi_{10}-\xi_{20}\rightarrow-\infty,\quad{\rm
as}\quad\rho_{0}\rightarrow 0+.$
Thus there exists a positive constant $\sigma_{0}$, such that if
$0\leq\rho_{0}\leq\sigma_{0}$, then
$(\Lambda(\rho_{0})-\xi_{10}-\xi_{20})_{+}\equiv 0.$
Now one has
$\begin{array}[]{ll}f(0)&\displaystyle=\Big{[}\Big{(}\int_{[0\leq\rho_{0}\leq\sigma_{0}]}+\int_{[\sigma_{0}\leq\rho_{0}\leq
M]}\Big{)}\rho_{0}(\Lambda(\rho_{0})-\xi_{10}-\xi_{20})_{+}^{2m}dx\Big{]}^{\frac{1}{2m}}\\\
&\displaystyle=\Big{[}\int_{[\sigma_{0}\leq\rho_{0}\leq
M]}\rho_{0}(\Lambda(\rho_{0})-\xi_{10}-\xi_{20})_{+}^{2m}dx\Big{]}^{\frac{1}{2m}}\\\
&\displaystyle\leq C(\sigma_{0},M)\Big{[}\|(\rho_{0}-\bar{\rho}){{\bf
1}_{\sigma_{0}\leq\rho_{0}\leq
M}}\|_{2m}+\|\xi_{10}+\xi_{20}\|_{2m}\Big{]}\leq
C(\sigma_{0},M)m^{\frac{3}{2}},\end{array}$ (3.43)
where the positive constant $C(\sigma_{0},M)$ is independent of $m$ and the
lower bound of the density.
It follows from (3.42) and (3.43) that for $t\in[0,T]$,
$f(t)\leq
Cm^{2}\Big{[}1+\int_{0}^{t}\big{(}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}+1\big{)}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}d\tau\Big{]}.$ (3.44)
For any $t\in[0,T]$, set
$\Omega_{1}(t)=\\{x\in\mathbb{R}^{2}|\rho(t,x)>2\bar{\rho}\\}$ and
$\Omega_{2}(t)=\\{x\in\Omega_{1}(t)|(\Lambda(\rho)-\xi_{1}-\xi_{2})(t,x)>0\\}$.
Then one has on $\Omega_{1}(t),$ $|\rho-\bar{\rho}|^{\beta}\leq
C\beta|\Lambda(\rho)|$ for some constant $C>0$, and on
$\Omega_{1}(t)\setminus\Omega_{2}(t),$ $0<\Lambda(\rho)\leq\xi_{1}+\xi_{2}.$
Thus it holds that
$\begin{array}[]{ll}\displaystyle\|\rho-\bar{\rho}\|_{2m\beta+1}^{\beta}(t)=\Big{(}\int_{\Omega_{1}(t)}|\rho-\bar{\rho}|^{2m\beta+1}dx+\int_{\mathbb{R}^{2}\setminus\Omega_{1}(t)}|\rho-\bar{\rho}|^{2m\beta+1}dx\Big{)}^{\frac{\beta}{2m\beta+1}}\\\
\displaystyle\leq\Big{(}\int_{\Omega_{1}(t)}|\rho-\bar{\rho}|^{2m\beta+1}dx+\bar{\rho}^{2m\beta-1}\int_{\mathbb{R}^{2}\setminus\Omega_{1}(t)}|\rho-\bar{\rho}|^{2}dx\Big{)}^{\frac{\beta}{2m\beta+1}}\\\
\displaystyle\leq\Big{[}\int_{\Omega_{1}(t)}\big{(}|\rho-\bar{\rho}|^{\beta}\big{)}^{\frac{2m\beta+1}{\beta}}dx\Big{]}^{\frac{\beta}{2m\beta+1}}+C\leq
C\Big{(}\int_{\Omega_{1}(t)}|\beta\Lambda(\rho)|^{\frac{2m\beta+1}{\beta}}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
\displaystyle\leq
C\Big{(}\int_{\Omega_{1}(t)}\Lambda(\rho)^{2m}\Lambda(\rho)^{\frac{1}{\beta}}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\leq
C\Big{(}\int_{\Omega_{1}(t)}\rho\Lambda(\rho)^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
\displaystyle=C\Big{(}\int_{\Omega_{2}(t)}\rho|\Lambda(\rho)-\xi_{1}-\xi_{2}+(\xi_{1}+\xi_{2})|^{2m}dx+\int_{\Omega_{1}(t)\setminus\Omega_{2}(t)}\rho|\Lambda(\rho)|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
\displaystyle\leq
C\Big{[}\int_{{}_{\Omega_{2}(t)}}\rho(\Lambda(\rho)-\xi_{1}-\xi_{2})^{2m}dx+\int_{{}_{\Omega_{2}(t)}}\rho|\xi_{1}+\xi_{2}|^{2m}dx+\int_{{}_{\Omega_{1}(t)\setminus\Omega_{2}(t)}}\rho|\xi_{1}+\xi_{2}|^{2m}dx\Big{]}^{\frac{\beta}{2m\beta+1}}+C\\\
\displaystyle\leq
C\Big{(}f(t)^{2m}+\int_{\mathbb{R}^{2}}\rho|\xi_{1}+\xi_{2}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
\displaystyle\leq
C\Big{[}f(t)+\Big{(}\int_{\mathbb{R}^{2}}\rho|\xi_{1}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+\Big{(}\int_{\mathbb{R}^{2}}\rho|\xi_{2}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+1\Big{]}.\end{array}$
(3.45)
Note that
$\begin{array}[]{ll}\displaystyle\Big{(}\int_{\mathbb{R}^{2}}\rho|\xi_{1}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}&\displaystyle\leq
C\Big{(}\int_{\mathbb{R}^{2}}|\rho-\bar{\rho}||\xi_{1}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\Big{(}\int_{\mathbb{R}^{2}}|\xi_{1}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}:=K_{11}+K_{12}.\end{array}$
(3.46) $\begin{array}[]{ll}K_{11}&\displaystyle\leq
C\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\||\xi_{1}|^{2m}\|^{\frac{\beta}{2m\beta+1}}_{\frac{2m\beta+1}{2m\beta}}=C\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\|\xi_{1}\|^{\frac{2m\beta}{2m\beta+1}}_{2m+\frac{1}{\beta}}\\\
&\displaystyle\leq\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\Big{[}C(m+\frac{1}{2\beta})^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m+\frac{1}{\beta}}\Big{]}^{\frac{2m\beta}{2m\beta+1}}\\\
&\displaystyle\leq
Cm^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\Big{[}\|(\rho-\bar{\rho}){\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}\|^{\frac{2m\beta}{2m\beta+1}}_{2}\\\
&\displaystyle\qquad\qquad\qquad\qquad\qquad+\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|^{\frac{2\gamma
m\beta(\beta-1)}{(2m\beta-\gamma+1)(2m\beta+1)}}_{\gamma}\|\rho-\bar{\rho}\|^{\frac{2m\beta(2m\beta-\gamma\beta+1)}{(2m\beta-\gamma+1)(2m\beta+1)}}_{2m\beta+1}\Big{]}\\\
&\displaystyle\leq
Cm^{\frac{1}{2}}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}+1\big{]},\end{array}$
(3.47)
and
$K_{12}=\|\xi_{1}\|^{\frac{2m\beta}{2m\beta+1}}_{2m}\leq\Big{(}Cm^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m}\Big{)}^{\frac{2m\beta}{2m\beta+1}}\leq
Cm^{\frac{1}{2}}\big{[}\|\rho-\bar{\rho}\|_{2m\beta+1}+1\big{]}.$ (3.48)
Furthermore, it holds that
$\begin{array}[]{ll}\displaystyle\Big{(}\int_{\mathbb{R}^{2}}\rho|\xi_{2}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}\leq
C\Big{(}\int_{\mathbb{R}^{2}}|\rho-\bar{\rho}||\xi_{2}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\Big{(}\int_{\mathbb{R}^{2}}|\xi_{2}|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}\\\
\displaystyle\qquad\qquad\leq
C\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\|\xi_{2}\|^{\frac{2m\beta}{2m\beta+1}}_{2m+\frac{1}{\beta}}+C\|\xi_{2}\|^{\frac{2m\beta}{2m\beta+1}}_{2m}\\\
\displaystyle\qquad\qquad\leq
Cm^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{\frac{2}{(m+\frac{1}{2\beta})\alpha}\frac{2m\beta}{2m\beta+1}}+Cm^{\frac{1}{2}}\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{\frac{2}{m\alpha}\frac{2m\beta}{2m\beta+1}}\\\
\displaystyle\qquad\qquad\leq
C\big{[}\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{2}+m^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m\beta+1}+m^{\frac{1}{2}}\big{]}.\end{array}$
(3.49)
Substituting and (3.46), (3.47), (3.48) and (3.49) into (3.45) yields that
$\begin{array}[]{ll}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\beta}(t)&\displaystyle\leq
C\Big{[}m^{\frac{1}{2}}+f(t)+m^{\frac{1}{2}}\|\rho-\bar{\rho}\|_{2m\beta+1}(t)+\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{2}(t)\Big{]}\\\
&\displaystyle\leq\frac{1}{2}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\beta}(t)+C\Big{[}f(t)+m^{\frac{\beta}{2(\beta-1)}}+\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|^{2}_{2}(t)\Big{]}.\end{array}$
(3.50)
Thus it follows from (3.44), (3.50) and the weighted estimates in Lemma 3.2
that
$\begin{array}[]{ll}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\beta}(t)\displaystyle\leq
C\Big{[}f(t)+m^{\frac{\beta}{2(\beta-1)}}+\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{2}(t)\Big{]}\\\
\displaystyle\leq
C\Big{[}m^{2}+m^{2}\int_{0}^{t}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}d\tau+\int_{0}^{t}\|\rho-\bar{\rho}\|_{2m\beta+1}^{\beta}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}d\tau\Big{]}.\end{array}$
Applying Gronwall’s inequality to the above inequality yields that
$\|\rho-\bar{\rho}\|_{2m\beta+1}^{\beta}(t)\leq
Cm^{2}\Big{[}1+\int_{0}^{t}\|\rho-\bar{\rho}\|_{2m\beta+1}^{1+\frac{1}{2m}}\big{(}\|\nabla
u\|_{2}^{2}(\tau)+1\big{)}d\tau\Big{]}.$
Denote
$y(t)=m^{-\frac{2}{\beta-1}}\|\rho-\bar{\rho}\|_{2m\beta+1}(t).$
Then it holds that
$\displaystyle y^{\beta}(t)\leq
C\Big{[}1+\int_{0}^{t}y(\tau)^{1+\frac{1}{2m}}\|\nabla
u\|_{2}^{2}(\tau)d\tau\Big{]}\leq
C\Big{[}1+\int_{0}^{t}\big{(}y^{\beta}(\tau)+1\big{)}\|\nabla
u\|_{2}^{2}(\tau)d\tau\Big{]}.$
So applying the Gronwall’s inequality to the above inequality yields that
$y(t)\leq C,\quad\forall t\in[0,T],$
that is, for sufficiently large $m>1$,
$\|\rho-\bar{\rho}\|_{2m\beta+1}(t)\leq Cm^{\frac{2}{\beta-1}},\quad\forall
t\in[0,T].$
Equivalently, (3.37) holds for sufficiently large $k$. Now by the elementary
energy estimate Lemma 3.1, if $\gamma\geq 2,$ then
$\|\rho-\bar{\rho}\|_{2}(t)\leq
C\|\Psi(\rho,\bar{\rho})\|_{1}^{\frac{1}{2}}\leq C,$ (3.51)
and if $1<\gamma<2$, then
$\|(\rho-\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}\|_{2}(t)\leq
C\|\Psi(\rho,\bar{\rho}){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}\|_{1}^{\frac{1}{2}}\leq C,$ (3.52)
and
$\|(\rho-\bar{\rho}){\bf 1}|_{\\{\rho>2\bar{\rho}\\}}\|_{\gamma}(t)\leq
C\|\Psi(\rho,\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|_{1}^{\frac{1}{\gamma}}\leq C.$
Therefore, for $1<\gamma<2$, it holds that
$\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|_{2}(t)\leq\|(\rho-\bar{\rho}){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}\|_{\gamma}^{\theta}\|\rho-\bar{\rho}\|^{1-\theta}_{k}\leq
C,$ (3.53)
where $k$ is sufficiently large such that (3.37) holds and $\theta\in(0,1)$
satisfying $\frac{1}{2}=\frac{\theta}{\gamma}+\frac{1-\theta}{k}.$ Thus by
(3.51), (3.52) and (3.53), it holds that for any $\gamma>1$ and $t\in[0,T]$,
$\|\rho-\bar{\rho}\|_{2}(t)\leq C.$
Thus Lemma 3.5 is proved for any $k\geq 2$. $\hfill\Box$
Step 4: First-order derivative estimates of the velocity.
Set
$Z^{2}(t)=\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx,$
$\varphi^{2}(t)=\int\rho(H^{2}+L^{2})dx=\int\rho|\dot{u}|^{2}dx$
and
$\Phi_{T}=\sup_{t\in[0,T]}\|\rho(\cdot,t)\|_{\infty}+1.$
The following Lemma is motivated by [38].
###### Lemma 3.6
For any $\varepsilon>0$, there exists a positive constant $C_{\varepsilon}$,
such that
$\sup_{t\in[0,T]}\log(e+Z^{2}(t))+\int_{0}^{T}\frac{\varphi^{2}(t)}{e+Z^{2}(t)}dt\leq
C_{\varepsilon}\Phi_{T}^{1+\varepsilon\beta}.$
Proof: Multiplying the equation $\eqref{F-omega}_{1}$ by $\mu\omega$, the
equation $\eqref{F-omega}_{2}$ by $\frac{F}{2\mu+\lambda(\rho)}$,
respectively, and then summing the resulted equations together, one has
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx+\frac{\mu}{2}\int\omega^{2}{\rm
div}udx-\frac{1}{2}\int\rho F^{2}(\frac{1}{2\mu+\lambda(\rho)})^{\prime}{\rm
div}udx\\\ \displaystyle-\frac{1}{2}\int F^{2}\frac{{\rm
div}u}{2\mu+\lambda(\rho)}dx-\int\rho F({\rm
div}u)(\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})^{\prime}dx+\int
F[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]dx\\\
\displaystyle=-\int\rho(H^{2}+L^{2})dx.\end{array}$ (3.54)
Notice that
$\begin{array}[]{ll}\displaystyle(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}\displaystyle=(u_{1x_{1}}+u_{2x_{2}})^{2}+2(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})\\\
\displaystyle\qquad=({\rm
div}u)^{2}+2(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})\\\
\displaystyle\qquad=({\rm
div}u)\left(\frac{F}{2\mu+\lambda(\rho)}+\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)}\right)+2(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}}),\end{array}$
then one has
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx+\int\rho(H^{2}+L^{2})dx\\\
\displaystyle\quad=-\frac{\mu}{2}\int\omega^{2}{\rm div}udx+\frac{1}{2}\int
F^{2}({\rm
div}u)\Big{[}\rho(\frac{1}{2\mu+\lambda(\rho)})^{\prime}-\frac{1}{2\mu+\lambda(\rho)}\Big{]}dx\\\
\displaystyle\quad+\int F({\rm
div}u)\Big{[}\rho(\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})^{\prime}-\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)}\Big{]}dx-\int
2F(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})dx.\end{array}$ (3.55)
Then
$\begin{array}[]{ll}\displaystyle\|\nabla u\|_{2}+\|\omega\|_{2}+\|{\rm
div}u\|_{2}+\|(2\mu+\lambda(\rho))^{\frac{1}{2}}{\rm div}u\|_{2}\\\
\displaystyle\leq
C\Big{[}Z(t)+\Big{(}\int\frac{|P(\rho)-P(\bar{\rho})|^{2}}{2\mu+\lambda(\rho)}dx\Big{)}^{\frac{1}{2}}\Big{]}\leq
C(Z(t)+1).\end{array}$ (3.56)
Now we estimate the four terms on the right hand side of (3.55). First, by the
interpolation inequality, Lemma 2.1 and (3.56), it holds that
$\begin{array}[]{ll}\displaystyle|-\frac{\mu}{2}\int\omega^{2}{\rm
div}udx|\leq C\|{\rm div}u\|_{2}\|\omega\|_{4}^{2}\leq
C(Z(t)+1)\|\omega\|_{2}\|\nabla\omega\|_{2}\\\ \displaystyle\qquad\qquad\leq
C(Z(t)+1)\|\omega\|_{2}\|\rho\dot{u}\|_{2}\leq
C(Z(t)+1)\|\rho\|^{\frac{1}{2}}_{\infty}\|\omega\|_{2}\|\sqrt{\rho}\dot{u}\|_{2}\\\
\qquad\qquad\leq\sigma\|\sqrt{\rho}\dot{u}\|_{2}^{2}+C_{\sigma}(Z(t)^{2}+1)\|\rho\|_{\infty}\|\omega\|_{2}^{2}.\end{array}$
(3.57)
Next, one has
$\begin{array}[]{ll}\displaystyle|\frac{1}{2}\int F^{2}{\rm
div}u\Big{[}\rho(\frac{1}{2\mu+\lambda(\rho)})^{\prime}-\frac{1}{2\mu+\lambda(\rho)}\Big{]}dx|\\\
\displaystyle\leq C\int|F|^{2}\frac{|{\rm
div}u|}{2\mu+\lambda(\rho)}dx\leq\|{\rm
div}u\|_{2}\|\frac{F^{2}}{2\mu+\lambda(\rho)}\|_{2},\end{array}$
while for any $\varepsilon>0$ suitably small,
$\begin{array}[]{ll}\displaystyle\|\frac{F^{2}}{2\mu+\lambda(\rho)}\|_{2}\leq\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{1-\varepsilon}\|F\|^{1+\varepsilon}_{\frac{2(1+\varepsilon)}{\varepsilon}}\\\
\displaystyle\leq
C\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{1-\varepsilon}\big{(}\|F\|_{2}^{\frac{\varepsilon}{1+\varepsilon}}\|\nabla
F\|_{2}^{\frac{1}{1+\varepsilon}}\big{)}^{1+\varepsilon}\\\ \displaystyle\leq
C\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{1-\varepsilon}\|F\|_{2}^{\varepsilon}\|\nabla
F\|_{2}\displaystyle\leq
C\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}\|\rho\|_{\infty}^{\frac{1+\beta\varepsilon}{2}}\|\sqrt{\rho}\dot{u}\|_{2}.\end{array}$
(3.58)
Then it holds that
$\begin{array}[]{ll}\displaystyle|\frac{1}{2}\int F^{2}{\rm
div}u\Big{[}\rho(\frac{1}{2\mu+\lambda(\rho)})^{\prime}-\frac{1}{2\mu+\lambda(\rho)}\Big{]}dx|\\\
\displaystyle\leq C\|{\rm
div}u\|_{2}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}\|\rho\|_{\infty}^{\frac{1+\beta\varepsilon}{2}}\|\sqrt{\rho}\dot{u}\|_{2}\\\
\displaystyle\leq\sigma\|\sqrt{\rho}\dot{u}\|_{2}^{2}+C_{\sigma}\|\rho\|_{\infty}^{1+\beta\varepsilon}\|{\rm
div}u\|^{2}_{2}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|^{2}_{2}.\end{array}$
(3.59)
On the other hand, it holds that
$\begin{array}[]{ll}\displaystyle|\int F({\rm
div}u)\Big{[}\rho(\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)})^{\prime}-\frac{P(\rho)-P(\bar{\rho})}{2\mu+\lambda(\rho)}\Big{]}dx|\\\
\displaystyle\leq C\int|F||{\rm
div}u|\frac{|P(\rho)-P(\bar{\rho})|+1}{2\mu+\lambda(\rho)}dx\\\
\displaystyle\leq C\|{\rm
div}u\|_{2}\Big{[}\|F\|_{\frac{2(2+\varepsilon)}{\varepsilon}}\|P(\rho)-P(\bar{\rho})\|_{2+\varepsilon}+\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}\Big{]}\\\
\displaystyle\leq C\|{\rm
div}u\|_{2}\Big{[}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{\varepsilon}{2+\varepsilon}}\|\rho\|_{\infty}^{\frac{\beta\varepsilon}{2(2+\varepsilon)}}\|\nabla
F\|_{2}^{\frac{2}{2+\varepsilon}}+\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}\Big{]}\\\
\displaystyle\leq C\|{\rm
div}u\|_{2}\Big{[}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{\varepsilon}{2+\varepsilon}}\|\rho\|_{\infty}^{\frac{1}{2}+\frac{\beta\varepsilon}{2(2+\varepsilon)}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{2}{2+\varepsilon}}+\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}\Big{]}\\\
\displaystyle\leq\sigma\|\sqrt{\rho}\dot{u}\|_{2}^{2}+C_{\sigma}\|\rho\|_{\infty}^{1+\frac{\beta\varepsilon}{2+\varepsilon}}\|{\rm
div}u\|_{2}^{\frac{2+\varepsilon}{1+\varepsilon}}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{\varepsilon}{1+\varepsilon}}+C\|{\rm
div}u\|_{2}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}\\\
\displaystyle\leq\sigma\|\sqrt{\rho}\dot{u}\|_{2}^{2}+C_{\sigma}(1+\|\rho\|_{\infty})^{1+\beta\varepsilon}\big{(}\|{\rm
div}u\|_{2}^{2}+1\big{)}\big{(}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{2}+1\big{)}.\end{array}$
(3.60)
Then due to [38], it holds that
$\begin{array}[]{ll}\displaystyle|-\int
2F(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})dx|=|-\int 2F\nabla
u_{1}\cdot\nabla^{\perp}u_{2}dx|\\\ \displaystyle\leq C\|F\|_{\rm BMO}\|\nabla
u_{1}\cdot\nabla^{\perp}u_{2}\|_{\mathcal{H}^{1}}\leq C\|\nabla
F\|_{2}\|\nabla u\|_{2}^{2}\leq
C\|\rho\|_{\infty}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{2}\|\nabla
u\|_{2}^{2}\\\
\displaystyle\leq\sigma\|\sqrt{\rho}\dot{u}\|_{2}^{2}+C_{\sigma}\|\rho\|_{\infty}\|\nabla
u\|_{2}^{4}\leq\sigma\|\sqrt{\rho}\dot{u}\|_{2}^{2}+C_{\sigma}\|\rho\|_{\infty}\|\nabla
u\|_{2}^{2}\Big{[}1+Z^{2}(t)\Big{]}.\end{array}$ (3.61)
In summary, substituting (3.57), (3.59), (3.60) and (3.61) into (3.55), one
can arrive at
$\displaystyle\frac{d}{dt}Z^{2}(t)+\varphi^{2}(t)\leq
C\Phi_{T}^{1+\beta\varepsilon}(1+\|\nabla u\|_{2}^{2})(1+Z^{2}(t))$
Multiplying the above inequality by $\frac{1}{e+Z^{2}(t)}$ and then
integrating over $[0,T]$ give the proof of Lemma 3.6 . $\hfill\Box$
Step 5: Upper and lower bound of the density:
The following Lemma comes from [17, 18]. With the following Lemma, the index
$\beta$ can be improved to $\beta>\frac{4}{3}$ as in [17, 18].
###### Lemma 3.7
There exists a positive constant $C$, such that
$\sup_{t\in[0,T]}\int\rho|u|^{2+\nu}dx\leq C,$
where
$\nu=\frac{\mu^{\frac{1}{2}}}{2(\mu+1)}\Phi_{T}^{-\frac{\beta}{2}}\in(0,\frac{1}{4}].$
Proof: Multiplying the momentum equation $\eqref{CNS}_{2}$ by
$(2+\nu)u|u|^{\nu}$ and integrating over $\mathbb{R}^{2}$ with respect to $x$
lead to
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\int\rho|u|^{2+\nu}dx+\mu(2+\nu)\int|\nabla
u|^{2}|u|^{\nu}dx+(2+\nu)\int(\mu+\lambda(\rho))({\rm div}u)^{2}|u|^{\nu}dx\\\
\displaystyle=(2+\nu)\int(P(\rho)-P(\bar{\rho})){\rm
div}(u|u|^{\nu})dx-\mu(2+\nu)\int\nabla\frac{|u|^{2}}{2}\cdot\nabla|u|^{\nu}dx\\\
\displaystyle\quad-(2+\nu)\int(\mu+\lambda(\rho))({\rm
div}u)u\cdot\nabla|u|^{\nu}dx.\end{array}$
Now we only estimate the first term on the right hand side of the above
equality, since the other terms can be done similarly as in [18]. Then it
holds that
$\begin{array}[]{ll}\displaystyle(2+\nu)|\int(P(\rho)-P(\bar{\rho})){\rm
div}(u|u|^{\nu})dx|\leq(2+\nu)(1+\nu)\int|P(\rho)-P(\bar{\rho})||\nabla
u||u|^{\nu}dx\\\ \displaystyle\leq\sigma(2+\nu)\int|\nabla
u|^{2}|u|^{\nu}dx+C_{\sigma}(2+\nu)(1+\nu)^{2}\int|P(\rho)-P(\bar{\rho})|^{2}|u|^{\nu}dx\\\
\displaystyle\leq\sigma(2+\nu)\int|\nabla
u|^{2}|u|^{\nu}dx+C_{\sigma}(2+\nu)(1+\nu)^{2}\|P(\rho)-P(\bar{\rho})\|_{2q_{1}}^{2}\|u\|_{q_{2}\nu}^{\nu}\\\
\displaystyle\leq\sigma(2+\nu)\int|\nabla
u|^{2}|u|^{\nu}dx+C_{\sigma}(2+\nu)(1+\nu)^{2}\big{(}\|\nabla
u\|_{2}^{2}+1\big{)}.\end{array}$
where $q_{1},q_{2}>1$ satisfying $\frac{1}{q_{1}}+\frac{1}{q_{2}}=1.$ Thus
Lemma 3.7 is proved.$\hfill\Box$
Now one can obtain the upper and lower bound of the density by using the
transport equation (3.36).
###### Lemma 3.8
There exists positive constants $C_{1}$ and $c_{1}$ such that
$c_{1}\leq\rho(t,x)\leq C_{1},\qquad\forall(t,x)\in[0,T]\times\mathbb{R}^{2}.$
Proof: First, for any $p>2$ and $q>1$ satisfying
$\frac{1}{p}=\frac{\frac{2}{p}}{2+\nu}+\frac{1-\frac{2}{p}}{q},$
it holds that
$\begin{array}[]{ll}\displaystyle\|\rho u\|_{p}\leq\|\rho
u\|_{2+\nu}^{\frac{2}{p}}\|\rho u\|_{q}^{1-\frac{2}{p}}\leq
C\Big{(}\|\rho^{\frac{1}{2+\nu}}u\|_{2+\nu}\|\rho\|_{\infty}^{\frac{1+\nu}{2+\nu}}\Big{)}^{\frac{2}{p}}\big{(}\|\rho\|_{\infty}\|u\|_{q}\big{)}^{1-\frac{2}{p}}\\\
\displaystyle\qquad\quad\leq
C\|\rho\|_{\infty}^{1-\frac{2}{p}+\frac{2(1+\nu)}{p(2+\nu)}}\Big{[}q^{\frac{1}{2}}(\|\nabla
u\|_{2}+1)\Big{]}^{1-\frac{2}{p}},\end{array}$
where in the last inequality one has used Lemma 3.4 (3). It can be computed
that
$q=(1+\frac{2}{\nu})(p-2)\leq C_{p}\Phi_{T}^{\frac{\beta}{2}}.$
Therefore, one has
$\|\rho u\|_{p}\leq
C\|\rho\|_{\infty}^{1-\frac{2}{p(2+\nu)}}\Phi_{T}^{\frac{\beta}{4}(1-\frac{2}{p})}(\|\nabla
u\|_{2}^{1-\frac{2}{p}}+1)\leq C\Phi_{T}^{1+\frac{\beta}{4}}(\|\nabla
u\|_{2}^{1-\frac{2}{p}}+1).$ (3.62)
Note that by the definition of $\xi_{i}~{}(i=1,2)$ from (3.31) and (3.32)
$u\cdot\nabla(\xi_{1}+\xi_{2})-\eta=[u,R_{i}R_{j}](\rho u),$ (3.63)
where $[\cdot,\cdot]$ is the usual commutator and $R_{i},R_{j}$ are the Riesz
operators. Thus from (3.36), it holds that
$D_{t}\Lambda(\rho)-D_{t}(\xi_{1}+\xi_{2})+(P(\rho)-P(\bar{\rho}))+[u,R_{i}R_{j}](\rho
u)=0,$ (3.64)
where the material derivative $D_{t}:=\partial_{t}+u\cdot\nabla$. Along the
particle path $\vec{X}(\tau;t,x)$ through the point
$(t,x)\in[0,T]\times\mathbb{R}^{2}$ defined by
$\left\\{\begin{array}[]{ll}\displaystyle\frac{d\vec{X}(\tau;t,x)}{d\tau}=u(\tau,\vec{X}(\tau;t,x)),\\\
\displaystyle\vec{X}(\tau;t,x)|_{\tau=t}=x,\end{array}\right.$
from the equation $\eqref{te1}$, there holds the following ODE
$\begin{array}[]{ll}\displaystyle\frac{d}{d\tau}(\Lambda(\rho)-\xi_{1}-\xi_{2})(\tau,\vec{X}(\tau;t,x))+(P(\rho)-P(\bar{\rho})){\bf
1}|_{\\{\rho>2\bar{\rho}\\}}(\tau,\vec{X}(\tau;t,x))\\\
\displaystyle\qquad=-\Big{(}(P(\rho)-P(\bar{\rho})){\bf 1}|_{\\{0\leq\rho\leq
2\bar{\rho}\\}}+[u,R_{i}R_{j}](\rho
u)\Big{)}(\tau,\vec{X}(\tau;t,x)),\end{array}$
and thus
$\frac{d}{d\tau}(\Lambda(\rho)-\xi_{1}-\xi_{2})(\tau,\vec{X}(\tau;t,x))\leq-\Big{(}(P(\rho)-P(\bar{\rho})){\bf
1}|_{\\{0\leq\rho\leq 2\bar{\rho}\\}}+[u,R_{i}R_{j}](\rho
u)\Big{)}(\tau,\vec{X}(\tau;t,x)).$
Integrating the above inequality over $[0,t]$ yields that
$\begin{array}[]{ll}\displaystyle
2\mu\ln\frac{\rho(t,x)}{\rho_{0}(\vec{X}_{0})}+\frac{1}{\beta}\big{(}\rho^{\beta}(t,x)-\rho_{0}^{\beta}(\vec{X}_{0})\big{)}-\big{(}(\xi_{1}+\xi_{2})(t,x)-(\xi_{10}+\xi_{20})(\vec{X}_{0})\big{)}\\\
\displaystyle\qquad\qquad\qquad\qquad\qquad\leq
C+\int_{0}^{t}\|[u,R_{i}R_{j}](\rho u)\|_{\infty}ds,\end{array}$ (3.65)
with $\vec{X}_{0}=\vec{X}(\tau;t,x)|_{\tau=0}$.
Then for any sufficiently large $p>4$, by the commutator estimates for (3.63)
and (3.62), it holds that
$\begin{array}[]{ll}\displaystyle\|[u,R_{i}R_{j}](\rho u)\|_{\infty}\leq
C\|[u,R_{i}R_{j}](\rho
u)\|_{p}^{1-\frac{4}{p}}\|\nabla\big{(}[u,R_{i}R_{j}](\rho
u)\big{)}\|_{\frac{4p}{p+4}}^{\frac{4}{p}}\\\ \displaystyle\leq
C\Big{[}\|u\|_{\rm BMO}\|\rho u\|_{p}\Big{]}^{1-\frac{4}{p}}\Big{[}\|\nabla
u\|_{4}\|\rho u\|_{p}\Big{]}^{\frac{4}{p}}\\\\[8.53581pt] \displaystyle\leq
C\|\nabla u\|_{2}^{1-\frac{4}{p}}\|\nabla u\|_{4}^{\frac{4}{p}}\|\rho
u\|_{p}\displaystyle\leq C\Phi_{T}^{1+\frac{\beta}{4}}\Big{(}\|\nabla
u\|_{2}^{1-\frac{2}{p}}+1\Big{)}\|\nabla u\|_{2}^{1-\frac{4}{p}}\|\nabla
u\|_{4}^{\frac{4}{p}},\end{array}$
while
$\begin{array}[]{ll}\displaystyle\|\nabla u\|_{4}\leq C\big{(}\|{\rm
div}u\|_{4}+\|\omega\|_{4}\big{)}\leq
C\big{(}\|\frac{F+(P(\rho)-P(\bar{\rho}))}{2\mu+\lambda(\rho)}\|_{4}+\|\rho\|_{\infty}^{\frac{1}{4}}\|\omega\|_{2}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{1}{2}}\big{)}\\\
\displaystyle\leq
C\big{(}\|\frac{F^{2}}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{1}{2}}+1+\|\rho\|_{\infty}^{\frac{1}{4}}\|\omega\|_{2}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{1}{2}}\big{)}\\\
\displaystyle\leq
C\big{(}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{1-\varepsilon}{2}}\|\rho\|_{\infty}^{\frac{1+\beta\varepsilon}{4}}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{\varepsilon}{2}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{1}{2}}+1+\|\rho\|_{\infty}^{\frac{1}{4}}\|\omega\|_{2}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{1}{2}}\big{)}\\\
\displaystyle\leq
C\big{(}\|\rho\|_{\infty}^{\frac{1+\beta\varepsilon}{4}}+1\big{)}\Big{[}\|\frac{F}{\sqrt{2\mu+\lambda(\rho)}}\|_{2}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{1}{2}}+\|\omega\|_{2}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{2}^{\frac{1}{2}}+1\Big{]}\\\
\displaystyle\leq
C\big{(}\|\rho\|_{\infty}^{\frac{1+\beta\varepsilon+\beta}{4}}+1\big{)}(e+\|\nabla
u\|_{2})\Big{(}\frac{\varphi^{2}(t)}{e+Z^{2}(t)}\Big{)}^{\frac{1}{4}},\end{array}$
where in the fourth inequality one has used the fact (3.58). Then it holds
that
$\begin{array}[]{ll}\displaystyle\|[u,R_{i}R_{j}](\rho u)\|_{\infty}\leq
C\big{(}\|\rho\|_{\infty}^{1+\frac{\beta}{4}+\frac{1+\beta\varepsilon+\beta}{p}}+1\big{)}(e+\|\nabla
u\|^{2}_{2})^{1-\frac{1}{p}}\Big{(}\frac{\varphi^{2}(t)}{e+Z^{2}(t)}\Big{)}^{\frac{1}{p}}\\\
\displaystyle\leq
C\Big{(}\frac{\varphi^{2}(t)}{e+Z^{2}(t)}+1\Big{)}+C\big{(}\|\rho\|_{\infty}^{\big{[}1+\frac{\beta}{4}+\frac{1+\beta\varepsilon+\beta}{p}\big{]}\frac{p}{p-1}}+1\big{)}(e+\|\nabla
u\|^{2}_{2}).\end{array}$
Thus it holds that for any $\varepsilon>0$, one can choose sufficiently large
$p>2$ such that
$\begin{array}[]{ll}\displaystyle\int_{0}^{T}\|[u,R_{i}R_{j}](\rho
u)\|_{\infty}(t)dt\leq C\Phi_{T}^{1+\frac{\beta}{4}+\varepsilon}.\end{array}$
(3.66)
By Lemma 3.4, it holds that for suitably large but fixed $m>1$,
$\|\xi_{1}+\xi_{2}\|_{2m}\leq
Cm^{\frac{1}{2}}\Big{[}\|\rho-\bar{\rho}\|_{2m}+\|\sqrt{\rho}u|x|^{\frac{\alpha}{2}}\|_{2}^{\frac{2}{m\alpha}}\Big{]}\leq
C_{m}.$
Then
$\|\nabla(\xi_{1}+\xi_{2})\|_{2}\leq C\|\rho u\|_{2}\leq
C\|\rho\|_{\infty}^{\frac{1}{2}}\|\sqrt{\rho}u\|_{2}\leq
C\|\rho\|_{\infty}^{\frac{1}{2}},$
and then
${\rm log}^{\frac{1}{2}}(e+\|\nabla(\xi_{1}+\xi_{2})\|_{2m})\leq C~{}{\rm
log}^{\frac{1}{2}}(e+\|\rho u\|_{2m})\leq C_{m}~{}{\rm
log}^{\frac{1}{2}}(e+\|\nabla u\|_{2})\leq
C\Phi_{T}^{\frac{1+\beta\varepsilon}{2}}.$
Therefore, it holds that
$\|\xi_{1}+\xi_{2}\|_{\infty}\leq
C\big{(}\|\xi_{1}+\xi_{2}\|_{2m}+\|\nabla(\xi_{1}+\xi_{2})\|_{2}\big{)}{\rm
log}^{\frac{1}{2}}(e+\|\nabla(\xi_{1}+\xi_{2})\|_{2m})\leq
C\Phi_{T}^{1+\frac{\beta\varepsilon}{2}}.$ (3.67)
Finally, substituting (3.66) and (3.67) into (3.65), it holds that
$\Phi_{T}^{\beta}\leq C\Phi_{T}^{1+\frac{\beta}{4}+\varepsilon}+C.$
Therefore, if $\beta>\frac{4}{3}$ and choose $\varepsilon$ suitably small,
then
$\sup_{t\in[0,T]}\|\rho\|_{\infty}(t)\leq C_{1},$ (3.68)
for some positive constant $C_{1}$. Again by (3.65), (3.66), (3.67) and
(3.68), it holds that
$\sup_{t\in[0,T]}\|\ln\rho(t,\cdot)\|_{\infty}\leq C,$
which implies that there exists some positive constant $c_{1}$ such that
$\rho(t,x)\geq c_{1}>0,\qquad\forall(t,x)\in[0,T]\times\mathbb{R}^{2}.$
Thus the proof of Lemma 3.8 is completed. $\hfill\Box$
## 4 Proof of main results
In this section, we give a sketch of proof of our main results.
Proof of Theorem 1.1:
Under the assumptions of the theorem, the local existence of the classical
solution can be proved in a similar way as in [34, 42] and we omit it for
simplicity. In view of the lower and upper bound of the density obtained in
Section 3, the compressible Navier-Stokes equations (1.3) are a hyperbolic-
parabolic coupled system. One can get the higher order a priori estimates.
Using these a priori estimates, one can extend the local solution to the
global one in a standard way(see [25, 26] for more details). The proof of
Theorem 1.1 is complete.
Proof of Theorem 1.2:
To use Theorem 1.1, we first construct the approximation of the initial data
in (1.10) as follows. Since
$\displaystyle\lim_{|x|\rightarrow+\infty}\rho_{0}(x)=\bar{\rho}>0$, there
exists a large number $M>0$ such that if $|x|\geq M$,
$\rho_{0}(x)\geq\frac{\bar{\rho}}{2}.$ Then for any
$0<\delta<\frac{\bar{\rho}}{2}$, we define
$\rho_{0}^{\delta}(x)=\left\\{\begin{array}[]{ll}\displaystyle\rho_{0}(x)+\delta,~{}~{}{\rm
if}~{}|x|\leq M,\\\ \displaystyle\rho_{0}(x)+\delta s(x),~{}~{}{\rm
if}~{}M\leq|x|\leq M+1,\\\ \displaystyle\rho_{0}(x),~{}{\rm if}~{}|x|\geq
M+1,\end{array}\right.$ (4.1)
where $s(x)=s(|x|)$ is a smooth and decreasing function satisfying $s(x)\equiv
1$ if $|x|\leq M$ and $s(x)=0$ if $|x|\geq M+1$. Similarly, one can construct
the approximation of the initial pressure denoted by $P_{0}^{\delta}(x)$. Then
it follows that $(\rho_{0}^{\delta},P_{0}^{\delta})(x)$ are regular functions
satisfying $\rho_{0}^{\delta}(x)>\delta,P_{0}^{\delta}(x)>P(\delta)$ for any
$x\in\mathbb{R}^{2}$ and
$(\rho_{0}^{\delta},P_{0}^{\delta})(x)=(\rho_{0},P_{0})(x)$ if $|x|\geq M+1$.
Moreover, one has
$(\rho_{0}^{\delta}-\bar{\rho},P_{0}^{\delta}-P(\bar{\rho}))\rightarrow(\rho_{0}-\bar{\rho},P(\rho_{0})-P(\bar{\rho}))~{}~{}{\rm
in}~{}W^{2,q}(\mathbb{R}^{2})\times W^{2,q}(\mathbb{R}^{2}),$
and
$\Psi(\rho_{0}^{\delta},\bar{\rho})(1+|x|^{\alpha})\rightarrow\Psi(\rho_{0},\bar{\rho})(1+|x|^{\alpha})~{}~{}{\rm
in}~{}L^{1}(\mathbb{R}^{2}),$
as $\delta\rightarrow 0$. To construct the approximation of the initial
velocity, we define $u_{0}^{\delta}$ as
$u_{0}^{\delta}=\left\\{\begin{array}[]{ll}\displaystyle\tilde{u}_{0}^{\delta},\qquad|x|\leq
M+1,\\\ \displaystyle u_{0},\qquad|x|\geq M+1,\end{array}\right.$ (4.2)
where $\tilde{u}_{0}^{\delta}$ is the unique solution to the following
elliptic problem
$\left\\{\begin{array}[]{ll}\displaystyle\mathcal{L}_{\rho_{0}^{\delta}}\tilde{u}_{0}^{\delta}=\nabla
P_{0}^{\delta}+\sqrt{\rho_{0}}g,\qquad{\rm in}~{}~{}\Omega_{M}:=\\{x|\
|x|<M+1\\},\\\
\displaystyle\tilde{u}_{0}^{\delta}|_{|x|=M+1}=u_{0}.\end{array}\right.$ (4.3)
From (4.3), one has
$\mathcal{L}_{\rho_{0}}\tilde{u}_{0}^{\delta}=-\nabla\big{[}(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})){\rm
div}\tilde{u}_{0}^{\delta}\big{]}+\nabla
P_{0}^{\delta}+\sqrt{\rho_{0}}g,~{}~{}{\rm in}~{}\Omega_{M}.$ (4.4)
By the elliptic regularity, one has
$\begin{array}[]{ll}\displaystyle\|\tilde{u}_{0}^{\delta}\|_{H^{2}(\Omega_{M})}\\\
\displaystyle\leq
C\Big{[}\|\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})\|_{\infty}\|\nabla({\rm
div}\tilde{u}_{0}^{\delta})\|_{2}+\|\nabla(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0}))\|_{\infty}\|{\rm
div}u_{0}^{\delta}\|_{2}+\|\nabla
P_{0}^{\delta}\|_{2}+\|\sqrt{\rho_{0}}g\|_{2}\Big{]}\\\ \displaystyle\leq
C\Big{[}\delta\|\nabla^{2}\tilde{u}_{0}^{\delta}\|_{2}+\|\nabla
P_{0}^{\delta}\|_{2}+\|\sqrt{\rho_{0}}\|_{L^{\infty}(\mathbb{R}^{2})}\|g\|_{2}\Big{]}\\\
\leq
C\Big{[}\delta\|\nabla^{2}\tilde{u}_{0}^{\delta}\|_{2}+1\Big{]}.\end{array}$
(4.5)
where the generic positive constant $C$ is independent of $\delta>0.$
Therefore, it follows from (4.5) that
$\|\tilde{u}_{0}^{\delta}\|_{H^{2}(\Omega_{M})}\displaystyle\leq C$ (4.6)
where the positive constant $C$ is independent of $0<\delta\ll 1.$
From the compatibility conditions (1.11), (4.3) and (4.4), it holds that
$\left\\{\begin{array}[]{ll}\displaystyle\mathcal{L}_{\rho_{0}}(\tilde{u}_{0}^{\delta}-u_{0})=-\nabla\big{[}(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})){\rm
div}u_{0}^{\delta}\big{]}+\nabla(P_{0}^{\delta}-P_{0}):=\Theta^{\delta},\quad{\rm
in}~{}~{}\Omega_{M},\\\
\displaystyle\displaystyle(\tilde{u}_{0}^{\delta}-u_{0})|_{|x|=M+1}=0.\end{array}\right.$
(4.7)
It follows from (4.1), (4.6) and (4.7) that
$\tilde{u}_{0}^{\delta}-u_{0}\in H_{0}^{1}(\Omega_{M})\cap H^{2}(\Omega_{M}),$
(4.8)
and
$\begin{array}[]{ll}\|\tilde{u}_{0}^{\delta}-u_{0}\|_{H^{2}(\Omega_{M})}\leq
C\|\Theta^{\delta}\|_{2}\\\ \displaystyle\leq\displaystyle
C\Big{[}\|\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})\|_{L^{\infty}(\Omega_{M})}\|\nabla^{2}\tilde{u}_{0}^{\delta}\|_{2}+\|\nabla(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0}))\|_{L^{\infty}(\Omega_{M})}\|{\rm
div}\tilde{u}_{0}^{\delta}\|_{2}+\|\nabla(P_{0}^{\delta}-P_{0})\|_{2}\Big{]}\\\
\displaystyle\leq
C\Big{[}\|\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})\|_{L^{\infty}(\Omega_{M})}+\|\nabla(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0}))\|_{L^{\infty}(\Omega_{M})}+\|\nabla(P_{0}^{\delta}-P_{0})\|_{2}\Big{]}\\\
\displaystyle\leq C\delta~{}\rightarrow 0,\end{array}$ (4.9)
as $\delta\rightarrow 0$. It follows from (4.2), (4.8) and (4.9) that
$u_{0}^{\delta}\in H^{2}(\mathbb{R}^{2})$ and
$u_{0}^{\delta}\rightarrow u_{0},~{}~{}{\rm in}~{}H^{2}(\mathbb{R}^{2}),$
and
$\sqrt{\rho_{0}^{\delta}}u_{0}^{\delta}|x|^{\alpha}\rightarrow\sqrt{\rho}_{0}u_{0}|x|^{\alpha},~{}~{}{\rm
in}~{}L^{2}(\mathbb{R}^{2}),$
as $\delta\rightarrow 0.$ By Theorem 1.1, there exists a unique classical
solution $(\rho^{\delta},u^{\delta})$ to the compressible Navier-Stokes
equations (1.3) with the initial data
$(\rho_{0}^{\delta},P_{0}^{\delta},u_{0}^{\delta})$ such that
$c_{\delta}\leq\rho^{\delta}\leq C$ for some positive constants $c_{\delta}$
depending on $\delta$ and $C>0$. It should be noted that the estimates
obtained in Section 3 are independent of the lower bound of the initial
density $\rho_{0}(x)$ except the lower bound of the density $\rho(t,x)$ in
Lemma 3.8. Then we can pass the limit $\delta\rightarrow 0$ to get the
classical solution satisfying (1.9). It is referred to [26] for more details
and the proof of Theorem 1.2 is finished.
## References
* [1] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238(1-2), 211-223 (2003).
* [2] D. Bresch, B. Desjardins, Chi-Kun Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28(3-4), 843-868 (2003).
* [3] Q. L. Chen, C. X. Miao, Z. F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63(9), 1173-1224, 2010.
* [4] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compositio Mathematica, 53, 259-275 (1984).
* [5] F. Catrina, Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., LIV, 229-258 (2001).
* [6] Y. Cho, H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscript Math., 120, 91-129 (2006).
* [7] M. DelPino, J. Dolbeault, Bestconstants for Gagliardo CNirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81, 847-875 (2002).
* [8] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141, 579-614 (2000).
* [9] B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Comm. PDEs, 22, 977-1008 (1997).
* [10] S. J. Ding, H. Y. Wen, C. J. Zhu, Global classical large solutions to 1D compressible Navier CStokes equations with density-dependent viscosity and vacuum, J. Diff. Equa., 251, 1696-1725, (2011).
* [11] E. Feireisl, Dynamics of viscous compressible fluids, Oxford University Press, Oxford, 2004.
* [12] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.
* [13] Z. H. Guo, Q. S. Jiu, Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39(5), 1402-1427 (2008).
* [14] D. Hoff, Discontinuous solution of the Navier-Stokes equations for multi-dimensional heat-conducting fluids, Arch. Rat. Mech. Anal., 193, 303-354 (1997).
* [15] D. Hoff, Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech., 7(3), 315-338 (2005).
* [16] D. Hoff, J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216(2), 255-276 (2001).
* [17] X. D. Huang, J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, preprint, arxiv:1205.5342.
* [18] X. D. Huang, J. Li, Global well-posedness of classical solutions to the Cauchy problem of two-dimesional baratropic compressible Navier-Stokes system with vacuum and large initial data, preprint, arxiv:1207.3746.
* [19] X. D. Huang, J. Li, Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (4), 549-585, (2012).
* [20] N. Itaya, On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids, Kodia Math. Sem. Rep., 23, 60-120 (1971).
* [21] S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropy Navier-Stokes with density-dependent viscosity, Methods and Applications of Analysis, 12 (3), 239-252 (2005).
* [22] S. Jiang, P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215, 559-581 (2001).
* [23] Q. S. Jiu, Y. Wang, Z. P. Xin, Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity, Comm. Part. Diff. Equ., 36, 602-634 (2011).
* [24] Q. S. Jiu, Y. Wang, Z. P. Xin, Vaccum behaviors around the rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity, http://arxiv.org/abs/1109.0871.
* [25] Q. S. Jiu, Y. Wang, Z. P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, http://arxiv.org/abs/1202.1382.
* [26] Q. S. Jiu, Y. Wang, Z. P. Xin, Global well-posedness of the Cauchy problem of 2D compressible Navier-Stokes equations in weighted spaces, http://arxiv.org/abs/1207.5874.
* [27] Q. S. Jiu, Z. P. Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, Kinet. Relat. Models, 1 (2), 313-330 (2008).
* [28] J. I. Kanel, A model system of equations for the one-dimensional motion of a gas (in Russian), Diff. Uravn., 4, 721-734 (1968).
* [29] A. V. Kazhikhov, V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. 41, 273-282 (1977); translated from Prikl. Mat. Meh. 41 , 282-291 (1977).
* [30] H. L. Li, J. Li, Z. P. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281(2), 401-444 (2008).
* [31] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998.
* [32] T. P. Liu, J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math., 1(4), 345-359 (1980).
* [33] T. P. Liu, Z. P. Xin, T. Yang, Vacuum states of compressible flow, Discrete Contin. Dyn. Syst., 4, 1-32 (1998).
* [34] Z. Luo, Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum, to appear at Comm. Math. Sci., 2011.
* [35] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20, 67-104 (1980).
* [36] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equation, Comm. Partial Differential Equations, 32(3), 431-452 (2007).
* [37] J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France, 90, 487-497 (1962).
* [38] M. Perepelitsa, On the global existence of weak solutions for the Navier-Stokes equations of compressible fluid flows. SIAM J. Math. Anal., 38 (4), 1126 C1153 (2006).
* [39] O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity, J. Differ. Eqs., 245, 1762-1774 (2008).
* [40] A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kytt Univ. , 13, 193-253 (1971).
* [41] V. A. Vaigant, A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. (Russian) Sibirsk. Mat. Zh., 36 (1995), no. 6, 1283–1316, ii; translation in Siberian Math. J., 36, no. 6, 1108-1141 (1995).
* [42] V. A. Solonnikov, On solvability of an initial-boundary value problem for the equations motion of a viscous compressible fluid, LOMI, 56 (1976), 128-142.
* [43] Z. P. Xin, Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density, Comm. Pure Appl. Math., 51, 229-240 (1998).
* [44] Z. P. Xin, W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, preprint, arxiv:1204.3169.
* [45] T. Yang, Z. A. Yao, C. J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (5-6), 965-981 (2001).
* [46] T. Yang, C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2), 329-363 (2002).
* [47] T. Zhang, D. Y. Fang, Compressible flows with a density-dependent viscosity coefficient. SIAM J. Math. Anal., 41, no. 6, 2453-2488 (2009/10).
|
arxiv-papers
| 2012-09-02T08:49:23 |
2024-09-04T02:49:34.686686
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quansen Jiu, Yi Wang and Zhouping Xin",
"submitter": "Yi Wang",
"url": "https://arxiv.org/abs/1209.0157"
}
|
1209.0228
|
# Linear-response theory of the longitudinal spin Seebeck effect
Hiroto Adachi [email protected] Sadamichi Maekawa Advanced Science
Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
###### Abstract
We theoretically investigate the longitudinal spin Seebeck effect, in which
the spin current is injected from a ferromagnet into an attached nonmagnetic
metal in a direction parallel to the temperature gradient. Using the fact that
the phonon heat current flows intensely into the attached nonmagnetic metal in
this particular configuration, we show that the sign of the spin injection
signal in the longitudinal spin Seebeck effect can be opposite to that in the
conventional transverse spin Seebeck effect when the electron-phonon
interaction in the nonmagnetic metal is sufficiently large. Our linear-
response approach can explain the sign reversal of the spin injection signal
recently observed in the longitudinal spin Seebeck effect.
Spin Seebeck effect, Spin caloritronics
###### pacs:
85.75.-d, 72.25.Mk, 75.30.Ds
## I INTRODUCTION
Because of the desire to deal with heating problems in modern spintronic
devices, there has been an increasing interest in investigating thermal
effects in spintronics. A new subfield “spin caloritronics” Bauer12 aims to
understand the basic physics behind the interplay of spin and heat. One of the
central issues in spin caloritronics is the newly discovered thermo-spin
phenomenon termed spin Seebeck effect Uchida08 , which enables the thermal
injection of spin currents from a ferromagnet into attached nonmagnetic metals
over a macroscopic scale of several millimeters. The spin Seebeck effect is
now established as a universal aspect of ferromagnets because this phenomenon
is observed in various materials ranging from the metallic ferromagnets
Ni81Fe19 Uchida08 and Co2MnSi Bosu11 , to the semiconducting ferromagnet
(Ga,Mn)As Jaworski10 , to the insulating magnets LaY2Fe5O12 Uchida10a .
It is important to note that the above experiments Uchida08 ; Bosu11 ;
Jaworski10 ; Uchida10a were performed in a configuration of the transverse
spin Seebeck effect, in which the direction of the thermal spin injection into
the attached nonmagnetic metal is perpendicular to the temperature gradient
[Fig. 1 (a)]. Recently, another type of spin Seebeck effect called the
longitudinal spin Seebeck effect Uchida10b ; Uchida10c is reported, in which
the direction of the thermal spin injection into the nonmagnetic metal is
parallel to the temperature gradient [Fig. 1 (b)]. Whereas the longitudinal
spin Seebeck effect is well defined only for the use of an insulating
ferromagnet due to the parasitic contribution from the anomalous Nernst effect
Huang11 ; Weiler12 , it has several attractive features: (i) it is substrate
free, (ii) the configuration is much simpler than that of the transverse spin
Seebeck effect, and (iii) it can be of wide application because it allows the
use of bulk samples.
Another pronounced feature of the longitudinal spin Seebeck effect is that the
sign of the spin injection signal is opposite to that in the transverse spin
Seebeck effect Uchida10b ; Uchida10c . Physically, the longitudinal spin
Seebeck effect is distinguished from the transverse spin Seebeck effect by the
fact that the attached nonmagnetic metal is in contact with the heat bath in
the longitudinal setup, while the attached nonmagnetic metal is out of contact
with the heat bath in the transverse setup. This brings about a clear
difference that the heat current intensely flows into the attached nonmagnetic
metal in the case of the longitudinal spin Seebeck effect, whereas it does not
in the case of the transverse spin Seebeck effect. It is obvious that theory
of magnon-driven spin Seebeck effect Xiao10 fails to explain the situation in
question.
Figure 1: (Color online) Schematic view of the experimental setup for (a) the
transverse spin Seebeck effect and (b) the longitudinal spin Seebeck effect.
In this paper, by employing linear-response theory of the spin Seebeck effect
Adachi11 and using the importance of the phonon-drag process in the spin
Seebeck effect Adachi10 , we show that the sign of the spin injection signal
in the longitudinal spin Seebeck effect can be opposite to that in the
conventional transverse spin Seebeck effect when the electron-phonon
interaction in the attached nonmagnetic metal is sufficiently large. The key
in our discussion is the aforementioned difference in the position of the
attached nonmagnetic metal between the longitudinal setup and the transverse
setup.
## II Phenomenology of the longitudinal spin Seebeck effect
Let us begin with the phenomenology of the longitudinal spin Seebeck effect.
In Fig. 1, a hybrid structure of a ferromagnet ($F$) and a nonmagnetic metal
($N$) is placed under a temperature gradient. The central quantity that
characterizes the spin Seebeck effect is the spin current $I_{s}$ injected
into $N$. As explained in detail in Ref. Adachi12 , the spin Seebeck effect is
a thermal spin injection by localized spins, and the injected spin current has
two contributions,
$I_{s}=I_{s}^{\rm pump}-I_{s}^{\rm back},$ (1)
where $I_{s}^{\rm pump}$ (the so-called pumping component) represents the spin
current pumped into $N$ by the thermal fluctuations of localized spins in $F$
, while $I_{s}^{\rm back}$ (the so-called backflow component) represents the
spin current coming back into $F$ by the thermal fluctuations of the spin
accumulations in $N$. We now focus on the spin current injected into $N$ which
is located close to the cold reservoir.
In the case of the conventional transverse spin Seebeck effect, the magnitude
of the pumping component $I_{s}^{\rm pump}$ is greater than that of the
backflow component $I_{s}^{\rm pump}$ [Fig. 1(a)]. In contrast, the magnitude
of $I_{s}^{\rm pump}$ is less than that of the backflow component $I_{s}^{\rm
pump}$ in the case of the longitudinal spin Seebeck effect [Fig. 1(b)]. Note
that, because magnons carry minus spin 1, both the pumping and backflow
components have a negative sign.
This difference can be explained phenomenologically on the basis of the
following conditions: (i) most of the heat current in the $F$/$N$ hybrid
system at room temperature is carried by phonons (see Ref. Slack71 in the
case of yttrium iron garnet), and (ii) the interaction between the phonons and
the spin accumulation in $N$ is much stronger than the magnon-phonon
interaction in $F$.
First, recall that the pumping and backflow components can be expressed as
follows Adachi12 :
$\displaystyle I_{s}^{\rm pump}$ $\displaystyle=$ $\displaystyle-
G_{s}k_{B}T_{F}^{*},$ (2) $\displaystyle I_{s}^{\rm back}$ $\displaystyle=$
$\displaystyle-G_{s}k_{B}T_{N}^{*},$ (3)
where $T_{F}^{*}$ and $T_{N}^{*}$ are the effective temperature of the magnon
in $F$ and the spin accumulation in $N$. Here, $G_{s}=J^{2}_{\rm
sd}\chi_{N}\tau_{\rm sf}/\hbar$ with $J_{\rm sd}$, $\chi_{N}$, and $\tau_{\rm
sf}$ being the $s$-$d$ interaction at the interface, the paramagnetic
susceptibility in $N$, and the spin-flip relaxation time in $N$, respectively.
The negative sign before $G_{s}$ arises from the fact that the magnon carries
spin $-1$. In the longitudinal spin Seebeck experiment, the nonmagnetic metal
$N$ is in direct contact with the heat bath, and thereby is exposed to the
flow of the phonon heat current due to condition (i). Then, because of
condition (ii), spin accumulation in $N$ is heated up faster than the magnons
in the ferromagnet $F$, and the resultant effective temperature of the spin
accumulation in $N$ increases above that of the magnons in $F$. In the
conventional transverse spin Seebeck setup, by contrast, the nonmagnetic metal
$N$ is out of contact with the heat bath and the phonon heat current does not
flow through the nonmagnetic metal $N$, while the ferromagnet $F$ is in
contact with the heat bath, resulting in an increase in the effective magnon
temperature in $F$. Therefore, in this case, the effective temperature of the
spin accumulation in $N$ is lower than that of the magnons in $F$. This
difference can explain the sign reversal of the spin Seebeck effect signal
between the longitudinal setup and the conventional transverse setup.
## III Linear-response Formulation
In this section we review the linear-response formalism of the spin Seebeck
effect developed in Ref. Adachi11 . In the next section, this formalism is
employed to evaluate the longitudinal spin Seebeck effect. We use a model
shown in Fig. 2, in which the localized spins in $F$ are interacting with the
spin accumulation in $N$ through the $s$-$d$ exchange interaction $J_{\rm sd}$
at the interface. In our approach, the spin accumulation is modeled as a
nonequilibrium itinerant spin density ${\bm{s}}$.
As in Ref. Adachi11 , the spin current $I_{s}$ injected into the nonmagnetic
metal $N$ is calculated as $I_{s}(t)=-\sum_{{\bm{q}},{\bm{k}}}\frac{4{\cal
J}^{{\bm{k}}+{\bm{q}}}_{\rm sd}\sqrt{S_{0}}}{\sqrt{2N_{F}N_{N}}\hbar}{\rm
Re}C^{<}_{{\bm{k}},{\bm{q}}}(t,t)$, where $N_{F}$ ($N_{N}$) is the number of
lattice sites in $F$ ($N$), $S_{0}$ is the size of the localized spins in $F$,
and ${\cal J}_{\rm sd}^{{\bm{k}}+{\bm{q}}}$ is the Fourier transform of the
$s$-$d$ interaction at the $F/N$ interface. Here,
$C^{<}_{{\bm{k}},{\bm{q}}}(t,t^{\prime})=-{\rm i}\langle
a^{+}_{\bm{q}}(t^{\prime})s^{-}_{\bm{k}}(t)\rangle$ measures the correlation
between the magnon operator $a_{\bm{q}}^{+}$ in $F$ and the itinerant spin-
density operator $s^{-}_{\bm{k}}=(s^{x}_{\bm{k}}-{\rm i}s^{y}_{\bm{k}})/2$ in
$N$. Note that the time dependence of $I_{s}(t)$ vanishes in the steady state
and it is hereafter discarded. Introducing the frequency representation
$C^{<}_{{\bm{k}},{\bm{q}}}(t-t^{\prime})=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}{C}^{<}_{{\bm{q}},{\bm{k}}}(\omega)e^{-{\rm
i}\omega(t-t^{\prime})}$ and adopting the representation Larkin75
$\check{C}=\left({{C^{R},C^{K}}\atop{0\;\;\;,C^{A}}}\right)$ as well as using
the relation $C^{<}=\frac{1}{2}[C^{K}-C^{R}+C^{A}]$, we obtain
$\displaystyle I_{s}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{q}},{\bm{k}}}\frac{-2{\cal
J}^{{\bm{k}}-{\bm{q}}}_{\rm
sd}\sqrt{S_{0}}}{\sqrt{2N_{F}N_{N}}\hbar}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}{\rm
Re}C^{K}_{{\bm{k}},{\bm{q}}}(\omega)$ (4)
for the spin current $I_{s}$ in the steady state com1 .
Up to the lowest order in the $s$-$d$ interaction $J_{\rm sd}$, the interface
correlation function $\check{C}$ appearing in Eq. (4) is generally expressed
as
$\displaystyle\check{C}_{{\bm{k}},{\bm{q}}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{{\cal J}^{{\bm{k}}-{\bm{q}}}_{\rm
sd}\sqrt{S_{0}}}{\sqrt{N_{N}N_{F}}\hbar}\check{\underline{\chi}}_{\bm{k}}(\omega)\check{\underline{X}}_{\bm{q}}(\omega),$
(5)
where
$\check{\underline{X}}_{\bm{q}}(\omega)=\check{X}_{\bm{q}}(\omega)+\delta\check{X}_{\bm{q}}(\omega)$
is the renormalized magnon propagator with the bare component
$\check{X}_{\bm{q}}(\omega)$, and
$\check{\underline{\chi}}_{{\bm{k}}}(\omega)=\check{\chi}_{{\bm{k}}}(\omega)+\delta\check{\chi}_{{\bm{k}}}(\omega)$
is the renormalized spin-density propagator with the bare component
$\check{\chi}_{{\bm{k}}}(\omega)$. The bare magnon propagator satisfies the
equilibrium condition:
$\displaystyle X^{A}_{\bm{q}}(\omega)=[X^{R}_{\bm{q}}(\omega)]^{*},$
$\displaystyle X^{K}_{\bm{q}}(\omega)=2{\rm i}\,{\rm
Im}X^{R}_{\bm{q}}(\omega)\coth(\tfrac{\hbar\omega}{2k_{\rm B}T}),$
where the retarded component is given by
$X^{R}_{\bm{q}}(\omega)=(\omega-\widetilde{\omega}_{\bm{q}}+{\rm
i}\alpha\omega)^{-1}$ with $\widetilde{\omega}_{\bm{q}}=\gamma
H_{0}+\omega_{\bm{q}}$ being the magnon frequency for uniform mode $\gamma
H_{0}$ and exchange mode $\omega_{\bm{q}}=D_{\rm ex}q^{2}/\hbar$. Likewise,
the bare spin-density propagator satisfies the local equilibrium condition:
$\displaystyle\chi^{A}_{\bm{k}}(\omega)=[\chi^{R}_{\bm{k}}(\omega)]^{*},$
$\displaystyle\chi^{K}_{\bm{k}}(\omega)=2{\rm i}\,{\rm
Im}\chi^{R}_{\bm{k}}(\omega)\coth(\tfrac{\hbar\omega}{2k_{\rm B}T}),$
where the retarded component of $\check{\chi}_{{\bm{k}}}(\omega)$ is given by
$\chi^{R}_{\bm{k}}(\omega)=\chi_{N}/(1+\lambda_{\rm sf}^{2}k^{2}-{\rm
i}\omega\tau_{\rm sf})$ with $\lambda_{\rm sf}$ being the spin diffusion
length.
Figure 2: (Color online) Diagrammatic representation of the spin Seebeck
effect in the longitudinal configuration. The double solid line (the dashed
line) represents the propagator of the itinerant spin density (the phonon).
The solid circle (solid triangle) denotes the interaction vertex
$\Gamma_{{{\bm{K}}},{\bm{q}}}$ ($\Upsilon_{{{\bm{K}}},{\bm{k}}}$) between the
magnon and the phonon (the itinerant spin density and the phonon). Here,
$T_{1}<T_{2}<T_{3}$.
## IV Calculation of the longitudinal spin Seebeck effect
In this section we present a linear-response calculation of the longitudinal
spin Seebeck effect and justify the phenomenological picture presented in Sec.
II. The spin current $I_{s}$ injected into $N$ due to the longitudinal spin
Seebeck effect is composed of three terms,
$I_{s}=I_{s}(a)+I_{s}(b)+I_{s}(c),$ (8)
where $I_{s}(a)$, $I_{s}(b)$, and $I_{s}(c)$ correspond to diagram (a), (b),
and (c) in Fig. 2. Below we show that each term has the following sign:
$I_{s}(a)<0,\;\;I_{s}(b)<0,\;\;I_{s}(c)>0.$ (9)
First, let us consider diagram (a). This can be calculated by setting
$\check{\underline{X}}_{\bm{q}}(\omega)\to\check{X}_{\bm{q}}(\omega)$ and
$\check{\underline{\chi}}_{{\bm{k}}}(\omega)\to\check{\chi}_{{\bm{k}}}(\omega)$
in Eq. (5), which was already done in Ref. Adachi11 with the injected spin
current given by (see Eq. (12) therein)
$\displaystyle I_{s}(a)$ $\displaystyle=$ $\displaystyle\frac{4N_{\rm
int}{J}_{\rm
sd}^{2}S_{0}^{2}}{\sqrt{2}\hbar^{2}N_{N}N_{F}}\sum_{{\bm{q}},{\bm{k}}}\int_{\omega}{\rm
Im}\chi_{{\bm{k}}}^{R}(\omega){\rm Im}X_{{\bm{q}}}^{R}(\omega)$ (10)
$\displaystyle\quad\times\left[\coth(\tfrac{\hbar\omega}{2k_{\rm
B}T_{2}})-\coth(\tfrac{\hbar\omega}{2k_{\rm B}T_{1}})\right],$
where we have introduced the shorthand notation
$\int_{\omega}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}$, and $N_{\rm int}$
is the number of localized spins at the $N$/$F$ interface. Note that
$I_{s}(a)$ has a negative value due to $T_{2}>T_{1}$.
Next, let us consider diagram (b). In this process, the localized spins in $F$
is excited by the nonequilibrium phonon driven by the temperature gradient in
$F$, hence this corresponds to the phonon-drag process Adachi10 . Evaluation
of the diagram was already given in Ref. Adachi10 , although the calculation
is lengthy and tedious (see the supplemental material therein). In short, this
term can be calculated by setting in Eq. (5)
$\check{\underline{X}}_{\bm{q}}(\omega)\to\delta\check{X}^{n\mathchar 45\relax
eq}_{\bm{q}}(\omega)=({0,\atop 0,}{{\delta X_{\bm{q}}^{n\mathchar 45\relax
eq,K}}\atop 0})$ with its Keldysh component given by
$\displaystyle\delta{X}^{n\mathchar 45\relax eq,K}_{\bm{q}}(\omega)$
$\displaystyle=$
$\displaystyle-2\sum_{{\bm{K}}}\frac{\Gamma_{{{\bm{K}}},{\bm{q}}}^{2}}{2N_{F}}\int_{\nu}\delta
D^{n\mathchar 45\relax eq,K}_{\bm{K}}(\nu){\rm
Im}X^{R}_{{\bm{q}}_{-}}(\omega_{-})$ $\displaystyle\times$
$\displaystyle|X^{R}_{\bm{q}}(\omega)|^{2}\left[\coth(\tfrac{\hbar\omega_{-}}{2k_{\rm
B}T_{2}})]-\coth(\tfrac{\hbar\omega}{2k_{\rm B}T_{2}})\right],$
where we have introduced shorthand notations $\omega_{-}=\omega-\nu$,
${\bm{q}}_{-}={\bm{q}}-{\bm{K}}$, and
$\int_{\nu}=\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}$, and
$\Gamma_{{{\bm{K}}},{\bm{q}}}={g}_{\rm
m-p}\sqrt{\frac{\hbar\nu_{\bm{K}}}{2M_{\rm ion}v_{\rm ph}^{2}}}$ (12)
is the magnon-phonon interaction vertex. Here, $\nu_{\bm{K}}$, $v_{\rm ph}$
and $M_{\rm ion}$ are the phonon frequency, phonon velocity and the ion mass,
respectively, and the strength of the magnon-phonon coupling is given by
$g_{\rm m-p}=|a_{S}{\bm{\nabla}}J_{\rm ex}|({\omega_{\bm{q}}}/{J_{\rm ex}})$
with the exchange interaction $J_{\rm ex}$. In Eq. (LABEL:Eq:dX_noneq02),
$\displaystyle\delta{D}_{\bm{K}}^{n\mathchar 45\relax eq,K}(\nu)$
$\displaystyle=$ $\displaystyle 2{\rm
i}\sum_{{\bm{K}}^{\prime}}\frac{|\Omega_{\rm
ph}^{{\bm{K}}+{\bm{K}}^{\prime}}|^{2}}{N_{F}^{2}}{\rm
Im}{D}^{R}_{{\bm{K}}^{\prime}}(\nu)|{D}^{R}_{\bm{K}}(\nu)|^{2}$ (13)
$\displaystyle\times$ $\displaystyle\big{[}\coth(\tfrac{\hbar\nu}{2k_{\rm
B}T_{3}})-\coth(\tfrac{\hbar\nu}{2k_{\rm B}T_{2}})\big{]}$
is the Keldysh component of the nonequilibrium phonon propagator. Here
${\Omega}_{\rm ph}^{{\bm{K}}+{\bm{K}}^{\prime}}$ is the Fourier transform of
${\Omega}_{\rm ph}({\bm{r}})=\Omega_{0}\sum_{{\bm{r}}_{0}\in F/F\mathchar
45\relax{\rm interface}}a_{S}^{3}\delta({\bm{r}}-{\bm{r}}_{0})$ with
$\Omega_{0}=\sqrt{2K_{\rm ph}/M_{\rm ion}}$, $K_{\rm ph}$ is the elastic
constant, and $a_{S}^{3}$ is the cell volume of the ferromagnet. Putting these
expressions into Eq. (5) and after some algebra, we finally obtain
$\displaystyle I_{s}(b)$ $\displaystyle=$
$\displaystyle\frac{-L}{N_{N}N_{F}^{3}}\sum_{{\bm{k}},{\bm{q}},{\bm{K}},{\bm{K}}^{\prime}}(\Gamma_{{{\bm{K}}},{\bm{q}}})^{2}\int_{\nu}A_{{\bm{k}},{\bm{q}}}(\nu)|{D}^{R}_{\bm{K}}(\nu)|^{2}$
(14) $\displaystyle\times{\rm
Im}{D}^{R}_{{\bm{K}}^{\prime}}(\nu)\big{[}\coth(\tfrac{\hbar\nu}{2k_{\rm
B}T_{3}})-\coth(\tfrac{\hbar\nu}{2k_{\rm B}T_{2}})\big{]},$
where $D^{R}_{\bm{K}}(\nu)=(\nu-\nu_{\bm{K}}+{\rm i}/\tau_{\rm
ph})^{-1}-(\nu+\nu_{\bm{K}}+{\rm i}/\tau_{\rm ph})^{-1}$ is the retarded
component of the phonon propagator with the phonon lifetime $\tau_{\rm ph}$,
$L=\sqrt{2}(J^{2}_{sd}S_{0})\Omega_{0}^{2}N_{\rm int}N^{\prime}_{\rm
int}/N_{F}$ with $N^{\prime}_{\rm int}$ being the number of lattice sites at
the $F$/$F$ interface, and $A_{{\bm{k}},{\bm{q}}}(\nu)$ is defined by
$\displaystyle A_{{\bm{k}},{\bm{q}}}(\nu)$ $\displaystyle=$
$\displaystyle\int_{\omega}{\rm Im}\chi^{R}_{\bm{k}}(\omega){\rm
Im}X^{R}_{{\bm{q}}_{-}}(\omega_{-})\hskip 85.35826pt$ (15)
$\displaystyle\times$
$\displaystyle|X^{R}_{\bm{q}}(\omega)|^{2}[\coth(\tfrac{\hbar\omega_{-}}{2k_{\rm
B}T_{2}})-\coth(\tfrac{\hbar\omega}{2k_{\rm B}T_{2}})].$
Note that only the even component of $A_{{\bm{k}},{\bm{q}}}(\nu)$ as a
function of $\nu$ gives a non-vanishing contribution to Eq. (14). Because the
even component of $A_{{\bm{k}},{\bm{q}}}(\nu)$ is negative definite as well as
${\rm Im}{D}_{\bm{K}}^{R}(\nu)[\coth(\tfrac{\hbar\nu}{2k_{\rm
B}T_{3}})-\coth(\tfrac{\hbar\nu}{2k_{\rm B}T_{2}})]$ in Eq. (14), $I_{s}(b)$
has a negative value.
Finally, let us consider diagram (c). Repeating essentially the same procedure
in evaluating diagram (b), we obtain
$\displaystyle I_{s}(c)$ $\displaystyle=$
$\displaystyle\frac{L^{\prime}}{N_{N}N_{F}^{3}}\sum_{{\bm{k}},{\bm{q}},{\bm{K}},{\bm{K}}^{\prime}}(\Upsilon_{{{\bm{K}}},{\bm{k}}})^{2}\int_{\nu}B_{{\bm{k}},{\bm{q}}}(\nu)|\widetilde{D}^{R}_{\bm{K}}(\nu)|^{2}$
(16) $\displaystyle\times{\rm
Im}{D}^{R}_{{\bm{K}}^{\prime}}(\nu)\big{[}\coth(\tfrac{\hbar\nu}{2k_{\rm
B}T_{2}})-\coth(\tfrac{\hbar\nu}{2k_{\rm B}T_{1}})\big{]},$
where $L^{\prime}=\sqrt{2}(J^{2}_{sd}S_{0})\Omega_{0}^{2}N^{2}_{\rm
int}/N_{F}$, $\widetilde{D}^{R}_{\bm{K}}(\nu)$ denotes the phonon propagator
in $N$. In the above equation, the coupling between the itinerant spin density
and the phonon in $N$ is given by
$\Upsilon_{{\bm{K}},{\bm{k}}}\approx g_{\rm
s-p}\sqrt{\frac{\hbar\nu_{\bm{K}}}{2M_{\rm ion}v_{\rm ph}^{2}}},$ (17)
where $g_{\rm s-p}\approx|a{\bm{\nabla}}t_{\rm hop}|U^{2}N^{\prime}(0)$ with
$a$ and $t_{\rm hop}$ being the lattice spacing and the hopping integral of
the nonmagnetic metal $N$. In Eq. (16), $B_{{\bm{k}},{\bm{q}}}(\nu)$ is
defined by
$\displaystyle B_{{\bm{k}},{\bm{q}}}(\nu)$ $\displaystyle=$
$\displaystyle\int_{\omega}{\rm Im}\chi^{R}_{\bm{k}}(\omega){\rm
Im}X^{R}_{{\bm{q}}_{-}}(\omega_{-})|\chi^{R}_{\bm{q}}(\omega)|^{2}$ (18)
$\displaystyle\times[\coth(\tfrac{\hbar(\omega_{-}}{2k_{\rm
B}T_{1}})-\coth(\tfrac{\hbar\omega}{2k_{\rm B}T_{1}})].$
Note that as in Eq. (14), only the even-in-$\nu$ component of
$B_{{\bm{k}},{\bm{q}}}(\nu)$ gives non-vanishing contribution to Eq. (16).
Then, because the even-in-$\nu$ component of $B_{{\bm{k}},{\bm{q}}}(\nu)$ is
negative definite, $I_{s}(c)$ has a positive value.
## V discussion
In the previous section, we proved that $I_{s}(a)$ and $I_{s}(b)$ have the
same sign, whereas $I_{s}(c)$ have the opposite sign [Eq. (9)]. Then, if
$I_{s}(c)$ is dominant in Eq. (8), it means that the sign of the longitudinal
spin Seebeck effect can be opposite to that of the transverse spin Seebeck
effect, since the sign of $I_{s}(a)$ and $I_{s}(b)$ is the same as that of the
transverse spin Seebeck effect (see Refs. Adachi11 and Adachi10 ). Because
$I_{s}(a)$ and $I_{s}(b)$ are considered to have the same magnitude at room
temperature (see Fig. 3 in Ref. Adachi10 ), we here compare the magnitude of
$I_{s}(b)$ and $I_{s}(c)$.
The key quantities determining the magnitude of $I_{s}(b)$ and $I_{s}(c)$ are
the interaction vertex $\Gamma_{{{\bm{K}}},{\bm{q}}}$ between magnons and
phonons [solid circle in Fig. 2 and Eq. (12)] and the interaction vertex
$\Upsilon_{{{\bm{K}}},{\bm{k}}}$ between spin accumulation and phonons [solid
triangle in Fig. 2 and Eq. (17)]. The magnitude of these couplings is roughly
given by $g_{\rm m-p}\approx|a_{S}{\bm{\nabla}}J_{\rm
ex}|(\omega_{\bm{q}}/J_{\rm ex})$ and $g_{\rm s-p}\approx|a{\bm{\nabla}}t_{\rm
hop}|U^{2}N^{\prime}(0)$, where $a$, $t_{\rm hop}$, $U$, $N^{\prime}(0)$ are
the lattice spacing, the hopping integral, the strength of the Coulomb
repulsion, and the strength of the particle-hole asymmetry in the nonmagnetic
metal $N$. For materials with a relatively large Coulomb repulsion $U$
Gunnarsson76 and moderate strength of particle-hole asymmetry $N^{\prime}(0)$
Moore73 such as Pt, we expect a situation $g_{\rm s-p}>g_{\rm m-p}$, which
then explains the sign reversal of the spin injection signal in the
longitudinal spin Seebeck effect due to Eqs. (8) and (9). From these
considerations, we conclude that this happens in the longitudinal spin Seebeck
effect reported in Refs. Uchida10b and Uchida10c .
## VI Conclusion
In this paper we have developed linear-response theory of the longitudinal
spin Seebeck effect. We have shown that the sign of the spin injection signal
in the longitudinal spin Seebeck effect can be opposite to that in the
conventional transverse spin Seebeck effect when the interaction between the
spin accumulation and the phonon in the attached nonmagnetic metal is
sufficiently stronger than the interaction between the magnon and the phonon
in the ferromagnet. The linear-response approach presented in this paper can
explain the sign reversal of the spin injection signal recently observed in
the longitudinal spin Seebeck effect Uchida10b ; Uchida10c .
Figure 3: Diagram corresponding to the interaction vertex
$\Upsilon_{{{\bm{K}}},{\bm{k}}}$ between the itinerant spin density and the
phonon. The solid line, the wavy line, the dashed line and the triple dashed
line represent the electron propagator, the Coulomb repulsion, the phonon
propagator, and the diffuson propagator, respectively. “R” or “A” means the
retarded or advanced component of the electron Green’s function.
###### Acknowledgements.
The authors would like to thank K. Uchida and E. Saitoh for helpful
discussions, and gratefully acknowledge support by a Grant-in-Aid for
Scientific Research from MEXT, Japan.
## Appendix A Calculation of the vertex $\Upsilon$
In this Appendix, we evaluate the interaction vertex
$\Upsilon_{{\bm{K}},{\bm{k}}}$ (solid triangle in Fig. 2) between the
itinerant spin density and the phonon in the nonmagnetic metal $N$. We assume
that the nonmagnetic metal $N$ has a moderately large Stoner enhancement
factor, and for conduction electrons in $N$ we use a model described by Ref.
Kawabata74 , and assume an elastic impurity scattering as well. The
interaction vertex before integrating out the fermionic degrees of freedom is
shown in Fig. 3. The building block of this diagram is given by a triangle
$\displaystyle{\cal T}$ $\displaystyle=$
$\displaystyle\int_{\epsilon}\left[\tanh(\tfrac{\hbar(\epsilon-\omega)}{2k_{\rm
B}T})-\tanh(\tfrac{\hbar\epsilon}{2k_{\rm B}T})\right]$ (19)
$\displaystyle\hskip 28.45274pt\times\int_{{\bm{p}}}{\rm
Im}G^{R}_{{\bm{p}}-{\bm{k}}}(\epsilon-\omega)G^{R}_{\bm{p}}(\epsilon)G^{A}_{\bm{p}}(\epsilon),$
where $G^{R/A}_{\bm{p}}(\epsilon)=(\epsilon-\epsilon_{\bm{p}}\pm{\rm
i}/\tau)^{-1}$ is the electron Green’s function with the electron’s lifetime
$\tau$. This diagram can be evaluated to be ${\cal T}\approx
N^{\prime}(0)\omega\tau$ as was done in Ref. Kamenev95 . After the inclusion
of a diffuson vertex correction (triple dashed ladder in Fig. 3) which is
important in a realistic diffusive situation, we obtain
${\cal T}\approx
N^{\prime}(0)\frac{(\omega\tau)^{2}}{(Dk^{2}\tau)^{2}+(\omega\tau)^{2}},$ (20)
where $D$ is the diffusion constant. The dominant contribution comes from the
dynamical region $\omega\gg Dk^{2}$ and in this case we approximately have
${\cal T}\approx N^{\prime}(0)$. By attaching two Coulomb repulsion $U$ and
one electron-phonon interaction $\approx|a{\bm{\nabla}}t_{\rm
hop}|\sqrt{\frac{\hbar\nu_{\bm{K}}}{2M_{\rm ion}v_{\rm ph}^{2}}}$ Walker01
coming from each vertex, we finally obtain Eq. (17).
## References
* (1) G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Materials 11, 391 (2012).
* (2) K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh Nature 455, 778 (2008).
* (3) S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota, E. Saitoh, and K. Takanashi, Phys. Rev. B 83, 224401 (2011).
* (4) C. M. Jaworski, J. Yang S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Mater. 9, 898 (2010).
* (5) K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Mater. 9, 894 (2010).
* (6) K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett 97, 172505 (2010).
* (7) K. Uchida, T. Nonaka, T. Ota, H. Nakayama, and E. Saitoh, Appl. Phys. Lett 97, 262504 (2010).
* (8) S. Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L. Chien, Phys. Rev. Lett. 107, 216604 (2011).
* (9) M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012).
* (10) J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010).
* (11) H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011).
* (12) H. Adachi, K. Uchida, E. Saitoh, J. Ohe, S. Takahashi, and S. Maekawa, Appl. Phys. Lett. 97, 252506 (2010).
* (13) H. Adachi and S. Maekawa, to appear in Handbook of Spintronics (Canopus Academic Publishing).
* (14) G. A. Slack and D. W. Oliver DW, Phys. Rev. B 4, 592 (1971).
* (15) A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 68, 1915 (1975) [Sov. Phys. JETP 41, 960 (1975)].
* (16) Definition of the magnon propagator in this paper differs from that in Ref. Adachi11 by a factor $S_{0}$.
* (17) A. Kawabata, J. Phys. F: Metal Phys. 4, 1477 (1974).
* (18) Platinum is known to be a metal with moderately large Stoner enhancement. See, e.g., O. Gunnarsson, J. Phys. F: Metal Phys. 6, 587 (1976).
* (19) The Seebeck coefficent can be a measure of the particle-hole assymmetry. See J. P. Moore and R. S. Graves, J. Appl. Phys. 44, 1174.
* (20) A. Kamenev and Y. Oreg, Phys. Rev. B 52, 7516 (1995).
* (21) M. B. Walker, M. F. Smith, and K. V. Samokhin, Phys. Rev. B 65, 014517 (2001).
|
arxiv-papers
| 2012-09-03T01:05:50 |
2024-09-04T02:49:34.697038
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hiroto Adachi and Sadamichi Maekawa",
"submitter": "Hiroto Adachi",
"url": "https://arxiv.org/abs/1209.0228"
}
|
1209.0278
|
# Dynamics aspect of subbarrier fusion reaction in light heavy ion systems
M. Huang1 F. Zhou2 R. Wada1,∗ X. Liu1,3 W. Lin1,3 M. Zhao1,3 J. Wang1 Z. Chen1
C. Ma4 Y. Yang1,3 Q. Wang1 J. Ma1 J. Han1 P. Ma1 S. Jin1,3 Z. Bai1,3 Q. Hu1,3
L. Jin1,3 J. Chen1,3 Y. Su1,3 and Y. Li1,3 1 Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou, 730000,China. 2 School of Nuclear
Science and Technology, University of Science and Technology of China, Hefei,
Anhui 230026, China. 3 Graduate University of Chinese Academy of Sciences,
Beijing, 100049, China. 4 Department of Physics, Henan Normal University,
Xinxiang, 453007, China. M. Huang, [email protected]*Corresponding
author: R. Wada, [email protected]
###### Abstract
Subbarrier fusion of the ${}^{7}Li+^{12}$C reaction is studied using an
antisymmetrized molecular dynamics model (AMD) with an after burner, GEMINI.
In AMD, ${}^{7}Li$ shows an $\alpha+t$ structure at its ground state and it is
significantly deformed. Simulations are made near the Coulomb barrier
energies, i.e., E${}_{cm}=3-8MeV$. The total fusion cross section of the AMD +
GEMINI calculations as a function of incident energy is compared to the
experimental results and both are in good agreement at E${}_{cm}>3MeV$. The
cross section for the different residue channels of the AMD + GEMINI at
$E_{cm}=5MeV$ is also compared to the experimental results.
## 1 Introduction
Availability of radioactive beam facilities has stimulated theoretical and
experimental interest in the structure of nuclei far from the stability line.
Nuclear fusion reactions near the Coulomb barrier are strongly affected by the
structure of the interacting nuclei, especially with weakly bound neutrons
[1]. Some theoretical calculations predict that the fusion cross section is
enhanced over well-bound nuclei because of the larger spatial extent of halo
nucleons [2]. On the other hand halo nuclei can easily break up in the field
of the other nucleus, due to their low binding energies, before the two nuclei
come close enough to fuse and carry away available energy. Early calculations
of Hussein et al. [3] indicate that the actual fusion cross section decreases
significantly at all energies. However recent couple channel calculations of
Hagino et al. [4] have concluded that the fusion cross section increases below
the Coulomb barrier because of the neutron halo whereas it decreases above the
barrier because of weak coupling of the halo nucleons. Experimentally this is
still a hot debate because of experimental difficulties. Another interest we
propose here is the influence of the cluster structure in the fusion
mechanism. The cluster structures of light nuclei have been studied
theoretically from stable to those far away from the stability line. Recent
calculations, using an antisymmetrized molecular dynamics model(AMD), indicate
that light nuclei exhibit variety of distinct cluster structures [5, 6, 7, 8].
The cluster structures are predicted even for nuclei with $Z\sim N$ of Li and
Be [6] (where Z and N are the charge and neutron number in a nucleus,
respectively). When nuclei with a well-developed cluster structure are
involved in fusion reactions near the barrier, it will be reflected on the
fusion cross section, especially in the variety of the exit channel
distribution or in the fusion residue distribution. The cluster structure
effect may be enhanced in the fusion reaction between light systems. In these
systems the Coulomb interaction becomes small and the proximity effect between
the two nuclei will be enhanced and therefore the structure of the projectile
and/or target may reflect the fusion cross section directly, especially as an
enhancement of particular incomplete fusion channels. In this paper we present
the calculated results in the study of the fusion reactions of the
${}^{7}Li+^{12}C$ system near the Coulomb barrier using AMD simulations.
## 2 AMD simulations
The initial nuclei of ${}^{7}Li$ and ${}^{12}C$ were produced by the AMD code
of Ono et al. in refs. [9], using the Gogny interaction. The binding energy
and root mean square radius of these initial nuclei are compared to the
experimental values in Table 1. All calculated values are in good agreement to
those of the experimental values, except for the root-mean square radius of
${}^{7}Li$ in which the calculated value is about $20\%$ larger than that of
the experiment. The more sophisticated calculation in ref. [6] of the
experimental root mean square radius of ${}^{7}Li$ is also well reproduced
with a distinct $\alpha+t$ structure.
Table 1: initial nuclei.
$\0\0$nucleus | AMD | AMD | $\m$Exp. | $\m$Exp.
---|---|---|---|---
$\0\0$ | Binding | rms (fm) | $\m$Binding | $\m$rms (fm)
$\0\0$ | energy (MeV) | | $\m$energy (MeV) | $\m$
7Li(t,$\alpha$) | 40.00 | 3.02 | $\m$39.24 | $\m$2.43
12C(3$\alpha$) | 92.24 | 2.53 | $\m$92.16 | $\m$2.47
| | | |
Figure 1: ${}^{7}Li$ initial nucleus with $\alpha$ \+ t structure.
Figure 2: ${}^{12}C$ initial nucleus with 3$\alpha$ structure.
The density plot of these nuclei are also shown in Fig.2 and Fig.2. Symbols
indicate the location of all nucleons. One can see in both figures that
nucleons are well clusterized in space. In the ${}^{7}Li$ case, two clusters
are observed, the larger one corresponds to an $\alpha$ and the other to a
triton, and the nucleus is very deformed. In ${}^{12}C$, $3\alpha$ clusters
are observed, but the nucleus is compact and much more spherical.
Using these initial nuclei, ${}^{7}Li+^{12}C$ reactions were simulated at
center of mass energies between 3 to 8 MeV. Calculations were performed in the
impact parameter range, b, from 0 to $7fm$. In $b>7fm$, no fusion reactions
are observed. In Fig.3 the time evolution of the density distributions is
shown as typical examples of the complete and incomplete fusion reactions. On
the left panel, a complete fusion reaction is observed. In the middle, only
the $\alpha$ particle is transferred into the ${}^{12}C$ nucleus and triton is
escaped as a spectator. On the right panel, only triton is absorbed and the
$\alpha$ particle becomes a spectator. The latter two cases are mainly
observed at larger impact parameters. In each incident energy, a few thousand
to ten thousand events are generated, depending on the fusion cross section,
proportional to the impact parameter in the given range.
Figure 3: Time evolution of the 2D density plots for typical fusion
reactions. Impact parameter, incident energy and reaction product at the
bottom of the simulations are indicated on the top of each figure. The density
plot is made by projecting that of all nucleons on the X-Z plane. The contour
lines are plotted on a linear scale.
The AMD calculations were performed up to times ranging from 3000 fm/c at
lower energies to 1000 fm/c at higher energy side and clusterized at the end
of the calculation, using a coalescence technique in phase space. The
coalescence radius, corresponding to 5 fm in the coordinate space, is used at
all energies. The Z, A, excitation energy, angular momentum and velocity
vector of each cluster were calculated. Even after such a relatively long
time, most clusters were in an excited state. In order to compare the
simulated results to those of the experiments, the excited fragments were
cooled down using the statistical decay code, GEMINI [10]. In this
calculation, the C++ version of GEMINI was used. These events are referred to
as the AMD + GEMINI events hereafter, whereas the events without the GEMINI
calculation are called the primary events and referred to as the AMD events.
The occurrence of the fusion reactions in the AMD + GEMINI events is defined
here by the emission of the fragments with $Z>6$ in a given event.
In Fig.4 the calculated fusion cross sections, indicated by closed triangles,
are compared to those of the experiments (open circles). The experimental data
are taken from ref. [11]. The experimental data are reproduced well within the
experimental errors above $E_{cm}>3MeV$ in the absolute scale. The absolute
cross sections predicted by the AMD simulations were calculated using the
number of events generated in the given impact parameter range. At $E_{cm}\leq
3MeV$ the AMD simulation underestimated the fusion cross sections. In this
energy range, the tunneling effect through the Coulomb barrier becomes
important and in the present AMD formulation, this process is not
incorporated. In the figure the formation cross sections of ${}^{19}F$ in the
primary AMD events are also plotted by open square symbols. As discussed
below, there are additional $20-30\%$ incomplete fusion contribution in the
primary fusion process.
Figure 4: Fusion cross section for the ${}^{7}Li+^{12}C$. Circles represent
experimental results and taken from [11]. Squares are the primary of AMD
results filtered by $Z=9$. Secondary values of the AMD + GEMINI events
filtered by $Z>6$ are showed as triangles. Figure 5: Primary major exit
channel distribution at different incident energies. Figure 6: Final exit
channel distribution of the fusion reactions for the ${}^{7}Li$ \+ ${}^{12}C$.
The blue histogram indicated the experimental values. The results of
AMD+GEMINI are shown by green histograms. pn and pnn channels also include d
and t decays.
In Fig.5, the fusion channel distribution at the primary stage is shown as a
probability distribution. Only the top three major channels are plotted. The
${}^{19}F$ formation and ${}^{15}N+\alpha$ channel dominate the fusion
reaction at all energies. Complete fusion occurs in about $80\%$ of the cases
at the lower incident energies and decreases to about $70\%$ at the higher
energies. The third channel contribution is from different reactions at
different incident energies, but their probabilities are only a few $\%$ at
most.
In Fig.6, the final exit channel distribution of the fusion reaction is
plotted from the AMD + GEMINI events and compared with the experimental
results of ref. [11]. The relative cross section of the major decay channels
is fairly well reproduced except the ${}^{15}N+\alpha$ channel. The
suppression of this channel in the experimental results is not yet fully
understood. Further study is now underway.
## 3 SUMMARY
The fusion cross section of the ${}^{7}Li+^{12}C$ reaction was studied using
the AMD and GEMINI codes. The AMD+GEMINI simulation reproduced the
experimental total fusion cross sections reasonably well at $E_{cm}>3MeV$ but
underestimated it below that energy. The relative experimental exit channel
distribution, except the ${}^{15}N+\alpha$ channel, was well reproduced by the
AMD+GEMINI simulation.
We thank A. Ono for helpful discussions and communications. And thank him and
R. Charity for letting us to use their calculation codes. One of us (R. Wada)
thanks ”the visiting professorship of senior international scientists” of the
Chinese Academy of Sciences for the support. This work is supported by
National Natural Science Foundation of China (Grant No. 11075189, 11075190 and
11005127) and Directed Program of Innovation Project of the Chinese Academy of
Science (Grant No. KJCX2-YW-N44). Z.Chen thanks the ”100 Person Project” of
the Chinese Academy of Science. And thank the high-performance computing
center of College of Physics and Information Engineering, Henan Normal
University.
## References
## References
* [1] L.F. Canto et al., Nucl. Phys. A424 1 (2006).
* [2] N. Takigawa et al., Phys. Rev. C47 R2470 (1993).
* [3] M.S. Hussein et al., Phys. Rev. C46, 377 (1992); M.S. Hussein et al., Phys. Rev. Lett. 72, 2693 (1994);M.S. Hussein et al., Nucl. Phys. A588, 85c (1995).
* [4] K. Hagino et al., Phys. Rev. C61 037602 (2000).
* [5] Y. Kaneda-En’yo et al., PRC52, 628 (1995)
* [6] Y. Kaneda-En’yo et al., PRC52, 647 (1995)
* [7] Y. Kanada-En’yo et al., PRC60, 064304 (1999);
* [8] Naoya Furutachi, Masaaki Kimura, Akinobu Dot$\acute{e}$ and Yoshiko Kanada-En’yo, Prog. Theor. Phys. 122, 865 (2009).
* [9] A. Ono, S. Hudan, A. Chbihi, J. D. Frankland, Phys. Rev. C66, 014603 (2002).
* [10] R. J. Charity et al., Nucl. Phys. A483, 371, 1988.
* [11] A. Mukherjee et al., Nucl. Phys. A596, (1996) 299.
|
arxiv-papers
| 2012-09-03T09:09:09 |
2024-09-04T02:49:34.702493
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Huang, F. Zhou, R. Wada, X. Liu, W. Lin, M. Zhao, J. Wang, Z. Chen,\n C. Ma, Y. Yang, Q. Wang, J. Ma, J. Han, P. Ma, S. Jin, Z. Bai, Q. Hu, L. Jin,\n J. Chen, Y. Su and Y. Li",
"submitter": "Meirong Huang",
"url": "https://arxiv.org/abs/1209.0278"
}
|
1209.0282
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-237 LHCb-PAPER-2012-015
Measurement of the fraction of $\mathchar 28935\relax{(1S)}$ originating from
$\chi_{b}{(1P)}$ decays in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration†††Authors are listed on the following pages.
The production of $\chi_{b}{(1P)}$ mesons in $pp$ collisions at a centre-of-
mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$ is studied using
$32\mbox{\,pb}^{-1}$ of data collected with the LHCb detector. The
$\chi_{b}{(1P)}$ mesons are reconstructed in the decay mode
$\chi_{b}{(1P)}\rightarrow\mathchar
28935\relax{(1S)}\gamma\rightarrow\mu^{+}\mu^{-}\gamma$. The fraction of
$\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$ decays in the
$\mathchar 28935\relax{(1S)}$ transverse momentum range $6<\mbox{$p_{\rm
T}$}^{\mathchar 28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
rapidity range $2.0<y^{\mathchar 28935\relax{(1S)}}<4.5$ is measured to be
$(20.7\pm 5.7\pm 2.1^{+2.7}_{-5.4})\%$, where the first uncertainty is
statistical, the second is systematic and the last gives the range of the
result due to the unknown $\mathchar 28935\relax{(1S)}$ and $\chi_{b}{(1P)}$
polarizations.
Submitted to JHEP
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38,
F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, C.
Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez
Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C.
Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M.
Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J.
Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R.
Gauld52, N. Gauvin36, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, T.
Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51,
P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P.
Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E. Hicks49, M.
Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, R.S. Huston12,
D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9, J.
Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S.
Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U.
Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52,
A. Martín Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z.
Mathe12, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8,
G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C.
Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro33,
W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I.
Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos
Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44,
S. Roiser35, V. Romanovsky32, A. Romero Vidal34, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A.
Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M.
Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6,
P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27,
T. Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T.
Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C.
Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R.
Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J.
Wicht35, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M.
Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z.
Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a,
F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The production of heavy quarkonium states at hadron colliders is a subject of
experimental and theoretical interest [1]. The non-relativistic QCD (NRQCD)
factorization approach has been developed to describe the inclusive production
and decay of quarkonia [2]. The LHCb experiment has measured the production of
inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ [3], $\psi{(2S)}$ [4] and $\mathchar
28935\relax(nS)\rightarrow\mu^{+}\mu^{-}$ $(n=1,2,3)$ [5] mesons in $pp$
collisions as a function of the quarkonium transverse momentum $p_{\rm T}$ and
rapidity $y$ over the range $0<\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$. A significant
fraction of the cross-section for both ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\mathchar 28935\relax(nS)$ production is expected to be due to
feed-down from higher quarkonium states. Understanding the size of this effect
is important for the interpretation of the quarkonia cross-section and
polarization data. A few experimental studies of hadroproduction of $P$-wave
quarkonia have been reported. In the case of the $\chi_{cJ}$ states, with spin
$J=0,1,2$, measurements from the CDF [6, 7], HERA-B [8] and LHCb [9, 10]
experiments exist, while $\chi_{bJ}$ related measurements have been reported
by the CDF [11], ATLAS [12] and D0 [13] experiments.
This paper reports studies of the inclusive production of the $P$-wave
$\chi_{b{}J}{(1P)}$ states, collectively referred to as $\chi_{b}{(1P)}$
throughout the paper. The $\chi_{b}{(1P)}$ mesons are reconstructed through
the radiative decay $\chi_{b}{(1P)}\rightarrow\mathchar
28935\relax{(1S)}\gamma$ in the $\mathchar 28935\relax{(1S)}$ rapidity and
transverse momentum range $2.0<y^{\mathchar 28935\relax{(1S)}}<4.5$ and
$6<\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $\chi_{b2}$
and $\chi_{b1}$ states differ in mass by
$20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the $\chi_{b1}$ and
$\chi_{b0}$ states by $33{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [14]. Since
these differences are comparable with the experimental resolution, the total
fraction of $\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$
decays is reported. The results presented here use a data sample collected at
the LHC with the LHCb detector at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$ and correspond to an integrated luminosity of
$32\mbox{\,pb}^{-1}$.
## 2 LHCb detector
The LHCb detector [15] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The nominal detector performance for photons and muons is described in [15].
The processes of radiative transitions of $\chi_{cJ}\rightarrow J/\psi\gamma$,
$J=1,2$ with similar kinematics of the photons are studied in [9, 10]. Another
physical analysis which uses $\pi^{0}\rightarrow\gamma\gamma$,
$\eta\rightarrow\gamma\gamma$ and $\eta^{\prime}\rightarrow\rho^{0}\gamma$ is
available as [16].
The trigger consists of a hardware stage followed by a software stage which
applies a full event reconstruction. The trigger used for this analysis
selects a pair of oppositely-charged muon candidates, where either one of the
muons has a $\mbox{$p_{\rm T}$}>1.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or
one of the pair has a $\mbox{$p_{\rm
T}$}>0.56{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and the other has a
$\mbox{$p_{\rm T}$}>0.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant
mass of the pair is required to be greater than
$2.9{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The photons are not used in the
trigger decision.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [17] with a
specific LHCb configuration [18]. Decays of hadronic particles are described
by EvtGen [19] in which final state radiation is generated using Photos [20].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [21, *Agostinelli:2002hh] as
described in Ref. [23]. The simulated signal events contain at least one
$\mathchar 28935\relax{(1S)}\rightarrow\mu^{+}\mu^{-}$ decay with both muons
in the LHCb acceptance. In this sample of simulated events the fraction of
$\mathchar 28935\relax{(1S)}$ mesons produced in $\chi_{b}{(1P)}$ decays is
47% and both the $\chi_{b}{(1P)}$ and $\mathchar 28935\relax{(1S)}$ mesons are
produced unpolarized.
## 3 Event selection
The reconstruction of the $\chi_{b}{(1P)}$ meson proceeds via the
identification of an $\mathchar 28935\relax{(1S)}$ meson combined with a
reconstructed photon. The $\mathchar 28935\relax(nS)$ candidates are formed
from a pair of oppositely-charged tracks that are identified as muons. Each
track is required to have a good track fit quality. The two muons are required
to originate from a common vertex with a distance to the primary vertex less
than $1\rm\,mm$.
Figure 1: Distribution of the $\mu^{+}\mu^{-}$ mass for selected $\mathchar
28935\relax(nS)$ candidates (black points), together with the result of the
fit (solid blue curve), including the background (dotted blue curve) and the
signal (dashed magenta curve) contributions.
The invariant mass distribution of the $\mu^{+}\mu^{-}$ candidates is shown in
Fig. 1. It is modelled with the sum of three Crystal Ball functions [24],
describing the $\mathchar 28935\relax{(1S)}$, $\mathchar 28935\relax{(2S)}$
and $\mathchar 28935\relax{(3S)}$ signals, and an exponential function for the
combinatorial background. The parameters of the Crystal Ball functions that
describe the radiative tail of the $\mathchar 28935\relax{(1S)}$, $\mathchar
28935\relax{(2S)}$ and $\mathchar 28935\relax{(3S)}$ mass distributions are
fixed to the values $a=2$ and $n=1$ [5]. The measured $\mathchar
28935\relax{(1S)}$ signal yield, mass and width are $N_{\mathchar
28935\relax{(1S)}}=39\,635\pm 252$, $m_{\mathchar 28935\relax{(1S)}}=9449.2\pm
0.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\sigma_{\mathchar
28935\relax{(1S)}}=51.7\pm 0.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where
the uncertainties are statistical only.
The $\mathchar 28935\relax{(1S)}$ candidates with a $\mbox{$p_{\rm
T}$}^{\mathchar 28935\relax{(1S)}}>6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
a $\mu^{+}\mu^{-}$ invariant mass in the range
$9.36-9.56{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are combined with photons
to form $\chi_{b}{(1P)}$ candidates. The photons are required to have
$\mbox{$p_{\rm T}$}^{\gamma}>0.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\cos\theta^{*}_{\gamma}>0$, where $\theta^{*}_{\gamma}$ is the angle of the
photon direction in the centre-of-mass frame of the $\mu^{+}\mu^{-}\gamma$
system with respect to the momentum of this system in the laboratory frame.
The $\chi_{b}{(1P)}$ signal peak observed in the distribution of the mass
difference, $x=m(\mu^{+}\mu^{-}\gamma)-m(\mu^{+}\mu^{-})$, is shown in Fig. 2
for the range $6<\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. It is modelled
with an empirical function given by
$\frac{dN}{dx}=A_{1}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\Delta
M)^{2}}{2\sigma^{2}}}+A_{2}(x-x_{0})^{\alpha}e^{-(c_{1}x+c_{2}x^{2}+c_{3}x^{3})},$
(1)
where $A_{1}$, $\Delta M$, $\sigma$, $A_{2}$, $x_{0}$, $\alpha$, $c_{1}$,
$c_{2}$ and $c_{3}$ are free parameters. The Gaussian function describes the
signal and the second term models the background. The number of
$\chi_{b}{(1P)}$ signal decays obtained from the fit is $201\pm 55$. The mean
value of the Gaussian function is $447\pm
4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and its width is $19.0\pm
4.2{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The expected values of the mass
differences for the three $\chi_{b{}J}{(1P)}$ states are $\Delta
M(\chi_{b2})=452{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $\Delta
M(\chi_{b1})=432{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\Delta
M(\chi_{b0})=399{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [14]. The peak
position in the data lies between $\Delta M(\chi_{b2})$ and $\Delta
M(\chi_{b0})$ as expected for any mixture of $\chi_{b{}J}{(1P)}$ states.
Figure 2: Distribution of the mass difference
$m(\mu^{+}\mu^{-}\gamma)-m(\mu^{+}\mu^{-})$ for selected $\chi_{b}{(1P)}$
candidates (black points), together with the result of the fit (solid blue
curve), including background (dotted blue curve) and signal (dashed magenta
curve) contributions. The solid (red) histogram is an alternative background
estimation using simulated events containing a $\mathchar 28935\relax{(1S)}$
that does not originate from a $\chi_{b}{(1P)}$ decay, normalized to the data.
It is used for evaluation of the systematic uncertainty due to the choice of
fitting model. The bottom insert shows the pull distribution of the fit. The
pull is defined as the difference between the data and fit value divided by
the data error.
## 4 Fraction of $\mathchar 28935\relax{(1S)}$ originating from
$\chi_{b}{(1P)}$ decays
The fraction of $\mathchar 28935\relax{(1S)}$ originating from
$\chi_{b}{(1P)}$ decays is determined using the following assumptions.
Firstly, all $\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$
arise from the radiative decay $\chi_{b}{(1P)}\rightarrow\mathchar
28935\relax{(1S)}\gamma$. Secondly, the total efficiency for $\mathchar
28935\relax{(1S)}\rightarrow\mu^{+}\mu^{-}$ as a function of $\mbox{$p_{\rm
T}$}^{\mathchar 28935\relax{(1S)}}$ is the same for directly produced
$\mathchar 28935\relax{(1S)}$ and for those from feed-down from
$\chi_{b}{(1P)}$. The total efficiency includes trigger, detection,
reconstruction and selection. Thirdly, the photon detection, reconstruction
and selection are independent of the $\mathchar
28935\relax{(1S)}\rightarrow\mu^{+}\mu^{-}$. Hence the total efficiency for
$\chi_{b}{(1P)}$ is factorized as
$\epsilon_{\mathrm{tot}}(\chi_{b})=\epsilon_{\mathrm{cond}}(\chi_{b})\cdot\epsilon_{\mathrm{tot}}(\mathchar
28935\relax)$, where $\epsilon_{\mathrm{tot}}(\mathchar 28935\relax)$ is the
total efficiency for $\mathchar 28935\relax{(1S)}$ and
$\epsilon_{\mathrm{cond}}(\chi_{b})$ is the conditional efficiency for
$\chi_{b}{(1P)}$ reconstruction and selection after the $\mathchar
28935\relax{(1S)}\rightarrow\mu^{+}\mu^{-}$ candidate has been selected.
The second assumption is tested by comparing the $\mathchar 28935\relax{(1S)}$
efficiencies obtained using simulated events for direct $\mathchar
28935\relax{(1S)}$ and for $\mathchar 28935\relax{(1S)}$ coming from decays of
$\chi_{b}{(1P)}$ states. These efficiencies for each $\mbox{$p_{\rm
T}$}^{\mathchar 28935\relax{(1S)}}$ interval agree within the statistical
error, which is less than 0.5%.
The conditional $\chi_{b}{(1P)}$ reconstruction and selection efficiency is
estimated from simulation as
$\epsilon_{\mathrm{cond}}(\chi_{b})=\frac{\epsilon_{\mathrm{tot}}(\chi_{b})}{\epsilon_{\mathrm{tot}}(\mathchar
28935\relax)}=\frac{N^{\mathrm{MC}}_{\mathrm{rec}}(\chi_{b})}{N^{\mathrm{MC}}_{\mathrm{gen}}(\chi_{b})}\cdot\frac{N^{\mathrm{MC}}_{\mathrm{gen}}(\mathchar
28935\relax)}{N^{\mathrm{MC}}_{\mathrm{rec}}(\mathchar 28935\relax)},$ (2)
where $N^{\mathrm{MC}}_{\mathrm{rec}}(\chi_{b})$ and
$N^{\mathrm{MC}}_{\mathrm{rec}}(\mathchar 28935\relax)$ are the number of
$\chi_{b}{(1P)}$ and $\mathchar 28935\relax{(1S)}$ mesons obtained from the
fit, and $N^{\mathrm{MC}}_{\mathrm{gen}}(\chi_{b})$ and
$N^{\mathrm{MC}}_{\mathrm{gen}}(\mathchar 28935\relax)$ are the number of
generated $\chi_{b}{(1P)}$ and $\mathchar 28935\relax{(1S)}$ mesons,
respectively. The value obtained is
$\epsilon_{\mathrm{cond}}(\chi_{b})=(9.4\pm 0.1)\%$ for $6<\mbox{$p_{\rm
T}$}^{\mathchar 28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y^{\mathchar 28935\relax{(1S)}}<4.5$.
The fraction of $\mathchar 28935\relax{(1S)}$ originating from
$\chi_{b}{(1P)}$ decays is determined from the ratio
$\frac{N_{\mathrm{prod}}(\chi_{b})}{N_{\mathrm{prod}}(\mathchar
28935\relax)}=\frac{N_{\mathrm{rec}}(\chi_{b})/\epsilon_{\mathrm{tot}}(\chi_{b})}{N_{\mathrm{rec}}(\mathchar
28935\relax)/\epsilon_{\mathrm{tot}}(\Upsilon)}=\frac{N_{\mathrm{rec}}(\chi_{b})/\epsilon_{\mathrm{cond}}(\chi_{b})}{N_{\mathrm{rec}}(\mathchar
28935\relax)},$ (3)
where $N_{\mathrm{prod}}(\chi_{b})$ and $N_{\mathrm{prod}}(\mathchar
28935\relax)$ are the total numbers of $\chi_{b}{(1P)}\rightarrow\mathchar
28935\relax{(1S)}\gamma$ and $\mathchar 28935\relax{(1S)}$ mesons produced,
and $N_{\mathrm{rec}}(\chi_{b})$ and $N_{\mathrm{rec}}(\mathchar 28935\relax)$
are the numbers of reconstructed $\chi_{b}{(1P)}$ and $\mathchar
28935\relax{(1S)}$ mesons obtained from the fits to the data, respectively. As
the muons from the $\mathchar 28935\relax{(1S)}$ are explicitly required to
trigger the event, the efficiency of the trigger cancels in this ratio. The
fraction of $\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$
decays for $6<\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y^{\mathchar 28935\relax{(1S)}}<4.5$ is found to be $(20.7\pm 5.7)\%$,
where the uncertainty is statistical only.
The procedure is repeated in four bins of $\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}$, giving the results shown in Table 1 and Fig. 3. No
significant $\mbox{$p_{\rm T}$}^{\mathchar 28935\relax{(1S)}}$ dependence is
observed. The mean of the measurements performed in the individual bins is
consistent with the measurement obtained in the whole $\mbox{$p_{\rm
T}$}^{\mathchar 28935\relax{(1S)}}$ range.
Table 1: Number of reconstructed $\chi_{b}{(1P)}$ and $\mathchar
28935\relax{(1S)}$ signal candidates, conditional efficiency and fraction of
$\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$ decays for
different $\mbox{$p_{\rm T}$}^{\mathchar 28935\relax{(1S)}}$ bins. The
uncertainties are statistical only.
$\mbox{$p_{\rm T}$}^{\mathchar 28935\relax{(1S)}}({\mathrm{\,Ge\kern-0.90005ptV\\!/}c})$ | $6-7$ | $7-8$ | $8-10$ | $10-15$ | $6-15$
---|---|---|---|---|---
$N_{\mathrm{rec}}(\chi_{b})$ | $41$$\,\pm\,$ | $39$ | $35$$\,\pm\,$ | $22$ | $91$$\,\pm\,$ | $30$ | $82$$\,\pm\,$ | $29$ | $201$$\,\pm\,$ | $55$
$N_{\mathrm{rec}}(\mathchar 28935\relax)$ | $2730$$\,\pm\,$ | $64$ | $2193$$\,\pm\,$ | $57$ | $2866$$\,\pm\,$ | $64$ | $2627$$\,\pm\,$ | $59$ | $10\,345$$\,\pm\,$ | $123$
$\epsilon_{\mathrm{cond}}(\chi_{b})$ in % | $6.7$$\,\pm\,$ | $0.2$ | $8.3$$\,\pm\,$ | $0.2$ | $10.0$$\,\pm\,$ | $0.2$ | $12.8$$\,\pm\,$ | $0.2$ | $9.4$$\,\pm\,$ | $0.1$
Fraction in % | $23$$\,\pm\,$ | $22$ | $20$$\,\pm\,$ | $12$ | $32$$\,\pm\,$ | $10$ | $25$$\,\pm\,$ | $9$ | $21$$\,\pm\,$ | $6$
## 5 Systematic uncertainties
Studies of quarkonium decays to two muons [3, 5, 9, 10, 4] show that the total
efficiency depends on the polarization of the vector meson. The effect of the
polarization has been studied by repeating the estimation of the efficiencies
$\epsilon_{\mathrm{tot}}(\chi_{b})$ and $\epsilon_{\mathrm{tot}}(\mathchar
28935\relax)$ for the extreme $\chi_{b}{(1P)}$ and $\mathchar
28935\relax{(1S)}$ polarization scenarios and taking the difference in
$\epsilon_{\mathrm{cond}}(\chi_{b})$ as the systematic uncertainty. The
largest variation is found for the cases of 100% transverse and longitudinal
polarization of the $\mathchar 28935\relax{(1S)}$. We assign this relative
variation of ${}^{+13}_{-26}$% as the range due to the unknown polarizations.
The systematic effect due to the unknown $\chi_{b{}J}{(1P)}$, $J=0,1,2$
relative contributions is estimated by varying these fractions in the
simulation in such a way that the peak position of the mixture is equal to the
peak position observed in the data plus or minus its statistical uncertainty.
The maximal relative variation of the result is found to be 7%. This value is
taken as a systematic uncertainty due to the unknown $\chi_{b{}J}{(1P)}$
mixture.
The systematic uncertainty due to the photon reconstruction efficiency is
determined by comparing the relative yields of the reconstructed
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(K^{*+}\rightarrow K^{+}\pi^{0})$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays in
data and simulated events. It is assumed that the reconstruction efficiencies
of the two photons from the $\pi^{0}$ are uncorrelated. The uncertainty on the
photon reconstruction efficiency is studied as a function of $\mbox{$p_{\rm
T}$}^{\gamma}$. The largest systematic uncertainty is found to be 6% for
photons in the range $0.6<\mbox{$p_{\rm
T}$}^{\gamma}<0.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and is dominated by
the uncertainties of the $B^{+}$ branching fractions.
The systematic uncertainty due to the choice of the background fit model is
estimated from simulated events containing an $\mathchar 28935\relax{(1S)}$
that does not originate from the decay of a $\chi_{b}{(1P)}$. The distribution
of the mass difference obtained with these events, using the same
reconstruction and selection as for data, is shown in Fig. 2, normalized to
the data below $0.38{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. It describes
rather well the background contribution above
$0.38{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, both in shape and level. The
difference between the number of data events and the normalized number of
simulated background events in the range
$0.38-0.50{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ gives an estimate of the
signal yield. For $6<\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ the signal yield
obtained using this method is $211$ to be compared with $201\pm 55$ obtained
from the fit. The procedure is repeated in each $\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}$ bin. We also study the variation of signal yield by
changing the normalization range to
$0.0-0.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ or
$0.7-1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The maximal relative
difference of 5% is taken as the uncertainty due to the choice of the signal
and background description. Systematic uncertainties are summarized in Table
2.
Table 2: Relative systematic uncertainties on the fraction of $\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$ decays. Source | Uncertainty (%)
---|---
Unknown $\chi_{b{}J}{(1P)}$ mixture | 7
Photon reconstruction efficiency | 6
Signal and background description | 5
Quadratic sum of the above | 10
## 6 Results and conclusions
The production of $\chi_{b}{(1P)}$ mesons is observed using data corresponding
to an integrated luminosity of $32\mbox{\,pb}^{-1}$ collected with the LHCb
detector in $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The
fraction of $\mathchar 28935\relax{(1S)}$ originating from $\chi_{b}{(1P)}$
decays in the kinematic range $6<\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y^{\mathchar 28935\relax{(1S)}}<4.5$ is measured to be
$(20.7\pm 5.7\pm 2.1^{+2.7}_{-5.4})\%,$
where the first uncertainty is statistical, the second is systematic and the
last gives the range of the result due to the unknown polarization of
$\mathchar 28935\relax{(1S)}$ and $\chi_{b}{(1P)}$ mesons.
Figure 3: Fraction of $\mathchar 28935\relax{(1S)}$ originating from
$\chi_{b}{(1P)}$ decays for different $\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}$ bins, assuming production of unpolarized $\mathchar
28935\relax{(1S)}$ and $\chi_{b}{(1P)}$ mesons, shown with solid circles. The
vertical error bars are statistical only. The result determined for the range
$6<\mbox{$p_{\rm T}$}<15{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ is shown with
the horizontal solid line, its statistical uncertainty with the dash-dotted
lines, and its total uncertainty (statistical and systematic, including that
due to the unknown polarization) with the shaded (light blue) band.
This result can be compared with the CDF measurement of $(27.1\pm 6.9\pm
4.4)\%$ [11], obtained in $p\bar{p}$ collisions at
$\sqrt{s}=1.8\mathrm{\,Te\kern-1.00006ptV}$ in the kinematic range
$\mbox{$p_{\rm T}$}^{\mathchar
28935\relax{(1S)}}>8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$|\eta^{\mathchar 28935\relax{(1S)}}|<0.7$.
The $\chi_{b}{(1P)}$ decays are observed to be a significant source of
$\mathchar 28935\relax{(1S)}$ mesons in $pp$ collisions. This will need to be
taken into account in the interpretation of the measured $\mathchar
28935\relax{(1S)}$ production cross-section and polarization.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] N. Brambilla et al., Heavy quarkonium: progress, puzzles, and opportunities, Eur. Phys. J. C71 (2011) 1534, arXiv:1010.5827
* [2] G. T. Bodwin, E. Braaten, and G. P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D51 (1995) 1125, arXiv:hep-ph/9407339, erratum ibid. D55 (1997) 5853
* [3] LHCb collaboration, R. Aaij et al., Measurement of $J/\psi$ production in $pp$ collisions at $\sqrt{s}=7$ TeV, Eur. Phys. J. C71 (2011) 1645, arXiv:1103.0423
* [4] LHCb collaboration, R. Aaij et al., Measurement of $\psi(2S)$ meson production in $pp$ collisions at $\sqrt{s}=7$ TeV, arXiv:1204.1258, to appear in Eur. Phys. J. C.
* [5] LHCb collaboration, R. Aaij et al., Measurement of $\mathit{\Upsilon}$ production in $pp$ collisions at $\sqrt{s}=7$ TeV, Eur. Phys. J. C72 (2012) 2025, arXiv:1202.6579
* [6] CDF collaboration, A. Abulencia et al., Measurement of $\sigma_{\chi_{c2}}\mathcal{B}(\chi_{c2}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)/\sigma_{\chi_{c1}}\mathcal{B}(\chi_{c1}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)$ in $p\overline{p}$ collisions at $\sqrt{s}=1.96$ TeV, Phys. Rev. Lett. 98 (2007) 232001, arXiv:hep-ex/0703028
* [7] CDF collaboration, F. Abe et al., Production of $J/\psi$ mesons from $\chi_{c}$ meson decays in $p\overline{p}$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 79 (1997) 578
* [8] HERA-B collaboration, I. Abt et al., Production of the charmonium states $\chi_{c1}$ and $\chi_{c2}$ in proton nucleus interactions at $\sqrt{s}=41.6$ GeV, Phys. Rev. D79 (2009) 012001, arXiv:0807.2167
* [9] LHCb collaboration, R. Aaij et al., Measurement of the cross-section ratio $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ for prompt $\chi_{c}$ production at $\sqrt{s}=7$ TeV, Phys. Lett. B 714 (2012) 215, arXiv:1202.1080
* [10] LHCb collaboration, R. Aaij et al., Measurement of the ratio of prompt $\chi_{c}$ to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in $pp$ collisions at $\sqrt{s}=7$ TeV, arXiv:1204.1462
* [11] CDF collaboration, T. Affolder et al., Production of $\mathit{\Upsilon}(1\mathit{S})$ mesons from $\chi_{b}$ decays in $p\overline{p}$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 84 (2000) 2094, arXiv:hep-ex/9910025
* [12] ATLAS collaboration, G. Aad et al., Observation of a new $\chi_{b}$ state in radiative transitions to $\mathit{\Upsilon}(1\mathit{S})$ and $\mathit{\Upsilon}(2\mathit{S})$ at ATLAS, Phys. Rev. Lett. 108 (2012) 152001, arXiv:1112.5154
* [13] D0 collaboration, V. Abazov et al., Observation of a narrow mass state decaying into $\mathchar 28935\relax{(1S)}+\gamma$ in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV, arXiv:1203.6034
* [14] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001
* [15] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [16] LHCb collaboration, R. Aaij et al., Evidence for the decay $B^{0}\rightarrow J/\psi\omega$ and measurement of the relative branching fractions of $B^{0}_{s}$ meson decays to $J/\psi\eta$ and $J/\psi\eta^{{}^{\prime}}$, arXiv:1210.2631
* [17] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [18] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [19] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [20] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [21] GEANT4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [22] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [23] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. of Phys. : Conf. Ser. 331 (2011) 032023
* [24] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
|
arxiv-papers
| 2012-09-03T09:25:10 |
2024-09-04T02:49:34.708363
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook,\n H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N.\n Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L.\n Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G. A. Cowan, D. Craik, R. Currie, C. D'Ambrosio, P.\n David, P. N. Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.\n M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El\n Rifai, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C.\n Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R.\n Gauld, N. Gauvin, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V.\n Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H.\n Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E.\n Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.\n C. Haines, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S. T. Harnew, J.\n Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, M. Hoballah, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M.\n Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y. M. Kim, M. Knecht, O. Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N.\n La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O.\n Leroy, T. Lesiak, L. Li, Y. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner,\n C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N.\n Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I. V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G.\n Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller,\n R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, B. K. Pal, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C. J.\n Parkinson, G. Passaleva, G. D. Patel, M. Patel, G. N. Patrick, C. Patrignani,\n C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S.\n Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D.\n Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A.\n Puig Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel,\n I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis,\n S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E.\n Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V.\n Romanovsky, A. Romero Vidal, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G.\n Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann,\n B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, M. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss},\n H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K.\n Watson, A. D. Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Mikhail Shapkin",
"url": "https://arxiv.org/abs/1209.0282"
}
|
1209.0313
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-247 LHCb-PAPER-2012-019 August 28, 2012
Measurement of the ratio of branching fractions ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)$/${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$ and the direct
$C\\!P$ asymmetry in $B^{0}\\!\rightarrow K^{*0}\gamma$
The LHCb collaboration†††Authors are listed on the following pages.
The ratio of branching fractions of the radiative $B$ decays
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$ has
been measured using an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ of $pp$
collision data collected by the LHCb experiment at a centre-of-mass energy of
$\sqrt{s}$ $=7$$\mathrm{\,Te\kern-1.00006ptV}$. The value obtained is
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=1.23\pm 0.06\mathrm{\,(stat.)}\pm
0.04\mathrm{\,(syst.)}\pm 0.10\,(f_{s}/f_{d})\,,$
where the first uncertainty is statistical, the second is the experimental
systematic uncertainty and the third is associated with the ratio of
fragmentation fractions $f_{s}/f_{d}$. Using the world average value for
${\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)$, the branching fraction ${\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$ is measured to be $(3.5\pm 0.4)\times
10^{-5}$.
The direct $C\\!P$ asymmetry in $B^{0}\\!\rightarrow K^{*0}\gamma$ decays has
also been measured with the same data and found to be
$\mathcal{A}_{C\\!P}(B^{0}\\!\rightarrow K^{*0}\gamma)=(0.8\pm
1.7\mathrm{\,(stat.)}\pm 0.9\mathrm{\,(syst.)})\%\,.$
Both measurements are the most precise to date and are in agreement with the
previous experimental results and theoretical expectations.
Submitted to Nuclear Physics B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, V.
Balagura28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, C. Bauer10, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N.
Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-Montells33, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D.
Craik45, S. Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5,
F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, D. Esperante Pereira34, A. Falabella14,e, C. Färber11, G. Fardell47,
C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10,
F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, R.S. Huston12, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9, J.
Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S.
Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U.
Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52,
A. Martín Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z.
Mathe12, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8,
G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C.
Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro33,
W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I.
Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos
Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues48,51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35,
V. Romanovsky32, A. Romero Vidal34, M. Rosello33,n, J. Rouvinet36, T. Ruf35,
H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48,
B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C.
Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R.
Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J.
Wicht35, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M.
Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z.
Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a,
F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
In the Standard Model (SM), the decays111Unless stated otherwise, charge
conjugated modes are implicitly included throughout this paper.
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$
proceed at leading order through the electromagnetic penguin transitions,
$b\\!\rightarrow s\gamma$. At one-loop level these transitions are dominated
by a virtual intermediate top quark coupling to a $W$ boson. Extensions of the
SM predict additional one-loop contributions that can introduce sizeable
changes to the dynamics of the transition [1, *Gershon:th-null-tests:2006,
*Mahmoudi:th-msugra:2006, *Altmannshofer:2011gn].
Radiative decays of the $B^{0}$ meson were first observed by the CLEO
collaboration in 1993 in the decay mode $B^{0}\\!\rightarrow K^{*0}\gamma$
[2]. In 2007 the Belle collaboration reported the first observation of the
analogous decay in the $B^{0}_{s}$ sector, $B^{0}_{s}\\!\rightarrow\phi\gamma$
[3]. The current world averages of the branching fractions of
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$ are
$(4.33\pm 0.15)\times 10^{-5}$ and $(5.7^{+2.1}_{-1.8})\times 10^{-5}$,
respectively [4, 5, *belle:exp-b2kstgamma:2004, *cleo:exp-excl-radiative-
decays:1999]. These results are in agreement with the latest theoretical
predictions from NNLO calculations using soft-collinear effective theory [8],
${\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)=(4.3\pm 1.4)\times 10^{-5}$ and
${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)=(4.3\pm 1.4)\times 10^{-5}$,
which suffer from large uncertainties from hadronic form factors. A better-
predicted quantity is the ratio of branching fractions, as it benefits from
partial cancellations of theoretical uncertainties. The two branching fraction
measurements lead to a ratio ${\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)/{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$=$0.7\pm 0.3$, while the SM prediction
is $1.0\pm 0.2$ [8]. When comparing the experimental and theoretical branching
fraction for the $B^{0}_{s}\\!\rightarrow\phi\gamma$ decay, it is necessary to
account for the large decay width difference in the $B^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system. This can give rise to
a correction on the theoretical branching fraction as large as $9\%$ as
described in [9].
The direct $C\\!P$ asymmetry in the $B^{0}\\!\rightarrow K^{*0}\gamma$ decay
is defined as $\mathcal{A}_{C\\!P}=[\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\overline{f})-\Gamma(B^{0}\rightarrow
f)]/[\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\overline{f})+\Gamma(B^{0}\rightarrow
f)]$. The SM prediction,
$\mathcal{A}^{\mathrm{SM}}_{C\\!P}(B^{0}\\!\rightarrow K^{*0}\gamma)=(-0.61\pm
0.43)\%$ [10], is affected by a smaller theoretical uncertainty from the
hadronic form factors than the branching fraction calculation. The precision
on the current experimental value, $\mathcal{A}_{C\\!P}(B^{0}\\!\rightarrow
K^{*0}\gamma)=(-1.6\pm 2.2\pm 0.7)\%$ [5, 11], is statistically limited and
more precise measurements would constrain contributions from beyond the SM
scenarios, some of which predict that this asymmetry could be as large as
$-15\%$ [12, *Aoki:SUSY, *Aoki:SUSY-CP, *Kagan:NP].
This paper presents a measurement of ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)$/${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$ using
1.0$\mbox{\,fb}^{-1}$ of data taken with the LHCb detector. The measured ratio
and the world average value of ${\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)$
are then used to determine ${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$. This
result supersedes a previous LHCb measurement based on an integrated
luminosity of 0.37$\mbox{\,fb}^{-1}$ of data at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [16]. A measurement of the direct
$C\\!P$ asymmetry of the decay $B^{0}\\!\rightarrow K^{*0}\gamma$ is also
presented.
## 2 The LHCb detector and dataset
The LHCb detector [17] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors (RICH). Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers. The trigger consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction.
Decay candidates are required to have triggered on the signal photon and the
daughters of the vector meson. At the hardware stage, the decay candidates
must have been triggered by an electromagnetic candidate with transverse
energy ($E_{\rm T}$) $>2.5\mathrm{\,Ge\kern-1.00006ptV}$. The software stage
is divided into two steps. The first one performs a partial event
reconstruction and reduces the rate such that the second can perform full
event reconstruction to further reduce the data rate. At the first software
stage, events are selected when a charged track is reconstructed with IP
$\chi^{2}$ $>16$. The IP $\chi^{2}$ is defined as the difference between the
$\chi^{2}$ of the $pp$ interaction vertex (PV) fit reconstructed with and
without the considered track. Furthermore, a charged track is required to have
either $p_{\rm T}$ $>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for a photon
with $E_{\rm T}$ $>2.5\mathrm{\,Ge\kern-1.00006ptV}$ or $p_{\rm T}$
$>1.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ when the photon has $E_{\rm T}$
$>4.2\mathrm{\,Ge\kern-1.00006ptV}$. At the second software stage, a track
passing the previous criteria must form a $K^{*0}$ or $\phi$ candidate when
combined with an additional track, and the invariant mass of the combination
of the $K^{*0}$ ($\phi$) candidate and the photon candidate that triggered the
hardware stage is required to be within
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ of the world average $B^{0}$
($B^{0}_{s}$) mass. The data used for this analysis correspond to
1.0$\mbox{\,fb}^{-1}$ of $pp$ collisions collected in 2011 at the LHC with a
centre-of-mass energy of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$.
Large samples of $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ Monte Carlo simulated events are used to
optimise the signal selection and to parametrise the invariant-mass
distribution of the $B$ meson. Possible contamination from specific background
channels has also been studied using dedicated simulated samples. For the
simulation, $pp$ collisions are generated using Pythia 6.4 [18] with a
specific LHCb configuration [19]. Decays of hadronic particles are described
by EvtGen [20] in which final state radiation is generated using Photos [21].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [22, *Agostinelli:geant4:2003] as
described in Ref. [24].
## 3 Offline event selection
The selection of $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ decays is designed to maximise the
cancellation of uncertainties in the ratio of their selection efficiencies.
The charged tracks used to build the vector mesons are required to have
$p_{\rm T}$ $>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, with at least one of
them having $p_{\rm T}$ $>1.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In
addition, a requirement of IP $\chi^{2}>25$ means that they must be
incompatible with coming from any PV. The charged tracks are identified as
either kaons or pions using information provided by the RICH system. This is
based on the comparison between the two particle hypotheses. Kaons (pions) in
the studied $B\\!\rightarrow V\gamma$ decays, where $V$ stands for the vector
meson, are identified with a $\sim 70\,(83)\,\%$ efficiency for a $\sim
3\,(2)\,\%$ pion (kaon) contamination.
Photon candidates are required to have $E_{\rm T}$
$>2.6\mathrm{\,Ge\kern-1.00006ptV}$. Neutral and charged clusters in the
electromagnetic calorimeter are separated based on their compatibility with
extrapolated tracks [25] while photon deposits are distinguished from
$\pi^{0}$ deposits using the shape of the showers in the electromagnetic
calorimeter.
Oppositely-charged kaon-pion (kaon-kaon) combinations are accepted as $K^{*0}$
($\phi$) candidates if they form a good quality vertex and have an invariant
mass within $\pm 50\,(\pm 10){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
world average $K^{*0}$ ($\phi$) mass [11]. The resulting vector meson
candidate is combined with the photon candidate to make a $B$ candidate. The
invariant-mass resolution of the selected $B$ candidate is
$\approx$100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the decays
presented in this paper.
The $B$ candidates are required to have an invariant mass within
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ of the world average $B$ mass [11]
and to have $p_{\rm T}$ $>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. They must
also point to a PV, with IP $\chi^{2}$ $<9$, and the angle between the $B$
candidate momentum direction and the $B$ line of flight has to be less than
20$\rm\,mrad$. In addition, the vertex separation $\chi^{2}$ between the $B$
meson vertex and its related PV must be larger than 100. The distribution of
the helicity angle $\theta_{\mathrm{H}}$, defined as the angle between the
momentum of any of the daughters of the vector meson and the momentum of the
$B$ candidate in the rest frame of the vector meson, is expected to follow a
$\sin^{2}\theta_{\mathrm{H}}$ function for $B\\!\rightarrow V\gamma$, and a
$\cos^{2}\theta_{\mathrm{H}}$ function for the $B\\!\rightarrow V\pi^{0}$
background. A requirement of $|\cos\theta_{\mathrm{H}}|<0.8$ is therefore made
to reduce $B\\!\rightarrow V\pi^{0}$ background, where the neutral pion is
misidentified as a photon. Background coming from partially reconstructed
$B$-hadron decays is reduced by requiring the $B$ vertex to be isolated: its
$\chi^{2}$ must increase by more than two units when adding any other track in
the event.
## 4 Signal and background description
The signal yields of the $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ decays are determined from an extended
unbinned maximum-likelihood fit performed simultaneously to the invariant-mass
distributions of the $B^{0}$ and $B^{0}_{s}$ candidates. A constraint on the
$B^{0}$ and $B^{0}_{s}$ masses is included in the fit which requires the
difference between them to be consistent with the LHCb measurement of $87.3\pm
0.4$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26]. The $K^{*0}$ and $\phi$
resonances are described by a relativistic $P$-wave Breit-Wigner distribution
[27] convoluted with a Gaussian distribution to take into account the detector
resolution. The natural width of the resonances is fixed to the world average
value [11]. A polynomial line shape is added to describe the background. The
resulting distribution is fitted to the vector meson invariant-mass
distribution, as shown in Fig. 1.
Figure 1: Invariant-mass distributions of the (a) $K^{*0}$ and (b) $\phi$
resonance candidates. The black points represent the data and the fit result
is represented as a solid blue line. The fit is described in the text. The
regions outside the vector meson invariant-mass window are shaded. The Poisson
$\chi^{2}$ residuals [28] are shown below the fits with the $\pm 2\,\sigma$
confidence-level interval delimited by solid red lines.
The fit to the invariant mass of the vector-meson candidates yields a
resonance mass of $895.7\pm 0.4$$\mathrm{\,Me\kern-1.00006ptV}$ and
$1019.42\pm 0.09$$\mathrm{\,Me\kern-1.00006ptV}$ for the $K^{*0}$ and $\phi$,
respectively, in agreement with the world average values [11]. The detector
resolution extracted from the fit is $5\pm 4$$\mathrm{\,Me\kern-1.00006ptV}$
for the $K^{*0}$ resonance and $1.3\pm 0.1$$\mathrm{\,Me\kern-1.00006ptV}$ for
the $\phi$. The effect of taking the value found in data or the world average
as the central value of the vector meson mass window is negligible. In
addition no systematic uncertainty due to the choice of the line shape of the
resonances is assigned.
Both $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ signal distributions are parametrised with
a two-sided Crystal Ball distribution [29]. In the low mass region, there can
be possible losses in the photon energy due to the fiducial volume of the
calorimeter. A tail at high masses is also observed and can be explained by
the spread in the error of the reconstructed $B$ mass and pile-up effects in
the photon deposition. The parameters describing the tails on both sides are
fixed to the values determined from simulation. The width of each signal peak
is left as a free parameter in the fit.
The reconstructed mass distribution of the combinatorial background has been
determined from the low-mass sideband of the $K^{*0}$ mass distribution as an
exponential function with different attenuation constants for the two decay
channels. Additional contamination from several exclusive background decays is
studied using simulated samples. The irreducible $B^{0}_{s}\rightarrow
K^{*}\gamma$ decays, the $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax^{*}(pK^{-})\gamma$ decays222$\mathchar 28931\relax^{*}$ stands for
$\mathchar 28931\relax(1520)$ and other b-baryon resonances promptly decaying
into a $pK^{-}$ final state., and the charmless $B^{0}_{(s)}\rightarrow
h^{+}h^{\prime-}\pi^{0}$ decays produce peaked contributions under the
invariant-mass peak of $B^{0}\\!\rightarrow K^{*0}\gamma$. As the experimental
branching fractions of the charmless $B^{0}_{s}$ and $\mathchar
28931\relax^{0}_{b}$ decays are unknown, the corresponding contamination rates
are estimated either using the predicted branching fraction in the case of
$B_{s}^{0}\rightarrow K^{*0}\gamma$ decays, assuming SU(3) symmetry for
$B^{0}_{s}\rightarrow h^{+}h^{\prime-}\pi^{0}$ decays, or by directly
estimating the signal yield from an independent sample as in $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{*}\gamma$ decays. The
overall contribution from these decays is estimated to represent $(2.6\pm
0.4)$% and $(0.9\pm 0.6)$% of the $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ yields, respectively. Each of these
contributions is modelled with a Crystal Ball function determined from a
simulated sample and their yields are fixed in the fit.
The partial reconstruction of the charged $B\rightarrow h^{+}h^{\prime-}\gamma
X$ or $B\rightarrow h^{+}h^{\prime-}\pi^{0}X$ decays gives a broad
contribution at lower candidate masses, with a high-mass tail that extends
into the signal region. The partially reconstructed $B^{+}\rightarrow
K^{*0}\pi^{+}\gamma$ and $B^{+}\rightarrow\phi K^{+}\gamma$ radiative decays
produce a peaking contribution in the low-mass sideband at around
5.0${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for $B^{0}\\!\rightarrow
K^{*0}\gamma$ and around 4.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for
$B^{0}_{s}\\!\rightarrow\phi\gamma$. The corresponding contamination has been
estimated to be $(3.3\pm 1.1)\%$ and $(1.8\pm 0.3)$% for the
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$
decays, respectively. The partially reconstructed neutral $B$ meson decays
also contribute at the same level and several other channels exhibit a similar
final state topology. These contributions are described by a Crystal Ball
function and the yields are left to vary in the fit. The parameters of the
Crystal Ball function are determined from the simulation. Additional
contributions from the partial reconstruction of multi-body charmed decays and
$B\rightarrow V\pi^{0}\mathrm{X}$ have been added to the simultaneous fit in
the same way. The shape of these contributions, again determined from the
simulation, follows an ARGUS function [30] peaking around
$4.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The various background
contributions included in the fit model are summarised in Table 1.
Table 1: Expected contributions to the $B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$ yields in the $\pm$1${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ mass window from the exclusive background channels: radiative decays, $h^{+}h^{\prime-}\gamma$ (top), charmless b decays involving energetic $\pi^{0}$, $h^{+}h^{\prime-}\pi^{0}$ (middle) and partially reconstructed decays (bottom). The average measurement (exp.) or theoretical (theo.) branching fraction is given where available. Each exclusive contribution above 0.1% is included in the fit model, with a fixed shape determined from simulation. The amplitude of the partially reconstructed backgrounds is left to vary in the fit while the $h^{+}h^{\prime-}\gamma$ and $h^{+}h^{\prime-}\pi^{0}$ contributions are fixed to their expected level. Decay | Branching fraction | Relative contribution to
---|---|---
| ($\times 10^{6}$) | $B^{0}\\!\rightarrow K^{*0}\gamma$ | $B^{0}_{s}\\!\rightarrow\phi\gamma$
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{*}\gamma$ | estimated from data | ($1.0\pm 0.3$)% | ($0.4\pm 0.3$)%
$B^{0}_{s}\\!\rightarrow K^{*0}\gamma$ | $1.26\pm 0.31$ (theo. [31]) | ($0.8\pm 0.2$)% | $\mathcal{O}(10^{-4})$
$B^{0}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ | $35.9^{\,+\,2.8}_{\,-\,2.4}$ (exp. [4]) | ($0.5\pm 0.1$)% | $\mathcal{O}(10^{-4})$
$B^{0}_{s}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ | estimated from SU(3) symmetry | ($0.2\pm 0.2$)% | $\mathcal{O}(10^{-4})$
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}\pi^{0}$ | estimated from SU(3) symmetry | $\mathcal{O}(10^{-4})$ | ($0.5\pm 0.5$)%
$B^{+}\\!\rightarrow K^{*0}\pi^{+}\gamma$ | $20^{\,+\,7}_{\,-\,6}$ (exp. [4]) | ($3.3\pm 1.1$)% | $<6\times 10^{-4}$
$B^{0}\\!\rightarrow K^{+}\pi^{-}\pi^{0}\gamma$ | $41\pm 4$ (exp. [4]) | $(4.5\pm 1.7)$% | $\mathcal{O}(10^{-4})$
$B^{+}\\!\rightarrow\phi K^{+}\gamma$ | $3.5\pm 0.6$ (exp. [4]) | $3\times 10^{-4}$ | $(1.8\pm 0.3)$%
$B\rightarrow V\pi^{0}\mathrm{X}$ | $\mathcal{O}(10\%)$ (exp. [4]) | $\mathrm{a~{}few}\%$ | $\mathrm{a~{}few}\%$
At the trigger level, the electromagnetic calorimeter calibration is different
from that in the offline analysis. Therefore, the
$\pm$1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ mass window requirement
imposed by the trigger causes a bias in the $B$ meson acceptance to appear
near the limits of this window. The inefficiency at the edges of the mass
window is modelled by including a three-parameter threshold function in the
fit model
$T(m_{B})=\left(1-\mathrm{erf}\left(\frac{m_{B}-t_{\mathrm{L}}}{\sqrt{2}\mathrm{\sigma_{d}}}\right)\right)\times\left(1-\mathrm{erf}\left(\frac{t_{\mathrm{U}}-m_{B}}{\sqrt{2}\mathrm{\sigma_{d}}}\right)\right)\,,$
(1)
where erf is the Gauss error function. The parameter
$\mathrm{t_{L}}$($\mathrm{t_{U}}$) represents the actual lower (upper) mass
threshold and $\mathrm{\sigma_{d}}$ is the resolution.
## 5 Measurement of the ratio of branching fractions
The ratio of branching fractions is measured as
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=\frac{N_{B^{0}\\!\rightarrow
K^{*0}\gamma}}{N_{B^{0}_{s}\\!\rightarrow\phi\gamma}}\times\frac{{\cal
B}(\phi\rightarrow K^{+}K^{-})}{{\cal B}(K^{*0}\rightarrow
K^{+}\pi^{-})}\times\frac{f_{s}}{f_{d}}\times\frac{\epsilon_{B^{0}_{s}\\!\rightarrow\phi\gamma}}{\epsilon_{B^{0}\\!\rightarrow
K^{*0}\gamma}}\,,$ (2)
where $N$ are the observed yields of signal candidates, $\mathrm{{\cal
B}(\phi\rightarrow K^{+}K^{-})/{\cal B}(K^{*0}\rightarrow
K^{+}\pi^{-})}=0.735\pm 0.008$ [11] is the ratio of branching fractions of the
vector mesons, $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$ [32] is the ratio of the
$B^{0}$ and $B_{s}^{0}$ hadronization fractions in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and
$\epsilon_{B^{0}_{s}\\!\rightarrow\phi\gamma}/\epsilon_{B^{0}\\!\rightarrow
K^{*0}\gamma}$ is the ratio of total reconstruction and selection efficiencies
of the two decays.
The results of the fit are shown in Fig. 2. The number of $B^{0}\\!\rightarrow
K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$ candidates is $5279\pm
93$ and $691\pm 36$, respectively, corresponding to a yield ratio of $7.63\pm
0.38$. The relative contamination from partially reconstructed radiative
decays is fitted to be $(15\pm 5)\%$ for $B^{0}\\!\rightarrow K^{*0}\gamma$
and $(5\pm 3)\%$ for $B^{0}_{s}\\!\rightarrow\phi\gamma$, in agreement with
the expected rate from $B^{+(0)}\rightarrow K^{*0}\pi^{+(0)}\gamma$ and
$B^{+(0)}\rightarrow\phi K^{+(0)}\gamma$, respectively. The contribution from
partial reconstruction of charmed decays at low mass is fitted to be $(5\pm
4)\%$ and $(0^{+9}_{-0})$% of the $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ yields, respectively.
Figure 2: Invariant-mass distributions of the (a) $B^{0}\\!\rightarrow
K^{*0}\gamma$ and (b) $B^{0}_{s}\\!\rightarrow\phi\gamma$ candidates. The
black points represent the data and the fit result is represented as a solid
blue line. The signal is fitted with a double-sided Crystal Ball function
(short-dashed green line). The combinatorial background is modelled with an
exponential function (long-dashed red line). In decreasing amplitude order,
the exclusive background contributions to $B^{0}\\!\rightarrow K^{*0}\gamma$
are $B^{+(0)}\\!\rightarrow K^{*0}\pi^{+(0)}\gamma$ (short-dotted black),
$B\rightarrow K^{*0}(\phi)\pi^{0}X$ (long-dashed blue),
$B^{0}_{s}\\!\rightarrow K^{*0}\gamma$ (dotted short-dashed green), $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{*}\gamma$ (double-
dotted dashed pink), $B^{0}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ (dotted long-
dashed black) and $B^{0}_{s}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ (long-dotted
blue). The background contributions to $B^{0}_{s}\\!\rightarrow\phi\gamma$ are
$B^{+(0)}\\!\rightarrow\phi K^{+(0)}\gamma$ (dotted black), $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{*}\gamma$ (double-
dotted dashed pink) and $B^{0}_{s}\\!\rightarrow K^{+}K^{-}\pi^{0}$ (dotted-
dashed black). No significant contribution to
$B^{0}_{s}\\!\rightarrow\phi\gamma$ is found from partially reconstructed
$B\rightarrow K^{*0}(\phi)\pi^{0}X$ decays. The Poisson $\chi^{2}$ residuals
[28] are shown below the fit with the $\pm 2\,\sigma$ confidence-level
interval delimited by solid red lines.
The systematic uncertainty from the background modelling is determined by
varying the parameters that have been kept constant in the fit of the
invariant-mass distribution within their uncertainty. The 95% CL interval of
the relative variation on the yield ratio is determined to be $[-1.2,+1.4]\%$
and is taken as a conservative estimate of the systematic uncertainty
associated with the background modelling. The relative variation is dominated
by the effect from the partially reconstructed background. This procedure is
repeated to evaluate the systematic uncertainty from the signal-shape
modelling, by varying the parameters of the Crystal-Ball tails within their
uncertainty. A relative variation of $[-1.3,+1.4]\%$ on the yield ratio is
observed and added to the systematic uncertainty. As a cross-check of the
possible bias introduced on the ratio by the modelling of the mass window
thresholds and the partially reconstructed background that populates the low
mass region, the fit is repeated in a reduced mass window of $\pm
700$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the world average $B$
meson mass. The result is found to be statistically consistent with the
nominal fit. Combining these systematic effects, an overall
$({}_{-1.8}^{+2.0})$% relative uncertainty on the yield ratio is found.
The efficiency ratio can be factorised as
$\frac{\epsilon_{B^{0}_{s}\\!\rightarrow\phi\gamma}}{\epsilon_{B^{0}\\!\rightarrow
K^{*0}\gamma}}=r_{\text{reco\&sel}}\times r_{\text{PID}}\times
r_{\text{trigger}}\,,$ (3)
where $r_{\text{reco\&sel}}$, $r_{\text{PID}}$ and $r_{\text{trigger}}$ are
the efficiency ratios due to the reconstruction and selection requirements,
the particle identification (PID) requirements and the trigger requirements,
respectively.
The correlated acceptance of the kaons due to the limited phase-space in the
$\phi\\!\rightarrow K^{+}K^{-}$ decay causes the $\phi$ vertex to have a worse
spatial resolution than the $K^{*0}$ vertex. This affects the
$B^{0}_{s}\\!\rightarrow\phi\gamma$ selection efficiency through the IP
$\chi^{2}$ and vertex isolation cuts, while the common track cut $p_{\rm T}$
$>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ is less efficient on the softer
pion from the $K^{*0}$ decay. These effects partially cancel and the
reconstruction and selection efficiency ratio is found to be
$r_{\text{reco\&sel}}=0.906\pm 0.007\,\text{(stat.)}\pm
0.017\,\text{(syst.)}$. The majority of the systematic uncertainties also
cancel, since the kinematic selections are almost identical for both decays.
The remaining systematic uncertainties include the hadron reconstruction
efficiency, arising from differences in the interaction of pions and kaons
with the detector and uncertainties in the description of the detector
material. The reliability of the simulation in describing the
$\text{IP}\,\chi^{2}$ of the tracks and the isolation of the $B$ vertex is
also included in the systematic uncertainty on the $r_{\text{reco\&sel}}$
ratio. The simulated samples are weighted for each signal and background
contribution to reproduce the reconstructed mass distribution seen in data. No
further systematic uncertainties are associated with the use of the
simulation, since kinematic properties of the decays are observed to be well
modelled. Uncertainties associated with the photon are negligible, because the
reconstruction is identical in both decays.
The PID efficiency ratio is determined from data by means of a calibration
procedure using pure samples of kaons and pions from $D^{*\pm}\\!\rightarrow
D^{0}(K^{+}\pi^{-})\pi^{\pm}$ decays selected without PID information. This
procedure yields $r_{\text{PID}}=0.839\pm 0.005\,\text{(stat.)}\pm
0.010\,\text{(syst.)}$.
The trigger efficiency ratio $r_{\text{trigger}}=1.080\pm
0.009\,\text{(stat.)}$ is obtained from the simulation. The systematic
uncertainty due to any difference in the efficiency of the requirements made
at the trigger level is included as part of the selection uncertainty.
Finally, the ratio of branching fractions is obtained using Eq. 2,
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=1.23\pm 0.06\,\mathrm{(stat.)}\pm
0.04\,\mathrm{(syst.)}\pm 0.10\,(f_{s}/f_{d})\,,$
where the first uncertainty is statistical, the second is the experimental
systematic uncertainty and the third is due to the uncertainty on
$f_{s}/f_{d}$. The contributions to the systematic uncertainty are summarised
in Table 2.
Table 2: Summary of the individual contributions to the relative systematic uncertainty on the ratio of branching fractions as defined in Eq. 2. Uncertainty source | Systematic uncertainty
---|---
$r_{\mathrm{reco\&sel.}}$ | 2.0%
$r_{\mathrm{PID}}$ | 1.3%
$r_{\mathrm{trigger}}$ | 0.8%
$\mathrm{{\cal B}(\phi\rightarrow K^{+}K^{-})/{\cal B}(K^{*0}\rightarrow K^{+}\pi^{-})}$ | 1.1%
Signal and background modelling | ${}^{+2.0}_{-1.8}$%
Total | 3.4%
## 6 Measurement of the $C\\!P$ asymmetry in $B^{0}\\!\rightarrow
K^{*0}\gamma$ decays
The $B^{0}\\!\rightarrow K^{*0}\gamma$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\gamma$ invariant mass
distributions are fitted simultaneously to measure a raw asymmetry defined as
$\mathcal{A}_{\mathrm{RAW}}=\frac{N(K^{-}\pi^{+}\gamma)-N(K^{+}\pi^{-}\gamma)}{N(K^{-}\pi^{+}\gamma)+N(K^{+}\pi^{-}\gamma)}\,,$
(4)
where $N(X)$ is the signal yield measured in the final state $X$. This
asymmetry must be corrected for detection and production effects to measure
the physical $C\\!P$ asymmetry. The detection asymmetry arises mainly from the
kaon quark content giving a different interaction rate with the detector
material depending on its charge. The $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons may also not be produced
with the same rate in the region covered by the LHCb detector, inducing the
$B^{0}$ meson production asymmetry. The physical $C\\!P$ asymmetry and these
two corrections are related through
${\cal A}_{CP}(B^{0}\\!\rightarrow K^{*0}\gamma)={\cal
A}_{\mathrm{RAW}}(B^{0}\\!\rightarrow K^{*0}\gamma)-{\cal
A}_{\mathrm{D}}(K\pi)-\kappa{\cal A}_{\mathrm{P}}(B^{0})\,,$ (5)
where ${\cal A}_{\mathrm{D}}(K\pi)$ and ${\cal A}_{\mathrm{P}}(B^{0})$
represent the detection asymmetry of the kaon and pion pair and $B^{0}$ meson
production asymmetry, respectively. The dilution factor $\kappa$ arises from
the oscillations of neutral $B$ mesons.
To determine the raw asymmetry, the fit keeps the same signal mean and width,
as well as the same mass-window threshold parameters for the $B^{0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal. The yields of the
combinatorial background and partially reconstructed decays are allowed to
vary independently. The relative amplitudes of the exclusive peaking
backgrounds, $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax^{*}\gamma$, $B^{0}_{s}\rightarrow K^{*0}\gamma$ and
$B^{0}_{(s)}\rightarrow K^{+}\pi^{-}\pi^{0}$, are fixed to the same values for
both $B$ flavours.
Figure 3: Invariant-mass distributions of the (a) $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ $\rightarrow$$\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ $\gamma$ and (b)
$B^{0}\\!\rightarrow K^{*0}\gamma$ decay candidates. The black points
represent the data and the fit result is represented as a solid blue line. The
different background components are also shown. The Poisson $\chi^{2}$
residuals [28] are shown below the fits with the $\pm 2\,\sigma$ confidence-
level interval delimited by solid red lines.
Figure 3 shows the result of the simultaneous fit. The yields of the
combinatorial background across the entire mass window are compatible within
statistical uncertainty. The number of combinatorial background candidates is
$2070\pm 414$ and $1552\pm 422$ in the full mass range for the
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\gamma$ decays, respectively. The
contribution from the charmless partially reconstructed decay
$B^{+}\\!\rightarrow K^{*0}\pi^{+}\gamma$ to $B^{0}\\!\rightarrow
K^{*0}\gamma$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\gamma$ is $(10\pm 6)\,\%$ and
$(24\pm 7)\,\%$ of the signal yield, respectively. Furthermore, the charmed
partially reconstructed decays $B\rightarrow K^{*0}\pi^{0}\mathrm{X}$
contribute with $(7\pm 8)\,\%$ and $(9\pm 8)\,\%$ of the signal yield to the
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\gamma$ decays, respectively. The
latter decays give contributions that are mainly located outside the signal
invariant-mass region, as can be seen from Fig. 3.
The value of the raw asymmetry determined from the fit is
$\mathcal{A}_{\mathrm{RAW}}=(0.3\pm 1.7)\,\%$, where the uncertainty is
statistical only.
The systematic uncertainty from the background modelling is determined as
explained in Sect. 4. To address the systematic uncertainty from the possible
$C\\!P$ asymmetry in the background, the yield of the $B^{0}\rightarrow
K^{+}\pi^{-}\pi^{0}$ decay is varied within its measured $C\\!P$ asymmetry
$\mathcal{A}_{C\\!P}(B^{0}\rightarrow K^{*0}\pi^{0})=(-15\pm 12)\%$ [4]. For
the other decays, a measurement of the $C\\!P$ asymmetry has not been made.
The variation is therefore performed over the full $\pm 100\%$ range. The
effect of these variations on $\mathcal{A}_{\mathrm{RAW}}$ gives rise to a
Gaussian distribution centred at $-0.2\%$ with a standard deviation of 0.7%,
thus a correction of $\Delta{\cal A}_{\mathrm{bkg}}=(-0.2\pm 0.7)\%$ is
applied. The systematic uncertainty from the signal modelling is evaluated
using a similar procedure and is found to be negligible. The possible double
misidentification ($K^{-}\pi^{+}\rightarrow\pi^{-}K^{+}$) in the final state
would induce a dilution of the measured raw asymmetry. This is evaluated using
simulated events and is also found to be negligible.
An instrumental bias can be caused by the vertical magnetic field, which
deflects oppositely-charged particles into different regions of the detector.
Any non-uniformity of the instrumental performance could introduce a bias in
the asymmetry measurement. This potential bias is experimentally reduced by
regularly changing the polarity of the magnetic field during data taking. As
the integrated luminosity is slightly different for the “up” and “down”
polarities, a residual bias could remain. This bias is studied by comparing
the $C\\!P$ asymmetry measured separately in each of the samples collected
with opposite magnet polarity, up or down. Table 3 summarises the $C\\!P$
asymmetry and the number of signal candidates for the two magnet polarities.
The asymmetries with the two different polarities are determined to be
compatible within the statistical uncertainties and the luminosity-weighted
average, ${\cal A}_{\mathrm{RAW}}=(0.4\pm 1.7)\%$, is in good agreement with
the $C\\!P$ asymmetry measured in the full data sample.
Table 3: $C\\!P$ asymmetry and total number of signal candidates measured for each magnet polarity. | Magnet Up | Magnet Down
---|---|---
$\int{\cal L}dt$ ($\mbox{\,pb}^{-1}$) | $432\pm 15$ | $588\pm 21$
${\cal A}^{\mathrm{RAW}}$ (%) | $1.3\pm 2.6$ | $-0.4\pm 2.2$
Signal candidates | $2189\pm 65$ | $3103\pm 71$
The residual bias can be extracted from the polarity-split asymmetry as
$\Delta{\cal
A}_{\mathrm{M}}=\left(\frac{\mathcal{L}^{\mathrm{up}}-\mathcal{L}^{\mathrm{down}}}{\mathcal{L}^{\mathrm{up}}+\mathcal{L}^{\mathrm{down}}}\right)\left(\frac{\mathcal{A}^{\mathrm{down}}_{\mathrm{RAW}}-\mathcal{A}^{\mathrm{up}}_{\mathrm{RAW}}}{2}\right)\,,$
(6)
which is found to be consistent with zero $\Delta{\cal
A}_{\mathrm{M}}=(+0.1\pm 0.2)\,\%$. The raw asymmetry obtained from the fit is
corrected by $\Delta{\cal A}_{\mathrm{bkg}}$ and $\Delta{\cal
A}_{\mathrm{M}}$.
The detection asymmetry can be defined in terms of the detection efficiencies
of the charge-conjugate final states by
${\cal
A}_{\mathrm{D}}(K\pi)=\frac{\epsilon(K^{-}\pi^{+})-\epsilon(K^{+}\pi^{-})}{\epsilon(K^{-}\pi^{+})+\epsilon(K^{+}\pi^{-})}\,.$
(7)
The related asymmetries have been studied at LHCb using control samples of
charm decays [33]. It has been found that for $K\pi$ pairs in the kinematic
range relevant for our analysis the detection asymmetry is ${\cal
A}_{\mathrm{D}}(K\pi)=(-1.0\pm 0.2)\%$.
The $B$ production asymmetry is defined in terms of the different production
rates
${\cal A}_{\mathrm{P}}(B^{0})=\frac{R(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})-R(B^{0})}{R(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})+R(B^{0})}$ (8)
and has been measured at LHCb to be ${\cal A}_{\mathrm{P}}(B^{0})=(1.0\pm
1.3)\%$ using large samples of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
[33]. The contribution of the production asymmetry to the measured $C\\!P$
asymmetry is diluted by a factor $\kappa$, defined as
$\kappa=\frac{\int^{\infty}_{0}\cos(\Delta
m_{d}t)e^{-\Gamma_{d}t}\epsilon(t)dt}{\int^{\infty}_{0}\cosh(\frac{\Delta\Gamma_{d}t}{2})e^{-\Gamma_{d}t}\epsilon(t)dt}\,,$
(9)
where $\Delta m_{d}$ and $\Delta\Gamma_{d}$ are the mass difference and the
decay width difference between the mass eigenstates of the $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ system, $\Gamma_{d}$ is the
average of their decay widths and $\epsilon(t)$ is the decay-time acceptance
function of the signal selection. The latter has been determined from data
using the decay-time distribution of background-subtracted signal candidates,
the known $B^{0}$ lifetime and assuming $\Delta\Gamma_{d}=0$. The dilution
factor is found to be $\kappa=0.41\pm 0.04$, where the uncertainty comes from
knowledge of the acceptance function parameters as well as $\Gamma_{d}$ and
$\Delta m_{d}$.
Table 4: Corrections to the raw asymmetry and corresponding systematic uncertainties. Correction to $A_{RAW}$ | | Value [%]
---|---|---
Background model | $\Delta{\cal A}_{bkg}$ | $-0.2\pm 0.7$
Magnet polarity | $\Delta{\cal A}_{\mathcal{M}}$ | $+0.1\pm 0.3$
Detection | $-{\cal A}_{\mathrm{D}}(K\pi)$ | $+1.0\pm 0.2$
$B^{0}$ production | $-\kappa{\cal A}_{\mathrm{P}}(B^{0})$ | $-0.4\pm 0.5$
Total | | $+0.5\pm 0.9$
Adding the above corrections, which are summarised in Table 4, to the raw
asymmetry, the direct $C\\!P$ asymmetry in $B^{0}\\!\rightarrow K^{*0}\gamma$
decays is measured to be
${\cal A}_{CP}(B^{0}\\!\rightarrow K^{*0}\gamma)=(0.8\pm
1.7\,(\mathrm{stat.})\pm 0.9\,(\mathrm{syst.}))\%\,.$
## 7 Results and conclusions
Using an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ of $pp$ collision data
collected by the LHCb experiment at a centre-of-mass energy of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, the ratio of branching fractions
between $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ has been measured to be
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=1.23\pm 0.06\,\mathrm{(stat.)}\pm
0.04\,\mathrm{(syst.)}\pm 0.10\,(f_{s}/f_{d})\,,$
which is the most precise measurement to date and is in good agreement with
the SM prediction of $1.0\pm 0.2$ [8].
Using the world average value ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)=(4.33\pm 0.15)~{}\times 10^{-5}$ [4], the
$B^{0}_{s}\\!\rightarrow\phi\gamma$ branching fraction is determined to be
${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)=(3.5\pm 0.4)\times 10^{-5}\,,$
in agreement with the previous measurement [3]. This is the most precise
measurement to date and is consistent with, but supersedes, a previous LHCb
result using an integrated luminosity of 0.37$\mbox{\,fb}^{-1}$ [16].
The direct $C\\!P$ asymmetry in $B^{0}\\!\rightarrow K^{*0}\gamma$ decays has
also been measured with the same data sample and found to be
${\cal A}_{C\\!P}(B^{0}\\!\rightarrow K^{*0}\gamma)=(0.8\pm
1.7\,\mathrm{(stat.)}\pm 0.9\,\mathrm{(syst.)})\%\,,$
in agreement with the SM expectation of $(-0.61\pm 0.43)$ % [10]. This is
consistent with previous measurements [5, *belle:exp-b2kstgamma:2004,
*cleo:exp-excl-radiative-decays:1999], and is the most precise result of the
direct $C\\!P$ asymmetry in $B^{0}\\!\rightarrow K^{*0}\gamma$ decays to date.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] S. Descotes-Genon, D. Ghosh, J. Matias, and M. Ramon, Exploring new physics in the $C_{7}$-$C_{7}^{\prime}$ plane, JHEP 06 (2011) 099, arXiv:1104.3342
* [2] CLEO collaboration, R. Ammar et al., Evidence for penguin-diagram decays: First observation of $B\rightarrow K^{*}(892)\gamma$, Phys. Rev. Lett. 71 (1993) 674
* [3] Belle collaboration, J. Wicht et al., Observation of $B^{0}_{s}\rightarrow\phi\gamma$ and search for $B^{0}_{s}\rightarrow\gamma\gamma$ decays at Belle, Phys. Rev. Lett. 100 (2008) 121801, arXiv:0712.2659
* [4] Heavy Flavor Averaging Group, Y. Amhis et al., Averages of b-hadron, c-hadron, and tau-lepton properties as of early 2012, arXiv:1207.1158
* [5] BaBar collaboration, B. Aubert et al., Measurement of branching fractions and CP and isospin asymmetries in $B\rightarrow K^{*}(892)\gamma$ decays, Phys. Rev. Lett. 103 (2009) 211802, arXiv:0906.2177
* [6] Belle collaboration, M. Nakao et al., Measurement of the $B\rightarrow{}K^{*}\gamma{}$ branching fractions and asymmetries, Phys. Rev. D69 (2004) 112001, arXiv:hep-ex/0402042
* [7] CLEO collaboration, T. Coan et al., Study of exclusive radiative B meson decays, Phys. Rev. Lett. 84 (2000) 5283, arXiv:hep-ex/9912057
* [8] A. Ali, B. D. Pecjak, and C. Greub, Towards $B\rightarrow V\gamma$ decays at NNLO in SCET, Eur. Phys. J. C55 (2008) 577, arXiv:0709.4422
* [9] K. de Bruyn et al., Branching ratio measurements of $B_{s}$ decays, Phys. Rev. D86 (2012) 014027, arXiv:1204.1735
* [10] M. Matsumori, A. I. Sanda, and Y. Y. Keum, $C\\!P$ asymmetry, branching ratios and isospin breaking effects of $B^{0}\rightarrow K^{*0}\gamma$ with perturbative QCD approach, Phys. Rev. D72 (2005) 014013, arXiv:hep-ph/0406055
* [11] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001
* [12] C. Dariescu and M.-A. Dariescu, $B^{0}\rightarrow K^{*0}\gamma$ decay within MSSM, arXiv:0710.3819
* [13] M. Aoki, G.-C. Cho, and N. Oshimo, Decay rate asymmetry in $B\rightarrow X_{s}\gamma$ as a signature of supersymmetry, Phys. Rev. D60 (1999) 035004, arXiv:hep-ph/9811251
* [14] M. Aoki, G.-C. Cho, and N. Oshimo, $C\\!P$ asymmetry for radiative $B$ meson decay in the supersymmetric standard model, Nucl. Phys. B554 (1999) 50, arXiv:hep-ph/9903385
* [15] A. L. Kagan and M. Neubert, Direct $C\\!P$ violation in $B\rightarrow X_{s}\gamma$ decays as a signature of new physics, Phys. Rev. D58 (1998) 094012, arXiv:hep-ph/9803368
* [16] LHCb collaboration, R. Aaij et al., Measurement of the ratio of branching fractions $\mathcal{B}(B^{0}\rightarrow K^{*0}\gamma)/\mathcal{B}(B_{s}^{0}\rightarrow\phi\gamma)$, Phys. Rev. D85 (2012) 112013, arXiv:1202.6267
* [17] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [18] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [19] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [20] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [21] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [22] GEANT4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [23] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [24] M. Clemencic et al., The LHCb simulation application, Gauss: Design, evolution and experience, J. Phys. Conf. Ser. 331 (2011) 032023
* [25] O. Deschamps et al., Photon and neutral pion reconstruction, LHCb-2003-091
* [26] LHCb collaboration, R. Aaij et al., Measurement of b-hadron masses, Phys. Lett. B708 (2012) 241, arXiv:1112.4896
* [27] HERA-B collaboration, I. Abt et al., $K^{*0}$ and $\phi$ meson production in proton-nucleus interactions at $\sqrt{s}$ =41.6$\mathrm{\,Ge\kern-1.00006ptV}$, Eur. Phys. J. C50 (2007) 315, arXiv:hep-ex/0606049
* [28] S. Baker and R. D. Cousins, Clarification of the use of chi-square and likelihood functions in fits to histograms, Nucl. Instrum. Meth. A221 (1984) 437
* [29] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [30] ARGUS collaboration, H. Albrecht et al., Search for hadronic $b$ $\rightarrow$ $u$ decays, Phys. Lett. B241 (1990) 278
* [31] P. Ball, G. W. Jones, and R. Zwicky, $B\rightarrow V\gamma$ beyond QCD factorization, Phys. Rev. D75 (2007) 054004, arXiv:hep-ph/0612081
* [32] LHCb collaboration, R. Aaij et al., Measurement of $b$ hadron production fractions in $7\,\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions, Phys. Rev. D85 (2012) 032008, arXiv:1111.2357
* [33] LHCb collaboration, Measurement of direct CP violation in charmless charged two-body B decays at LHCb using 2010 data, LHCb-CONF-2011-042
|
arxiv-papers
| 2012-09-03T11:41:25 |
2024-09-04T02:49:34.715661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.\n A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P. N. Y.\n David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H.\n Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van\n Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S. T. Harnew,\n J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M.\n Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, R. S.\n Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, O. Kochebina, I.\n Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y.\n Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.\n von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier,\n A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J.\n Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave,\n U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z.\n Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus,\n F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi,\n A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, M.\n Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K. Sobczak, F. J.\n P. Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U.\n Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R.\n Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Ricardo Vazquez Gomez",
"url": "https://arxiv.org/abs/1209.0313"
}
|
1209.0410
|
∎
11institutetext: G. Teodoro(🖂) and J.H. Saltz 22institutetext: Center for
Comprehensive Informatics, Emory University, GA, USA
22email: {gteodor,jhsaltz}@emory.edu 33institutetext: E. Valle
44institutetext: Recod Lab / DCA / FEEC, State University of Campinas, SP,
Brazil
44email: [email protected] 55institutetext: N. Mariano and W. Meira
Jr 66institutetext: Department of Computer Science, Universidade Federal de
Minas Gerais, MG, Brazil
66email: {nathanr,meira}@dcc.ufmg.br 77institutetext: R. Torres
88institutetext: Recod Lab / DSI / IC, State University of Campinas, SP,
Brazil
88email: [email protected]
# Approximate Similarity Search for Online Multimedia Services on Distributed
CPU–GPU Platforms††thanks: E. Valle and R. Torres thank FAPESP for the
financial support to this work. Preprint — submitted for peer review.
George Teodoro Eduardo Valle Nathan Mariano Ricardo Torres Wagner Meira Jr
Joel H. Saltz
(Received: date / Accepted: date)
###### Abstract
Similarity search in high-dimentional spaces is a pivotal operation found a
variety of database applications. Recently, there has been an increase
interest in similarity search for online content-based multimedia services.
Those services, however, introduce new challenges with respect to the very
large volumes of data that have to be indexed/searched, and the need to
minimize response times observed by the end-users. Additionally, those users
dynamically interact with the systems creating fluctuating query request
rates, requiring the search algorithm to adapt in order to better utilize the
underline hardware to reduce response times. In order to address these
challenges, we introduce hypercurves, a flexible framework for answering
approximate k-nearest neighbor (kNN) queries for very large multimedia
databases, aiming at online content-based multimedia services. Hypercurves
executes on hybrid CPU–GPU environments, and is able to employ those devices
cooperatively to support massive query request rates. In order to keep the
response times optimal as the request rates vary, it employs a novel dynamic
scheduler to partition the work between CPU and GPU. Hypercurves was throughly
evaluated using a large database of multimedia descriptors. Its cooperative
CPU–GPU execution achieved performance improvements of up to 30$\times$ when
compared to the single CPU-core version. The dynamic work partition mechanism
reduces the observed query response times in about 50% when compared to the
best static CPU–GPU task partition configuration. In addition, Hypercurves
achieves $superlinear$ scalability in distributed (multi-node) executions,
while keeping a high guarantee of equivalence with its sequential version —
thanks to the proof of probabilistic equivalence, which supported its
aggressive parallelization design.
###### Keywords:
Descriptor indexing Multimedia databases Information retrieval Hypercurves
Filter-stream GPU
## 1 Introduction
Similarity search is the process of finding among objects stored in a
reference database, those nearest to a query object. In multimedia processing,
both the query and the database objects are represented by a feature vector in
a high-dimensional space. Several choices are available to establish the
notion of distance, Euclidean distance being the most common. That operation
is of fundamental importance for several applications in content-based
multimedia retrieval services, which include not only search engines for web
images Penatti:2012:CSG:2109696.2110003 but also image recognition on mobile
devices springerlink:10.1007/s11263-011-0506-3 , real-time song identification
DBLP:conf/ismir/ChandrasekharSR11 , photo tagging in social networks 4562956 ,
recognition of copyrighted material Valle:2008:FIV:1410140.1410175 and many
others. Nowadays, those services instigate an exciting scientific frontier and
impel a multimillionaire consumer market.
Though those services may appear extremely diverse, they are all founded upon
the use of descriptors, which extract feature vectors from multimedia
documents, thus giving them a perceptually meaningful geometry. The
descriptors allow us to bridge the so called “semantic gap”: the disparity
between the amorphous low-level coding of multimedia, e.g., image pixels or
audio samples, and the complex high-level tasks, e.g. classification or
document retrieval, we need to perform. Looking for similar documents becomes
equivalent to looking for similar vectors. The actual query processing may be
complex, consisting of several phases, but similarity search will often be the
first step and, because of the (in-)famous “curse of dimensionality”, one of
the most expensive.
The success of current content-based multimedia retrieval services depends on
their ability to handle extremely large and increasing volumes of data, and
keep the response times observed by the end-user low. The databases needed to
process even a tiny fractions of the images available in the Web are larger
than the storage capacity of most commodity single-user machines. The great
majority of indexing methods for similarity search, however, were designed to
execute sequentially, and are not able to take advantage of the aggregate
power in distributed environments. Moreover, classical distributed algorithms
tend to ignore the response-time of processing each individual query, and to
concentrate in providing maximum throughput for batches of queries. That
strategy clashes with the online nature of content-based multimedia retrieval
services, because just like on other search engines, the waiting time observed
by individual user requests is critical. Moreover, online interaction between
user and services creates large variations in the query rates submitted to the
system, requiring those systems to adapt continuously to better exploit the
available hardware and lower the response times whenever possible.
In order to address these challenges, in this work, we propose Hypercurves, a
concurrent index built upon the sequential multidimensional index Multicurves
Valle:2008:HDI:1458082.1458181 ; Valle:2010:10.1109:TCE.2010.5606242 and the
concurrent execution environment Anthill anthillII ; hpdc10george .
Multicurves addresses the challenges of approximate similarity queries for
multimedia services, including an optimizing scheduler that adapts the
parallelization regimen online to minimize query response-times under
fluctuating request loads. Hypercurves near-linear speedups and super-linear
scaleup on distributed environments rest upon the fact that Multicurves design
fits extremely well into the filter–stream execution model implemented in
Anthill. Nevertheless, the transition from sequential to parallel indexing
remains very challenging, and depends crucially on the ability of accessing
independently each partition of the data. We demonstrate that this can be done
efficiently, while keeping the algorithms equivalent with very high
probability (Section 4.2).
Hypercurves was published in preliminary form in
Teodoro:2011:APA:2063576.2063651 , where we evaluated its performance in CPU-
only multi-core distributed machines. Though it performed extremely well as
compared to the sequential version, its performance remained constrained by
the compute-intensive task of evaluating distances between the query feature
vector and hundreds of candidate vectors. In this paper, we address that
shortcoming by redesigning Hypercurves for execution on heterogeneous
environments, comprising both CPUs and GPUs (graphical processing units). GPUs
are massively parallel and power-efficient processors, which have found a
niche as accelerators for regular compute-intensive applications. The
utilization of GPUs with Hypercurves, however, is very challenging, since we
are interested in providing low response times under online workloads that
vary throughout execution. GPUs, on the other hand, are fundamentally
throughput-oriented, because they are built as a large collection of low-
frequency computing cores, which are able to process a very large number of
simple operations in parallel.
In Hypercurves, therefore, queries dispatched for execution with a GPU may
observe higher average response times, as they are using a collection of less
powerful GPU computing cores as compared to a CPU core. Thus, a carefully
scheduling is employed to decide the best partition of queries between CPUs
and GPUs in order to minimize the average query response times during the
execution. This problem is specially critical in Hypercurves as the best
partition is affected by the load of requests submitted to the system
throughout the execution.
In this paper, we address these challenges, obtaining a dramatic improvement
on top of the former version of Hypercurves Teodoro:2011:APA:2063576.2063651 .
The techniques we use to schedule tasks for Hypercurves are also generalizable
to other online applications that benefit from GPU accelerations. The key
contributions include:
1. 1.
An improved Hypercurves, able to employ GPUs concurrently to answer a massive
number of requests in very large databases;
2. 2.
A careful design and implementation of optimizations for the parallelization
in hybrid CPU–GPU environments, including cooperation CPU–GPU execution and
asynchronous execution between these devices, as detailed in Section 5. This
version achieved speedups of about 30$\times$ as compared to the CPU-only
single core version of the application;
3. 3.
A dynamic scheduler algorithm for hybrid CPU–GPU environments, which employs
both devices cooperatively in order to minimize response times, and is able to
adapt the application automatically under fluctuating request loads to
optimize response times. When compare to the best static partition, the
scheduler obtained average query response times up to 48% smaller.
All contributions are throughly evaluated in a comprehensive set of
experiments.
The remainder of the text is organized as follows. In the next section, we
discuss content-based multimedia services, examining how the problem of
similarity search is critical to their success. Section 3 summarizes the
algorithmic foundations of this work, by presenting the sequential index
Multicurves, and the parallel framework Anthill, which is used to build our
parallel index Hypercurves. Hypercurves parallelization strategy is detailed
in Section 4 along with an analytical proof of the probabilistic equivalence
between Multi- and Hypercurves. Section 5 introduces progressively
sophisticated execution plans for Hypercurves on heterogeneous CPU–GPU
environments. Section 6 discusses scheduling considering heterogeneous CPU–GPU
environments, under online time constrained applications as Hypercurves. In
Section 7 we present an experimental evaluation of the proposed scheme in many
stress scenarios, proceeding to the conclusions in Section 8.
## 2 Related work
In textual data, low-level representation is strongly coupled with semantic
meaning because the correlation between textual words and high-level concepts
is strong.
In multimedia, by contrast, the low-level coding (pixels, samples, frames) is
extremely distant from the high-level semantic concepts needed to answer the
user queries, precipitating the much debated “semantic gap”. In order to
overcome that difficulty, it is necessary to embed the multimedia documents in
a space where distances represent perceptual dissimilarities: that is the task
of descriptors. The descriptors are an essential first step towards bridging
the gap between the amorphous low-level coding and the high-level semantic
concepts.
Multimedia descriptors are very diverse, including a large choice of
representations for perceptual properties that may help to understand the
documents. Those properties include shape, color and texture for visual
documents; tone, pitch and timbre for audio documents; flow and rhythm of
movement for moving pictures; and many others. The descriptor gives these
perceptual properties a precise representation, by encoding them into a
feature vector. That induces a geometric organization where perceptually
similar documents are given vectors near in the space, while perceptually
distinct documents are given vectors further apart. To establish those
distances, often a simple metric is employed, like the Euclidean or the
Manhattan, but sometimes more complex metrics are chosen
Penatti:2012:CSG:2109696.2110003 .
Especially in what concerns images and videos, the last decade witnessed the
ascent of descriptors inspired by Computer Vision, especially the so-called
_local descriptors_ 10.1109/TPAMI.2005.188 ;
Tuytelaars:2008:LIF:1391081.1391082 , with the remarkable success of SIFT
Lowe:2004:DIF:993451.996342 . As their name suggests, local descriptors
represent the properties of small areas of the images or videos (in opposition
to the traditional _global descriptors_ , which attempt to represent the
entire document in a single feature vector). Their success was followed by the
idea of using compact representations based on their quantization using
codebooks, in the so-called “bag of visual words” model, which became one of
the main tools in the literature 10.1109/CVPR.2010.5539963 .
Regardless of the specific choice, the retrieval of similar feature vectors
becomes a cornerstone operation to almost all systems. That operation can be
used either directly (many early CBIR systems were little more than a
similarity search engine attached to a descriptor space
Smeulders:2000:PAMI:895972 ), either indirectly (similarity search may be part
of a kNN classifier, it can retrieve a preliminary set of candidates to be
refined by a more computationally intensive classifier, etc.). In one way or
another, it remains a critical component, if the system is to be used in real-
world, large-scale databases Liu:2007:PR:262 .
We can formalize the problem of search with multimedia descriptors in the
framework of feature-based processing of similarity queries of Böhm et al.
Bohm:2001:SHS:502807.502809 . The multimedia description algorithm corresponds
to the feature vector extraction, which, formally is a function $F$ that maps
a space of multimedia objects ${Obj}$ into $d$-dimensional real vectors:
$F:{Obj}\rightarrow\mathbb{R}^{d}$ (1)
Now, the dissimilarity between two objects ${obj}_{i}\in{Obj}$ can be
determined by establishing the distance (e.g., Euclidean) between their
feature vectors:
$\operatorname{\Delta}({obj}_{1},{obj}_{2})=\lVert
F(obj_{1}),F(obj_{2})\rVert$ (2)
Given that dissimilarity between objects, we can establish several types of
similarity queries Bohm:2001:SHS:502807.502809 (range, nearest neighbor, $k$
nearest neighbors, inverse $k$ nearest neighbors, etc.). In this work, we are
especially interested in $k$ nearest neighbors queries (kNN, for short). Given
a database $B\subseteq{Obj}$ and a query $q\in{Obj}$, the $k$ nearest
neighbors to $q$ in $B$ are the indexed set of the $k$ objects in $B$ closest
to $q$:
$\begin{split}&\operatorname{kNN}(B,q,k)=\bigl{\\{}b_{1},\ldots,b_{k}\in
B\;\bigl{|}\;\forall i\leq k\\\ &\forall b\in
B\backslash\\{b_{1}\ldots,b_{i}\\},\operatorname{\Delta}(q,b_{i})\leq\operatorname{\Delta}(q,b)\bigr{\\}}\end{split}$
(3)
That defines the _exact_ version of kNN search. As we will see, for large-
scale multimedia services, that definition will have to be relaxed to account
for approximate answers, which allows for dramatic gains in speed.
### 2.1 Prior Art
Efficient query processing for multidimensional data has been pursued for at
least four decades, with a myriad of applications that go far beyond
multimedia feature vector matching. Those include satisfying multi-criteria
searches, and searches with spatial and spatiotemporal constraints
DuMouza:2009:LIS:1612760.1612764 ; Fagin:2001:OAA:375551.375567 ;
Pang:2010:EPE:1825238.1825263 ; Yiu:2009:MTD:1553321.1553325 .
An exhaustive review would be overwhelming and beyond the scope of this
article. The most comprehensive reference to the subject is the textbook of
Samet Samet:2005:FMM:1076819 . The book chapters of Castelli CastelliChapter
and Faloutsos FaloutsosChapter provide a less daunting introduction, more
focused on content-based based retrieval for images. Another comprehensive, if
somewhat old, reference is the survey of Böhm et al.
Bohm:2001:SHS:502807.502809 , which also provides an excellent introduction to
the theme, with a good formalization of similarity queries, the principles
involved in their indexing and their cost models. The book edited by
Shakhnarovich et al. Shakhnarovich:2006:NML:1197919 focuses on computer
vision and machine applications. In what concerns metric methods, which are
able to process non-vector features, as long as they are embedded in a metric
space, the essential reference is the textbook of Zezula et al.
Zezula:2010:SSM:1951721 . Although already decade-old, the survey of Chávez et
al. Chavez:2001:SMS:502807.502808 is also an excellent, comprehensive
introduction to similarity search in metric spaces.
Despite the huge assortment of methods available, those of practical interest
in the context of large-scale content-based multimedia services are
surprisingly few. Because of the “curse of dimensionality” (explained below),
methods that insist on exact solutions are only adequate for low-dimensional
spaces, while multimedia feature vectors often have hundreds of dimensions.
Most methods assume that the implementation uses shared main memory (with
cheap uniform random access), which cannot be the case on the very large
databases we want to address. Other methods, such as the ones based on
clustering, have prohibitively high index building times (with a forced
rebuilding if the index changes too much), being adequate only for moderate-
size static databases.
The focus of multimedia retrieval and classification on approximate techniques
is not just a result of the technical challenge of treating high
dimensionalities. Multimedia descriptors are always intrinsically approximate,
due to the fact the relationship between the visual properties they encode and
the high-level semantic concepts remains limited. In addition, descriptors
almost always employ quantization and averaging to various degrees, making
them approximate also in a numerical and statistical sense. Therefore,
insisting on exact techniques makes no sense. What is needed is a good trade-
off between precision and speed.
Approximation in kNN search may imply different compromises: sometimes it
means finding elements not too far from the exact answers, i.e., guaranteeing
that the distance to the elements returned will bet up to a factor from the
distance to the correct elements; sometimes it means a bounded probability of
missing the correct elements. Sometimes, the guarantee offered is more complex
than that, for example, a bounded probability of finding the correct answer,
provided it is sufficiently closer to the query than the closest incorrect
answer DBLP:conf/stoc/IndykM98 .
Approximation on a bounded factor is formalized as following: given a database
$B\subseteq{Obj}$ and a query $q\in{Obj}$, the $(1+\epsilon)$ $k$ nearest
neighbors to $q$ in $B$ are an indexed set of objects in $B$ whose distance to
the true kNN is at most a $(1+\epsilon)$ factor higher:
$\begin{split}\operatorname{\epsilon-
kNN}&(B,q,k)=\Bigl{\\{}b_{1},\ldots,b_{k}\in B\;\Bigl{|}\\\ &\forall i\leq
k,\;\bigl{[}\forall b\in B\backslash\\{b_{1},\ldots,b_{i}\\},\\\
&\operatorname{\Delta}(q,b_{i})\leq(1+\epsilon)\operatorname{\Delta}(q,b)\bigr{]}\Bigr{\\}}\end{split}$
(4)
Some methods might only guarantee such results with a probability bounded by
some constant. More often than not, however, practical approximative methods
offer no formal guarantees, but just good empirical performance.
If perfect accuracy can be excused, the efficiency requirements remain very
challenging: the method should perform well for high-dimensional data
(hundreds of dimensions) in very large databases (at least millions of
records); it must adapt well to secondary-memory storage, which in practice
means that few random accesses should be performed; it should be dynamic,
i.e., allow data insertion and deletion without performance degradation.
A common pattern found in methods useful for large-scale multimedia is a
strategy of projecting the data onto different subspaces and creating
subindexes for each of those subspaces. The subindexes can be queried more or
less independently, and the results aggregated to find the final answer.
MEDRANK is one of those methods, which projects the data into several random
straight lines. The one-dimensional position in the line is used to index the
data Fagin:2003:ESS:872757.872795 . The method has an interesting theoretical
analysis, establishing that under certain hypotheses, rank aggregation on
straight line projections offers some (lax) bounds on approximation error. The
techniques employed by the algorithm were extremely well succeeded in
moderately-dimensional multi-criteria databases, for which it is still
feasible to search for exact solutions. In those cases, many of the choices
are provably optimal Fagin:2001:OAA:375551.375567 . For high-dimensional
multimedia information, however, the technique fails, mainly due to the lack
of correlation between the distance in the straight lines and the distance in
the high-dimensional space Valle:2008:FIV:1410140.1410175 .
Locality-sensitive hashing (LSH) uses locality-aware hashing functions to
index the data. The method uses several of those “hash tables” at once, to
improve reliability DBLP:conf/stoc/IndykM98 . LSH is backed by an interesting
theoretical background, which allows predicting the approximation bounds for
the index, for a given set of parameters. The well-succeeded family of pStable
locality sensitive hash functions Datar:2004:LHS:997817.997857 has allowed
LSH to directly index Euclidean spaces, and its geometric fundament is also
strongly based on the idea of projection onto random straight lines. LSH works
extremely well when one wants to minimize the number of distances to be
evaluated, and can count on uniform cost random access to the data. However,
in situations where the cost of accessing the data dominates the cost of
computation, its efficiency is compromised. The parameterization of LSH tends
to favor the use of a large number of hash functions (and thus subindexes),
which also poses a challenge for scalability.
An interesting family of solutions employs the fractal space-filling curves.
Like MEDRANK, those methods reduce multidimensional indexing to one-
dimensional indexing, but using more sophisticated projections. The method
upon which we build our work, Multicurves, is one of those methods and it is
explained in more detail on Section 3.1. Since that family of methods is
particularly related to our work, we focus our review on them.
### 2.2 Indexing with Space-filling Curves
Space-filling curves are maps from the unit interval to a hypercube of any
dimensionality Sagan:1994 . Most of those curves are constructed by fractal,
self-similar, recursive procedures. Although the curves are fascinating in
themselves, here we are interested in their ability to induce a “vicinity-
sensitive” total order in the data. With good probability, they preserve
neighborhood relations: if point A is closer to point B than to point C in the
space, that relationship tends to remain in the curve (Figure 1).
Space-filling curves have been implicitly used to perform similarity searches
in multidimensional spaces for a very long time. Indeed, one of the first
multidimensional indexes ever proposed MortonIndexing employed them hidden in
the idea of “bit shuffling”, “bit interlacing” or “bit interleaving”, which
consisted in interleaving the bits of the individual space coordinates to
generate a search key. Interleaving the bits, in fact, induces a type of
space-filling curve called Z-order curve, which explains why the method works
well. However, it was Faloutsos Faloutsos:1988:GCP:53064.53065 the first to
explicitly refer to the concept of curves, and were Faloutsos and Roseman
Faloutsos:1989:FSK:73721.73746 the first to suggest the use of curves other
than the Z-order, first proposing the Gray-code curve and then the Hilbert
curve.
Those pioneering methods worked in a very simple way, using the curve to map
the multidimensional vector onto a one-dimensional key representing the
position in the curve (which we call here _extended-key_). That position was
then employed to perform the search by similarity. For example, when
performing kNN search, a good heuristic is to take the nearest elements in the
curve as the nearest elements in the space, because of the “vicinity-
sensitiveness” explained above.
Unfortunately, points near in the space are not always near in the curve. In
fact, the biggest problem when employing the curves is the existence of
boundary regions where the neighborhood-relation preserving properties are
violated, and points closer in space are placed further apart in the curve
(Figure 1). That issue worsens dramatically as dimensionality grows
Liao:2001:HDS:645484.656398 ; Shepherd99afast .
Figure 1: Space-filling curves provide a “vicinity-sensitive” map: relative
closeness in the space tends to be preserved in the curve (points A, B and C).
However, in some boundary regions, those properties are violated (points A, B
and D).
In order to conquer the boundary effects, Megiddo and Shaft ms-ennib-97
suggested the use of several curves at once. As is done for the multiple
straight lines of MEDRANK, or for the multiple hash-tables of LSH, we build an
independent subindex for each curve. The query is then sought on all
subindexes, in the hope that in at least one of them, it will not fall close
to a boundary region. Megiddo and Shaft present the idea in very general
terms, without describing which types of space-filling curves should be used
and what had to be done to make them different. Therefore, Shepherd et al.
Shepherd99afast developed that idea, specifically recommending the use of
several identical Hilbert curves, where different copies of the vectors are
transformed by random rotations and translations. Whether or not those
transformations could be optimized was left unanswered. Finally, Liao et al.
Liao:2001:HDS:645484.656398 solved the problem of choosing the
transformations, by devising the necessary number of curves and an optimal set
of translations to obtain (lax) bounds on the approximation error in the case
of kNN search.
A depart from those methods was suggested by Mainar-Ruiz and Pérez-Cortés
Mainar-Ruiz:2006:ANN:1170748.1172124 . Instead of using multiple curves, they
propose using multiple instances of the same element in only one curve. Before
inserting those instances in the curve, the algorithm disturbs randomly their
position, to give them the opportunity of falling into different regions of
the curve. In that way even if the query falls in a problematic region,
chances are it will be reasonably near to at least one of the instances. Akune
et al. improved on that method, by proposing a more careful placement of the
instance copies on the curve 5597727 , obtaining thus a significant
improvement in precision.
Another depart was suggested by Valle et al. in Multicurves
Valle:2008:HDI:1458082.1458181 ; Valle:2010:10.1109:TCE.2010.5606242 , which
also employed several curves, but with the important difference that each
curve maps a projection of the vectors onto a moderate-dimensional subspace.
That dimensionality reduction makes for an efficient implementation, reducing
the effects of the “curse of dimensionality”. Because of the exponential
nature of the “curse” it is more efficient to process several low or moderate-
dimensional indexes than a single high-dimensional one. That is explained by
the fact that we do not only gain the intrinsic advantages of using multiple
curves (i.e., elements that are incorrectly separated in one curve will
probably stay together in another), but also, we lower the boundary effects
inside each one of the curves. Multicurves is explained in detail in Section
3.1.
#### 2.2.1 Technical Details
Space-filling curves are fractal curves introduced by G. Peano and D. Hilbert
Sagan:1994 , which provide a continuous surjective map
$C:[0,1]\rightarrow[0,1]^{d}$ from the unit interval to a hypercube of any
dimensionality. Most of those curves are constructed by recursive procedures,
where, in the limit, the curve fills the entire space (Figure 2).
Figure 2: Space-filling curves are usually obtained by a recursive refinement
procedure, resulting in a fractal, self-similar curve. Each refinement step is
called a _order_ of the curve. The curve fills the Continuum at the infinite
limit of that procedure, but for working on digital data, it is not necessary
to reach that limit. The figure shows three successive steps of the Hilbert
curve (left) and the Z-order curve (right).
It was already known (due to a result of E. Netto) that such a mapping $C$
could not be at once bijective and continuous. Dropping injectivity, Peano was
able to construct the first known continuous surjective map from the line to
the space. Interestingly, in the recursive procedure to build the curve, all
finite steps are bijective, but the limiting (and thus, effectively space-
filling) curve becomes self-intersecting. In our applications with digital
data, we can always consider $C$ bijective, because we never reach the limit
needed to deal with the true Continuum.
When using the space-filling curve map in indexing, we are interested in the
pre-images of the query and data points. Using the same notation as before,
for $b_{i}\in B$ and $q\in{Obj}$, we are interested in those
$C^{-1}(F(b_{i}))$ which are close to $C^{-1}(F(q))$ (remember that $F$ is the
function that maps multimedia objects into feature vectors). We call the pre-
image $C^{-1}(x)$ the extended-key of feature vector $x$.
There is a direct relationship between the number of refinements we need to go
through in the recursive curve (called the _order_ of the curve) and the
precision of the data we want to index. If we are employing dyadic curves
(like the Z-order, Gray-order or Hilbert curves), we need $m^{th}$ order
curves to index coordinates of $m$ bits. Remark that the bijective map
$C^{-1}$ preserves the number of bits: from $d$ coordinates of $m$ bits each
to a single extended-key with $d\times m$ bits.
It is important to emphasize that the curve does not have a concrete
representation in the indexes. That is a common source of confusion for those
who get acquainted for the first time with indexing based on space-filling
curves. The curve is an useful abstraction, employed to create the map
$C^{-1}$, which generates “neighborhood-sensitive” extended-keys. Then, the
extended-keys are used in conventional, one-dimensional, indexing structures
(a hash-table, a B-tree, etc.).
The actual computation of $C^{-1}$ depends, of course, on the type of space-
filling curve being employed. For the Hilbert curve, several recursive
algorithms have been proposed, but the most efficient scheme is an iterative
one ButzHilbert . As we have mentioned, for the Z-order curve, the computation
is extremely simple: it suffices to intercalate the bits of the coordinates.
It is interesting to analyze which kind of data can be indexed by the curves.
The curves are able to organize vectors of any fixed-length ordinal data,
provided that the order is the “natural” one: the order of the data is the
same order of the numbers (binary codes) in which they are encoded. Otherwise,
a transformation must be used to translate the vector of data into a vector of
orders.
In the case of multimedia descriptors, we are mainly interested in vectors of
numeric data. When the coordinates are integer, it is easy to see the scheme
works, although the programmer must ensure to deal correctly with negative
numbers in $C^{-1}$. Although less obvious to see, the scheme works with
almost no modification for (IEEE 754) floating-point numbers. Indeed, because
in that encoding the bits of the exponent are in more significant positions
than the bits of the mantissa, the order of the encoded numbers is “natural”.
Again, the only caveat is to deal correctly with the most significant sign
bit, used for negative numbers.
## 3 Background
This section presents an introduction to Multicurves, the algorithmic
foundation, upon which our parallel solution is built; and details Anthill,
the dataflow-based framework employed in the parallelization.
### 3.1 The Sequential Index Multicurves
Figure 3: Multicurves in action. (In black:) The index is created by
projecting the database feature vectors (small dots) onto different subspaces
and mapping each projection in a space-filling curve to obtain the extended-
keys. Each subspace induces an independent subindex, where the vectors are
stored, sorted by extended-keys. (In red:) Searching is performed by
projecting the query feature vector (red star) onto the same subspaces and
computing the extended-keys of the projections. A number (probe-depth) of
candidates closest to the query’s extended-key is retrieved from each
subindex. Finally, the true distance of the candidates to the query is
evaluated and the $k$ closest are returned.
Multicurves Valle:2008:HDI:1458082.1458181 ;
Valle:2010:10.1109:TCE.2010.5606242 is an index for accelerating kNN queries
based on space-filling curves. Its properties make it especially adapted for
large-scale multimedia databases.
As we have seen in Section 2.2, the greatest problem in using space-filling
curves comes from boundary effects brought by the existence of regions where
their neighborhood-relation preserving properties are violated. Different
methods propose different solutions, usually through the simultaneous use of
multiple curves. As we have mentioned, Multicurves is also based on the use of
multiple curves, but with the important improvement that each curve is only
responsible for a subset of the dimensions. Because of the exponential nature
of the “curse”, it is more efficient to process several low-dimensional
queries than a single high-dimensional one.
Algorithm 1 Multicurves index construction
Multicurves index construction is simple (Algorithm 1). The feature vector for
each database element is obtained (almost always, it will be computed
beforehand, so the operation in line 3 just retrieves the corresponding
field). The dimensions of the feature vectors are divided among a certain
number of subindexes based on a space-filling curve. Geometrically, that can
be understood as projecting the feature vector onto a subspace and then
mapping it using a curve that fills the subspace. For didactical reasons, the
algorithm is presented as a “batch” operation, but nothing prevents the index
from being built incrementally, as long as the structure used to back the
sorted lists allows so.
Algorithm 2 Multicurves search phase
The search is conceptually similar: the query is decomposed into projections
(whose subspaces must be the same used during the index construction) and each
projection has its extended-key computed. Then, from each subindex, we obtain
a certain number of candidate elements (probe-depth), whose extended-keys are
the nearest to the extended-key of the corresponding projection of the query.
In the end, we compute the actual distance from those elements to the query
and keep the $k$ nearest (Algorithm 2).
The index creation and search processes are illustrated in Figure 3.
It should be noted that in the scheme shown above, for simplicity sake, we
supposed that both the query and the database elements are associated with a
single feature vector by the description function $F()$. The extension of the
algorithm is trivial for descriptors (like local descriptors) that generate
several vectors per multimedia object, but bear in mind that the each vector
is indexed and queried independently (for example, if a query object generates
10 feature vectors, the kNN search will produce 10 sets of $k$ nearest
neighbors, one for each query vector). The task of taking a final decision
(classification result, retrieval ranking) from those multiple answers is very
application-dependent and beyond the scope of our article, which is concerned
with the basic infra-structure. Here, we are concerned in achieving
efficiently good results for each individual query vector.
In an experimental evaluation Valle:2008:HDI:1458082.1458181 ;
Valle:2010:10.1109:TCE.2010.5606242 on high-dimensional feature vectors,
Multicurves compared favorably to the state of the art, represented by the
methods of Liao et al. Liao:2001:HDS:645484.656398 and Mainar-Ruiz and Pérez-
Cortes Mainar-Ruiz:2006:ANN:1170748.1172124 , presenting a better compromise
between precision and speed. It also performed well
Valle:2010:10.1109:TCE.2010.5606242 , when compared to LSH
Datar:2004:LHS:997817.997857 , presenting an equivalent compromise between
precision and number of distances computed, but performing fewer random
accesses.
### 3.2 The Parallel Environment Anthill
Anthill hpdc10george ; anthillII ; cluster09george ;
springerlink:10.1007/s10586-010-0151-6 ; anthill is a run-time system based
on the filter–stream programming model datacutter and, as such, applications
are decomposed into processing stages, called filters, which communicate with
each other using unidirectional streams. At run time, Anthill spawns, on the
nodes of the cluster, instances of each filter, which are called transparent
copies, and automatically handles communication and state partitioning among
those copies anthillII .
When developing an application using the filter–stream model, both task and
data parallelism are exploited. Task parallelism is achieved as the
application is broken up into a set of filters, which perform independently,
accomplishing the application functionality in a pipeline fashion. Data
parallelism, on the other hand, is obtained by creating transparently multiple
copies of each filter and distributing the data to be processed among them
(Figure 4).
Figure 4: The architecture of an Anthill application. Filters (columns)
cooperate to process the data. Their communication is mediated by
unidirectional streams (arrows). The filters are instantiated in transparent
copies (circles) automatically by Anthill’s runtime. The non-blocking I/O flow
and event scheduling is also handled by Anthill.
Anthill provides an event-oriented filter programming abstraction, deriving
heavily from the message-oriented programming model coyote ; xkernel ;
welsh01sosp . The streams that establish the communication between filters
generate input events, which must be handled. The programmer provides handling
functions for those events. Anthill runtime instantiates those functions and
controls the non-blocking I/O flow to keep the system running. It is a
dataflow model, where event handling amounts to asynchronous and independent
tasks. Because the filters are multithreaded, multiple tasks can be spawned
when there are enough pending events and computational resources. That feature
is essential both in exploiting the full capability of current multi-core
architectures, and in spawning tasks on multiple devices in heterogeneous,
CPU–GPU equipped platforms. That flexibility is accomplished by allowing the
programmer to provide, for the same event, handler functions targeting
different devices which can be invoked by the scheduler to use the appropriate
processor.
Figure 5: The architecture of a single filter. Input streams (top blocks)
generate events that must be handled by the filter. Different handler
functions (dashed round boxes) can be provided by the programmer for each type
of event and processing unit. The event scheduler coordinates the filter
operation, dequeuing the input events and invoking the handling functions
according to the available processing units (round boxes). As processing
progresses, data is sent to the output streams (bottom blocks), generating
events on the next filter (not shown).
Figure 5 illustrates the architecture of a typical filter (a single
application will be composed of several of those). It receives data from
multiple input streams (In1, In2, and In3), each generating its own event
queue, with handler functions associated to each of them. As shown, those
functions are implemented targeting different types of processors. The Event
Scheduler, depicted in the picture is responsible for consuming events from
the queues, invoking appropriate handlers according to the availability of
computational resources. As events are consumed, eventually some data is
generated on the filter that will be forwarded to the next filter. As those
data arrive in the next filter, they will trigger input events there. All
filters run in parallel. Communication between filters, although not shown in
the figure, is also managed by the run-time system.
When events are queued, they are not immediately assigned to a processor.
Rather, that occurs on-demand, as devices become idle and request new events
to process. In the current implementation, the demand-driven, first-come,
first-served (DDFCFS) task assignment policy is used as default strategy of
the Event Scheduler.
The first decision for the DDFCFS policy is to select from which queue to
execute events; this decision is made in a round-robin fashion, provided the
event has a handling function compatible with the available processor. The
oldest event on the selected queue is dispatched for processing. That simple
approach guarantees assignment to different devices according to their
relative performance in a transparent way, as processors will consume events
in proportion to their capacity to process them.
## 4 The Distributed Index Hypercurves
In this section, we discuss how Multicurves (Section 3.1) has been redesigned
for efficient execution on distributed environments, focusing on the CPU-only
version of the application (details of the GPU-based presented in Section 5).
Further, we present a proof of the probabilistic equivalence between
Multicurves and Hypercurves (Section 4.2). The proof is essential for the
efficiency of the scheme, because in Hypercurves, the database is partitioned
without overlapping among the nodes in the execution environment. Search is
performed locally in the subindexes managed by each node, and a reduction
stage merges the results. The cost of the algorithm is dominated by the local
searches, which are further dependent on the probe-depth used (the number of
candidates to retrieve from each subindex). When using the same probe-depth of
the sequential algorithm for each local index of the distributed environment,
the answer of Hypercurves is guaranteed at least as good as the sequential
algorithm. However, that is an extremely pessimistic and costly choice for the
local indexes probe-depth: we show that the quality of Hypercurves is
equivalent to that of Multicurves with very high probability, when using a
probe-depth slightly higher than the original probe-depth divided by the
number of nodes.
The ability to avoid data replication improves the scalability of the
solution. The user can further modify the probe-depth of the parallel
algorithm according to Equation 5 (Section 4.2), to guarantee that the quality
of Hypercurves is equivalent to that of Multicurves with any desired
probability.
### 4.1 Hypercurves Parallelization Strategy
Hypercurves Teodoro:2011:APA:2063576.2063651 is a concurrent index built upon
the sequential multidimensional index Multicurves
Valle:2010:10.1109:TCE.2010.5606242 and the concurrent execution environment
Anthill anthillII , in order to provide approximate similarity search support
for large-scale online multimedia services. Therefore, Hypercurves addresses
both the need to scale the database to sizes beyond the capability of a single
machine, and the need to keep the answer times as short as possible.
Hypercurves strategy is to partition the database among the nodes (_filter
copies_ in the nomenclature of Anthill) of the distributed environment. The
queries are broadcast to all filter copies, which find a local answer in their
database subsets. The local answers are then reduced to a global answer in a
later merge step.
To better exploit Anthill execution environment, Hypercurves employs four
types of filter, organized in two parallel computation pipelines (Figure 6).
Figure 6: Hypercurves parallelization design. Four filter types are involved:
IRR, which reads data from the database and dispatches them to the IHLS to be
indexed; QR, which reads queries from the user and dispatches them to the IHLS
to be processed; IHLS, which provides a “local” index and query processing,
for a subset of the data; Aggregator, which collects local kNN answers to the
queries and aggregates them into a global kNN answer. Transparent copies of
those filters are instantiated as needed by Anthill’s runtime. Several types
of streams are used in the communication between those copies: for example,
during search, a query is broadcast from QR to all copies of IHLS; then all
local answers relative to that query are sent to the same Aggregator filter,
using the “labeled stream” facility.
The first pipeline is conceptually an index builder/updater, with the filters
_Input Reader_ (IRR) and _Index Holder/Local Searcher_ (IHLS). IRR reads the
feature vectors from the input database and partitions them among the copies
of IHLS, which add the vectors received to their local index, according to
Algorithm 1. The filters execute concurrently, and after the input is
exhausted, interact to update the database.
The second pipeline, which is conceptually the query processor, contains three
filters: (i) _Query Receiver_ (QR); (ii) IHLS (shared with the first
pipeline); and (iii) _Aggregator_. QR is the entry point to the search server,
receiving and broadcasting the queries to all IHLS copies. For each query,
IHLS instances independently perform the search on their local index
partition, retrieving $k$ nearest _local_ feature vectors just like the
sequential Multicurves (Algorithm 2). The final answer is obtained by the
Aggregator filter, which reduces the local answers into $k$ _global_ nearest
vectors. Since several Aggregator filter copies may exist, it is crucial that
the messages related to a particular query (same query-id) be sent to the same
Aggregator instance. That is guaranteed by making full use of Anthill Labeled-
Stream communication policy, which computes the particular copy of the
Aggregator filters that will receive a given message sent from IHLS based on a
$hash$ computed in the query-id. Therefore, in this context, query-id
corresponds to the label of the message. The transaction between IHLS and
Aggregator is very similar to a generalized parallel data reduction
Yu:2000:ARP:335231.335238 , except that it outputs a list of values for each
output, and that an arbitrary number of reductions are executed in parallel.
Hypercurves exploits all four dimensions of parallelism: task, data, pipeline,
and intra-filter. Task parallelism occurs as the two pipelines are executed in
parallel (e.g., index updates and searches). Data parallelism is achieved as
the database is partitioned among the IHLS filters copies. Pipeline
parallelism results from Anthill ability to execute in parallel the filters of
a single computational pipeline (e.g., IRR and IHLS for updating the index).
Intra-filter parallelism refers to a single filter copy being able to process
events in parallel, thus, efficiently exploiting modern multi- and many-core
computers.
The broadcast from QR to IHLS has little impact on performance, because the
cost is dominated by the local searches. Therefore, the communication latency
is offset by the computation speedups. The disproportionate cost of those
searches has prompted a GPU-based implementation (Section 5), which in turn
raised interesting challenges to the scheduling of the pipelined events,
leading to another important contribution of this work (Sections 6).
The cost of local searches depends critically on the probe-depth used (the
number of candidates to retrieve from each subindex). Hypercurves can be made
assuredly equivalent to Multicurves, by employing on each parallel node a
probe-depth at least as large as the one used in the sequential algorithm.
However, this over-pessimistic choice is unnecessarily costly and can be
significantly improved, as we will see in the next section.
### 4.2 Probabilistic Equivalence Multicurves–Hypercurves
Multicurves is based upon the ability of space-filling curves giving a total
order to data. That makes each subindex a sorted list where a number of
candidates can be retrieved and then verified against the query in order to
obtain the $k$ nearest (Algorithm 2).
In Hypercurves, the index is fragmented, with each IHLS filter copy having
available only a subset of the database: a single filter cannot warrant the
equivalent approximate $k$ nearest neighbors. That is the role of the
Aggregator filter: collecting the local best answers and returning a final
solution.
In terms of equivalence between Multi- and Hypercurves it matters little how
the candidates are distributed among the IHLS instances, because the reduction
steps performed after the candidates are selected are conservative: they will
never discard one of the “good” answers once it is retrieved. Either Multi- or
Hypercurves will only miss a correct answer if they fail to retrieve it from
the subindexes. Therefore, Hypercurves can be made guaranteedly at least as
good as Multicurves by employing on each IHLS filter copy the same probe-depth
used on the sequential Multicurves. However, that is costly, and, as we will
shortly see, over-pessimistic.
Consider the same database, either in a Multicurves’ subindex with probe-depth
$=2\Phi$, or partitioned among $\ell$ Hypercurves’ IHLS filter copies, each
with probe-depth $=2\varphi$ (even probe-depths make the analyses more
symmetric, although the argument is essentially the same for odd values). For
any query, the candidates that would be in a single sorted list in Multicurves
are now distributed among $\ell$ sorted lists in Hypercurves. In more general
terms, we start with a single sorted list and retrieve the $2\Phi$ elements
closest to a query vector. If we distribute randomly that single sorted list
into $\ell$ sorted lists, how many elements must we retrieve from each of
those new lists (i.e. which value for $2\varphi$ must we employ) to ensure
that none of the originally retrieved elements is missed? Note that: (i) due
to the sorted nature of the list, the elements before the query cannot
exchange positions with the elements after the query; (ii) no element of the
original list can be lost as long as all those $2\ell$ “half-lists” are
shorter than $\varphi$. Those observations, which are essential to understand
the equivalence proof, are illustrated in Figure 7.
Figure 7: The probabilistic equivalence between Multi- and Hypercurves
corresponds to the following model. In a sorted list, for the query $q$, we
retrieve $\Phi$ elements $<q$ and $\Phi$ elements $\geq q$. If we distribute
the elements of that list randomly into $\ell$ sorted lists, how many
$2\varphi$ elements must we retrieve in each of those new lists, in order to
ensure missing none of the original ones. Because the elements $<q$ and $\geq
q$ cannot exchange positions, each “half-list” can be analyzed independently.
In the example shown, the equivalence is not guaranteed, because some elements
“spill over” the $\varphi$ limit in two of the half-lists.
Due to (i), we can analyze each half of the list independently. The
distribution of the elements among the $\ell$ lists is given by a Multinomial
distribution with $\Phi$ trials and all probabilities equal to $\ell^{-1}$.
The exact probability of no list being longer than $\varphi$ involves
computing a truncated part of the distribution, but the exact formulas are
exceedingly complex and little elucidative. We can, however, bound it from
below Mallows:1968:Biometrika:55 with:
$P\left(List_{max}\leq\varphi\right)\geq 1-\left(\Phi\times
P\left(List_{i}>\varphi\right)\right)$ (5)
where $List_{i}$ is an arbitrary single component of the equiprobable
Multinomial, which, by construction has a Binomial distribution for $\Phi$
trials and success rate of $\ell^{-1}$. Thus, the probability of any miss on
any of the $2\ell$ half-lists is bounded from above by:
$\begin{split}&1-\operatorname{Max}\left[0;1-\Phi\sum\limits_{k=\varphi+1}^{\Phi}\binom{\Phi}{k}\left(\frac{1}{\ell}\right)^{k}\left(1-\frac{1}{\ell}\right)^{\Phi-k}\right]^{2}\\\
&=1-\operatorname{Max}\left[0;1-\Phi\left(1-I_{1-\ell^{-1}}\left(\Phi-\varphi,\varphi+1\right)\right)\right]^{2}\end{split}$
(6)
where $I()$ is the regularized incomplete Beta function. That probability
tends to zero for very reasonable values of $\varphi$, still much lower than
$\Phi$. That is more easily seen if we make
$\varphi=\left(1+\varsigma\right)\lceil\Phi/\ell\rceil$, i.e., if we
“distribute” the probe-depth among the filters, adding a “slack factor” of
$\varsigma$. For all reasonable scenarios, the probability tends to zero very
fast, even for small $\varsigma$ (Figure 8).
Figure 8: Equivalence between sequential Multicurves with a probe-depth of
$2\Phi=256$ and parallel Hypercurves with distributed probe-depth of
$2\varphi$, with $\varphi=\left(1+\varsigma\right)\lceil\Phi/\ell\rceil$ and
$\ell$ = the number of filter copies. The probability of missing any of the
candidate vectors drops sharply to zero, for values of $\varsigma$ that are
still very small.
## 5 Hypercurves in Heterogeneous CPU–GPU Environments
The use of GPUs as general computing processors is a strong trend in high
performance computing, and represents a major paradigm shift towards massively
parallel and power efficient systems. GPus have an impressive computing power,
but taking advantage of them is challenging, especially for online services
like Hypercurves.
In this section, we introduce the design and implementation of Hypercurves for
heterogeneous, CPU–GPU environments, with a set of optimizations to maximize
its performance. We anticipate that the use of GPUs in this context raises
important challenges, especially in what concerns the optimization of response
times under fluctuating request loads. Those dynamic aspects are discussed in
Section 6.
### 5.1 GPU-based IHLS Implementation
Figure 9: The three progressively sophisticated proposed schemes for
implementing Hypercurves in CPU–GPU heterogeneous environments. (a) Grouping
queries to fully utilize the GPU. (b) Additionally, employing a double-buffer
to avoid idleness either on CPUs and GPU. (c) Additionally, using CPUs
unutilized capacity to perform kNN in an alternative processing path.
In the Hypercurves pipeline, the IHLS filter is responsible for performing the
compute-intensive operations of the application. Consequently, it is the phase
of Hypercurves to be accelerated using GPUs. The computation performed by IHLS
has granularity per user request (query), and the query execution depends on
the probe-depth (the number of candidates returned from the subindexes for
further kNN computation, explained on Section 3.1) as the distances from the
query to all candidates are calculated.
However, a single query is insufficient to fully utilize a GPU, because probe-
depths assume small values, around a few hundred elements. Therefore, the
first step towards using GPUs to accelerate this filter was to modify IHLS to
group an arbitrary number of queries (group-size), which are then dispatched
together for execution in a GPU. Our parallelization uses the CPU to perform
the operations related to query grouping, while the GPU is employed during the
kNN search. The execution of the IHLS filter is divided into stages (Figure
9–a), explained below:
Retrieve candidates:
returns the probe-depth vectors closest to the query from each subindex (Lines
3 to 8 in Algorithm 2). Those candidates are accumulated in a continuous block
of memory (the _CPU buffer_). That operation is repeated for each query in the
group. At the end, the buffer will contain group-size sets of candidates, each
set with probe-depth vectors.
Copy to GPU buffer:
copies the buffer from the system memory (_CPU buffer_) to the GPU memory (the
_GPU buffer_);
Perform kNN search:
computes the kNN search for all group-size sets of candidates in parallel,
comparing each query to its own candidate set. At the end, returns the results
in _kNN results_ , which will have group-size sets, each with $k$ answers;
Send to aggregator:
copies the results sets from the GPU memory and sends them downstream to the
Aggregator filter, which is the next stage in processing pipeline.
The operation performed in _retrieve candidates_ is a binary search on each
subindex, with very irregular access patterns, dependent on both the query and
the database. Since it is not realistic to assume that an entire subindex
would fit into the GPU memory, the implementation of that stage in the
accelerator is not worthwhile: it would require intensive data transfer
between CPU and GPU. Fortunately, its computational cost is low, due to the
logarithmic growth of the binary search with respect to database size. It can
be executed fairly efficiently on CPU.
_Perform kNN search_ is a special version of the traditional kNN, which
compares several queries against the same database Garcia_2008_CVGPU . Here,
however, each query is independently compared to a different subset of the
data, which consists of the candidates just retrieved from the subindexes. In
addition, the number of queries available to execute (group-size) will vary
between executions. The group-size can be optimized according to a metric of
interest, for instance, average response time as is detailed in the following
sections.
The kNN search itself is implemented using two GPU computing $kernels$: (i)
_CalcDist_ , which calculates the distance between each query and its
candidates; (ii) _FindTopK_ , which selects the $k$ nearest vectors among the
candidates, and moves them to the top positions in the list.
The other operations, _copy to GPU buffer_ and _send to aggregator_ involve
data transfers between the system memory and the GPU memory. The latter also
involves dispatching the results for further processing downstream, in the
Aggregator filter.
### 5.2 Overlapping CPU and GPU phases
In the design just discussed, the IHLS steps are performed sequentially,
resulting in no overlapping between CPU and GPU computations. However, this
approach creates idle periods in both devices. The GPU has to wait until its
data buffers are filled by the CPU with the candidates retrieved from the
subindexes; and the CPU has to wait until the kNN execution is completed on
the GPU to transfer the next batch.
In order to reduce idleness, we propose to overlap those operations by
pipelining them and using a double-buffer scheme for the communication between
operations composing the IHLS filter (Figure 9–b). It then allows the CPU to
accumulate the sets of candidates from incoming queries, while the GPU may be
asynchronously processing the kNN search for the previous batch of queries.
Additionally, this design employs several CPU threads to retrieve candidates
from the subindexes in parallel. That strategy significantly reduces
Hypercurves execution times, because it maximizes the utilization of the GPU.
Similar double-buffer schemes Sancho08-doublebuffering have being used in
multi-core CPU architectures, for instance, to overlap useful computation with
data transfers among different levels of the memory hierarchy. Here, on the
other hand, it is employed to enable pipelining and, consequently, to overlap
execution of different code sections on different devices of the hybrid
environment.
### 5.3 Cooperative execution on CPU and GPU
In the design of Hypercurves described so far, the CPU cores perform only
operations which are not appropriate for GPUs: the retrieval of candidates,
the data transfers, and the task of coordinating the GPU execution. However,
as discussed, the cost of those CPU computations is low, and may not be
sufficient to completely occupy all CPU cores available, especially in current
multi-core architectures. To better utilize the CPUs, we propose to employ
them also to perform the kNN computations. However that must occur only when
they would otherwise be idle, meaning that the next buffer with query
candidates is ready for the GPU execution.
In our cooperative CPU/GPU scheduling solution (Figure 9–c), the CPU execute
both its ordinary tasks (main processing path) and the kNN search (alternative
processing path), giving the former higher priority. Thus, the CPU will follow
the first path until the buffers used by the GPU are completely full, becoming
afterwards available to process queries on the second path. The double-
buffering scheme employed reduces the possibility of the GPU be kept waiting:
even when the CPU is momentarily held on kNN search tasks, as the CPU will
usually have enough time to fill the next buffer before the GPU is done with
the current one.
The use of hybrid CPU–GPU computation, as well as the task scheduling problem
that arises of employing those processors, has received increasing attention
in the last few years Ding:2008:UGP:1367497.1367732 ; qilin09luk ; mars ;
merge ; hpdc10george ; Teodoro-IPDPS2012 ; hartley ; ravi2010compiler ;
Diamos:2008:HEM:1383422.1383447.Harmony . Those works, however, assume that
implementations of all stages of the computation are available for both
devices, and try to minimize the execution times by employing CPU–GPU tasks
partition using either static offline mars ; Ding:2008:UGP:1367497.1367732 ;
qilin09luk or online hpdc10george ; ravi2010compiler strategies. Those
strategies may work well in several contexts, including ones with divisible
workloads, such as generalized reductions or MapReduce computations. However,
they are restricted to cases where both CPU and GPU implementations are
available for each stage. That contrasts to Hypercurves, where the CPU is used
to assist the GPU in tasks that are not appropriate for acceleration, besides
perform its own compute-intensive tasks during periods of idleness. Therefore,
Hypercurves’ task partitioning has to assign dynamically the priorities to
types of tasks the CPU performs.
The next sections will further elaborate on that problem of task partition to
minimize response times in online applications. To the best of our knowledge,
ours is the first work to address that problem on CPU–GPU computation
environments.
## 6 Response Time Aware Task Partition in Heterogeneous CPU–GPU Platforms
As discussed, the online nature of Hypercurves poses the interesting challenge
of optimizing the response time of the individual queries, while using GPUs,
which are throughput-oriented devices. In order to reduce response times, it
is necessary to perform a dynamic partition of the load among CPU and GPU
under a fluctuating user request load. That partition in Hypercurves is
directly related to the group-size used by IHLS, which will determine the
number of tasks queued for GPU execution and, consequently, those remaining
for CPU computation. The optimal configuration for each load intensity depends
on complex factors, including the hardware architecture, application
parameters, and dataset properties. Such complex optimization is beyond the
ability of any static configuration.
Formally, the problem can be defined as such: given a set of $n$ tasks
$t_{1}\dots t_{n}$ within a filter, $m$ processors $p_{1}\dots p_{m}$
allocated to that filter, the execution time of each task in each processor
denoted by $e_{ij}$, and the wait time of each task to be processed by each
processor as $w_{ij}$, we want to determine the schedule $x_{ij}$ where $1\leq
i\leq n$ and $1\leq j\leq m$. Each $x_{ij}$ is a binary variable indicating
whether the tasks $i$ is executed by processor $j$. Notice that, for a given
task $t_{i}$, there exists just one $x_{ij}$, $1\leq j\leq m$ that is nonzero.
Since there is no task reordering within a processor, the $w_{ij}$ is defined
as the sum of the processing time of tasks $e_{xj}$ computed by a given
processor $j$ before it gets to execute $x$.
Our goal is to find a schedule that minimize $E$, the average execution time
of the tasks within each application filter. For simplicity, and since the
filters execute independently, we state the problem in terms of a single
filter. In general, the execution time takes the form:
$E=\operatorname{avg}_{i=0...n}\sum_{i=1}^{m}x_{ij}\times(e_{ij}+w_{ij})$ (7)
The execution time $e_{ij}$ varies according to the processor used. For the
GPU, it has two components: a buffer-waiting time, in which the task remains
buffered in the CPU memory; and the computation, which involves both the data
transfer between CPU and GPU, and the execution itself. The cost of both
components further depends on the task granularity (group-size).
Scheduling in such environment is difficult for various reasons: (i) the
problem is NP-hard, since the bin packing problem, widely known to be NP-
hards, is a very simplified version of the scheduling problem discussed (the
equivalence is true when waiting times are zero); (ii) the tasks are created
at run time, making static scheduling impossible; (iii) estimating the
execution time ($e_{ij}$) of a task has been an open challenge for decades
582071 .
The optimization is not only complex, but also dynamic, varying
instantaneously as the request load changes. Thus, exact solutions would be
too costly to be practical. We propose instead a light, greedy algorithm that
calculates the scheduling throughout the execution, optimizing the granularity
of tasks assignment to GPU (group-size), and the tasks assignment between CPU
and GPU.
The solution we propose is driven by the idea that GPU use is advantageous
only when the aggregate throughput delivered by the available CPU computing
cores is insufficient to promptly process the demand. That approach diverges
from the usual parallelizations for heterogeneous environments, where GPUs are
systematically preferred for their capacity to achieve highest throughputs.
However, as discussed before, the use of GPUs improve throughput at the cost
of increasing the average tasks processing time.
Therefore, it is not worthwhile using GPUs in online application unless the
tasks response times are dominated by waiting time, which occurs when the
request rates submitted to the system are above the CPU throughput. On the
other hand, under high request loads, it becomes beneficial using GPUs to
increase the system throughput: though the processing time of a single task
will be higher, due to the overheads of starting up a GPU computation, the
average response times of the system will be reduced, because the better
throughput will lower or even eliminate waiting times.
Algorithm 3 Dynamic Task Assigner for Heterogeneous Environments — DTAHE.
The solution we propose is called Dynamic Task Assigner for Heterogeneous
Environments (DTAHE) and is presented in Algorithm 3. The DTAHE is executed in
parallel with the computing threads in each filter copy, and loops until the
upstreams filters notify that execution has finished — the
$\mathit{EndOfWorks}$ message. Line 2 of the algorithm will check the number
of events/requests queued in the filter ($\mathit{ready}$), and further decide
whether the thread should follow the event processing using the CPU or the GPU
(line 3).
If the GPU buffer is full or if the CPU computing cores available are enough
to process all queued events, an event is dequeued and dispatched to the CPU
(line 4). Otherwise, an event is dequeued, the items to be compared to that
query are retrieved from the subindexes and stored in the GPU data buffer
(Lines 6–7). The scheduler decides, then, if the buffer should be sent for the
execution in the GPU, which happens when it becomes full, when the GPU becomes
idle (which we want to avoid), or if the number of ready events becomes low
enough so that the remaining ones can be processed in the CPUs. Each GPU has
one CPU computing core reserved for managing purposes (coordination, buffer
copying, etc.), but, when that GPU has no buffers ready to be processed, its
dedicated CPU core become available to perform other tasks. The motivation,
again, is to get the GPU running as often and as early as possible, avoiding
as much as possible to keep it waiting, while keeping the CPU available to
process some queries when the GPU is busy.
DTAHE solves at once the problem of dynamically distributing the tasks among
CPUs and GPUs and determining the optimal group-size. The latter is done
implicitly, as buffers are dispatched for GPU computation either when they are
full (maximum group-size, corresponding to situations of high loads and
maximum throughput) either when the GPU becomes idle (smaller group-sizes, as
the load becomes lighter). When the load becomes light enough to be dealt with
just the CPU cores, the GPU is not employed, again keeping response times as
short as possible. On the other hand, as the load increases, processing
migrates increasingly to the GPU.
The problem of optimizing group-sizes is similar to optimizing the parallelism
granularity in parallel loops or _doall_ -like operations, which have been
extensively studied Chen:1990:ISG:325096.325150 . Most works on that area,
however, focus on applying loop transformations according to the available
resources, in order to adjust the granularity, reducing the synchronization
overhead and do not take into account load variability. The best parallelism
is often beyond any static tuning. That has motived several recent works,
which focus on runtime transformations Aleen:2010:IDE:1693453.1693494 ;
Blagojevic:2007:RSD:1316088.1316319 ;
springerlink:10.1007/978-3-540-31832-3-12 . Those interesting works aim at
parallelism tuning for a lower level and are only concerned with CPU-based
machines, thus being complementary to the strategy we propose.
## 7 Experimental results
In this section, we evaluate the impact of our propositions on Hypercurves’
performance. The experiments have been performed using two setups of machines.
The first setup consisted of 2 PCs connected through a Gigabit Ethernet, each
with two quad-core AMD Opteron 2.0 GHz 2350, 16 GB of main memory, and one
NVidia GeForce GTX260 GPU. The second setup was a 8-node cluster connected
with Gigabit Ethernet, each node being a PC with two quad-core hyper-threaded
Intel Xeon E5520, 24 GB of main memory, and one NVidia GeForce GTX470. All
machines run Linux.
The main database used to evaluate our algorithm contained 130,463,526 SIFT
local feature vectors Lowe:2004:DIF:993451.996342 , with 128 dimensions each.
Those feature vectors have been computed from 233,852 background images from
the Web, and 225 foreground images from our personal collections. The
foreground images have been used to compute sets of feature vectors that must
be matched, while the background images have generated the feature vectors
used to confound the method. The foreground images, after strong
transformations (rotations, changes in scale, non-linear photometric
transformations, noise, etc.) have also been used to create 187,839 query
feature vectors. Due to the number of evaluations performed, we have also
employed smaller partitions of that main database, in order both to achieve
feasible experimentation running times, and to emphasize certain aspects
(e.g., overhead) in specific experiments.
The experiments concentrate on issues of efficiency, since, as demonstrated in
Section 4.2, Hypercurves, with very high probability, has the same results of
Multicurves. Thus, by construction, it inherits the good trade-off between
precision and speed of Multicurves Valle:2010:10.1109:TCE.2010.5606242 .
### 7.1 The Impact of Task Granularity
As discussed in Section 5.1, task granularity has an important impact on the
GPU acceleration achieved. In Hypercurves, it is dictated by the group-size:
the number of queries aggregated to be processed concurrently within each GPU
execution (Section 5.1).
This study has employed a subsample of 1,000,000 feature vectors randomly
selected from the main database, and 30,000 queries. A smaller dataset has
been used in order to provide shorter execution times, which are more
appropriate to highlight any potential overheads in our solution. The entire
set of queries has been dispatched to the filter QR at the beginning of the
execution, creating a high number of concurrent queries ready to execute in
the IHLS filter throughout execution (check the filter enchainment in Figure
6). In addition, in this initial evaluation, a single machine has been
employed.
Figure 10(d) presents Hypercurves throughput (in queries per second), for 3
typical choices of probe-depth, as group-size varies up to the limit of
queries that can be accommodated within the GPU memory. Group-size is kept
fixed during each single execution. We focus on the impact of task grouping,
expressed on the “Grouping only” curves in the graphs. The other results
presented in the same graphs are discussed on the following sections.
(a) 2-node AMD — Probe-depth=250
(b) 8-node Intel — Probe-depth=250
(c) 2-node AMD — Probe-depth=350
(d) 8-node Intel — Probe-depth=350 Figure 10: Hypercurves performance as
group-size varies, for multiple probe-depth values, in both machine
configurations (2 nodes of AMD/GTX260, and 8 nodes of Intel/GTX450). Dynamic
scheduling is _not_ employed in those experiments: the parameters are kept
fixed on each execution (each data point). The speedup brought by the use of
the GPU is dramatic, and each successive improvement (from group-only to
double-buffering to cooperative CPU/GPU scheduling) is considerable.
(e) 2-node AMD — Probe-depth=450
(f) 8-node Intel — Probe-depth=450
As expected, a small number of queries is insufficient to completely utilize
the GPU, but the GPU performance is consistently improved as group-size
increases. Also, when 50% of the maximal group-size is employed, the speedups
achieved are about 90% of the best acceleration. In addition, the best
speedups are similar for all probe-depths: 13.34$\times$ (probe-depth=250),
13.52$\times$ (probe-depth=350), and 13.73$\times$ (probe-depth=450). The
small impact of the probe-depth over the speedup is an important, positive
property of the method, since it allows the user to calibrate that parameter
more or less freely, provided that there are enough queries to group and
process in parallel.
The gains attained by task grouping are promising, enhancing a very efficient
approximate search algorithm — whose sequential CPU implementation is already
orders of magnitude faster than the exact search. In the following sections,
we will demonstrate increasing performance gains, obtained by the
optimizations proposed on top of that GPU parallelization (See Section 5).
### 7.2 The Effect of Overlapping CPU and GPU
Figure 10(d) presents the performance of Hypercurves when using the double-
buffering scheme intended to maximize the GPU utilization by reducing the
waiting time between batches of queries. That enhancement, discussed in
Section 5.2, is built on top of the grouping mechanism demonstrated in the
previous section. The results are shown in the curve labeled “Double buffer”.
Similarly to the “Grouping only” case, the performance grows as the value of
group-size increases, nearly doubling the throughput of the grouping-only
version for most of the group-size values. When compared to the single CPU
core execution, the maximum speedups of the double-buffered GPU-accelerated
version are 25.81, 26.02, and 26.75, for probe-depths 250, 350, and 450
respectively.
During the evaluation, a complete overlapping of the CPU and the GPU was
achieved, except at the very beginning, during the preparation of the first
batch of queries. We have also observed that, in all experiments, the GPU was
always occupied processing queries, while the CPU experienced idle periods for
the threads responsible for retrieving candidates from the subindexes and copy
them to the buffers. The CPU underutilization motivated its use in the kNN
phase of the application, in addition to the preparation of the buffers for
the GPU. That cooperative strategy is evaluated in the next section.
### 7.3 Maximizing CPU–GPU Cooperation
Since the CPUs present in our systems prepare the buffers faster than the GPU
is able to consume them, they experience idle periods. To maximize the system
performance under those circumstances, we also employed the CPU cores in the
compute-intensive kNN tasks during that idleness, as discussed in Section 5.3.
In the remainder of this section, the benefits of that strategy are evaluated
in addition to double-buffering and grouping. Figure 10(d) shows the results,
in the curves labeled “CPU/GPU sched.”.
As shown, the gains with that technique are very relevant, and speedups of
1.23$\times$, 1.22$\times$ and 1.22$\times$ for, respectively, probe-depths of
250, 350 and 450 were achieved on top of the previous version of Hypercurves.
Figure 11: Fraction of the tasks computed by each type of device (CPU vs. GPU)
as group-size varies, illustrating how GPU utilization is favored by large
group-sizes. Those experiments were executed on an AMD node using probe-depth
of 350.
Interestingly, the speedup obtained here slightly decreases as the group-size
increases. This behavior is consequence of the higher efficiency of the GPU
for large group-sizes, which tends to reduce the idle time of the CPU cores.
Figure 11 illustrates that as the group-size grows, more queries tend to be
processed by the GPU instead of the using CPU for the kNN phase. Even so, the
improvements achieved were rewarding, with more than 1.2$\times$ average
speedup across the group-size configurations used. As the number of available
CPU cores also tend to increase in new architectures, the potential of using
those devices in cooperation with the GPU cannot be neglected.
### 7.4 Granularity vs. Response Times
So far, in the experimental evaluation, we have analyzed Hypercurves
performance in scenarios where a very large number of queries is submitted at
once, thus assessing the application throughput capabilities. However, in
real-world operation, the query rate submitted to an online application is
generated by users, and is subjected to variability throughout the execution.
Moreover, under those circumstances, the most important metric is the response
time observed by the users. Therefore, in this section, we analyze how the
grouping technique and cooperative CPU–GPU execution impact the average query
response time.
(a) 2-node AMD — GPU-only
(b) 2-node AMD — CPU–GPU
(c) 8-node Intel — CPU–GPU
Figure 12: Average query response times as query rate (% of the maximum) and
group-size vary, for a probe-depth of 350. The parameters are kept fixed on
each execution (each data point). The graphs demonstrate the complex task of
optimizing group-size as the load (query-rate) fluctuates.
In this analysis, we vary both the number of queries submitted per second and
group-size across experiments, but _keep them fixed within each run_. The
capacity of adjusting the group-size dynamically is evaluated in the next
section.
We employ the same 1,000,000-vectors database, and the 30,000 queries used in
previous sections. First, in Figure 12(a), we present the average response
times of Hypercurves when the GPU is used to compute all the tasks, and the
group-size is varied.
It is noticeable that a single, fixed, group-size value is unable to deliver
the best response times for all request loads. The response-time function we
seek to optimize has a complex behavior and its minimum moves up the group-
size axis as the query rate increases. Hypercurves has been able to answer
queries in reasonably low response times until the load reached about 80% of
the maximum supported by the configuration, but after that point response
times have grown steeply.
The comparison of Hypercurves using only the GPU for kNN, versus the
cooperative CPU–GPU configuration is possible contrasting Figures 12(a) and
12(b). Visual inspection of those figures show that the CPU–GPU cooperation
resulted in a systematic reduction of the average response times. Moreover,
across all the experiments, the CPU–GPU version reduces the response times in
58%, on average. Those gains are much more impacting that the previous
improvements in throughput achieved with the same cooperation (Section 7.3).
The reason is that although the CPU has a lower throughput, the response times
of the queries processed by this device are much smaller: the queries routed
to the GPU have a longer processing path, having to be queued, copied to the
GPU, processed using lower frequency cores, and copied back to the system main
memory.
We also present the performance of Hypercurves using CPU–GPU collaboratively
for the Intel/GTX470 node in Figure 12(c). When compared to the AMD/GTX260
machine, the performance of the Intel node is superior, and much better
average response-times are achieved. That is mainly a consequence of the
improvements in design achieved by the GTX470 GPU as compared to the GTX260
GPU, as the first has more computing cores, better bandwidth, etc.
### 7.5 Response times on variable request rates
In this section, we analyze the DTAHE dynamic scheduler (Section 6) capacity
to adapt Hypercurves’ work partition among CPU and GPU under scenarios with
stochastically variable workloads. During this evaluation, the load/request
rate follows a Poisson distribution, with expected average rate ($\lambda$)
varying from 20 to 100% of the maximum throughput of the application in each
configuration. That maximum throughput is computed in a preliminary run, where
all the queries are sent for computation in the beginning of the execution.
This set of experiments employ the cooperative CPU–GPU computation, and a
probe-depth of 350.
Scheduling | Poisson $\lambda$ (% of max. throughput)
---|---
20 | 40 | 60 | 80 | 100
Best static | 0.11 | 0.4 | 0.42 | 0.61 | 0.98
DTAHE | 0.06 | 0.13 | 0.14 | 0.22 | 1.02
(a) First setup (2-node AMD)
Scheduling | Poisson $\lambda$ (% of max. throughput)
---|---
20 | 40 | 60 | 80 | 100
Best static | 0.054 | 0.098 | 0.12 | 0.25 | 0.65
DTAHE | 0.034 | 0.089 | 0.1 | 0.16 | 0.68
(b) Second setup (8-node Intel)
Table 1: Average query response times (in s) for static and dynamically-tuned
scheduling configurations, under stochastic loads. Unless the system is
completely saturated, the dynamic scheduling always wins, usually by a
considerable margin.
The results are summarized in Table 1. The best static configuration refers to
the minimum response times achieved from an exhaustive search in the group-
size space for each query rate employed. It is noticeable that DTAHE strongly
outperform the best static configuration for most of the cases. On average,
its response times are 52% (2-node AMD) and 81% (8-node Intel) of the best
static configuration.
The only configurations where DTAHE falls slightly behind static scheduling
have Poisson rates $\lambda$ equal to 100% of the maximum throughput bearable
by the machines. In that extreme scenario not much can be accomplished by
dynamic scheduling, since the goal of maximum throughput coincides with the
one of minimum response times, and a simple static configuration manages that
with slightly less overhead. The differences, however, are small, and dynamic
scheduling is, of course, much more flexible.
### 7.6 Evaluating Hypercurves’ scalability
The distributed memory analysis in this section has focused on evaluating
Hypercurves scaleup. We consider the compromises between performance of the
parallelism and conservation of the results precision, as the database is
partitioned among the computing nodes. The scaleup evaluation is appropriate
in our application scenario because we expect to have an abundant volume of
data for indexing, thus the speedup evaluation starting with a single node
holding the entire database might not be realistic. The experiments executed
in this section used the 8-node Intel cluster, employing all CPU cores and
GPUs available on the nodes. The main database with 130,463,526 local feature
vectors is used proportionally, with $n/8$ of the database being used for the
experiment with $n$ nodes.
Number of nodes | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
# of cores / # of GPUs | 16 / 1 | 32 / 2 | 48 / 3 | 64 / 4 | 70 / 5 | 86 / 6 | 102 / 7 | 118 / 8
Optimist — queries per s | 964 | 1904 | 2649 | 3598 | 4490 | 5397 | 6297 | 7483
Pessimist — queries per s | 964 | 1683 | 2197 | 2849 | 3416 | 3968 | 4498 | 5135
Table 2: Scaleup evaluation: query rate as database and number of nodes
increase proportionally (probe-depth=350).
The query rate delivered by the algorithm considers two parameterization
scenarios named Optimist and Pessimist (Table 2), which differ in their
guarantees of equivalence (in terms of precision of the kNN search) to the
sequential Multicurves algorithm. The Optimist parameterization divides the
probe-depth equally among the nodes, without any slack — it will only be
equivalent to Multicurves in the unlikely case that all candidates of that
query are equally distributed on the nodes. The Pessimist parameterization
uses a slack that guarantees a probability smaller than 2% that a candidate
vector selected by the sequential algorithm will be missing from the
distributed version (see Section 4.2 for details). Note that that choice is
extremely conservative, because in order to effectively affect the answer, the
missed feature vectors from the candidate set have to be among the actual
top-$k$ set, and $k$ is much smaller than the probe-depth.
Table 2 presents Hypercurves query rates on the scaleup evaluation. As shown,
the scalability of the algorithm is impressive for both Optimist and Pessimist
configurations, achieving _superlinear scaleup_ in all setups. That strong
performance of Hypercurves is observed because the application is only
affected by the size of the database during the phase where candidates are
retrieved from subindexes and the cost of that stage grows only
logarithmically with the size of the database. The costly phase of computing
the distances from the query to the retrieved candidates can, thanks to the
probabilistic equivalence (Section 4.2), be efficiently distributed among the
nodes, with a relatively small overhead.
Not only is the scalability of the algorithm very good, but its raw response
rates (queries per s) are very high. For instance, number of queries that the
algorithm would be able to answer a per day are: 646 and 443 million,
respectively, for the Optimist and the Pessimist configurations. Those rates
indicate that by employing the technology proposed, a large-scale image search
system could be built at reasonably low hardware and power costs per request,
since GPU accelerators are very computational- and power-efficient platforms.
## 8 Conclusions and future work
This work has proposed and evaluated Hypercurves, an online similarity search
engine for very large multimedia databases. Hypercurves has been designed to
fully exploit massively parallel machines, with both CPUs and GPUs. Its use in
a CPU–GPU environment, along with a set of optimizations, resulted in
accelerations of about 30$\times$ on top of the single-core CPU version.
We have also studied the problem of response-time aware (DTAHE) partition of
tasks between CPU and GPU, under request load fluctuations, which occurs as a
result of the variating number of queries submitted by the user to the
application. DTAHE has been able to reduce the average query response times in
about 50% and 80% (respectively for both machine configurations used in the
experiments), when compared to the best static partition in each case.
Furthermore, Hypercurves achieved _superlinear_ scaleups in all experiments,
while keeping a high guarantee of equivalence with the sequential Multicurves
algorithm, as asserted by the proof of probabilistic equivalence.
We are currently interested in the complex interactions between algorithmic
design and parallel implementation for services such as Hypercurves. We are
also investigating how a complete system for content-based image retrieval can
be built upon our indexing services, and optimized using our techniques and
scheduling algorithms. We consider it a promising direction for future, as
Hypercurves subindexes implementation in heterogeneous environments offers
very good reply rate.
## References
* (1) Akune, F., Valle, E., Torres, R.: Monorail: A disk-friendly index for huge descriptor databases. In: 2010 20th International Conference on Pattern Recognition (ICPR), pp. 4145 –4148 (2010). DOI 10.1109/ICPR.2010.1008
* (2) Aleen, F., Sharif, M., Pande, S.: Input-driven dynamic execution prediction of streaming applications. In: Proc. of the 15th ACM SIGPLAN symposium on Principles and practice of parallel programming, PPoPP ’10. ACM (2010)
* (3) Beynon, M., Ferreira, R., Kurc, T.M., Sussman, A., Saltz, J.H.: DataCutter: Middleware for filtering very large scientific datasets on archival storage systems. In: IEEE Symposium on Mass Storage Systems, pp. 119–134 (2000)
* (4) Bhatti, N.T., Hiltunen, M.A., Schlichting, R.D., Chiu, W.: Coyote: a system for constructing fine-grain configurable communication services. ACM Trans. Comput. Syst. 16(4), 321–366 (1998). DOI 10.1145/292523.292524
* (5) Blagojevic, F., Nikolopoulos, D.S., Stamatakis, A., Antonopoulos, C.D., Curtis-Maury, M.: Runtime scheduling of dynamic parallelism on accelerator-based multi-core systems. Parallel Comput. 33, 700–719 (2007). DOI 10.1016/j.parco.2007.09.004
* (6) Böhm, C., Berchtold, S., Keim, D.A.: Searching in high-dimensional spaces: Index structures for improving the performance of multimedia databases. ACM Comput. Surv. 33, 322–373 (2001). DOI 10.1145/502807.502809
* (7) Boureau, Y.L., Bach, F., LeCun, Y., Ponce, J.: Learning mid-level features for recognition. Computer Vision and Pattern Recognition, IEEE Computer Society Conference on 0, 2559–2566 (2010). DOI 10.1109/CVPR.2010.5539963
* (8) Butz, A.R.: Alternative algorithm for hilbert’s space-filling curve. IEEE Trans. on Computers C-20 (1971)
* (9) Castelli, V.: Multidimensional Indexing Structures for Content-Based Retrieval, pp. 373–433. John Wiley & Sons, Inc. (2002). DOI 10.1002/0471224634.ch14
* (10) Chandrasekhar, V., Sharifi, M., Ross, D.A.: Survey and evaluation of audio fingerprinting schemes for mobile query-by-example applications. In: A. Klapuri, C. Leider (eds.) ISMIR, pp. 801–806. University of Miami (2011)
* (11) Chávez, E., Navarro, G., Baeza-Yates, R., Marroquín, J.L.: Searching in metric spaces. ACM Comput. Surv. 33(3), 273–321 (2001). DOI 10.1145/502807.502808
* (12) Chen, D.K., Su, H.M., Yew, P.C.: The impact of synchronization and granularity on parallel systems. SIGARCH Comput. Archit. News 18, 239–248 (1990). DOI 10.1145/325096.325150
* (13) Datar, M., Immorlica, N., Indyk, P., Mirrokni, V.S.: Locality-sensitive hashing scheme based on p-stable distributions. In: Proceedings of the twentieth annual symposium on Computational geometry, SCG ’04, pp. 253–262. ACM, New York, NY, USA (2004). DOI 10.1145/997817.997857
* (14) Diamos, G.F., Yalamanchili, S.: Harmony: an execution model and runtime for heterogeneous many core systems. In: Proceedings of the 17th international symposium on High performance distributed computing, HPDC ’08, pp. 197–200. ACM, New York, NY, USA (2008). DOI http://doi.acm.org/10.1145/1383422.1383447
* (15) Ding, S., He, J., Yan, H., Suel, T.: Using graphics processors for high-performance ir query processing. In: Proceeding of the 17th international conference on World Wide Web, WWW ’08, pp. 1213–1214. ACM, New York, NY, USA (2008). DOI 10.1145/1367497.1367732
* (16) Du Mouza, C., Litwin, W., Rigaux, P.: Large-scale indexing of spatial data in distributed repositories: the sd-rtree. The VLDB Journal 18, 933–958 (2009). DOI 10.1007/s00778-009-0135-4
* (17) Duran, A., Silvera, R., Corbalán, J., Labarta, J.: Runtime adjustment of parallel nested loops. In: B.M. Chapman (ed.) Shared Memory Parallel Programming with Open MP, _Lecture Notes in Computer Science_ , vol. 3349. Springer Berlin / Heidelberg (2005)
* (18) Fagin, R., Kumar, R., Sivakumar, D.: Efficient similarity search and classification via rank aggregation. In: Proceedings of the 2003 ACM SIGMOD international conference on Management of data, SIGMOD ’03, pp. 301–312. ACM, New York, NY, USA (2003). DOI 10.1145/872757.872795
* (19) Fagin, R., Lotem, A., Naor, M.: Optimal aggregation algorithms for middleware. In: Proc. of the 20th ACM SIGMOD-SIGACT-SIGART Symp. on Principles of database systems, PODS ’01, pp. 102–113. ACM (2001). DOI 10.1145/375551.375567
* (20) Faloutsos, C.: Gray codes for partial match and range queries. IEEE Trans. Softw. Eng. 14, 1381–1393 (1988). DOI 10.1109/32.6184
* (21) Faloutsos, C.: Multimedia Indexing, pp. 435–464. John Wiley & Sons, Inc. (2002). DOI 10.1002/0471224634.ch15
* (22) Faloutsos, C., Roseman, S.: Fractals for secondary key retrieval. In: Proceedings of the eighth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems, PODS ’89, pp. 247–252. ACM, New York, NY, USA (1989). DOI 10.1145/73721.73746
* (23) Ferreira, R., Jr., W.M., Guedes, D., Drummond, L., Coutinho, B., Teodoro, G., Tavares, T., Araujo, R., Ferreira, G.: Anthill:a scalable run-time environment for data mining applications. In: Symposium on Computer Architecture and High-Performance Computing (SBAC-PAD) (2005)
* (24) Garcia, V., Debreuve, E., Barlaud, M.: Fast k nearest neighbor search using GPU. In: CVPR Workshop on Computer Vision on GPU (CVGPU). Anchorage, Alaska, USA (2008)
* (25) Hartley, T.D., Catalyurek, U.V., Ruiz, A., Ujaldon, M., Igual, F., Mayo, R.: Biomedical Image Analysis on a Cooperative Cluster of GPUs and Multicores. In: 22nd ACM Intl. Conference on Supercomputing (2008)
* (26) He, B., Fang, W., Luo, Q., Govindaraju, N.K., Wang, T.: Mars: A mapreduce framework on graphics processors. In: Parallel Architectures and Compilation Techniques (2008)
* (27) Hua, G., Fu, Y., Turk, M., Pollefeys, M., Zhang, Z.: Introduction to the special issue on mobile vision. International Journal of Computer Vision 96, 277–279 (2012). DOI 10.1007/s11263-011-0506-3. 10.1007/s11263-011-0506-3
* (28) Indyk, P., Motwani, R.: Approximate nearest neighbors: Towards removing the curse of dimensionality. In: STOC, pp. 604–613 (1998). DOI 10.1145/276698.276876
* (29) Kerbyson, D.J., Alme, H.J., Hoisie, A., Petrini, F., Wasserman, H.J., Gittings, M.: Predictive performance and scalability modeling of a large-scale application. In: Supercomputing ’01: Proceedings of the 2001 ACM/IEEE conference on Supercomputing (CDROM), pp. 37–37 (2001). DOI 10.1145/582034.582071
* (30) Liao, S., Lopez, M.A., Leutenegger, S.T.: High dimensional similarity search with space filling curves. In: Proc. of the 17th Int. Conf. on Data Engineering, pp. 615–622 (2001)
* (31) Linderman, M.D., Collins, J.D., Wang, H., Meng, T.H.: Merge: a programming model for heterogeneous multi-core systems. SIGPLAN Not. 43(3), 287–296 (2008). DOI 10.1145/1353536.1346318
* (32) Liu, Y., Zhang, D., Lu, G., Ma, W.Y.: A survey of content-based image retrieval with high-level semantics. Pattern Recognition 40(1), 262 – 282 (2007). DOI 10.1016/j.patcog.2006.04.045
* (33) Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vision 60, 91–110 (2004). DOI 10.1023/B:VISI.0000029664.99615.94
* (34) Luk, C.K., Hong, S., Kim, H.: Qilin: Exploiting parallelism on heterogeneous multiprocessors with adaptive mapping. In: 42nd International Symposium on Microarchitecture (MICRO) (2009)
* (35) Mainar-Ruiz, G., Perez-Cortes, J.C.: Approximate nearest neighbor search using a single space-filling curve and multiple representations of the data points. In: Proceedings of the 18th International Conference on Pattern Recognition - Volume 02, ICPR ’06, pp. 502–505. IEEE Computer Society, Washington, DC, USA (2006). DOI 10.1109/ICPR.2006.275
* (36) Mallows, C.L.: An inequality involving multinomial probabilities. Biometrika 55(2), pp. 422–424 (1968)
* (37) Megiddo, N., Shaft, U.: Efficient nearest neighbor indexing based on a collection of space filling curves. Tech. Rep. IBM Research Report RJ 10093 (91909), IBM Almaden Research Center, San Jose California (1997)
* (38) Mikolajczyk, K., Schmid, C.: A performance evaluation of local descriptors. IEEE Transactions on Pattern Analysis and Machine Intelligence 27, 1615–1630 (2005). DOI 10.1109/TPAMI.2005.188
* (39) Morton, G.M.: A computer oriented geodetic data base and a new technique in file sequencing. Tech. Rep. Technical Report, IBM Ltd., Ottawa, Ontario, Canada (1966)
* (40) O’Malley, S.W., Peterson, L.L.: A dynamic network architecture. ACM Trans. Comput. Syst. 10(2) (1992). DOI 10.1145/128899.128901
* (41) Pang, H., Ding, X., Zheng, B.: Efficient processing of exact top-k queries over disk-resident sorted lists. The VLDB Journal 19, 437–456 (2010). DOI 10.1007/s00778-009-0174-x
* (42) Penatti, O.A.B., Valle, E., Torres, R.d.S.: Comparative study of global color and texture descriptors for web image retrieval. J. Vis. Comun. Image Represent. 23(2), 359–380 (2012). DOI 10.1016/j.jvcir.2011.11.002
* (43) Ravi, V., Ma, W., Chiu, D., Agrawal, G.: Compiler and runtime support for enabling generalized reduction computations on heterogeneous parallel configurations. In: Proceedings of the 24th ACM International Conference on Supercomputing, pp. 137–146. ACM (2010)
* (44) Sagan, H.: Space-Filling Curves. Springer-Verlag, New York, NY, USA (1994)
* (45) Samet, H.: Foundations of Multidimensional and Metric Data Structures (The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2005)
* (46) Sancho, J.C., Kerbyson, D.J.: Analysis of Double Buffering on two Different Multicore Architectures: Quad-core Opteron and the Cell-BE. In: International Parallel and Distributed Processing Symposium (IPDPS) (2008)
* (47) Shakhnarovich, G., Darrell, T., Indyk, P.: Nearest-Neighbor Methods in Learning and Vision: Theory and Practice (Neural Information Processing). The MIT Press (2006)
* (48) Shepherd, J., Zhu, X., Megiddo, N.: A fast indexing method for multidimensional nearest neighbor search. In: SPIE Conference on Storage and Retrieval for Im-age and Video Databases VII, pp. 350–355 (1999)
* (49) Smeulders, A., Worring, M., Santini, S., Gupta, A., Jain, R.: Content-based image retrieval at the end of the early years. Pattern Analysis and Machine Intelligence, IEEE Trans. on 22(12), 1349 –1380 (2000). DOI 10.1109/34.895972
* (50) Stone, Z., Zickler, T., Darrell, T.: Autotagging facebook: Social network context improves photo annotation. In: Computer Vision and Pattern Recognition Workshops, 2008. CVPRW ’08. IEEE Computer Society Conference on, pp. 1 –8 (2008). DOI 10.1109/CVPRW.2008.4562956
* (51) Teodoro, G., Fireman, D., Guedes, D., Jr., W.M., Ferreira, R.: Achieving multi-level parallelism in filter-labeled stream programming model. In: The 37th International Conference on Parallel Processing (ICPP) (2008)
* (52) Teodoro, G., Hartley, T., Catalyurek, U., Ferreira, R.: Optimizing dataflow applications on heterogeneous environments. Cluster Computing 15, 125–144 (2012). URL http://dx.doi.org/10.1007/s10586-010-0151-6. 10.1007/s10586-010-0151-6
* (53) Teodoro, G., Hartley, T.D.R., Catalyurek, U., Ferreira, R.: Run-time optimizations for replicated dataflows on heterogeneous environments. In: Proc. of the 19th ACM International Symposium on High Performance Distributed Computing (HPDC) (2010)
* (54) Teodoro, G., Kurc, T.M., Pan, T., Cooper, L.A., Kong, J., Widener, P., Saltz, J.H.: Accelerating Large Scale Image Analyses on Parallel, CPU-GPU Equipped Systems. In: 26th IEEE International Parallel and Distributed Processing Symposium (IPDPS) (2012)
* (55) Teodoro, G., Sachetto, R., Sertel, O., Gurcan, M., Jr., W.M., Catalyurek, U., Ferreira, R.: Coordinating the use of GPU and CPU for improving performance of compute intensive applications. In: IEEE Cluster (2009)
* (56) Teodoro, G., Valle, E., Mariano, N., Torres, R., Meira Jr., W.: Adaptive parallel approximate similarity search for responsive multimedia retrieval. In: Proceedings of the 20th ACM international conference on Information and knowledge management, CIKM ’11, pp. 495–504. ACM, New York, NY, USA (2011). DOI 10.1145/2063576.2063651
* (57) Tuytelaars, T., Mikolajczyk, K.: Local invariant feature detectors: a survey. Found. Trends. Comput. Graph. Vis. 3, 177–280 (2008). DOI 10.1561/0600000017
* (58) Valle, E., Cord, M., Philipp-Foliguet, S.: Fast identification of visual documents using local descriptors. In: Proceeding of the eighth ACM symposium on Document engineering, DocEng ’08, pp. 173–176. ACM (2008). DOI 10.1145/1410140.1410175
* (59) Valle, E., Cord, M., Philipp-Foliguet, S.: High-dimensional descriptor indexing for large multimedia databases. In: Proceeding of the 17th ACM conference on Information and knowledge management, CIKM ’08, pp. 739–748. ACM, New York, NY, USA (2008). DOI 10.1145/1458082.1458181
* (60) Valle, E., Cord, M., Phillip-Folliguet, S., Gorisse, D.: Indexing personal image collections: A flexible, scalable solution. IEEE Trans. Consumer Elect. 56, 1167–1175 (2010). DOI 10.1109/TCE.2010.5606242
* (61) Welsh, M., Culler, D., Brewer, E.: Seda: an architecture for well-conditioned, scalable internet services. SIGOPS Oper. Syst. Rev. 35(5), 230–243 (2001). DOI 10.1145/502059.502057
* (62) Yiu, M.L., Mamoulis, N.: Multi-dimensional top-k dominating queries. The VLDB Journal 18, 695–718 (2009). DOI 10.1007/s00778-008-0117-y
* (63) Yu, H., Rauchwerger, L.: Adaptive reduction parallelization techniques. In: Proc. of the 14th Int. Conf. on Supercomputing, ICS ’00, pp. 66–77. ACM, New York, NY, USA (2000). DOI 10.1145/335231.335238
* (64) Zezula, P., Amato, G., Dohnal, V., Batko, M.: Similarity Search: The Metric Space Approach, 1st edn. Springer Publishing Company, Incorporated (2010)
|
arxiv-papers
| 2012-09-03T17:12:59 |
2024-09-04T02:49:34.725116
|
{
"license": "Public Domain",
"authors": "George Teodoro, Eduardo Valle, Nathan Mariano, Ricardo Torres, Wagner\n Meira Jr, Joel H. Saltz",
"submitter": "George Teodoro",
"url": "https://arxiv.org/abs/1209.0410"
}
|
1209.0493
|
# Temperature dependent graphene suspension due to thermal Casimir interaction
Anh D. Phan$,^{1}$ Lilia M. Woods$,^{1}$ D. Drosdoff$,^{1}$ I. V.
Bondarev$,^{2}$ and N. A. Viet3 1Department of Physics, University of South
Florida, Tampa, Florida 33620, USA 2Physics Department, North Carolina
Central, Durham, North Carolina 27707, USA 3Institute of Physics, 10 Daotan,
Badinh, Hanoi, Vietnam
###### Abstract
Thermal effects contributing to the Casimir interaction between objects are
usually small at room temperature and they are difficult to separate from
quantum mechanical contributions at higher temperatures. We propose that the
thermal Casimir force effect can be observed for a graphene flake suspended in
a fluid between substrates at the room temperature regime. The properly chosen
materials for the substrates and fluid induce a Casimir repulsion. The balance
with the other forces, such as gravity and buoyancy, results in a stable
temperature dependent equilibrium separation. The suspended graphene is a
promising system due to its potential for observing thermal Casimir effects at
room temperature.
###### pacs:
Valid PACS appear here
Casimir interactions between objects arise due to electromagnetic
fluctuations. The Casimir force is universal and it is present at all length
scales. Investigations in the past several years directed towards
understanding the fundamental nature of this interaction and its role
especially in miniature devices have been particularly intense 1 . In most
cases, the Casimir force is attractive, and as a result close proximity
between materials can lead to unwanted effects due to stiction in micro and
nanomechanical systems 2 . Thus, finding ways to reduce the magnitude of the
force or even make it repulsive is an important field of research 3 ; 4 ; 5 ;
28 ; 29 ; 30 .
In most vacuum separated materials, the Casimir interaction arise due to
quantum mechanical effects ($T=0$ $K$). As the temperature is elevated, the
force changes due to changes in the photon thermal distribution, but such
effects are usually small at submicron scales and room temperature, and they
are difficult to observe 12 ; 6 . Recently, it has been proposed to utilize
objects suspended in a fluid, in which the corrections due to thermal
fluctuations in the Casimir force can be observed 7 ; 8 . This method is
particularly intriguing since it relies on the balance between attractive and
repulsive contributions to the force arising from the dielectric response of
the materials.
Graphene is an atomically thin planar sheet of carbon atoms arranged in a
honeycomb lattice. Its recent isolation has generated much scientific interest
9 . The Casimir interaction involving graphene has also been considered 27 ;
10 ; 11 ; 16 ; 17 ; 24 . The unique properties of this material, originating
from the linear in wave vector low energy dispersion together with its 2D
geometry have given rise to several unusual features. In particular, it was
found that the graphene Casimir interaction is strongly temperature dependent.
In the quantum mechanical limit, the force has the same distance dependence
($~{}d^{-4}$) as compared to the one between perfect metals, but with a
significantly reduced strength 4 , which is also distinct from 2D metals and
insulators 24 . At room temperature the interaction is dominated by thermal
fluctuations in contrast to regular metals and dielectrics in vacuum, in which
case it is dominated by the quantum fluctuations 27 ; 11 .
In this Letter, we investigate a graphene sheet suspended in a fluid between
substrates. We study how the Casimir interaction is influenced by the
substrates. In addition, by using the balance between the Casimir force,
gravity and bouyancy, we demonstrate the existence of a stable temperature
dependent equilibrium of the suspended graphene. The chosen materials are
suitable for the detection of purely thermal effects in the micron and
submicron scale. This is particluarly encouraging for future experimental
developments utilizing strong temperature dependent Casimir interactions.
Figure 1: (Color online) Dielectric function $\epsilon(i\xi)$ as a function of
frequency $\xi$ ($eV$) for several materials. Insert: Schematics of a planar
sheet (graphene) immersed in fluid (BB) between two slab substrates (top slab
- $Au$; bottom slab - $SiC$, $SiO_{2}$, or teflon).
The studied system is shown schematically in the insert of Fig. 1. It consists
of a graphene sheet immersed in a fluid sandwiched between a lower (of
thickness $D$) and an upper semi-infinite substrate. For such a layered
configuration, the Casimir force per unit area can be calculated in each layer
4 ; 10 ; 26 . The force per unit area in layer $2^{\prime}$ is effectively the
one between graphene and the bottom substrate given by
$\displaystyle
F(d_{1})=-\dfrac{k_{B}T}{\pi}\sum_{n=0}^{\infty}{{}^{\prime}}\int_{0}^{\infty}q(i\xi_{n})k_{\perp}dk_{\perp}$
$\displaystyle\times\left(\dfrac{R_{TE}^{+}(i\xi_{n})R_{TE}^{-}(i\xi_{n})}{e^{2q(i\xi_{n})d_{1}}-R_{TE}^{+}(i\xi_{n})R_{TE}^{-}(i\xi_{n})}\right.$
$\displaystyle\left.+\dfrac{R_{TM}^{+}(i\xi_{n})R_{TM}^{-}(i\xi_{n})}{e^{2q(i\xi_{n})d_{1}}-R_{TM}^{+}(i\xi_{n})R_{TM}^{-}(i\xi_{n})}\right),$
(1)
where $k_{\perp}$ is the 2D wave vector,
$q(i\xi_{n})=\sqrt{k_{\perp}^{2}+\varepsilon_{m}(i\xi_{n})(\xi_{n}/c)^{2}}$,
and $\xi_{n}=2n\pi k_{B}T/\hbar$ are the Matsubara frequencies. The prime in
the sum of Eq. (1) indicates that $n=0$ term is multiplyed by 1/2.
$R_{TM}(i\xi_{n})$ and $R_{TE}(i\xi_{n})$ are the effective reflection
coefficients for the transverse ellectric (TE) and transverse magnetic (TM)
polarizations of the electromagnetic field.
$R_{TE,TM}^{+,-}$ correspond to the effective boundary conditions due to the
objects above layer $2^{\prime}$ ($+$ subscipt) and below layer $2^{\prime}$
($-$ subscipt)4 ; 26 . For the system given in Fig. 1 they are derived as
follows:
$\displaystyle
R_{TM,TE}^{-}=r_{TM,TE}^{-}\frac{1-e^{-2k_{1}D}}{1-(r_{TM,TE}^{-})^{2}e^{-2k_{1}D}},$
$\displaystyle
r^{-}_{TM}=\frac{{\varepsilon_{1}q-\varepsilon_{m}k_{1}}}{{\varepsilon_{1}q+\varepsilon_{m}k_{1}}},r^{-}_{TE}=\frac{q-k_{1}}{q+k_{1}},$
$\displaystyle
R^{+}_{TM}=\frac{r_{TM}^{\sigma}+r_{TM}^{t}(1-2r_{TM}^{\sigma})e^{-2qd_{2}}}{1-r_{TM}^{\sigma}r_{TM}^{t}e^{-2qd_{2}}},$
$\displaystyle
R^{+}_{TE}=\frac{r_{TE}^{\sigma}+r_{TE}^{t}(1+2r_{TE}^{\sigma})e^{-2qd_{2}}}{1-r_{TE}^{\sigma}r_{TE}^{t}e^{-2qd_{2}}},$
$\displaystyle r_{TM}^{\sigma}=\frac{2\pi\sigma
q/\xi_{n}}{\varepsilon_{m}+2\pi\sigma
q/\xi_{n}},r_{TE}^{\sigma}=-\frac{2\pi\xi_{n}\sigma/c^{2}}{q+2\pi\xi_{n}\sigma/c^{2}},$
$\displaystyle
r_{TM}^{t}=\frac{{\varepsilon_{3}q-\varepsilon_{m}k_{3}}}{{\varepsilon_{3}q+\varepsilon_{m}k_{3}}},r_{TE}^{t}=\frac{q-k_{3}}{q+k_{3}},$
(2)
where
$k_{1,3}=\sqrt{k_{\perp}^{2}+\varepsilon_{1,3}(i\xi_{n})(\xi_{n}/c)^{2}}$ and
$\sigma$ is 2D graphene conductivity.
For this study, the material for the bottom substrate is $SiO_{2}$, $SiC$, or
teflon (PTFE), the top one is $Au$, while the medium fluid is bromobenzene
(BB). We assume that the thickness of the $Au$ substrate to be much greater
than the $Au$ skin depth ($\approx 22$ $nm$). Under this condition, the
Casimir interaction is not influenced by the thickness, as shown by others 31
, thus we take it to be semi-infinite. $\varepsilon_{1,3,m}(i\xi)$ are taken
from available experiments with Lorentz and Drude (for $Au$) models fitted
parameters 3 ; 5 ; 13 .
The dielectric function for graphene is found as $\varepsilon_{g}({\bf
k},i\xi)=1+2\pi k\sigma(i\xi)/\xi$ 25 . The graphene conductivity is
calculated from the Kubo formalism using a two-band Dirac model 14 .
$\sigma(i\xi)=\frac{2e^{2}k_{B}T\ln
2}{\pi\hbar^{2}\xi}+\frac{e^{2}\xi}{\pi}\int_{0}^{\infty}\frac{\tanh[\epsilon/2k_{B}T]d\epsilon}{\epsilon^{2}+(\hbar\xi)^{2}},$
(3)
The first term corresponds to intraband and the second term - to interband
transitions. In the infrared and low optical regime ($\leq$ 3 eV), $\sigma$
takes a universal value $\sigma_{0}=e^{2}/4\hbar$ characterizing the optical
tranparency of graphene. This has been demonstrated experimentally even in
room temperatures 15 .
Figure 2: (Color online) $F/F_{0}$ as a function of $d_{1}$ for (a) silica;
(b) teflon; (c) $SiC$; (d) the BB liquid phase temperature regime for the
three substrates. $F_{0}=\pi^{2}\hbar c/\left(240d_{1}^{4}\right)$ is the
Casimir force between two perfect metal plates. The bottom substrate is semi-
infinite. The BB liquid phase exists for $T=[242,429]K$.
Using Eq.(1-3), we calculate the Casimir force between the submerged in BB
graphene and the bottom substrate. The results are shown in Fig. 2 (a,b,c) for
several separations of the top $Au$ substrate. For $SiO_{2}$ and PTFE, the
force is found to be repulsive, while for $SiC$ \- it is attarctive for all
$d_{1}$ and $d_{2}$ distances. This is understood by examining the dielectric
response of the materials and their relative contribuion in the reflection
coefficients. For silica and teflon and $\xi<10$ $eV$,
$\varepsilon_{1}(i\xi)<\varepsilon_{m}(i\xi)<\varepsilon_{eff}(i\xi)$, where
$\varepsilon_{eff}$ is the effective dielectric function for graphene,
bromobenzene in layer $2"$ and the $Au$ substrate. Since $\varepsilon_{g}$ and
$\varepsilon_{BB}$ are much smaller than $\varepsilon_{Au}$, the effective
materials response above layer $2^{\prime}$ is reduced as compared to the one
for $Au$. This particular ordering in $\varepsilon$ 5 ; 6 results in
$R_{TM}^{-}<0$, thus $F$ changes sign. Stronger repulsion is achieved by
making $R_{TM}^{-}$ more negative either by bringing the $Au$ substrate closer
and/or by choosing PTFE instead of $SiO_{2}$. For $SiC$, however, this
ascending in dielectric response ordering is not maintained as
$\varepsilon_{SiC}>\varepsilon_{BB}$, and one obtains attraction for all
separations. We also find that for $d_{2}>80$ $nm$, the influence of the $Au$
is minimal and the force is changed little upon taking away the top substrate.
Graphene has an important effect on the temperature dependence of the Casimir
interaction. Thermal effects in most materials become important at separations
larger than the characteristic length $\lambda=\hbar c/(k_{B}T)$. For
graphene, however, $\lambda$ is reduced by the fine structure constant
$\alpha=1/137$ 27 , thus at room temperature thermal fluctuations effects are
seen at distances greater than $25-30$ $nm$. In that case, the interaction is
dominated by the $n=0$ term in Eq.(1). This is typical for graphene/graphene
10 or graphene/other materials interactions 4 ; 11 . The results shown in
Fig.2 were performed using the full expression Eq.(1), although for
$d_{2}>0.1$ $\mu m$, the $n=0$ dominates the force. In Fig.2 (d), we also show
how the force changes for the temperature region where BB liquid phase exists.
The force modulations are not very significant, although $F$ at higher $T$ is
more repulsive for PTFE and $SiO_{2}$ and more attractive for $SiC$.
The repulsive interaction together with the strong thermal effects in the
Casimir force can be used to create equilibrium separations of the submerged
in BB graphene sheet above PTFE or $SiO_{2}$ substrates. Besides the Casimir
interaction for the configuration in Fig. 1 (insert), the graphene experiences
other forces - buoyancy force and gravity attraction. To illustrate the
equilibrium configuration, we focus on the calculation of the total energy of
the suspended graphene:
$\displaystyle E(d_{1})=E_{c}+E_{g}+E_{b}$ $\displaystyle
E_{c}=\frac{k_{B}T}{2\pi}\sum_{n=0}^{\infty}{{}^{\prime}}\int_{0}^{\infty}k_{\perp}dk_{\perp}\left(\ln\left[1-R_{TM}^{+}R_{TM}^{-}e^{-2qd_{1}}\right]\right.$
$\displaystyle\left.+\ln\left[1-R_{TE}^{+}R_{TE}^{-}e^{-2qd_{1}}\right]\right).$
(4)
where $E_{c}$ is the Casmir energy found from Eq. (10. The energy due to
gravity is $E_{g}=\rho_{g}gd_{1}$ with $\rho_{g}=7.6\times 10^{-7}$ $kg/m^{2}$
being the surface mass density of graphene18 . The buoyancy energy is
$E_{b}=-\rho_{b}gN_{0}Vd_{1}$, where $\rho_{b}=1.5\times 10^{-3}$ $kg/m^{3}$
is the BB volume mass density, $g$ is the gravitational acceleration, $N_{0}$
is the number of carbon atoms per unit area and $V$ is volume of a carbon
atom.
We consider the case when the top substrate is not present. For distances at
the submicron scale, gravity becomes important attracting the graphene sheet
downward. The buoyancy acts upward together with the Casimir repulsion. The
balance between these forces is captured by calculating the total energy from
Eq.(4). Results for a $50\times 50$ $\mu m^{2}$ graphene sheet are shown in
Fig.3, where the equilibrium graphene/substrate separation $a_{0}$ corresponds
to the minimum of $E/(k_{B}T)$ vs $d_{1}$. $a_{0}$ is in the submicron range
and it becomes larger as the thikness of the bottom substrate is increased.
This behavior is also shown in the insert of Fig.3 for different temperatures.
One finds that for PTFE, $a_{0}=1.1-2.0$ $\mu m$ , while for $SiO_{2}$
$a_{0}=0.8-1.5$ $\mu m$ for 242-420 $K$. We note that the Brownian motion will
cause random fluctuations around $a_{0}$. As a result, it is possible for
stiction to occur. The probability for such a process is $\sim
e^{-E/(k_{B}T)}$ 7 ; 8 . For the suspended graphene, the probablity is small,
thus the Brownian motion is not expected to be important.
Figure 3: (Color online) (Top) Total energy of the suspended graphene in terms
of $k_{B}T$ as a function of $d_{1}$ for various thicknesses of the bottom
substrate. Inserts show the equilibrium distance $a_{0}$ vs the thickness $D$
for silica and teflon for different temperatures.
For graphene separations in the submicron and micron scales and large
temperatures, the contribution to the Casimir energy $E_{C}$ comes almost
entirely from the $n=0$ term in the Matsubara summation due to the reduced
characteristic length $\lambda$, as discussed earlier. We consider a bottom
substrate with width $D=800$ $nm$. Evaluating the $n=0$ term in Eq.(4), we
find $E_{c}=k_{B}T\gamma/(4\pi d_{1}^{2})$, where $\gamma\approx 0.028$ is a
constant. Note that $\gamma=0.038$ for $D=\infty$. Using this dominant term
together with Eq.(4), we can describe the graphene/substrate energy around the
equilibrium distance $a_{0}$ by a Taylor series of $E$.
In Fig.4, we show $E/(k_{B}T)$ as a function of the separation $d_{1}$
calculated via the full expression (Eq.(4)) and the Taylor series expansion by
retaining the first several terms. Fig.4 shows that in a rather wide range
around the equilibrium, the total energy can be described by just the first
term in the Taylor series as
$E(d_{1})-E(a_{0})\approx\frac{3k_{B}T\gamma}{4\pi
a_{0}^{4}}(d_{1}-a_{0})^{2}$, where $E(a_{0})$ is the total energy at the
equilibrium distance, found to be
$a_{0}=\left({k_{B}T\gamma}/{2\pi(\rho_{g}g-\rho_{b}gN_{0}V)}\right)^{1/3}$.
This result is especially useful in relating the Casimir interaction to a
simple harmonic-like oscillatory behavior of the submerged graphene. From
Eq.(5), one finds that the frequency of oscillations is
$\omega\sim\sqrt{\frac{3k_{B}T\gamma}{2\pi\rho_{g}a_{0}^{4}}}.$ (5)
The frequency is in the typical for a mechanical system kHz regime (about 2 to
5 kHz) for the studied temperature regime and substrates.
As the graphene/substrate distance is decreased, the Casimir repulsion becomes
dominant and $E$ rapidly increases. As the separation increases, the gravity
attraction dominates and $E$ increases again. The balance between these three
forces results in an equilibrium distance between the submerged graphene with
respect to the substrate.
Figure 4: (Color online) Total energy of the $50\times 50$ $\mu m^{2}$
suspeded graphene in terms of $k_{B}T$ vs $d_{1}$ calculated by the full
expression from Eq.(4) and by the first four terms in its Taylor series for
$D=800$ $nm$.
This study shows that graphene experiences suspension in bromobenzene above
substrates providing the dielectric functions of the layers (insert Fig. 1)
are in ascending (discending) order. The suspension is characterized by a
temperature-dependent equilibrium separation in the micron range. The balance
between gravity, buoyancy, and the Casimir force yields one equilibrium
$a_{0}$ which changes at a rate $~{}2-3$ $nm/K$ \- a substantial rate at the
room temperature.
We compare this setting to the sphere/plate geometry in Ref.7 ; 8 , proposed
for achieving tempereture-dependent Casimir effect. It was shown that a
temperature dependent stable equilibrium of the suspended sphere due to the
Casimir force alone is obtained by making the integrand in Eq. (1)
oscillatory. This is done by choosing materials with dielectric crossings
occuring at sufficiently small Matsubara frequencies close to room
temperature. Note that here temperature fluctuations appear together with the
quantum mechanical contributions. Inlcuding gravity and buoyancy results in
the appearance of additional stable or unstable equilibriua, which maybe
difficult to measure experimentally due to the possibility of sticktion. At
the same time, for suspended graphene the Casimir force alone does not result
in an equlibrium sepration. The inlcusion of gravity and buoyancy is necessary
to ensure the desired balance. Thus we find only one stable equilibrium, which
is $\sim T^{1/3}$. Another important point is that due to the significantly
reduced thermal characteristic length of graphene, only thermal fluctuations
at the micron and submicron range are relavant for room temperatures. This is
truly unique since in regular dielectrics and metals such effects are seen at
much larger $T$ and $d_{1}$. We note that the suspended in fluid sphere as
well as the suspended in fluid graphene attest to the richness of the
temperature dependent Casimir phenomenon, which is yet to be explored
experimentally.
Furthermore, the fact that the graphene Casimir force is essentially thermal
for such separations and small $T$ yields a relatively simple estimate for the
oscillatory-like frequency around the stable separation. Around the
equilibrium the suspended graphene behaves like a strongly temperature-
dependent Hooke-like spring with characteristics directly related to the
thermal Casimir force. This system is a example, which can be used by
experimentalists to measure submicron separation changes over a few hundred
Kelvin temperature regimes entirely due to thermal Casimir effects. Such
distance scale measurements are feasible with available techniques, such as
traditional setups for Casimir measurements 19 , setups using suspended
Michelson interferrometers 20 , total internal reflection microscopy in
suspensions 21 , or other techniques based on transition electron microscopy,
suited for oscillatory measurements 22 ; 23 .
The authors are indebted to Dr. Alexandro Rodriguez (MIT) for discussions.
Financial support from the Department of Energy under Contract No. DE-
FG02-06ER46297 is also acknowledged. I.V.B. was supported by the NSF-
HRD-0833184 and ARO-W911NF-11-1-0189 grants. N.A.V. was supported by the
Nafosted Grant No. 103.06-2011.51.
## References
* (1) G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Rev. Mod. Phys. 81, 1827 (2009).
* (2) H.B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and Federico Capasso, Science 291, 1941 (2001).
* (3) Anh D. Phan and N. A. Viet, Phys. Rev. A 84, 062503 (2011).
* (4) D. Drosdoff and Lilia M. Woods, Phys. Rev. A 84, 062501 (2011).
* (5) P. J. van Zwol and G. Palasantzas, Phys. Rev. A 81, 062502 (2010).
* (6) J. N. Munday, F. Capasso, and V. A. Parsegian, Nature 457, 170 (2009).
* (7) M. Bostrom, Bo E. Sernelius, G. Baldissera, C. Persson, and Barry W. Ninham, Phys. Rev. A 85, 044702 (2012).
* (8) M. Bostrom and Bo E. Sernelius, Phys. Rev. A 85, 012508 (2012).
* (9) J.S Hoye, I. Brevik, J. B. Aarseth, and K. A. Milton, J. Phys. A 39, 6031 (2006).
* (10) S. K. Lamoreaux, Rep. Prog. Phys. 68, 201-236 (2005).
* (11) A. W. Rodriguez, D. Woolf, Alexander P. McCauley1, F. Capasso, John D. Joannopoulos, and Steven G. Johnson, Phys. Rev. Lett. 105, 060401 (2010).
* (12) A. W. Rodriguez, Alexander P. McCauley, D. Woolf, F. Capasso, J. D. Joannopoulos, and Steven G. Johnson, Phys. Rev. Lett. 104, 160402 (2010).
* (13) A. K. Geim and K. S. Novoselov, Nature Matter. 6, 183 (2007).
* (14) G. Gomez-Santos, Phys. Rev. B 80, 245424 (2009).
* (15) D. Drosdoff and Lilia M. Woods, Phys. Rev. B 82, 155459 (2010).
* (16) I. V. Fialkovsky, V. N. Marachevsky, and D. V. Vassilevich, Phys. Rev. B 84, 035446 (2011).
* (17) B. E. Sernelius, EPL 95, 57003 (2011).
* (18) V. Svetovoy, Z. Moktadir, M. Elwenspoek, and H. Mizuta, EPL 96, 14006 (2011).
* (19) John F. Dobson, Angela White, and Angel Rubio, Phys. Rev. Lett. 96, 073201 (2006).
* (20) K. Mehrany and S. Khorasani, J. Opt. A: Pure Appl. Opt. 4, 624 (2002).
* (21) M. Lisanti, D. Iannuzzi, and F. Capasso, PNAS 102, 11989 (2005).
* (22) E. Rousseau, M. Laroche, and J. J. Greffet, J. Appl. Phys. 111, 014311 (2012).
* (23) Frank Stern, Phys. Rev. Lett. 18, 546 (1967).
* (24) L. A. Falkovsky and A. A. Varlamov, Eur. Phys. J. B 56, 281-284 (2007).
* (25) R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, Science 320, 1308 (2008).
* (26) J. H. Chen, C. Jang, S. Xiao, M. Ishigami, Michael S. Fuhrer, Nature Nanotechnology 3, 206 (2008).
* (27) M. Bordag, U. Mohideen, and V.M. Mostepanenko, Physics Reports 353, 1-205 (2001).
* (28) A. Grado, E. Calloni, and L. Di Fiore, Phys. Rev. D 59, 042002 (1999).
* (29) C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger, Nature 451, 172-175 (2008).
* (30) John Cumings and A. Zettl, Science 289, 602-604 (2000).
* (31) A. Popescu, L. M. Woods, and I. V. Bondarev, Nanotechnology 19, 435702 (2008).
|
arxiv-papers
| 2012-09-03T22:14:50 |
2024-09-04T02:49:34.736673
|
{
"license": "Public Domain",
"authors": "Anh D. Phan, Lilia M. Woods, D. Drosdoff, I. V. Bondarev, and N. A.\n Viet",
"submitter": "Anh Phan",
"url": "https://arxiv.org/abs/1209.0493"
}
|
1209.0502
|
Tsemo Aristide
College Boreal
1, Yonge Street, Toronto, ON
[email protected]
Some properties of schemes in groups theory and Top couples.
Abstract.
Let $G$ be a group, and $H$ a $G$-group defined by an imbedding map
$G\rightarrow H$; in [12] we have defined a topology on a subset of normal
subgroups of $H$, the so-called prime ideals. In this work, we generalize this
topology to other classes of groups. We hope that such topologies define new
effective tools to tackle well-known problems in groups theory. We study
properties of the topology defined in [12], for example we define the notions
of functor of points, dimension and outline a Galois theory.
1\. Introduction.
Classical algebraic geometry is the study of solutions of polynomial equations
defined on commutative rings. The nature of the objects involved here can be
very different when one allows the base ring to vary; for example varieties
defined over the fields of complex numbers are complex manifolds with
singularities, thus, inherit a separated topology from their complex
structure, this is not the case of a variety defined over a finite field. The
first tools developed in classical algebraic geometry where related to
topology or complex geometry on the purpose to study complex schemes. In fact,
Serre has shown in the G.A.G.A [8] deep relations between the objects of these
theories.
To study the variety defined on more general rings, some tools issued from
complex algebraic geometry have been adapted. The most efficient work in that
direction has been done by A. Grothendieck with the theory of schemes; he has
introduced topological concepts to objects which does not look a priori
topologically by generalizing the Zariski topology. This approach has allowed
to give new interpretations of many conjectures and often led to their
solutions. We can for examples quote the Weil conjectures and more recently
the Milnor conjecture.
Many generalizations of the theory of schemes have been achieved for non
commutative algebras. In group theory, many classical sets are defined by
algebraic-like equations. Thus formally, they are similar to the objects
studied in classical algebraic geometry. Recently, Baumslag, Myasnikov and
Remeslennikov [2] have initiated the study of varieties of groups. They
consider a group $G$, and $G[X_{1},...,X_{n}]$ the free product of $G$ and the
free group generated by a set of cardinality $n$. For any group $H$ endowed
with an injective morphism $G\rightarrow H$, the elements of
$G[X_{1},...,X_{n}]$ define polynomial functions on $H^{n}$ which took their
values in $H$. The solutions of these equations define a Zariski-like
topology. These authors have adapted tools from the theory of algebraic
varieties like Zariski topology and irreducible subsets in groups theory.
In our paper [12], we have generalized the construction of Baumslag, Myasnikov
and Remeslennikov by defining a theory of schemes in this context, The purpose
of this work is to continue to study this notion; we also generalize it by
extending it to others subcategories of the category of groups. We hope that
these new concepts will also lead to the solutions of some problems in groups
theory by using the topological intuition.
Plan.
1\. Introduction
2\. Some properties of groups
3\. Topology in groups theory
4\. The geometry of $G$-groups
5\. The functor of points
6\. Noetherian $G$-schemes
7\. Dimension of affine schemes
8\. Galois theory.
2\. Some properties of groups.
Let $G$ be a group, the comma category over $G$ is the category whose objects
$(H,\phi_{H})$ are morphisms of groups $\phi_{H}:G\rightarrow H$. A morphism
$f:(H,\phi_{H})\rightarrow(H^{\prime},\phi_{H^{\prime}})$ of the comma
category is a morphism of groups $f:H\rightarrow H^{\prime}$ such that
$f\circ\phi_{H}=\phi_{H^{\prime}}$.
We denote by $C(G)$ the subcategory of the comma category such that each
object $(H,\phi_{H})$ of $C(G)$ is an object of the comma category such that
$\phi_{H}$ is injective.
Let $(H,\phi_{H})$ be an object of $C(G)$ and $x$ an element of $H$. For every
element $g\in G$, we denote $\phi_{H}(g)x\phi_{H}(g^{-1})$ by $x^{g}$. We
denote by $G(x)$ the subgroup of $H$ generated by $\\{x^{g},g\in G\\}$. We say
that the element $x\in H$ is invertible if and only if
$G(x)\cap\phi_{H}(G)\neq 1$.
Let $H$ be a group, and $g$, $g^{\prime}$ elements of $H$; the commutator
$[g,g^{\prime}]$ of $g$ and $g^{\prime}$ is the element
$gg^{\prime}g^{-1}{g^{\prime}}^{-1}$. If $U$ and $V$ are subgroups of $H$,
$[U,V]$ is the normal subgroup generated by $[u,v],u\in U,v\in V$.
Let $H$ be a group, we denote by $D^{0}H$ the group $H$, suppose defined
$D^{n}H$, for an integer $n$, then $D^{n+1}H=[D^{n}H,D^{n}H]$.
We say that an element $x$ of $H$ distinct of the neutral element is a divisor
of zero if and only if there exists an element $y$ of $H$ distinct of the
neutral element such that the group of commutators $[G(x),G(y)]=1$. Remark
that the element $y$ in this situation is also divisor of zero.
Proposition 2.1.
The element $x$ of $H$ is a divisor of zero if and only if there exists a non
trivial element $y$ of $H$ such that for every element $g\in G$,
$[x,y^{g}]=1$.
Proof.
Let $x$ be an element of $H$ which is a divisor of zero, and $y$ an element of
$H$ such that $[G(x),G(y)]=1$. The set $U=\\{[x,y^{g}],g\in G\\}$ is contained
in $[G(x),G(y)]$; we deduce that all elements of $U$ are equal to the neutral
element.
Conversely, suppose that all elements of $U$ are equal to the neutral element.
Let $u,v\in G$,
$[x^{u},y^{v}]=uxu^{-1}vyv^{-1}ux^{-1}u^{-1}vy^{-1}v^{-1}.$
We deduce that:
$u^{-1}[x^{u},y^{v}]u=xu^{-1}vyv^{-1}ux^{-1}u^{-1}vy^{-1}v^{-1}u=[x,y^{u^{-1}v}]$
Since $[x,y^{u^{-1}v}]=1$,we deduce that $[x^{u},y^{v}]=1$.
Recall that if $a,b,c$ are elements of a group, if $a$ commutes with $b$ and
$c$, then $a$ commutes with $b^{-1}$ and $bc$, this implies that $G(x)$
commutes with $G(y)$.
Remark.
Let $N_{G}(x)$ be the normal subgroup generated by $G(x)$, we can find an
example of a $G$-group $H$ such that $[G(x),G(y)]=1$ and
$[N_{G}(x),N_{G}(y)]\neq 1$:
Consider $H=SL(4,C)$ the group of linear automorphisms of the $4$-dimensional
complex vector space $C^{4}$ whose determinant is $1$. Let
$e_{1},e_{2},e_{3},e_{4}$ be a basis of $C^{4}$. Let $G=SL(2,C)$ be the
subgroup of $H$ such that for every $f\in G$, $f(e_{3})=e_{3},f(e_{4})=e_{4}$
and $f(e_{1})$ and $f(e_{2})$ are elements of the vector subspace
$Vect(e_{1},e_{2})$ generated by $e_{1}$ and $e_{2}$. Let $g$ be an element of
$H$ such that $g(e_{1})=e_{1},g(e_{2})=e_{2}$ and $g(Vect(e_{3},e_{4}))\subset
Vect(e_{3},e_{4})$, $[g,G]=1$, but $[N_{G}(g),G]$ is not the trivial subgroup:
if $f:C^{4}\rightarrow C^{4}$ is defined by
$f(e_{1})=e_{3},f(e_{2})=e_{4},f(e_{3})=e_{1}$ and $f(e_{4})=e_{2}$,
$fgf^{-1}\in G$, and we can choose $g$ such that $fgf^{-1}$ is not in the
center of $G$.
Proposition 2.2.
Suppose that $G$ is a solvable $G$-group different of $\\{1\\}$, then any
$G$-group $H$ has a divisor of zero. If $H$ is a finite dimensional simply
connected Lie group which does not have zero divisors, then $H$ is simple.
Proof.
Let $G$ be a solvable group different of $\\{1\\}$; there exists $n$ such that
$D^{n}G$ is not trivial and $D^{n+1}G$ is trivial. For every element $x\in
D^{n}G$ different of its neutral element we have $[G(x),G(x)]=1$, thus $x$ is
a divisor of zero; if $H$ is a $G$-group, $x$ is also a zero divisor of $H$.
Suppose that $H$ is a finite dimensional simply connected Lie group which does
not have divisors of zero and its radical is not trivial. let $R$ be the
radical of $H$, recall that $R$ is a maximal solvable normal subgroup of $H$.
Let $n$ be an integer such that $D^{n}R$ is not trivial, and $D^{n+1}R$ is
trivial. Since $R$ is a normal subgroup of $H$, for every element $x\in
D^{n}R$ and every element $g\in G$, $x^{g}\in D^{n}R$, this implies that
$[G(x),G(x)]=1$ for every element $x\in D^{n}R$. In particular, if $x$ is not
the neutral element it is a divisor of zero. This is in contradiction with the
fact that we have supposed that $G$ does not have divisors of zero. Thus the
radical of $H$ is trivial and $H$ is semi-simple. If $H$ is not simple, then
it is the product of two non trivial groups $H_{1}$ and $H_{2}$. Let $h_{1}$
and $h_{2}$ be respectively elements of $H_{1}$ and $H_{2}$ distinct of the
neutral element, we have $[G((h_{1},1));G((1,h_{2}))]=1$. Thus $H$ has
divisors of zero.
Remark.
The previous proposition shows that a non commutative nilpotent group $G$ has
divisors of zero. In fact, every element in the center of $G$ is a divisor of
zero.
Proposition 2.3.
Let $k$ be a field, and $G$ a subgroup of $PSL(n,k)$ suppose that the action
of $G$ on the $n$-dimensional $k$-projective space $Pk^{n}$ is reducible, then
$G$ has divisors of zero. If $G$ is a connected semi-simple Lie group defined
over the complex numbers imbedded in $PSL(n,C)$ or if $G$ is a finite group
imbedded in $PSL(n,k)$ where $k$ is a field whose characteristic does not
divide the order of $G$, then $G$ does not preserves the image of a proper
linear vector space in $P^{n}C$ (resp. $P^{n}k)$ if $G$ does not have zero
divisors.
Proof.
Suppose that the action of $G$ is reducible. Let $G^{\prime}$ be the inverse
image of the group $G$ by the projection map $p:Sl(n,k)\rightarrow PSL(n,k)$.
We can write $k^{n}=U\oplus V$, where $U$ and $V$ are non trivial vector
subspaces of $k^{n}$ and are stable by $G^{\prime}$. Let $u^{\prime}$ be a non
trivial invertible linear map of $U$ and $v^{\prime}$ be a non trivial
invertible linear map of $V$. Consider the linear morphism $u$ of $k^{n}$
defined by $u(x)=u^{\prime}(x)$ if $x\in U$, $u(x)=x$ if $x\in V$. Let $v$ be
the linear morphism of $k^{n}$ defined by $v(x)=x$ if $x\in U$,
$v(x)=v^{\prime}(x)$ if $x\in V$. We have $[G^{\prime}(u),G^{\prime}(v)]=1$.
It results that the image of $u$ and $v$ in $PSL(n,k)$ are divisors of zero.
Suppose that $G$ is a connected semi-simple subgroup of $PSL(n,C)$, (resp. $G$
is a finite subgroup of $PSl(n,k)$ such that the order of $G$ and the
characteristic of $k$ are prime each other), $p^{-1}(G)$ is semi-simple (resp.
finite subgroup) and $C^{n}$ (resp $k^{n}$) under the action of $G$ can be
decomposed in the direct summand of irreducible vector spaces. Thus $PSL(n,C)$
(resp. $PSL(n,k)$ has zero divisors if $p^{-1}(G)$ preserves a proper linear
subspace.
Definitions 2.1.
A $G$-domain $(H,\phi_{H})$ of $C(G)$ is an object of $C(G)$ such that $H$
does not have divisors of zero.
A normal subgroup $I$ of $H$ is an ideal of $H$ if and only if $I\cap
G=\\{1\\}$.
A normal subgroup $P$ of $H$ is a prime ideal if and only if:
\- $P$ is an ideal distinct of $H$ and
\- $G/P$ is a $G$-domain, this fact is equivalent to saying that for every
$x,y\in H$, $[G(x),G(y)]\in P$ if and only if $x$ is in $P$ or $y$ is in $P$.
\- The $G$-group $H$ is $G$-simple if $\\{1\\}$ is the the only ideal of $H$.
Proposition 2.4.
Suppose that $G$ is a connected simple Lie group over a field of
characteristic zero, and $H$ a connected Lie group in $C(G)$, if $H$ is
$G$-simple then $H$ is simple Lie group.
Proof.
We can write $H$ has a semi-product of a semi-simple group $U$ and a solvable
group $L$ by using the Levi decomposition. Without restricting the generality,
we can suppose that $G$ is contained in $U$ since $G$ is simple. Since $G\cap
L=\\{1\\}$, $L$ is an ideal. We deduce that $L$ is trivial since the only
prime of $H$ is $1$. If the subgroup $U$ is not simple, there will exist a
simple component of $U$ which intersection with $G$ is trivial since the
simple Lie group $G$ is contained in a simple component of $H$. This fact is
impossible since all ideals of $H$ are trivial.
Remark.
The previous result is not true if we don’t assume that the groups $G$ and $H$
are connected as shows the following example: Let $S_{n}$ be the group of
permutations of the set of $n$ elements, $n>4$. The subgroup $A_{n}$ of
permutations of even signature is a simple group. The group $S_{n}$ is endowed
with an $A_{n}$-structure. It is well-known that the only normal subgroups of
$S_{n}$ are the trivial subgroup, $A_{n}$ and $S_{n}$. Thus $S_{n}$ is an
$A_{n}$-simple group, and $S_{n}$ is not simple.
Proposition 2.5.
Let $k$ be a field and $G$ a Zariski dense subgroup of $PSl(n,k)$, the
$G$-group $PSl(n,k)$ does not have divisors of zero.
Proof.
Let $x$ be a divisor of zero and $y$ a non trivial element of $PSl(n,k)$ such
that for every element $g\in G$, $[x,y^{g}]=1$. The equation $[x,y^{u}]=1$ is
an algebraic equation in $PSl(n,k)$. This implies that the Zariski adherence
of $G$ satisfies this equation. Since $G$ is Zariski dense in $PSL(n,k)$, this
implies that for every $g\in PSl(n,k)$, $[x,y^{g}]=1$. Let $x^{\prime}$ (resp.
$y^{\prime}$) an element of $Sl(n,k)$ above $x$ (resp. above $y$). Thus
$[x^{\prime},{y^{\prime}}^{g}]=cI_{n}$ is in the center of $SL(n,k)$ for every
$g\in SL(n,k)$. Recall that the elements of the center of $SL(n,k)$ are
$c_{1}I_{1},...,c_{n}I_{n}$ where $c_{1},...,c_{n}$ are the $n$-roots of
unity. The subset $G_{i}$ of $SL(n,k)$ such that
$[x^{\prime},{y^{\prime}}^{g}]=c_{i}^{-1}I_{n}$ is closed, and $SL(n,k)$ is a
finite union of the $G_{i}$. Since $SL(n,k)$ is irreducible, there exists a
$i$ such that $G_{i}=SL(n,k)$. It results that
$[{x^{\prime}}^{n},{y^{\prime}}^{g}]=I_{n}$ for every $g\in SL(n,k)$. Let
$E_{h}$ be an eigenvector space of $y^{\prime}$ associated to the eigenvalue
$h$. Let $p$ be the dimension of $E_{h}$, for every vector subspace $E$ of
dimension $p$, there exists an element of $SL(n,k)$ such that $g(E_{h})=E$,
this implies that $E$ is the eigenvector space of ${y^{\prime}}^{g}$ and
${x^{\prime}}^{n}$ preserves $E$. We deduce that ${x^{\prime}}^{n}=cI$, $c\in
k$. The same argument shows also that ${y^{\prime}}^{n}=dI_{n}$, $d\in k$.
Thus $x^{\prime}$ and $y^{\prime}$ are diagonalizable maps. If $g$ is an
element of $Sl(n,k)$, $g(E_{h})$ is an eigenvector space of ${y^{\prime}}^{g}$
associated to $h$. This implies that ${x^{\prime}}(g(E))$ is the eigenvector
space of ${y^{\prime}}^{g}$ associated to $c_{i}^{-1}h$ if
$[x^{\prime},{y^{\prime}}^{g}]=c_{i}I_{n}$. Thus $x^{\prime}$ permutes the
eigenvector spaces of $y^{\prime}$. Since $x^{\prime}$ and $y^{\prime}$ are
not homothetic maps since $x$ and $y$ are not trivial, $y^{\prime}$ has two
distinct eigenvector spaces $E_{1}$ and $E_{2}$ associated respectively to the
eigenvalue $c_{1}$ and $c_{2}$; without restricting the generality, since
$x^{\prime}$ permutes the eigenvector space of ${y^{\prime}}^{g}$ for every
$g\in SL(n,k)$, we can suppose that $x^{\prime}(g(E_{1}))=g(E_{2})$. This last
equality is not possible for every element of $SL(n,k)$.
Proposition 2.6.
Let $H$ be an element of $C(G)$, suppose that $G$ is a normal subgroup of $H$,
then for every ideal ideal $I$, $[I,G]=1$.
Proof.
Let $g$ be an element of $G$, and $i$ and element of $I$, we have
$[u,i]=uiu^{-1}i^{-1}$ is an element of $G\cap I=\\{1\\}$.
Definitions 2.2.
Let $(H,\phi_{H})$ be an object of $C(G)$, an element $x$ of $H$ is nilpotent
of length $n$ if and only if the group $G(n)(x)=[G(x),[G(x)[,...,G(x)]=1$
where in the formula $G(x)$ appears $n$ times.
Let $H$ be an element of $C(G)$, the radical $rad_{G}(H)$ of $H$ is the
intersection of all prime ideals of $H$. We denote by $Com_{H}(G)$ the
subgroup of $H$ such that for every element $h\in Com_{H}(G)$ and $g\in G$, we
have $[g,h]=1$. The group $Com_{H}(G)$ is included in every element $P\in
Spec_{G}(H)$ and so is its normalizer $N(Com_{H}(G))$. This implies that
$N(Com_{H}(G))$ is a subset of $Rad_{G}(H)$.
Proposition 2.7.
Let $P$ be a prime and $x$ an element of $H$, suppose that $G(n)(x)$ is
contained in $P$, then $x$ is an element of $P$. In particular The normal
subgroup $Nil(H)$ generated by nilpotent elements of $H$ is contained in $P$.
Proof.
We show the result recursively. Let $P$ be a prime, suppose that every element
$u$ such that $G(n-1)(u)\in P$ is in $P$. Let $x$ be an element of $H$ such
that, $G(n)(x)$ is contained in $P$. Suppose that $x$ is not in $P$. Let $u$
be an element of $G(n-1)(x)$, we have $[G(x),u]\subset G(n)(x)\subset P$.
Since $P$ is prime, $u$ is an element of $P$, we deduce that $G(n-1)(x)$ is
contained in $P$, the iteration hypothesis implies that $x$ is an element of
$P$.
Proposition 2.8.
Suppose that $G$ is a simple group, and $H$ is an element of $C(G)$, then
every normal subgroup of $H$ is an ideal or contain $G$.
Proof.
Let $I$ be a normal subgroup of $H$, $I\cap G$ is a normal subgroup of $G$;
since $G$ is simple we deduce that $I\cap G=G$ or $I\cap G=\\{1\\}$.
3\. Topologies in group theory.
In this part, we are going to generalize the topology introduced in [12]. Let
$C$ be a subcategory of the category of groups, we denote by $D$ a subclass of
the class of objects of $C$ which satisfies the following properties:
T1 If $G$ is an object of $C$, $G^{\prime}$ is an object of $D$ such that
there exists an injective $C$-morphism $i:G\rightarrow G^{\prime}$, then $G\in
D$.
T2 Let $G$ be an element of $D$, and $I$, $J$ normal subgroups of $G$,
$[I,J]=1$ implies that $I=1$ or $J=1$.
Definitions 3.1.
Let $H$ be an element of $C$, an ideal of $H$ is a normal subgroup $I$ of $H$
such that $H/I\in C$ and the projection $H\rightarrow H/I$ is a morphism of
$C$.
An ideal $P$ of $H$ is a prime if and only if $H/P$ is an object of $D$ and
$P$ is distinct of $H$.
T3 We suppose that the inverse image of an ideal by a morphism of $C$ is an
ideal.
A couple $(C,D)$ which satisfies the properties T1, T2 and T3 will be called a
Top couple.
Let $H$ be an element of $C$ and $I$ a normal subgroup of $H$, we denote by
$V(I)$ the set of prime ideals of $H$ which contain $I$.
Proposition 3.1.
Let $H$ be an element of $C$, and $I$, $J$ normal subgroups of $H$, we have:
$V([I,J])=V(I)\bigcup V(J)$
Let $(I_{a})_{a\in A}$ a family of normal subgroups of $H$, and $I_{A}$ the
normal subgroup generated by $(I_{a})_{a\in A}$ we have:
$V(I_{A})=\cap_{a\in A}V(I_{a})$
Proof.
Let $P$ be an element of $V(I)\bigcup V(J)$, $P$ contains $I$ or $J$. This
implies that $P$ contains $[I,J]$.
Let $P$ be an element of $V(I,J])$ and $p:H\rightarrow H/P$ the natural
projection map, we have $[p(I),p(J)]=1$. Since $H/P$ is an object of $D$, we
deduce that $p(I)=1$ or $p(J)=1$. This is equivalent to saying that $I\subset
P$ of $J\subset P$.
Let $P$ be an element of $V(I_{A})$, for every $a\in A$, we have $I_{a}\subset
I_{A}\subset P$ this implies $P\in\cap_{a\in A}V(I_{a})$.
Let $P$ be an element of $\cap_{a\in A}V(I_{a})$, for every $a\in
A,I_{a}\subset P$ this implies that $I_{A}\subset P$ and $P\in V(I_{A})$, we
deduce that $V(I_{A})=\cap_{a\in A}V(I_{a})$.
Remarks.
Let $(C,D)$ be a Top couple and $H$ be an object of $C$, and $Spec(H)$ the set
of prime ideals of $H$. The set $V(I)$ are the closed subsets of a topology
defined on $Spec(H)$; $V(H)$ is the empty subset, and $V(\\{1\\})=Spec(H)$.
Proposition 3.2.
Let $(C,D)$ be a Top couple and $H$, $H^{\prime}$ two objects of $C$. A
$C$-morphism $f:H\rightarrow H^{\prime}$ induces a continuous morphism
$c(f):Spec(H^{\prime})\rightarrow Spec(H)$ by setting $c(f)(P)=f^{-1}(P)$.
Proof.
Let $P$ be an ideal of $H^{\prime}$, the property T3 implies that $f^{-1}(P)$
is an ideal subgroup of $H$ since the morphism $H/f^{-1}(P)\rightarrow
H^{\prime}/P$ is injective, the property T1 implies that $H/f^{-1}(P)\in D$.
This implies that $f^{-1}(P)$ is a prime ideal.
The morphism $c(f)$ is continuous since $c(f)^{-1}(V(I))=V(f(I))$.
Examples of Top couples.
We are going to present examples of Top couple, we start by the following
example which is already presented in Tsemo [12]:
Proposition 3.3.
Let $G$ be a group, and $D$ the class of $G$-groups which do not have divisors
of zero, the couple $(C(G),D)$ is a Top couple.
proof.
We have to show that the couple $(C(G),D)$ satisfies the properties $T1$, $T2$
and $T3$.
If $H$ is a $G$-group which does not have zero divisors, any $G$-subgroup of
$H$ does not have zero divisors of zero, thus the property $T1$ is satisfied.
We show now that the couple $(C(G),D)$ satisfies the property T2. Let $H$ be a
group which does not have divisors of zero, and $I,J$ two normal subgroups of
$H$ such that $[I,J]=1$. Suppose that $I\neq 1$ and $J\neq 1$. This implies
the existence of elements $x\in I$ and $y\in J$ different of the neutral
elements. We have $[G(x),G(y)]\subset[I,J]=1$. This is equivalent to saying
that $x$ and $y$ are divisors of zero. This fact is a contradiction to the
hypothesis; thus $I=1$ or $J=1$.
It remains to show that the couple $(C,D)$ satisfies the property $T3$. Recall
that an ideal $I$ of the element $H$ of $C(G)$ is a normal subgroup $I$ of $H$
such that $I\cap G=1$, if $f:H\rightarrow H^{\prime}$ is a $G$-morphism, and
$I$ an ideal of $H^{\prime}$, $f^{-1}(I)$ is an ideal of $H$ since the
restriction of $f$ on $G$ is injective.
We present now another example of a Top couple. One of the most interesting
problem in groups theory is to determine if a group has a finite subgroup
which is isomorphic to the free group generated by two elements. Tits Jacques
has shown that a subgroup of a linear group is virtually solvable or he
contains a free subgroup isomorphic to $F_{2}$. The Top couple that we are
presenting now is related to this subject.
Here $C$ is $C(F_{2})$ the category whose objects are injective morphisms
$F_{2}\rightarrow H$. An object $F_{2}\rightarrow H$ of $C$ is an object of
$D$ if and only if $H$ is a free group. Remark that the rank of an object of
$D$ is greater or equal to $2$. Let $(H,\phi_{H})$ be an object of $C(F_{2})$,
an ideal $I$ of $H$ is a normal subgroup $I$ such that
$I\cap\phi_{H}(F_{2})=\\{1\\}$. From this definition, we deduce that the ideal
$P$ of $H$ is a prime if and only if $H/P$ is a free group.
Proposition 3.4.
The couple $(C(F_{2}),D)$ is a Top couple.
Proof.
Let show that $(C(F_{2}),D)$ satisfies $T1$. Let $i:H\rightarrow H^{\prime}$
be an injective $F_{2}$-morphism, the fact that $H^{\prime}$ is an object of
$D$ is equivalent to saying that $H^{\prime}$ is free, this implies that $H$
is free since a subgroup of a free group is free.
Property T2. Let $H$ be an object of $D$ and $I,J$ ideal of $H$ such that
$[I,J]=1$. Since $H$ is a free group, we deduce that $I=1$ or $J=1$.
Property T3 Let $f:H\rightarrow H^{\prime}$ be an $F_{2}$-morphism, and $P$ an
ideal of $H^{\prime}$, $f^{-1}(P)$ is an ideal of $H$ since the restriction of
$f$ on $F_{2}$ is injective, the canonical map $H/f^{-1}(P)\rightarrow
H^{\prime}/P$ is injective and a subgroup of a free subgroup is free.
4\. The geometry of $G$-groups.
In the sequel we are going to consider only the topology mentioned at
proposition 3.3.
Proposition 4.1.
Let $H$ be an element of $C(H)$, for every normal subgroups $I$ and $J$ of
$H$, we have $V([I,J])=V(I\cap J)$.
Proof.
We know that $V([I,J])=V(I)\bigcup V(J)$. Since $[I,J]\subset I\cap J$, we
have $V(I\cap J)\subset V([I,J])$. Let $P$ be an element of $V([I,J])$,
suppose that $P$ is not an element of $V(I\cap J)$, this implies that there
exists an element $x$ of $I\cap J$ which is not an element of $P$. For every
element $g\in G$, $[x,x^{g}]\in[I,J]$ since $x\in I\cap J$. Since $P$ is a
$G$-domain, we deduce that $x$ is an element of $P$. This is a contradiction,
thus $P$ contains $I\cap J$.
Proposition 4.2.
Suppose that $G$ is a solvable group distinct of $\\{1\\}$, for every object
$H$ of $C(G)$, $Spec_{G}(H)$ is empty.
Proof.
Let $P$ be a prime ideal of an object $H$ of $V(I)$. Since $G$ is a solvable
group distinct of $1$, the proof of the proposition 2.2 shows that there
exists an element $x\in G$ which is a divisor of zero. This implies that $x\in
P$. This is impossible since $P\cap G=\\{1\\}$.
Remark.
Let $G$ be the trivial subgroup $\\{1\\}$, any group $H$ is an element of
$C(G)$. Let $P$ be a prime ideal of $H$, for any element $x\in H$, $x\in P$
since $[x,x]=1$. This implies that $Spec_{G}(H)$ is empty since a prime of $H$
is distinct of $H$.
Let $H$ be an element of $C(G)$, the proposition 3.3 shows that the subsets
$V(I)$ where $I$ is a normal subgroup of $H$ are the closed subsets of a
topology on the set $Spec_{G}(H)$ of prime ideals of $H$. The empty closed
subset is $V(H)$, and the $Spec_{G}(H)=V(\\{1\\})$.
We will say that $Spec_{G}(H)$ is an affine scheme.
The structural sheaf.
Let $(H,\phi_{H})$ be an element of $C(G)$, and $U$ an open subset of
$Spec_{G}(H)$, we denote by $Rad(U)$ the intersection of the elements of $U$.
We define on $Spec_{G}(U)$ the structural presheaf:
$U\longrightarrow P_{H}(U)=H/Rad(U)$
Let $U$ and $V$ be two open subsets of $Spec_{G}(H)$ such that $U\subset V$;
$Rad(V)\subset Rad(U)$, this implies the existence of a map
$r_{U,V}:P_{H}(V)=H/Rad(V)\rightarrow P_{H}(U)=H/Rad(U)$ which is the
restriction map of the presheaf $P_{H}$.
The structural sheaf of $O_{H}$ of $Spec_{G}(H)$ is the sheaf associated to
the presheaf $P_{H}$.
Recall that if $x$ is an element of $Spec_{G}(H)$ and $U(x)$ the set of open
subsets of $Spec_{G}(H)$ which contains $x$, the stalk ${P_{H}}_{x}$ of $x$ is
the inductive limit ${P_{H}}_{x}=lim_{V\in U(x)}P_{H}(V)$. The etale space
$E_{P_{H}}$ of $P_{H}$ is the topological space which is the disjoint union
$\\{{P_{H}}_{x},x\in Spec_{G}(H)\\}$ endowed with the topology generated by
$\\{(s_{x})_{x\in U},s\in P_{H}(U)\\}$ where $U$ is any open subset of
$Spec_{G}(H)$. The sheaf $O_{H}$ is a the sheaf of continuous sections of the
canonical morphism $p:E_{P_{H}}\rightarrow Spec_{G}(H)$ which sends an element
of ${P_{H}}_{x}$ to $x$.
Definition 4.1.
Let $(X,O_{X})$ and $(Y,O_{Y})$ be two topological spaces $X$ and $Y$
respectively endowed with the sheaves $O_{X}$ and $O_{Y}$. A morphism
$(f,f^{\prime}):(X,O_{X})\rightarrow(Y,O_{Y})$ is defined by a continuous map
$f:X\rightarrow Y$ and a morphism of sheaves $f^{\prime}:O_{Y}\rightarrow
f^{-1}O_{X}$ which commutes with the restriction maps.
Proposition 4.3.
Let $(H,\phi_{H})$ and $(L,\phi_{L})$ elements of $C(G)$ with trivial
radicals, there exists a natural bijection between morphisms:
$(f,f^{\prime}):(Spec_{G}(H),O_{H})\rightarrow(Spec_{G}(L),O_{L})$ and
$Hom_{G}(L,H)$.
Proof.
Let $(f,f^{\prime}):(Spec_{G}(H),O_{H})\rightarrow Spec_{G}(L),O_{L})$ be a
morphism. We associate to $(f,f^{\prime})$ the $G$-morphism
$f^{\prime}(Spec_{G}(L)):O_{L}(Spec_{G}(L))=L\rightarrow
O_{H}(Spec_{G}(H))=H$.
Let $u:L\rightarrow H$ be a $G$-morphism of $C(G)$, then $u$ induces a
continuous map
$(f_{u},f^{\prime}_{u}):(Spec_{G}(H),\phi_{H})\rightarrow(Spec_{G}(L),\phi_{L})$
defined as follows:
Let $P$ be an element of $Spec_{G}(H)$, then $f_{u}(P)=u^{-1}(P)$. The
proposition 3.3 shows that $f_{u}$ is well defined. Let $V$ be an open subset
of $Spec_{G}(L)$, the composition by $u$ induces a morphism
$f^{\prime}_{u}(V):O_{L}(V)\rightarrow f^{-1}(O_{H})(V)$.
The correspondences defined above between $Hom_{G}(L,H)$ and the set of
morphisms between $(Spec(H),O_{H})$ and $(Spec(L),O_{L})$ are inverse each
others.
Definition 4.2.
A topological manifold $X$ is an $G$-scheme if and only if there exists a
covering $(U_{i})_{i\in I}$ of $X$ and a sheaf $O_{X}$ on $X$, such that for
every $i\in I$, there exists an affine $G$-scheme
$(Spec_{G}(H_{i}),O_{H_{i}})$ and an homeomorphism of sheafed spaces
$f_{i}:(Spec_{G}(H_{i}),O_{H_{i}})\rightarrow(U_{i},O_{U_{i}})$, where
$O_{U_{i}}$ is the restriction of $O_{X}$ to $U_{i}$.
Proposition 4.4.
Let $(H,\phi_{H})$ be an element of $C(H)$, and $Nil(H)$ the normal subgroup
of $H$ generated by its nilpotent elements. The projection morphism
$p^{\prime}:H\rightarrow H/Nil(H)$ induces an homeomorphism between
$p:Spec_{G}(H/Nil(H))\rightarrow Spec_{G}(H)$.
Proof.
Let $P$ be a prime ideal of $H/Nil(H)$, $p^{\prime}({p^{\prime}}^{-1})(P)=P$.
This implies that $p$ is injective. Let $Q$ be a prime ideal of $H$,
${p^{\prime}}^{-1}(p^{\prime}(Q))=Q$ since $Q$ contains $Nil(H)$. This implies
that $p$ is surjective thus bijective. Let $I$ be an ideal of $H/Nil(H)$,
$p(V(I))=V({p^{\prime}}^{-1}(I))$, this implies that $p$ is open. We conclude
that $p$ is an homeomorphism.
Definition 4.3.
Let $G$ and $L$ be two elements of $C(G)$ with trivial radical. A morphism
$(f,f^{\prime}):(Spec_{G}(H),O_{H})\rightarrow(Spec_{G}(L),O_{L})$ is of
finite type if and only if the $L$-group $H$ is the quotient of a free group
$L[x_{1},...,x_{n}]$ by an ideal generated by a finite subset for the
$L$-structure of $H$ induced by $(f,f^{\prime})$.
The smallest $n$ such that $H$ is the quotient of $L[x_{1},...,x_{n}]$ by an
ideal of $L[x_{1},...,x_{n}]$ generated by a finite subset is called the
$L$-rank of $G$.
Suppose that $G$ is a domain and the radical of $H$ is trivial, we say that
$(Spec_{G}(H),)O_{H})$ is of finite type if and only if the canonical morphism
$(Spec_{G}(H),O_{H})\rightarrow(Spec_{G}(G),O_{G})$ is a morphism of finite
type.
We endow now the category of affine schemes with a product:
Proposition 4.5.
Let $H$ and $K$ be two elements of $C(G)$, the affine scheme
$Spec_{G}(H*_{G}K)$ is the categorical product of $Spec_{G}(H)$ and
$Spec_{G}(K)$.
Proof.
A couple of morphisms $f:Spec_{G}(L)\rightarrow Spec_{G}(H)$ and
$g:Spec_{G}(L)\rightarrow Spec_{G}(K)$ is defined by a couple of morphisms of
$G$-groups $f^{\prime}:H\rightarrow L$ and $g^{\prime}:K\rightarrow L$ which
induces a unique morphism $(f^{\prime},g^{\prime}):H*_{G}K\rightarrow L$ such
that $f^{\prime}=(f^{\prime},g^{\prime})\circ i_{H}$ and
$g^{\prime}=(f^{\prime},g^{\prime})\circ i_{K}$ where $i_{H}:H\rightarrow
H*_{G}K$ is the map defined by $i_{H}(h)=h*_{G}1$. This is equivalent to
saying that $Spec_{G}(H*_{G}K)$ is the product of $Spec_{G}(H)$ and
$Spec_{G}(K)$.
Proposition 4.6.
The category of affine $G$-schemes has sums.
Proof.
Let $H$ and $K$ be two elements of $C(G)$, we can endow the product of these
groups $H\times K$ with the diagonal action of $G$; Let
$f:Spec_{G}(H)\rightarrow Spec_{G}(L)$ and $g:Spec_{G}(K)\rightarrow
Spec_{G}(L)$ be morphisms of $G$-schemes. These morphisms defined respectively
by $G$-morphisms: $f^{\prime}:L\rightarrow H$ and $g^{\prime}:L\rightarrow K$
which induce a $G$-morphism $(f^{\prime},g^{\prime}):L\rightarrow H\times K$,
which defines the sum $(f,g):Spec_{G}(H\times K)\rightarrow Spec_{G}(L)$.
Definitions 4.4.
Let $H$ be an element of $C(G)$, we define a functor $c_{H}:C(G)\rightarrow
C(H)$ which associates to $L$ the amalgamated sum $H*_{G}L$. We say that the
$H$-scheme $Spec_{H}(H*_{G}L)$ is obtained from $L$ by a change of the basis.
In particular if $H$ is an algebraic closure $G_{al}$ of the group $G$,
$Spec_{G_{al}}(G_{al}*_{G}L)$ will be called the geometric scheme associated
to $H$.
Definitions 4.5.
Recall that an irreducible topological set is a set which cannot be the union
of two proper distinct closed subsets.
Let $(H,\phi_{H})$ be an element of $C(G)$. The normal subgroup $I$ of $H$ is
a radical, if $I$ is the intersection of all the primes ideals which contains
$I$.
Proposition 4.7,
Let $(H,\phi_{H})$ be an element of $C(H)$, if $I$ a prime ideal of $H$, then
$V(I)$ is irreducible. Conversely, suppose that $I$ is a radical ideal, then
if $V(I)$ is irreducible $I$ is a prime ideal.
Proof.
Suppose that $I$ is a prime ideal and $V(I)$ is the union of the proper closed
subsets $V(J)$ and $V(K)$. Since $I$ is a prime ideal, $I$ is an element of
$V(I)$; thus $I$ is an element of $V(J)$ or $I$ is an element of $V(K)$. If
$I$ belongs to $V(J)$; this implies that $I$ contains $J$, thus $V(I)\subset
V(J)$. This is a contradiction with the fact that $V(J)$ is a proper closed
subset of $V(I)$. The same argument shows that $I$ cannot belong to $V(K)$.
Suppose now that $V(I)$ is an irreducible closed subset and $I$ is a radical
ideal. Let $x$ and $y$ be elements of $H$ such that $[G(x),G(y)]\subset I$.
For every element $P\in V(I)$, $[G(x),G(y)]\subset P$, this implies that $x\in
P$ or $y\in P$. Let $V_{x}$ the subset of elements of $V(I)$ which contains
$x$ and $V_{y}$ the subset of elements of $V(I)$ which contains $y$, $V(I)$ is
the union of $V(x)$ and $V(y)$. Let $N(x)$ be the minimal subgroup of $H$
which contains $x$, if $P$ is an ideal which contains $x$, $N(x)\subset P$.
This implies that $V(x)=V(I)\cap V(N(x))$ and $V(y)=V(I)\cap V(N(y))$. Since
$V(I)$ is irreducible, we deduce that $V(I)\cap V(N(x))=V(I)$ or $V(I)\cap
V(N(y))=V(I)$. Suppose that $V(I)\cap N(x)=V(I)$, this implies that
$V(I)\subset V(N(x))$ and for every prime ideal $P\in V(I)$, $x\in P$, since
$I$ is a radical ideal, $x\in I$. By the same argument, we deduce that if
$V(I)\cap V(N(y))=V(I)$, $y\in I$.
Corollary 4.1.
Suppose that $G$ does not have divisors of zero, $Spec_{G}(H)$ is irreducible
if and only if the radical of $H$ is a prime ideal.
Proof.
Recall that the radical $Rad_{G}(H)$ is the intersection of all the prime
ideals of $Spec_{G}(H)$. Thus by definition, $Rad_{G}(H)$ is a radical ideal,
we can apply proposition 4.7.
Remark.
If $Rad_{G}(H)$ is a prime, it is a generic point of $Spec_{G}(H)$, that is,
its adherence is $Spec_{G}(H)$.
5\. Functor of points.
Let $H$ be an element of $C(G)$. Consider the $n$-uple $[h]=(h_{1},...,h_{n})$
of $H^{n}$; it induces a $G$-morphism:
$u_{[h]}:G[X_{1},...,X_{n}]\rightarrow H$
defined by $u_{[h]}(X_{i})=h_{i}$, for every element $f\in
G[X_{1},...,X_{n}]$, we set $f_{H}([h])=u_{[h]}(f)$, we have thus defined a
morphism $f_{H}:H^{n}\rightarrow H$.
Let $S$ be a subset of $G[X_{1},...,X_{n}]$ in their paper [2], Baumslag and
al defined the subset:
$V_{H}(S)=\\{[h]=(h_{1},...,h_{n})\in H^{n}:\forall f\in S,f_{H}([h])=1.\\}$
We are going to present this notion with the concept of functor of points.
Definition 5.1.
We denote by $Spec_{G}^{0}$ opposite to the category $Spec_{G}$ of affine
$G$-schemes. For every object $H$ of $C(G)$, the functor of points
$h_{Spec_{G}(H)}:Spec_{G}^{0}\rightarrow Sets$ is the functor defined by:
$h_{Spec_{G}(H)}(Spec_{G}(K))=mor_{G}(Spec_{G}(K),Spec_{G}(H))=Mor_{G}(H,K)$.
In particular, if $H$ is a scheme of finite type, $H$ is the quotient of
$G[X_{1},...,X_{n}]$ by an ideal $I$, an element $f\in Mor_{G}(H,K)$ is
defined by a $n$-uple $(k_{1},...,k_{n})\in K^{n}$ such that for every $f\in
I$, $f(k_{1},...,k_{n})=1$. Thus the elements of $Mor_{G}(H,K)$ in this
situation correspond to the elements of the closed subset $V_{G}(K)$ defined
in [2], where $S$ is replaced by $I$.
6.1 Noetherian $G$-schemes.
The elements of $G[X_{1},...,X_{n}]$ are often called equations in
$n$-variables over $G$. Every set of equations $E$ generates a normal subgroup
$I_{E}$ of $G[X_{1},...,X_{n}]$.
Definition 6.1.
We say that the group $G$ is Noetherian if and only if for every set of
equations $E$, there exists a finite subset $E_{0}$ of $E$ such that
$V_{G}(I_{E})=V_{G}(I_{E_{0}})$.
The problem of the existence of a solution of a finite set of equations
defined on a group $G$ has been studied by many authors. This situation is
analog the study of the roots of polynomial equations in field theory. In
fact, Scott [7] has defined a notion of algebraically closed group that we
recall.
Definition 6.2.
A set $E$ of equations and inequations defined on a group $G$ is consistent,
if there exists an embedding $G\rightarrow H$ such that $E$ has a solution in
$H$.
An example of a set of equations and inequations which is not consistent is:
$x^{2}=1,x^{3}=1,x\neq 1$.
Definitions 6.3.
A group is algebraically closed if and only if every finite set of equations
consistent defined on $G$ has a solution in $G$.
Scott has shown that every group can be imbedded in an algebraically closed
group. We show now the following proposition (See also Baumslag and al [2] p.
62)
Proposition 6.1.
Let $G$ be an algebraically closed group, then for every finitely generated
maximal $M$ ideal of $G[X_{1},...,X_{n}]$, there exists elements
$a_{1},...,a_{n}$ in $G$ such that $M$ is generated by
$\\{X_{1}a_{1}^{-1},...,X_{n}a_{n}^{-1}\\}$.
Proof.
Let $H$ be the quotient $H=G[X_{1},...,X_{n}]/M$, and $x_{1},...,x_{n}$, the
image of $X_{1},...,X_{n}$ by the natural projection
$G[X_{1},...,X_{n}]\rightarrow H$. Then $(x_{1},...x_{n})$ is a solution of
every equation defined by elements of $M$. Since $G$ is algebraically closed
and $M$ is finitely generated, there exists an element $(a_{1},...,a_{n})$ in
$G^{n}$ which is a solution of the equations defined by elements of $M$. This
implies that the ideal generated by $X_{1}a_{1}^{-1},...,X_{n}a_{n}^{-1}$
contains $M$ henceforth this ideal is equal to $M$ since $M$ is maximal.
Definition 6.4.
A topological set is Noetherian if and only if every descending chain of
closed subspaces $Y_{1}\supseteq Y_{2}....\supseteq Y_{i}....$ stabilizes.
This means that there exists $n$ such that $Y_{i}=Y_{n}$ for every $n\geq i$.
Let $H$ be an object of $C(G)$, $H$ is a Noetherian group if and only if every
family of ideals $I_{1}\subset I_{2}\subset...\subset I_{n}...$ stabilizes,
this is equivalent to saying that there exists an $i$ such that for every
$n\geq i$, $I_{n}=I_{i}$.
Remark that $Spec_{G}(H)$ is Noetherian if and only if $H$ is a Noetherian
group.
Proposition 6.2.
The object $H$ of $C(G)$ is Noetherian if and only if every ideal of $H$ is
finitely generated.
Proof.
Suppose that $H$ is Noetherian, let $I$ be an ideal of $H$, consider the set
$A$ whose objects are finitely generated ideals contained in $I$ ordered by
the inclusion. Every totally ordered family $(I_{l})_{l\in L}$ of $A$ has a
maximal element. If not we can construct a subfamily $(I_{n})_{n\in N}$ of
$(I_{l})_{l\in L}$ such that $I_{n}$ is strictly contained in $I_{n+1}$. This
is in contradiction with the fact that $H$ is Noetherian. Let $M$ be a maximal
element of $A$, $M=I$, if not there exists an element $x$ of $I$ which is not
in $M$ and the ideal generated by $M$ and $x$ contains strictly $M$.
Suppose that every ideal of $H$ is finitely generated. Let $I_{0}\subset
I_{2}...\subset I_{n}...$ be an ascending sequence of ideal, $\bigcup_{n\in
N}I_{n}=I$ is a finitely generated ideal by $x_{1},...,x_{p}$. The element
$x_{i}$ belongs to $I_{m(i)}$. Let $m$ be the maximum of $m(i)$, $I=I_{m}$.
Proposition 6.3.
Suppose that $Spec_{G}(H)$ is a Noetherian topological space, then the radical
of $H$ is the intersection of a finite number of primes ideals. In particular
if the intersection of all the elements of $Spec_{G}(H)$ is the neutral
element, then there exists a finite number of elements of $Spec_{G}(H)$ whose
intersection is the neutral element.
Proof.
We know that a Noetherian topological space is the union of a finite number of
irreducible closed subsets. This implies that there exists a finite number of
primes ideals $P_{1},...,P_{n}$ such that
$Spec_{G}(H)=\bigcup_{i=1,...,n}V(P_{i})$. The proposition 4.1 implies that
$Spec_{G}(H)=V(\bigcap_{i=1,...,n}P_{i})$. This implies that every prime ideal
$P$ of $H$ contains $\bigcap_{i=1,...,n}P_{i}$, thus
$\bigcap_{i=1,...,n}\subset Rad_{G}(H)$. It results that
$Rad_{G}(H)=\bigcap_{i=1,...,n}P_{i}$ since $Rad_{G}(H)$ is the intersection
of all the elements of $Spec_{G}(H)$.
7\. Dimension of affine schemes.
We want to introduce the notion of dimension of an affine schemes. In
commutative algebra, the dimension of a ring can be defined to be its height
that is, the maximal length of chains $0\subset P_{1}...\subset P_{n}$ of
prime ideals such that $P_{i}$ is distinct of $P_{i+1}$. We can adopt the same
definition here for the prime ideals of elements of $H$ for every object $H$
of $C(G)$. But the following proposition shows that the height of $G*Z$ can be
infinite.
Proposition 7.1.
Let $G$ be a group which does not have divisors of zero, then the height of
$G*Z$ is infinite.
Proof.
Let $I_{n}$ be the ideal of $G*Z$ generated by $2^{n}$,
$(G*Z)/I_{n}=G*(Z/2^{n})$. Suppose that $G*Z/2^{n}$ have divisors of zero
$x,y$. For every $g\in G$, we have $[x,y^{g}]=1$, $x$ and $y$ cannot be in a
subgroup conjugated to $G*1$ since $G$ does not have zero divisors; $x,y$
cannot be in a subgroup conjugated to $1*Z$ in this case, $[x,y^{g}]$ is not
trivial for every non trivial element of $g\in G$. If $x$ is not in a
conjugated of $G*1$ and $1*Z$, then [5] page 187 shows that for every $g\in
G$, $y^{g}$ and $x$ are the power of the same element; this is impossible. We
conclude that $I_{n}$ is a prime, thus the sequence $I_{n}\subset
I_{n-1}...\subset I_{0}$ is an infinite sequence of distinct prime ideals.
In classical algebraic geometry, the dimension of a scheme is also the the
transcendence dimension of its field of fraction. The corresponding notion
here is the $G$-rank:
Definition 7.1.
Let $H$ be an object of $C(G)$, a family of $G$-generators of $H$ is a family
of elements $(x_{i})_{i\in I}$ of $H$ such that $G$ and $(x_{i})_{i\in I}$
generate $H$. The $G$-rank of $H$ is the minimal cardinal of the set $I$ such
that there exists a family of generators $(x_{i})_{i\in I}$ of $H$.
We have the following result:
Proposition 7.2.
Let $G$ be a simple group, and $G^{n}$ the product of $n$ copies of $G$
endowed with the $G$-structure defined by the embedding of $G$ as the first
factor, $(G,1,...,1)$ then $Spec_{G}(H)$ contains only one element the prime
$(1,G,...,G)$ and the $rank_{G}(G^{n})=n-1$.
Proof.
Let $I$ be an ideal of $G^{n}$, suppose that $I$ contains an element
$u=(g_{1},...,g_{n})$ such that $g_{1}$ is different of the neutral element of
$G$. For every $g\in G$, let $i_{g}=(g,1,...,1)$, we have:
$i_{g}ui_{g}^{-1}=(gg_{1}g^{-1},g_{2},...,g_{n})\in I$. Since $G$ is simple,
there exists $g\in G$ such that $gg_{1}g^{-1}\neq g_{1}$. This implies that
$u^{-1}i_{g}ui_{g}^{-1}$ is a non trivial element of $G$. This is a
contradiction since $I$ is an ideal, thus $I$ is contained in $\\{1\\}\times
G^{n-1}$. If $I$ is distinct of $\\{1\\}\times G^{n-1}$, there exists an
element $v\in\\{1\\}\times G^{n-1}$ who does not belong to $I$. Let
$p:G^{n}\rightarrow G^{n}/I$ be the natural projection, for every element
$g\in G$, we have $[g,v]=1$ thus $[p(g),p(v)]=1$. This implies that $G^{n}/I$
is not a domain, thus $I$ is not a prime.
Remark.
Suppose that the center of $G$ is not trivial, then for every element of
$C(G)$, $Spec_{G}(H)$ is empty and the $G$-rank of $H$ needs not to be $0$ or
$-\infty$.
Definition 7.2.
Let $H$ be an object of $C(G)$:
\- if $Spec_{G}(H)$ is empty, the dimension of $Spec_{G}(H)$ is $-\infty$.
if $Spec_{G}(H)$ is not empty, the dimension of $Spec_{G}(H)$ is the $G$-rank
of $H/Rad_{G}(H)$ the quotient of $H$ by the intersection $Rad_{G}(H)$ of all
the elements of $Spec_{G}(H)$.
Remarks.
If $H$ is a $G$-domain, then $Rad_{G}(H)=\\{1\\}$ and the dimension of
$Spec_{G}(H)$ is $rank_{G}(H)$.
Let $G$ be a group without zero divisors, and $H=G[x_{1},...,x_{n}]$, $H$ is a
$G$-domain and the dimension of $H$ is $n$.
If $H$ is a $G$-domain, then dimension of $Spec_{G}(H)=0$ is equivalent to
saying that $H$ is isomorphic to $G$.
Let $H=G^{n}$ endowed with the $G$-structure defined by the imbedding
$(G,1,...,1)$ of $G$ in $H$, then $Rad_{G}(H)=(1,G,...,G)$ and
$H/Rad_{G}(H)\simeq G$; this implies that the dimension of $Spec_{G}(H)=0$.
We know that a if $H$ is $G$-simple, then $Spec_{G}(H)$ contains only one
element, but the $G$-rank of $H$ is not necessarily $1$. But if $G$ is
algebraically closed, we have the following result:
Proposition.
Suppose that $G$ is algebraically closed and $H$ a $G$-domain of finite type
which is $G$-simple, then $H=G$, thus $dim_{G}(H)=0$.
Proof.
There exists a positive integer $n$ and an ideal $I$ of $G[x_{1},...,x_{n}]$
such that $H=G[x_{1},...,x_{n}]/I$. Since $G$ is algebraically closed, there
exists a maximal ideal $M$ (defining by a solution of the equations defined by
a finite set of generators of $I$ containing $I$) such that
$G[x_{1},...,x_{n}]/M=G$. Let $M^{\prime}$ be the image of $M$ by the
projection map $G[x_{1},...,x_{n}]\rightarrow H$, $M^{\prime}$ is an ideal of
$H$, since $H$ is $G$-simple, we conclude that $M^{\prime}=\\{1\\}$ thus
$M=I$. We deduce that $H=G$ since $H=G[x_{1},...,x_{n}]/M$.
Let $H$ be a group, the co-rank of $n$ is the maximal integer $n$ such that
there exists a surjective morphism: $H\rightarrow F[x_{1},...,x_{n}]$. We say
that the $G$-co-rank of the object $n$ in $C(G)$ is $n$ if $n$ is the maximal
integer such that there exists a $G$-morphism: $H\rightarrow
G[x_{1},...,x_{n}]$.
Contrary to the rank, a class of ideals allows to express geometrically the
idea of the co-rank,
Definition.
Let $H$ be an element of $C(G)$, a $G$-free ideal $I$ of $H$ is an ideal such
that $H/I=G[x_{1},...,x_{n}]$.
The co-dimension of $Spec_{G}(H)$ is the maximum integer $n$ such that there
exists an ideal $I$ such that $H/I=G[x_{1},...,x_{n}]$.
Proposition 7.4.
Let $H$ be an element of $C(G)$, the co-dimension of $Spec_{G}(H)$ is the
length of the maximal sequence $P_{0}\subset P_{2}\subset...\subset P_{n}$
such that for $i=1,...,n$ $P_{i}$ is $G$-free and $P_{i+1}$ contains strictly
$P_{i}$.
Proof.
Suppose that the co-dimension of $Spec_{G}(H)$ is $n$, there exists a
surjective $G$-morphism $f:H\rightarrow G[x_{1},...,x_{n}]$. We denote by
$P_{0}$ the ideal$f^{-1}(1)$ of $H$. Let $I_{l}$ be the normal subgroup of
$G[x_{1},...,x_{n}]$ generated by $x_{1},...,x_{l}$, and for $i=1,...,n$,
$P_{i}=f^{-1}(I_{i})$. The family of ideal $(P_{i})_{i\in\\{0,...,n\\}}$ is a
chain of free ideals of length $n$ such that $P_{i}$ is strictly included in
$P_{i+1}$.
Let $P_{0}\subset P_{2}\subset...\subset P_{n}$ be a maximal chain of $G$-free
ideals such that $P_{i+1}$ contains strictly $P_{i}$. Remark that the $G$-co-
rank of $H$ is inferior or equal to $n$, since if the co-rank is superior to
$n$ the first stage of the proof shows that we can construct a chain of
$G$-free prime ideals of length strictly greater than $n$ and whose $G$-co-
rank is strictly decreasing. Thus it is enough to show that $G/P_{0}\simeq
G[x_{1},...,x_{n}]$.
Suppose that the $G$-rank of $G/P_{0}$ is strictly inferior to $n$; this
implies that there exists, $i\leq n$ such that co-rank$(P_{i})$ =co-
rank$(P_{i+1})$. Let $l_{i}:H\rightarrow H/P_{i}$ be the natural projection,
$H/P_{i+1}$ is the quotient of $H/P_{i}$ by $l_{i}(P_{i+1})$. Write
$H/P_{i}=G[x_{1},...x_{n_{i}}]$ and $H/P_{i+1}=G[y_{1},...,y_{n_{i}}]$. The
quotient of $G[x_{1},...,x_{n_{i}}]$ by the normal subgroup generated by $G$
is isomorphic to the free group $F[x_{1},...,x_{n_{i}}]$. The quotient of
$G[y_{1},...,y_{n_{i}}]$ by the normal subgroup generated by $G$ is isomorphic
to the quotient of $F[x_{1},...,x_{n_{i}}]$ by the normal subgroup generated
by the image of $l_{i}(P_{i+1})$ in $F[x_{1},...,x_{n_{i}}]$. Since this group
is isomorphic to the free group of rank $n_{i}$ $F[y_{1},...,y_{n_{i}}]$, we
deduce that $l_{i}(P_{i+1})$ is included in the normal subgroup of
$G[x_{1},...,x_{n_{i}}]$ generated by $G$. This in contradiction with the fact
that the quotient of $G[x_{1},...,x_{n_{i+1}}]$ by the image of
$l_{i}(P_{i+1})$ is a free product of $G$ and a free group.
The $G$-free ideals of $H$ generate a topology as shows the following
proposition:
Proposition 7.5.
Let $G$ be a group which does no have divisors of zero, and Let $H$ be an
element of $C(G)$, for every normal subgroup $I$ of $H$, denote
$V_{F}(I)=\\{P:$ $P$ is a $G$-free prime ideal and $I\subset P\\}$, then for
every normal subgroups $I,J$ of $H$, $V_{F}(I)\bigcup V_{F}(J)=V_{F}([I,J])$.
Let $(I_{a})_{a\in A}$ be a family of normal subgroups of $H$. Denote by
$I_{A}$ the normal subgroup generated by $(I_{a})_{a\in A}$, then
$V_{F}(I_{A})=\bigcap_{a\in A}V_{F}(I_{a})$.
Proof.
Firstly we show that $V_{F}(I)\bigcup V_{F}(J)=V_{F}([I,J])$. Let $P$ be an
element of $V_{F}(I)\bigcup V_{F}(J)$, $P$ contains $I$ or $J$ this implies
that $P$ contains $[I,J]$. Let $P$ be an element of $V_{F}([I,J])$. Suppose
that $P$ is neither an element of $V_{F}(I)$ nor an element of $V_{F}(J)$.
This implies that there exists an element $x\in I$ and $y\in J$ which are not
in $P$. For every $g\in G$, $[x,y^{g}]\in P$. Let $p:H\rightarrow H/P$ be the
natural projection, we have $[p(x),p(y)^{g}]=1$. Since $H/P$ is $G$-free, we
deduce that $x$ and $y$ are elements of $G$, this is in contradiction with the
fact that $G$ does not have divisors of zeros.
Now we show that $V(I_{A})=\bigcap_{a\in A}V(I_{a})$. Let $P$ be an element of
$V(I_{A})$, since $I_{a}\subset I_{A}\forall a\in A,$ $P\subset
V(I_{a})\forall a\in A$. Let $P$ be an element of $\bigcap_{a\in A}V(I_{a})$,
$\forall a\in A,I_{a}\subset P$, this implies that $I_{A}\subset P$ and $P\in
V(I_{A})$.
Remark.
Let $G$ be a group which does not have zero divisors, and $H$ an object of
$C(G)$, the previous proposition shows that the family of subspaces $V_{F}(I)$
where $I$ is any normal subgroup in $H$ defines a topology on the space
$Spec_{G}^{F}(H)$ of $G$-free prime ideals of $H$. But if $f:H\rightarrow
H^{\prime}$ is a morphism between the elements $H$ and $H^{\prime}$ of $G$ and
$P$ is an element of $Spec_{G}^{F}(H)$, $f^{-1}(P)$ is not necessarily a
$G$-free prime of $H$. For example let $G=PSL(n,Z)$ and
$H^{\prime}=PSL(n,Z)[X]$. Let $H$ be the subgroup of $H^{\prime}$ generated by
$PSL(n,Z)$ and $XNX^{-1}$ where $N$ is a nilpotent subgroup of $PSL(n,Z)$. The
canonical imbedding $i:H\rightarrow H^{\prime}$ is a $G$-morphism. The normal
trivial subgroup $I=\\{1\\}$ of $H^{\prime}$ is a prime ideal of $H$, but the
quotient of $H$ by $i^{-1}(I)$ is not $G$-free, since $H$ is not $G$-free.
The theorem of Kurosh says that a subgroup of a free product of groups
$H_{1}*H_{2}$ is the free product of a free group and groups isomorphic to
subgroups of $H_{1}$ and $H_{2}$. Thus if we choose $G=F_{n}n\geq 2$, a
$G$-morphism $f:H\rightarrow H^{\prime}$ induces a morphism
$Spec_{G}^{F}(H^{\prime})\rightarrow Spec_{G}^{F}(H)$ of $G$-free spectra. In
fact in this case the topology is defined by a Top couple (see definition
3.1).
Definition 7.4.
Let $G$ be a group without divisors of zero, and $H$ an element of $C(G)$; an
element $P$ of $Spec_{G}(H)$ is a $G$-point if and only if $H/P=G$.
Remark.
If $P$ is a $G$-point, the restriction of the projection $p:H\rightarrow H/P$
to $G$ is an isomorphism, thus this extension splits.
Proposition 7.6.
let $H$ be a be a $G$-domain if $G$ is a normal subgroup of $H$, for every
prime $P$ of $H$, we have of $[G,P]=1$. If there exists a $G$-point $P$ in
$H$, then $H=P\times G$.
Proof.
Let $P$ be a prime ideal. We have $[P,G]\subset P\cap G=\\{1\\}$ since $P$ is
an ideal and $G$ is a normal subgroup of $H$.
If $P$ is a $G$-point of $H$, then $H$ is an extension of $G$ by $P$ which
splits, since $[P,G]=1$ we deduce the result.
8\. Galois theory.
Let $G$ be a group, a polynomial of $G$ is an element of $G[X]$. The Galois
theory in classical field theory enables to study the solutions of polynomial
functions defined on a field. One of the property essential in that study is
the existence of factorization: let $P$ be a polynomial defined on a field,
and $a$ a root of $P$, we can write $P=(X-a)Q$; if $deg(P)$ is the degree of
$P$, we have: $deg(Q)=deg(P)-1$. Here we do not have factorization system, in
fact we can have a polynomial which has an infinite numbers of roots: Let
$G=Gl(2,C)$ be the group of automorphisms of the complex plan; the solutions
of the equation defined on $G$ by $X^{2}=1$ is the set of symmetries of the
complex plan. We can introduce anyway some notions related to Galois theory.
Definitions 8.1.
Let $G$ be a group, and $H$ an element of $C(G)$, an element $a\in H$ is
algebraic over $G$ if $a$ is the root of a polynomial whose coefficients are
in $G$.
An element $H$ of $C(G)$ is an algebraic extension of $G$ if and only if every
element of $H$ is algebraic over $G$.
We say that an algebraic extension of $G$ is finite if and only if the action
of $G$ on $H$ by left multiplications has a finite number of orbits
Proposition 8.1.
Suppose that $G=\\{1\\}$, then a group $H$ is an algebraic extension of $G$ if
and only if every element of $H$ has a finite order. In particular a group is
a finite extension of $\\{1\\}$ if and only if it is a finite group.
Proof.
If $G=\\{1\\}$, then $G[X]=F[X]=Z$ and the polynomials of $\\{1\\}$ are
$X^{n}$. A group $H$ is algebraic over $\\{1\\}$ if and only if for every $h$
in $H$, there exists an element $P$ of $F[X]$ such that $P(h)=1$. This is
equivalent to the existence of an integer $n$ such that $h^{n}=1$.
Suppose that $H$ is a finite group. Let $n_{H}$ be the order of $H$, for every
element $h\in H$, we have $h^{n_{H}}=1$; this implies that $H$ is algebraic
over $\\{1\\}$. The left action of $\\{1\\}$ has finite orbit since $H$ is
finite. Conversely, if the left action of $\\{1\\}$ on $H$ has finite orbits,
then $H$ is a finite group.
Remark.
An algebraic extension $H$ of $\\{1\\}$ is not necessarily finite even if $H$
is finitely generated. In 1904 Burnside asks the question wether every
finitely generated group for which every element has a finite order is finite.
There are groups which have been constructed answering negatively the problem
of Burnside.
Proposition 8.2.
Let $H$ be an element of $C(G)$, suppose that $H$ is the union of a finite
number of orbits for the action of $G$ on $H$ defined by the left
multiplication, then $H$ is algebraic over $G$. In particular if $G$ is a
normal subgroup of $H$ such that $G/H$ is finite, $H$ is algebraic over $G$.
Proof.
Let $x$ be an element of $H$; if $H$ is the union of a finite number of orbits
for the left action of $G$, there exists distinct integers $n$ and $m$ with
$n<m$ such that $x^{n}$ and $x^{m}$ are in the same orbit. This is equivalent
to saying that there exists $h\in H$, $g_{1},g_{2}\in G$ such that
$x^{n}=g_{1}h$ and $x^{m}=g_{2}h$, we deduce that
$h={g_{1}}^{-1}x^{n}={g_{2}}^{-1}x^{m}$. It results that
$g_{1}{g_{2}}^{-1}x^{m-n}=1$. Thus $x$ is algebraic over $G$.
Proposition 8.3.
Let $H$ be an algebraic extension of $G$, then the $G$-co-rank of $H$ is
strictly inferior to $1$.
Proof.
Suppose that the $G$-co-rank of $H$ is superior or equal to $1$. There exists
a $G$-surjection: $s:H\rightarrow G[x_{1},...,x_{n}]$. Let $y_{1},...,y_{n}$ a
family of algebraic elements of $H$ which generate $H$. There exists
$i\in\\{1,...,n\\}$ such that $s(y_{i})$ is not in $G$. Since $y_{i}$ is
algebraic, there exists a $G$-polynomial $P$ such that $P(y_{i})=1$. We also
have $P(s(y_{i}))=1$. This is impossible since $s(y_{i})$ is a non trivial
words written with elements of $G$ and with elements of the free group
generated by $x_{1},...,x_{n}$.
Proposition 8.4.
Let $H$ be an element of $C(G)$ and $K$ an element of $C(H)$, suppose that $H$
is a finite algebraic extension of $G$ and $K$ is a finite algebraic extension
of $H$; then $K$ is a finite algebraic extension of $G$.
Proof.
To show this proposition it is enough to show that the action of $G$ on $K$
has a finite number of orbits by applying proposition 8.2.
Since $H$ is a finite algebraic extension of $G$ and $K$ a finite algebraic
extension of $H$, we know that the orbit spaces $H/G$ and $K/H$ are finite; we
denote by $h_{1},...,h_{m}$ elements of $H$ such that the union of
$Gh_{1},...,Gh_{m}$ is $H$, and by $k_{1},...,k_{n}$ elements of $K$ such that
the union of $Hk_{1},...,Hk_{n}$ is $K$. Let $x$ be an element of $K$, there
exists $1\leq i\leq n$ such that $x\in Hk_{i}$, since $H$ is the union of
$Gh_{1},...,Gh_{m}$, we deduce that $x\in\\{Gh_{1}k_{i},...,Gh_{m}k_{i}\\}$.
This is equivalent to saying that the union of $Gh_{j}k_{i}$ $1\leq j\leq m$,
$1\leq i\leq n$ is $K$.
Definition 8.2.
Let $H$ be an algebraic extension of $G$, the Galois group $Gal_{G}(H)$ of $H$
is the group of automorphisms of $H$ whose restriction to $G$ is the identity.
References.
1\. Amaglobeli. M.G Algebraic sets and coordinate groups for a free nilpotent
group of nilpotency class 2. Sibirsk. Mat. Zh. Volume 48 p. 5-10.
2\. Baumslag, G, Miasnikov, A. Remeslennikov, V.N. Algebraic geometry over
groups I. Algebraic sets and ideal theory. J. Algebra. 1999, 219, 16 79.
3\. M. Chabashvili. Lattice isomorphisms of nilpotent of class 2 Hall W powers
groups. Bulletin of the Georgian National Academic of Sciences, volume3 2 2008
4\. A.Grothendieck, Él’ements de géométrie algébrique I.Publications
mathématiques de l’I.H.E.S 4, 5-228
5\. A. Karass, W. Magnus, D. Solitar, combinatorial group theory, Dover
publication 1976
6\. Micali, A. Algèbres génétiques, Cahiers de mathématiques de Montpellier
1985
7\. Scott W.R Algebraically closed groups Proc. Amer. Math. Soc. 2 (1951)
118-121
8\. Serre J-P. Géométrie algébrique, géométrie analytique, Annales de
l’Institut Fourier, Grenoble t. 6 1955-1956 1-42.
9\. J-P. Serre, Arbres, amalgames, $SL_{2}$ Astérisque 46 1977.
10\. Jacques Tits; Sous-algèbres des algèbres de Lie semi-simples. Séminaire
Bourbaki 1954-1956 p. 197-214
11\. Tits, J; Free groups in linear groups, Journal of Algebra 20 (2) 250-270
(19720
12\. Tsemo, A. Scheme theory for groups and Lie algebra, International Journal
of Algebra 5. 2011 139-148
13\. Tsemo, A. Algebraic geometry on groups: the theory of curves In
preparation.
|
arxiv-papers
| 2012-09-03T23:27:54 |
2024-09-04T02:49:34.742202
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aristide Tsemo",
"submitter": "Aristide Tsemo",
"url": "https://arxiv.org/abs/1209.0502"
}
|
1209.0628
|
# Some new identities of Genocchi numbers and polynomials involving Bernoulli
and Euler polynomials
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY [email protected],
[email protected] , Mehmet Acikgoz University of Gaziantep, Faculty of
Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY
[email protected] and Erdoğan Şen Department of Mathematics, Faculty of
Science and Letters, Namık Kemal University, 59030 Tekirdağ, TURKEY
[email protected]
###### Abstract.
In this paper, we deal with some new formulae for product of two Genocchi
polynomials together with both Euler polynomials and Bernoulli polynomials. We
get some applications for Genocchi polynomials. Our applications possess a
number of interesting properties to study in theory of Analytic numbers which
we express in the present paper.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Genocchi numbers and polynomials, Bernoulli numbers and
polynomials, Euler numbers and polynomials, Application.
## 1\. Introduction
The history of Genocchi numbers can be traced back to Italian mathematician
Angelo Genocchi (1817-1889). From Genocchi to the present time, Genocchi
numbers have been extensively studied in many different context in such
branches of Mathematics as, for instance, elementary number theory, complex
analytic number theory, Homotopy theory (stable Homotopy groups of spheres),
differential topology (differential structures on spheres), theory of modular
forms (Eisenstein series), $p$-adic analytic number theory ($p$-adic
$L$-functions), quantum physics (quantum Groups). The works of Genocchi
numbers and their combinatorial relations have received much attention [1],
[2], [4], [5], [6], [7], [24], [25], [10], [12]. For showing the value of this
type of numbers and polynomials, we list some of their applications.
In the complex plane, the Genocchi numbers, named after Angelo Genocchi, are a
sequence of integers that defined by the exponential generating function:
(1) $\frac{2t}{e^{t}+1}=e^{Gt}=\sum_{n=0}^{\infty}G_{n}\frac{t^{n}}{n!},\text{
}\left(\left|t\right|<\pi\right)$
with the usual convention about replacing $G^{n}$ by $G_{n}$, is used. When we
multiply with $e^{xt}$ in the left hand side of the Eq. (1), then we have
(2)
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{2t}{e^{t}+1}e^{xt},\text{
}\left(\left|t\right|<\pi\right)$
where $G_{n}\left(x\right)$ called Genocchi polynomials. It follows from (2)
that
$G_{1}=1,G_{2}=-1,G_{3}=0,G_{4}=1,G_{5}=0,G_{6}=-3,G_{7}=0,G_{8}=17,\cdots,$
and $G_{2n+1}=0$ for $n\in\mathbb{N}$ (for details, see [1], [2], [4], [5],
[24], [25], [10], [12]).
Differentiating both sides of (1), with respect to $x$, then we have the
following:
(3) $\frac{d}{dx}G_{n}\left(x\right)=nG_{n-1}\left(x\right)\text{ {and} }\deg
G_{n+1}\left(x\right)=n.$
On account of (1) and (3), we can easily derive the following:
(4)
$\int_{b}^{a}G_{n}\left(x\right)dx=\frac{G_{n+1}\left(a\right)-G_{n+1}\left(b\right)}{n+1}\text{.}$
By (1), we get
(5) $G_{n}\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}G_{k}x^{n-k}.$
Thanks to (4) and (5), we acquire the following equation (6):
(6) $\int_{0}^{1}G_{n}\left(x\right)dx=-2\frac{G_{n+1}}{n+1}\text{.}$
It is not difficult to see that
$\displaystyle e^{tx}$ $\displaystyle=$
$\displaystyle\frac{1}{2t}\left(\frac{2t}{e^{t}+1}e^{\left(1+x\right)t}+\frac{2t}{e^{t}+1}e^{xt}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2t}\sum_{n=0}\left(G_{n}\left(x+1\right)+G_{n}\left(x\right)\right)\frac{t^{n}}{n!}\text{.}$
By expression of (1), then we have
(8) $2x^{n}=\frac{G_{n+1}\left(x+1\right)+G_{n+1}\left(x\right)}{n+1}$
(see [10], [25]).
Let
$\mathcal{P}_{n}\mathcal{=}\left\\{p\left(x\right)\in\mathbb{Q}\left[x\right]\mid\deg
p\left(x\right)\leq n\right\\}$ be the $\left(n+1\right)$-dimensional vector
space over $\mathbb{Q}$. Probably, $\left\\{1,x,x^{2},\cdots,x^{n}\right\\}$
is the most natural basis for $\mathcal{P}_{n}$. From this, we note that
$\left\\{G_{1}\left(x\right),G_{2}\left(x\right),\cdots,G_{n+1}\left(x\right)\right\\}$
is also good basis for space $\mathcal{P}_{n}$.
In [17], Kim $et$ $al$. introduced the following integrals:
(9) $I_{m,n}=\int_{0}^{1}B_{m}\left(x\right)x^{n}dx\text{ \ and
}J_{m,n}=\int_{0}^{1}E_{m}\left(x\right)x^{n}dx$
where $B_{m}\left(x\right)$ and $E_{n}\left(x\right)$ are called Bernoulli
polynomials and Euler polynomials, respectively. Also, they are defined by the
following generating functions:
(10) $\displaystyle e^{B\left(x\right)t}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}B_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{t}{e^{t}-1}e^{xt},\text{
}\left|t\right|<2\pi,$ (11) $\displaystyle e^{E\left(x\right)t}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{2}{e^{t}+1}e^{xt},\text{
}\left|t\right|<\pi$
with $B^{n}\left(x\right):=B_{n}\left(x\right)$ and
$E^{n}\left(x\right):=E_{n}\left(x\right)$, symbolically. By substituting
$x=0$ in (10) and (11), then we readily see that,
(12) $\displaystyle\frac{t}{e^{t}-1}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}B_{n}\left(0\right)\frac{t^{n}}{n!},$ (13)
$\displaystyle\frac{2}{e^{t}+1}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}E_{n}\left(0\right)\frac{t^{n}}{n!}.$
Here $B_{n}\left(0\right):=B_{n}$ and $E_{n}\left(0\right):=E_{n}$ are called
Bernoulli numbers and Euler numbers, respectively. Thus, Bernoulli and Euler
numbers and polynomials have the following identities:
(14) $B_{n}\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}B_{k}x^{n-k}\text{ and
}E_{n}\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}E_{k}x^{n-k}.$
(for details, see [8], [21], [22], [23], [9], [11], [27]). By (12) and (13),
we have the following recurrence relations of Euler and Bernoulli numbers, as
follows:
(15) $B_{0}=1,\text{ }B_{n}\left(1\right)-B_{n}=\delta_{1,n}\text{ and
}E_{0}=1,\text{ }E_{n}\left(1\right)+E_{n}=2\delta_{0,n}$
where $\delta_{n,m}$ is the Kronecker’s symbol which is defined by
(16) $\delta_{n,m}=\left\\{\begin{array}[]{cc}1,&\text{if }n=m\\\ 0,&\text{if
}n\neq m.\end{array}\right.$
In the complex plane, we can write the following:
(17)
$\sum_{n=0}^{\infty}G_{n}\frac{\left(it\right)^{n}}{n!}=it\frac{2}{e^{it}+1}=it\sum_{n=0}^{\infty}E_{n}\frac{\left(it\right)^{n}}{n!}\text{.}$
By (17), we have
$\sum_{n=0}^{\infty}\left(\frac{G_{n+1}}{n+1}\right)\frac{\left(it\right)^{n}}{n!}=\sum_{n=0}^{\infty}E_{n}\frac{\left(it\right)^{n}}{n!},$
by comparing coefficients on the both sides of the above equatlity, then we
have
(18) $\frac{G_{n+1}}{n+1}=E_{n},\text{ (see \cite[cite]{[\@@bibref{}{Kimm
8}{}{}]}).}$
Via the equation (18), our results in the present paper can be extended to
Euler polynomials.
From of Eqs (9-16), Kim $et$ $al$. derived some new formulae on the product
for two and several Bernoulli and Euler polynomials (for details, see
[21-26]). In [26], He and Wang also gave formulae of products of the Apostol-
Bernoulli and Apostol-Euler Polynomials.
By the same motivation of the above knowledge, we write this paper. We give
some interesting properties which are procured from the basis of Genocchi.
From our methods, we obtain some new identities including Bernoulli and Euler
polynomials. Also, by using (18), we derive our results in terms of Euler
polynomials.
## 2\. On the Genocchi numbers and polynomials
In this section, we introduce the following integral equation: For $m,n\geq
1,$
(19) $T_{m,n}=\int_{0}^{1}G_{m}\left(x\right)x^{n}dx\text{.}$
By (19), becomes:
$T_{m,n}=-\frac{G_{m+1}}{m+1}-\frac{n}{m+1}\int_{0}^{1}G_{m+1}\left(x\right)x^{n-1}dx\text{.}$
Thus, we have the following recurrence formulas, as follows:
$T_{m,n}=-\frac{G_{m+1}}{m+1}-\frac{n}{m+1}T_{m+1,n-1}$
by continuing with the above recurrence relation, then we derive that
$T_{m,n}=-\frac{G_{m+1}}{m+1}+\left(-1\right)^{2}\frac{n}{\left(m+1\right)\left(m+2\right)}G_{m+2}+\left(-1\right)^{2}\frac{n\left(n-1\right)}{\left(m+1\right)\left(m+2\right)}T_{m+2,n-2}.$
Now also, we develop the following for sequel of this paper:
(20)
$T_{m,n}=\frac{1}{n+1}\sum_{j=1}^{n}\left(-1\right)^{j}\frac{\binom{n+1}{j}}{\binom{m+j}{m}}G_{m+j}+2\frac{\left(-1\right)^{n+1}G_{n+m+1}}{\left(n+m+1\right)\binom{n+m}{m}}.$
Let us now introduce the polynomial
$p\left(x\right)=\sum_{l=0}^{n}G_{l}\left(x\right)x^{n-l}\text{, with
}n\in\mathbb{N}.$
Taking $k$-th derivative of the above equality, then we have
$\displaystyle p^{\left(k\right)}\left(x\right)$ $\displaystyle=$
$\displaystyle\left(n+1\right)n\left(n-1\right)\cdots\left(n-k+2\right)\sum_{l=k}^{n}G_{l-k}\left(x\right)x^{n-l}$
$\displaystyle=$
$\displaystyle\frac{\left(n+1\right)!}{\left(n-k+1\right)!}\sum_{l=k}^{n}G_{l-k}\left(x\right)x^{n-l}\text{
}\left(k=0,1,2,\cdots,n\right).$
###### Theorem 2.1.
The following equality holds true:
$\displaystyle\sum_{l=0}^{n}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=\sum_{k=1}^{n-1}\left(\sum_{j=1}^{n-k}\left(-1\right)^{j}\frac{\binom{n-k+1}{j}}{\left(n-k+1\right)\binom{k+j}{k}}G_{k+j}+2\frac{\left(-1\right)^{n-k+1}G_{n+1}}{\left(n+1\right)\binom{n}{k}}-2\frac{G_{k+1}}{k+1}\right)$
$\displaystyle+\sum_{k=1}^{n}\left(\frac{\binom{n+2}{k}}{n+2}\sum_{l=k-1}^{n-1}\left(2-G_{l-k+1}-G_{n-k+1}\right)\right)B_{k}\left(x\right)\text{.}$
###### Proof.
On account of the properties of the Genocchi basis for the space of
polynomials of degree less than or equal to $n$ with coefficients in
$\mathbb{Q}$, then $p\left(x\right)$ can be written as follows:
(22)
$p\left(x\right)=\sum_{k=0}^{n}a_{k}B_{k}\left(x\right)=a_{0}+\sum_{k=1}^{n}a_{k}B_{k}\left(x\right)\text{.}$
Therefore, by (22), we obtain
(23) $\displaystyle a_{0}$
$\displaystyle=\int_{0}^{1}p\left(x\right)dx=\sum_{k=1}^{n}\int_{0}^{1}G_{k}\left(x\right)x^{n-k}dx=\sum_{k=1}^{n}T_{k,n-k}=\sum_{k=1}^{n-1}T_{k,n-k}+T_{k,0}$
$\displaystyle=\sum_{k=1}^{n-1}\frac{1}{n-k+1}\sum_{j=1}^{n-k}\left(-1\right)^{j}\frac{\binom{n-k+1}{j}}{\binom{k+j}{k}}G_{k+j}+2\frac{\left(-1\right)^{n-k+1}G_{n+1}}{\left(n+1\right)\binom{n}{k}}-2\frac{G_{k+1}}{k+1}\text{.}$
From expression of (2), we get
$\displaystyle a_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{k!}\left(p^{\left(k-1\right)}\left(1\right)-p^{\left(k-1\right)}\left(0\right)\right)$
$\displaystyle=$
$\displaystyle\frac{\left(n+1\right)!}{k!\left(n-k+2\right)!}\left(\sum_{l=k-1}^{n}G_{l-k+1}\left(1\right)-0^{n-l}G_{n-k+1}\right)$
$\displaystyle=$
$\displaystyle\frac{\binom{n+2}{k}}{n+2}\sum_{l=k-1}^{n-1}\left(2-G_{l-k+1}-G_{n-k+1}\right)\text{.}$
Substituting equations (23) and (2) into (22), we arrive at the desired
result.
By using (18) and theorem 2.1, we get the following corollary, which has been
stated in terms of Euler polynomials.
###### Corollary 2.2.
For any $n\in\mathbb{N}$, then we have
$\displaystyle\sum_{l=0}^{n}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=\sum_{k=1}^{n-1}\left(\sum_{j=1}^{n-k}\left(-1\right)^{j}\frac{\left(k+j\right)\binom{n-k+1}{j}}{\left(n-k+1\right)\binom{k+j}{j}}E_{k+j-1}+2\frac{\left(-1\right)^{n-k+1}E_{n}}{\binom{n}{k}}-2E_{k}\right)$
$\displaystyle+\sum_{k=1}^{n}\left(\frac{\binom{n+2}{k}}{n+2}\sum_{l=k-1}^{n-1}\left(2-\left(l-k+1\right)E_{l-k}-\left(n-k+1\right)E_{n-k}\right)\right)B_{k}\left(x\right)\text{.}$
###### Theorem 2.3.
The following nice identity
$\displaystyle\sum_{l=0}^{n}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=\sum_{k=0}^{n}\left(\left(n+1\right)\binom{n}{k}-\frac{\binom{n+1}{k}}{2}\sum_{l=k}^{n-1}\left(G_{l-k}-G_{n-k}\right)\right)E_{k}\left(x\right)$
is true.
###### Proof.
Let us now consider the polynomial $p\left(x\right)$ in terms of Euler
polynomials as follows:
$p\left(x\right)=\sum_{k=0}^{n}b_{k}E_{k}\left(x\right)\text{.}$
In [17], Kim $et$ $al$. gave the coefficients $b_{k}$ by utilizing from the
definition of Bernoulli polynomials. Now also, we give the coefficients
$b_{k}$ by using the definition of Genocchi polynomials, as follows:
$\displaystyle b_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2k!}\left(p^{\left(k\right)}\left(1\right)+p^{\left(k\right)}\left(0\right)\right)$
$\displaystyle=$
$\displaystyle\frac{\left(n+1\right)!}{2k!\left(n-k+1\right)!}\sum_{l=k}^{n}\left(G_{l-k}\left(1\right)+0^{n-l}G_{l-k}\right)$
$\displaystyle=$
$\displaystyle\left(n+1\right)\binom{n}{k}-\frac{\binom{n+1}{k}}{2}\sum_{l=k}^{n-1}\left(G_{l-k}-G_{n-k}\right)\text{.}$
After the above applications, we complete the proof of theorem.
By employing (18) and theorem 2.3, we have the following corollary, which is
sums of products of two Euler polynomials.
###### Corollary 2.4.
For each $n\in\mathbb{N}$, then we have
$\displaystyle\sum_{l=0}^{n}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=\sum_{k=0}^{n}\left(\left(n+1\right)\binom{n}{k}-\frac{\binom{n+1}{k}}{2}\sum_{l=k}^{n-1}\left(\left(l-k\right)E_{l-k-1}-\left(n-k\right)E_{n-k-1}\right)\right)E_{k}\left(x\right)\text{.}$
We now discover the following theorem, which will be interesting and
worthwhile theorem for studying in Analytic numbers theory.
###### Theorem 2.5.
The following equality holds:
$\displaystyle\sum_{l=0}^{n}\frac{1}{l!\left(n-l\right)!}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=\sum_{l=1}^{n}\frac{2^{l-2}}{l!}\sum_{j=l-1}^{n}\frac{\left(2-G_{l-j+1}\right)G_{l}\left(x\right)}{\left(j-l+1\right)!\left(n-j\right)!}+\frac{2^{l-2}}{l!\left(n-l+1\right)!}G_{n-l+1}G_{l}\left(x\right)\text{.}$
###### Proof.
It is proved by using the following polynomial $p\left(x\right):$
(25)
$p\left(x\right)=\sum_{l=0}^{n}\frac{1}{l!\left(n-l\right)!}G_{l}\left(x\right)x^{n-l}=\sum_{l=0}^{n}a_{l}G_{l}\left(x\right)\text{.}$
It is not difficult to indicate the following:
(26)
$p^{\left(k\right)}\left(x\right)=2^{k}\sum_{l=k}^{n}\frac{1}{\left(l-k\right)!\left(n-l\right)!}G_{l-k}\left(x\right)x^{n-l}\text{.}$
Then, we see that for $k=1,2,\cdots,n,$
(27) $\displaystyle a_{l}$
$\displaystyle=\frac{1}{2l!}\left(p^{\left(l-1\right)}\left(1\right)+p^{\left(l-1\right)}\left(0\right)\right)$
$\displaystyle=\frac{2^{l-2}}{l!}\sum_{j=l-1}^{n}\frac{1}{\left(j-l+1\right)!\left(n-j\right)!}\left(G_{j-l+1}\left(1\right)+0^{n-j}G_{j-l+1}\right)$
$\displaystyle=\frac{2^{l-2}}{l!}\sum_{j=l-1}^{n}\frac{\left(2-G_{l-j+1}\right)}{\left(j-l+1\right)!\left(n-j\right)!}+\frac{2^{l-2}}{l!\left(n-l+1\right)!}G_{n-l+1}.$
By (25) and (27), we arrive at the desired result.
###### Theorem 2.6.
The following identity
(28)
$\displaystyle\sum_{l=0}^{n}\frac{1}{l!\left(n-l\right)!}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=-2\frac{G_{n+1}}{n+1}+\sum_{l=1}^{n-1}\sum_{j=1}^{n-l}\frac{\left(-1\right)^{j}}{l!\left(n-l+1\right)!}\frac{\binom{n-l+1}{j}}{\binom{l+j}{l}}G_{l+j}+2\frac{\left(-1\right)^{n-l+1}G_{n+1}}{\left(n+1\right)\binom{n}{l}}$
$\displaystyle+\sum_{k=1}^{n}\left(\frac{2^{k-1}}{k!}\sum_{l=k-1}^{n}\frac{\left(2-G_{l-k+1}\right)}{\left(l-k+1\right)!\left(n-l\right)!}-\frac{2^{k-1}}{k!\left(n-k+1\right)!}G_{n-k+1}\right)B_{k}\left(x\right)$
is true.
###### Proof.
Now also, let us take the polynomial in terms of Bernoulli polynomials as
(29) $p\left(x\right)=\sum_{k=0}^{n}a_{k}B_{k}\left(x\right).$
By using the above identity, we develop as follows:
(30) $\displaystyle a_{0}$
$\displaystyle=\int_{0}^{1}p\left(x\right)dx=\sum_{l=0}^{n}\frac{1}{l!\left(n-l\right)!}\int_{0}^{1}G_{l}\left(x\right)x^{n-l}dx$
$\displaystyle=\sum_{l=0}^{n}\frac{1}{l!\left(n-l\right)!}T_{l,n-l}=T_{n,0}+\sum_{l=1}^{n-1}\frac{1}{l!\left(n-l\right)!}T_{l,n-l}$
$\displaystyle=-2\frac{G_{n+1}}{n+1}+\sum_{l=1}^{n-1}\sum_{j=1}^{n-l}\frac{\left(-1\right)^{j}}{l!\left(n-l+1\right)!}\frac{\binom{n-l+1}{j}}{\binom{l+j}{l}}G_{l+j}+2\frac{\left(-1\right)^{n-l+1}G_{n+1}}{\left(n+1\right)\binom{n}{l}}.$
By (26), we compute $a_{k}$ coefficients, as follows:
$\displaystyle a_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{k!}\left(p^{\left(k-1\right)}\left(1\right)-p^{\left(k-1\right)}\left(0\right)\right)$
$\displaystyle=$
$\displaystyle\frac{2^{k-1}}{k!}\sum_{l=k-1}^{n}\frac{1}{\left(l-k+1\right)!\left(n-l\right)!}\left(G_{l-k+1}\left(1\right)-0^{n-l}G_{l-k+1}\right)$
$\displaystyle=$
$\displaystyle\frac{2^{k-1}}{k!}\sum_{l=k-1}^{n}\frac{\left(2-G_{l-k+1}\right)}{\left(l-k+1\right)!\left(n-l\right)!}-\frac{2^{k-1}}{k!\left(n-k+1\right)!}G_{n-k+1}\text{.}$
When we substituted (30) and (2) into (29), the proof of theorem will be
completed.
By using equation (18) and theorem 2.6, we procure the following corollary.
###### Corollary 2.7.
For any $n\in\mathbb{N},$ then we have
$\displaystyle\sum_{l=0}^{n}\frac{1}{l!\left(n-l\right)!}G_{l}\left(x\right)x^{n-l}$
$\displaystyle=-2E_{n}+\sum_{l=1}^{n-1}\sum_{j=1}^{n-l}\frac{\left(-1\right)^{j}}{l!\left(n-l+1\right)!}\frac{\left(l+j\right)\binom{n-l+1}{j}}{\binom{l+j}{l}}E_{l+j-1}+2\frac{\left(-1\right)^{n-l+1}E_{n}}{\binom{n}{l}}$
$\displaystyle+\sum_{k=1}^{n}\left(\frac{2^{k-1}}{k!}\sum_{l=k-1}^{n}\frac{\left(\frac{2}{l-k+1}-E_{l-k}\right)}{\left(l-k\right)!\left(n-l\right)!}-\frac{2^{k-1}}{k!\left(n-k\right)!}E_{n-k}\right)B_{k}\left(x\right)$
In [10], it is well-known that
(32)
$G_{n}\left(x+y\right)=\sum_{k=0}^{n}\binom{n}{k}G_{k}\left(x\right)y^{n-k}\text{.}$
For $x=y$ in (32), then we have the following
(33)
$\frac{1}{n!}G_{n}\left(2x\right)=\sum_{k=0}^{n}\frac{1}{k!\left(n-k\right)!}G_{k}\left(x\right)x^{n-k}.$
By comparing the equations of (28) and (33), then we readily derive the
following corollary.
###### Corollary 2.8.
$\frac{1}{n!}G_{n}\left(2x\right)=\text{the right-hand-side of equation in
Theorem 2.4.}$
###### Theorem 2.9.
The following equality
$\displaystyle\sum_{k=1}^{n-1}\frac{1}{k\left(n-k\right)}G_{k}\left(x\right)x^{n-k}$
$\displaystyle=\sum_{k=0}^{n}\left(\frac{\binom{n}{k}}{2\left(n-k+1\right)}\left(H_{n-1}-H_{n-k}\right)-\frac{\binom{n}{k}}{2n}\sum_{l=k}^{n-1}\frac{\left(2-G_{l-k+1}\right)}{\left(n-l\right)\left(l-k+1\right)}\right)G_{k}\left(x\right)$
holds true.
###### Proof.
To prove this theorem, we introduce the following polynomial
$p\left(x\right):$
$p\left(x\right)=\sum_{k=1}^{n-1}\frac{1}{k\left(n-k\right)}G_{k}\left(x\right)x^{n-k}\text{.}$
Then, we derive $k$-th derivative of $p\left(x\right)$ is given by
(34)
$p^{\left(k\right)}\left(x\right)=C_{k}\left(x^{n-k}+G_{n-k}\left(x\right)\right)+\left(n-1\right)\left(n-2\right)\cdots\left(n-k\right)\sum_{l=k+1}^{n-1}\frac{G_{l-k}\left(x\right)x^{n-l}}{\left(n-l\right)\left(l-k\right)},$
where
$C_{k}=\frac{\sum_{j=1}^{k}\left(n-1\right)...\left(n-j+1\right)\left(n-j-1\right)...\left(n-k\right)}{n-k}\text{
}\left(k=1,2,...,n-1\right)\text{, }C_{0}=0\text{.}$
We want to note that
$p^{\left(n\right)}\left(x\right)=\left(p^{\left(n-1\right)}\left(x\right)\right){\acute{}}=C_{n-1}\left(x+G_{1}\left(x\right)\right)=C_{n-1}=\left(n-1\right)!H_{n-1},$
where $H_{n-1}$ are called Harmonic numbers, which are defined by
$H_{n-1}=\sum_{j=1}^{n-1}\frac{1}{j}\text{.}$
With the properties of Genocchi basis for the space of polynomials of degree
less than or equal to $n$ with coefficients in $\mathbb{Q}$, $p\left(x\right)$
is introduced by
(35) $p\left(x\right)=\sum_{k=0}^{n}a_{k}G_{k}\left(x\right)\text{.}$
By expression of (35), we obtain that
$\displaystyle a_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2k!}\left(p^{\left(k-1\right)}\left(1\right)+p^{\left(k-1\right)}\left(0\right)\right)$
$\displaystyle=$
$\displaystyle\frac{C_{k-1}}{2k!}\left(1+2\delta_{1,n-k+1}\right)+\frac{\left(n-1\right)!}{2k!\left(n-k\right)!}\sum_{l=k}^{n-1}\frac{\left(G_{l-k+1}\left(1\right)+0^{n-l}G_{l-k+1}\right)}{\left(n-l\right)\left(l-k+1\right)}$
$\displaystyle=$
$\displaystyle\frac{C_{k-1}}{2k!}-\frac{\binom{n}{k}}{2n}\sum_{l=k}^{n-1}\frac{\left(2-G_{l-k+1}\right)}{\left(n-l\right)\left(l-k+1\right)}.$
As a result,
$a_{n}=\frac{1}{2n!}\left(p^{\left(n\right)}\left(1\right)+p^{\left(n\right)}\left(0\right)\right)=\frac{C_{n-1}}{n!}=\frac{H_{n-1}}{n}\text{.}$
In [17], it is well-known that
(36)
$\frac{C_{k-1}}{k!}=\frac{\binom{n}{k}}{\left(n-k+1\right)}\left(H_{n-1}-H_{n-k}\right)\text{.}$
By (34), (35) and (36), then we arrive at the desired result.
From (18) and (2.9), we acquire the following.
###### Corollary 2.10.
The following identity holds:
$\displaystyle\sum_{k=1}^{n-1}\frac{1}{k\left(n-k\right)}G_{k}\left(x\right)x^{n-k}$
$\displaystyle=\sum_{k=1}^{n}\left(\frac{\binom{n}{k}}{2\left(n-k+1\right)}\left(H_{n-1}-H_{n-k}\right)-\frac{\binom{n}{k}}{2n}\sum_{l=k}^{n-1}\frac{\left(\frac{2}{l-k+1}-E_{l-k}\right)}{\left(n-l\right)}\right)kE_{k-1}\left(x\right)$
## 3\. Further Remarks
Let $\mathcal{P}_{n}=\left\\{\sum_{j=0}a_{j}x^{j}\mid
a_{j}\in\mathbb{Q}\right\\}$ be the space of polynomials of degree less than
or equal to $n.$ In this final section, we will give the matrix formulation of
Genocchi polynomials. Let us now consider the polynomial
$p\left(x\right)\in\mathcal{P}_{n}$ as a linear combination of Genocchi basis
polynomials with
$p\left(x\right)=C_{1}G_{1}\left(x\right)+C_{2}G_{2}\left(x\right)+\cdots+C_{n+1}G_{n+1}\left(x\right)\text{.}$
We can write the above as a product of two variables
(37)
$p\left(x\right)=\left(\begin{array}[]{cccc}G_{1}\left(x\right)&G_{2}\left(x\right)&\cdots&G_{n+1}\left(x\right)\end{array}\right)\left(\begin{array}[]{c}C_{1}\\\
C_{2}\\\ \vdots\\\ C_{n+1}\end{array}\right).$
From expression of (37), we consider the following equation:
$p\left(x\right)=\left(\begin{array}[]{ccccc}1&x&x^{2}&\cdots&x^{n}\end{array}\right)\left(\begin{array}[]{cccc}g_{1,1}&g_{1,2}&\cdots&g_{1,n+1}\\\
0&g_{2,2}&\cdots&g_{2,n+1}\\\ 0&0&\cdots&g_{3,n+1}\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&0&g_{n+1,n+1}\end{array}\right)\left(\begin{array}[]{c}C_{1}\\\ C_{2}\\\
C_{3}\\\ \vdots\\\ C_{n+1}\end{array}\right)$
where $g_{i,j}$ are the coefficients of the power basis that are used to
determine the respective Genocchi polynomials. We now list a few Genocchi
polynomials as follows:
$G_{1}\left(x\right)=1,\text{ }G_{2}\left(x\right)=2x-1,\text{
}G_{3}\left(x\right)=3x^{2}-3x,\text{
}G_{4}\left(x\right)=4x^{3}-6x^{2}-1,\cdots.$
In the quadratic case ($n=2$), the matrix representation is
$p\left(x\right)=\left(\begin{array}[]{ccc}1&x&x^{2}\end{array}\right)\left(\begin{array}[]{ccc}1&-1&0\\\
0&2&-3\\\ 0&0&3\end{array}\right)\left(\begin{array}[]{c}C_{1}\\\ C_{2}\\\
C_{3}\end{array}\right)\text{.}$
In the cubic case ($n=3$), the matrix representation is
$p\left(x\right)=\left(\begin{array}[]{cccc}1&x&x^{2}&x^{3}\end{array}\right)\left(\begin{array}[]{cccc}1&-1&0&-1\\\
0&2&-3&0\\\ 0&0&3&-6\\\ 0&0&0&4\end{array}\right)\text{.}$
Throughout this paper, many considerations for Genocchi polynomials seem to be
useful for a matrix formulation.
## References
* [1] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [2] S. Araci, J. J. Seo, D. Erdal, New construction weighted $(h,q)$-Genocchi numbers and polynomials related to zeta type functions, Discrete Dyn. Nat. Soc. 2011, Art. ID 487490, 7 pp.
* [3] S. Araci, M. Acikgoz and J. J. Seo, Explicit formulas involving $q$-Euler numbers and polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 298531, 11 pages.
* [4] S. Araci, M. Acikgoz and F. Qi, On the $q$-Genocchi numbers and polynomials with weight zero and their interpolation function, arXiv:1202.2643v1 [math.NT].
* [5] S. Araci, D. Erdal, and D. J. Kang, Some new properties on the $q$-Genocchi numbers and polynomials associated with $q$-Bernstein polynomials, Honam Mathematical J. 33 (2011) no. 2, pp. 261-270.
* [6] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, Bulletin of the Malaysian Mathematical Sciences and Society (accepted for publication).
* [7] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $log$ gamma type functions associated with $q$-extensions of Genocchi and Euler numbers with weight $\alpha$, Bulletin of the Korean Mathematical Society (In press).
* [8] M. Acikgoz and Y. Simsek, On multiple interpolation functions of the Nörlund-type $q$-Euler polynomials, Abstract and Applied Analysis, Volume 2009, Article ID 382574, 14 pages.
* [9] T. Kim, Symmetry of power sum polynomials and multivariate fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys. 16 (2009), no. 1, 93-96.
* [10] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20, 23–28 (2010).
* [11] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys. 16 (2009), No. 4, 484-491.
* [12] T. Kim, On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465.
* [13] T. Kim, Identities involving Frobenius–Euler polynomials arising from non-linear differential equations, Journal of Number Theory, Volume 132, Issue 12, Pages 2854-2865, 2012.
* [14] D. S. Kim, T. Kim, S. H. Lee, Y. H. Kim, Some identities for the product of two Bernoulli and Euler polynomials, Advances in Difference Equations 2012 2012:95.
* [15] T. Kim and D. S. Kim, Extended Laguerre polynomials associated with Hermite, Bernoulli and Euler numbers and polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 95730, 13 pages.
* [16] T. Kim, D. S. Kim and D. V. Dolgy, Some identities on Bernoulli and Hermite polynomials associated with Jacobi polynomials, Discrete Dynamics in Nature and Society, Volume 2012, Article ID 584643, 10 pages.
* [17] D. S. Kim, D. V. Dolgy, T. Kim, and S. H. Rim, Some formulae for the product of two Bernoulli and Euler polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 784307, 15 pages.
* [18] D. S. Kim and T. Kim, Euler basis, identities and their applications, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article ID 343981, 13 pages.
* [19] D. S. Kim and T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article ID 463659, 13 pages.
* [20] D. S. Kim, T. Kim, W. J. Kim and D. V. Dolgy, A note on Eulerian polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 269640, 10 pages.
* [21] A. Bayad, T. Kim, Identities involving values of Bernstein, $q$-Bernoulli, and $q$-Euler polynomials, Russ. J. Math. Phys. 18 (2011), no. 2, 133-143.
* [22] E. Cetin, M. Acikgoz, I. N. Cangul and S. Araci, A note on the ($h,q$)-Zeta type function with weight $\alpha$, arXiv:1206.5299v1 [math.NT].
* [23] I. N. Cangul, H. Ozden, and Y. Simsek, Generating functions of the ($h,q$) extension of twisted Euler polynomials and numbers, Acta Mathematica Hungarica, vol. 120, no. 3, pp. 281–299, 2008.
* [24] H. Jolany and H. Sharifi, Some results for the Apostol-Genocchi polynomials of higher order, Bulletin of Malaysian Mathematical Sciences Society (in press).
* [25] H. Jolany, R. E. Alikelaye and S. S. Mohamad, Some results on the generalization of Bernoulli, Euler and Genocchi polynomials, Acta Universitatis Apulensis, No. 27, 2011, pp. 299-306.
* [26] Y. He and C. Wang, Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials, Discrete Dynamics in Nature and Society, vol. 2012, Article ID 927953, 11 pages, 2012. doi:10.1155/2012/927953.
* [27] Q. M. Luo, B. N. Guo, F. Qi, and L. Debnath, Generalization of Bernoulli numbers and polynomials, IJMMS, Vol. 2003, Issue 59, 2003, 3769-3776.
|
arxiv-papers
| 2012-09-04T12:31:26 |
2024-09-04T02:49:34.752978
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Erdo\\u{g}an \\c{S}en",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1209.0628"
}
|
1209.0648
|
# Fukushima plutonium effect and blow-up regimes in neutron-multiplying media
V.D. Rusov1111Corresponding author: Vitaliy D. Rusov, E-mail: [email protected],
V.A. Tarasov1, V.M. Vaschenko2, E.P. Linnik1,
T.N. Zelentsova1, M.E. Beglaryan1, S.A. Chernegenko1, S.I. Kosenko1,
P.A. Molchinikolov1, V.P. Smolyar1, E.V. Grechan1
###### Abstract
It is shown that the capture and fission cross-sections of 238U and 239Pu
increase with temperature within 1000-3000 K range, in contrast to those of
235U, that under certain conditions may lead to the so-called blow-up modes,
stimulating the anomalous neutron flux and nuclear fuel temperature growth.
Some features of the blow-up regimes in neutron-multiplying media are
discussed.
1Department of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University, Odessa, Ukraine
2State Ecological Academy for Postgraduate Education, Kiev, Ukraine
## 1 Introduction
It is known that after the loss of coolant at three nuclear reactors during
Fukushima nuclear accident its nuclear fuel melted. It means that the
temperature in the active zone at some moments reached the melting point of
uranium-oxide fuel222Let us note that the third block partially used MOX-fuel
enriched with plutonium., i.e. $\sim$3000∘C.
Surprisingly enough, scientific literature today contains absolutely no either
experimental or even theoretically calculated data on behavior of the
${}^{238}U$ and ${}^{239}Pu$ capture and fission cross-sections depending on
temperature at least in 1000-3000∘C range. At the same time there are serious
reasons to believe that the cross-section values of the mentioned elements
increase with temperature. At least we may point, for example, to the
qualitative estimates by Ukraintsev [1], Obninsk Institute of Atomic
Energetics (Russia), that confirm the possibility for ${}^{239}Pu$ cross-
sections growth in 300-1500∘C range.
Obviously, such anomalous temperature dependence of ${}^{238}U$ and
${}^{239}Pu$ capture and fission cross-sections may change the neutron and
heat kinetics of nuclear reactors drastically. This is also true for the
perspective new generation fast reactors (uranium-plutonium of Feoktistov [2]
and thorium-uranium of Teller [3] type), that we classify as fast TWR
reactors. Hence it is very important to know the anomalous temperature
behavior of ${}^{238}U$ and ${}^{239}Pu$ capture and fission cross-sections,
and furthermore it becomes critically important to know their influence on the
heat transfer kinetics, since it may become a reason of the positive
feedback333Positive Feedback is a type of feedback when a change in the output
signal leads to the change in the input signal, which in its turn leads to a
further deviation of the output signal from its original value. In other
words, PF leads to instability and appearance of qualitatively new (often
self-oscillation) systems. (PF) with neutron kinetics leading to undesirable
solution stability loss (the nuclear burning wave) as well as to a trivial
reactor runaway with a subsequent nontrivial catastrophe.
A special case of PF is a non-linear PF, which leads to the system evolution
in so-called blow-up mode [4, 5, 6, 7, 8, 9], or in other words, in such a
dynamic mode when one or several modeled values (e.g. temperature and neutron
flux) grows to infinity at a finite time. In reality, a phase transition is
observed instead of the infinite values in this case, and this can in its turn
become a first stage or a precursor of the future technogenic disaster.
Investigation of the temperature dependence of ${}^{238}U$ and ${}^{239}Pu$
capture and fission cross-sections in 300-3000∘C range, and correspondingly,
the heat transfer kinetics and its influence on neutron kinetics in TWR, is
the main goal of the present paper.
## 2 Temperature blow-up regimes in neutron-multiplying media
Heat transfer equation for uranium-plutonium fissile medium is:
$\displaystyle\rho\left(\vec{r},T,t\right)\cdot$ $\displaystyle
c\left(\vec{r},T,t\right)\cdot\dfrac{\partial
T\left(\vec{r},t\right)}{\partial t}=$
$\displaystyle=\aleph\left(\vec{r},T,t\right)\cdot\Delta
T\left(\vec{r},t\right)+\nabla\aleph\left(\vec{r},T,t\right)\cdot\nabla
T\left(r,t\right)+q_{T}^{f}\left(\vec{r},T,t\right),$ (1)
where the effective medium density is
$\rho\left(\vec{r},T,t\right)=\sum\limits_{i}N_{i}\left(\vec{r},T,t\right)\cdot\rho_{i},$
(2)
$\rho_{i}$ are tabulated values, $N_{i}\left(\vec{r},T,t\right)$ are the
concentrations of the medium components, while the effective specific heat
capacity (accounting for the medium components heat capacity values $c_{i}$)
and fissile material heat conductivity coefficient (accounting for the medium
components heat conductivity coefficients $\aleph_{i}(T)$) respectively are:
$c\left(\vec{r},T,t\right)=\sum\limits_{i}c_{i}(T)N_{i}\left(\vec{r},T,t\right),$
(3)
$\aleph\left(\vec{r},T,t\right)=\sum\limits_{i}\aleph_{i}(T)N_{i}\left(\vec{r},T,t\right).$
(4)
Here $q_{T}^{f}\left(\vec{r},T,t\right)$ is the heat source density produced
by nuclear fissions $N_{i}$ of fissile metal components which vary in time.
Theoretical temperature dependence of heat capacity $c(T)$ for metals is
known: at low temperatures $c(t)\sim T^{3}$, and at high temperatures
$c(T)\rightarrow const$, where the constant value ($const\approx
6~{}Cal/(mol\cdot deg)$) is determined by Dulong-Petit law. At the same time
it is known that thermal expansion coefficient is small for metals, therefore
the specific heat capacity at constant volume $c_{v}$ almost equals to the
specific heat capacity at constant pressure $c_{p}$. On the other hand, the
theoretical dependence of heat conductivity $\aleph_{i}(T)$ at high
temperature of ”fissile” metals is not known, while it is experimentally
determined that the heat conductivity coefficient $\aleph(T)$ of fissile
medium is a non-linear function of temperature (e.g. see [10], where heat
conductivity coefficient is given for $\alpha$-uranium 238 and for metallic
plutonium 239, and also [11]).
Further for solving thermal conductivity equation we used the following
initial and boundary conditions:
$T(r,t=0)=300~{}K~{}~{}~{}and~{}~{}j_{n}=\aleph\left[T(r\in\Re,t)-T_{0}\right],$
(5)
where $j_{n}$ is the normal (to the fissile medium boundary) heat flux density
component, $\aleph(T$) is the thermal conductivity coefficient, $\Re$ is the
fissile medium boundary, $T_{0}$ is the temperature of the medium adjacent to
the active zone.
Obviously, if the cross-sections of some fissile nuclides increase with
temperature, then due to exothermic nature of the nuclei fission reaction, the
significantly non-linear kinetics of mother and daughter nuclides in the
nuclear reactor will immediately result in an autocatalytic increase of
generated heat, just like in the autocatalytic processes of exothermic
chemical reactions. The heat source density
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ which characterizes the amount of
generated heat in this case will be:
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)=\Phi\left(\vec{r},T,t\right)\sum\limits_{i}Q_{i}^{f}\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)N_{i}\left(\vec{r},T,t\right),~{}~{}[W/cm^{3}],$
(6)
where
$\Phi\left(\vec{r},T,t\right)=\int\limits_{0}^{E^{max}_{n}}\Phi\left(\vec{r},E,T,t\right)dE$
is the total neutron flux density; $\Phi\left(\vec{r},E,T,t\right)$ is the
density of a neutron flux with energy $E$; $Q_{i}^{f}$ is the mean heat
released in one fission event of the $i$-th nuclide;
$\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)=\int\limits_{0}^{E_{n}^{max}}\sigma_{f}^{i}(E,T)\rho\left(\vec{r},E,T,t\right)dE$
is the fission cross-section of the $i$-th nuclide averaged over the neutron
spectrum;
$\rho\left(\vec{r},E,T,t\right)=\Phi\left(\vec{r},E,T,t\right)/\Phi\left(\vec{r},T,t\right)$
is the probability density function of the neutron energy distribution;
$\sigma_{f}^{i}(E,T)$ is the microscopic fission cross-section of the i-th
nuclide, which is known to depend on the neutron energy and fissile medium
temperature (Doppler effect [12]); $N_{i}\left(\vec{r},T,t\right)$ is the
density of the $i$-th nuclide nuclei.
As follows from (6), in order to build a density function of the heat source,
it is necessary to solve a problem related to the construction of a
theoretical dependence of the cross-sections
$\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)$ averaged over the neutron
spectrum on the temperature of reactor fuel (fissile medium). As is known, the
impact of the nuclei thermal motion in the medium comes to a broadening and
lowering of the resonances. By optical analogy this phenomenon is usually
referred to as Doppler effect [12]. Since the resonance levels in the low
energy region are observed for heavy nuclei only, the Doppler effect is
noticeable only during the interaction of neutrons with such nuclei. And the
higher is the environment temperature, the stronger is the effect.
Therefore a program was developed using Microsoft Fortran Power Station 4.0
(MFPS 4.0) that allows at the first stage to calculate the cross-sections of
the resonance neutron reactions depending on neutron energy taking into
account the Doppler effect. The cross-sections dependence on neutron energy
for reactor nuclides from ENDF/B-VII database [13] corresponding to 300K
environment temperature were taken as the input data for the calculations. For
example, the results for radioactive neutron capture cross-sections dependence
on neutron energy for 235U are given in Fig. 1 for different temperatures of
the fissile medium in 300K-3000K temperature range. Using this program, the
dependence of scattering, fission and radioactive neutron capture cross-
sections for essential reactor fuel nuclides ${}_{92}^{235}$U,
${}_{92}^{238}$U, ${}_{92}^{239}$U and ${}_{94}^{239}$Pu were obtained for
different temperatures in 300K-3000K range.
Figure 1: Calculated dependency of radioactive neutron capture reaction on its
energy for ${}^{235}_{92}U$ at different temperatures within 300K to 3000K.
At the second stage a program was developed to calculate the dependence of the
cross-sections $\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)$ averaged
over the neutron spectrum for main reactor nuclides and for main reactor
nuclear neutron reactions for the specified temperatures. The averaging of
neutron cross-sections for the Maxwell distribution was performed using the
following expression:
$\left\langle\sigma\left(E_{lim},T\right)\right\rangle=\dfrac{\int\limits_{0}^{E_{lim}}E^{1/2}e^{-E/kT}\sigma(E,T)dE}{\int\limits_{0}^{E_{lim}}E^{1/2}e^{-E/kT}dE},$
where $E_{lim}$ is the the upper limit of neutrons thermalization, while for
the procedure of neutron cross-sections averaging over Fermi spectrum the
following expression was used:
$\left\langle\sigma\left(E_{lim},T\right)\right\rangle=\dfrac{\int\limits_{E_{lim}}^{\infty}\sigma(E,T)E^{-1}dE}{\int\limits_{E_{lim}}^{\infty}E^{-1}dE},$
During further calculations in our programs we used the results obtained at
the first stage i.e. the dependence of reaction cross-sections on neutron
energy and medium temperature (Doppler effect). The neutron spectrum was
specified in a combined way – below the limit of thermalization $E_{lim}$ the
neutron spectrum was described by Maxwell spectrum
$\Phi_{M}\left(E_{n}\right)$; above $E_{lim}$ but below $E_{F}$ (upper limit
for Fermi neutron energy spectrum) the neutron spectrum was described by Fermi
spectrum $\Phi_{F}(E)$ for a moderating medium with absorption; above $E_{F}$,
but below maximal neutron energy $E_{n}^{max}$ the spectrum was described by
${}^{239}Pu$ fission spectrum [14, 15]. Here the neutron gas temperature for
Maxwell distribution was given by (7), described in [12]. According to this
approach [12], the drawbacks of standard slowing-down theory for
thermalization area may be formally reduced if a variable $\xi(x)=\xi(1-2/z)$
is introduced instead of the average logarithmic energy loss $\xi$, which is
almost independent of neutron energy (as is known, for environment consisting
of nuclei with $A>10$ the statement $\xi\approx 2/A$ is true). Here
$z=E_{n}/kT$, $E_{n}$ is the neutron energy, $T$ is the environment
temperature. Then within such a framework the following expression may be used
for the temperature of the neutron gas in Maxwell spectrum of thermal
neutrons444A very interesting expression revealing a hidden connection between
the temperature of the neutron gas and the medium (fuel) temperature.:
$T_{n}=T_{0}\left[1+\eta\cdot\dfrac{\Sigma_{a}(kT_{0})}{\langle\xi\rangle\Sigma_{S}}\right],$
(7)
where $T_{0}$ is the fuel medium temperature, $\Sigma_{a}(kT_{0})$ is the
absorption cross-section for energy $kT_{0}$, $\eta=1.8$ is the dimensionless
constant, $\langle\xi\rangle$ is averaged over the whole energy interval of
Maxwell spectrum $\xi(z)$ at $kT=1~{}eV$.
Fermi neutron spectrum for a moderating medium with absorption (we considered
carbon as a moderator and 238U, 239U and 239Pu as absorbers) was set in the
form [12, 16]:
$\Phi_{Fermi}\left(E,E_{F}\right)=\dfrac{S}{\langle\xi\rangle\Sigma_{t}E}\exp{\left[-\int\limits_{E_{lim}}^{E_{f}}\dfrac{\Sigma_{a}\left(E^{\prime}\right)dE^{\prime}}{\langle\xi\rangle\Sigma_{t}\left(E^{\prime}\right)E^{\prime}}\right]},$
(8)
where $S$ is the total volume neutron generation rate,
$\langle\xi\rangle=\sum\limits_{i}\left(\xi_{i}\Sigma_{S}^{i}\right)/\Sigma_{S}$,
$\xi_{i}$ is the average logarithmic decrement of energy loss,
$\Sigma_{S}^{i}$ is the macroscopic scattering cross-section of the $i$-th
nuclide, $\Sigma_{t}=\sum\limits_{i}\Sigma_{S}^{i}+\Sigma_{a}^{i}$ is the
total macroscopic cross-section of the fissile material,
$\Sigma_{S}=\sum\limits_{i}\Sigma_{S}^{i}$ is the total macroscopic scattering
cross-section of the fissile material, $\Sigma_{a}$ is the macroscopic
absorption cross-section, $E_{F}$ is the upper neutron energy for Fermi
spectrum.
The upper limit of neutron thermalization $E_{lim}$ in our calculation was
considered as a free parameter, setting the neutron fluxes of Maxwell and
Fermi spectra at a common energy limit $E_{lim}$ equal:
$\Phi_{Maxwell}\left(E_{lim}\right)=\Phi_{Fermi}\left(E_{lim}\right).$ (9)
The high energy neutron spectrum part ($E>E_{F}$) was defined by fission
spectrum [16, 17, 18] in our calculations. Therefore for the total volume
neutron generation rate $S$ in the expression for the Fermi spectrum (8) the
following expression may be written:
$S\left(\vec{r},T,t\right)=\int\limits_{E_{F}}^{E_{n}^{max}}\tilde{P}\left(\vec{r},E,T,t\right)\left[\sum\limits_{i}\nu_{i}(E)\cdot\Phi\left(\vec{r},E,T,t\right)\cdot\sigma_{f}^{i}(E,T)\cdot
N_{i}\left(\vec{r},T,t\right)\right]dE,$ (10)
where $E_{n}^{max}$ is the maximum energy of neutron fission spectrum (usually
taken as $E_{n}^{max}\approx 10~{}MeV$), $E_{F}$ is the neutron energy, below
which the moderating neutrons spectrum is described as Fermi spectrum (usually
taken as $E_{F}\approx 0.2~{}MeV$); $\tilde{P}\left(\vec{r},E,T,t\right)$ is
the probability for the neutron not to leave the boundaries of the fissile
medium, which depends on the fissile material geometry and conditions at its
border as well (e.g. presence of a reflector).
The obtained calculation results show that the cross-sections averaged over
the spectrum may increase (Fig. 2 for ${}^{239}Pu$ and Fig. 4 for ${}^{238}U$)
as well as decrease (Fig. 3 for ${}^{235}U$) with fissile medium temperature
increase. As follows from the obtained results, the arbitrariness in selection
of the limit energy for joining Maxwell and Fermi spectra does not alter the
character of these dependences evolution significantly.
a)
b)
Figure 2: Temperature dependences for the fission cross-section (a) and
radioactive capture cross-section (b) for ${}^{239}Pu$, averaged over the
Maxwell spectrum, on the Maxwell and Fermi spectra joining energy and
$\eta=1.8$ (see (7)).
a)
b)
Figure 3: Temperature dependences for the fission cross-section (a) and
radioactive capture cross-section (b) for ${}^{235}U$, averaged over the
Maxwell spectrum, on the Maxwell and Fermi spectra joining energy and
$\eta=1.8$ (see (7)).
a)
b)
Figure 4: Temperature dependences for the fission cross-section (a) and
radioactive capture cross-section (b) for ${}^{238}_{92}$U, averaged over the
combined Maxwell and Fermi spectra depending on Maxwell and Fermi spectra
joining energy and $\eta=1.8$ (see (7)).
This can be justified by the fact that 239Pu resonance region starts from
significantly lower energies than that of 235U and with fuel temperature
increase, the neutron gas temperature increases producing Maxwell’s neutron
distribution maximum shift to higher neutron energies. In other words, the
neutron gas spectrum hardening, when more neutrons fit into resonance area of
239Pu, is the cause of the averaged cross-sections growth.
For 235U this process in not as significant because its resonance region is
located at higher energies. As a result, the 235U neutron gas spectrum
hardening related to fuel temperature increase (in the considered interval)
does not result in a significant increase of the number of neutrons fitting
into the resonance region. Therefore according to the known dispersion
relations for 235U giving the neutron reactions cross-sections behaviour
depending on their energy $E_{n}$ for non-resonance areas, we observe a
dependence for the averaged cross-sections $\sigma_{nb}\sim 1/\sqrt{E_{n}}$.
The data on the averaged fission and capture cross-sections of 238U presented
at Fig. 4 show that the averaged fission cross-section for 238U is almost
insensitive to the neutron spectrum hardening caused by the fuel temperature
increase, because of the high fission threshold $\sim$1 MeV (see fig. 4a). On
the other hand, they confirm the dependence of the capture cross-section on
temperature, because its resonance region is located as low as for 239Pu.
Obviously, fuel enrichment with 235U essentially makes no difference in this
case, because the averaged cross-sections for 235U, as described above, behave
in a standard way.
And finally we performed a computer estimate of the heat source density
dependence $q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ (6) on temperature for
different compositions of uranium-plutonium fissile medium with a constant
neutron flux density, presented at fig. 5. We used the dependences presented
above at Fig.2-4 for these calculations. Let us note that our preliminary
calculations were made not taking into account the change in composition and
density of the fissile uranium-plutonium medium that is a direct consequence
of the constant neutron flux assumption.
The necessity of such assumption is caused by the following. It is obvious
that for a reasonable description of the neutron source density
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ (6) dependence on temperature, a
system of three equations must be solved. Two of them correspond to the
neutron kinetics equation (flux and fluence) and to the system of equations
for the parental and child nuclides nuclear density kinetics (e.g. see [19,
20]), while the third equation corresponds to a heat transfer equation of (1)
type. However, some serious difficulties arise at this point, associated with
the limited computational capabilities available. And here is why.
One of the principal physical peculiarities of TWR is the fact [21] that
fluctuation residuals of plutonium (or 233U in Th-U cycle) over its critical
concentration burn out for the time comparable with the reactor lifetime of a
neutron $\tau_{n}(x,t)$ (not considering delayed neutrons), or at least
comparable with the reactor period555The reactor period by definition equals
to $T(x,t)=\tau_{n}(x,t)/\rho(x,t)$, i.e. is a ratio of the reactor neutron
lifetime to reactivity. $T(x,t)$ (considering delayed neutrons). Meanwhile,
the new plutonium (or 233U in Th-U cycle) is formed in a few days (or a month)
and not at once. It means [21] that numerical calculation must be performed
with a temporal step around 10-6-10-7 for the case of not taking into account
the delayed neutrons and $\sim$10-1-100 otherwise. At first glance, taking
into account the delayed neutrons, according to [21], really ”saves the day”,
however it is not always true. If the heat transfer equation contains a
significantly non-linear source, then in the case of a blow-up mode, the
temperature at some conditions may grow extremely fast and in 10-20 steps
(with time step 10-6-10-7 s) reaches the critical amplitude that may lead to
(as a minimum) a solution stability loss or (as a maximum) blow-up bifurcation
of the phase state, almost unnoticeable with a rough time step.
Here we should also mention that modern scientific literature has absolutely
no theoretical or experimental data on heat capacity $c_{p}$ and heat
conductivity $\aleph$ of the fissile material for the temperatures over 1500K,
which makes the further model calculations quite problematic.
According to these remarks, and considering the goal and format of this paper,
we didn’t aim at finding the exact solution of some specific system of three
joined equations described above. However, we found it important to illustrate
– at the qualitative level – the consequences of the possible blow-up modes in
case of non-linear heat source existence in the heat transfer equation. As it
was described above, we performed the estimate computer calculations of the
heat source density $q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ (6) dependence on
temperature in 300-1400K range for selected compositions of uranium-plutonium
fissile medium at a constant neutron flux (Fig. 5).
Figure 5: Dependence of the heat source density
$q_{T}^{f}\left(r,\Phi,T,N_{i},t\right)~{}[eV]$ on the fissile medium
temperature (300-1400K) for several compositions of uranium-plutonium medium
(1 - 10% Pu; 2 - 5% Pu; 3 - 1% Pu) at a constant neutron flux density
$\Phi=10^{13}~{}n/(cm^{2}\cdot s)$.
The obtained dependences for the heat source density
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ were successfully approximated by a
power function of temperature with an exponent of 4 (Fig. 5). In other words,
we obtained a heat transfer equation with a significantly nonlinear heat
source in the following form:
$q_{T}(T)=const\cdot T^{(1+\delta)},$ (11)
where $\delta>1$ in the case of a non-linear heat conductivity dependence on
temperature [4, 5, 6, 7, 8]. The latter means that the solutions of the heat
transfer equation (1) describe the so-called Kurdyumov blow-up modes [4, 5, 6,
7, 8, 9], i.e. such dynamic modes when one of the modeled values (e.g.
temperature) turns into infinity for a finite time interval. As noted before,
in reality instead of reaching infinite values, a corresponding phase
transition is observed (a final phase of the parabolic temperature growth),
that requires a separate model and is a basis for an entirely new problem.
Mathematical modeling of blow-up modes was performed mainly using Mathematica
5.2-6.0, Maple 10, Matlab 7.0, utilizing multiprocessor calculations for
effective application. Runge-Kutta method of 8-9th order and the numerical
methods of lines [22] were also applied to the calculations. The numerical
error estimate did not exceed 0.01%. The coordinate and temporal step were
variable and chosen by the program in order to fit the given error at every
calculation step.
Below we give the solutions for the heat transfer equation (1) with nonlinear
exponential heat source (11) in uranium-plutonium fissile medium for the
boundary and initial parameters corresponding to those of the technical
reactors. The calculations were done for a cube of fissile material with
different spatial size, boundary and initial temperature values. Since the
temperature dependences of the heat source densities were obtained without
account for changing composition and density of the uranium-plutonium fissile
medium, different blow-up modes can take place (HS-mode, S-mode, LS-mode)
depending on the ratio between the exponents of the temperature dependences of
thermal conductivity and heat source according to [4, 5, 6, 7, 8, 9].
Therefore we considered cases for 1${}^{\text{st}}$, 2${}^{\text{nd}}$ and
4${}^{\text{th}}$ temperature order sources. Here the power of the source also
varied by varying the proportionality factor in (11)
($const=1.00J/(cm^{3}\cdot s\cdot K$) for 1${}^{\text{st}}$ temperature order
source; $0.10J/(cm^{3}\cdot s\cdot K^{2})$, $0.15~{}J/(cm^{3}\cdot s\cdot
K^{2})$ and $1.00~{}J/(cm^{3}\cdot s\cdot K^{2})$ for 2${}^{\text{nd}}$
temperature order source; $1.00~{}J/(cm^{3}\cdot s\cdot K^{4})$ for
4${}^{\text{th}}$ temperature order source). While calculating the the heat
capacity $c_{p}$ (Fig. 6a) and thermal conductivity $\aleph$ (Fig. 6b)
dependences on the fissile medium temperature in 300-1400K range, the
specified parameters were given by the analytic expressions, obtained by
approximation of the experimental data for238U based on the polynomial
progression:
$\displaystyle c_{p}(T)\approx-7.206+0.64T$
$\displaystyle-0.0047T^{2}+0.0000126T^{3}+$ $\displaystyle+2.004\cdot
10^{-8}T^{4}-1.60\cdot 10^{-10}T^{5}-2.15\cdot 10^{-13}T^{6},$ (12)
$\aleph(T)\approx 21.575+0.0152661T.$ (13)
Figure 6: Temperature dependence of the fissile material heat capacity $c_{P}$
and thermal conductivity $\chi$. Points represent the experimental values for
the heat capacity and thermal conductivity of 238U.
And finally a solution of the heat transfer equation (1) was obtained for the
constant thermal conductivity ($27.5~{}W/(m\cdot K)$) and heat capacity
($11.5~{}J/(K\cdot mol)$) values, presented in Fig. 7a, as well as the
solutions of the heat transfer equation considering their temperature
dependences (Fig. 7b-d).
Figure 7: Heat transfer equation (1) solution for 3D case (crystal sizes
0.001$\times$0.001$\times$0.001 mm; initial and boundary temperatures equal to
100K): a) The source is proportional to the 4${}^{\text{th}}$ order of
temperature; $const$ =1.00 $J/(cm^{3}\cdot s\cdot K^{4})$, heat capacity and
thermal conductivity are constant and equal to 11.5 $J/(K\cdot mol)$ and 27.5
$W/(m\cdot K)$ respectively; b) the source is proportional to the
4${}^{\text{th}}$ order of temperature; $const$ =1.00 $J/(cm^{3}\cdot s\cdot
K^{4})$; c) The source is proportional to the 2${}^{\text{nd}}$ order of
temperature; $const$ =1.00 $J/(cm^{3}\cdot s\cdot K^{2})$; d) the source is
proportional to the 2${}^{\text{nd}}$ order of temperature; $const$ = 0.10
$J/(cm^{3}\cdot s\cdot K^{2})$. Note: in cases b)- d) the heat capacity and
thermal conductivity were determined by (12) and (13) respectively.
Preliminary results point directly to a possibility of the local melting of
the uranium-plutonium fissile medium, having melting temperature almost
identical to that of 238U, that equals 1400K (Fig. 6a-d). Moreover, these
regions of the local melting are not the areas of the so-called thermal spikes
[23], and probably are the anomalous areas of the uranium surface melting
observed by Laptev and Ershler [24] that were also mentioned in [25]. More
detailed analysis of the probable temperature scenario associated with the
blow-up modes is discussed below.
## 3 Physical peculiarities of the blow-up regimes in neutron-multiplying
media
Earlier we noted the fact that due to the coolant loss at nuclear reactors
during Fukushima nuclear accident the fuel was melted, or in other words,
temperature inside the active zone at some moment reached the melting
temperature of the uranium-oxide fuel, i.e. $\sim$3000∘C.
On the other hand, we already know that the coolant loss may become a cause of
the nonlinear heat source formation inside the nuclear fuel and thus become a
cause of the temperature and neutron flux blow-up mode occurrence. A natural
question arises as to whether it is possible to use such blow-up mode (in
terms of temperature and neutron flux) for the initiation of certain
controlled physical conditions, under which the nuclear burning wave would
regularly ”experience” the so-called controlled blow-up regime. It is quite
difficult to answer this question definitely, because such fast process has
some physical vaguenesses, any of which can become experimentally
unsurmountable for the its controlling.
Nevertheless such process is very elegant and beautiful from the physical
point of view and therefore requires more detailed phenomenological
description. Let us try to make it in short.
As one can see from the plots of the capture and fission cross-sections
evolution for 239Pu (Fig. 2), the blow-up mode may develop actively at
$\sim$1000-2000K (depending on the real value of Fermi and Maxwell spectra
joining boundary), but it returns to almost the initial cross-sections values
at temperatures over 2500-3000K. If we turn on some effective heat sink at
that point, the fuel may return to its initial temperature state. However,
while the blow-up mode develops, the fast neutrons already penetrate to the
adjacent fuel areas, where the new fissile material starts accumulating and so
on (see cycles (1) and (2)). After some time the similar blow-up mode will
develop in this adjacent area and everything starts over again. In other
words, such hysteresis blow-up mode, closely time-conjugated to a heat takeoff
procedure, will appear against the background of a stationary nuclear burning
wave in a form of the periodic impulse bursts.
In order to demonstrate the marvelous power of such process, we investigated
the heat transfer equation with non-linear exponential heat source in uranium-
plutonium fissile medium with the boundary and initial parameters emulating
the heat takeoff process. In other words, we investigated the blow-up modes in
the fast Feoktistov-type uranium-plutonium reactor [2] where the temperature
was deliberately fixed at 6000K inside and outside the boundary.
This temperature is defined by the following important question: ”Is it
possible to obtain a solution, i.e. the spacio-temporal temperature
distribution not in a form of a $\delta$-function at some local spatial area,
but as some kind of a stationary and limited by amplitude solitary wave under
such conditions (6000K), which emulate the time-conjugated heat takeoff (see
Fig. 2)?” As shown below, such suggestion proved productive.
Below we present some calculation characteristics and parameters. During these
calculations we used the following expression for the heat conductivity
coefficient:
$\aleph=0.18\cdot 10^{-4}\cdot T,$
which was obtained using Wiedemann–Franz law and the data on electric
conductivity of metals at temperature 6000K [26]. Specific heat capacity at
constant pressure was set to $c_{p}\approx 6~{}cal/(mol\cdot deg)$ according
to Dulong and Petit law.
The fissile uranium-plutonium medium was modeled as a cube of the size
10.0$\times$10.0$\times$10.0 m (Fig.8) during the calculations. Here we used
the 2nd order temperature dependence for the heat source (see (11)).
And finally Fig. 8a-d present a set of solutions of the heat transfer equation
(1) with nonlinear exponential heat source (11) in uranium-plutonium fissile
medium with boundary and initial conditions emulating such process of heat
takeoff that initial and boundary temperatures remain constant and equal to
6000K.
Figure 8: Heat transfer equation solution for a model reactor (source
$\sim$2nd order temperature dependence, $const$ = 4.19 $J/(cm^{3}\cdot s\cdot
K^{2})$; Initial and boundary temperatures equal to 6000K; fissile medium is a
cube 10$\times$10$\times$10 m. The presented results correspond to the
following times of the temperature field evolution: (a) (1-10)$\cdot 10^{-7}$
s, (b) $10^{-6}$ s, (c) 0.5 s, (d) 50 s.
It is important to note here, that the solution set presented at Fig. 8
demonstrates the temporal evolution of the solution to its ”stationary” state
quite clearly. This is achieved using the so-called ”magnifying glass”
approach when the solutions of the same problem are deliberately investigated
at the different timescales. For example, Fig. 8a shows the solution at the
time scale $t\in[0,10^{-6}~{}s]$, while Fig. 8b describes the spatial solution
of the problem (temperature field) for $t=10^{-6}~{}s$. The Fig. 8c-d present
the solution (the spatial temperature distribution) at $t=0.5~{}s$ and
$t=50~{}s$.
As one can see, the solution (Fig. 8d) is completely identical to the previous
(Fig.8c), i.e. to the distribution established in the medium in 0.5 seconds,
which allows us to make a conclusion about the temperature field stability,
starting from some moment. It is interesting that the established temperature
field creates the conditions enough for thermonuclear synthesis reaction, i.e.
reaching 108K, and the time of such temperature field existence is not less
than 50 s. These conditions are highly favorable for a stable thermonuclear
burning given the necessary nuclei concentration to enter the thermonuclear
synthesis reaction.
That said, one should remember that the results of the current chapter are
only demonstrative, because their accuracy is very relative and requires
careful investigations involving the necessary computational resources.
However, qualitative peculiarities of these solutions should attract the
researchers’ attention to the nontrivial properties of the blow-up modes, at
least, with respect to the obvious problem of internal TWR safety violation.
## 4 Conclusion
Below we give short conclusions stimulated by significant problems, that can
be formulated in the following form.
1. 1.
The consequences of the anomalous 238U and 239Pu cross-sections behavior with
temperature increase. It is shown that the capture and fission cross-sections
of 238U and 239Pu manifest a monotonous growth in 1000-3000K range. Obviously,
such anomalous temperature dependence of 238U and 239Pu cross-sections changes
the neutron and heat kinetics drastically in the nuclear reactors, and in TWR
in particular. It becomes crucial to know their influence on kinetics of heat
transfer because it may become the cause of a positive feedback with neutron
kinetics, which may lead not only to undesirable loss of solution stability
(the nuclear burning wave), but also to a trivial reactor runaway with a
subsequent nontrivial disaster.
2. 2.
Blow-up modes and the problem of nuclear burning wave stability. One of the
causes of a possible fuel temperature growth may lie, for instance, in a
deliberate or spontaneous coolant loss, analogous to what happened during the
Fukushima nuclear accident. As shown above, the coolant loss may become a
cause of the nonlinear heat source formation in the nuclear fuel, and
consequently emergence of the mode with the temperature and neutron flux blow-
up. In our opinion, the preliminary investigation of the heat transfer
equation with nonlinear heat source points to an extremely important
phenomenon of the anomalous of the temperature and neutron flux blow-up modes
behavior. This result poses a natural nontrivial problem of fundamental
nuclear burning wave stability and, orrespondingly, of a physically reasonable
application of Lyapunov method to this problem, which is a basis for the
motion stability theory, and thus a reliable basis for justification of the
Lyapunov functional minimum existence.
It is shown that some variants of solution stability loss are caused by
anomalous nuclear fuel temperature evolution. They can be not only the cause
of TWR internal safety loss, but can lead to a new stable mode when nuclear
burning wave would periodically ”experience” the so-called controlled blow-up
regime through a bifurcation of states (which is very important!). At the same
time, it is noted that such fast (blow-up regime) process has a number of
physical vaguenesses, any of which may happen to be experimentally
insurmountable for the control of such process.
3. 3.
On-line remote neutrino diagnostics of intra-reactor processes. Due to the
fact that a high-power TWR or a nuclear fuel transmutation reactor are the
projects of the single-load and fuel burn-up with a subsequent burial of the
reactor apparatus, there is an obvious necessity for remote system of neutrino
monitoring of the neutron-nuclear burning wave in normal operation mode and
control of the neutron kinetics in emergency situation. The calculation of the
spatio-temporal distribution of the isotope composition in the TWR active zone
in the framework of the inverse problem of neutrino diagnostics of intra-
reactor processes is presented in [27, 28, 29] in detail.
## References
* Ukraintsev [2000] V.F. Ukraintsev. _Reactivity effects in energetic installations (in Russian). Handbook._ Obninsk Institute for Nuclear Power Engineering, Obninsk, 2000.
* Feoktistov [1989] L.P. Feoktistov. Neutron-fission wave. _Dokl. Akad. Nauk SSSR_ , (309):4–7, 1989.
* Teller et al. [1996] E. Teller, M. Ishikawa, L. Wood, R. Hyde, and J. Nuckolls. Completely automated nuclear reactors for long-term operation ii: Toward a concept-level point-design of a high-temperature, gas-cooled central power station system, part ii. In _Proceedings of the International Conference on Emerging Nuclear Energy Systems, ICENES’96, Obninsk, Russian Federation_ , pages 123–127, Obninsk, Russian Federation, 1996. Obninsk, Russian Federation. also available from Lawrence Livermore National Laboratory, California, publication UCRL-JC-122708-RT2.
* Akhromeeva et al. [1992] T.S. Akhromeeva, S.P. Kurdyumov, G.G. Malinetskii, and A.A. Samarskii. _Non-stationary structures and diffusive chaos (in Russian)_. Nauka, Moscow, 1992.
* Samarskii et al. [1995] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov. _Blow-up in Quasilinear Parabolic Equations_. Walter de Gruyter, Berlin, 1995.
* ref [1999] Blow-up modes. evolution of the idea. the laws of co-evolution of complex systems. Moscow: Nauka, 1999. A collection of papers by S.P. Kurdyumov and his followers published on the occasion of his 70th birthday.
* Kurdyumov [2006] S.P. Kurdyumov. _Blow-up modes_. Fizmatlit, Moscow, 2006.
* Knyazeva and Kurdyumov [2007] E.N. Knyazeva and S.P. Kurdyumov. _Synergetics: nonlinearity of time and landscape of co-evolution_. KomKniga, Moscow, 2007.
* Rusov et al. [2011a] V.D. Rusov, V.A. Tarasov, and S.A. Chernegenko. Blow-up modes in uranium-plutonium fissile medium in technical nuclear reactors and georeactor (in russian). _Problems of Atomic Science and Technology_ , 97:123–131, 2011a.
* Skorov et al. [1979] D.M. Skorov, Yu.F. Bychkov, and A.I. Dashkovskii. _Reactor material science (in Russian)_. Atomizdat, Moscow, 1979.
* Nadykto [2003] B.A. Nadykto, editor. _Plutonium. Fundamental problems (in Russian)_. RFNC-AREPRI, Sarov, 2003.
* Bartolomey et al. [1989] G.G. Bartolomey, G.A Bat’, V.D. Babaykov, and M.S. Altukhov. _Basic theory and methods of nuclear power installations calculation_. Energoatomizdat, Moscow, 1989.
* Los [1998] _ENSDF/B-VI_. Los Alamos National Laboratory, 1998.
* Pavlovich et al. [2008a] V.N. Pavlovich, V.N Khotyaintsev, and E.N. Khotyaintseva. Physical basics of the nuclear burning wave reactor. 2. specific models. _Nuclear Physics and Energetics_ , (3):39–48, 2008a.
* Hyde et al. [2008] R. Hyde, M. Ishikawa, N. Myhrvold, J. Nuckolls, and L. Wood. Nuclear fission power for 21st century needs: Enabling technologies for large-scale, low-risk, affordable nuclear electricity. _Progress in Nuclear Energy_ , 50:32–91, 2008.
* Shirokov [1992] S.V. Shirokov. _Nuclear reactor physics (in Russian)_. Naukova dumka, Kiev, 1992.
* Fedorov [1961] N.D. Fedorov. _Short reference book for engineer-physicist. Nuclear physics and atomic physics (in Russian)_. State publishing company for atomic science and technology literature, Moscow, 1961.
* Vladimirov [1986] V.I. Vladimirov. _Practical problems on nuclear reactors operation (in Russian)_. Energoatomizdat, Moscow, 1986.
* Rusov et al. [2011b] Vitaliy D. Rusov, Elena P. Linnik, Victor A. Tarasov, Tatiana N. Zelentsova, Igor V. Sharph, Vladimir N. Vaschenko, Sergey I. Kosenko, Margarita E. Beglaryan, Sergey A. Chernezhenko, Pavel A. Molchinikolov, Sergey I. Saulenko, and Olga A. Byegunova. Traveling wave reactor and condition of existence of nuclear burning soliton-like wave in neutron-multiplying media. _Energies_ , 4(9):1337–1361, 2011b. ISSN 1996-1073. doi: 10.3390/en4091337. URL http://www.mdpi.com/1996-1073/4/9/1337.
* Pavlovich et al. [2010] V.N. Pavlovich, V.N Khotyaintsev, and E.N. Khotyaintseva. Nuclear burning wave reactor: wave parameter control. _Nuclear Physics and Energetics_ , (11):49–56, 2010.
* Pavlovich et al. [2008b] V.N. Pavlovich, V.N Khotyaintsev, and E.N. Khotyaintseva. Physical basics of the nuclear burning wave reactor. 1. _Nuclear Physics and Energetics_ , (2):39–48, 2008b.
* Samarskii and Gulin [2003] A.A. Samarskii and A.V. Gulin. _Numerical methods in mathematical physics (in Russian)_. Nauchnyi mir, Moscow, 2003.
* Kinchin and Pease [1955] G.H. Kinchin and R.S. Pease. Displacement of atoms in solid by radiation. _Rep. Prog. Phys._ , 18:590–615, 1955.
* Ershler and Lapteva [1957] B.V. Ershler and F.S. Lapteva. The evaporation of metals by fission fragments. _Journal of Nuclear Energy (1954)_ , 4(4):471 – 474, 1957. ISSN 0891-3919. doi: 10.1016/0891-3919(57)90075-X. URL http://www.sciencedirect.com/science/article/pii/089139195790075X.
* Lifshits et al. [1960] I.M. Lifshits, M.I. Kaganov, and L.V. Tanatarov. On the theory of the changes produced in metals by radiation. _Atomic energy_ , pages 261–270, 1960.
* Zharkov [1983] V.N. Zharkov. _The inner structure of Earth and planets_. Nauka, Moscow, 1983.
* Rusov et al. [2004] V. D. Rusov, T. N. Zelentsova, V. A. Tarasov, , and D. A. Litvinov. Inverse problem of remote neutrino diagnostics of intrareactor processes. _J. Appl. Phys._ , 96:1734, 2004.
* Rusov et al. [2007] V.D. Rusov, V.N. Pavlovich, V.N. Vaschenko, V.A. Tarasov, T.N. Zelentsova, V.N. Bolshakov, D.A. Litvinov, S.I. Kosenko, and O.A. Byegunova. Geoantineutrino spectrum and slow nuclear burning on the boundary of the liquid and solid phases of the earth’s core. _J. Geophys. Res._ , 112, 2007. B09203, doi:10.1029/2005JB004212.
* Rusov et al. [2008] V.D. Rusov, V.A. Tarasov, and D.A. Litvinov. _Reactor antineutrino physics (in Russian)_. URSS, Moscow, 2008.
|
arxiv-papers
| 2012-08-05T15:24:52 |
2024-09-04T02:49:34.758995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.D. Rusov, V.A. Tarasov, V.M. Vaschenko, E.P. Linnik, T.N.\n Zelentsova, M.E. Beglaryan, S.A. Chernegenko, S.I. Kosenko, P.A.\n Molchinikolov, V.P. Smolyar, E.V. Grechan",
"submitter": "Vladimir Smolyar",
"url": "https://arxiv.org/abs/1209.0648"
}
|
1209.0690
|
# Origin of the structural phase transition in Li7La3Zr2O12
N. Bernstein M. D. Johannes Center for Computational Materials Science,
Naval Research Laboratory, Washington, DC 20375 Khang Hoang Resident at the
Center for Computational Materials Science, Naval Research Laboratory,
Washington, DC 20375. Computational Materials Science Center, George Mason
University, Fairfax, VA 22030
###### Abstract
Garnet-type Li7La3Zr2O12 (LLZO) is a solid electrolyte material with a low-
conductivity tetragonal and a high-conductivity cubic phase. Using density-
functional theory and variable cell shape molecular dynamics simulations, we
show that the tetragonal phase stability is dependent on a simultaneous
ordering of the Li ions on the Li sublattice and a volume-preserving
tetragonal distortion that relieves internal structural strain. Supervalent
doping introduces vacancies into the Li sublattice, increasing the overall
entropy and reducing the free energy gain from ordering, eventually
stabilizing the cubic phase. We show that the critical temperature for cubic
phase stability is lowered as Li vacancy concentration (dopant level) is
raised and that an activated hop of Li ions from one crystallographic site to
another always accompanies the transition. By identifying the relevant
mechanism and critical concentrations for achieving the high conductivity
phase, this work shows how targeted synthesis could be used to improve
electrolytic performance.
The garnet structure material Li7La3Zr2O12 (LLZO) has an ionic conductivity
that varies by two orders of magnitude depending on whether synthesis produces
a cubic ($\sigma_{\rm cubic}$=1.9$\times$10-4 S/cm) Murugan et al. (2007) or
tetragonal ($\sigma_{\rm tetra}$=1.63$\times$10-6 S/cm) Awaka et al. (2009)
phase. With the higher conductivity, LLZO has excellent solid electrolyte
material characteristics. Its stability against both Li metal and standard
cathode LiCoO2 Ohta et al. (2011), combined with a $\sim$5 eV band gap Murugan
et al. (2007) and high ionic conductivity, make it suitable for exploiting the
full voltage difference between anode and cathode while circumventing the
safety concerns inherent to all current liquid electrolytes. Initially, the
process by which the cubic ground state could be stabilized against the
tetragonal was unknown and uncontrolled. The relevant factor was eventually
traced down to the addition of Al in the structure, whether via accidental
uptake from Al-containing crucibles Geiger et al. (2011) or via intentional
doping Jin and McGinn (2011); Allen et al. (2012). Inclusion of other
supervalent ions, including Ta, Nb, and Ga Allen et al. (2012); Adams and Rao
(2011), has also been successful in producing the high conductivity phase.
However, the underlying mechanism that controls the transition has so far
remained a mystery, hindering further progress toward improving the material
for practical usage.
Here we use our recently developed variable cell shape density-functional
theory (DFT) plus molecular dynamics (MD) method to investigate the driving
force behind the tetragonal to cubic transition which subsequently raises the
conductivity. We find that in the cubic phase, the Li sublattice is always
disordered (all Li symmetry sites partially occupied), while in the tetragonal
phase it is always ordered (all Li sites either full or empty). We find that
the energy is lowered by a simultaneous ordering of the Li atoms that relieves
Li$-$Li Coulomb repulsion but unfavorably distorts the ZrO6 octahedra and a
lattice distortion that restores the preferred Zr$-$O bonds. The two symmetry-
breaking but volume-preserving phenomena always occur in conjunction and
either one alone actually raises the total energy of the system. When Al3+ is
doped into the system, charge compensation takes place through creation of Li+
vacancies that reduce the free energy advantage of complete ordering on the Li
sublattice, eventually leading to disorder and a transition to cubic symmetry.
The transition is always signaled by a sudden shift of Li occupation which can
be used to pinpoint the critical temperature and vacancy concentration. Using
these criteria, we estimate that the critical concentration of Al dopants
necessary to achieve the high-conductivity cubic phase of Li7-2xAlxLa3Zr2O12
is $x=0.2$ (vacancy number $n_{\rm vac}=2x=0.4$ per formula unit) at zero
temperature, in good agreement with experiment. We show that the cubic phase
can be reached at some temperature, regardless of Li content, but that the
critical temperature drops as a function of vacancy number. The understanding
uncovered in this work will be useful for choosing more efficient dopants and
further raising the ionic conductivity.
The Li sublattices in the cubic and tetragonal LLZO phases are shown in Fig.
1. Li positions in each structure are generally referred to as Li(1) if they
are tetrahedrally coordinated to oxygen, and Li(2) and Li(3) if they are
octahedrally coordinated. To avoid confusion, we refer to each site using its
crystallographic notation, with superscript $c$ or $t$ to designate cubic and
tetragonal lattices, respectively. Each tetrahedral cubic 24$d^{\rm c}$ site
is surrounded by four pairs of octahedrally coordinated 96$h^{\rm c}$ sites,
and all sites are partially occupied (see Fig. 1). Because of Coulomb
repulsion, it is energetically prohibited for both members of each pair of
96$h^{\rm c}$ sites to be occupied, and if a particular 24$d^{\rm c}$ site is
occupied, the adjacent 96$h^{\rm c}$ sites are consistently unoccupied Xie et
al. (2011). The tetragonal distortion transforms the cubic 24$d^{\rm c}$ sites
into fully occupied 8$a^{\rm t}$ sites, often denoted as Li(1), and unoccupied
16$e^{\rm t}$ sites. The 96$h^{\rm c}$ sites are transformed into fully
occupied 16$f^{\rm t}$ and 32$g^{\rm t}$ sites, often denoted as Li(2) and
Li(3), respectively, with the rest fully unoccupied. As we will show, a shift
from 8$a^{\rm t}$ sites to 16$e^{\rm t}$ sites, both subsets of 24$d^{\rm c}$,
always accompanies the tetragonal to cubic transition.
Figure 1: (Color online) Li sublattice in the cubic (left) and tetragonal
(right) phases of LLZO. All Li positions are included, although in the cubic
phase not all are occupied. The Li(1) atoms (8$a^{\rm t}$ and 24$d^{\rm c}$)
are large gray (gold online), Li(2) or 16$f^{\rm t}$ are white, and Li(3) or
32$g^{\rm t}$ are dark gray. The cubic Li(1) positions that become vacant upon
transition to the ordered tetragonal structure (16$e^{\rm t}$) are indicated
by small (gold online) spheres.
All of our simulations use density-functional theory, with the Perdew-Burke-
Ernzerhoff exchange-correlation functional Perdew et al. (1996, 1997) as
implemented in the Vienna Ab-initio Simulation Package (VASP) projector-
augmented-wave software version 5.2.12 Kresse and Hafner (1993); Kresse and
Furthmuller (1996). The calculations used an energy cutoff of 400 eV, except
for the relaxations used to explain the microscopic basis of the structural
transition, which used a cutoff of 600 eV. Molecular dynamics (MD) time
evolution is carried out using the velocity Verlet algorithm, as implemented
in the libAtoms lib package, with a time step of 1 fs and using forces and
stresses calculated at each time step with VASP. All MD simulations use
constant temperature and stress. The time propagation algorithm, specified in
Supplementary Information, is a rediscretization, using the ideas from Ref.
Jones and Leimkuhler, 2011 and Ref. Leimkuhler et al., 2011, of the Langevin
constant pressure algorithm in Ref. Quigley and Probert, 2004, for a modified
version of the equations of motion of Ref. Tadmor and Miller, 2011. Each
simulation starts from a unit cell of the experimental tetragonal structure of
Ref. Awaka et al., 2009, for a total of 8 formula units. The structure is
relaxed with respect to ionic positions and unit cell size and shape using
VASP, and then simulated at finite temperature and zero stress. While the MD
simulation results implicitly include the effects of entropy, all energies we
calculate explicitly and quote here include only DFT total internal energy.
The system is allowed to evolve for at least 24 ps with ionic motion as well
as cell shape and volume changes, so it can spontaneously transform into other
structures. To determine the effect of composition and temperature we simulate
the stoichiometric system, with 56 Li atoms, as well as systems with 1, 2, and
4 Li+ ion vacancies per simulated cell, at temperatures ranging from 300 K to
1300 K. The net negative charge of the vacancies is compensated by a uniform
positive background charge. The vacancies are formed by removing Li+ ions
randomly selected from the tetragonal 16$f^{\rm t}$ and 32$g^{\rm t}$ sites,
where the vacancy formation energy is about 100 meV lower than at the 8$a^{\rm
t}$ sites. The crystallographic site identity of each atom is determined
during each MD trajectory (see Supplementary Information).
An example of the time evolution of one system, with $n_{\rm vac}=0.25$ and
$T=600$ K, is plotted in Fig. 2. The ratios of the lattice constants along $x$
and $y$ to that along $z$ initially show the tetragonal structure with one
axis ($a_{z}$) about 3% shorter than the others. The system transforms into a
cubic phase where both axis ratios fluctuate around 1 at $t\sim 5$ ps. It
fluctuates back to a tetragonal phase at $t\sim 15$ ps, and then again to
cubic at $t\sim 30$ ps where it remains for the rest of the simulation. The
volume is not affected by the unit cell shape change. We also plot the
occupancies of various crystallographic sites of the cubic and tetragonal
structures, computed as described above. We see a perfect correlation between
the symmetry of the unit cell parameters and the occupancy of the tetragonal
sites. The 8$a^{\rm t}$ occupancy is initially near 1, as it is in the
experimental structure, and most of the remaining atoms are identified as
16$f^{\rm t}$ and 32$g^{\rm t}$. When projected into the cubic structure the
8$a^{\rm t}$ atoms are identified as 24$d^{\rm c}$ and the 16$f^{\rm t}$ and
32$g^{\rm t}$ atoms are identified as 96$h^{\rm c}$, as expected by symmetry.
Whenever the system transforms into the tetragonal structure the occupancies
of 8$a^{\rm t}$ and 16$f^{\rm t}$+32$g^{\rm t}$ sites drop. Since we do not
see a corresponding change in 24$d^{\rm c}$ and 96$h^{\rm c}$, we conclude
that the atoms are moving from the subset of the cubic sites that are occupied
in the tetragonal ordering to the other cubic structure sites. This is indeed
seen in the increased occupancy of 16$e^{\rm t}$ sites, which correspond to
24$d^{\rm c}$ sites that are unoccupied in the experimental tetragonal
structure.
Figure 2: (Color online) Evolution over time of structural and site occupation
quantities for a sample system with $n_{\rm vac}=2$ at $T=600$ K. Top: unit
cell shape ($a_{x}/a_{z}$ blue, $a_{y}/a_{z}$ red) and volume (black). Middle:
96$h^{\rm c}$ (black) and 16$f^{\rm t}$+32$g^{\rm t}$ (red) lattice site
occupation. Bottom: 24$d^{\rm c}$ (black), 8$a^{\rm t}$ (red) and 16$e^{\rm
t}$ (blue with symbols) lattice site occupations.
The unit cell shape, 8$a^{\rm t}$ occupancy, and unit cell volume, averaged
over the last 5 ps of each run, are listed in Table 1. At each vacancy
concentration the system transforms from an ordered tetragonal structure at
low $T$ to a cubic structure at higher $T$, and the transition temperature
goes down with increasing vacancy concentration. Note that at $n_{\rm
vac}=0.5$ and $T=300$ K our simulations predict an ordered tetragonal
structure that is different from the experimentally observed low $T$
tetragonal structure for the stoichiometric composition. The structure we find
has one long and two short axes, an occupation of the original tetragonal
8$a^{\rm t}$ sites of 0.5, and an occupation of 1 of the original tetragonal
16et sites (which are unoccupied in the stoichiometric tetragonal structure).
The volume is much more sensitive to temperature than to vacancy number, with
a linear thermal expansion coefficient (from a linear fit of $V^{1/3}$ vs. $T$
to the $n_{\rm vac}=0.5$ results, where we have the widest range of
temperatures) of $2.2\times 10^{-4}$ K-1. The vacancy number dependence (from
a linear fit of $V$ vs. $n_{\rm vac}$ to the $T=600$ K and $T=800$ K results),
which is the vacancy formation volume, is 10$\pm$1.5 Å3/vacancy. Note that
this value is dependent on the charge compensation mechanism, which is a
uniform background charge in this calculation, compared to any of a variety of
supervalent dopants in experiment. In all cases, a transition to cubic
symmetry is preceded by a sudden decrease in the 8$a^{\rm t}$ occupation and
any return to tetragonal symmetry is characterized by a reoccupation of these
sites.
Table 1: Mean unit cell distortion ($a_{\alpha}/a_{z}-1$ for $\alpha$=$x$ and $y$, in %) (top) and occupancy of 8$a^{\rm t}$ sites per formula unit (bottom), averaged over the last 5 ps of each run, for different vacancy numbers $n_{\rm vac}$ and temperatures. Transition temperature $T_{c}$ is also indicated for each quantity with a clear signal of a structural transition (see text). $n_{\rm vac}$ | 300 K | 450 K | 600 K | 800 K | 1000 K | 1300 K | $T_{c}$
---|---|---|---|---|---|---|---
$a_{x}/a_{z}$, $a_{y}/a_{z}$ |
0.00 | | | 3.2, 3.3 | 2.3, 2.8 | 0.3, $-$0.6 | $-$0.1, $-$0.2 | 800 K$\leq T_{c}\leq$1000 K
0.12 | | | 3.0, 2.7 | 1.7, 2.1 | $-$0.2, $-$1.8 | | 800 K$\leq T_{c}\leq$1000 K
0.25 | 3.3, 3.2 | 0.5, $-$2.2 | $-$0.5, $-$1.6 | $-$0.6, 0.7 | | | 300 K$\leq T_{c}\leq$450 K
0.50 | 1.3, $-$0.3 | 0.8, 0.3 | $-$0.0, $-$0.4 | $-$0.2, $-$0.5 | $-$0.2, $-$0.4 | 0.3, 0.0 | $T_{c}\leq$300 K
8$a^{\rm t}$ occupancy |
0.00 | | | 1.0 | 0.9 | 0.3 | 0.3 | 800 K$\leq T_{c}\leq$1000 K
0.12 | | | 0.9 | 0.8 | 0.2 | | 800 K$\leq T_{c}\leq$1000 K
0.25 | 1.0 | 0.2 | 0.2 | 0.3 | | | 300 K$\leq T_{c}\leq$450 K
0.50 | 0.5 | 0.4 | 0.4 | 0.3 | 0.3 | 0.4 | ?
Figure 3: (Color online) Energy difference between tetragonal and cubic
structures as a function of vacancy number, for minimum energy configuration
(solid red line) and mean configuration energy (dashed blue line).
We have also computed the energy difference between the tetragonal and cubic
structures as a function of vacancy concentration. For the tetragonal
structure we relaxed 10 structures with random vacancies at 16$f^{\rm t}$ and
32$g^{\rm t}$ sites. For the cubic structure, which is disordered even at the
stoichiometric composition, we use 10 configurations from a uniform sampling
of all configurations that obey the restriction that no pairs of nearest
neighbors (adjacent 96$h^{\rm c}$ pairs, or a 24$d^{\rm c}$ and its nearest
neighbor 96$h^{\rm c}$) are simultaneously occupied. We plot the energy
difference between the minima and means of each of the ten structures in Fig.
3. The tetragonal structure is energetically favored for $n_{\rm vac}\leq
0.25$, and the cubic is favored for $n_{\rm vac}\geq 0.5$. Entropy effects
will shift the equivalent curves for the free energies. We expect that this
shift will reduce the free energy of the tetragonal phase more than the cubic
phase, because the tetragonal phase starts out ordered and therefore gains
more entropy with the introduction of vacancies than the already disordered
cubic phase. The transition vacancy concentration will therefore shift to
higher values as the temperature increases, although we expect this shift to
be small at $T=300$ K.
Figure 4: Relaxed energies (closed symbols) of 10 disordered configurations
and one at the experimentally observed tetragonal phase order, for a cubic
unit cell and for a fully relaxed one, relative to the lowest energy fully
relaxed ordered tetragonal cell. The energy of a model including only ZrO6
distortions is also plotted (open symbol).
To understand the relationship between Li order/disorder and the
tetragonal/cubic lattice parameters, we perform total energy calculations for
the stoichiometric system in 10 disordered configurations (again from a
uniform sampling of configurations obeying first-neighbor exclusion) and in
the experimentally observed Li order in the tetragonal phase. We first relax
each one with respect to atomic positions and unit cell volume but constrain
the unit cell shape to remain cubic, and then relax with full freedom for the
unit cell size and shape as well as atomic positions. The total energies are
plotted in Fig. 4. The energies of the disordered system are generally
highest, with only a small gain (and correspondingly small change in unit cell
shape) from relaxing the cubic unit cell constraint. Imposing the Li order
while maintaining a cubic unit cell actually increases the energy slightly as
compared with the mean disordered energy. Only once the ordered cell is
allowed to relax to the tetragonal shape does its energy become lower than the
disordered system, and the ordered system emerges as the $T=0$ K ground state.
To understand the source of the coupling between energy and structure, we
calculated the pair distribution functions (PDF) of various ion pairs in the
system for the cubic and tetragonal ordered systems, along with a simple point
charge model, based on the nominal ionic charges. The point charge model shows
that the overall Coulombic energy is actually increased upon relaxation to a
tetragonal unit cell, indicating that the energy gain is elsewhere. The PDFs
show that the La3+$-$O2- distances do not vary, presumably since these are the
two largest ionic charges in the system. Li$-$O and Li$-$Li distances do
change, but the highly ionic nature of these interactions means that such
changes are nearly entirely accounted for in the point charge model. The
shifting of Li$-$Li and Li$-$O distances, combined with the rigidity of the
La$-$O spacing, results in some distortion of the ZrO6 octahedra. Unlike the
other pairs, the Zr$-$O interaction is at least partially covalent. In the
cubic cell, the Zr$-$O bond lengths are 2.130$\pm$0.020 Å and O$-$Zr$-$O bond
angles are 180$\pm$4.0∘. The relaxation to a tetragonal cell relieves this
distortion, restoring the octahedra to a uniform Zr$-$O bond length of
2.125$\pm$0.005 Å and a O$-$Zr$-$O bond angle of 180$\pm$0.01∘. To estimate
the energetic contribution of ZrO6 octahedra distortions we parametrize the
calculated total energies of the ordered tetragonal cell as a quadratic
function of Zr$-$O bond length and O$-$Zr$-$O bond angle by computing the
energies for small displacements of an O atom. This parametrization predicts
an energy for the ordered cubic structure of 0.083 eV/formula unit relative to
the tetragonal structure (Fig. 4), very nearly equal to the DFT energy
difference. We therefore conclude that lithium ordering to relieve Li-Li
Coulomb repulsion leads to internal rearrangements of ions to maintain
La3+$-$O2+ distances, which leads to a lattice distortion that allows the ZrO6
octahedra to preserve their preferred shape.
In summary, our variable cell shape DFT MD simulations of LLZO show that the
DFT ground state at low temperatures is the experimental ordered tetragonal
structure, and that at higher temperatures the system transforms into the
experimental disordered cubic structure. The transition temperature decreases
with increasing Li+ vacancy concentration, and the disordered cubic structure
has lower energy when the number of vacancies per formula unit is larger than
about 0.4. These results are in agreement with the experimental observations
of a transition as a function of Li+ vacancy concentration Allen et al.
(2012). The microscopic cause of the structural transition is the coupling
between the unit cell shape and the hopping of atoms from the subset of the
disordered cubic sites occupied in the ordered tetragonal structure to the
remaining cubic sites, which are unoccupied in the tetragonal structure. The
relative stability of the ordered tetragonal low temperature structure is
driven by the ordering, which reduces Li$-$Li Coulomb repulsion but distorts
the ZrO6 octahedra, and the tetragonal distortion which allows these octahedra
to return to their preferred high-symmetry shape. Our simulations enable us to
explain the atomistic mechanism behind this finite temperature structural
transformation in a complex material, and predict the number of vacancies
necessary to achieve the high conductivity material. This should enable better
doping schemes that optimize ionic conductivity by providing the requisite
number of vacancies with as few artificial dopants as possible, thereby
realizing the potential of LLZO as a truly stable solid electrolyte material.
This work was supported by the Naval Research Laboratory core 6.1 research
program and the Nanoscience Institute. Computer time was provided through the
DOD HPCMPO at the ERDC and AFRL MSRCs.
## References
* Murugan et al. (2007) R. Murugan, V. Thangadurai, and W. Weppner, Angew Chem Int Edit 46, 7778 (2007).
* Awaka et al. (2009) J. Awaka, N. Kijima, H. Hayakawa, and J. Akimoto, J. Solid State Chem. 182, 2046 (2009).
* Ohta et al. (2011) S. Ohta, T. Kobayashi, and T. Asaoka, J Power Sources 196, 3342 (2011).
* Geiger et al. (2011) C. A. Geiger, E. Alekseev, B. Lazic, M. Fisch, T. Armbruster, R. Langner, M. Fechtelkord, N. Kim, T. Pettke, and W. Weppner, Inorg Chem 50, 1089 (2011).
* Jin and McGinn (2011) Y. Jin and P. J. McGinn, J Power Sources 196, 8683 (2011).
* Allen et al. (2012) J. L. Allen, J. Wolfenstine, E. Rangasamy, and J. Sakamoto, J Power Sources 206, 315 (2012).
* Adams and Rao (2011) S. Adams and R. P. Rao, J Mater Chem 22, 1426 (2011).
* Xie et al. (2011) H. Xie, J. A. Alonso, Y. Li, M. T. Fernandez-Diaz, , and J. Goodenough, Chem. Mat. 23, 3587 (2011).
* Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys Rev Lett 77, 3865 (1996).
* Perdew et al. (1997) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys Rev Lett 78, 1396 (1997).
* Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys Rev B 47, 558 (1993).
* Kresse and Furthmuller (1996) G. Kresse and J. Furthmuller, Phys Rev B 54, 11169 (1996).
* (13) http://libatoms.org/.
* Jones and Leimkuhler (2011) A. Jones and B. Leimkuhler, J Chem Phys 135, 084125 (2011).
* Leimkuhler et al. (2011) B. Leimkuhler, E. Noorizadeh, and O. Penrose, J Stat Phys 143, 921 (2011).
* Quigley and Probert (2004) D. Quigley and M. I. J. Probert, J Chem Phys 120, 11432 (2004).
* Tadmor and Miller (2011) E. B. Tadmor and R. E. Miller, in _Modeling Materials_ (Cambridge University Press, Cambridge, 2011), pp. 520–533.
|
arxiv-papers
| 2012-09-04T16:20:09 |
2024-09-04T02:49:34.765932
|
{
"license": "Public Domain",
"authors": "N. Bernstein, M.D. Johannes, Khang Hoang",
"submitter": "Michelle Johannes",
"url": "https://arxiv.org/abs/1209.0690"
}
|
1209.0692
|
# Hierarchical configurations for cross-correlation interferometers with many
elements
Eric Keto1 1Harvard-Smithsonian Center for Astrophysics, 60 Garden St,
Cambridge, MA 02138, [email protected]
((May 29, 2012); (to be inserted by publisher); (to be inserted by publisher))
###### Abstract
Array configurations built on a hierarchy of simple elements have excellent
properties for cross-correlation imaging interferometers including a smooth
distribution of measured Fourier components, high angular resolution, low side
lobes, and compact array size. Compared to arrays with a Gaussian distribution
of antenna separations, hierarchical arrays (H-arrays) produce beams with
higher angular resolution and a tighter concentration of the total power
(encircled energy) within a smaller area around the main beam. An attractive
feature of H-arrays is their simplicity. The relationships between the Fourier
coverage and the array configuration are easy enough to understand that they
can be adjusted to achieve different design goals without the need for
numerical optimization. H-arrays will be useful for future multi-element
interferometers.
###### keywords:
Instrumentation: interferometers, Telescopes, Techniques: interferometric
; ; ;
## 1 Introduction
The design or configuration of a cross-correlation imaging interferometer, the
placement of the antennas, presents a challenge because the properties desired
for imaging cannot all be optimized simultaneously. Imaging interferometers
such as those used in radio astronomy construct a picture of the sky from a
set of measured Fourier components. Each pair of antennas in a multi-antenna
array measures one Fourier component whose scale or spatial wavelength is
proportional to the distance between the pair and whose direction is aligned
with their orientation. The entire set of Fourier components is given by all
the possible pairs, and the point source response or the beam of the telescope
is given by their Fourier transform. Thus the relative locations of the
antennas determines the quality of the images obtained by the telescope
[Bracewell, 1958].
The impossibility of obtaining an infinite set of Fourier components requires
choices as trade-offs among different qualities desired for imaging. For
example, the extent or size of the array should be small because the
atmospheric phase coherence decreases with distance and also because the
construction is easier. At the same time, the width of the beam or point-
source response should also be small to achieve high angular resolution.
Because the sizes of the array and the beam are related through the Fourier
transform of the antenna separations, these two goals amount to minimizing
both the extent of a function and the extent of its Fourier transform. The
challenge is that the two generally have an inverse relationship, one big the
other small. Another challenge lies in the trade-off between the angular
resolution and the level of the side lobes around the main beam. The highest
angular resolution requires as many long spatial wavelength Fourier components
(antenna separations) as possible. This requires a sharp cut-off in the number
of components at the maximum spatial wavelength (antenna separation) because a
smooth transition to the maximum necessarily implies a decreasing number of
components nearing the limit. The sharper the transition, the larger the side
lobes so that generally, the higher the angular resolution, the higher the
side lobe levels. A further design goal is complete sampling of the Fourier
plane. Although imaging requirements might push the distribution to emphasize
shorter or longer wavelength spatial frequencies, gaps in the distribution
represent missing information and allow unfaithful imaging. The assumption is
that the likelihood of a significant error in the estimate of a Fourier
component increases with the distance in Fourier space from the component to
the measurement. Therefore, the best estimate of the strength of a Fourier
component is a nearby measurement, and the distribution of separations should
minimize the maximum distance from any point in the Fourier plane to the
nearest measurement.
In summary, four design goals:
1. 1.
High angular resolution
2. 2.
Concentrated power, low side lobes
3. 3.
Compact array size
4. 4.
Uniform or smooth distribution of Fourier components
How do we design arrays to fulfill these goals? How do we measure success? Do
the trade-offs define a continuous space with the different goals at the
vertices of a polyhedral boundary? Can we position our design within this
space to best meet the imaging requirements of a particular application? Must
we rely on numerical optimization that might provide a configuration but no
explanation leaving doubt that we have found the best possible design?
This paper introduces array configurations built on a hierarchy made by
repeating one simple array configuration on different scales. In this
construction, the relationships between the antenna locations, their
separations, and the beam are simple enough to provide answers to these
questions. For example, hierarchical or H-arrays can be designed to achieve
either higher angular resolution or a more concentrated beam with lower side
lobes by simply changing the scaling between the levels of the hierarchy to
emphasize shorter or longer spatial wavelength Fourier components. Because
H-arrays are nested and scaled, the choice can be made in the design stage by
scaling the separation between the hierarchies, or for arrays designed with
more possible locations than antennas, the beam can be varied by populating
different levels of the hierarchy. This makes H-arrays particularly well
suited for the strategy employed by most radio astronomy interferometers of
periodically rearranging a smaller number of antennas among a larger number of
locations to observe different angular scales. H-arrays will be useful for
future interferometers with many antennas.
## 2 Classic pairs
Two well-known Fourier transform pairs [Bracewell, 1999] illustrate the trade-
offs and suggest that there is a functional form for a continuously varying
aperture distribution resulting in a beam pattern that can be varied smoothly
between the design goals. For the moment, suppose that we have complete
control over the aperture distribution, and can assume an ideal continuous
distribution. We will come back to the question of how an array can be
configured to provide the required aperture.
First, consider a uniform distribution across the aperture. In one-dimension
this would be a boxcar function or in two dimensions a uniform disk. The beam
pattern is the Fourier transform of the aperture distribution, which for one
and two dimensions respectively, is the sinc function, $\sin{x}/x$, and the
function $J_{1}(r)/r$ where $J_{1}$ is a Bessel function. In the two-
dimensional case, the power pattern, or the square of the beam, is the Airy
function. The beam and power patterns are shown in figure 1. The angular
resolution of the beam power is $\lambda/D$, the wavelength of the observing
frequency divided by the diameter of the aperture. For example, if the
aperture is 1000m in diameter and the observing frequency is 230 GHz, then the
angular resolution, measured by the width of the power in the main beam, is
0.27 arc seconds, full-width at half-maximum (FWHM). However, a significant
amount of the power is outside the main beam in the side lobes. The encircled
energy, or the percentage of the total beam power as a function of radius is a
measure of the concentration of the power in the main beam. The beam of a
1000m uniform aperture at 230 GHz contains 98% of the power within a radius of
1.53 arc seconds.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{sinc_eeplot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{gauss_eeplot}\\\
\end{array}$
Figure 1: Left: One dimension of the beam (solid line) and the beam power
(dashed line) of a uniform circular aperture. The beam pattern is normalized
to one. The beam power is multiplied by 10 so that the scale shows the
percentage of the peak power. The scale in arc seconds assumes a hypothetical
uniform aperture of 1000m diameter and an observing frequency of 230 GHz. The
main beam is relatively narrow, 0.27 arc seconds FWHM. Because of the high
side lobes, the radius that contains 98% of the total power is relatively
wide, 1.53 arc second, indicated by the red horizontal bar. Right: One
dimension of the beam and power pattern of a 2D Gaussian aperture with
$\sigma=250{\rm m}$ ($2.35\sigma={\rm FWHM}$) truncated at 1000m. The main
beam is relatively wide, 0.40 arc seconds FWHM. Because the side lobes, caused
by the truncation, are relatively low, 98% of the energy is encircled within a
relatively narrow radius of 0.34 arc second. The figures of merit are listed
in table 15.
In contrast, consider a Gaussian distribution whose Fourier transform is also
a Gaussian with an inverse relation in widths, $\sigma$ and $\lambda/\sigma$.
Because the Gaussian has no side lobes other than its own extended wings, all
the power is concentrated in the main beam. However, the inverse relation
between the widths in the real and Fourier domains means that a large array
size is required to achieve a small beam size. Furthermore a Gaussian aperture
must be truncated at a finite radius. A Gaussian aperture of 1000m diameter
with a width of one-half the radius or $\sigma=250{\rm m}$ has an angular
resolution of $\lambda/\sigma=0.40$ arc seconds. This is significantly broader
than the beam of a uniform aperture, but the encircled energy of the Gaussian
is more concentrated. The radius encircling 98% of the total power in 2.15 arc
sec is 0.35 arc seconds. The beam and power patterns are shown in figure 1.
The figures of merit are listed in table 15.
These two Fourier transform pairs illustrate two different trade-offs: first,
between the angular resolution and the size of the array; and second between
the angular resolution and the encircled energy which is sensitive to the
amount of power in side lobes. In general, the best apertures are a compromise
between these two extremes, a uniform distribution with a smoothed or apodized
boundary.
For example, a sigmoid such as the logistic function
$y=1/(1+\exp{(-(x-a)/b}))$ defines a curve which is uniform, equal to one in
its interior, and has a smooth boundary at $a$ whose transition width, $b$,
can be continuously varied. Figure 2 (left) shows two aperture distributions
made with the logistic function with different values for the width of the
boundary, $b$ = 50 and 42 m, and its location, $a$ = 250 and 333 m. Both
apertures are truncated at a radius of 500 m. Both beams (figure 2 right) have
broader FWHM and lower side lobes than the beam of the idealized uniform
aperture with a sharp boundary (figure 1). The comparison here shows
additionally that the side lobes decrease with the increasing width or
smoothness of the boundary while the beam width increases, and the angular
resolution decreases.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{logistic_uv}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{logistic_beam}\\\
\end{array}$
Figure 2: Left: Two examples of a uniform distribution with a smoothed
boundary using the logistic function with different constants to create two
different boundary widths. Right: Beam patterns corresponding to these two
aperture distributions. The comparison shows that the side lobe levels
decrease with the smoothness of the distribution while the beam width
increases and the angular resolution decreases
Finally, we must come back to the constraint that the aperture distribution is
not arbitrary but given by the separations of the individual antennas. The
relationship between the pattern of antenna locations and the pattern of their
separations is not always obvious. Particularly with Earth rotation synthesis
(ERS) [Ryle, 1962], it is a challenge to arrange the individual antennas to
create a desired distribution from the sum of their separations. The fourth
design goal indicates how well the antenna separations approximate a
continuous distribution.
## 3 Previous array designs
One of the first arrays for radio astronomy, the Mills cross [Mills & Little,
1953] had regularly-spaced dipole antennas aligned in the shape of a cross
which provided a square aperture with a regularly-spaced, two-dimenensional
grid of Fourier components. Ryle & Hewish [1960] developed an interferometer
with more sampling efficiency by eliminating one of the arms, which changed
the cross to a T-shape, and by locating the antennas with non-uniform spacing
in such a way as to minimize the number of separations of the same length.
Minimum redundancy is an advantange particularly for interferometers that use
expensive parabolic-dish antennas rather than dipoles. The theory of minimum
redundancy was further explored by Moffett [1968] and extended to a large
number of antennas by Ishiguro [1980]. Interferometers such as the Hat Creek
Observatory/BIMA [Welch et al., 1996], the Owens Valley Radio Observatory
Millimeter Array [Padin it et al., 1991], and the Plateau de Bure
Interferometer [Guilloteau et al., 1991], among others, are T-shaped arrays
designed for Fourier sampling with minimum redundancy.
The Very Large Array (VLA) [Thompson et al., 1980] improves on the T-shape by
arranging the 3 arms in a Y-shape which provides better sampling when used in
ERS. A typical astronomical observation is about 8 hours, or one-third of a
complete rotation. The antennas are distributed along the arms according to a
power-law spacing. This creates a non-uniform sampling that allows a higher
dynamic range between the smallest and the largest measured Fourier component.
In contrast, the Submillimeter Array interferometer is designed to to provide
uniform and non-redundant Fourier sampling for high image fidelity [Keto,
1997]. Uniform coverage is discussed more immediately below (§4) and power-law
spacings in §9.
Yet another design goal, a Gaussian-shaped beam, was used to design the
configurations for the Combined Array for Research in Millimeter Astronomy
(CARMA) [Helfer, 2004]. The large number of antennas of the recently built
Atacama Large Millimeter Array (ALMA) [Wooten & Thompson, 2009] allows more
flexibility to shape the beam, and there are three different design goals for
configurations of different scale. The most compact configuration seeks to
minimize side lobe levels, the intermediate is a spiral pattern with a
Gaussian beam in mind, and the most extended is a ”Y” shape modified to reduce
the side lobes of the beam [Conway, 2006; Holdaway, 2007]. Examples of the
three configurations are shown in §14.
The Long Wavelength Array (LWA) distributes 256 antennas in a uniform random
pattern that is modified to reduce the side lobe levels [Kogan, 2000]. Random
arrays are discussed in §10.
One proposed design for the Square Kilometer Array (SKA) puts the antennas
along logarithmic spirals and seeks to minimize large gaps in the Fourier
sampling [Millenaar et al., 2011]. Spiral arrays are discussed in §9 and the
minimum gap criterion in §4.2.
The study presented here in this article does not discuss the merits of
different design goals. These depend on the particular scientific aims of the
observatories and their particular constraints, for example, the number of
antennas. The different designs of the observatories listed above represent a
significant diversity of goals, and the list does not even include all radio
astronomy arrays. Rather, this paper shows how array design can be understood
in terms of trade-offs between different goals and how the design process can
be simplified by a hierarchical strategy that allows the designer to place the
array at a desired point between the trade-offs.
## 4 Uniform sampling by curves of constant width
The hierarchical arrays introduced in this paper are constructed by repeating
a simple configuration, a subarray, on multiple scales. A good choice for the
repeating pattern is one of the configurations that approximates uniform
sampling in the Fourier plane. This property ensures that for each scale the
gaps in the Fourier coverage, or more precisely the separations between
sampling points are all about the same size. Since they are a basis for the
H-arrays, it is worth reviewing the properties of the configurations that
provide the best uniform coverage.
A previous paper showed that arrays designed on the figures known as curves of
constant width (CW) come closest to approximating a uniform aperture
distribution [Keto, 1997]. These figures are in the shape of closed rings. The
name, constant-width, refers to the diameter. These curves have the property
that from any point on the curve, the maximum distance to the opposite side,
effectively the diameter, is always the same. This guarantees that the
boundary of the aperture and its Fourier transform, the beam, are both
circular. For example, the circle, which by definition has a constant
diameter, is a curve of constant width. The circle can be thought of as a CW-
curve with an infinite number of sides. The Reuleaux triangle is the limiting
case of a CW-curve with the least (three) number of sides. Of the two, the
Reuleaux triangle is preferred for array design and was adopted for the design
of the Submillimeter Array.
The uniform aperture distribution with its $J_{1}(r)/r$ beam approximated by a
CW-array lies toward one end of our space of design trade-offs accomplishing
three of the design goals, high angular resolution, compact array size, and
uniform sampling but at the expense of rather high side lobes. The uniform
sampling has another attractive feature. In imaging applications, the uniform
sampling allows the measured Fourier components to be combined with equal
weights (”natural weighting” in radio astronomy) and therefore achieves the
highest sensitivity simultaneously with uniform sampling. CW-arrays are a good
choice for interferometers that have relatively few antennas and therefore
deeply cherish each individual Fourier component and regret weighting down any
of them. Furthermore, a small number of antennas limits the flexibility to
shape the beam, and the small number of measured Fourier components
necessarily results in high side lobes which, unavoidable, are therefore less
of a design concern. Imaging with high side lobes relies on numerical
techniques to deconvolve the beam pattern from the image [Hogbom, 1974].
CW-arrays are also the best choice for dithering patterns for flat-fielding
multi-pixel imaging arrays [Arendt et al., 2000]. In this application the goal
is to measure the relative sensitivity of all the pixels in a sensor array
such as a CCD. One possibility is to point the telescope repeatedly until each
pixel has viewed the same patch of sky. This requires many observations, one
for each pixel in the array, but each comparison to determine the relative
sensitivity of two pixels requires the use of only two measurements. Another
possibility is to point the telescope only twice in such a way that the view
is shifted by one pixel. Each pixel then views the patch of sky previously
viewed by its neighbor, and the sensitivity of neighboring pixels may be
measured relative to each other. The relative sensitivity of all the pixels is
found by working across the array comparing neighboring pixels. This requires
only two observations, but the number of comparisons needed to determine the
sensitivity of any two pixels increases with their separation. The noise in
each intermediate measurement and comparison reduces the accuracy of the flat-
fielding. The optimal strategy uses as few observations and as few
intermediate comparisons as possible. This is achieved by distributing the
pointing shifts uniformly across the space of all possible shifts, exactly
analagous to the problem of distributing the separations of antennas uniformly
across their separation space. In their study, Arendt et al. [2000]
objectively compared a number of different array designs and verified that the
CW-arrays provide the best approximation to uniform sampling. The Spitzer
Space Telescope uses an observing pattern based on the Reuleaux triangle for
imaging with its infrared cameras.
### 4.1 A 6-element CW-array
The simplest example of CW-array is the six-element pattern in figure 3 whose
separations are uniformly distributed on an hexagonal grid. The antenna
coordinates are given in table 1. Visually there appears to be a hole in the
center that is missing a separation. This is a separation of zero, or
equivalently the Fourier component of zero spatial wavelength corresponding to
the total power. All interferometers are missing this measurement. Figure 3
shows that the distance between all the separations, including the distance
from the smallest separations to the zero point, is exactly one. In this
sense, the configuration is not missing short spacings. The shortest spacing
is set by the distance between the pairs of antennas on each side of the
Reuleaux triangle which of course can be made arbitrarily small. The reason
the central hole in the coverage looks larger in this array than in other
arrays with non-uniform coverage is because the uniform coverage limits the
size of the largest separation. For example, arrays with a power-law
distribution of antenna separations achieve a higher ratio of the largest to
smallest separations, higher spatial dynamic range, but only by leaving larger
holes in the coverage at larger spatial wavelengths. However, in these non-
uniform arrays, the central hole looks smaller with respect to the total
extent of the non-uniform coverage. The next section, §7, shows that one
advantage of hierarchical arrays is that this central hole can be filled by a
smaller scale in the hierarchy.
While the UV coverage in ERS is no longer strictly uniform, the mid-points of
the ERS tracks are still on the original uniform grid. This means that arrays
that have good snapshot coverage generally have good coverage in ERS even
though the coverage in snapshot and ERS is not the same. Figure 3 shows the
tracks of the separations (baselines in radio astronomy) in an 8.2 hr ERS
assuming the array is located at a latitude of 23∘ and the target transits
through the zenith. As shown in the figure, ERS generates a pattern of tracks,
or an effective aperture, slightly larger in extent north-south than east-
west. This is because the target is low on the horizon at the beginning and
end of the track when it is rising and setting. At these times from the
vantage point of the target, the array and its antenna separations appear
shortened by projection. This results in UV coverage that is narrower and a
beam pattern that is wider in the east-west direction than north-south. It is
trivial to make the beam circular by adjusting the aspect ratio of the array,
but the correction is also a function of the declination of the target.
$\begin{array}[]{cc}\hskip 46.97505pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{s6_ant_pos}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6_snapshot}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6_uvtracks}\\\ \end{array}$
Figure 3: Top: Antenna locations of a simple 6-element array whose separations (UV coverage) (middle) are uniformly spaced on a hexagonal grid. Bottom: Tracks of the antenna separations (baselines) in an 8.2 hr Earth rotation synthesis (ERS). The uniform distribution of separations (middle) produces a reasonably uniform distribution of baseline tracks. Table 1: Antenna positions for 6-element subarray X | Y
---|---
0 | 0
1 | 0
1 | $\sqrt{3}$
1/2 | $3\sqrt{3}/2$
-1 | $\sqrt{3}$
-3/2 | $\sqrt{3}/2$
### 4.2 The minimax metric
CW-arrays produce the most uniform sampling across a circular aperture, but
aside from the particular 6-element pattern in snapshot observing (discussed
above), their separations do not lie exactly on a uniform grid. How do we
measure uniformity? For imaging arrays, the most important feature of uniform
coverage is that no part of the relevant Fourier space is very far from a
measurement. This decreases the possibility of error which increases with the
distance in Fourier space between the component and a nearby measurement,
assuming that there is some smoothness or coherence in the image itself. With
this in mind, uniformity can be defined in a statistical sense with a
particular figure of merit, a minimax that measures the maximum separation
from every point in the Fourier plane to the closest measured Fourier
component. With the usual distance as the metric, the best approximation is
the one that minimizes the maximum distance,
$\min\sum(\max_{i}[(U-u_{i})^{2}+(V-v_{i})^{2}])$ where $U$ and $V$ are
probability densities that represent the desired distribution across the
entire Fourier plane within the circular boundary allowed by the maximum
antenna separation, and $u_{i}$ and $v_{i}$ are the coordinates of each
antenna separation. A particularly simple way to compute this figure of merit
is with a Monte Carlo algorithm. To measure how closely the separations of an
array approximate a uniform distribution, generate a set of random points
$U,V$, uniformly distributed inside the boundary of separations and compute
the distance between all pairs $U,V$ and $u_{i},v_{i}$. The best approximation
minimizes the maximum distance or the average maximum distance. Figure 4
illustrates the minimax for a 9-antenna array. The antenna locations for this
array are listed in table 2. In the right panel the antenna separations are
shown as asterisks and one particular realization of a set of random UV points
is shown as red dots. An actual evaluation would use many more random points
than the ones shown here for illustration. The number of UV points necessary
for an accurate measure is about 10 to 100 times the number of separations. It
is easy to determine the accuracy of the algorithm by running a few trials
with different numbers of points and different realizations of the random
distribution. This figure of merit was used in [Keto, 1997] to show that the
arrays that best approximate a uniform distribution of separations are based
on the Reuleaux triangle. The minimax figure of merit is different from other
figures of merit that consider the antenna separations only with respect to
one another.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska9_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{minimax}\\\
\end{array}$
Figure 4: Left: Antenna locations of a 9-antenna array whose separations approximate a uniform distribution. Right: Minimax test. Crosses show the antenna separations and red dots show the random UV points generated by a Monte Carlo algorithm. Table 2: Antenna positions for 9-element array X | Y
---|---
-1.02847 | -0.955366
-0.471921 | -1.22493
0.195772 | -1.03746
1.29924 | -0.459679
1.27441 | 0.187321
0.775329 | 0.631558
-0.308142 | 1.38484
-0.751473 | 1.05465
-0.984755 | 0.419072
## 5 Figures of merit
What if we want a more concentrated beam and power pattern with lower side
lobes and a smaller radius of encircled energy than provided by a uniform
aperture distribution? Hierarchical arrays are one solution. First we need a
complete set of figures of merit to quantitatively evaluate different array
designs in terms of each of the four design goals. The first two goals can be
combined into the product of the array size and the beam size, leaving three.
1. 1.
$K_{nn}$ = maximum diameter of the array times the diameter that contains $nn$
percent of the encircled energy. This measures the degree to which the design
is compact in both real and Fourier space. Because the example arrays all have
a maximum separation of 1000m the $K$-product is sensitive to the distribution
of the power between the main beam and the side lobes. For radio frequency
arrays of 1000 m with arc second resolution, the units are meter arc seconds.
This measurement depends on the angular size scale used to define 100% of the
total power. Since there are a limited number of Fourier components, they
never completely cancel, and the side lobes are infinite in extent. A good
practice is to integrate out until the decrease in side lobe level slows. For
our example arrays of 1000m diameter and observing frequency of 230 GHz, we
use a radius of 2.15 arc seconds.
2. 2.
The angular resolution, the full width at half maximum (FWHM) of the main
beam. We use the root of the product of the widths in the north-south and
east-west directions.
3. 3.
A smooth distribution of UV points. There are several ways to measure the
smoothness. Some experimentation shows that a simple and satisfactory measure
is the variance off a polynomial fit to the density of antenna separations as
a function of separation. We use a third order polynomial and the reduced
$\chi^{2}$ of this fit.
To generate the figures of merit, the hypothetical observatory is located at a
latitude of 23 degrees tracking a source through zenith in Earth rotation
synthesis. The UV tracks are calculated for 8.2 hours between Hour Angles $\pm
4.1$ with points recorded every 0.25 hours. The UV tracks and the beams are
calculated assuming that the arrays have a maximum diameter of 1000m, and that
the observing frequency is 230 GHz.
The maps of the beam pattern are made with a particularly punishing color
scale that emphasizes the low-level side lobes. The beam is normalized by its
peak, then subject to an asymmetric sigmoid function, the Gompertz function
$y=\exp{(-\exp{(-ax)})}$ with constant, $a=50$, large enough to saturate the
beam at a few percent revealing the side lobes. The color table, inspired by
Tang sancai (three colors) pottery, shows the negative and positive side lobes
in blue and red. The asymmetry of the Gompertz function compensates in part
for the visual bias that red appears slightly brighter than blue.
## 6 Hierarchical configurations
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6p6_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6p6_center_tracks.ps}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6p6_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6p6_off_center_tracks.ps}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{s6p6_uvtracks}\\\ \end{array}$
Figure 5: Top left: Antenna locations in a 2-level s6p6 array. The subarrays
are rotated relative to the subarray at the bottom of the figure by 50, 10,
40, 20, and 30 degrees moving counterclockwise around the pattern. The
subarrays are also scaled by factors of 1.03, or from the bottom, by 1.03,
1.06, 1.09, 1.13, 1.16. The entire pattern is rotated 60 degrees with respect
to the 6 positions in table 1. Middle left: Antenna separations. The
$180=30\times 6$ short separations between antennas within each of the 6
subarrays occupy the center of the UV plane in green and in detail top right.
The 30 other patches around the center are the separations between the
subarrays. The set of patches replicate the uniform spacing of the basic
6-element array (figure 3). Detail of one patch middle right. The exact
pattern is different in each patch. (bottom) UV tracks in an 8.2 hr ERS. Each
of the 30 patches moves across the UV plane like a paint brush.
To construct a hierarchical array, repeat the subarray pattern on a larger
scale with copies of the subarray distributed on a pattern equal to itself. If
we call the first level the s-level, the second the p-level, the third the
d-level, and the fourth the f-level, a two-level array with 6 elements in the
first level and 6 in the second is indicated by s6p6. Figure 5 shows this
configuration with the second level scaled to 12 times the first. The two
hierarchies of antennas create the two hierarchies of separations. In this
example, the scaling between the hierarchical levels is large enough that the
separations in different levels do not overlap.
In the very center marked in green are the $30\times 6$ shortest separations
within each of the 6 subarrays. To eliminate the redundancy in the separations
that would occur if the subarrays were identical, the subarrays are rotated
and scaled very slightly with respect to each other. Relative to the subarray
at the bottom of the figure, the subarrays are rotated by 50, 10, 40, 20, and
30 degrees moving counterclockwise around the pattern, and scaled by factors
of 1.03, or from the bottom, by 1.03, 1.06, 1.09, 1.13, and 1.16. This
produces 180 unique short separations rather than 6 copies of the same 30
separations (figure 5 top right). The entire pattern is rotated 60 degrees
with respect to the 6 positions in table 1.
The second level of the hierarchy of separations is made up of all possible
separations of the antennas of different subarrays. These larger scale
separations show the same uniform pattern as in figure 3, but with patches
made of the 30 possible pairs of the 6 subarrays instead of the 30 UV points
from the 6 individual antennas. In each of the 30 patches there are 36 points
from the 36 possible pairs of antennas between 2 subarrays. Figure 5 (middle
right) shows one set of these 36 separations. Each patch has a slightly
different pattern. Finally, figure 5 (bottom) shows how ERS drags each of the
30 patches across the UV plane like a wide paint brush across a white canvas.
Because the separations of the 6-element subarray are uniformly distributed,
so are the midpoints of the tracks of the second level of the hierarchy.
The extension to further hierarchical levels is obvious. The entire two-level
array is repeated on the same basic pattern of the subarray. To keep the
figures from becoming too crowded, the following examples refer to just two
levels or three levels.
## 7 The flexibility of a 2-level H-array
The simple 36-element 2-level hierarchy pattern can be scaled to select a
point source response along the continuum of choices in the trade-off between
high angular resolution and low side lobes. First, to design a high angular
resolution beam, scale the second level just large enough to place the 30
patches of separations side-by-side. Next, to design a beam with low side
lobes, scale the second level so that the patches overlap and create a
distribution that tapers gracefully to the boundary.
### 7.1 A high angular resolution beam
Scaling the second level by a factor of 5.5 relative to the first (figure 6
top left) produces a tiered distribution of separations with a nearly uniform
density in two regions. Figure 6 (top right) shows the two-dimensional pattern
of separations and the UV coverage obtained in snapshot imaging at zenith.
Figure 6 (middle left) shows the corresponding density of separations as a
function of radius averaged over angle. In ERS, the elliptical arcs across the
UV plane improve the UV coverage, particularly as a function of angle. Because
the UV points of the shorter separations move more slowly across the UV plane,
ERS also affects the radial distribution by increasing the density of the
shorter separations. Nonetheless, the radial distribution in ERS still
maintains its essential character (figure 6 middle right). The smooth red line
shows the fit of a third order polynomial to the distribution in the outer
zone. In snapshot the reduced $\chi^{2}$ is 0.00094. The better UV coverage in
ERS fills in gaps and improves the smoothness to 0.00017.
The two-dimensional beam pattern in ERS (figure 6 bottom) shows the high side
lobes characteristic of arrays with uniform UV coverage, but the angular
resolution is exquisite, 0.17 arc seconds FWHM. The radius encircling 98% of
the beam power is 1.38 arc seconds and the K-product, $K_{98}=1378$ (figure 6
bottom right). The figures of merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_sinc_revB_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_sinc_revB_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_sinc_revB_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_sinc_revB_density_ERS}\\\ {\hskip
25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_sinc_revB_beam_ERS}}{\hskip-21.68121pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_sinc_revB_eeplot_ERS}}\\\ \end{array}$
Figure 6: A 36-element s6p6 array designed for high angular resolution. Top
left: Antenna positions. Moving counterclockwise around the figure the
subarrays are rotated with respect to the left subarray of the two on the
bottom by 20, 60, 30, 100, and 20 degrees. The subarrays are also scaled by
factors of 1.05 with respect to each other, or counterclockwise from the left
bottom, by 1.05, 1.10, 1.16, 1.22, and 1.28. Top right: Separations and UV
coverage in snapshot at zenith. Middle left: Radial distribution of the
density of separations in snapshot at zenith. The smooth red line shows a
polynomial fit to the outer part of the distribution. The variance from this
fit is a measure of the smoothness of the coverage in the outer zone. Middle
right: Radial distribution of the density of separations in ERS. Bottom left:
Beam pattern in ERS. The black contours are at 10, 50, and 90% of the peak.
From the peak, the second black contour shows the FWHM. The white contours are
at 1, 2, and 3% of the peak. The negative contours are dashed. The zero
contour is white and dotted. Bottom right: Beam and power pattern in the same
format as figure 1. The beam power (dashed line) is multiplied by 10. The
figures of merit are listed in table 15.
### 7.2 A beam with low side lobes
To reduce the side lobes, reduce the scaling of the second level of the
hierarchy to pull the 30 patches of separations inward until they are
overlapping and create a smoothly tapered distribution. Figure 7 shows the
same plots as in the previous example but for an array with the second level
only 1.5 times larger than the first. The radial distribution of the
separation density is now shifted to shorter separations and gradually tapers
to the maximum. The beam has lower angular resolution, 0.21 arc seconds, but
also lower side lobes and more concentrated beam power. The radius encircling
98% of the energy is 0.29 arc seconds, and the K-product is $K_{98}=285$. This
beam has better angular resolution than the 0.35 arc sec Gaussian beam of our
idealized example discussed in the introduction, and the smaller $K$-product
indicates that the side lobes of this beam contain less energy than the
extended wings of the Gaussian beams of either the truncated or idealized
Gaussian apertures.
The concentrated UV distribution leaves fewer points to cover the outer zone
resulting in wider gaps than in the more uniform distribution of the array in
figure 6. The $\chi^{2}$ figure of merit measures the local smoothness and is
accordingly larger even though the distribution is globally tapered to the
boundary. For snapshot and ERS, $\chi^{2}$ is 0.0098 and 0.0041, respectively.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36fix_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36fix_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36fix_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36fix_density_ERS}\\\ \hskip
25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36fix_beam_ERS}\hskip-21.68121pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36fix_eeplot_ERS}\\\ \end{array}$
Figure 7: A 36-element s6p6 array designed for a high concentration of power
in the main beam and low side lobes. Top left: Antenna positions. The 6
subarrays overlap and each is drawn with a different number. The 6 positions
marked with the number 0 belong to the same subarray. Moving counterclockwise
around the figure, relative to the subarray marked by zeroes, the subarrays
are rotated by 40, 20, 60, 20, 100, and 20 degrees. The subarrays are also
scaled by factors of 1.075 with respect to each other, or counterclockwise
from the upper left by 1.075, 1.163, 1.24, 1.34, 1.44. The entire pattern is
rotated 30 degrees with respect to the locations in table 1. Same format as
figure 6 except for the panel in the lower right. Here the solid line shows
the bottom of the beam which is normalized to one. The dashed line shows the
beam power multiplied by 100. The scale then shows the power as a percentage
of the peak power. The figures of merit are listed in table 15.
### 7.3 Design principles
These examples show how H-arrays allow control over different scales in the UV
plane. A change in the first-level subarrays affects the UV distribution of
the smallest separations in the center of the UV plane and also the
distribution inside the 30 patches or brush strokes in ERS of the second level
but does not much affect the distribution of these patches across the UV
plane. This is controlled by the placement of the subarrays in the second
level. In this way we have simple and understandable control over the aperture
distribution. We can shape the aperture distribution by scaling the levels to
emphasize Fourier components of different scales.
The conceptual separation of scales possible with H-arrays can be thought of
as accomplishing more than one observation at the same time. Arrays for
astronomical imaging are often built with multiple configurations of different
sizes to measure Fourier components on different angular scales. The
configurations are nested so that the separations of each configuration are
just larger than the next smaller size, usually with some overlap. The full
range of Fourier components is measured in multiple observations by moving the
antennas to each of the configurations between observations. The multiple
scales of the H-arrays accomplish this in one observation.
## 8 Sparse H-arrays
Is the uniform pattern of separations provided by the 6-element array of
figure 3 essential? What if the number of available antennas is not a power of
six? Two examples with a minimal subarray of 3 elements show that it is
possible to obtain good coverage with sparse arrays built with a subset (half)
of the basic 6-element subarray. The lower density of UV coverage leaves some
gaps, but there is a positive trade-off as well. Sparse arrays can achieve a
larger dynamic range, the difference between the smallest and largest
separations, with fewer antennas.
### 8.1 A sparse array for high angular resolution
As with our earlier example, a larger scaling between the hierarchical levels
can be used to create approximately uniform UV coverage in separate zones.
Figure 8 shows an array with 3 hierarchical levels. The first two levels use
3-element subarrays made by dropping every other element of the basic
6-element array. The third level uses the full 6-element pattern. This array
has 54 antennas in an s3p3d6 configuration (figure 8). The second and third
levels are scaled by 3.4 and 22.0 times the basic subarray. The subarrays
within the two levels are scaled by 6% and 8% to eliminate redundancy. The
details describing the rotation and placement of the subarrays are listed in
table 3. With a maximum separation of 1000m, the angular resolution is 0.18
arc seconds at 230 GHz. The $K$-product is $K_{98}=891$ indicating that 98% of
the encircled energy is within a radius of 0.89 arc seconds. In snapshot
imaging, the coverage of this s3p3d6 sparse array has less uniform coverage
than the 2-level s6p6 array but still maintains the character of uniform
coverage tiered in separate zones. Figure 8 shows the separations (snapshot
coverage at zenith), the radial density of the separations in snapshot and
ERS, and the beam and its trace in ERS. With more UV points than the
36-antenna s6p6 array, the $\chi^{2}$ measure of local smoothness is lower,
0.00027 and 0.00020 in both snapshot and ERS, respectively. The figures of
merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54_flat_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54_flat_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54_flat_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54_flat_density_ERS}\\\ \hskip
25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54_flat_beam_ERS}\hskip-21.68121pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{ska54_flat_eeplot}\\\
\end{array}$
Figure 8: Top left: Antenna locations, UV distribution, and beam for an
example configuration of 54 antennas weighted toward longer baselines. Same
format as figure 6. The rotations and scalings of the subarrays are listed in
table 3 The beam power (dashed line) in the lower right panel is multiplied by
10. The figures of merit are listed in table 15. Table 3: Subarray rotations
and scalings for figures 8, 9
Subarray | Rotation | Exponent of the
---|---|---
| | scaling factor
s0 111Corresponding author. | 100 | 0
s1 | 80 | 4
s2 | 60 | 1
s3 | 20 | 3
s4 | 20 | 2
s5 | 0 | 5
p0 22footnotemark: 2 | 0 | 0
p1 | 180 | 4
p2 | 60 | 1
p3 | 240 | 3
p4 | 120 | 2
p5 | 300 | 5
* 1
1 The s-level subarrays for the s3p3d6 configuration for high angular
resolution shown in figure 8 are scaled by 1.06 raised to the power shown in
the table. For the s3p3d6 and s6p6d6 arrays shown in figures 9 the scaling
factor is 1.03.
* 2
2 The p-level subarrays for the s3p3d6 configuration shown in figure 8 are
scaled by powers of 1.08. For the s3p3d6 and s6p6d6 arrays shown in figures 9
the scaling factor is 1.03.
### 8.2 A sparse array with a concentrated beam
Figure 9 shows an example with a factor of 2 scaling between the levels and 3%
scaling of the subarrays within each level to eliminate redundant separations.
The details describing the rotation and placement of the subarrays are listed
in table 3. The smoothly tapered distribution of separations from the
overlapping subarrays creates a beam with an angular resolution of 0.23 arc
seconds FWHM and a K-product, $K_{98}=294$ indicating that 98% of the
encircled energy is within 0.29 arc seconds. Both the angular resolution and
encircled energy are better than the Gaussian aperture in §2. Compared to the
sparse H-array with uniform UV coverage, here the concentrated UV coverage
results in less local smoothness and higher $\chi^{2}$ measures, 0.0036 and
0.0018 in snapshot and ERS respectively. The figures of merit are listed in
table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54-n_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54-n_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54-n_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54-n_density_ERS}\\\ \hskip
25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54-n_beam_ERS}\hskip-21.68121pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{ska54-n_eeplot_ERS}\\\
\end{array}$
Figure 9: Top left: Antenna locations, UV distribution, and beam for a
54-antenna s3p3d6 designed for low side lobes. Same format as figure 7. The
beam power (dashed line) in the lower right panel has been multiplied by 10.
The figures of merit are listed in table 15.
## 9 Hierarchical spirals, H-spirals
The technique of scaling and rotating subarrays with good uniform coverage can
also be used to create intelligently designed high performance spiral arrays
with many antennas. These hierarchical spirals or H-spiral arrays are
aesthetically attractive and also have some practical advantages useful in
construction. (The process of scaling and rotating produces spirals that are
mathematically equivalent to logarithmic spirals with different constants.)
Because of their concentric rather than hierarchical structure, spiral arrays
do not produce the highest angular resolution beams with a compact array size.
Nevertheless, these H-spirals share the property of the H-arrays that a simple
change in scale again moves the design between the trade-offs of high angular
resolution and a more concentrated beam.
The Square Kilometer Array might be based on a spiral pattern. The primary
design goal is to minimize gaps in the UV coverage within the constraints of
the spiral design [Millenaar et al., 2011]. The uniform coverage of the CW-
arrays by definition provides minimal gaps (§4.2). A spiral array built by
scaling a CW-subarray according to a power law has larger gaps at larger
scales, but at each scale, the CW-subarray provides coverage with the minimum
possible gaps at that scale.
### 9.1 Rotation in H-spirals
First consider the effect of rotation. The first two examples are H-spiral
patterns based on the 9-element CW-array in figure 4 but with different
amounts of rotation between the subarrays. The power law scaling, 1.25, is the
same in both cases. The first pattern rotates the subarrays by 164∘ to create
a tight, wrapping spiral resembling a late-type galaxy (figure 10). In the
second, a smaller rotation of 113∘ produces a 9-ray pattern resembling the
nine-armed sea stars of the Florida Keys (figure 11). Both have essentially
identical performance metrics. In ERS, the H-spiral galaxy has a angular
resolution of 0.23 arc seconds and a K-product, $K_{98}=285$. The $\chi^{2}$
measure of smoothness is 0.0029. The H-spiral sea star has the same angular
resolution, 0.23 arc seconds and a K-product, $K_{98}=285$, but the $\chi^{2}$
indicates that the distribution of separations is slightly smoother at 0.0026.
In snapshot, the $\chi^{2}$ measures of local smoothness for the galaxy and
sea star H-spirals are 0.0039 and 0.0024 respectively. Both these arrays have
better angular resolution and tighter encircled energy than the Gaussian beam
in §2. The figures of merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_164_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_164_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_164_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_164_density_ERS}\\\ \hskip
25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_164_beam_ERS}\hskip-21.68121pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_164_eeplot_ERS}\\\ \end{array}$
Figure 10: Antenna locations, UV distribution, and beam for a 54-antenna
spiral array designed for a high concentration of power in the main beam. The
pattern uses the 9-element CW-array in figure 4 as the simple subarray. A
power law scaling by a factor of 1.25 and a constant rotation of 164∘ between
subarrays produces the 3-armed pattern resembling a spiral galaxy. The
difference between this array and the one in figure 11 is the amount of
rotation between the scaled subarrays. Same format as figure 7. The beam power
(dashed line) in the lower right panel has been multiplied by 100. The figures
of merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_113_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_113_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_113_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_113_density_ERS}\\\ \hskip
25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_113_beam_ERS}\hskip-21.68121pt\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral_113_eeplot_ERS}\\\ \end{array}$
Figure 11: Antenna locations, UV distribution, and beam for a 54-antenna
spiral array designed for a high concentration of power in the main beam. The
pattern uses the 9-element CW-array in figure 4 as the simple subarray. A
power law scaling by a factor of 1.25 and a constant rotation of 113∘ between
subarrays produces the 9-armed pattern resembling sea stars. The difference
between this array and the one in figure 10 is the amount of rotation between
the scaled subarrays. Same format as figure 7. The beam power (dashed line) in
the lower right panel has been multiplied by 100. The figures of merit are
listed in table 15.
### 9.2 Scaling in H-spirals
Differences in the amount of rotation between the levels of an H-spiral do not
much affect the properties of the beam; however, their relative scaling is
quite important as the next four examples show (figures 12 and 13). These
H-spirals are built from the basic 6-antenna pattern (figure 3) scaled 9
times. Different values are used in each example for the power law exponent,
1.05, 1.15, 1.25, and 1.35. The figures of merit are listed in table 15. At
the lower value, the UV coverage and beam resemble those of a CW-array with
its raptor resolution. At the high end, the scaling produces a soft beam with
wide Lorentzian wings similar to the beam of VLA with its scaling factor of
1.716. However, the curved arms of the spiral produce a rounder beam with less
concentrated sidelobes than the straight arms of the VLA. Similarly, amateur
astronomers sometimes use curved instead of straight vanes in their Newtonian
telescopes as supports for the secondary in order to spread out the
diffraction pattern of the supports.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_105_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_115_ant_pos}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_105_density}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_115_density}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_105_eeplot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_115_eeplot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_105_beam}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{ispiral6_115_beam}\\\
\end{array}$
Figure 12: Antenna locations, UV distribution, and beam for 54-antenna spiral
arrays showing the effect of the power law scaling between subarrays. The
pattern uses the 6-element CW-array in figure 3 as the simple subarray. In the
left column the configuration is built by scaling successive subarrays by a
factor of 1.05. The right column shows the results for a scaling of by 1.15.
The same arrays but with power law scalings of 1.25 and 1.35 are shown in
figure 13. The lower scalings here produce beams of higher angular resolution.
The panels have the same format as figure 7, except that here the UV coverage
is not shown and the UV density is shown for ERS only. The beam power (dashed
line) in the third panel down in both columns has been multiplied by 100. The
figures of merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_125_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_135_ant_pos}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_125_density}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_135_density}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_125_eeplot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_135_eeplot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ispiral6_125_beam}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{ispiral6_135_beam}\\\
\end{array}$
Figure 13: Antenna locations, UV distribution, and beam for 54-antenna spiral
arrays showing the effect of the power law scaling between subarrays. The
pattern uses the 6-element CW-array in figure 3 as the simple subarray. In the
left column the successive subarrays are scaled by a factor of 1.25, and the
right by 1.35. The same arrays but with power law scalings of 1.05 and 1.15
are shown in figure 12. The higher scalings here produce beams of lower
angular resolution. Same format as figure 12. The beam power (dashed line) in
the third panel down in both columns has been multiplied by 100. The figures
of merit are listed in table 15.
## 10 Random arrays
Successful array designs represent a curious blend of symmetry and randomness.
Configurations with too much symmetry concentrate their separations in
specific patterns in the UV space resulting in the equivalent of diffraction
patterns in the beam. Symmetry in circular and triangular arrays is discussed
in Keto [1997] where the better performance of the triangular array is
attributed to its lower degree of symmetry. Better performance for both arrays
is obtained by slightly perturbing the antenna locations off of a regular
distribution around the circle or Reuleaux triangle. Keto [1997] used a
numerical algorithm to optimize the perturbations. Arendt et al. [2000] simply
used random perturbations. Their slightly randomized CW-arrays look to be at
least nearly as good as those designed by optimization, but a direct
comparison has not been done.
In contrast, fully random patterns, rather than randomly perturbed patterns,
generally do not have as good performance as more thoughtfully designed
patterns. The reason is that random patterns may by chance position antennas
too close to one another resulting in a concentration of separations and
correspondingly higher side lobes as well as larger gaps elsewhere in the UV
distribution.
However, it is easy to create arrays with random patterns, and there is an
attractiveness to the savage simplicity of the process. For example, a
Gaussian distribution of antenna locations produces a Gaussian distribution of
separations because the auto-correlation function of two Gaussians is another
Gaussian. This in turn provides a Gaussian beam through the Fourier transform
relationship. A Monte Carlo algorithm can generate a large number of possible
arrays from which the best can be selected according to whatever criteria.
Figure 14 shows an example selected for the best K-product from a very small
number of trials, specifically 100 trials.
In contrast, a uniform distribution of antenna locations results in an array
at the other end of our trade-off between high resolution and a concentrated
beam. In one dimension, the auto-correlation function of a uniform
distribution is a triangular function whose Fourier transform is the sinc
function squared. The power pattern of the beam then follows the fourth power
of the sinc. The beam of a 2-dimensional uniform distribution with a circular
boundary is conceptually the same, approximately a sinc2 as a function of
radius. An example of a two-dimensional array based on a uniform random
distribution of antennas is shown in figure 14. This array had the best
angular resolution out of 100 trials. The figures of merit are listed in table
15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{gauss48_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{unifor51_ant_pos}\\\
\hskip 25.29494pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{gauss48_density}\hskip
21.68121pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{unifor51_density}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{gauss48_eeplot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{unifor51_eeplot}\\\
\hskip 46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{gauss48_beam}\hskip
7.22743pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{unifor51_beam}\\\ \end{array}$
Figure 14: Antenna locations, UV distributions, and beams for two 54-antenna
arrays with different random antenna locations. The left column shows an array
created from a Gaussian or normal distribution of antenna locations. The right
column shows an array with a uniform distribution of antennas. Same format as
figure 7 except that the second panel in the left column plots a trace across
the UV distribution (solid blue) to show the Gaussian distribution of antenna
separations resulting from the Gaussian distribution of locations. The dashed
red line shows a Gaussian fit and the dotted green line the residuals. The
figures of merit are listed in table 15.
## 11 Outriggers
Interferometers are sometimes built with a few outrigger antennas around a
main array to obtain a limited sample of longer Fourier components. What do
the H-arrays say about the placement of outriggers? H-arrays may be designed
with concentric or asymmetric outriggers. Each has different properties.
Figure 15 shows a concentric configuration with the main array at the center
of 6 outriggers. In this example, the main array is the 36-antenna s6p6 array
designed for high angular resolution; the outriggers use the locations of the
6-antenna subarray to make a 42-antenna s6p6+d6 array. Figure 15 shows four
concentric zones of UV coverage. First, on the largest scale, the 30
separations between the outriggers uniformly cover the UV space within the
boundary. The next zone includes the separations between the outriggers and
the main array, also colored red in figure 15. The two inner zones, blue and
green in figure 15 (top right), are the separations of the main s6p6 H-array
as described in section §7.
In most applications, the inner UV points from the main array would be used in
imaging and the outer UV points from the outriggers would be used separately
to locate point sources to high angular resolution. The outer UV points would
generally not be used for imaging because they create uncancelled sine waves
that do not improve the image fidelity. If the inner array, not including the
outriggers, is scaled to 1000m and the outriggers are not included in the
Fourier transform of the beam, then the array design is the same as shown in
figure 6, and the beam pattern and figures of merit are also the same. The UV
coverage of the outriggers alone would be as shown in figure 3.
Alternatively, we can design an asymmetric outrigger array with the main array
placed at the one of the 6 locations of the 6-element pattern as shown in
figure 16. This creates a 41 antenna array, one less than the concentric
design. This array again shows the four zones of UV coverage but has higher
angular resolution, FWHM = 0.30 arc seconds because the location of the main
array on the border of the 6-element CW pattern creates more longer spacings
than the concentric placement. However, the UV coverage and the beam are
asymmetric with more of the longer spacings in the north-south direction. The
asymmetric outrigger array results in better signal to noise on some of the
longer separations whereas the concentric design improves some of the shorter.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{outrigger2_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{outrigger2_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{outrigger2_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{outrigger2_density_ERS}\\\ \end{array}$
Figure 15: An s6p6-array with 6 concentric outriggers. The s6p6-array is the
same as shown in figure 6, and the 6 outriggers take the positions of the
6-element subarray in figure 3 scaled by a factor of 5.5 times the first
s-level subarray. Same format as figure 6 except the plots of the beam are not
shown.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska216thin_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska216thin_tracks_snapshot}\\\ \hskip
46.97505pt\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska216thin_density_snapshot}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska216thin_density_ERS}\\\ \end{array}$
Figure 16: An s6p6-array with 5 asymmetric outriggers. The s6p6-array is the
same as shown in figure 6, and the 5 outriggers take 5 of the positions of the
6-element subarray in figure 3 scaled by a factor of 5.5 times the first
s-level subarray. The main s6p6-array takes the sixth position of the s-level
subarray. Same format as figure 15.
## 12 Beam patterns with different observations
How does the performance of an array change for different types of
observations, for example when tracking targets at different declinations that
do not pass directly overhead? In astronomical imaging, the beams of good
arrays, those that have a good distribution of antenna separations, generally
maintain their good properties when observing sources in either instantaneous
(snapshot) imaging or earth rotation synthesis and also when observing at
different astronomical declinations.
The previous examples show the beam patterns for a source that transits
through zenith. Figure 17 shows the beam patterns for two low side lobe
designs, the 36-antenna s6p6, and the 54-antenna s3p3d6 configurations,
tracking sources at declinations of $-23^{\circ}$ and $+46^{\circ}$ assuming
that the array is at a latitude of $+23^{\circ}$. The FWHM changes with
declination as the baselines are shortened by projection, but the arrays
generally maintain their good characteristics.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska36_beam_south}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{ska36_beam_north}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{ska54_beam_south}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{ska54_beam_north}\\\
\end{array}$
Figure 17: Beam patterns for 3 H-arrays tracking targets to the north (right)
and south (left). Top and bottom show the beam patterns for the 36 antenna
configuration s6p6 of figure 7 and the 54 antenna configuration s3p3d6 of
figure 9, respectively.
## 13 Numerical optimization
These examples show how easy it is to build H-arrays with excellent
performance without numerical optimization, but the arrays could still be
improved. The optimization problem is much simpler with H-arrays because the
number of parameters that describe the arrays, the relative scaling of the
levels and the subarrays, and the rotation of the subarrays, is fewer than the
number of antennas. For example, the description of a three-level array with
216 antennas requires 14 numbers: two scalings between the levels, and six
scalings and six rotations for the subarrays. An H-spiral can be described by
as few two numbers. In contrast, the description of the antenna locations
individually requires 432 numbers for the 216 X,Y pairs. Optimizing the
antenna positions with respect to each other is a combinatorially explosive
problem with the number of possible configurations increasing exponentially
with the number of antennas. It is difficult for algorithms that optimize the
antenna positions to demonstrate convergence to a global minimum. One
difficulty is that quasi-random configurations with quasi-Gaussian
distributions represent particularly seductive local minima. Another problem
is that ordered patterns such as the Reuleaux triangle require a degree of
coherence that is difficult to obtain for algorithms that move one antenna at
a time.
## 14 Comparison to other designs
It is worth comparing these example H-arrays to other designs. The recently
constructed Atacama Large Millimeter Array (ALMA) uses a variety of different
configurations listed in their CASA simulation software. Three of them,
configurations ALMA-02, ALMA-14, and ALMA-28, are shown in figures 18 to 20.
Aside from their different diameters, here normalized to 1000 m, these three
configurations are quite different from one another reflecting their different
design goals as mentioned in the §3. Configuration ALMA-02 is designed to
minimize side lobe levels. ALMA-14 is a spiral modified to approximate a
Gaussian beam. ALMA-28 is a Y-pattern similar to the VLA. Both ALMA-02 and
ALMA-14 have side lobe levels comparable to the 36 or 54 antenna H-arrays, but
the H-arrays, the appropriately scaled H-spirals, as well as the Gaussian
random array, all have better angular resolution and a tighter concentration
of beam energy encircled in a smaller radius. The figures of merit for the
three ALMA configurations are listed in table 15
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{alma02_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{alma02_uvrange}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{alma02_eeplot98}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{alma02_beam_map}\\\
\end{array}$
Figure 18: Top left: Antenna locations, UV-distribution, encircled energy, and
beam pattern for the ALMA-02 configuration. Same format as figure 12 The
figures of merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{alma14_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{alma14_uvrange}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{alma14_eeplot98}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{alma14_beam_map}\\\
\end{array}$
Figure 19: Top left: Antenna locations, UV-distribution, encircled energy, and
beam pattern for the ALMA-14 configuration. Same format as figure 12 The
figures of merit are listed in table 15.
$\begin{array}[]{cc}\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{alma28_ant_pos}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{alma28_uvrange}\\\
\includegraphics[trim=7.22743pt 21.68121pt 57.81621pt
14.45377pt,clip,width=198.7425pt]{alma28_eeplot98}\includegraphics[trim=7.22743pt
21.68121pt 57.81621pt 14.45377pt,clip,width=198.7425pt]{alma28_beam_map}\\\
\end{array}$
Figure 20: Top left: Antenna locations, UV-distribution, encircled energy, and
beam pattern for the ALMA-28 configuration. Same format as figure 12 The
figures of merit are listed in table 15.
## 15 Conclusions
1. 1.
Different designs for cross-correlation interferometer arrays exist in a
continuous space of trade-offs between competing goals of high angular
resolution, concentrated beam power, compact array size, and the smoothness of
the distribution of the measured Fourier components.
2. 2.
Three figures of merit are useful in assessing the imaging performance of an
array.
1. (a)
The $K_{nn}$ product of the array size and the radius which encompasses $nn$
of the total beam energy.
2. (b)
The angular resolution as measured by the beam FWHM.
3. (c)
The smoothness of the distribution of antenna separations.
3. 3.
A technique of building a large array by scaling and rotating one simple
pattern of a small number of antenna locations can be used to construct
hierarchical arrays (H-array) or spiral arrays (H-spirals) with a large number
of antennas.
4. 4.
Hierarchical arrays (H-arrays) provide an excellent framework for locating the
antennas of a cross-correlation imaging interferometer to provide optimal
imaging qualities.
5. 5.
By changing the scaling between the subarrays, both the H-arrays and the
H-spirals can produce beams at a particular point along the trade-off between
angular resolution and concentrated beam power.
6. 6.
Sparsely populated H-arrays also have excellent performance and are useful for
interferometers designed with more antenna locations than antennas. This
allows the angular resolution to be changed by populating different subsets of
the antenna locations.
7. 7.
The construction procedure of scaling and rotating a basic subarray can also
be used to design high performance, intelligent spiral arrays, H-spirals.
Spirals are good for array designs emphasizing low side lobe designs rather
than high angular resolution.
8. 8.
The H-arrays are easy to optimize because they can be described by a few
parameters, much fewer than the number of antennas themselves.
9. 9.
H-arrays are useful for future multi-element interferometers.
[ht] Figures of merit for example arrays in earth rotation synthesis Array
name number of shown as FWHM radius of 98% $K_{98}$ product Normalized
$\chi^{2}$ antennas figure encircled energy (arc sec) (arc sec) (m arc sec)
($\times 1000$) $J_{1}(r)/r$ … 1 left 0.27 1.53 1530 … Gauss … 1 right 0.40
0.35 353 … s6p6 36 6 0.17 1.38 1379 1.70 s6p6 36 7 0.21 0.29 285 41.5 s3p3d6
54 8 0.17 0.89 891 0.20 s3p3d6 54 9 0.23 0.29 294 1.79 H-spiral galaxy 54 10
0.23 0.29 285 1.47 H-spiral sea star 54 11 0.23 0.29 285 0.81 Gauss random 50
14 0.26 0.33 327 7.80 unif. random 50 14 0.20 0.37 369 6.11 ALMA-02 50 18 0.26
0.35 353 1.74 ALMA-14 50 19 0.29 0.44 437 0.79 ALMA-28 50 20 0.25 0.97 967
2.36
[ht] Figures of merit for spirals based on a 6-element subarray with different
scaling Scaling number of shown as FWHM radius of 98% $K_{98}$ product
Normalized $\chi^{2}$ factor antennas figure encircled energy (arc sec) (arc
sec) (m arc sec) ($\times 1000$) 1.05 54 12 left 0.18 0.52 521 5.20 1.15 54 12
right 0.23 0.26 260 0.88 1.25 54 13 left 0.27 0.37 370 0.84 1.35 54 13 right
0.32 0.66 664 0.43
## References
* Arendt et al. [2000] Arendt, R. G., Fixsen, D. J., Moseley, S. H. [2000] Ap. J. 536, 500.
* Bracewell [1958] Bracewell, R., 1958, Proc. of the IRE, 46, 97
* Bracewell [1999] Bracewell, R., 1999 The Fourier Transform and its Applications, McGraw-Hill
* Conway [2006] Conway, J., 2006, ALMA Document 2006-10-17-ALMA-90.02.00.00-006-A-SPE
* Guilloteau et al. [1991] Guilloteau, S., Delannoy, J., Downes, D., Greve, A., Guelin, M., Lucas, R., Morris, D., Radford, S. J. E., Wink, J., Cernicharo, J., Forveille, T., Garcia-Burillo, S., Neri, R., Blondel, J., Perrigourad, A., Plathner, D., Torres, M., 1991, A&A, 262, 624
* Ishiguro [1980] Ishiguro, M., Radio Science, 15, 1163
* Helfer [2004] Helfer, T., 2004, CARMA Memorandum Series number 20, version 2
* Hogbom [1974] Hogbom, 1974, A&A Suppl., 15, 417
* Holdaway [2007] Holdaway, M., 2007, ALMA Document 2006-10-17-ALMA-90.02.00.00-006-A-SPE
* Keto [1997] Keto, E., [1997] Ap. J. 475, 843.
* Kogan [2000] Kogan, L., 2000, IEEE Trans. Antennas & Propagation, 48, 1075
* Mills & Little [1953] Mills, B.Y. & Little, A.G., 1953, Aust. J. Phys., 6, 272
* Millenaar et al. [2011] Millenaar, R.P., Bolton, R.C., & Lazio, J., 2011, SKA document number WP3-050.020.010-TR-001, Array configurations for candidate SKA sites: design and analysis
* Moffett [1968] Moffett, A.T., 1968, IEEE Trans. Ant. Prop. 16, 172
* Padin it et al. [1991] Padin, S., Scott, S.L., Woody, D.P., Scoville, N.Z., Seling, T.V, Finch, R.P., Giovanine, C.J., Lawrence, R.P., 1991, PASP, 103, 461
* Ryle [1962] Ryle, M., [1962] Nature 194, 517.
* Ryle & Hewish [1960] Ryle, M., Hewish, A., [1960] MNRAS 120, 220.
* Thompson et al. [1980] Thompson, A.R., Clark, B.G., Wade, C.M., & Napier, P.J., 1980, ApJS 44, 151
* Welch et al. [1996] Welch, W. J., Thornton, D. D., Plambeck, R. L., Wright, M. C. H., Lugten, J., Urry, L., Fleming, M., Hoffman, W., Hudson, J., Lum, W. T., Forster, J.. R., Thatte, N., Zhang, X., Zivanovic, S., Snyder, L., Crutcher, R., Lo, K. Y., Wakker, B., Stupar, M., Sault, R., Miao, Y., Rao, R., Wan, K., Dickel, H. R., Blitz, L., Vogel, S. N., Mundy, L., Erickson, W., Teuben, P. J., Morgan, J., Helfer, T., Looney, L., de Gues, E., Grossman, A., Howe, J. E., Pound, M., Regan, M., 1996, PASP, 108,93
* Wooten & Thompson [2009] Wooten, A. & Thompson, A.R., 2009, Proc. IEEE, 97, 1463
|
arxiv-papers
| 2012-09-04T16:25:28 |
2024-09-04T02:49:34.771769
|
{
"license": "Public Domain",
"authors": "Eric Keto",
"submitter": "Eric Keto",
"url": "https://arxiv.org/abs/1209.0692"
}
|
1209.0735
|
# Lambert W Function for Applications in Physics
Darko Veberič [email protected] Laboratory for Astroparticle Physics,
University of Nova Gorica, Slovenia Department of Theoretical Physics, J.
Stefan Institute, Ljubljana, Slovenia
###### Abstract
The Lambert $\operatorname{W}(x)$ function and its possible applications in
physics are presented. The actual numerical implementation in C++ consists of
Halley’s and Fritsch’s iterations with initial approximations based on branch-
point expansion, asymptotic series, rational fits, and continued-logarithm
recursion.
###### keywords:
Lambert W function , computational physics , numerical methods and algorithms
, C++
††journal: Computer Physics Communications
## Program summary
Program title: LambertW
Catalogue identifier: AENC_v1_0
Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENC_v1_0.html
Program obtainable from: CPC Program Library, Queen’s University, Belfast, N.
Ireland
Licensing provisions: GNU General Public License version 3
No. of lines in distributed program, including test data, etc.: 1335
No. of bytes in distributed program, including test data, etc.: 25 283
Distribution format: tar.gz
Programming language: C++ (with suitable wrappers it can be called from C,
Fortran etc.), the supplied command-line
012 utility is suitable for other scripting languages like sh, csh, awk, perl
etc.
Computer: All systems with a C++ compiler.
Operating system: All Unix flavors, Windows. It might work with others.
RAM: Small memory footprint, less than 1 MB
Classification: 1.1, 4.7, 11.3, 11.9.
Nature of problem: Find fast and accurate numerical implementation for the
Lambert W function.
Solution method: Halley’s and Fritsch’s iterations with initial approximations
based on branch-point expansion,
012 asymptotic series, rational fits, and continued logarithm recursion.
Additional comments: Distribution file contains the command-line utility
lambert-w. Doxygen comments, included in
012 the source files. Makefile.
Running time: The tests provided take only a few seconds to run.
## 1 Introduction
The Lambert W function is defined as the inverse function of the
$x\mapsto x\,e^{x}$ (1)
mapping and thus solves the
$y\,e^{y}=x$ (2)
equation. This solution is given in the form of the Lambert W function,
$y=\operatorname{W}(x),$ (3)
and according to Eq. (2) it satisfies the following relation,
$\operatorname{W}(x)\,e^{\operatorname{W}(x)}=x.$ (4)
Since the mapping in Eq. (1) is not injective, no unique inverse of the
$x\,e^{x}$ function exists (except at the minimum). As can be seen in Fig. 1,
the Lambert W function has two real branches with a branching point located at
$(-e^{-1},\,-1)$. The bottom branch, $\operatorname{W}_{-1}(x)$, is defined in
the interval $x\in[-e^{-1},\,0]$ and has a negative singularity for $x\to
0^{-}$. The upper branch $\operatorname{W}_{0}(x)$ is defined for
$x\in[-e^{-1},\,\infty]$.
The earliest mention of the problem of Eq. (2) is attributed to Euler [2].
However, Euler himself credited Lambert for his previous work in this subject,
Lambert’s transcendental equation [3]. The $\operatorname{W}(x)$ function
started to be named after Lambert only recently, in the last 20 years or so;
nevertheless, the first usage of the letter $\operatorname{W}$ appeared much
earlier [4].
Recently, the $\operatorname{W}(x)$ function amassed quite a following in the
mathematical community [5]. Its most faithful proponents are suggesting
elevating it among the present set of elementary functions, such as $\sin(x)$,
$\cos(x)$, $\exp(x)$, $\ln(x)$, etc. The main argument for doing so is the
fact that W is the root of the simplest exponential polynomial function given
in Eq. (2). We will acknowledge these efforts by strict usage of a _roman_
symbol W as its name.
Figure 1: The two branches of the Lambert W function,
$\operatorname{W}_{-1}(x)$ in blue and $\operatorname{W}_{0}(x)$ in red. The
branching point at $(-e^{-1},\,-1)$ is indicated with a green dash.
While the Lambert W function is called LambertW in the mathematics software
tool _Maple_ [6] and lambertw in the Matlab programming environment [7], in
the _Mathematica_ computer algebra framework this function [8] is implemented
under the name ProductLog (in the recent versions an alias LambertW is also
supported). In open source format the Lambert W function is available in the
special-functions part of the GNU Scientific Library (GSL) [9], unfortunately
implemented using only the slower Halley’s iteration (see discussion in
Section 6).
There are numerous, well documented applications of $\operatorname{W}(x)$,
certainly abundant in mathematics (like linear delay-differential equations
[10]), numerics [11], computer science [12] and engineering [13], but
surprisingly many also in physics, just to mention a few (without preference):
quantum mechanics (solutions for double-well Dirac-delta potentials [14]),
quantum statistics [15], general relativity (solutions to (1+1)-gravity
problem [16], inverse of Eddington-Finkelstein (tortoise) coordinates [17]),
statistical mechanics [18], fluid dynamics [19], optics [20] etc.
The main motivation for the implementation in this work comes from the
research in cosmic ray physics where it has been in use already for several
years [21] and new interesting applications are appearing frequently [22]. In
the next sections let us describe two new examples that arise from this field
of astrophysics.
### 1.1 Inverse of the Moyal function
Figure 2: _Top:_ The Moyal function $\operatorname{M}(x)$. _Bottom:_ A family
of one-parametric Gaisser-Hillas functions $g(x;\,x_{\text{max}})$ for
$x_{\text{max}}$ in the range from 1 to 10 with step 1.
The Moyal function and the related distribution was proposed as a good
approximation for the more complicated Landau distribution [23] used in
descriptions of energy loss of charged particles passing through matter [24].
The un-normalized Moyal function is defined as
$\operatorname{M}(x)=\exp\left[-\tfrac{1}{2}\left(x+e^{-x}\right)\right]$ (5)
and can be seen in Fig. 2 (top).
Its inverse can be written in terms of the two branches of the Lambert W
function,
$\operatorname{M}^{-1}_{\pm}(x)=\operatorname{W}_{0,-1}(-x^{2})-2\ln x.$ (6)
In experimental astrophysics the Moyal function is used for phenomenological
recovery of the saturated signals from photomultipliers [25], where the
largest parts of the saturated signals are obscured by the limited range of
the analog-to-digital converters.
### 1.2 Inverse of the Gaisser-Hillas function
In astrophysics the Gaisser-Hillas function is used to model the longitudinal
particle density in a cosmic-ray air shower [26]. We can show that the inverse
of the three-parametric Gaisser-Hillas function,
$G(X;\,X_{0},X_{\text{max}},\lambda)=\left[\frac{X-X_{0}}{X_{\text{max}}-X_{0}}\right]^{\frac{X_{\text{max}}-X_{0}}{\lambda}}\exp\left(\frac{X_{\text{max}}-X}{\lambda}\right),$
(7)
is intimately related to the Lambert W function. Using rescale substitutions,
$x=\frac{X-X_{0}}{\lambda}\qquad\text{and}\qquad
x_{\text{max}}=\frac{X_{\text{max}}-X_{0}}{\lambda},$ (8)
the Gaisser-Hillas function is modified into a function of one parameter only,
$g(x;\,x_{\text{max}})=\left[\frac{x}{x_{\text{max}}}\right]^{x_{\text{max}}}\exp(x_{\text{max}}-x).$
(9)
The family of one-parametric Gaisser-Hillas functions is shown in Fig. 2
(bottom). The problem of finding an inverse,
$g(x;\,x_{\text{max}})\equiv y$ (10)
for $0<y\leqslant 1$, can be rewritten into
$-\frac{x}{x_{\text{max}}}\exp\left(-\frac{x}{x_{\text{max}}}\right)=-y^{1/x_{\text{max}}}\,e^{-1}.$
(11)
According to the definition (2), the two (real) solutions for $x$ are obtained
from the two branches of the Lambert W function,
$x_{1,2}=-x_{\text{max}}\operatorname{W}_{0,-1}(-y^{1/x_{\text{max}}}\,e^{-1}).$
(12)
Note that the branch $-1$ or $0$ simply chooses the right or left side
relative to the maximum, respectively.
The Gaisser-Hillas function is intimately related to the gamma distribution
which was successfully used somewhat earlier [27] in an approximate
description of the mean longitudinal profile of the energy deposition in
electromagnetic cascades. It is trivial to show that the inverses of the gamma
and inverse-gamma distributions [28] are expressible in terms of the Lambert W
function as well.
## 2 Numerical methods
Before describing the actual implementation let us review some of the possible
numerical and analytical approaches to find $\operatorname{W}(x)$.
### 2.1 Recursion
For $x>0$ and $\operatorname{W}(x)>0$ we can take the natural logarithm of Eq.
(4) and rearrange it into
$\operatorname{W}(x)=\ln x-\ln\operatorname{W}(x).$ (13)
From here it is clear that an analytical expression for $\operatorname{W}(x)$
will exhibit a certain degree of self similarity. The $\operatorname{W}(x)$
function has multiple branches in the complex domain. Due to the $x>0$ and
$\operatorname{W}(x)>0$ conditions, Eq. (13) represents the positive part of
the $\operatorname{W}_{0}(x)$ principal branch and in this form it is suitable
for evaluation when $\operatorname{W}_{0}(x)>1$, i.e. when $x>e$.
Unrolling the self-similarity in Eq. (13) as a recursive relation, the
following curious expression for $\operatorname{W}_{0}(x)$ is obtained,
$\operatorname{W}_{0}(x)=\ln x-\ln(\ln x-\ln(\ln x-\,\ldots\,)),$ (14)
or in the shorthand of a continued logarithm,
$\operatorname{W}_{0}(x)=\ln\frac{x}{\ln\frac{x}{\ln\frac{x}{\cdots}}}.$ (15)
The above expression clearly has the form of successive approximations, the
final result given by the limit, if it exists.
For $x<0$ and $\operatorname{W}(x)<0$ we can multiply both sides of Eq. (4)
with $-1$, take logarithm, and rewrite it into a similar expansion for the
$\operatorname{W}_{-1}(x)$ branch,
$\operatorname{W}(x)=\ln(-x)-\ln(-\operatorname{W}(x)).$ (16)
Again, this leads to the similar recursion,
$\operatorname{W}_{-1}(x)=\ln(-x)-\ln(-(\ln(-x)-\ln(-\ldots))),$ (17)
or as a continued logarithm,
$\operatorname{W}_{-1}(x)=\ln\frac{-x}{-\ln\frac{-x}{-\ln\frac{-x}{\cdots}}}.$
(18)
For this continued logarithm we will use the symbol $R_{-1}^{[n]}(x)$ where
$n$ denotes the depth of the recursion in Eq. (18).
Starting from yet another rearrangement of Eq. (4),
$\operatorname{W}(x)=\frac{x}{\exp\operatorname{W}(x)},$ (19)
we can obtain a recursion relation for the $-e^{-1}<x<e$ part of the principal
branch $\operatorname{W}_{0}(x)<1$,
$\operatorname{W}_{0}(x)=\frac{x}{\exp\frac{x}{\exp\frac{x}{\ldots}}}.$ (20)
### 2.2 Halley’s iteration
We can apply Halley’s root-finding method [29] to derive an iteration scheme
for $f(y)=W(y)-x$ by writing the second-order Taylor series
$f(y)=f(y_{n})+f^{\prime}(y_{n})\,(y-y_{n})+\tfrac{1}{2}f^{\prime\prime}(y_{n})\,(y-y_{n})^{2}+\cdots$
(21)
Since root $y$ of $f(y)$ satisfies $f(y)=0$ we can approximate the left-hand
side of Eq. (21) with 0 and replace $y$ with $y_{n+1}$. Rewriting the obtained
result into
$y_{n+1}=y_{n}-\frac{f(y_{n})}{f^{\prime}(y_{n})+\tfrac{1}{2}f^{\prime\prime}(y_{n})\,(y_{n+1}-y_{n})}$
(22)
and using Newton’s method $y_{n+1}-y_{n}=-f(y_{n})/f^{\prime\prime}(y_{n})$ on
the last bracket, we arrive at the expression for Halley’s iteration for the
Lambert W function
$W_{n+1}=W_{n}+\frac{t_{n}}{t_{n}\,s_{n}-u_{n}},$ (23)
where
$\displaystyle t_{n}$ $\displaystyle=W_{n}\,e^{W_{n}}-x,$ (24) $\displaystyle
s_{n}$ $\displaystyle=\frac{W_{n}+2}{2(W_{n}+1)},$ (25) $\displaystyle u_{n}$
$\displaystyle=(W_{n}+1)\,e^{W_{n}}.$ (26)
This method is of the third order, i.e. having
$W_{n}=\operatorname{W}(x)+\mathcal{O}(\varepsilon)$ will produce
$W_{n+1}=\operatorname{W}(x)+\mathcal{O}(\varepsilon^{3})$. Supplying this
iteration with a sufficiently accurate first approximation of the order of
$\mathcal{O}(10^{-4})$ will thus give a machine-size floating point precision
$\mathcal{O}(10^{-16})$ in at least two iterations.
### 2.3 Fritsch’s iteration
For both branches of the Lambert W function a more efficient iteration scheme
exists [30],
$W_{n+1}=W_{n}(1+\varepsilon_{n}),$ (27)
where $\varepsilon_{n}$ is the relative difference of successive
approximations at iteration $n$,
$\varepsilon_{n}=\frac{W_{n+1}-W_{n}}{W_{n}}.$ (28)
The relative difference can be expressed as
$\varepsilon_{n}=\left(\frac{z_{n}}{1+W_{n}}\right)\left(\frac{q_{n}-z_{n}}{q_{n}-2z_{n}}\right),$
(29)
where
$\displaystyle z_{n}$ $\displaystyle=\ln\frac{x}{W_{n}}-W_{n},$ (30)
$\displaystyle q_{n}$
$\displaystyle=2(1+W_{n})\left(1+W_{n}+\tfrac{2}{3}z_{n}\right).$ (31)
The error term in this iteration is of a fourth order, i.e. with
$W_{n}=\operatorname{W}(x)+\mathcal{O}(\varepsilon_{n})$ we obtain
$W_{n+1}=\operatorname{W}(x)+\mathcal{O}(\varepsilon_{n}^{4})$.
Supplying this iteration with a sufficiently reasonable first guess, accurate
to the order of $\mathcal{O}(10^{-4})$, will therefore deliver machine-size
floating point precision $\mathcal{O}(10^{-16})$ in only one iteration and
excessive $\mathcal{O}(10^{-64})$ in two! The main goal now is to find
reliable first order approximations that can be fed into Fritsch’s iteration.
Due to the lively landscape of the Lambert W function properties, the
approximations will have to be found in all the particular ranges of the
function’s behavior.
## 3 Initial approximations
The following section deals with finding the appropriate initial
approximations in the whole definition range of the two branches of the
Lambert W function.
### 3.1 Branch-point expansion
Figure 3: Successive orders of the branch-point expansion for
$\operatorname{W}_{-1}(x)$ on the top and $\operatorname{W}_{0}(x)$ on the
bottom.
The inverse of the Lambert W function, $\operatorname{W}^{-1}(y)=y\,e^{y}$,
has two extrema located at $\operatorname{W}^{-1}(-1)=-e^{-1}$ and
$\operatorname{W}^{-1}(-\infty)=0^{-}$. Expanding $\operatorname{W}^{-1}(y)$
to the second order around the minimum at $y=-1$ we obtain
$\operatorname{W}^{-1}(y)\approx-\frac{1}{e}+\frac{(y+1)^{2}}{2e}.$ (32)
The inverse $\operatorname{W}^{-1}(y)$ is thus approximated in the lowest
order by a parabolic term implying that the Lambert W function will have
square-root behavior in the vicinity of the branch point $x=-e^{-1}$,
$\operatorname{W}_{-1,0}(x)\approx-1\mp\sqrt{2(1+ex)}.$ (33)
To obtain the additional terms in Eq. (33) we proceed by defining an inverse
function, centered and rescaled around the minimum, i.e.
$f(y)=2(e\operatorname{W}^{-1}(y-1)+1)$. Due to this centering and rescaling,
the Taylor series around $y=0$ becomes particularly neat,
$f(y)=y^{2}+\tfrac{2}{3}y^{3}+\tfrac{1}{4}y^{4}+\tfrac{1}{15}y^{5}+\cdots$
(34)
Using this expansion we can derive coefficients [31] of the corresponding
inverse function
$\displaystyle f^{-1}(z)$
$\displaystyle=1+\operatorname{W}\left(\frac{z-2}{2e}\right)=$ (35)
$\displaystyle=z^{1/2}-\tfrac{1}{3}z+\tfrac{11}{72}z^{3/2}-\tfrac{43}{540}z^{2}+\cdots$
(36)
From Eq. (35) we see that $z=2(1+ex)$. Using $p_{\pm}(x)=\pm\sqrt{2(1+ex)}$ as
an independent variable we can write this series expansion as
$\operatorname{W}_{-1,0}(x)\approx
B_{-1,0}^{[n]}(x)=\sum_{i=0}^{n}b_{i}p_{\mp}^{i}(x),$ (37)
where the lowest few coefficients $b_{i}$ are
$i$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
$b_{i}$ | -1 | 1 | $-\tfrac{1}{3}$ | $\tfrac{11}{72}$ | $-\tfrac{43}{540}$ | $\tfrac{769}{17\,280}$ | $-\tfrac{221}{8\,505}$ | $\tfrac{680\,863}{43\,545\,600}$
and more of them are given in the accompanying code (see Fig. 3 for succesive
orders of the series).
### 3.2 Asymptotic series
Another useful tool is the asymptotic expansion [32]. Using
$A(a,b)=a-b+\sum_{k}\sum_{m}C_{km}a^{-k-m-1}b^{m+1},$ (38)
with $C_{km}$ related to the Stirling numbers of the first kind, the Lambert W
function can be expressed as
$\operatorname{W}_{-1,0}(x)=A(\ln(\mp x),\,\ln(\mp\ln(\mp x))),$ (39)
with $a=\ln x$, $b=\ln\ln x$ for the $\operatorname{W}_{0}$ branch and
$a=\ln(-x)$, $b=\ln(-\ln(-x))$ for the $\operatorname{W}_{-1}$ branch. The
first few terms are
$\displaystyle A(a,b)$
$\displaystyle=a-b+\frac{b}{a}+\frac{b(-2+b)}{2a^{2}}+\frac{b(6-9b+2b^{2})}{6a^{3}}+$
$\displaystyle\,+\frac{b(-12+36b-22b^{2}+3b^{3})}{12a^{4}}+$ (40)
$\displaystyle\,+\frac{b(60-300b+350b^{2}-125b^{3}+12b^{4})}{60a^{5}}+\cdots$
### 3.3 Rational fits
A useful quick-and-dirty approach to the functional approximation is to
generate a large enough sample of data points
$\\{w_{i}\,e^{w_{i}},\,w_{i}\\}$, which evidently lie on the Lambert W
function. Within some appropriately chosen range of $w_{i}$ values, the points
are fitted with a rational approximation
$Q(x)=\frac{\sum_{i}a_{i}x^{i}}{\sum_{i}b_{i}x^{i}},$ (41)
varying the order of the polynomials in the nominator and denominator, and
choosing the one that has the lowest maximal absolute residual in a particular
interval of interest.
For the $\operatorname{W}_{0}(x)$ branch, the first set of equally-spaced
$w_{i}$ components was chosen in a range that produced $w_{i}\,e^{w_{i}}$
values in an interval $[-0.3,\,0]$. The optimal rational fit turned out to be
$Q_{0}(x)=x\frac{1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}{1+b_{1}x+b_{2}x^{2}+b_{3}x^{3}+b_{4}x^{4}}$
(42)
where the coefficients111The ’ symbol in coefficient values denotes truncation
in this presentation; the full machine-size sets of decimal places are given
in the accompanying code. for this first approximation $Q_{0}^{[1]}(x)$ are
$i$ | 1 | 2 | 3 | 4
---|---|---|---|---
$a_{i}$ | 5.931375’ | 11.392205’ | 07.338883’ | 0.653449’
$b_{i}$ | 6.931373’ | 16.823494’ | 16.430723’ | 5.115235’
For the second fit of the $\operatorname{W}_{0}(x)$ branch a $w_{i}$ range was
chosen so that the $w_{i}\,e^{w_{i}}$ values were produced in the interval
$[0.3,\,2e]$, giving rise to the second optimal rational fit $Q_{0}^{[2]}(x)$
of the same form as in Eq. (42) but with coefficients
$i$ | 1 | 2 | 3 | 4
---|---|---|---|---
$a_{i}$ | 2.445053’ | 1.343664’ | 0.148440’ | 0.000804’
$b_{i}$ | 3.444708’ | 3.292489’ | 0.916460’ | 0.053068’
For the $\operatorname{W}_{-1}(x)$ branch a single rational approximation of
the form
$Q_{-1}(x)=\frac{a_{0}+a_{1}x+a_{2}x^{2}}{1+b_{1}x+b_{2}x^{2}+b_{3}x^{3}+b_{4}x^{4}+b_{5}x^{5}}$
(43)
with the coefficients
$i$ | 0 | 1 | 2
---|---|---|---
$a_{i}$ | -7.814176’ | 253.888101’ | 657.949317’
$b_{i}$ | | -60.439587’ | 99.985670’
$i$ | 3 | 4 | 5
$b_{i}$ | 682.607399’ | 962.178439’ | 1477.934128’
is enough.
## 4 Implementation
To quantify the accuracy of a particular approximation
$\widetilde{\operatorname{W}}(x)$ of the Lambert function
$\operatorname{W}(x)$ we can introduce a quantity $\Delta(x)$ defined as
$\Delta(x)=\log_{10}|\operatorname{W}(x)|-\log_{10}|\widetilde{\operatorname{W}}(x)-\operatorname{W}(x)|,$
(44)
so that it gives us a number of correct decimal places the approximation
$\widetilde{\operatorname{W}}(x)$ is producing for some value of the parameter
$x$.
In Fig. 4 all approximations for the $\operatorname{W}_{0}(x)$ mentioned above
are shown in the linear interval $[-e^{-1},\,0.3]$ on the left and the
logarithmic interval $[0.3,\,10^{5}]$ on the right. For each of the
approximations an optimal interval is chosen so that the number of accurate
decimal places is maximized over the whole definition range. For the
$\operatorname{W}_{0}(x)$ branch the resulting piecewise approximation
$\widetilde{\operatorname{W}}_{0}(x)=\begin{cases}B_{0}^{[9]}(x)&;\,-e^{-1}\leqslant
x<a\\\ Q_{0}^{[1]}(x)&;\,a\leqslant x<b\\\ Q_{0}^{[2]}(x)&;\,b\leqslant x<c\\\
A_{0}(x)&;\,c\leqslant x<\infty\end{cases}$ (45)
with $a=-0.323581^{\prime}$, $b=0.145469^{\prime}$, and $c=8.706658^{\prime}$,
is accurate in the definition range $[-e^{-1},\,7]$ to at least 5 decimal
places and to at least 3 decimal places in the whole definition range
$[-e^{-1},\,\infty]$. Note that $B_{0}^{[9]}(x)$ comes from Eq. (37),
$Q_{0}^{[1]}(x)$ and $Q_{0}^{[2]}(x)$ are from Eq. (42), and $A_{0}(x)$ is
from Eq. (40).
Figure 4: Combining different approximations of $\operatorname{W}_{0}(x)$ into
a final piecewise function. The number of accurate decimal places $\Delta(x)$
is shown for two ranges, linear interval $[-e^{-1},\,0.3]$ (top) and
logarithmic interval $[0.3,\,10^{5}]$ (bottom). The approximations are branch-
point expansion $B_{0}^{[9]}(x)$ from Eq. (37) (blue), rational fits
$Q_{0}^{[1]}(x)$ and $Q_{0}^{[2]}(x)$ from Eq. (42) in black and red,
respectively, and asymptotic series $A_{0}(x)$ from Eq. (40) (green).
Figure 5: Final values of the combined approximation
$\widetilde{\operatorname{W}}_{0}(x)$ (black) from Fig. 4 after one step of
Halley’s iteration (red) and one step of Fritsch’s iteration (blue).
The accuracy of the final piecewise approximation
$\widetilde{\operatorname{W}}_{0}(x)$ is shown in Fig. 5 (black line). Using
this approximation, a single step of Halley’s iteration (red) and Fritsch’s
iteration (blue) is performed and the resulting number of accurate decimal
places is shown. As can be seen from the plots, both iterations produce
machine-size accurate floating point numbers in the whole definition interval
except for the interval between 6.5 and 190 where the Halley’s method requires
another step of the iteration. For that reason we have decided to use only
(one step of) Fritsch’s iteration in the C++ implementation of the Lambert W
function.
Figure 6: _Top:_ Approximations of the $\operatorname{W}_{-1}(x)$ branch. The
branch-point expansion $B_{-1}^{[9]}(x)$ is shown in blue, the rational
approximation $Q_{-1}(x)$ in black, and the logarithmic recursion
$R_{-1}^{[9]}$ in red. _Bottom:_ The combined approximation is accurate to at
least 5 decimal places in the whole definition range. The results after
applying one step of Halley’s iteration are shown in red and after one step of
Fritsch’s iteration in blue.
In Fig. 6 (top) the same procedure is shown for the $\operatorname{W}_{-1}(x)$
branch. The final approximation
$\widetilde{\operatorname{W}}_{-1}(x)=\begin{cases}B_{-1}^{[9]}(x)&;\,-e^{-1}\leqslant
x<a\\\ Q_{-1}(x)&;\,a\leqslant x<b\\\ R_{-1}^{[9]}(x)&;\,b\leqslant
x<0\end{cases}$ (46)
with $a=-0.302985^{\prime}$ and $b=-0.051012^{\prime}$, is accurate to at
least 5 decimal places in the whole definition range $[-e^{-1},\,0]$. Note
that $B_{-1}^{[9]}(x)$ is taken from Eq. (37), $Q_{-1}(x)$ is from Eq. (43),
and $R_{-1}^{[9]}(x)$ is from Eq. (18).
In Fig. 6 (bottom) the combined approximation
$\widetilde{\operatorname{W}}_{-1}(x)$ is shown (black line). The values after
one step of Halley’s iteration are shown in red and after one step of
Fritsch’s iteration in blue. Similarly as for the previous branch, Fritsch’s
iteration turns out to be superior, yielding machine-size accurate results in
the whole definition range, while Halley’s iteration is accurate to at least
13 decimal places.
## 5 Source availability, installation and usage
The most recent version of the sources of this implementation with some
additional material and examples are available at
http://www.ung.si/~darko/LambertW/ and are released under the GPL license.
Apart from the special functions in GSL [9], this is the only freely available
implementation of the Lambert W function in C++ and to the best of our
knowledge the only implementation using the superior Fritsch’s version of the
iteration.
The supplied C++ code implements the following set of functions222Which can be
found in the files LambertW.h and LambertW.cc.
* 1.
double LambertWApproximation<$b$>(double x);
* 2.
double LambertW<$b$>(double x);
* 3.
double LambertW(int branch, double x);
where $b$ in the first two functions should be replaced with the compile-time
branch integer value, e.g. LambertW<-1>(x) or LambertW<0>(x). Apart from the
slightly increased efficiency, the main reason for implementing the first two
functions with the branch $b$ as a compile time parameter is to force the
users to think about the two possible solutions to the problem in Eq. (2).
Just as for solutions to the quadratic equation where the $\pm$ sign has to be
chosen based on the desired solution, the Lambert W function offers two
possibilities that need careful consideration.
The initial approximations $\widetilde{\operatorname{W}}_{b}(x)$ from Eqs.
(46) and (45), that are used to kick-start the iterations, are also directly
available as LambertWApproximation<$b$>(x), as they might be useful in
applications for which it is sufficient to have a limited number of accurate
decimal places (see the discussion above).
The supplied code does not need any kind of special installation procedures.
In the case that your analysis needs an evaluation of the Lambert W function,
the two source files, LambertW.h and LambertW.cc, should be simply bundled
with your project structure and compiled with a suitable C++ compiler.
The source distribution also includes a command-line utility implemented by
the lambert-w.cc source file. The included Makefile can, with the request make
lambert-w, build an executable command. It can be invoked through a shell as
./lambert-w [branch] x, taking an optional branch number (0 by default) and a
parameter x. The output of the command is equivalent to the
$\operatorname{W}_{\texttt{branch}}(\texttt{x})$ return value and can thus be
simply used in shell scripts (sh, bash, or csh) or other programming languages
with easy access to shell invocations (awk, perl etc.).
Also included in the distribution is a test suite which can perform a
correctness check on your build by comparing obtained and expected return
values of the Lambert W function on your system. It is invoked by the command
make tests. Any potential discrepancies larger than the double machine
precision ($\gtrsim 10^{-14}$) will be reported in the output.
## 6 Timing
Figure 7: Execution speedup $t^{\prime}_{\text{gsl}}/t^{\prime}$, relative to
the GSL implementation [9] for the $\operatorname{W}_{0}(x)$ branch (red) and
the $\operatorname{W}_{-1}(x)$ branch (blue). For $x>2$ the ratio slowly
decreases and is $\sim 2$ for $x>8$. Time of the respective driver loops and
function calls was subtracted in order to measure only differences between the
pure numerical parts of the two implementations (see text for details).
We have decided to compare the execution time of our code relative to the GNU
GSL library (implemented in the C language) since comparisons to interpreted
code (like Maple, Matlab or Mathematica) would, besides common availability
problems, not be fair in terms of speed.
In Fig. 7 relative speedup, $t_{\text{gsl}}/t$, is shown as a function of the
Lambert W parameter $x$. Timing accuracy of several % was achieved by running
3 000 000 function calls in a loop, taking special care that the compiler did
not optimize code away by slightly modifying $x$ on each call and then
gathering all results into a summed variable reported at the end. For each of
the two implementations an identical code was also run with the Lambert W
function call replaced with a simple identity function (just returning its
input parameter) in order to estimate the overhead of the surrounding timing
code. This $t_{\text{overhead}}$ is then consequently subtracted from the time
of the Lambert W runs $t$, giving an approximation to the time taken by the
pure function call, $t^{\prime}=t-t_{\text{overhead}}$. The ratio
$t^{\prime}_{\text{gsl}}/t^{\prime}$ is then plotted in Fig. 7.
As can be clearly seen from Fig. 7, our implementation is at least $2\times$
faster than GSL for a broad range of input parameters $x$, but the largest
efficiency gains (up to $5\times$) are observed in the ranges where rational
fits $Q_{b}$ from Eqs. (45) and (46) are used. Although Fritsch’s iteration is
in general more complex than Halley’s, at the end it pays off, yielding
machine-size accuracy with a single step where Halley’s might need more, also
due to poor initial approximations used in GSL. GSL performs better only in
the small regions where branch-point and asymptotic expansions are used
without the consequent Halley’s iteration refinements. The comparisons were
made on the Ubuntu 12.04 x86_64 Linux operating system running on a 2.2 GHz
AMD Opteron 275 processor, using the GCC 4.6.3 compiler with optimization
option -O2.
## 7 Conclusions
We have shown that Fritsch’s iteration scheme coupled with accurate initial
approximations can deliver significant efficiency gains in the numerical
evaluation of the real branches of the Lambert W function. The open-source
release of our C++ implementation is suitable for inclusion in various
analysis packages used in all fields of physics.
## Acknowledgments
The author wishes to thank Matej Horvat, Michael Unger, Martin O’Loughlin, and
Amir Malekpour for useful discussions and suggestions. This work was partially
supported by the Slovenian Research Agency.
## References
* [1]
* [2] L. Euler, Acta Acad. Scient. Petropol. 2 (1783) 29–51.
* [3] J.H. Lambert, Acta Helvitica 3 (1758) 128–168.
* [4] G. Pólya and G. Szegö, _Aufgaben und Lehrsätze aus der Analysis_ , J. Springer, Berlin, 1925.
* [5] http://www.orcca.on.ca/LambertW/.
* [6] R.M. Corless, G.H. Gonnet, D.E.G. Hare, and D.J. Jeffrey, Maple Technical Newsletter 9 (1993) 12.
* [7] http://www.mathworks.com/help/toolbox/symbolic/
lambertw.html; some open source versions for Matlab: P. Getreuer and D.
Clamond, http://www.getreuer.info/home/
lambertw; N.N. Schraudolph and D. Ross,
http://www.cs.toronto.edu/~dross/code/LambertW.m.
* [8] E.W. Weisstein, _Lambert W-Function_ , MathWorld Wolfram Web Resource, http://mathworld.wolfram.com/LambertW-Function.html.
* [9] G. Jungman and K. Briggs, GNU Scientific Library, gsl-1.15, specfunc/lambert.c, http://www.gnu.org/software/gsl/.
* [10] F.M. Asl and A.G. Ulsoy, J. Dyn. Sys. Meas. Control 125 (2003) 215; S. Yi, P.W. Nelson, and A.G. Ulsoy, IEEE Trans. Autom. Control 53 (2008) 854–860.
* [11] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, Adv. Comput. Math. 5 (1996) 329; R.M. Corless, D.J. Jeffrey, and D.E. Knuth, 1997, _A Sequence of Series for the Lambert Function_ , in: Proc. ISSAC ’97, Maui, (1997) 197–204; R.M. Corless and D.J. Jeffrey, _The Wright $\omega$ function_, in: J. Calmet, B. Benhamou, O. Caprotti, L. Henocque, V. Sorge (Eds.), _Artificial Intelligence, Automated Reasoning, and Symbolic Computation_ , in: Proc. AISC-Calculemus 2002, Springer (2002) 76–89.
* [12] J. Bustos-Jimenez, N. Bersano, S.E. Schaeffer, J.M. Piquer, A. Iosup, and A. Ciuffoletti, in _Grid Computing: Achievements and Prospects_ , eds. S. Gorlatch, P. Fragopoulou, and T. Priol, Springer (2008); doi:10.1007/978-0-387-09457-1_6.
* [13] S. Yi, P.W. Nelson, and A.G. Ulsoy, Math. Biosci. Engrg. 4 (2007) 355–368.
* [14] T.C. Scott, M. Aubert-Frécon, and J. Grotendorst, Chem. Phys. 324 (2006) 323–338; T.C. Scott, A. Lüchow, D. Bressanini, and J.D. Morgan III, Phys. Rev. A 75 (2007) 060101R; T.C. Scott, J.F. Babb, A. Dalgamo, and J.D. Morgan III, Chem. Phys. Lett. 203 (1993) 175–183.
* [15] S.R. Valluri, M. Gil, D.J. Jeffrey, and S. Basu, J. Math. Phys. 50 (2009) 102103.
* [16] T.C. Scott, R. Mann, and R.E. Martinez II, Appl. Algebra Engrg. Comm. Comput. 17 (2006) 41–47.
* [17] A.S. Eddington, Nature 113 (1924) 192; T. Regge and J.A. Wheeler, Phys. Rev. 108 (1957) 1063–1069.
* [18] J.-M. Caillol, J. Phys. A 36 (2003) 10431–10442.
* [19] S.P. Pudasaini, Phys. Fluids 23 (2011) 043301.
* [20] O. Steinvall, Appl. Optics 48 (2009) B1–B7.
* [21] S. Argirò _et al._ , Nucl. Instr. and Meth. A 580 (2007) 1485.
* [22] K.-H. Kampert and M. Unger, Astropart. Phys. 35 (2012) 660–678; arXiv:1201.0018.
* [23] J.E. Moyal, Phil. Mag. 46 (1955) 263.
* [24] L.D. Landau, J. Phys. USSR 8 (1944) 201–205.
* [25] I.C. Mariş, M. Roth, and T. Schmidt, _A Phenomenological Method to Recover the Signal from Saturated Stations_ , P. Auger Collaboration internal note GAP-2006-012.
* [26] T.K. Gaisser and A.M. Hillas, Proc. of the 15th Internation Cosmic-Ray Conference, Plavdiv, 8 (1977) 353.
* [27] E. Longo and I. Sestili, Nucl. Instrum. Methods 128 (1975) 283\.
* [28] C. Walck, _Hand-book on statistical distributions for experimentalists_ , Stockholms Universitet, Internal Report SUF-PFY/96-01, 10 September 2007, pp. 69–74.
* [29] T.R. Scavo and J.B. Thoo, Amer. Math. Monthly 102 (1995) 417.
* [30] F.N. Fritsch, R.E. Shafer, and W.P. Crowley, Commun. ACM 16 (1973) 123.
* [31] P.M. Morse and H. Feshbach, _Methods of Theoretical Physics, Part I_ , McGraw-Hill, New York (1953) 411.
* [32] N.G. de Bruijn, _Asymptotic Methods in Analysis_ , New York, Dover (1981) 27.
|
arxiv-papers
| 2012-08-31T21:07:48 |
2024-09-04T02:49:34.780900
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Darko Veberic",
"submitter": "Darko Veberic",
"url": "https://arxiv.org/abs/1209.0735"
}
|
1209.0788
|
# Signatures of unconventional pairing in near-vortex electronic structure of
LiFeAs
Kyungmin Lee, Mark H. Fischer, Eun-Ah Kim Department of Physics, Cornell
University, Ithaca, New York 14853, USA [email protected]
###### Abstract
A major question in Fe-based superconductors remains the structure of the
pairing, in particular whether it is of unconventional nature. The electronic
structure near a vortex can serve as a platform for phase-sensitive
measurements to answer this question. By solving the Bogoliubov-de Gennes
equations for LiFeAs, we calculate the energy-dependent local electronic
structure near a vortex for different nodeless gap-structure possibilities. At
low energies, the local density of states (LDOS) around a vortex is determined
by the normal-state electronic structure. At energies closer to the gap value,
however, the LDOS can distinguish an anisotropic $s$-wave gap from a
conventional isotropic $s$-wave gap. We show within our self-consistent
calculation that in addition, the local gap profile differs between a
conventional and an unconventional pairing. We explain this through admixing
of a secondary order parameter within Ginzburg-Landau theory. In-field
scanning tunneling spectroscopy near a vortex can therefore be used as a real-
space probe of the gap structure.
###### pacs:
74.25.Ha, 74.55.+v, 74.70.Xa
## 1 Introduction
The gap structure in the Fe-based superconductors and its possible
unconventional nature is still a key issue in the field four years after their
discovery. In most compounds, the pairing is believed to be of the so-called
$s^{\pm}$ type, for which the order parameter changes sign between the
electron-like and the hole-like Fermi surfaces [1, 2]. Some experimental
results are consistent with this prediction[3, 4, 5, 6, 7]. However, a major
difficulty in distinguishing such an unconventional pairing state from a
trivial $s$-wave gap is that both states are nodeless and transform trivially
under all the symmetry operations of the material’s point group. As the
experimental probes that are usually used to distinguish various gap
structures, such as phase-sensitive probes, are not Fermi pocket specific, an
unambiguous evidence of the unconventional $s^{\pm}$ pairing remains evasive.
One route to accessing phase information using a phase-insensitive probe would
be through vortex bound states, as a vortex introduces a spatial texture to
the superconducting order parameter. Advancements in in-field scanning
tunneling spectroscopy (STS) have enabled the study of vortex bound states.
Indeed, a recent STS experiment on LiFeAs under a magnetic field has shown an
intriguing energy dependence in the spatial distribution of the local density
of states (LDOS) near a vortex[8]. The remaining question is whether the
observed LDOS distribution near vortex can be instrumental in selecting one of
the proposed gap structures: $s^{\pm}$-wave [1], $s^{++}$-wave [9, 10], and
(spin-triplet) $p$-wave [11, 12]. At zero bias, the LDOS shows a four-fold
star shape with high-intensity ‘rays’ along the Fe-As direction. Similar
features in NbSe2 [13] were interpreted as a sign of gap minima along this
direction. However, a quasi-classical analysis by Wang et al.[14] pointed out
that the normal-state band structure of LiFeAs – namely a highly anisotropic
hole pocket around the $\Gamma$ point – could be producing these rays
irrespective of gap structure. By contrast, little attention has been given to
the high energy LDOS distribution observed in Ref. [8]: hot spots appearing at
the intersection of split rays.
Motivated by these observations, we present a study of the near-vortex
electronic structure and signatures of unconventional pairing therein within
the Bogoliubov-de Gennes (BdG) framework. By (non-self-consistently) imposing
a gap structure and solving the BdG Hamiltonian, we first show that the
isotropic $s$-wave and $s^{\pm}$-wave pairing result in different spatial
distributions of the LDOS at energies approaching the gap value. In
particular, we find $s^{\pm}$-wave pairing to yield the observed hot spots.
Then we solve the BdG equations self-consistently, and based on our results
propose detecting the spatial distribution of the gap around a vortex for a
more direct evidence of unconventional $s^{\pm}$-wave pairing. A vortex not
only suppresses the order-parameter amplitude at its core and introduces a
singular point in space around which the phase of the order parameter winds,
but it also induces a secondary order parameter in its vicinity [15, 16, 17,
18, 19]. Due to the induced secondary order parameter near the vortex, the gap
recovery should show a strong angular dependence. Detection of such anisotropy
will be an unambiguous evidence of unconventional pairing.
The remainder of this paper is organized as follows: In sections 2 and 3, we
introduce the microscopic model and describe the Bogoliubov-de Gennes
calculations, respectively. In section 4, we present the results of the BdG
calculations and discuss them within Ginzburg-Landau theory. In section 5, we
summarize our findings and remark on future directions. Throughout the paper
we focus on the large hole pocket and study the single band problem. However,
we also present results from non-self-consistent BdG calculations on a five-
band model in section 4, which show good agreement with observations from
single-band model calculations in the energy range of our interest.
## 2 Model
Figure 1: (a) Comparison of two tight-binding models for LiFeAs used in this
paper in the 1-Fe Brillouin zone. The dashed lines indicate the Fermi surfaces
of the five-band model from Ref. [20]. For the most part of this work, we
focus on the $\gamma$ band that is around the $\Gamma$ point, whose Fermi
surface is shown as a solid line. (b) Sketch of the three gap functions with
$s$-, $s^{\pm}$-, and $d_{xy}$-wave momentum structure around the
$\gamma$-band Fermi surface.
We describe LiFeAs in the superconducting state with the (mean-field) BdG
Hamiltonian
$\displaystyle\mathcal{H}^{\rm BdG}$
$\displaystyle=\sum_{ij}\Psi_{i}^{\dagger}\begin{pmatrix}-t_{ij}&\Delta_{ij}\\\
\Delta_{ij}^{*}&t_{ij}^{*}\end{pmatrix}\Psi_{j}.$ (1)
Here,
$\Psi_{i}\equiv(c^{\phantom{\dagger}}_{i\uparrow},c^{\dagger}_{i\downarrow})^{T}$
is a Nambu spinor, and $c_{is}$ ($c^{{\dagger}}_{is}$) annihilates (creates)
an electron at lattice site $i$ with spin $s$ within a single-band model for
the large hole pocket around the $\Gamma$ point: the so-called $\gamma$ band.
However, Eq. (1) can easily be generalized for a multi-band model. In this
paper, we focus on the single-band model for the most part since the
superconducting gap is the smallest on the $\gamma$ band[21, 22] and hence we
expect low energy physics to be dominated by this band. Moreover, this band
mainly stems from the (in-plane) $d_{xy}$ orbitals, and thus shows little
$k_{z}$ dependence[23]. It is therefore a natural choice for LiFeAs. Note that
previous BdG calculations on different Fe-pnictides focused on two-band models
for the $d_{xz}$ / $d_{yz}$ orbitals [24, 25, 26, 27, 28, 29]. Our choice of
the hopping matrix $t_{ij}$ is guided by the experimental observations on the
$\gamma$ pocket[30, 21, 22] to be $t=-0.25$eV for nearest-neighbor hopping,
$t^{\prime}=0.082$eV for next-nearest-neighbor hopping, and
$t_{ii}=\mu=0.57$eV for the chemical potential. Figure 1(a) shows the
resulting Fermi surface in solid red line. Though we stay within this single-
band model for the self-consistent BdG studies, we have also used a five-band
model for the non-self-consistent calculation with tight-binding parameters
from Ref. [20] to test the validity of focussing on the $\gamma$ band for the
energy range of our interest (see section 4.2). Figure 1(a) shows the Fermi
surface of the five-band model in dashed lines.
The $\Delta_{ij}$ are the (bond) gap functions. For a self-consistent solution
of $\mathcal{H}^{\rm BdG}$, we require the gap functions to satisfy
$\displaystyle\Delta_{ij}=$ $\displaystyle\frac{1}{2}V_{ij}\left\langle
c_{i\downarrow}c_{j\uparrow}+c_{j\downarrow}c_{i\uparrow}\right\rangle,$ (2)
where $V_{ij}<0$ is the attractive interaction between sites $i$ and $j$ in
the singlet channel, and $\langle\cdot\rangle$ denotes the thermal expectation
value. Restricting the interaction $V_{ij}$ to a specific form constrains the
momentum structure of the gap function, since $\Delta_{ij}\neq 0$ only if
$V_{ij}\neq 0$. In the uniform case, an on-site attraction
$V_{ij}=U\delta_{ij}$ leads to a spin-singlet $s$-wave gap
$\Delta(\mathbf{k})=\Delta_{s}^{0}$, while a next-nearest-neighbor (NNN)
attraction $V_{ij}=V^{\prime}\delta_{\langle\\!\langle i,j\rangle\\!\rangle}$
allows for the singlet gap functions of $s^{\pm}$ form,
$\Delta(\mathbf{k})=4\Delta_{s^{\pm}}^{0}\cos k_{x}\cos k_{y}$, and $d_{xy}$
form, $\Delta(\mathbf{k})=4\Delta_{d_{xy}}^{0}\sin k_{x}\sin k_{y}$. Figure
1(b) shows sketches of these gap functions. We restrict our calculations in
the following to these ‘pure’ gap structures. Even though the true gap
function is a (symmetry-allowed) mixture of such gap functions, the dominant
channel (on-site or NNN interactions) will determine whether an $s^{\pm}$\- or
an $s^{++}$-wave gap is realized in the presence of the electron pockets.
For the non-self-consistent BdG study, the vortex will be imposed through the
gap-function configuration of
$\displaystyle\Delta_{ij}=$
$\displaystyle\Delta^{0}\tanh(|\mathbf{r}_{ij}|/\xi)e^{i\theta_{ij}},$ (3)
where the vector $\mathbf{r}_{ij}$ points to the midpoint of sites $i$ and
$j$, and $\theta_{ij}$ is its azimuthal angle measured from the Fe-Fe
direction. This corresponds to a single vortex located at the origin
suppressing locally the order-parameter amplitude. In addition, the order-
parameter phase winds around the vortex core.
For the self-consistent BdG study, we induce the vortices by applying a
magnetic field $H\hat{\mathbf{z}}$. Assuming minimal coupling between an
electron and the field, the hopping between sites $i$ and $j$ acquires a
Peierls phase
$\displaystyle t_{ij}\;\longrightarrow\;$ $\displaystyle
t_{ij}e^{i\varphi(\mathbf{r}_{i},\mathbf{r}_{j})},$
$\displaystyle\varphi(\mathbf{r}_{i},\mathbf{r}_{j})\equiv$
$\displaystyle-\frac{\pi}{\Phi_{0}}\int_{\mathbf{r}_{j}}^{\mathbf{r}_{i}}\mathbf{A}(\mathbf{r})\cdot
d\mathbf{r},$ (4)
where $\Phi_{0}=h/2e$ is the magnetic fluxoid and $\mathbf{r}_{i}$ is the
vector pointing to the site $i$. We assume a uniform magnetic field $H$ and
write the vector potential in the Landau gauge
$\mathbf{A}(\mathbf{r})=-Hy\hat{\mathbf{x}}$. From the self-consistent
solution $\Delta_{ij}$, we can define local gap order parameters of different
symmetries. For an on-site interaction, the local $s$-wave order parameter is
defined as $\Delta_{s}(\mathbf{r})=\Delta_{\mathbf{r},\mathbf{r}}$. Note that
from here on, we use $\mathbf{r}$ without any site index to denote both a
lattice site and the vector pointing to it in units of the lattice constant
$a_{0}$. With NNN interaction, we define local order parameters of $s^{\pm}$
form
$\displaystyle\Delta_{s^{\pm}}(\mathbf{r})=\frac{1}{4}[\widetilde{\Delta}_{\mathbf{r}+(1,1),\mathbf{r}}+\widetilde{\Delta}_{\mathbf{r}+(1,-1),\mathbf{r}}+\widetilde{\Delta}_{\mathbf{r}+(-1,-1),\mathbf{r}}+\widetilde{\Delta}_{\mathbf{r}+(-1,1),\mathbf{r}}]$
(5)
and $d_{xy}$ form
$\displaystyle\Delta_{d_{xy}}(\mathbf{r})=\frac{1}{4}[\widetilde{\Delta}_{\mathbf{r}+(1,1),\mathbf{r}}-\widetilde{\Delta}_{\mathbf{r}+(1,-1),\mathbf{r}}+\widetilde{\Delta}_{\mathbf{r}+(-1,-1),\mathbf{r}}-\widetilde{\Delta}_{\mathbf{r}+(-1,1),\mathbf{r}}],$
(6)
where
$\widetilde{\Delta}_{\mathbf{r}\mathbf{r}^{\prime}}\equiv\Delta_{\mathbf{r}\mathbf{r}^{\prime}}\exp[-i\varphi(\mathbf{r},\mathbf{r}^{\prime})]$
ensures that order parameters of different symmetries do not mix under
magnetic translations. Note that for the uniform case,
$\Delta_{s}(\mathbf{r})=\Delta_{s}^{0}$,
$\Delta_{s^{\pm}}(\mathbf{r})=\Delta_{s^{\pm}}^{0}$, and
$\Delta_{d_{xy}}(\mathbf{r})=\Delta_{d_{xy}}^{0}$ as defined above.
## 3 Method
In this section, we elaborate on our two approaches to solve the BdG equations
and obtain the LDOS near a vortex. For both, diagonalizing the Hamiltonian
$\mathcal{H}^{\mathrm{BdG}}$ in Eq. (1) for a system of size $(N_{x},N_{y})$
is computationally the most expensive part.
### 3.1 Non-Self-Consistent Approach
For the non-self-consistent calculation, we impose a gap function in the form
given by Eq. (3) and find the low lying eigenvalues and eigenstates of
$\mathcal{H}^{\mathrm{BdG}}$ using the Lanczos algorithm111We suppress low
energy states from forming at the boundary by imposing an on-site potential of
10 eV to the sites at the boundary.. The LDOS can be calculated from the
eigenenergies $E^{n}$ and eigenstates $[u^{n}(\mathbf{r}),v^{n}(\mathbf{r})]$
as
$\displaystyle
N(\mathbf{r},E)=\\!\sum_{n}|u^{n}(\mathbf{r})|^{2}\delta(E-E^{n})+|v^{n}(\mathbf{r})|^{2}\delta(E+E^{n}).$
(7)
Since we are not interested in the absolute value of the LDOS but rather in
the spatial profile at a given energy, we normalize the LDOS such that for a
given energy $E$, the maximum value of $N(\mathbf{r},E)$ is unity.
### 3.2 Self-Consistent Approach
For the self-consistent calculation, we assume initial gap functions and use
the eigenvalues and eigenvectors of Eq. (1) to calculate the gap functions
given by Eq. (2). We proceed iteratively until self-consistency is achieved.
In diagonalizing $\mathcal{H}^{\mathrm{BdG}}$, we can no longer make use of
the crystal momentum basis to simplify the problem since the Peierls phase
factor prevents the kinetic part of the Hamiltonian from commuting with the
ordinary lattice translation operator $T_{\mathbf{R}}$. However, the kinetic
part commutes with the magnetic translation operator
$\displaystyle\hat{T}_{\mathbf{R}}$ $\displaystyle\equiv
e^{-i\frac{\pi}{\Phi_{0}}\mathbf{A}(\mathbf{R})\cdot\mathbf{r}}T_{\mathbf{R}}$
(8)
for a magnetic lattice vector $\mathbf{R}$ whose unit cell contains two
magnetic fluxoids.
The pairing term in general does not commute with $\hat{T}_{\mathbf{R}}$.
Nevertheless, when vortices form a lattice, $\hat{T}_{\mathbf{R}}$ commutes
with the pairing term when $\mathbf{R}$ is a vector of a vortex sublattice
containing every other vortex. Since we focus on the electronic structure near
a single vortex, we expect the shape of the vortex lattice to have little
influence on our results. Therefore, we make an arbitrary choice for its
primitive vectors to be $L_{x}\hat{\mathbf{x}}$ and $L_{y}\hat{\mathbf{y}}$,
such that $\mathbf{R}$ forms a rectangular lattice
$\mathbf{R}=(m_{x}L_{x},m_{y}L_{y})$, where $m_{\alpha}=0\cdots M_{\alpha}-1$
and $M_{\alpha}\equiv N_{\alpha}/L_{\alpha}$222This choice yields an oblique
vortex lattice, since there are two vortices in each (rectangular) magnetic
unit cell, trying to form a triangular vortex lattice as a self-consistent
solution.. Note that periodic boundary conditions in the Landau gauge
$\mathbf{A}(\mathbf{r})=-Hy\hat{\mathbf{x}}$ require the total magnetic flux
through the system to be an integer multiple of $2\Phi_{0}N_{x}$. In addition,
one magnetic unit cell contains a magnetic flux of $2\Phi_{0}$, i.e.
$H=2\Phi_{0}/L_{x}L_{y}$. We satisfy these two requirements by choosing
$M_{x}=L_{y},M_{y}=L_{x}$.
Working with the magnetic Bloch states
$\displaystyle\Psi_{\mathbf{k}}(\mathbf{r})$
$\displaystyle=\sum_{\mathbf{R}}e^{-i\mathbf{k}\cdot\mathbf{R}}\;\hat{T}_{\mathbf{R}}\Psi(\mathbf{r})\hat{T}_{\mathbf{R}}^{-1}$
(9)
allows us to block diagonalize the Hamiltonian
$\displaystyle\mathcal{H}^{BdG}$
$\displaystyle=\frac{1}{M_{x}M_{y}}\sum_{\mathbf{k}}\sum_{\mathbf{r},\mathbf{r}^{\prime}}\Psi_{\mathbf{k}}^{\dagger}(\mathbf{r})H_{\mathbf{k}}(\mathbf{r},\mathbf{r}^{\prime})\Psi_{\mathbf{k}}(\mathbf{r}^{\prime}).$
(10)
The indices $\mathbf{k}$ and $\mathbf{r}$ from here on are defined in the
magnetic Brillouin zone and magnetic unit cell, respectively, that is
$\displaystyle\mathbf{k}$
$\displaystyle=\left(2\pi\frac{m_{x}}{L_{x}M_{x}},2\pi\frac{m_{y}}{L_{y}M_{y}}\right),$
$\displaystyle m_{\alpha}$ $\displaystyle=0\cdots M_{\alpha}-1,$ (11a)
$\displaystyle\mathbf{r}$ $\displaystyle=\left(\ell_{x},\ell_{y}\right),$
$\displaystyle\ell_{\alpha}$ $\displaystyle=0\cdots L_{\alpha}-1.$ (11b)
By diagonalizing the matrices $H_{\mathbf{k}}$ of dimension $2L_{x}L_{y}\times
2L_{x}L_{y}$ in Eq. (10), we can compute the eigenstates and eigenenergies of
$\mathcal{H}^{BdG}$. These are then used to calculate $\Delta_{ij}$ with Eq.
(2) closing the self-consistency cycle. Finally, we use the self-consistent
solution $\Delta_{ij}$ to calculate the local order parameters of $s$-,
$s^{\pm}$\- and $d_{xy}$-wave symmetry and also the LDOS of the electronic
degrees of freedom, as defined in Eq. (7).
## 4 Results
### 4.1 Non-Self-Consistent Approach on Single Band Model
Figure 2: Local density of states near a vortex for the non-self-consistent
calculation with the gap function given by Eq. (3). The value
$N(\mathbf{r},E)$ has been normalized such that the maximum value in each map
is unity. (a) shows the LDOS at the lowest bound state energy with on-site
pairing with $\Delta^{0}=3\mathrm{meV}$, and (e) is at higher energy. (b) and
(f) are with NNN pairing with $\Delta^{0}=1.5\mathrm{meV}$. The left insets in
(a),(b),(e) and (f) indicate the local structure of the pairing, and the right
insets are LDOS after gaussian filtering ($\sigma=3a_{0}$) reducing spatial
resolution for better comparison with experiment[8]. (c) and (g) are the near-
vortex LDOS maps observed in Ref. [8]. (d) is the LDOS as a function of energy
at the vortex core for the on-site pairing, Gaussian-filtered in both energy
($\sigma=0.15\mathrm{meV}$) and position ($\sigma=a_{0}$). (h) shows the
experimental tunneling spectra from Ref. [8] for comparison.
Figure 2 shows the near-vortex LDOS calculated by diagonalizing
$\mathcal{H}^{\mathrm{BdG}}$ of Eq. (1) with fixed gap functions as given by
Eq. (3) on a system of dimension $(N_{x},N_{y})=(301,301)$. We choose
realistic values of the parameters for the coherence length
$\xi=16.4a_{0}$[31, 32], as well as gap values $\Delta_{s}^{0}=3\mathrm{meV}$
for on-site pairing and $\Delta_{s^{\pm}}^{0}=1.5\mathrm{meV}$ for NNN
pairing[21, 22, 30].
We can interpret the vortex bound states in this non-self-consistent BdG
calculation as bound states in a potential well given by Eq. (3), where only
states around the normal-state Fermi surface constitute the bound states.
There are then two sources of anisotropy: anisotropic, quasi-one-dimensional
low-energy properties of the normal state, and an anisotropic gap, both
defined in the momentum space. The geometric distribution of LDOS will be
dominated by one or the other source of anisotropy at different energies.
At low energies, the normal state properties dominate the distribution of LDOS
[Figs. 2(a) and (b)]. Hence irrespective of pairing structure, the bound state
is located at the center of the potential well. Since the Bloch states making
up this bound state have two main velocities due to the quasi-one-dimensional
parts of the Fermi surface, the bound state mainly extends in these two
directions out of the well, resulting in the rays in Figs. 2(a) and (b). The
gap is suppressed near the vortex center, and its anisotropy is of little
importance. Hence the flat (quasi-one-dimensional) parts of the electronic
structure in Fig. 1(a) (solid line) dominate over the small anisotropy of the
$s^{\pm}$ gap [see Fig. 1(b)]. For a better comparison with experiment, we
present results of reduced spatial resolution by gaussian filtering
($\sigma=3a_{0}$) in the insets. The low resolution result is consistent with
results of the quasi-classical analysis by Wang et al. [14] and in good
agreement with experiment shown in Fig. 2(c).
At higher energies on the other hand, the bound state is located away from the
vortex core. The quasi-one-dimensionality of the Fermi surface allows for
localization in one direction and extension in the other. This leads to a
square-like inner ring in the LDOS for both pairings [Figs. 2(e) and (f)]. The
difference, however, results from the anisotropy of the gap function. While
the isotropic $s$-wave gap is analogous to a potential that is independent of
momentum, the anisotropic gap is one for which different states around the
Fermi surface experience different potentials depending on their momenta. With
the gap function of $s^{\pm}$ form, the quasi-one-dimensional portion of the
Fermi surface experiences a stronger trap potential, leading to a suppression
of its contribution to the bound-state wave function. As a result, the bound
state exhibits pronounced isolated segments, ‘hot spots,’ within the inner
ring that point in the Fe-Fe direction, as shown in Fig. 2(f). We again
gaussian filter the images and show them in the insets. Note the ’hot spot’
are robust and even more pronounced in the low resolution insets in Fig. 2(f)
in good agreement with the experimental data Fig. 2(g).
We now turn to the LDOS at the core of the vortex and its particle-hole
asymmetry. This turns out to be largely insensitive to anisotropy of pairing.
The LDOS at the core of the vortex for the on-site pairing shown in Fig 2(d)
exhibits particle-hole asymmetry with the highest peak at negative energy.
Such asymmetry appears in the so-called ‘quantum-limit’ vortex bound
state[33], whose highest LDOS peak is at energy $\Delta^{2}/2E_{F}$
above(below) the Fermi energy for an electron(hole)-like band, where $E_{F}$
is the energy difference between the Fermi energy and the bottom(top) of the
band. The energy of the LDOS peak being $0.05\mathrm{meV}$ below the Fermi
energy is expected given $E_{F}=98\mathrm{meV}$ and $\Delta=3\mathrm{meV}$
within our input bandstructure. Though similar particle-hole asymmetry has
been observed in Ref. [8] [see Fig. 2(h)] the energy at which the peak was
observed suggests that other hole pockets with larger gap values may be
responsible.
### 4.2 Non-Self-Consistent Approach on Five Band Model
Figure 3: LDOS near a vortex from the non-self-consistent calculation with
the five-band model from Ref. [20] and NNN pairing of
$\Delta^{0}_{s^{\pm}}=15\mathrm{meV}$, (a) at the lowest bound state energy,
(b) at an energy where the electron-band contribution dominates, and (c) at an
energy where the $\gamma$-band contribution dominates.
Now, we check whether the single-band model is sufficient to describe vortex
bound states within the energy range of interest. A simple insight can be
gained by treating each band independently and estimating the energy of its
lowest bound state to be $\Delta^{2}/2E_{F}$ following Caroli et al.[33] for
the gap size $\Delta$ and the Fermi energy $E_{F}$ specific to each band.
Using measured Fermi energies and gap parameters[21, 10, 22, 30], we estimate
the energies of the lowest bound states of the $\gamma$ pocket and the
electron pockets to be of the same order. However, the lowest bound state
energies of the two smaller hole pockets are an order of magnitude larger.
This rough estimate implies that the LDOS within the energy below
$1\mathrm{meV}$ should be dominated by bound states coming from the $\gamma$
band and those coming from the two electron bands. If indeed each bound state
comes from a single band, we expect to find bound states with LDOS
distribution resembling what we predicted in section 4.1.
For concreteness, we carry out a non-self-consistent BdG calculation using the
band structure given by Ref. [20] with five bands. Unfortunately, the
$\gamma$-pocket Fermi surface of this band structure [dashed line in Fig 1(a)]
is far more isotropic compared to what has been measured in Ref.[30] and
guided the band structure we use in the rest of this paper. Hence we do not
expect as pronounced ‘ray’ features at low energies compared to what is shown
in Fig. 2 from our (single-band) calculations and experiment. Another issue we
face with a five-band calculation is the limitation on the accessible system
size. For a system of size $(101,101)$, we impose NNN pairing that is trivial
in the orbital space having magnitude $\Delta_{s^{\pm}}=15\mathrm{meV}$ in
order to fit the vortex bound states within the system and minimize the
boundary effect. As in the single-band calculation, we create a vortex at the
center of the form given in Eq. (3), however with $\xi=10a_{0}$. Figure 3
shows the resulting LDOS at different bound state energies. At the lowest
energy there is no clear sign of ‘rays’ though a small amount of anisotropy is
still present, as expected from the smaller $\gamma$-band anisotropy [see Fig.
3(a)]. Figures 3(b) and (c) show typical LDOS images of vortex bound states at
higher energies. Figure 3(b) looks very different from the LDOS distribution
obtained in section 4.1 and we hence assign the corresponding bound state to
the electron pockets. However, the LDOS shown in Fig. 3(c) shows the same ‘hot
spots’ as obtained within our single-band calculation and shown in Fig. 2(f).
Focussing on the $\gamma$ band should thus suffice to capture the features
observed in Ref. [8].
### 4.3 Self-Consistent Approach
Figure 4: Local density of states near a vortex within our self-consistent
calculation. Again, $N(\mathbf{r},E)$ is normalized within each image. (a) is
the LDOS at the lowest bound state energy with on-site attraction
$U=-0.35\mathrm{eV}$, and (c) is at higher energy. (b) and (d) are with NNN
attraction $V^{\prime}=-0.3\mathrm{eV}$. The inset in each figure represents
the local attractive interaction in the singlet pairing channel.
Figure 4 shows the results from the (single-band) self-consistent calculation.
We compare two pairing interactions – on-site attraction $U=-0.35\mathrm{eV}$,
and NNN attraction $V^{\prime}=-0.3\mathrm{eV}$ – for a system with magnetic
unit cell of dimensions $(L_{x},L_{y})=(19,38)$. This corresponds to a full
lattice size of $(N_{x},N_{y})=(38\times 19,19\times 38)$. In zero field, the
two cases lead to a uniform superconducting gap of
$\Delta_{s}^{0}=27\mathrm{meV}$ and $\Delta_{s^{\pm}}^{0}=10\mathrm{meV}$,
respectively. We have chosen $U$ and $V^{\prime}$ such that the coherence
length $\xi\propto\Delta^{-1}$ is small compared to the inter-vortex spacing.
This allows us to focus on a nearly isolated vortex within the computationally
feasible size of the magnetic unit cell. Although the resulting gap values are
an order of magnitude larger than what is known experimentally, this should
not affect the validity of the results in a qualitative manner. Both at low
energy and at higher energy close to the gap value, we observe features that
qualitatively agree with the results obtained in the previous section.
Figure 5: Spatial distribution of different symmetry components of order
parameters. (a) $\Delta_{s}(\mathbf{r})$ for on-site attraction
$U=-0.35\mathrm{eV}$. (b) $\Delta_{s^{\pm}}(\mathbf{r})$ and (c)
$\Delta_{d_{xy}}(\mathbf{r})$ for NNN pairing $V^{\prime}=-0.3\mathrm{eV}$.
The values have been normalized by the value of the dominant order parameter
in the absence of magnetic field for each case: $\Delta_{s}^{0}$ for (a), and
$\Delta_{s^{\pm}}^{0}$ for (b), (c). The equal-amplitude contours in red go
from 0.825 for the innermost to 0.925 for the outermost contours (after
normalization) with equal intervals between the contours in between. The
insets again indicate the structure of the local order parameter. Note that
the color-scale for $\Delta_{d_{xy}}(\mathbf{r})$ is much smaller than for
$\Delta_{s}(\mathbf{r})$.
The self-consistent calculation also allows us to study the local order
parameters of a given structure near a vortex. Unlike for the on-site
attraction, where $\Delta_{s}(\mathbf{r})$ is the only allowed gap function,
order parameters of different symmetries can mix near a vortex for NNN
attraction. A near-vortex map of $\Delta_{s}(\mathbf{r})$ for on-site pairing
shown in Fig. 5(a) indeed shows almost isotropic healing of the order
parameter away from the vortex core. However, for the NNN attraction which
leads to uniform $s^{\pm}$-wave pairing in zero-field, the secondary order
parameter $\Delta_{d_{xy}}(\mathbf{r})$ is induced near the vortex. Coupling
between this secondary order parameter and the primary
$\Delta_{s^{\pm}}(\mathbf{r})$ leads to a strong angular variation of both
components as can be seen in Figs. 5(b) and (c).
To gain further insight into the admixing of a secondary order parameter near
a vortex for the anisotropic pairing, we analyze the Ginzburg-Landau free-
energy density. The free-energy density for $s$-wave and $d$-wave order
parameters reads
$\displaystyle f=$
$\displaystyle\alpha_{s}|s|^{2}+\alpha_{d}|d|^{2}+\beta_{1}|s|^{4}+\beta_{2}|d|^{4}+\beta_{3}|s|^{2}|d|^{2}$
$\displaystyle+\beta_{4}(s^{*2}d^{2}+\mathrm{c.c.})+\gamma_{s}|\vec{D}s|^{2}+\gamma_{d}|\vec{D}d|^{2}$
$\displaystyle+\gamma_{v}(D_{x}sD_{y}d^{*}+D_{y}sD_{x}d^{*}+\mathrm{c.c.}),$
(12)
where $s$ and $d$ are shorthands for $s(\mathbf{r})$ and $d(\mathbf{r})$, the
order-parameter fields for the $s^{\pm}$ and $d_{xy}$ gaps, respectively, and
$D_{i}=\partial_{i}-ieA_{i}$ is the covariant derivative. The fields
$s(\mathbf{r})$ and $d(\mathbf{r})$ can be thought of as
$\Delta_{s/s^{\pm}}(\mathbf{r})$ and $\Delta_{d_{xy}}(\mathbf{r})$ after
coarse graining. This type of admixing near a vortex has previously been
studied in the context of cuprates, leading to the prediction of a fourfold-
anisotropic order parameter around a vortex [15, 16, 17, 18]333The microscopic
model we consider is related to the single-band model of cuprates through
rotation by 45∘, the roles played by $s$-wave and $d$-wave order parameters
are reversed and our $d$-wave order parameter is of $d_{xy}$ form rather than
$d_{x^{2}-y^{2}}$.. As the large halo around vortices in cuprates[34] hindered
the observation of this admixing, LiFeAs presents an opportunity for this
observation.
The spatial variation of the secondary component $d_{xy}$ in Fig. 5(c) is
largely due to the derivative coupling, the term proportional to $\gamma_{v}$
in Eq. (12). This intermixing term is expected to be large when the $s$-wave
order parameter is of $s^{\pm}$ type, since the same NNN pairing interaction
is reponsible for both $s$-wave and $d$-wave order parameter. For $|s|\gg|d|$
and $|\vec{D}s|\gg|\vec{D}d|$ the spatial structure of the $d_{xy}$ component
is determined largely by the structure of the $s$-wave component. Minimizing
Eq. (12) with respect to $d(\mathbf{r})$ and keeping only terms up to linear
order in $d(\mathbf{r})$, we find
$\displaystyle-\gamma_{d}\vec{D}^{2}d+\alpha_{d}d+\beta_{3}|s|^{2}d+\beta_{4}s^{2}d^{*}=$
$\displaystyle\gamma_{v}(D_{x}D_{y}+D_{y}D_{x})s.$ (13)
Hence, the curvature in the leading $s$-wave component will induce the
secondary ($d_{xy}$) component. Now, consider a single isolated vortex. As
$s(\mathbf{r})$ is recovered at the length scale of the coherence length $\xi$
away from the core of the vortex, we expect a large $d(\mathbf{r})$ due to
coupling to the large curvature of $s(\mathbf{r})$ at this distance. Since
$\xi=\hbar v_{F}/\pi\Delta\sim 3.0a_{0}$ for the uniform gap value with
$V^{\prime}=-0.3\mathrm{eV}$, this is in agreement with the positions of the
maxima of $d(\mathbf{r})$ in Fig. 5(c) as a function of $|\mathbf{r}|$ setting
the vortex core at the origin. We can also explain the angular variation and
the form of the anisotropy of $d(\mathbf{r})$ in this framework. If we assume
$s(\mathbf{r})=f(r)e^{i\theta}$ with a slowly changing $f(r)$ and the
azimuthal angle $\theta$ measured from the Fe-Fe direction, we find from Eq.
(13)
$\displaystyle d(\mathbf{r})$
$\displaystyle\sim\partial_{x}\partial_{y}s(\mathbf{r})\sim
e^{-i\theta}(1+3e^{4i\theta}),$ (14)
ignoring the phase due to the magnetic field. The structure of the derivative
hence gives rise to a four-fold anisotropy, which explains the fact that
$|d(\mathbf{r})|$ is maximum in the Fe-Fe direction, while it is suppressed
along the 45∘ direction. Coupling to $d(\mathbf{r})$ gives then in turn cause
for the four-fold anisotropy in $s(\mathbf{r})$.
## 5 Conclusion
We have contrasted the effects of anisotropic $s^{\pm}$-wave (NNN) pairing and
isotropic $s$-wave (on-site) pairing on the near-vortex local density of
states in LiFeAs by solving Bogoliubov-de Gennes equations both non-self-
consistently and self-consistently. We have found qualitative changes in the
geometric distribution of the density of states as a function of energy. At
low energies, the anisotropy of the vortex bound state, and hence the LDOS, is
determined by the normal state low energy electronic structure, independent of
the gap structure. Different pairing structures, however, lead to
qualitatively different LDOS distributions at higher energies: While the
isotropic $s$-wave shows a square-like feature of roughly equal intensity,
four ‘hot spots’ develop in the case of an (anisotropic) $s^{\pm}$-wave gap.
Indeed, our results for the latter case qualitatively agree with recent
experiments[8].
From the self-consistent treatment we have further found a difference in the
recovery of the order parameter away from the vortex core: a pronounced
angular dependence of the $s^{\pm}$-wave gap compared to isotropic behavior
for the $s$-wave gap. Employing a Ginzburg-Landau analysis, we have explained
this difference through admixing of a secondary order parameter supported by
the NNN interaction. Note that such intermixing is negligible for an $s$-wave
pairing with a dominant on-site pairing interaction, as no other pairing
instabilities are nearby. For the NNN interaction, however, $s^{\pm}$\- and
$d_{xy}$-wave instabilities have comparable transition temperatures. Detection
of the anisotropy or even the secondary order parameter would be a strong
proof of the unconventional nature of the pairing.
In this work, we focused on the $\gamma$ band with interest in low energy
properties, as this is the band with the smallest gap[21, 22]. Hence, for
features at energies less than the gap scale, we expect our calculation to
capture salient features of in-field STS experiments. The comparison between
the calculated LDOS for the single- and the five-band models and the results
in Ref. [8] supports this conjecture.
In closing we note that our calculation captures Friedel-like oscillations,
frequently referred to as quasi-particle interference (QPI), due to vortices.
QPI in the presence of vortices was successfully used to access phase
information with STS in cuprates[35]. Recent in-field QPI experiments on FeSe
have been interpreted to be consistent with an $s^{\pm}$ scenario when a
vortex is treated as a magnetic scatterer for BdG quasiparticles[4]. However,
a vortex is at once a point of gap suppression, a point with magnetic flux,
and a point around which the order-parameter phase winds. While we treated
vortices faithfully in the self-consistent calculation, we could not
investigate effects of inter-pocket sign change as we only considered one
pocket. An extension of the present work with the full band structure would be
necessary to work out what to expect for different order-parameter
possibilities, especially how the phase difference between different pockets
affects in-field QPI.
We are grateful to M. Allan, J. Berlinsky, J.C. Davis, I. Firmo, T. Hanaguri,
C. Kallin, A.W. Rost, and Z. Tesanovic for helpful discussions. We acknowledge
support from NSF Grant DMR-0955822. MHF and E-AK additionally acknowledge
support from NSF Grant DMR-1120296 to the Cornell Center for Materials
Research.
## References
## References
* [1] P J Hirschfeld, M M Korshunov, and I I Mazin. Gap symmetry and structure of Fe-based superconductors. Rep. Prog. Phys., 74(12):124508, 2011.
* [2] G. R. Stewart. Superconductivity in iron compounds. Rev. Mod. Phys., 83:1589–1652, Dec 2011.
* [3] C. T. Chen, C. C. Tsuei, M. B. Ketchen, Z. A. Ren, and Z. X. Zhao. Integer and half-integer flux-quantum transitions in a niobium-iron pnictide loop. Nat Phys, 6(4):260–264, 04 2010.
* [4] T. Hanaguri, S. Niitaka, K. Kuroki, and H. Takagi. Unconventional s-wave superconductivity in Fe(Se,Te). Science, 328(5977):474–476, 2010.
* [5] A. D. Christianson, E. A. Goremychkin, R. Osborn, S. Rosenkranz, M. D. Lumsden, C. D. Malliakas, I. S. Todorov, H. Claus, D. Y. Chung, M. G. Kanatzidis, R. I. Bewley, and T. Guidi. Unconventional superconductivity in Ba0.6K0.4Fe2As2 from inelastic neutron scattering. Nature, 456(7224):930–932, 12 2008.
* [6] M. D. Lumsden, A. D. Christianson, D. Parshall, M. B. Stone, S. E. Nagler, G. J. MacDougall, H. A. Mook, K. Lokshin, T. Egami, D. L. Abernathy, E. A. Goremychkin, R. Osborn, M. A. McGuire, A. S. Sefat, R. Jin, B. C. Sales, and D. Mandrus. Two-dimensional resonant magnetic excitation in ${\mathrm{BaFe}}_{1.84}{\mathrm{Co}}_{0.16}{\mathrm{As}}_{2}$. Phys. Rev. Lett., 102:107005, Mar 2009.
* [7] Shiliang Li, Ying Chen, Sung Chang, Jeffrey W. Lynn, Linjun Li, Yongkang Luo, Guanghan Cao, Zhu’an Xu, and Pengcheng Dai. Spin gap and magnetic resonance in superconducting ${\text{BaFe}}_{1.9}{\text{Ni}}_{0.1}{\text{As}}_{2}$. Phys. Rev. B, 79:174527, May 2009.
* [8] T. Hanaguri, K. Kitagawa, K. Matsubayashi, Y. Mazaki, Y. Uwatoko, and H. Takagi. Scanning tunneling microscopy/spectroscopy of vortices in lifeas. Phys. Rev. B, 85:214505, Jun 2012.
* [9] Hiroshi Kontani and Seiichiro Onari. Orbital-fluctuation-mediated superconductivity in iron pnictides: Analysis of the five-orbital hubbard-holstein model. Phys. Rev. Lett., 104:157001, Apr 2010.
* [10] Sergey V. Borisenko, Volodymyr B. Zabolotnyy, Alexnader A. Kordyuk, Danil V. Evtushinsky, Timur K. Kim, Igor V. Morozov, Rolf Follath, and Bernd Büchner. One-sign order parameter in iron based superconductor. Symmetry, 4(1):251–264, 2012.
* [11] P. M. R. Brydon, Maria Daghofer, Carsten Timm, and Jeroen van den Brink. Theory of magnetism and triplet superconductivity in lifeas. Phys. Rev. B, 83:060501, Feb 2011.
* [12] Torben Hänke, Steffen Sykora, Ronny Schlegel, Danny Baumann, Luminita Harnagea, Sabine Wurmehl, Maria Daghofer, Bernd Büchner, Jeroen van den Brink, and Christian Hess. Probing the unconventional superconducting state of lifeas by quasiparticle interference. Phys. Rev. Lett., 108:127001, Mar 2012.
* [13] Nobuhiko Hayashi, Masanori Ichioka, and Kazushige Machida. Star-shaped local density of states around vortices in a type-II superconductor. Phys. Rev. Lett., 77:4074–4077, Nov 1996.
* [14] Yan Wang, Peter J. Hirschfeld, and Ilya Vekhter. Theory of quasiparticle vortex bound states in iron-based superconductors: Application to scanning tunneling spectroscopy of LiFeAs. Phys. Rev. B, 85:020506, Jan 2012.
* [15] Robert Joynt. Upward curvature of ${\mathit{H}}_{\mathit{c}2}$ in high-${\mathit{T}}_{\mathit{c}}$ superconductors: Possible evidence for s \- d pairing. Phys. Rev. B, 41:4271–4277, Mar 1990.
* [16] A. J. Berlinsky, A. L. Fetter, M. Franz, C. Kallin, and P. I. Soininen. Ginzburg-Landau theory of vortices in $\mathit{d}$-wave superconductors. Phys. Rev. Lett., 75:2200–2203, Sep 1995.
* [17] Yong Ren, Ji-Hai Xu, and C. S. Ting. Ginzburg-Landau equations and vortex structure of a ${d}_{{x}^{2}{-y}^{2}}$ superconductor. Phys. Rev. Lett., 74:3680–3683, May 1995.
* [18] M. Ichioka, N. Enomoto, N. Hayashi, and K. Machida. s \- and ${\mathit{d}}_{\mathit{xy}}$-wave components induced around a vortex in ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{-}{\mathit{y}}^{2}}$-wave superconductors. Phys. Rev. B, 53:2233–2236, Feb 1996.
* [19] R. Heeb, A. van Otterlo, M. Sigrist, and G. Blatter. Vortices in d -wave superconductors. Phys. Rev. B, 54:9385–9398, Oct 1996.
* [20] Helmut Eschrig and Klaus Koepernik. Tight-binding models for the iron-based superconductors. Phys. Rev. B, 80:104503, Sep 2009.
* [21] S. V. Borisenko, V. B. Zabolotnyy, D. V. Evtushinsky, T. K. Kim, I. V. Morozov, A. N. Yaresko, A. A. Kordyuk, G. Behr, A. Vasiliev, R. Follath, and B. Büchner. Superconductivity without nesting in LiFeAs. Phys. Rev. Lett., 105:067002, Aug 2010.
* [22] K. Umezawa, Y. Li, H. Miao, K. Nakayama, Z.-H. Liu, P. Richard, T. Sato, J. B. He, D.-M. Wang, G. F. Chen, H. Ding, T. Takahashi, and S.-C. Wang. Unconventional anisotropic $s$-wave superconducting gaps of the LiFeAs iron-pnictide superconductor. Phys. Rev. Lett., 108:037002, Jan 2012.
* [23] T. Hajiri, T. Ito, R. Niwa, M. Matsunami, B. H. Min, Y. S. Kwon, and S. Kimura. Three-dimensional electronic structure and interband nesting in the stoichiometric superconductor LiFeAs. Phys. Rev. B, 85:094509, Mar 2012.
* [24] Xiang Hu, C. S. Ting, and Jian-Xin Zhu. Vortex core states in a minimal two-band model for iron-based superconductors. Phys. Rev. B, 80:014523, Jul 2009.
* [25] Hong-Min Jiang, Jian-Xin Li, and Z. D. Wang. Vortex states in iron-based superconductors with collinear antiferromagnetic cores. Phys. Rev. B, 80:134505, Oct 2009.
* [26] Tao Zhou, Z. D. Wang, Yi Gao, and C. S. Ting. Electronic structure around a vortex core in iron-based superconductors: Numerical studies of a two-orbital model. Phys. Rev. B, 84:174524, Nov 2011.
* [27] Yi Gao, Huai-Xiang Huang, Chun Chen, C. S. Ting, and Wu-Pei Su. Model of vortex states in hole-doped iron-pnictide superconductors. Phys. Rev. Lett., 106:027004, Jan 2011.
* [28] Hsiang-Hsuan Hung, Can-Li Song, Xi Chen, Xucun Ma, Qi-kun Xue, and Congjun Wu. Anisotropic vortex lattice structures in the FeSe superconductor. Phys. Rev. B, 85:104510, Mar 2012.
* [29] Xiao-Shan Ye. Bound states and vortex core shrinking effects in iron-based superconductors. Physica C: Superconductivity, 485(0):6 – 9, 2013.
* [30] M. P. Allan, A. W. Rost, A. P. Mackenzie, Yang Xie, J. C. Davis, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, and T.-M. Chuang. Anisotropic energy gaps of iron-based superconductivity from intraband quasiparticle interference in LiFeAs. Science, 336(6081):563–567, 2012.
* [31] Bumsung Lee, Seunghyun Khim, Jung Soo Kim, G. R. Stewart, and Kee Hoon Kim. Single-crystal growth and superconducting properties of LiFeAs. EPL (Europhysics Letters), 91(6):67002, 2010.
* [32] Nobuyuki Kurita, Kentaro Kitagawa, Kazuyuki Matsubayashi, Ade Kismarahardja, Eun-Sang Choi, James S. Brooks, Yoshiya Uwatoko, Shinya Uji, and Taichi Terashima. Determination of the upper critical field of a single crystal LiFeAs: The magnetic torque study up to 35 Tesla. J. Phys. Soc. Jpn., 80:013706, 2012.
* [33] C. Caroli, P.G. De Gennes, and J. Matricon. Bound fermion states on a vortex line in a type II superconductor. Phys. Lett., 9(4):307 – 309, 1964.
* [34] J E Hoffman, E W Hudson, K M Lang, V Madhavan, H Eisaki, S Uchida, and J C Davis. A four unit cell periodic pattern of quasi-particle states surrounding vortex cores in Bi2Sr2CaCu2O8+delta. Science, 295(5554):466–9, January 2002.
* [35] T. Hanaguri, Y. Kohsaka, M. Ono, M. Maltseva, P. Coleman, I. Yamada, M. Azuma, M. Takano, K. Ohishi, and H. Takagi. Coherence factors in a high-Tc cuprate probed by quasi-particle scattering off vortices. Science, 323(5916):923–926, 2009.
|
arxiv-papers
| 2012-09-04T20:08:12 |
2024-09-04T02:49:34.788395
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kyungmin Lee, Mark H. Fischer, Eun-Ah Kim",
"submitter": "Kyungmin Lee",
"url": "https://arxiv.org/abs/1209.0788"
}
|
1209.0803
|
11institutetext: 1: Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
2: Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
3: Department of Physics, Washington University, St. Louis, Missouri 63160,
USA
4: Department of Electrophysics, National Chiao Tung University, Hsinchu
30013, Taiwan
# Defects and glassy dynamics in solid 4He: Perspectives and current status
A. V. Balatsky1,2 M. J. Graf1 Z. Nussinov3 J.-J. Su4
(09.04.2012)
###### Abstract
We review the anomalous behavior of solid 4He at low temperatures with
particular attention to the role of structural defects present in solid. The
discussion centers around the possible role of two level systems and
structural glassy components for inducing the observed anomalies. We propose
that the origin of glassy behavior is due to the dynamics of defects like
dislocations formed in 4He. Within the developed framework of glassy
components in a solid, we give a summary of the results and predictions for
the effects that cover the mechanical, thermodynamic, viscoelastic, and
electro-elastic contributions of the glassy response of solid 4He. Our
proposed glass model for solid 4He has several implications: (1) The anomalous
properties of 4He can be accounted for by allowing defects to freeze out at
lowest temperatures. The dynamics of solid 4He is governed by glasslike
(glassy) relaxation processes and the distribution of relaxation times varies
significantly between different torsional oscillator, shear modulus, and
dielectric function experiments. (2) Any defect freeze-out will be accompanied
by thermodynamic signatures consistent with entropy contributions from
defects. It follows that such entropy contribution is much smaller than the
required superfluid fraction, yet it is sufficient to account for excess
entropy at lowest temperatures. (3) We predict a Cole-Cole type relation
between the real and imaginary part of the response functions for rotational
and planar shear that is occurring due to the dynamics of defects. Similar
results apply for other response functions. (4) Using the framework of glassy
dynamics, we predict low-frequency yet to be measured electro-elastic features
in defect rich 4He crystals. These predictions allow one to directly test the
ideas and very presence of glassy contributions in 4He.
###### Keywords:
Thermodynamics torsional oscillator shear modulus dielectric function glass
viscoelastic electro-elastic supersolid solid helium
###### pacs:
67.80.B-, 64.70.Q-, 67.80.bd
††journal: Journal of Low Temperature Physics
## 1 Introduction
The discovery of anomalous frequency and dissipation behavior seen in
torsional oscillators (TOs) Kim04a ; Kim04b at low temperatures in 4He has
stimulated numerous investigations. These anomalies have been argued to
demonstrate a non-classical rotational inertia (NCRI) of the long ago
predicted supersolid quantum stateAndreev69 ; Chester67 ; Reatto69 ; Leggett70
; Anderson84 . Successive TO experiments Rittner06 ; Kondo07 ; Aoki07 ;
Clark07 ; Penzev07 ; Hunt09 ; Pratt11 ; Gadagkar2012 confirmed the finding of
the reported anomalous behavior. Hysteresis behavior and long equilibration
times have been observed Aoki07 ; Hunt09 ; Kim09 , which depend strongly on
growth history and annealing Rittner06 . In the same temperature range,
experiments including shear modulus Beamish05 ; Beamish06 , ultrasonic
Goodkind02 ; Burns93 and heat propagation Goodkind02 have also shown various
anomalies. The character and the existence of mass flow is still a matter of
intense investigation. Experiments designed to probe for mass flow by
squeezing the lattice report no such flow Beamish05 ; Beamish06 ; Greywall77 ;
Paalanen81 ; Sasaki06 ; Ray08 ; Bonfait89 ; Balibar08 . However, experiments
in which a chemical potential gradient was created via coupling to a
superfluid reservoir, suggest mass flow of an unusual type Ray08 ; Ray09 ;
Ray10 ; Ray11 ; Vekhov2012 . These and many other results have led to a flurry
of activity. Structural measurements Burns08 ; Blackburn07 suggest that solid
4He may be composed of a dynamic mosaic of crystals, highlight the importance
of defects, and report the absence of notable structural change in the
vicinity of the putative supersolid transition. The intricate structure and
dynamics of sold 4He make it a fascinating system. We note that when a mosaic
of 4He crystals with intervening liquid channels is placed in a metallic
container Ray08 ; BalibarComment , thermal expansion effects (in particular
the larger thermal expansion coefficient of helium vis a vis that of the
metallic container) may be at play in blocking a remnant superleak. Such a
thermal compression will effectively shut down any mass flow and close the
open channels as the temperature is raised. This scenario remains a viable
explanation for the anomalous mass flow and fountain effect reported by
Hallock’s group Ray08 ; Ray09 ; Ray10 ; Ray11 ; Vekhov2012 , until a superleak
can be ruled out.
After eight years of intense experimental and theoretical investigations one
must ask what are the established facts, what are the current expectations and
hypotheses, and what are the future directions of research in solid 4He. There
are numerous reviews and progress updates available describing the field
ProkofevAdvances ; Balibar2011 ; Boninsegni2012 and addressing some of these
questions. Most of the literature reviewing the subject deals with the notion
of supersolidity in 4He as established and proceeds with the discussion on the
status and future experiments with the goal of further “proving” the existence
of a supersolid phase transition at very low temperatures. Amongst the many
exciting proposals concerning 4He as well as supersolids, we briefly mention
Andreev’s proposal for superglass Andreev07 ; Andreev09 ; Korshunov09 ,
Anderson’s suggestion of vortex proliferation and flow Anderson2007 ; Kubota
and supersolid dislocation cores dislocation_core1 ; dislocation_core2 ;
dislocation_core3 ; Rossi . As will become evident in later sections, our
analysis centers on the dynamical effects of defects and as such may include
vortices, dislocations, or any other defects. Estimating the product of
typical dislocation core sizes ($\sim 3$ nm) and their density ($\sim 10^{10}$
m-2), it is seen that a direct NCRI origin from superfluid dislocation cores
in 4He cannot account for the magnitude of the TO anomaly (it is orders of
magnitude too small).
We believe that at this mature stage of research in 4He a somewhat more
general view on possible options is needed, where one asks the question: what
are the options for possible states that might form at low temperatures in 4He
and by implication in other solid bosonic matter? By taking a broader view one
explicitly allows for states other than pure supersolidity and includes
possible coexistence phases to form as well. This review is a contribution in
the spirit of broadening the conversation by explicitly allowing for other
components or “active ingredients” to be present in addition to, or perhaps
instead of, supersolidity in 4He. Thus, if one accepts the presence of defects
in a solid, then naturally the question arises about the dynamic signatures of
such crystal defects and whether they can dominate the response to an external
stimulus.
Figure 1: The phase diagram of 4He with the putative supersolid phase
transition below 0.3 K Kim04b or an alternate order-disorder dominated
crossover region governed by impedance matching between the applied frequency
and internal relaxation processes.
We provide a brief critical analysis of some of the existing data and point
out that a significant fraction of the data on mechanical, thermodynamic and
dielectric properties of 4He can be analyzed in terms of the emergence of a
dissipative viscous component that we shall call glassy component Balatsky07 ;
Nussinov07 ; Graf08 ; Su10a ; Su10b . This extra component by itself is
sufficient to modify numerous properties of solid 4He and can be responsible
for anomalous thermodynamic, elastic and dielectric properties in solid 4He
observed in experiments. The glassy component is not a supersolid in the
classical sense ProkofevAdvances , yet it can coexist and couple to a
supersolid component as some of the proposed superglass phases indicate Hunt09
; Boninsegni06 ; wu2008 ; Biroli08 .
The precise nature of the state of solid 4He at lowest temperatures remains a
puzzle. In the past, investigation of 4He has played an important role in the
development of basic concepts in modern condensed matter physics like
superfluidity, order parameter, topological excitations, and critical
exponents. Given such a prominent role it played in the past it is paramount
to come up with the resolution of the puzzles that are clearly seen in
experiments at lowest temperatures. Yet there is one important difference that
might be key to a solution this time. We propose that precisely because the
compound is clean and very well characterized the enabling component for the
anomalies at lowest temperature are defects that undergo freeze-out and
constitute a glass like component.
To sharpen this point of the discussion, any supersolid component would imply
some sort of two-fluid hydrodynamics that schematically equates the total mass
current of helium atoms as a sum of normal component and superfluid component
determined by the normal and superfluid mass fractions $\rho_{n}$ and
$\rho_{s}$ and respective velocities. In this two-fluid picture the total mass
current is given as
${\bf j}=\rho_{n}{\bf v_{n}}+\rho_{s}{\bf v_{s}}$ (1)
In any superfluid or supersolid phase the coefficients $\rho_{n}(T,P)$ and
$\rho_{s}(T,P)$ are functions of thermodynamic variables like temperature $T$
and pressure $P$. There is no direct evidence of either dc or ac supersolid
mass current at lowest temperatures, where the putative supersolid state sets
in Rittner06 ; Beamish05 , except for the experiments by Hallock Ray08 ; Ray09
; Ray10 ; Vekhov2012 . However, it remains to be seen whether Hallock’s
results of mass flow and fountain effect cannot be explained through the
presence of superleaks connecting the superfluid leads through solid helium
with channels. Therefore, some even proposed that one cannot “squeeze a
superfluid component out of a stone” Dorsey06 . We know that these supersolid
expectations do not apply in the case of solid 4He at lowest temperatures,
because TO experiments clearly indicate that the period and damping exhibit
significant hysteresis effects and strong dependence on the history of sample
preparations and annealing. All these experimental observations combined would
suggest to an impartial observer that there is at the very least, in addition
to supersolidity, another physical component at play in 4He. Our analysis of
various experiments suggests that the crossovers, seen in the specific heat,
TO, shear modulus and dielectric function experiments Aoki07 ; LinChan07 ;
Beamish10 ; Yin11 are a reflection of the physics whose origin is not due to
supersolidity, but a consequence of the dynamics of defects in solid 4He .
Specifically, we propose the presence of a glassy component in solid 4He at
low temperatures in order to explain the observed anomalous linear temperature
dependence in the specific heat of an otherwise perfect Debye solid 4He
Balatsky07 ; Graf08 . We used similar ideas when we analyzed the TO, shear
modulus, and dielectric properties by assuming the presence of a glassy
component at the parts-per-million (ppm) concentration and asked what the
dynamic consequences should be. With the wealth of data available we do not
attempt to provide a complete overview of the field, but give a summary of the
work centered on the role of a glassy component in an otherwise nearly perfect
crystal.
The exact nature of the glassy component is not known. For example, it may be
caused by two-level systems of pinned dislocation lines, vortex excitations,
etc. It is however important to point out that the amplitude of period shift
can be changed dramatically and depends on growth history and annealing
procedures of the crystal. To explain the puzzling features of solid 4He we
_conjecture_ that structural defects like localized dislocation segments or
groups of displaced (out-of-equilibrium) atoms effectively form a set of two
level systems (TLSs) which are present at low temperatures. These immobile
crystalline defects will affect the thermodynamics of solid 4HeBalatsky07 ;
Graf08 , the mechanics of TOs Nussinov07 and shear modulus Su10a ; Su10b ,
and dielectric properties Su11 . Other mechanisms for TLS and glassy dynamics,
e.g., due to point defects and grain boundaries are possible as well. We are
at a stage where phenomenology allows us to make progress with testable
predictions, while a microscopic picture of crystal defects and interactions
is still missing.
Early on it was recognized that pinned vibrating dislocation lines can account
for a plethora of anelastic damping phenomena in the ultrasound, TO, and shear
modulus experiments. Since most experiments are believed to be in the linear
or elastic strain-stress regime any plastic deformation due to the motion of
gliding dislocation loops has been neglected. However, this is not necessarily
so. We proposed Caizhi2012 that dislocation-induced anomalous softening of
solid 4He is possible due to the classical motion of gliding dislocation lines
in slip planes. This picture of dislocation motion is widely accepted in
conventional metals. Similar effects are at play in the quantum arena where
mobile dislocations (dislocation currents) lead to a screening of applied
shear via a Meissner-Higgs type effect Zaanen04 ; Cvetkovic08 . Such unpinned
dislocations that screen shear render the system more fluid and may, in line
with the framework that we advance in this article, trigger the 4He anomalies
as the mobile dislocations become quenched at low temperatures
dislocation_core3 ; Caizhi2012 . Recent experiments provide further impetus
for such a picture eyal12a ; eyal12b .
The technique of choice for interrogation of solid 4He has been the TO with
varying degrees of complexity. However, the TO does not provide any direct
information on the microstructure of samples. More direct structural x-ray and
neutron measurements do not have the adequate resolution to detect any changes
at the ppm level in the structure of 4He at lowest temperatures. Additional
challenges arise given how small the volume fraction of the glassy component
can be. We estimate it to be in the range of few hundreds of ppm in the
specific heat and pressure contribution. Therefore the precise
characterization of the microstructure of solid 4He, growth history,
annealing, and 3He dependence remain pressing issues in resolving the
hypothesis of the presence of a glassy component and TLS in solid 4He. The
notion of importance of the role of disorder in solid 4He received further
support over the years in observations of the strong dependence of TO results
on the specific design of sample cells, see work by the group of Chan
DYKim2012 . It is hard to imagine that an intrinsic material property like
supersolidity should depend strongly on the stiffness or geometry of the
torsional oscillator apparatus, while crystalline disorder can easily be
affected by those design properties.
In the analysis of the observed excess specific heat, we used a model of
independent TLS, which gives the canonical signature of a linear in
temperature contribution at lowest temperatures. Building on the presence of
TLS we evaluated the mechanical properties of the TO using a model of quenched
defects. This approach allows us to make predictions on the viscoelastic
properties of 4He and on the electro-elastic coupling that can be tested in a
setup that does not require the TO and hence can be tested over a much broader
frequency range. These predictions allow to directly verify the very presence
of quenched defects in 4He. We therefore would welcome any direct tests of our
ideas.
The remainder of this paper is organized as follows. We present the general
discussion on the role of quenched defects and disorder in solid 4He with
particular attention to the consequence of defects on the dynamics in Section
2. This is followed in Section 3 by an analysis of the thermodynamic
properties of solid 4He. In Section 4, we present a unified framework to
analyze dynamical response function that invokes arbitrarily high order
backaction effects of defects onto the solid bulk. We then invoke, in Section
5 this approach and summarize our analysis of the torsional oscillator. In
Section 6 we discuss the shear modulus analysis and the strain-stress
relations using a viscoelastic model designed to capture the anelastic
contribution from defects. In Section 7 we discuss predictions for the
dielectric properties that follow from the viscoelastic model with locally
frozen-in defect dynamics. Finally we conclude with a discussion and give our
view on future perspectives in the field.
## 2 Defect Quenching and Its Implications
In this review, we will examine the general consequences of a transition from
mobile defects (or dynamic fluid-like components) at high temperatures to
quenched immobile defects at low temperatures. Microscopically, these defects
may be dislocations, grain boundaries, vortices, or others. It may be posited
that in quantum solids such as 4He , zero point motion leads to larger
dynamics than is common in classical solids. To put the discussion in
perspective, we recall a few rudiments concerning annealing and quenching.
When quenched, systems fall out of equilibrium. En route to non-equilibrium
states, relaxation times increase until the dynamic components essentially
“freeze” (on pertinent experimental time scales) into an amorphous state. In
materials with (sufficiently large) external disorder, quenching may lead to
“spin-glass” characteristics. By contrast, in the absence of imposed external
disorder, when fluid components (either classical or quantum) fall out of
equilibrium by sufficient rapid cooling (so-called “super-cooling”) to low
enough temperatures, the resultant state is termed a “glass”. As liquids are
supercooled, their characteristic relaxation times and viscosity may increase
dramatically. If, instead, the temperature is lowered at a sufficiently slow
rate, the system does not quench and instead remains an “annealed” system in
thermal equilibrium. Notwithstanding exciting progress, exactly how the
dynamic components evolve in 4He crystals as the temperature is lowered is,
currently, an open problem. An initially surprising and undeniable feature
that has, by now, been seen in many experiments is that the solid 4He
anomalies depend dramatically on the growth history of the crystal and
diminish as the system is cooled down slowly and defect quenching is thwarted.
Memory effects reminiscent to those in glasses are further present. These
observations imply that “there is more to life” than static NCRI and other
annealed supersolid properties on their own– quenching plays a definitive role
in triggering the observed effects. To understand the experimental
observations and build a predictive theoretical framework, we invoke general
physical principles allowing a computation of response functions and using as
input known characteristics of quenching. Physically, as alluded to in the
Introduction, the quenching is that of the mobile defects (which may
constitute only a tiny fraction of the system) as they become arrested against
a crystalline background. Our analysis does not rule out the presence of a
small supersolid component. The observed large change in the TO dissipation
cannot be solely described by uniform Bose-Einstein condensation Nussinov07 ;
Graf08 ; Huse07 . It remains to be seen if nonuniform Bose-Einstein
condensation alone either along grain boundaries Clark06 or along the axis of
screw dislocations Boninsegni06 ; Pollet07 ; Boninsegni07 can explain the
dynamic response of TOs. Note that simple estimates of the supersolid fraction
of dislocation cores are orders of magnitude too small.
There exists a vast literature on defect quenching in systems that range from
vortices in superfluid Helium to cosmic strings Kibble76 ; Zurek85 and
countless others. Our particular focus is, of course, on defects in a
crystalline system (solid 4He). Quenched dynamics of such defects is
associated with a change of plasticity and related internal dissipation.
Dielectric (and other) response functions in systems of varying plasticity,
such as various glass formers as their temperature is lowered, indicate, in a
nearly universal fashion, the presence of a distribution of local relaxation
times. These lead to the canonical Cole-Cole or Cole-Davidson distribution
functions and related forms as we briefly elaborate. In an overdamped
dissipative system, an impulse (e.g., an external electric field or an elastic
deformation) at time $t=0$ leads a response $g(t)$ which at later times scales
as $g_{single}\sim\exp(-t/\tau)$ where $\tau$ is the (single) relaxation time.
When Fourier transformed to the complex frequency ($\omega$) plane, this leads
to the Debye relaxor $g_{single}(\omega)=g_{0}/(1-i\omega\tau)$. Now, in
systems that exhibit a distribution $P(\tau^{\prime})$ of local relaxation
events, the response functions attain the form $\int
d\tau^{\prime}P(\tau^{\prime})\exp(-t/\tau^{\prime})$. Empirically, in
dissipative plastic systems, relaxations scale as $(\exp(-t/\tau)^{c})$ with a
power $0<c<1$ that leads to a “stretching” (slower decay) of the response
function as compared to its single overdamped mode form of $\exp(-t/\tau)$.
This stretched exponential and other similar forms of the response function
capture the quintessence of the distribution of relaxation times. Two widely
used relaxation time distributions are the Cole-Cole (CC) and Davidson-Cole
(DC) functions that describe a superposition of overdamped oscillators (Debye
relaxors) Phase1 ; Phase2 . With $g(\omega)=g_{0}G(\omega)$, where $g_{0}$ is
a material specific constant, these two forms are given by different choices
for the function $G$,
$\displaystyle G_{CC}(\omega)=1/[1-(i\omega\tau)^{\alpha}],$ $\displaystyle
G_{DC}(\omega)=1/[1-i\omega\tau]^{\beta}.$ (2)
Values of $\alpha$ and $\beta$ that differ from unity qualitatively play the
role of the real-time stretching exponent $c$. In the dc limit the mechanical
motion of any mobile component will have ceased and there will be no relative
motion and no transients. Therefore the coefficient $g_{0}$ is generally a
function of frequency. In the case of the single TO its value is set by the
resonance, $g_{0}\approx g_{0}(\omega_{0})$.
These relaxation times can be associated with a distribution of TLSs
describing viable low temperature configurations of the defects. The simple
TLS analysis can account for thermodynamic measurements. Recent work Vural11
obtained results beyond the TLS model with fewer parameters for generic non-
uniform systems irrespective of specific microscopic origin. We will, for the
sake of simplicity, review our work on the low temperature properties of 4He
assuming TLSs. We conjectured Balatsky07 that structural defects, e.g.,
localized dislocation segments, form such a set of TLSs at low temperatures.
These immobile crystal defects affect the thermodynamics Balatsky07 ; Graf08
of bulk 4He and the mechanics Nussinov07 of the TO loaded with 4He. For the
analysis of the specific heat, we used independent TLS to obtain the universal
signature of a linear-in-temperature specific heat term at low temperatures.
## 3 Thermodynamics
Any true phase transition, including supersolid, is accompanied by a
thermodynamic signature. Therefore it was anticipated that thermodynamic
measurements will resolve the existing puzzles of supersolidity. The search
for such thermodynamic signatures proved to be challenging so far, see e.g.,
measurements of the specific heat Swenson62 ; Frank64 ; ClarkChan05 ;
LinChan07 ; LinChan09 ; WestChan09 , measurements of the pressure dependence
of the melting curve Todoshchenko06 ; Todoshchenko07 , and pressure-
temperature measurements of the solid phase Grigorev07 ; Grigorev07b ;
Lisunov2011 . The main difficulties lie in measuring small signals at low
temperatures in the presence of large backgrounds. With improving experiments
measurements were conducted down to 20 mK. While there is still no clear
evidence of a phase transition in the melting curve experiments, recent
pressure measurements and specific heat measurements have both shown
deviations from the expected pure Debye lattice behavior. Early on we proposed
that these deviations might be related to a glass transition and be described
by the contributions of two-level systems (TLS). Balatsky07 ; Graf08
We model the system of noninteracting TLS with a compact distribution of two-
level excitation spacings, see Fig. 2. Our results show that the low-
temperature deviations in the measured specific heat can be explained by
contributions from a glassy fraction and/or TLS of the solid.
### 3.1 Two level system model for the specific heat
Figure 2: Density of states (DOS) of the two-level tunneling system. The
black-dashed line represents the DOS for the standard glass model Anderson72 ;
Phillips72 ; Balatsky07 , while the blue-solid line is the truncated DOS used
to describing the TLS with a cutoff energy.
To avoid complications due to the presence of 3He atoms, we will compare the
effect of different growth processes on ultrapure 4He containing at most
(nominally) 1 ppb of 3He atoms. At such low levels of impurities, we expect to
see the intrinsic properties of solid.
We postulate that at temperature much below the lattice Debye temperature, the
specific heat of solid 4He is described by
$\displaystyle C(T)=C_{L}(T)+C_{g}(T),$ (3)
where the lattice contribution to the molar specific heat is given by
$C_{L}(T)=B_{L}T^{3}$, with coefficient $B_{L}=12\pi^{4}R/5\Theta_{D}^{3}$,
$R=8.314$ J/(mol K) is the gas constant, and $\Theta_{D}$ is the Debye
temperature. The second term describes the glass contribution due to the TLS
subsystem and is given by
$C_{g}(T)=k_{B}R\frac{d}{dT}\int_{0}^{\infty}dE\,{\cal D}_{g}(E)\,f(E),$ with
$k_{B}$ the Boltzmann constant and $f(E)$ the Fermi function. The density of
states (DOS) of the TLS may be modeled by the box distribution function (Fig.
2):
$\displaystyle{\cal D}_{g}(E)=\frac{1}{2}{\cal
D}_{0}\left[1-\tanh((E-E_{c})/W)\right].$ (4)
Here ${\cal D}_{0}$ is the zero-energy DOS, $E_{c}$ is a characteristic cutoff
energy, and $W$ is the width of the truncated density of states. For
$E_{c}\to\infty$, one obtains the standard hallmark result of glasses at low
temperatures: $C_{g}(T)=B_{g}T$, where $B_{g}=k_{B}R{\cal D}_{0}$. As we will
elaborate in the next section, the glass coefficient $B_{g}$ has an intrinsic
finite value at low temperature even for the purest 4He samples, independent
of 3He concentration.
As shown in (4), our model goes beyond the standard glass model by introducing
a cutoff in the DOS of the TLS (Fig. 2). The cutoff could be due to the finite
barrier height of double-well potentials giving rise to the TLS because in
real materials the tunneling barrier has an upper bound set by lattice and
dislocation configurations Jaeckle72 . At high temperatures, the TLS
contribution is less important since the thermal energy easily overcomes the
barrier and effectively resembles a single harmonic degree of freedom.
Figure 3: $\delta C/T$ for experiments (squares) LinChan09 and the modified
glass model with a cutoff energy in the TLS DOS (blue line) of four samples
with different structural quality where $\delta C=C-C_{L}$. Note the large
deviation of data points at high temperatures in the highest purity crystal
SL34 of solid-liquid coexistence, which makes the subtraction of the Debye
contribution in this sample questionable.
#### 3.1.1 Specific Heat
We compare our calculated specific heat with the experimental data by the Penn
State group LinChan09 ; LinChan07 for four different growth processes: BC20,
BC04, SL34 and SL31. BC20 (04) is the sample grown by blocked capillary (BC)
method over 20 (4) hours. SL 34 (31) represents the samples in solid-liquid
coexistence state with 34 (31) percents of solid ratio. Notice that sample
SL34 actually corresponds to their reported 75% solid-liquid coexistence
sample and SL31 corresponds to their constant pressure sample (CP) Su10a ;
LinChan09 . The data are described with three parameters: ${\cal D}_{0}$,
$E_{c}$ and the Debye temperature $\Theta_{D}$. We first determine
$\Theta_{D}$, or the lattice contribution, from the high-temperature data. The
lattice contribution is then subtracted from $C$ to obtain the difference
$\delta C=C-C_{L}$. We fit $\delta C/T$ with our specific heat formula for
TLS.
Next we plot $\delta C/T$ in Fig. 3. The TLS model with cutoff describes well
the data. In these plots we fixed the width of the cutoff to $W=1\,{\rm\mu
eV}$ since there is no qualitative difference when varying $W$ within
reasonable range. Notice that the shape of $\delta C(T)$ depends strongly on
the subtraction of the high-temperature lattice contribution.
The TLS behavior is mainly characterized by the zero-energy DOS and the cutoff
energy of the TLS, which are both noticeably larger in BC04, see Table 1. This
may be explained by the rapid growth process of a strained crystal, which
gives rise to both a larger TLS concentration, $n_{\rm TLS}$, and a smaller
cutoff energy, $E_{c}$, i.e., a smaller maximum tunneling barrier height.
Since the TLS concentrations of these samples range from $3.7$ to $21.5$ ppm,
which are at least 1000 times larger than the nominal 3He concentration, we
believe that 3He has no effect on the observed intrinsic heat capacity of
ultrapure solid 4He.
| $P$ | $V_{m}$ | $\Theta_{D}$ | ${\cal D}_{0}\times 10^{4}$ | $E_{c}\times 10^{2}$ | $n_{\rm TLS}$ | $\Delta S$
---|---|---|---|---|---|---|---
| (bar) | (cm3/ mol) | (K) | (1/meV) | (meV) | (ppm) | ($\mu$J/(mol K))
SL34 | 25 | 21.25 | 24.5 | 2.2 | 1.7 | 3.7 | 21.3
SL31 | 25 | 21.25 | 24.8 | 2.9 | 2.2 | 6.4 | 36.9
BC20 | 33 | 20.46 | 29.7 | 3.0 | 2.3 | 6.9 | 39.5
BC04 | 33 | 20.46 | 28.9 | 6.5 | 3.3 | 21.5 | 115.0
Table 1: Physical and model parameters: Debye temperature $\Theta_{D}$, zero
energy TLS DOS ${\cal D}_{0}$, cutoff energy $E_{c}$, concentration of TLS
$n_{\rm TLS}={\cal D}_{0}\times E_{c}$ and excess entropy $\Delta S$.
#### 3.1.2 Entropy Analysis
Our analysis of the excess entropy supports the existence of a glassy
component or TLS. The excess entropy,
$\displaystyle\Delta S(T)=\int_{0}^{T}dT^{\prime}\,\delta
C(T^{\prime})/T^{\prime},$ (5)
is associated with an excess specific heat. We find consistently for specific
heat measurements ClarkChan05 ; LinChan07 ; LinChan09 that the obtained
entropy values $\Delta S\sim 10^{-4}$ J/(K mol) are 5 to 6 orders of magnitude
smaller compared to the theoretical prediction for a homogeneous supersolid if
the entire sample actually underwent Bose-Einstein condensation (BEC). In the
limit of a non-interacting BEC bosons with a quadratic dispersion one finds
the standard result $\Delta
S_{BEC}=15/4(\zeta(5/2)/\zeta(3/2))\,R(T/T_{c})^{3/2}\sim(5/4)\,R\sim 10.4$
J/(K mol). This means that if $\Delta S$ is indeed due to supersolidity, then
the supersolid volume fraction is at most 11 ppm or 0.0011% in the most
disordered or quenched sample of the four ultrapure samples studied in this
work, i.e., sample BC04. Such a supersolid fraction in the specific heat is
more than 100 to 1000 times smaller than is usually reported for the non-
classical rotational inertia fraction (NCRIF) in TO experiments. This enormous
discrepancy between supersolid fractions in specific heat and TO experiments
was already noticed in Refs. Balatsky07 ; Graf08 , while Lin et al. LinChan09
keep invoking a hyperscaling mechanism of unknown origin. Until to date, this
discrepancy remains a major puzzle that is hard to reconcile within a
supersolid scenario.
The validity of the analysis of the entropy in terms of a non-interacting BEC
is repeatedly questioned on grounds of how robust it is in the presence of
interactions. We discussed in our original study Balatsky07 the effects of
interactions on the entropy. Here it is important to realize that the entropy
is a total count of all low-energy states irrespective of a particular model
for the specific heat. Hence, we concluded that strong-coupling effects cannot
change the order of the effect. They may only change the magnitude. For
example, the well-known strongly correlated superfluid system 4He possesses a
superfluid entropy of $\sim 0.6R\sim 4.6$ J/(K mol) Ahlers1973 , which is only
half the value of the non-interacting BEC. With hindsight this justifies the
neglect of strong-coupling effects in the order of magnitude analysis. Clearly
the entropy is a reliable measure of any phase transition in 4He. No matter
how one looks at this puzzle, the reported excess entropy is either far too
small to explain observed NCRIF effects in torsional oscillators or far too
large to describe the boiling off of 3He atoms from dislocation lines, when
the nominal concentration of 3He is less than 1 ppb. For those reasons, we
argued in favor of two-level systems of low-lying states until a better
microscopic understanding of solid 4He emerges.
#### 3.1.3 Comparison with Pressure Measurements
Figure 4: ($P-P_{0})/T^{2}$ vs. $T^{2}$ for $P\sim 33$ bar. The squares
represent the data reported by Grigor’ev et al. Grigorev07 ; Grigorev07b The
blue-thick and black-thin lines are predictions for BC04 and BC20,
respectively. The black-dotted line has been shifted vertically by a constant
compared to BC04 to illustrate the capability of describing Grigor’ev’s data
by the TLS model. Predicted curves for BC04 and BC20 are shown for comparison.
Figure 5: Low-temperature pressure deviation from lattice contribution of
Debye solid (data by Yin et al. Yin11 ). The intercept of $\Delta P/T^{2}$ vs.
$T^{2}$ extrapolated from low-temperature data points is in agreement with the
TLS contribution of order 100 ppm ($\Delta P=P-P_{0}$). The arrow marks the
deviation from the Debye solid behavior. The large scatter in data at lowest
$T$ is due to the subtraction of $P_{0}$. Dashed lines are guides to the eyes.
Next we relate the pressure measurements with the specific heat measurement
through our model. The quantities to characterize the pressure measurement in
the combined lattice and glass models are $a_{L}$ and $a_{g}$ defined by
$\displaystyle P(T)\equiv
P_{0}+P_{L}(T)+P_{g}(T)=P_{0}+a_{L}T^{4}+a_{g}T^{2},$ (6)
where $P(T)$ is the pressure at temperature $T$. $P_{0}$, $P_{L}$, $P_{g}$ are
the corresponding pressure contributions of the ions at zero temperature,
lattice vibrations, and two-level excitations of the glass. On the other hand,
the thermodynamic Maxwell relations between pressure and specific heat give
$\displaystyle\left(\frac{\partial P}{\partial
T}\right)_{V}=\frac{\gamma_{g}}{V_{m}}\,C_{g,V}+\frac{\gamma_{L}}{V_{m}}\,C_{L,V},$
(7)
where $\gamma_{i}$ are the Grüneisen coefficients of the glass ($g$) and
lattice ($L$). Literature values for the Grüneisen coefficient of phonons in
solid hcp 4He range between $2.6<\gamma_{L}<3.0$ Grigorev07b ; Driessen86 ,
while nothing is known about $\gamma_{g}$ of glassy 4He. For simplicity we
assume $\gamma_{g}\sim\gamma_{L}=2.6$. Equation (7) is related to the first
Ehrenfest relation involving the compressibility, which was shown to be always
satisfied in glasses Nieuwenhuizen1997 . In Figs. 4 and 5 we show the
temperature dependence of the $(P-P_{0})/T^{2}$ data reported by Grigor’ev et
al. Grigorev07 ; Grigorev07b and Yin et al.Yin11 . The curves for samples
BC04 and BC20 are derived from our specific heat analysis of the data by Lin
et al.LinChan09 The key result is the dependence $\sim T^{2}$ with finite
intercept at $T=0$. In the TLS model, the finite intercept describes the
glassy contribution, whereas the $T^{2}$ behavior is attributed to phonons. In
conclusion, the data by Grigor’ev et al. and Yin et al. are in agreement with
predicted $(P-P_{0})/T^{2}$ curves for a system of TLS.
## 4 General response function formalism and physical origin of dynamical
anomalies
To rationalize the TO experiments, we developed a general phenomenological
formalism in Ref. Nussinov07 . We have since extended and applied it to visco-
elastic and dielectric properties. With simple modifications, this predictive
approach can be used to study all measurable dynamical response functions.
Here, we summarize the essence of our approach. In later sections, we will
apply it in a self-contained way to the various quantities that we wish to
interrogate. We start with the equation of motion for generalized coordinates
$q_{i}$. These coordinates may be an angle, $q=\theta$, in the case of the TO
experiments, Cartesian components of atomic displacements , $q_{i}=u_{i}$ with
${\bf u}$ the local atomic displacement (as in our analysis of the visco-
elastic and dielectric response functions), or any other. Associated with
these coordinates are conjugate generalized momenta $p_{i}$ (e.g., angular
momentum in the TO analysis, linear momentum for Cartesian coordinates) and
their associated generalized forces $F_{i}$ (torques in the case of the TO,
and rectilinear forces for atomic displacements). Physically, these forces
correspond to a sum of two contributions: (i) Direct forces ($F_{direct}$).
These may originate from either externally applied forces on the solid
($F_{ext}$) or lowest order “direct” internal forces $(F_{int}$). By direct
forces we allude to forces on the coordinate $q$ that do not involve the
response of the system on $q$ as a result of its change.
(ii) Indirect “backaction” forces ($F_{BA}$). These allude to higher order
effects wherein a variation in the coordinate $q$ can lead to displacements in
other parts of the medium (e.g., those involving plastic regions or nearby
atoms) which then act back on the original coordinate $q$. To linear order
$\displaystyle
F_{BA}(t)=\int_{-\infty}^{t}g(t-t^{\prime})q(t^{\prime})dt^{\prime}.$ (8)
The backaction function $g(t-t^{\prime})$ captures how a displacement $q$ at
time $t^{\prime}$ can lead to a perturbation in the solid which then acts back
on the coordinate $q$ at time $t$. With these preliminaries in tow, we now
outline our standard linear response formalism which further takes into
account all higher order backaction effects.
$\bullet$ (1) Write down the Newtonian equation(s) of motion for the
generalized coordinate(s) $q$ that we wish to study. With $\chi_{0}$ a
suitable differential operator $\chi_{0}^{-1}$, these can be cast as
$\displaystyle\chi_{0}^{-1}q(t)=F_{direct}(t)+F_{BA}(t).$ (9)
This equation might seem a bit formal. To make it concrete, we note that in
the simplest case that we will discuss- that of the compact scalar angular
coordinate $\theta$ for the TO orientation, the operator
$\displaystyle\chi_{0}^{-1}=I_{osc}\frac{d^{2}}{dt^{2}}+\gamma_{osc}\frac{d}{dt}+\alpha_{osc}$
(10)
where $I_{osc},\gamma_{osc},$ and $\alpha_{osc}$ are, respectively, the
oscillator moment of inertia, dissipation, and torsion rod stiffness. A more
complicated tensorial operator involving the elastic modulus appears when
writing the equations of motion for the Cartesian components of the atomic
displacements.
$\bullet$ (2) When Fourier transformed to the complex frequency ($\omega$)
plane, $\chi_{0}(\omega)$ corresponds to the bare susceptibility. We will
denote the Fourier transforms of the various quantities (forces, backaction
function) by simply making it clear that the argument of the various
quantities is now the frequency $\omega$ and not the time $t$. We trivially
recast Eq. (9) as
$\displaystyle\chi^{-1}(\omega)q(\omega)=F_{direct}(\omega),$ (11)
where
$\displaystyle\chi^{-1}(\omega)=\chi_{0}^{-1}(\omega)-g(\omega).$ (12)
Equation (12) will be used in all of our upcoming analysis. The physical
content of Eq. (12) can be seen by writing its inverse as a geometric (or
Dyson) series
$\displaystyle\chi(t)=\chi_{0}(t)+\int
dt^{\prime}\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})$ $\displaystyle+\int
dt^{\prime}\int
dt"\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})g(t^{\prime}-t")\chi_{0}(t")+....$
(13)
The terms in this series correspond to (a) the direct contribution
($\chi_{0}$), (b) a lowest order backaction effect wherein a displacement at
an earlier time $t^{\prime}<t$ leads to a deformation of the solid which then
acts back on the coordinate at time $t$ (the term $\int
dt^{\prime}\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})$), (c) a higher
order process in which a deformation at time $t^{\prime\prime}$ leads to a
backaction from the solid on the coordinate at time $t^{\prime}>t"$ which in
turn then acts back on its surroundings which then acts back on the original
coordinate at time $t$ (the term $\int dt^{\prime}\int
dt"\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})g(t^{\prime}-t")\chi_{0}(t")$),
and so on ad infinitum. In Fourier space, the convolution integrals become
products and Eq. (4) becomes
$\displaystyle\chi(\omega)=\chi_{0}(\omega)+\chi_{0}(\omega)g(\omega)\chi_{0}(\omega)+\chi_{0}(\omega)g(\omega)\chi_{0}(\omega)g(\omega)\chi_{0}(\omega)+....$
(14)
The geometric series of Eq. (14) sums to Eq. (12). As simple as it is, Eq.
(12) combining standard linear response theory (step 1) with backaction
effects (or the Dyson equation) of step 2, leads to a very powerful tool that
allows us to investigate numerous systems while accounting for arbitrarily
high order backaction effects. As is well known and we will expand on and
employ in later sections, the real and imaginary parts of the poles of the
susceptibility $\chi(\omega)$ allow us to probe typical oscillation times and
dissipation. This will allow us to connect with TO and other measurements and
make precise statements about the backaction function $g$ which affords
information about the dynamics within the solid. It is important to stress
that Eq. (12) is general. In the derivation above no assumptions need to be
made concerning the precise physical origin of the backaction function $g$.
The adduced function $g$ captures all effects not present in the direct
equations of motion for the normal solid. If supersolid effects would be
present, they will directly appear in the function $g$.
Thus far, our sole assumption was that the deformations $q$ are small enough
to justify the linear (in $q$) order analysis of Equations (8, 9) for
measurements on the rigid solid. We now invoke additional assumptions (which
have been partially vindicated in a growing number of experiments since our
original proposal Nussinov07 ). These assumption relate to the form of
$g(\omega)$ and its dependence on temperature. They are motivated by our view
of defect quenching and characteristic relaxation times as the origin of the
4He anomalies.
$\bullet$ (3) As we will elaborate on in later sections, data for disparate
susceptibilites $\chi$ taken at different frequencies $\omega$ or temperatures
$T$collapse onto one curve. This indicates that $g$ is a function of only one
dimensionless argument ($\omega\tau(T)$) instead of both $\omega$ and $T$
independently. That is, there is only one dominant temperature dependent
relaxation time scale $\tau(T)$ for the backaction of the quenched solid on
the original coordinate $q$. As is well known and alluded to in Section (2),
exponential damping with a single relaxation time $\tau$, leads to a function
$g_{single}(\omega)=g_{0}(1-i\omega\tau)/(1+\omega^{2}\tau^{2})$ which, when
plotted with the real and imaginary parts of $g$ along the horizontal and
vertical axes describes a semi-circle as a function of the dimensionless
quantity $(\omega\tau)>0$. However, when plotted in this way, the 4He data for
the complex susceptibility measured by TO and other probes lead to a collapsed
curve which is more like that of a skewed semi-circle and there is a
distribution of relaxation times about a characteristic time scale $\tau$. For
ease of analysis, we will approximate the complex response functions by Eqs.
(2). The curve collapse allows for information about how the characteristic
transient relaxation times $\tau(T)$ increase as $T$ is decreased. We further
invoke the Vogel-Fulcher-Tammann form for glasses Rault00 ,
$\displaystyle\tau(T)=\left\\{\begin{array}[]{ll}\tau_{0}\,e^{\Delta/(T-T_{0})}&\mbox{
for $T>T_{0}$},\\\ \infty&\mbox{ for $T\leq T_{0}$}.\end{array}\right.$ (17)
Here, $T_{0}$ is the temperature at which the relaxation times would truly
diverge and $\Delta$ is an energy scale. In fitting the data in this way,
negative values $T_{0}<0$ were often found. That is, the typical dynamics as
adduced from the collapsed $\tau(T)$ is faster than simple activated dynamics
(one in which $T_{0}=0$ in Eq. (17)). This is consistent with the intrinsic
quantum character of the solid 4He crystal with large zero point motion as
compared to classical activated dynamics.
What is the physical content of this general formalism vis a vis the putative
supersolid transtition? Given perturbations of a typical frequency $\omega$,
the backaction response $g$ from the plastic components acting on $q$ may
either be sufficiently rapid or slow to respond. Just at the tipping point
when $\omega\tau(T_{X})\simeq 1$ different components will be maximally out of
synchrony with each other in being able to respond to the perturbation or not
and the dissipation (given by the reciprocal of the imaginary part of the zero
of $\chi^{-1}(\omega)$) is maximal. Similarly, there will be a change in the
typical periods of the system between a system which at high $T$ (i.e.,
$T<T_{X}$) contains rapidly equilibrating plastic components to those at low
$T$, which are too slow to respond and thus the system appears to have
undergone “a transition”.
In the sections that follow, we will apply and replicate anew and at great
length the considerations outlined above to the particular set of physical
quantities that we wish to investigate. We start, in Section 5 with the
investigation of the TO (for which the above formalism was first developed)
and then move to explore other arenas- the viscoelastic (Section 6) and
dielectric response (Section 7) functions where the above formalism leads to
experimentally testable predictions.
## 5 General susceptibility and response function of torsional oscillators
A formulation of the rotational susceptibility of the TO was given in Ref.
Nussinov07 . It is now often used as a basis for the linear response
discussion Dorsey08 ; Pratt11 ; Gadagkar2012 . In Section (4) we outlined the
key points of this formulation when written in its general form. Since its
derivation it has been applied by us and others to study other response
functions. The result of Eq. (20) below is none other than Eq. (12) when the
generalized coordinate $q$ corresponds to the TO angle. The bulk of this
section will be devoted to analyzing the experimental consequences of this
relation and its related counterpart for the double TO.
It is important to realize that the TO experiment measures the period and
dissipation of the entire apparatus by reporting the relationship between the
force and displacement (angle). Therefore a model is needed to determine the
relation between observable period and dissipation and the moment of inertia,
damping and effective stiffness of the media. We start with the general
equation of motion for a harmonic TO defined by an angular coordinate $\theta$
in the presence of an external and internal torque,Nussinov07
$\displaystyle
I_{osc}\ddot{\theta}(t)+\gamma_{osc}\dot{\theta}(t)+\alpha_{osc}\theta(t)={M}_{ext}(t)+{M}_{int}(t).$
(18)
Here, $I_{osc}$ is the moment of inertia of the (empty) TO apparatus,
$\alpha_{osc}$ is the restoring constant of the torsion rod, and
$\gamma_{osc}$ is its damping coefficient. $M_{ext}(t)$ is the externally
imposed torque by the drive.
${M}_{int}(t)=\int{g}(t-t^{\prime})\theta(t^{\prime})dt^{\prime}$ is the
internal torque exerted by solid 4He on the oscillator for a system with time
translation invariance. In general, the backaction $g(t-t^{\prime})$ is
temperature, $T$, dependent.
The external torque, $M_{ext}(t)=\dot{L}(t)$, is the derivative of the total
angular momentum of a rigid body, $L(t)=\frac{d}{dt}\int
d^{3}x~{}\rho(\vec{x})r^{2}~{}\dot{\theta}(\vec{x})$, where $r$ is the
distance to the axis of rotation, $\rho(\vec{x})$ is the mass density and
$\dot{\theta}(\vec{x})$ the local angular velocity about the axis of
rotation.Anderson08 The experimentally measured quantity is the angular
motion of the TO - not that of bulk helium, which is enclosed in it. Ab
initio, we cannot assume that the medium moves as one rigid body. If the non-
solid subsystem “freezes” into a glass, the medium will move with greater
uniformity and speed. This leads to an effect similar to that of the
nonclassical rotational moment of inertia, although its physical origin is
completely different. We argue for an alternate physical picture, namely that
of softening of the oscillator’s stiffness.
The angular coordinate $\theta(t)$ of the oscillator is a convolution of the
applied external torque ${M}_{ext}(t)$ with the response function
${\chi}(t,t^{\prime})$. Causality demands
${\chi}(t,t^{\prime})=\theta(t-t^{\prime}){\chi}(t,t^{\prime})$. Under Fourier
transformation, this leads to the Kramers-Kronig relations. In any time
translationally invariant system, the Fourier amplitude of the angular
response of the TO is
$\displaystyle\chi_{0}^{-1}(\omega)\theta(\omega)=M_{ext}(\omega)+M_{int}(\omega).$
(19)
Defining the total angular susceptibility as
$\chi^{-1}=\chi_{0}^{-1}-M_{int}$, we write the effective inverse
susceptibility as
$\displaystyle\chi^{-1}(\omega)=\alpha_{osc}-i\gamma_{osc}\omega-
I_{osc}\omega^{2}-g(\omega),$ (20)
where $g(\omega)$ is the Fourier transform of the backaction due to the added
solid 4He. In what follows, we will treat the backaction as a small
perturbation to the TO chassis.
We will now apply our formalism to the study of the ingle TO, which is
described by Eq. (20), and then turn to the double TO. Very recently, Beamish
Beamish2012 and Maris Maris2012 employed the same general linear response
formalism to explain some of the TO results in terms of purely mechanical
effects due to either the changing stiffness of the torsion rod or floppiness
of the sample cell flange (lid).
### 5.1 A model for the single torsional oscillator
In what follows, we analyze the experimental consequences of Eq. (20).
#### 5.1.1 Rotational susceptibility - period and dissipation
We can now calculate specific consequences of the phenomenological model
introduced above. The effective oscillator parameters are defined as the sum
of parameters describing the apparatus, $\chi_{0}^{-1}$, and the added solid
4He given by
$g(\omega)=i\gamma_{He}\omega+I_{He}\omega^{2}+g_{0}G(\omega).$ (21)
It is convenient to introduce a net moment of inertia $I=I_{osc}+I_{He}$ and
net dissipation $\gamma=\gamma_{osc}+\gamma_{He}$. The transient dynamic
response function $G(\omega)$ can be approximated by a distribution of
overdamped oscillators with relaxation time $\tau$ as discussed in Section (2)
[see Eqs. (2), in particular].
The resonant frequency of the TO with a backaction is given by the root of
$\displaystyle\chi^{-1}(\omega)=\alpha-i\gamma\omega-I\omega^{2}-g_{0}G(\omega)\equiv
0.$ (22)
As discussed in Section 4, we anticipate that when the relaxation time is
similar to the period of the underdamped oscillator, the dissipation will be
maximal, sometimes referred to as “$\omega\tau=1$” physics.
In linear response theory, the homogeneous Eq. (22) is scale invariant. Thus,
we normalize all oscillator quantities by the effective moment of inertia $I$,
i.e., $\bar{\alpha}=\alpha/I$, $\bar{\gamma}=\gamma/I$, and
$\bar{g}_{0}=g_{0}/I$. As can be seen from Eq. (22), for an ideal
dissipationless oscillator ($\bar{\gamma}=0$), the resonant frequency
$\omega_{0}=\sqrt{\bar{\alpha}}$ is the pole of $\chi(\omega)$ in the limit
$1/\tau\to 0$. If we expand $\chi^{-1}$ about this root,
$\omega=\omega_{0}+\delta\omega$, we find to leading order in $\delta\omega$
$\displaystyle\delta\omega\approx-\frac{i\bar{\gamma}\omega_{0}+\bar{g}_{0}G(\omega_{0})}{i\bar{\gamma}+2\omega_{0}}.$
(23)
It follows that the shift in dissipation with respect to high temperatures is
$\displaystyle\Delta Q^{-1}\equiv
Q^{-1}-Q^{-1}_{0}\approx\frac{\bar{g}_{0}}{\omega_{0}^{2}}{\rm Im\
}G(\omega_{0}),$ (24)
whereas the shift in resonant frequency is
$\displaystyle\Delta\omega\equiv 2\pi(f_{0}-f)$ $\displaystyle\approx$
$\displaystyle\frac{\bar{g}_{0}}{4\omega_{0}^{2}}\Big{(}2\omega_{0}\,{\rm Re\
}G(\omega_{0})+\bar{\gamma}\,{\rm Im\ }G(\omega_{0})\Big{)},$ (25)
which increases monotonically when $T$ is lowered. Combining Eqns. (24) and
(25) for the strongly underdamped oscillator, we arrive at
$\frac{\Delta Q^{-1}}{\Delta\omega}=\frac{4{\rm Im\
}G(\omega_{0})}{2\omega_{0}{\rm Re\ }G(\omega_{0})+\bar{\gamma}{\rm Im\
}G(\omega_{0})}\approx\frac{2}{\omega_{0}}\frac{{\rm Im\ }G(\omega_{0})}{{\rm
Re\ }G(\omega_{0})}.$ (26)
It is this general relationship for the response function of the damped
oscillator that was successfully applied in the TO analysis by Pratt et al.
Pratt11 ; Gadagkar2012 to demonstrate the interplay of rotational,
relaxation, and shear dynamics in solid 4He. For a Debye relaxor the ratio of
Eq. (26) reduces to $2\tau$ and provides a direct means to measure the
relaxation time. Similar results for the ratio were obtained for other
phenomenological models.Huse07 ; Dorsey08 For example, Huse and Khandker
Huse07 assumed a simple phenomenological two-fluid model, where the
supersolid is dissipatively coupled to a normal solid resulting in a ratio of
${\Delta Q^{-1}}\frac{\omega_{0}}{\Delta\omega}\approx 1$, while Yoo and
DorseyDorsey08 developed a viscoelastic model, and KorshunovKorshunov09
derived a two-level system glass model that captures the results of the
general model originally proposed by Nussinov et al.Nussinov07 To make
further progress we assume that $\tau$ follows the phenomenological Vogel-
Fulcher-Tammann (VFT) equation of Eq. (17).
Figure 6: The resonant frequency (black, left axis) and dissipation (red,
right axis) vs. temperature. The experimental data from Hunt et al.Hunt09 are
well described by a Cole-Cole (CC) distribution function. Graf10 ; Graf11
Figure 6 provides a fit to the measured data by Hunt et al.Hunt09 assuming a
CC distribution of relaxation times. As shown, an excellent fit is obtained.
For comparison, we also tried a DC distribution for relaxation times, but
found only fair agreement. It is worth mentioning that unlike in the Debye
relaxation analysis by Hunt and coworkers, i.e., a single overdamped mode, we
do not require a supersolid component to simultaneously account for frequency
shift and dissipation peak Graf10 ; Graf11 . Our model leads to a universal
scaling of period change vs. dissipation in a Cole-Cole or Davidson-Cole plot
as seen in Ref. Pratt11 ; Gadagkar2012 . Indeed similar viscoelastic behavior
may have already been observed in solid hydrogen Clark2006 .
### 5.2 A model for the double oscillator
Figure 7: Cartoon of the double torsional oscillator modeled in Eqns. (5.2).
The upper moment of inertia ($I_{1}$) is the dummy bob, while the lower moment
of inertia ($I_{2}$) is the cylindrical pressure cell that can be loaded with
4He. The stiffness of the torsion rods is given by $k_{1}$ and $k_{2}$ with
$k_{1}\approx k_{2}$ by design.
The double TO results of the Rutgers group by Kojima have proved difficult to
explain when simply extrapolating from the single TO model Dorsey08 . Here we
model the coupled double TO, sketched in Fig. 7, by the following system of
equations:
$\displaystyle\left(-I_{1}\omega^{2}-i\gamma_{1}\omega+k_{1}+k_{2}\right)\Theta_{1}(\omega)-k_{2}\Theta_{2}(\omega)$
$\displaystyle=$ $\displaystyle F(\omega),$
$\displaystyle\left(-I_{2}\omega^{2}-i\gamma_{2}\omega+k_{2}-g(\omega)\right)\Theta_{2}(\omega)-k_{2}\Theta_{1}(\omega)$
$\displaystyle=$ $\displaystyle 0.$ (27)
where $\Theta_{i}$ are torsion angles, $\gamma_{i}$ are damping coefficients,
$k_{i}$ are torsion rod stiffnesses, $g(\omega)$ is the glass backaction term,
and $F(\omega)$ is the applied external torque. The subindex “$1$” refers to
the upper or dummy bob in the experiment, while “$2$” refers to the lower
oscillator with the pressure cell that can be loaded with solid 4He. For a
strongly underdamped oscillator and a small backaction, it suffices to solve
first for the bare resonant frequencies and later include perturbatively
damping and backaction terms, for details see Ref. Graf10 . More recently this
approach has been extended to a triple TO Mi2011 .
Figure 8: Frequency and dissipation in double TO by Aoki et al.Aoki07
(symbols) compared with glass theory (lines). Panel (a): Temperature
dependence of resonant frequency shifts $\Delta f_{1}$ (black, left axis) and
$\Delta f_{2}$ (blue, right axis). Panel (b): Temperature dependence of
dissipation $Q_{1}^{-1}$ (black, left axis) and $Q_{2}^{-1}$ (blue, right
axis). The experimental data were corrected for the significant temperature-
dependent background of the empty cell.
Figure 8 shows good agreement between our phenomenological model of the
coupled double TO and experiment. The TO parameters $I_{i}$ and $k_{i}$ can be
determined from the bare resonant frequencies $f_{i}^{0}=\omega_{i}^{0}/2\pi$.
In addition, the damping coefficients $\gamma_{i}$ can be extracted from the
high-temperature dissipation $Q_{i,\infty}^{-1}$. Finally, the backaction
${g}(\omega)$ accounts through $\tau(T)$ for the temperature dependence of
$\Delta f_{i}$ and $Q_{i}^{-1}$. Our phenomenological theory of the double
oscillator explained for the first time both frequency shift and dissipation
peak for in-phase and out-of-phase torsional response Aoki07 . Data for in-
phase frequency $f_{1}=496$ Hz and out-of-phase $f_{2}=1173$ Hz are shown in
Fig. 8, plotted against the temperature. The obtained values for moment of
inertia and rod stiffness agree well with other estimates Aoki08 .
Remarkably an anomalous damping coefficient $\gamma_{1}\sim-\gamma_{2}$ is
required to explain the behavior of increased dissipation with increased
frequency. Such anomalous damping is already required to describe the unloaded
pressure cell. Thus it is unrelated to the properties of solid 4He and an
intrinsic property of the composite TO. After loading the cell with solid 4He
the dissipation ratio becomes $Q_{2}^{-1}/Q_{1}^{-1}=2.5$ at 300 mK with
frequency ratio $f_{2}/f_{1}=2.37$. Our fit results in a negative parameter
$T_{0}=-32.73$ mK. This effective negative value of $T_{0}$ is in line with
earlier comment in Section 4, concerning the quantum character of solid 4He.
This value may be indicative of strong zero point quantum fluctuations that
thwart a glass transition.
Finally, the comparison in Fig. 8 shows that an explicit frequency-dependent
backaction must be used with
$g_{0}(\omega)=g_{0}\left({\omega}/{\omega_{1}^{0}}\right)^{p}$ and $p=1.77$
to account for the experimental fact of $\Delta f_{1}/f_{1}\approx\Delta
f_{2}/f_{2}$, i.e., the relative frequency shift is unaffected by the changing
resonant frequency. In contrast, various theories describing solid 4He in
torsional oscillators as viscoelastic material Dorsey08 or two-level systems
moving through a solid matrix Andreev07 ; Andreev09 ; Korshunov09 predict an
exponent of $p=4$ for the backaction term.
## 6 Shear and stiffness of a viscoelastic medium
Another aspect of the dynamic response of 4He crystals was revealed through a
series of elasticity studies.Paalanen81 ; Goodkind02 ; Burns93 ; Beamish07 ;
Beamish09 ; Beamish10 In particular, Beamish and coworkers demonstrated the
qualitative similarity between shear modulus and the TO measurements. In the
shear modulus experiment solid helium is grown in between and around two
closely spaced sound transducers. When one of the transducers applies an
external strain, the other transducer measures the induced stress from which
the shear modulus of the sample is deduced.Beamish07 In this way, the
experiment provides a direct measurement of the elastic response to the
applied force within a broad and tunable frequency range. The frequency
dependence is especially crucial in determining the nature of the relaxation
processes and complements current TO experiments with their limited frequency
range.
Similar to the TO analysis in the previous section, we analyzed the shear
modulus within the general linear response function framework, where the
amplitude of the shear modulus increases (stiffens) upon lowering $T$, because
of the freezing out of low-energy excitations. This change is accompanied by a
prominent dissipation peak, indicative of anelastic relaxation processes. We
calculated the complex shear modulus $\mu(\omega;T)$ of a viscoelastic
material and predicted: (a) the maximum of the shear modulus change and the
height of the dissipation peak are independent of frequency and (b) the
inverse crossover temperature $1/T_{X}$ vs. the applied frequency $\omega$
obeys the form $\omega\tau(T_{X})=1$ characteristic of dynamic crossover.
### 6.1 Model of dynamic shear modulus
As in our analysis of the TO, we start with the same general linear response
function formulation outlined in Section 4. Our final result of Eq. (31) will,
once again, reflect the general relation of Eq. (12). Here we replace the
angular coordinate of the TO with a displacement coordinate and the restoring
force with a stress tensor Su10b . For ease, our notation in the below will
differ slightly from that in Section 4. The equation of motion for
displacement $u_{i}$ in the $i$-th direction of a volume element in the
presence of an external driving force is
$\displaystyle-\rho\,\omega^{2}u_{i}+\partial_{j}\,\sigma_{ij}^{\rm He}\ \
=f_{i}^{\rm EXT}(\omega)+f_{i}^{\rm BA}(\omega),$ (28)
where $\rho$ is the mass density, $f_{i}^{\rm EXT}$ and $f_{i}^{\rm BA}$ are
the external force density and the backaction force density, and
$\sigma_{ij}^{\rm He}$ is the elastic stress tensor of solid helium. In
general, $\sigma_{ij}^{\rm He}=\lambda_{ijkl}\,u_{kl}$, with the elastic
modulus tensor $\lambda_{ijkl}$ LL_elastic . In the case of a homogeneous
solid with shear wave propagation along the $z$ axis and wave polarization in
the $x$-$y$ plane, the backaction takes on the form
$\displaystyle f_{i}^{\rm
BA}(\omega)={\overline{G}}(\omega;T)\,\partial_{z}^{2}\,u_{i}(\omega),$ (29)
where $\overline{G}$ describes the strength of the backaction on the solid
(viscoelastic response) and $i=x,y$. Although $f_{i}^{\rm BA}(\omega)$ is much
smaller than the purely elastic restoring force
$\partial_{j}\,\sigma_{ij}^{\rm He}$, it is this term that is responsible for
the stiffening of shear modulus with decreasing temperature.
Polycrystalline and amorphous materials are nearly isotropic, hence the
elastic modulus tensor becomes
$\lambda_{ijkl}=\lambda_{0}\delta_{ij}\delta_{kl}+\tilde{\mu}_{0}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$.
Note the stress tensor in Eq. (28) is finite only for orientations $j=z$ and
either $k$ or $l$ equal to $z$. With $k,l$ being interchangeable, the relevant
element will be $\lambda_{iziz}$, which gives the purely elastic shear modulus
$\tilde{\mu}_{0}$. Finally the fully dressed shear modulus (dressed by the
backaction) relates the displacement to an external force, or
$[-\rho\,\omega^{2}+\mu\,\partial_{z}^{2}]u_{i}(\omega)=f_{i}^{\rm
EXT}(\omega)$. Comparing this expression with Eq. (28), we obtain for the
dynamic shear modulus in a viscoelastic material the general response function
$\displaystyle\mu(\omega;T)=\tilde{\mu}_{0}(T)-\overline{G}(\omega;T).$ (30)
Next we employ for $\overline{G}(\omega;T)$ the Cole-Cole distribution
function [specifically, by reference to Eq. (2), we set
$\overline{G}(\omega;T)=g_{0}\mu_{0}G(\omega,T)$ to obtain the specific form
$\displaystyle\mu(\omega;T)$ $\displaystyle=$
$\displaystyle\tilde{\mu}_{0}\left[1-\frac{g_{0}}{1-(i\omega\tau)^{\alpha}}\right],$
(31)
with the sample dependent parameter $g_{0}$. The experimentally measurable
quantities are the amplitude of the shear modulus, $|\mu|$, and the phase
delay between the input and read-out signal, $\phi\equiv{\rm arg}\,(\mu)$;
$\phi$ measures the dissipation of the system, which is related to the inverse
of the quality factor $Q^{-1}\equiv\tan\phi$.
Several interesting results follow immediately from the general response
function in Eq. (31): (1) The change in shear modulus $\Delta\mu$ between zero
and infinite relaxation time is $\Delta\mu/\tilde{\mu}_{0}=g_{0}$. It measures
the strength of the backaction as well as the concentration of defects. (2) At
fixed $T$, the shear modulus amplitude $|\mu(\omega;T)|$ decreases with
increasing $\omega$. (3) The parameter $\omega\tau$ is the only scaling
parameter. (4) The peak height $\Delta\phi$ of the phase angle is proportional
to $g_{0}$. When $\omega\tau\sim 1$, then $\Delta\phi\approx
g_{0}\,{\cot(\alpha\pi/4)/(2-g_{0})}$ for $g_{0}\ll 1$. In the limit
$1<\alpha\leq 2$, this simplifies further to
$\displaystyle\Delta\phi\approx(1-\alpha/2)(\Delta\mu/\tilde{\mu}_{0})\ll 1,$
(32)
where $\Delta\phi$ is in units of radians. Quite remarkably, the peak height
$\Delta\phi$ depends only on the phenomenological parameters $\alpha$ and
$g\propto\Delta\mu$. At fixed temperature the maximum change of both
$\Delta\mu$ and $\Delta\phi$ is independent of frequency.
### 6.2 Results
Let us compare our model calculations with the experimental shear modulus
measurements by Beamish and coworkers Beamish10 for a transducer driven at
2000 Hz, 200 Hz, 20 Hz, 2 Hz, and 0.5 Hz Comment1 . Specifically, for the
$T$-dependent $\tau(T)$ in Eq. (31) we consider Vogel-Fulcher-Tammann (VFT)
and power-law (PL) relaxation processes. In our model parameter search, we are
not constraining $T_{0}$ to be positive (where $\tau$ diverges). Fair
agreement between calculations with a single set of parameters and experiments
is obtained with $T_{0}=-69.3$ mK, see Fig. 9. We refer to these calculations
as “VFT<”. Similar to the TO results, a negative $T_{0}$ means that no real
phase transition occurs in solid 4He. The agreement between theory and
experiment at highest (2000 Hz) and lowest (0.5 Hz) frequencies is of poorer
quality. This discrepancy may be related to very different backgrounds at
these frequencies and may not be intrinsic to 4He. Another reason may be the
presence of additional dissipation mechanisms, not accounted for. Furthermore,
our calculations confirm that the change of amplitude and peak height of phase
angle are nearly frequency independent between 2 Hz and 200 Hz. By defining
the crossover temperature $T_{X}$ as the point where the phase angle peaks, we
find as predicted that $T_{X}$ decreases with decreasing $\omega$.
Figure 9: Experimental data and theoretical calculations of the shear modulus
vs. temperature assuming a VFT relaxation time. The red and blue squares are
the experimental data for the amplitude and dissipation of shear modulus. The
black-solid lines show the theoretical calculations. We used the set of
parameters $\alpha=1.31$, $g_{0}=1.44\times 10^{-1},\tilde{\mu}_{0}=0.47$ pA
Hz-1, $\tau_{0}=50.0$ ns, $\Delta=1.92$K, and $T_{0}=-69.3$ mK. Notice a
negative $T_{0}$ means that there is no true phase transition occurring at
finite temperatures, probably because of strong quantum fluctuations of helium
atoms. The shear modulus amplitude and the phase angle were shifted by 0.1 pA
Hz-1 and 3 degrees with respect to the 2 Hz data.
Finally, when we set $T_{0}=0$ K the VFT expression (“VFT0”) reduces to an
activated Arrhenius form. While it describes reasonably well the position
$T_{X}$, it gives a much narrower linewidth for $\Delta\phi$ than does VFT<,
which is not in accord with the data and thus not shown. Notice that the VFT
relaxation is not the only possible relaxation process that can describe the
data; power-law or other types of relaxation can give similar level of
agreement to the experiment Beamish10 . Iwasa proposed a relaxation process
Iwasa10 similar to our phenomenological one, which is based on the theory of
pinned vibrating dislocation lines by Granato and Lücke GranatoLuecke56 .
Clearly further experiments at lower frequencies and lower temperatures are
required to determine the exact type of relaxation processes in solid 4He.
Figure 10: The Cole-Cole plots for experimental data and for VFT calculation.
For given form of $\tau$, all different frequency curves collapse onto one
single master curve reflecting that $\omega\tau$ is the only scaling
parameter. The Cole-Cole plots show reflection symmetry about
Re[$|\mu-\tilde{\mu}_{0}|/\Delta\mu$]=0.5, which is a consequence of the Cole-
Cole distribution function.
Figure 10 shows the Cole-Cole plot for experiments and calculations. The
experimental curves for different frequencies collapse roughly onto one curve
confirming our theoretical assumption that $\omega\tau(T)$ is a universal
scaling parameter. This behavior was also seen in TO experiments Hunt09 . In
addition, the data are symmetric about
Re$[|\mu-\tilde{\mu}_{0}|/\Delta\mu]=0.5$ as expected for the Cole-Cole
distribution. A more detailed data analysis may resolve the remaining
discrepancy between theory and experiment. The discrepancy may be due to
either the presence of additional relaxation processes at temperatures above
$T_{X}$, i.e., a more complicated form for $\tau(T)$, or by a modified
functional form of the defect distribution function.
Figure 11: Prediction for the inverse crossover temperature vs. applied
frequency. The green squares correspond to $T_{X}$ in Ref. Beamish10 . For the
power-law process with phase transition occurring at 40 mK, we used
$\tau=\tau_{0}(|T_{g}|/(T-T_{0}))^{p}$ for $T>T_{0}$ and $\tau=\infty$ for
$T\leq T_{0}$.
The pertinent question about a true phase transition at zero frequency vs.
crossover dynamics can be addressed by investigating the asymptotic limit of
$\omega\tau(T)=1$. From this expression we estimate $T_{X}$ as a function of
the applied frequency $f=\omega/2\pi$. Figure. 11 shows $1/T_{X}$ vs. $f$. The
VFT< and VFT0 calculations give significantly better agreement than the PL
calculations with phase transitions occurring at 0 K (PL0) and 40 mK (PL40).
For positive $T_{0}$ (see PL40), we find a true freeze-out transition, which
would indicate arrested dynamics for $f\to 0$ Hz. For both VFT and PL
relaxation times our calculations demonstrate that in the low-frequency limit
the existence of a phase transition shows clear signatures of $T_{X}$
converging toward the ideal glass temperature $T_{0}$. Therefore the absence
of arrested behavior may serve as experimental evidence against a true phase
transition.
### 6.3 Viscoelastic model
The viscoelastic model successfully describes composite materials with
anelastic properties. In fact, Yoo and Dorsey Dorsey08 applied the
viscoelastic model to the TO experiments. More generally, a distribution of
viscous components, embedded in an otherwise elastic solid, may be treated
through a generalized Maxwell model Su10b ; Su11 . Conceptually one may
subdivide the material into many elements and solve the coupled materials
equations for stress and strain. Here we use constitutive materials equations
to show that our results for a glass are equivalent to the generalized Maxwell
model with parallel connections of an elastic component with an infinite set
of Debye relaxors, see Fig. 12.
Figure 12: Lump circuit of the generalized Maxwell model. The elastic spring
$\mu_{0}$ represents the purely elastic shear modulus at high temperature.
Each Debye relaxor is made out of a series of elastic spring $\mu_{\rm RS}$
and dissipative dash-pot $\eta^{(n)}$.
The anomalous stiffening of the shear modulus can be described within the
viscoelastic model, though other defect mechanisms like dislocation glide are
possible too Friedel ; Caizhi2012 . The equivalent lump circuit model is
sketched in Fig. 12, where the basic dissipative element is the Debye relaxor.
It is comprised of a serial connection of a rigid solid (RS), characterized by
an elastic shear modulus $\mu_{\rm RS}$, and a Newtonian liquid (NL) or dash-
pot, characterized by a viscosity $\eta$. The RS component describes the ideal
elastic solid helium of this volume element, while NL represents the anelastic
component, which gives rise to viscous damping. The two parts are connected in
series, so that both share the same magnitude of stress, while the net strain
is additive. The strain rate equation for both constituents of the Debye
relaxor is
$\displaystyle\dot{\epsilon}={\dot{\sigma}}/{\mu_{\rm RS}}+{\sigma}/{\eta},$
(33)
where $\epsilon$ is the net strain of the Debye relaxor (DR) and $\sigma$ is
the magnitude of stress shared by the components RS and NL. In order to obtain
the above equation, we used the constitutive materials equations for strain
and stress: $\epsilon_{\rm RS}=\sigma_{RS}/\mu_{RS}$ and $\dot{\epsilon}_{\rm
NL}=\sigma_{NL}/\eta$. After performing the Fourier transformation we obtain
for a single Debye relaxor (DR) the shear modulus $\mu_{\rm
DR}=\sigma/\epsilon$,
$\displaystyle\mu_{DR}(\omega)=\frac{\mu_{\rm RS}}{1+\frac{i}{\omega\tau_{\rm
DR}}}=\mu_{\rm RS}\left[1-\frac{1}{1-i\omega\tau_{\rm DR}}\right],$ (34)
with relaxation time $\tau_{\rm DR}\equiv\eta/\mu_{\rm RS}$. For a
viscoelastic material with a single relaxation time, the solid behaves like a
parallel connection between the elastic part and the Debye relaxor
$\displaystyle\mu(\omega)$ $\displaystyle\equiv$
$\displaystyle\mu_{0}+\mu_{\rm
DR}(\omega)=\tilde{\mu}_{0}\left[1-\frac{g_{0}}{1-i\omega\tau_{\rm
DR}}\right],$ (35)
with $g_{0}=\mu_{\rm RS}/\tilde{\mu}_{0}$ and
$\tilde{\mu}_{0}=\mu_{0}+\mu_{\rm RS}$ is the dressed elastic shear modulus.
To consider the general case of many components, we introduce Debye relaxors
with different relaxation times connected in parallel as shown in Fig. 12. The
total anelastic contribution from $n$ such constituents is given by a weighted
sum. The corresponding continuous version of this expression with a
distribution $P(s)$ of relaxation times is
$\displaystyle\mu_{ae}(\omega)=\mu_{\rm RS}\int_{0}^{\infty}ds\
P(s)\,\left[1-\frac{1}{1-i\omega\tau s}\right]=\mu_{\rm RS}-\frac{\mu_{\rm
RS}}{1-(i\omega\,\tau)^{\alpha}}.$ (36)
Here the Cole-Cole distribution,
$\displaystyle
P(s)=\frac{t^{-(1-\alpha)}\sin\alpha\pi}{1+s^{2\alpha}+2s^{\alpha}\cos\alpha\pi},$
(37)
was used. Similar to our response functions elsewhere, the net shear modulus
of the composite system is given by the sum of two terms- the purely elastic
component and the anelastic contribution
$\displaystyle\mu(\omega)$ $\displaystyle=$
$\displaystyle\mu_{0}+\mu_{ae}(\omega)=\tilde{\mu}_{0}\left[1-\frac{g_{0}}{1-(i\omega\tau)^{\alpha}}\right].$
(38)
Indeed, this expression is identical to the one obtained previously using the
general response function formalism with a backaction (12). This is no
coincidence, since the backaction term accounts for damping and thus describes
the anelastic response of defects to the external stress.
## 7 Dielectric properties of a viscoelastic medium
The measurements of $\epsilon(\omega,T)$ by Yin et al. Yin11 showed the
anomalous increase of the dielectric function of solid 4He at low
temperatures. A similar test experiment in liquid helium showed no such
effect. We propose that these results may be explained by an electro-elastic
coupling of a quenched solid with frozen-in internal stress. Such behavior
cannot be described by the standard Clausius-Mossotti equation via a change in
mass density or polarizability (due to, e.g., dipole-induced dipole
interactions). Neither the measured change of the mean mass density
$\delta\rho/\rho\sim 10^{-6}$, nor the predicted correction in polarizability,
which actually leads to a decrease of $\epsilon(\omega,T)$ at low temperatures
Kirkwood36 ; Chan77 , can account for the reported anomalous change of the
dielectric function of order $\delta\epsilon/\epsilon\sim 10^{-5}$, while a
model with frozen-in stress and electro-elastic coupling can explain the data.
### 7.1 Model for electro-elastic properties
The relation of Eq. (47) that we derive below constitutes yet another
realization of our general relation of Eq. (12). We now turn to the specifics
of the electro-elastic coupling describing the interaction between the
electromagnetic and the strain fields. The coupling may be obtained by
expanding the dipole moment, $\bf p(r_{\it a})$, of a nonpolar atom around its
equilibrium value:
$\displaystyle{\bf p(r_{\it a})\approx p(R_{\it a})+(u_{\it a}\cdot\nabla)\
p\left.\right|_{R_{\it a}}},$ (39)
where $\bf r_{\it a}=R_{\it a}+u_{\it a}$ is the position of atom $a$, $\bf
R_{\it a}$ is its equilibrium position, and $\bf u_{\it a}$ is its
displacement. The polarization is obtained by averaging ${\bf p}$ over a
macroscopic volume element $v$, ${\bf P(r)}=({1}/{v})\sum_{\it a}{\bf
p(r_{a})}$. In the continuum limit (when the macroscopic length scale is far
larger than the atomic length) $\sum_{a}{\bf p(r_{\it
a})}\approx(1/v)\int_{v}d{\bf r}^{\prime}\ {\bf p(r^{\prime})}$ . In the
presence of a local strain field, to linear order in the displacement, Eq.
(39) reads
$\displaystyle{\bf P(r)}\approx(1/v^{2})\int_{v}d\,{\bf R\
[\,p(R)+(u\cdot\nabla)\ p(R)\,]}.$ (40)
Integration of the first term yields the macroscopic polarization for zero
internal strain (a solid in equilibrium), ${\bf P}_{0}$. It is related to the
macroscopic electric field ${\bf E}$ by ${\bf P}_{0}\equiv\chi_{0}{\bf E}$
where $\chi_{0}=(\epsilon_{0}-1)/4\pi$ is the zero-strain susceptibility and
$\epsilon_{0}$ is the permittivity. The second term in the integration
describes the polarization change $\bf\delta P$ due to atomic displacements.
Neglecting surface contributions the second term modifies the polarization
${\bf P}={\bf P}_{0}+\delta{\bf P}$ by
$\displaystyle\delta{\bf P}=-(1/v^{2})\int d{\bf R\ (\nabla\cdot u)\
p(R)}\approx-e_{ii}\,{\bf P}_{0},$ (41)
with $e_{ii}=(1/v)\int_{v}d{\bf R}\,(\nabla\cdot\bf u)$ the macroscopic
frozen-in dilatory strain (we use the repeated indices summation convention
$e_{ii}\equiv e_{11}+e_{22}+e_{33}$). In obtaining Eq. (41), we assumed that
${\bf P}$ is slowly varying. This long-wave length approximation holds for
wavelengths of microwave ($\lambda>10^{-3}$m) and above. The polarization
change alters the dielectric function
$\epsilon_{ii}=(\epsilon_{0})_{ii}+\delta\epsilon_{ii}$ by
$\displaystyle\delta\epsilon_{ii}\,(\omega,T)=-4\pi\chi_{0}\,e_{ii}\,(\omega,T)$
(42)
with $\epsilon_{ii}-1=4\pi P_{i}/E_{i}$. Only the diagonal components of the
strain tensor play a role in a leading order expansion of the electro-elastic
coupling. In principle, the shear strain can couple to the electric field by
considering dipole-induced dipole interactions (van der Waals), which is a
higher order effect. The same coupling mechanism between photon (electric
field) and phonons (strain) gives rise to the acousto-optical effect. Our
derivation of Eq. (42) is, to leading order, equivalent to the Pockels
coefficient for acousto-optical coupling in isotropic dielectrics,
$\delta\epsilon_{ij}=-\epsilon^{2}\,[2P_{44}\,e_{ij}+P_{12}\,e_{kk}\delta_{ij}],$
where $P_{kl}$ are the reduced Pockels coefficients (of order unity)
Werthamer69 ; Kisliuk91 ; Landau_Lifshitz .
We next turn to the locally frozen-in strains and write the equation of motion
within the general response function theory. As before in an isotropic medium
the elastic stress tensor $\sigma_{ij}^{\rm He}=\lambda_{ijkl}\,\partial
u_{k}/\partial x_{l}$ with
$\lambda_{ijkl}=\lambda_{0}\delta_{ij}\delta_{kl}+\mu_{0}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$.
If the electric field couples to local density fluctuations only through
dilatory strain, then the important matrix element is the Lamé parameter
$\lambda_{0}$. We write the displacement to an out-of-equilibrium internal
(INT) force in the presence of the backaction as
$\displaystyle-\rho\omega^{2}\,u_{i}(\omega)+\lambda_{0}\,\partial_{i}^{2}u_{i}(\omega)=f_{i}^{\rm
INT}(\omega)+f_{i}^{\rm BA}(\omega),$ (43)
where $u_{i}$ is a displacement in the $i$th direction and $\rho$ is the mass
density. The backaction force density of nearby atoms is given by
$\displaystyle f_{i}^{\rm
BA}(\omega)=\overline{G}(\omega;T)\,\partial_{i}^{2}\,u_{i}(\omega),$ (44)
where ${\overline{G}}$ is the strength of the pertinent backaction.
$f_{i}^{\rm INT}$ is the out-of-equilibrium (frozen-in) internal force density
in the $i$th direction at the defect. Integrating Eq. (43) yields the strain
due to the internal dilatory stress $\sigma_{ii}$:
$\displaystyle{e}_{ii}(\omega,T)={\sigma}_{ii}\left(\lambda_{0}-\overline{G}(\omega;T)\right)^{-1}.$
(45)
Again, we assume that the backaction can be described by a Cole-Cole
distribution of Debye relaxors,
$\overline{G}(\omega;T)={g_{0}}\lambda_{0}/{\big{[}1-(i\omega\,\tau)^{\alpha}\big{]}}$.
The corresponding local dilatory strain reads
$\displaystyle{e}_{ii}(\omega,T)$ $\displaystyle=$ $\displaystyle
e_{0}\left(1-{g_{0}}/\big{[}{1-(i\omega\tau)^{\alpha}}\big{]}\right)^{-1},$
(46)
where $e_{0}\equiv\sigma_{ii}/\lambda_{0}$ at $T=0$. From Eqns. (42) and( 46)
the change in the dielectric function due to local strain fluctuations is
$\displaystyle\delta\epsilon_{ii}$ $\displaystyle=$
$\displaystyle-4\pi\chi_{0}\,e_{0}\left(1-{g_{0}}/\big{[}{1-(i\omega\tau)^{\alpha}}\big{]}\right)^{-1}.$
(47)
This result is similar to the one for the TO and shear modulus discussed in
the earlier sections. At low temperatures, $\tau\rightarrow\infty$ and
$e_{ii}\rightarrow e_{0}$, hence the strain is minimal and the reduction of
the dielectric function due to local strain fluctuations is small. At high
temperatures, $\tau\rightarrow 0$ and $e_{ii}\rightarrow e_{0}(1-g_{0})^{-1}$
reaches its maximum resulting in the largest reduction of strain, where solid
4He is softest. The main result is that the dielectric function reflects the
arrested dynamics of the glassy components at low temperatures through the
acousto-optical (electro-elastic) coupling.
The derivation of Eq. (47) for the dielectric response is extremely general.
We illustrate how a Cole-Cole form for the elastic relaxation implies a
similar dielectric response (and vice versa). An identical result holds for
other forms of the local elastic relaxation dynamics- these will leave a
similar imprint on the dielectric response. Historically (since the 1940s) the
Cole-Cole response function Phase1 was found to be valid for the dielectric
response. In our initial works on 4He in trying to capture the local quenched
dynamics, we first assumed this form for the TO and then for the general
elastic response. By virtue of their inter-relation and coupling, the local
relaxation dynamics is the same for all of these quantities. Therefore
measurements of the dielectric response may inform about local dynamics and
vice versa. Practically, our prediction of Eq. (47) applies to any
nonpolarized system with a local distribution of stress relaxations. In
polarized materials, detecting a change in the dielectric function due to
atoms sensing different local fields is challenging because the large
intrinsic polarizability of the material will overwhelm the contributions
derived above. Among the nonpolarized solids, solid 4He favors the observation
of this phenomenon especially because of its softness. Rapid cooling of solid
helium allows a large local strain build-up, which is proportional to the size
of the effect. Similarly, we expect the effect to be seen via delicate effects
in solid 3He, hydrogen or xenon.
### 7.2 Results
The results of our electro-elastic predictions for the dielectric function are
shown in Fig. 13. We obtain excellent agreement with experiment Yin11 for an
applied alternating voltage at 500 Hz. Our analysis predicts, that similar to
the TO and shear modulus, a dissipation peak appears in the dielectric
function. The phase lag angle $\phi=arg(\epsilon)$ records the lag between the
real and imaginary part of $\epsilon(\omega;T)$. Future observation of the
dissipation peak will provide an important test of our picture concerning
quenched dynamics in solid 4He. Consistent with earlier sections in this
review, we assumed a Vogel-Fulcher-Tammann (VFT) form for the defect
relaxation time $\tau(T)=\tau_{0}\,e^{\Delta/(T-T_{0})}$. As in the previous
sections, we obtain from our fits a negative $T_{0}=-119$ mK, in accordance
with an avoided dynamic arrest of defect motion.
Figure 13: Experimental data and calculations of the dielectric function vs.
temperature. The red circles are the experimental data of the dielectric
function (data by Yin et al. Yin11 ). The black lines are the calculated
amplitude and phase lag (dissipation) of $\epsilon(\omega;T)$. We used
parameters $\alpha=1.49$, $e_{0}=8.88\times 10^{-4},g=0.21$, $\tau_{0}=10.4$
ns, $\Delta=1.92$ K, and $T_{0}=-119$ mK.
A rough estimate of the electro-elastic coupling can be obtained from mass
flow measurements in bulk solid 4He, Ray10 where a pressure difference of
$\Delta P_{L}\sim 0.1$ bar across a centimeter-sized pressure cell was
reported. The estimated local strain, with a bulk modulus $B=320$ bar, is
accordingly $\Delta P_{L}/B=3\times 10^{-4}$. This is consistent with the
value we used for the fit in Fig. 13, namely $e_{0}=8.88\times 10^{-4}$. In
addition, the $P(T)$ measurement by Yin clearly deviates from a purely Debye
lattice behavior at around $T=0.4$ K with a large positive intercept
corresponding to the order of 100 ppm of TLS, see Fig. 5. This number is
roughly five times larger than the most disordered sample in Lin’s Lin09
specific heat experiments on ultrapure 4He with less than 1 ppb of 3He
impurities Lin09 ; Su10 . Thus the crystals grown by Yin are strongly
disordered and harbor sufficiently many defects to support centers of local
strain fields.
Thus far, we assumed that both local and global stress are constant at low
temperatures. From Fig. 5 we can read off that the global pressure change
between 300 mK and 40 mK is less than $\Delta P_{T}=0.18$ mbar. This is more
than three orders of magnitude smaller than the local stress
$\sigma_{L}=8.88\times 10^{-4}\times 320\ {\rm bar}\sim 250$ mbar inferred
from the dilatory strain $e_{0}$ used in the fit, as well as the static
pressure difference $\Delta P_{L}=0.1$ bar measured at two pressure gauges in
mass flow experiments Ray10 . Putting all the pieces together, we find that
the change of the dielectric function based on global density changes in the
Clausius-Mossotti equation is negligible. This is because the corresponding
density change, $\Delta\rho/\rho=\Delta P_{T}/B<10^{-6}$, leads to a change in
the dielectric function of only
$\delta\epsilon\approx(\epsilon-1)\Delta\rho/\rho=0.065\Delta
P_{T}/B<10^{-7}$, which is more than two orders of magnitude too small to
account for the observed effect of order $10^{-5}$. Whereas the model of local
stress and electro-elastic coupling can account for the magnitude and
temperature dependence of the dynamic dielectric function. More recently Yin
et al. Yin2012 redesigned their dielectric function experiment with a
simplified capacitor geometry and found no measurable anomaly at low
temperatures. Within our theory of disorder such a null result is consistent
with a negligible amount of frozen-in stress in the solid.
## 8 Conclusions and future directions
In this review we provided a general overview of the role of defects in solid
4He. We suggested that defect dynamics and freeze-out provides a rich ground
to account for a significant fraction of the data, while at the same time they
allow enough flexibility to accomodate sample and history dependence and
hysteretic behavior ubiquitously seen in experiments. The general response
function approach presented covers a wide range of observed effects. We
provide a brief synopsis of these below.
### 8.1 Thermodynamics
We started our discussion with the notion that any true phase transition,
including supersolid, is accompanied by a thermodynamic signature. Therefore
thermodynamic measurements like specific heat can reveal the signature one
would naturally associate with a phase transition. However, we found that the
excess specific heat and corresponding entropy are consistent with the
contribution from noninteracting two level systems (TLSs). The estimated
fraction of these TLS to the specific heat is at the level of tens to hundreds
of parts-per-million. Consequently, the corresponding entropy contribution
$\Delta S\sim 10^{-4}$ J/(K mol) is inconsistent by several orders of
magnitude with reported values of the superfluid fraction (NCRIF) in the TO
and shear modulus experiments. We also point out that some of the most
disordered samples demonstrated ”supersolid” fraction up to 20%, while there
is little evidence for any excess entropy on the scale of the gas constant $R$
times the fraction of defects in any measured sample.
### 8.2 Single and double torsional oscillators
We have shown that a phenomenological glass (viscoelastic) model for quenched
defects accounts for the experimentally observed change in resonant frequency
and the concomitant peak in dissipation. Our analysis of torsional oscillator
(TO) experiments revealed that most are well described by a Cole-Cole
distribution for relaxation times. In addition, we derived a simple relation
for the ratio of change in dissipation and change in resonant frequency that
can explain the large ratios observed in experiments. The values for the glass
exponents in the distribution function of the backaction required to fit the
experimental data point toward broad distributions of glassy relaxation times,
which clearly invalidate any attempt to describe these experiments by a single
overdamped mode, i.e., a single Debye relaxation process.
We also applied these ideas to understand the double oscillator data. Our
studies of the coupled oscillator showed that the observed shifts in resonant
frequencies and dissipation are in agreement with a glassy backaction
contribution provided one includes anomalous damping in the dummy bob and an
explicit frequency dependence in the backaction term. As a side comment, it
came as a surprise that already the unloaded double TO (no 4He) required a
negative damping coefficient for the dummy bob to accurately describe resonant
frequencies and dissipation at 300 mK.
Finally, one should keep in mind that a significant difference between glassy
and supersolid dynamics is that a glassy contribution to the TO grows with
frequency, while a superfluid component decreases with frequency. This could
be another differentiating factor for separating very different relaxation
mechanisms.
### 8.3 Shear modulus
We showed that the shear modulus anomaly of solid 4He is strongly affected by
the dynamics of defects. The freezing out of defects leads to stiffening of
the solid concomitant with a peak in dissipation. By studying the glass
susceptibility due to the backaction, we found that both the amplitude change
and $T$-dependence of the shear modulus are well captured by this model. An
important consequence of the dynamic response analysis was the description of
the dissipation or phase angle. We found that the peak height of the
dissipation is independent of the applied frequency and linearly proportional
to the Cole-Cole exponent $\alpha$ as well as the backaction strength $g_{0}$.
As $g_{0}$ depends on the concentration of the TLS, we predicted that
increasing disorder will result in larger amplitude changes of the shear
modulus. Additionally, we extracted a universal scaling behavior proportional
to $\omega\tau(T)$ using the Cole-Cole plot. In this plot all curves of the
shear modulus collapsed onto a single curve over a wide range of frequencies.
Furthermore, we have shown that the glass contribution can be described by a
viscoelastic model through the incorporation of anelastic elements in
constitutive equations of stress and strain.
### 8.4 Dielectric function
We have shown that the arrested glass dynamics causes the low-temperature
anomaly in strained solid 4He through the acousto-optical (electro-elastic)
coupling and proposed that the temperature behavior of the dielectric function
is coupled to local strain fields near crystal defects. It records the glassy
dynamics and freeze-out of the hypothesized TLS excitations, which also lead
to a stiffening of the solid with decreasing temperature. This effect is not
captured by the standard Clausius-Mossotti relation, which attributes
dielectric function changes to a change in mass density or polarizability of
the nonpolar 4He atom. An important consequence of the phenomenological glass
susceptibility is the decrease of the dielectric function at high
temperatures, accompanied by a broad dissipation peak that can be measured by
the imaginary part of the dielectric function. We hypothesized that the
cooperative motion of atoms forming the TLS along dislocation segments is the
relevant process contributing to the reported anomaly. In our model, both the
change in $\epsilon(\omega;T)$ and the dissipation peak are to leading order
independent of the applied frequency. Since the coefficient $g_{0}$ of the
backaction depends on the concentration of defects, we predicted that the
change in dielectric function will be larger in quench-cooled or shear-
stressed samples, while it vanishes in defect-free single crystals. Beyond the
specific application to solid 4He, which we invoked here, our formalism
allowed for a direct link between elastic and dielectric properties which
could prove fruitful in many other systems.
In summary we encourage and welcome experiments that will allow a more precise
structural characterization of 4He. Also one needs to sharpen and provide a
more detailed analysis of the structural aspects of solid 4He in order to be
able to investigate any arrested dynamics of the postulated glassy regions to
separate it from a simple crossover phenomenon. Finally, we believe that more
dynamic studies probing the frequency or time response to a stimulus are
necessary to investigate the differences between small subsystems of glassy,
supersolid or superglassy origin in solid 4He.
###### Acknowledgements.
We are grateful to colleagues and collaborators who provided encouragement and
constructive criticism of the ideas presented here, A.F. Andreev, P. W.
Anderson, I. Beyerlein, J. C. Davis, A. Dorsey, C. Reichardt, B. Hunt, E.
Pratt, V. Gadagkar, J. Reppy, M. Chan, N. Prokof’ev, B. Svistunov, D.
Schmeltzer, A. Kuklov and E. Rudavsky. This work was supported by the US Dept.
of Energy at Los Alamos National Laboratory under contract No. DE-
AC52-06NA25396 and the Basic Energy Sciences Office. Z. N. was partially
supported by the NSF grant Award No. DMR-1106293. We also acknowledge steady
and generous support by the Aspen Center for Physics where parts of this
review were written.
## References
* (1) E. Kim and M. H. W. Chan, Nature (London) 427, 225 (2004).
* (2) E. Kim and M. H. W. Chan, Science 305, 1941 (2004).
* (3) A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP 29, 1107 (1969).
* (4) G. V. Chester and L. Reatto, Phys. Rev. 155, 88 (1967).
* (5) L. Reatto, Phys. Rev. 183, 334 (1969).
* (6) A. J. Leggett, Phys. Rev. Lett. 25, 2543 (1970).
* (7) P.W. Anderson, Basic Notions of Condensed Matter Physics, (Benjamin, Menlo Park, CA), Ch. 4, 143 (1984).
* (8) A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97, 165301 (2006); ibid. 98, 175302 (2007).
* (9) M. Kondo, S. Takada, Y. Shibayama, and K. Shirahama, J. Low Temp. Phys. 148, 695 (2007).
* (10) Y. Aoki, J. C. Graves, and H. Kojima, Phys. Rev. Lett. 99, 015301 (2007); J. Low Temp. Phys. 150, 252 (2008).
* (11) A. C. Clark, J. T. West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007).
* (12) A. Penzev, Y. Yasuta, and M. Kubota, J. Low Temp. Phys. 148, 677 (2007); Phys. Rev. Lett. 101, 065301 (2008).
* (13) B. Hunt, E. J. Pratt, V. Gadagkar, M. Yamashita, A. V. Balatsky, and J. C. Davis, Science 324, 632 (2009).
* (14) E. J. Pratt, B. Hunt, V. Gadagkar, M. Yamashita, M. J. Graf, A. V. Balatsky, and J. C. Davis, Science 332, 821 (2011).
* (15) V. Gadagkar, E. J. Pratt, B. Hunt, M. Yamashita, M. J. Graf, A. V. Balatsky, and J. C. Davis, THIS ISSUE (XXX).
* (16) D. Y. Kim, S. Kwon, H. Choi, H. C. Kim, E. Kim, arXiv:0909.1948 (2009).
* (17) J. Day, T. Herman, and J. Beamish, Phys. Rev. Lett. 95, 035301 (2005).
* (18) J. Day and J. Beamish, Phys. Rev. Lett. 96, 105304 (2006).
* (19) J.M. Goodkind, Phys. Rev. Lett. 89, 095301 (2002).
* (20) C. A. Burns and J.M. Goodkind, J. Low Temp. Phys. 93, 15 (1993).
* (21) D. S. Greywall, Phys. Rev. B 16, 1291 (1977).
* (22) M. A. Paalanen, D. J. Bishop, and H. W. Dail, Phys. Rev. Lett. 46, 664 (1981).
* (23) S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, and S. Balibar, Science 313, 1098 (2006).
* (24) M. W. Ray and R. B. Hallock, Phys. Rev. Lett. 100, 235301 (2008); ibid. 101, 189602 (2008)
* (25) G. Bonfait, H. Godfrin, and S. Balibar, J. Phys. (Paris) 50, 1997 (1989).
* (26) S. Balibar and F. Caupin, Phys. Rev. Lett. 100, 235301 (2008).
* (27) M. W. Ray and R. B. Hallock, Phys. Rev. B 79, 224302 (2009).
* (28) M. W. Ray and R. B. Hallock, Phys. Rev. Lett. 105,145301 (2010).
* (29) M. W. Ray and R. B. Hallock, Phys. Rev. B 84, 144512 (2011)
* (30) Y. Vekhov and R. B. Hallock, arXiv:1205.5751.
* (31) C. A. Burns, N. Mulders, L. Lurio, M. H. W. Chan, A. Said, C. N. Kodituwakku, and P. M. Platzman, Phys. Rev B 78, 224305 (2008).
* (32) E. Blackburn, J. M. Goodkind, S. K. Sinha, J. Hudis, C. Broholm, J. van Duijn, C. D. Frost, O. Kirichek, and R. B. E. Down, Phys. Rev. B 76, 024523 (2007).
* (33) S. Balibar and F. Caupin, Phys. Rev. Lett. 101, 189601 (2008).
* (34) N. Prokof’ev, Adv. Phys., 56, 381 (2007).
* (35) S. Balibar, A. Haziot , and X. Rojas, Proceedings of the SPIE - The International Society for Optical Engineering 7945, 794502 (2011).
* (36) M. Boninsegni and N. V. Prokof’ev, Rev. Mod. Phys. 84, 759 (2012)
* (37) A. F. Andreev, JETP Lett. 85, 585 (2007).
* (38) A. F. Andreev, JETP 109, 103 (2009).
* (39) S. E. Korshunov, JETP Lett. 90, 156 (2009).
* (40) P. W. Anderson, Nature Phys. 3, 160 (2007).
* (41) M. Yagi, A. Kitamura, N. Shimizu, Y. Yasuta, and M. Kubota, J. of Low Temp. Phys. bf 162, 492 (2011)
* (42) M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Physical Review Letters, 99, 035301 (2007)
* (43) J. Toner, Phys. Rev. Lett. 100, 035302 (2008)
* (44) T. Arpornthip, A. V. Balatsky, M. J. Graf, and Z. Nussinov, Phys. Rev. B 84, 174304 (2011)
* (45) M. Rossi, E. Vitali, D.E. Galli, and L. Reatto, J. Phys.: Condens. Matter 22, 145401 (2010)
* (46) A. V. Balatsky, M. J. Graf, Z. Nussinov, S. A. Trugman, Phys. Rev. B 75, 094201 (2007).
* (47) Z. Nussinov, M. J. Graf, A. V. Balatsky, and S. A. Trugman, Phys. Rev. B 76, 014530 (2007).
* (48) M. J. Graf, A. V. Balatsky, Z. Nussinov, I. Grigorenko, and S. A. Trugman, J. Phys.: Conf. Ser. 150, 032025 (2009).
* (49) J.-J. Su, M. J. Graf, and A. V. Balatsky, J. Low Temp. Phys. 159, 431 (2010).
* (50) J.-J. Su, M. J. Graf, and A. V. Balatsky, Phys. Rev. Lett. 105, 045302 (2010).
* (51) M. Boninsegni, N. Prokof’ev, and B. Svistunov, Phys. Rev. Lett. 96, 105301 (2006).
* (52) J. Wu and P. Phillips, Phys. Rev. B 78, 014515 (2008).
* (53) G. Biroli, C. Chamon, and F. Zamponi, Phys. Rev. B 78, 224306 (2008).
* (54) A. T. Dorsey, P. M. Goldbart, and J. Toner, Phys. Rev. Lett. 96, 055301 (2006)
* (55) X. Lin, A. C. Clark, and M. H. W. Chan, Nature 449, 1025 (2007).
* (56) O. Syshchenko, J. Day, and J. Beamish, Phys. Rev. Lett. 104, 195301 (2010).
* (57) L. Yin, J. S. Xia, C. Huan, N. S. Sullivan, and M. H. W. Chan , J. Low Temp. Phys. 162, 407 (2011).
* (58) J.-J. Su , M. J. Graf, and A. V. Balatsky, J. Low Temp. Phys. 162, 433 (2011).
* (59) C. Zhou, J.-J. Su, M. J. Graf, C. Reichhardt, A. V. Balatsky, and I. J. Beyerlein, Philos. Mag. Lett. (2012), doi: 10.1080/09500839.2012.704415.
* (60) J. Zaanen, Z. Nussinov, and S. I. Mukhin, Ann. Phys. 310, 181 (2004)
* (61) V. Cvetkovic, Z. Nussinov, S. Mukhin, and J. Zaanen, Europhys. Lett. 81, 27001 (2008).
* (62) A. Eyal and E. Polturak, arXiv:1202.1392 (2012)
* (63) A. Eyal, E. Livne, and E. Polturak, arXiv:1206.0938 (2012)
* (64) D. Y. Kim and M. H. W. Chan, arXiv:1207.7050.
* (65) D. A. Huse and Z. U. Khandker, Phys. Rev. B 75, 212504 (2007).
* (66) B. K. Clark and D. M. Ceperly, Phys. Rev. Lett. 96, 105302 (2006).
* (67) L. Pollet, , M. Boninsegni, A. B. Kuklov, N. V. ProkofÕev, B. V. Svistunov, and M. Troyer Phys. Rev. Lett., 98, 135301 (2007).
* (68) M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. ProkofÕev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett. 99, 035301 (2007).
* (69) T. W. B. Kibble, J. Phys. A 9, 1387 (1976)
* (70) W. H. Zurek, Nature 317, 505 (1985)
* (71) K. S. Cole and R. H. Cole, J. of Chem. Phys. 9, 341 (1941).
* (72) D. W. Davidson and R. H. Cole, Journal of Chem. Phys. 18, 1417 (1950); ibid. 19, 1484 (1951).
* (73) D. C. Vural and A. J. Leggett, Journal of Non-Crystalline Solids, 357, 3528 (2011)
* (74) E. C. Helmets and C. A. Swenson, Phys. Rev. 128, 1512 (1962).
* (75) J. P. Frank, Phys. Lett. 11, 208 (1964).
* (76) A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).
* (77) X. Lin, A. C. Clark, Z. G. Cheng, and M. H. W. Chan, Phys. Rev. Lett. 102, 125302 (2009); Phys. Rev. Lett. 103, 259903(E) (2009).
* (78) J. T. West and X. Lin, Z. G. Cheng, M. H. W. Chan, Phys. Rev. Lett. 102, 185302 (2009).
* (79) I. A. Todoshchenko, H. Alles, J. Bueno, H. J. Junes, A. Ya. Parshin, and V. Tsepelin, Phys. Rev. Lett 97, 165302 (2006).
* (80) I. A. Todoshchenko, H. Alles, H. J. Junes, A. Ya. Parshin, and V. Tsepelin, JETP Lett. 85, 454 (2007).
* (81) V. N. Grigor’ev, V. A. Maidanov, V. Yu. Rubanskii, S. P. Rubets , E. Ya. Rudavskii , A. S. Rybalko , Ye. V. Syrnikov , and V. A. Tikhii , Phys. Rev. B 76, 224524 (2007).
* (82) V. N. Grigor’ev and Ye. O. Vekhov, J. Low Temp. Phys. 149, 41 (2007).
* (83) A. A. Lisunov, V. A. Maidanov, V. Yu. Rubanskyi, S. P. Rubets, E. Ya. Rudavskii, A. S. Rybalko, and V. A. Tikhiy, Phys. Rev. B 83, 132201 (2011).
* (84) P. W. Anderson and B. I. Halperin and C. M. Varma, Philos. Mag. 25, 1 (1972).
* (85) W. A. Phillips, J. Low Temp. Phys 11, 757 (1972).
* (86) J. Jäckle, Z. Physik 257, 212 (1972).
* (87) G. Ahlers, Phys. Rev. A 8, 530 (1973).
* (88) A. Driessen, E. van der Poll, and I. F. Silvera, Phys. Rev. B 33, 3269 (1986).
* (89) Th. M. Nieuwenhuizen, Phys. Rev. Lett. 79, 1317 (1997).
* (90) J. Rault, J. of Non-Crystalline Solids 271, 177 (2000)
* (91) C.-D. Yoo and A. T. Dorsey, Phys. Rev. B 79, 100504(R) (2009).
* (92) P. W. Anderson, Phys. Rev. Lett. 100, 215301 (2008).
* (93) J. R. Beamish, A. D. Fefferman, A. Haziot, X. Rojas, and S. Balibar, Phys. Rev. B 85, 180501(R) (2012).
* (94) H. J. Maris, Phys. Rev. B 86, 020502 (2012).
* (95) M. J. Graf, Z. Nussinov, and A. V. Balatsky, J. Low Temp. Phys. 158, 550 (2010)
* (96) M. J. Graf, J.-J. Su, H. P. Dahal, I. Grigorenko, and Z. Nussinov, J. Low Temp. Phys. 162, 500 (2011).
* (97) A. C. Clark, X. Lin, and M. H. W. Chan, Phys. Rev. Lett. 97, 245301 (2006).
* (98) X. Mi, E. Mueller, and J. D. Reppy, arXiv:1109.6818.
* (99) Y. Aoki, J. C. Graves, and H. Kojima, J. Low Temp. Phys. 150, 252 (2008).
* (100) J. Day and J. Beamish, Nature 450, 853 (2007).
* (101) J. Day, O. Syshchenko, and J. Beamish, Phys. Rev. B 79, 214524(2009).
* (102) L. D. Landau, L. P. Pitaevskii, E. M. Lifshitz, and A. M. Kosevich, Theory of Elasticity, Ed. 3 (Oxford, New York, Pergamon Press, 1986).
* (103) The 2000 Hz and 0.5 Hz data sets are not discussed in this paper, because they show qualitatively different behavior. This may be due to different relaxation processes or experimental difficulties.
* (104) I. Iwasa, Phys. Rev. B 81, 104527 (2010).
* (105) A. Granato and K. Lücke, J. App. Phys. 27, 583 (1956); ibid. 27. 789 (1956); ibid. 52, 7136 (1981).
* (106) J. Friedel, Dislocations, (International series of monographs on solid state physics) [U.S.A. ed. distributed by Addison-Wesley Pub. Co., Reading, Mass.]; [1st English ed.] edition (1964)
* (107) J. G. Kirkwood, J. Chem. Phys. 4, 592 (1936).
* (108) M. Chan, M. Ryschkewitsch, and H. Meyer, J. Low Temp. Phys. 26, 211 (1977).
* (109) N. R. Werthamer , Phys. Rev. 185, 348 (1969).
* (110) A. Kisliuk, A. Kudlik, J. Sukmamowski, S. Loheider, M. Soltwisch, and D. Quitmann, J. Phys. Cond. Mat. 3, 9831 (1991).
* (111) J. S. Bell, M. J. Kearslev, L. P. Pitaevskii, L. D. Landau, E. M. Lifshitz, and Sykes J B, Electrodynamics of Continuous Media, Ed. 2 (Oxford, New York, Pergamon Press, 1984).
* (112) X. Lin, A. C. Clark, Z. G. Cheng, and M. H. W. Chan, Phys. Rev. Lett. 102, 125302 (2009).
* (113) J-J. Su, M. J. Graf, and A. V. Balatsky, J. Low Temp. Phys. 159, 431 (2010).
* (114) L. Yin, J. S. Xia, C. Huan, N. S. Sullivan, and M. H. W. Chan, J. Low Temp. Phys. 168, 251 (2012).
|
arxiv-papers
| 2012-09-04T21:30:45 |
2024-09-04T02:49:34.795980
|
{
"license": "Public Domain",
"authors": "A. V. Balatsky, M. J. Graf, Z. Nussinov, J.-J. Su",
"submitter": "Matthias J. Graf",
"url": "https://arxiv.org/abs/1209.0803"
}
|
1209.1030
|
# Climatology of Mid-latitude Ionospheric Disturbances from the Very Large
Array Low-frequency Sky Survey
###### Abstract
The results of a climatological study of ionospheric disturbances derived from
observations of cosmic sources from the Very Large Array (VLA) Low-frequency
Sky Survey (VLSS) are presented. We have used the ionospheric corrections
applied to the 74 MHz interferometric data within the VLSS imaging process to
obtain fluctuation spectra for the total electron content (TEC) gradient on
spatial scales from a few to hundreds of kilometers and temporal scales from
less than one minute to nearly an hour. The observations sample nearly all
times of day and all seasons. They also span latitudes and longitudes from
$28^{\circ}$N to $40^{\circ}$N and $95^{\circ}$W to $114^{\circ}$W,
respectively. We have binned and averaged the fluctuation spectra according to
time of day, season, and geomagnetic ($K_{p}$ index) and solar ($F10.7$)
activity. These spectra provide a detailed, multi-scale account of seasonal
and intraday variations in ionospheric activity with wavelike structures
detected at wavelengths between about 35 and 250 km. In some cases, trends
between spectral power and $K_{p}$ index and/or $F10.7$ are also apparent. In
addition, the VLSS observations allow for measurements of the turbulent power
spectrum down to periods of 40 seconds (scales of $\sim\\!0.4$ km at the
height of the $E$-region). While the level of turbulent activity does not
appear to have a strong dependence on either $K_{p}$ index or $F10.7$, it does
appear to be more pronounced during the winter daytime, summer nighttime, and
near dusk during the spring.
HELMBOLDT ET. AL VLSS-BASED TEC DISTURBANCE CLIMATOLOGY J. F. Helmboldt, US
Naval Research Laboratory, Code 7213, 4555 Overlook Ave. SW, Washington, DC
20375 ([email protected]) W. M. Lane Peters, US Naval Research
Laboratory, Code 7213, 4555 Overlook Ave. SW, Washington, DC 20375
([email protected]) W. D. Cotton, National Radio Astronomy
Observatory, 520 Edgemont Rd, Charlottesville, VA 22903 ([email protected])
11affiliationtext: US Naval Research Laboratory, Washington, DC,
USA.22affiliationtext: National Radio Astronomy Observatory, Charlottesville,
VA, USA.
## 1 Introduction
The ionosphere is known to contain a host of transient ionospheric
disturbances. They range from fine-scale, turbulent like density fluctuations
(e.g., Cohen and Röttgering, 2009) to large-scale plasma waves (e.g., Borries
et al., 2009) and large areas of storm-enhanced density (e.g., Foster et al.,
2002). Some of the most common among these are so called traveling ionospheric
disturbances (TIDs) which are known to be plasma density waves with a range of
sizes and speeds. TIDs are often divided into two classes, large-scale TIDs
(LSTIDs) and medium-scale TIDs (MSTIDs). LSTIDs are typically associated with
geomagnetic storms and have sizes $\sim\\!2000$ km, periods of $\sim\\!\\!1$
hour, and speeds of about 700 m s-1 (Borries et al., 2009). MSTIDs have
periods between 10 minutes and $\sim\\!1$ hour, speeds between 50 and a few
hundred m s-1, and sizes $\sim\\!100$ km (e.g., Jacobson and Erickson, 1992;
Jacobson et al., 1995).
Unlike their large-scale counterparts, MSTIDs are not typically associated
with geomagnetic storms and are likely generated by more localized processes.
While the climatology of these relatively commonly occurring waves has been
studied in the past (see, e.g., Hernández-Pajares et al., 2006; Tsugawa et
al., 2007), the mechanisms responsible for generating them are still not
clear. They are most frequently seen at dawn and dusk, implying a relationship
to the dynamics associated with the transition between the daytime and
nighttime ionospheres. In the northern hemisphere, daytime MSTIDs are seen
most often in the winter, usually propagating toward the southeast (Hernández-
Pajares et al., 2006). MSTIDs that appear during the night are most prevalent
in the summer, mostly directed toward the southwest (Hernández-Pajares et al.,
2006; Tsugawa et al., 2007). The nighttime MSTIDs in particular remain
something of a mystery, especially regarding their orientation and direction
of motion. Their formation may be related to the Perkins instability (Perkins,
1973), and/or to coupling between instabilities within the $E$ and $F$ regions
(see, e.g., Cosgrove and Tsunoda, 2004). Different mechanisms have been
proposed to explain their southwestward motion (Kelley and Makela, 2001;
Cosgrove, 2007; Yokoyama et al., 2009), however, none have been conclusively
verified by observations.
In addition, MSTIDs have been observed to propagate in a variety of other
directions, attributed to changes in the $F$-region neutral wind and/or
orographic gravity waves (e.g., Shiokawa et al., 2008; Dymond et al., 2011;
Helmboldt et al., 2012b) and to deposition of material from the plasmasphere
(Helmboldt and Intema, 2012). Both of these types of MSTIDs in particular can
have relatively weak amplitudes, between $10^{-3}$ and $10^{-2}$ TECU (1
TECU$=\\!10^{16}$ m-2). The presence (or lack thereof) of MSTIDs may also be
linked to increased levels of turbulent activity within the ionosphere (e.g.,
Helmboldt et al., 2012b).
A detailed understanding of the generation, maintenance, and evolution of
MSTIDs requires a climatological study which is sensitive to the full range of
observable MSTIDs and to smaller-scale density fluctuations to which they may
be related. Recently, Helmboldt et al. (2012a, b) have demonstrated that radio
interferometers, especially those observing in the VHF regime, may be powerful
tools for producing such a study. With the capability to measure differential
total electron content (TEC) with a precision $<\\!10^{-3}$ TECU on spacings
of 1 km or less when observing a bright cosmic source (Cohen and Röttgering,
2009; Helmboldt et al., 2012a), such interferometers are capable of observing
small TEC fluctuations such as those related to turbulence. Using long
temporal baselines, Helmboldt et al. (2012a) showed that one can also detect
and characterize the properties of MSTIDs as they pass over the
interferometer, some of which have relatively small fluctuation amplitudes
($\sim\\!10^{-3}$ TECU). In addition, Helmboldt and Intema (2012) have
demonstrated that by using simultaneous observations of several moderately
bright VHF sources, one can characterize fluctuations on much larger scales.
For instance, at 74 MHz, the $15^{\circ}$ field of view of the Karl G. Jansky
Very Large Array (VLA) in New Mexco corresponds to an area 80 km in diameter
at a height of 300 km.
Here, we seek to apply these radio interferometer-based techniques to a large,
74 MHz survey conducted with the VLA to obtain a statistical, climatological
picture of ionospheric disturbances spanning a wide range of scales in size,
oscillation period, and amplitude. In terms of amplitude and size, this study
constitutes one of the largest dynamic range investigations of its kind, and
it contains one of the first climatological studies of ionospheric turbulence.
In Section 2, we briefly describe the characteristics of the survey used, the
VLA Low-frequency Sky Survey (VLSS; Cohen et al., 2007). In the next section,
we detail the analysis used and ionospheric data products that were generated.
In Section 4, we show the results of our climatological study and discuss them
further in Section 5.
## 2 Experimental Detail
In order to create high quality astronomical images, the VLSS standard data
processing corrects the data for ionospheric effects. In this study, we re-
analyze the data products from that processing to extract information about
the dynamics of the ionosphere.
### 2.1 The Survey
The VLSS is a survey of the northern sky covering declinations north of about
$-30^{\circ}$. It was conducted between 2001 and 2007 using the 74 MHz system
of the VLA (Kassim et al., 2007) with full dual polarization (VLA program
numbers AP397, AP441, AP452, and AP509). The VLA was used in its
B-configuration, an inverted “Y” of 27 antennas spanning roughly 11 km.
Observations of sources with low declinations were conducted with the VLA in
the hybrid “BnA” configuration which has the northern arm extended to its
A-configuration size of nearly 20 km. Hybrid configurations such as BnA are
routinely used when observing sources at low elevations to improve the
uniformity of the antenna positions projected onto the sky. Approximately 20%
of the VLSS observations were conducted in the BnA configuration.
Other VLA configurations, which are available roughly four months at a time,
are the A, C, and D configurations spanning roughly 40, 4, and 1 km,
respectively. Observations made at 74 MHz with the two most compact
configurations have a substantially larger effective system temperature due to
the fact that their shorter baselines make them more sensitive to diffuse
emission from the Galaxy, generally referred to as “sky noise.” The Galactic
emission is dominated by synchrotron emission that is much brighter at lower
frequencies and on relatively large angular scales. Consequently, the A and B
configurations have lower sky noise contributions due to the fact that their
smallest baselines are too big to be sensitive to the large-angular-scale
Galactic emission. The B configuration was chosen for the VLSS because its
angular resolution, 80 arcseconds, is similar to that of other northern sky
surveys at 325 MHz (54 arcseconds; Rengelink et al., 1997) and 1400 MHz (45
arcseconds; Condon et al., 1998) to facilitate the compilation of radio-
frequency spectra for detected sources.
The details of the survey and associated data processing are discussed in
detail by Cohen et al. (2007). Recently, improvements in imaging and
calibration techniques as well as new algorithms for mitigating radio
frequency interference (RFI) were combined to re-process all of the VLSS data.
The results of this effort are referred to as “VLSS redux,” or VLSSr, and the
improvements over the previously used processing techniques are discussed by
Lane et al. (2012).
Briefly, the VLSS mapped the sky using 523 overlapping pointings and imaging
the entire $15^{\circ}$ wide, circular field of view of the VLA at 74 MHz at
each pointing. Each pointing was observed for a total of about 1.25 hours or
more, broken into four or more 20-minute segments, or three or more 25-minute
segments. For each pointing, these segments were spaced by hours or days. This
amounted to a total of 1862 separate $\sim\\!20$-minute observations. These
observations cover nearly all times of day and season and the associated
ionospheric pierce-points span a range in latitude and longitude, which we
have illustrated in Fig. 1.
### 2.2 Field-based Calibration
The effect of the ionosphere on the VLSS observations varied significantly
across the $15^{\circ}$ field of view, which corresponds to a width of 80 km
at an altitude of 300 km. The first-order effect of the ionosphere is to shift
the positions of observed sources by an amount proportional to the mean TEC
gradient over the area spanned by the array along the line of sight to each
source. The VLSS used a procedure referred to as “field-based” calibration
(Cotton et al., 2004) to account for these shifts as a function of time and
position within the field of view. This calibration scheme uses “snapshot”
images of relatively bright sources using 1–2 minutes of data for each
snapshot. The positions of these sources are then measured within the
snapshots and compared to a reference catalog. The original VLSS used a VLA-
based, 1400 MHz catalog (Condon et al., 1998) as its reference. This produced
a catalog of 74 MHz sources whose positions are tied to the bright sources
contained within the 1400 MHz reference catalog.
Substantial improvements were obtained in ionospheric calibration by Lane et
al. (2012) for the VLSSr by using the original VLSS as the reference catalog.
In other words, Lane et al. (2012) essentially produced a second iteration of
the field-based imaging scheme where the catalog of 74 MHz sources generated
by the first iteration was used as the reference catalog during the second
iteration. Since there is a variety of spectral shapes among cosmic radio
sources, not all bright 1400 MHz sources are also bright at 74 MHz, and vice
versa. Consequently, having a reference catalog of sources known to be bright
(here, $>\\!2.5$ Jy; 1 Jy $=10^{-26}$ W m-2 Hz-1) at 74 MHz made the field-
based calibration process more robust in that a larger area could be searched
within each snapshot for the shifted source position.
### 2.3 Self-calibration
While not used as part of the VLSS, a separate calibration technique referred
to as “self-calibration” (Cornwell and Fomalont, 1999) can be used to derive
ionospheric corrections toward a single bright cosmic source if the intensity
of that source dominates the field of view. Self-calibration uses a model for
the brightness distribution on the sky (often a simple point-source) to solve
for antenna-based phase corrections. For $N$ antennas, there are $N(N-1)/2$
redundant pairs or baselines, making this an over-determined problem. The
derived phase corrections are proportional to the difference in TEC, or
“$\delta\mbox{TEC}$”, along the lines of sight of the antennas and an
arbitrarily chosen reference antenna. Helmboldt et al. (2012a) showed that
using a bright source with the VLA at 74 MHz, these $\delta\mbox{TEC}$
measurements can be made to a precision of about $3\times 10^{-4}$ TECU,
allowing for an analysis of very fine-scale, small amplitude fluctuations
along the lines of sight toward a particular source.
To make such fine-scale ionospheric measurements, we sought to apply self-
calibration whenever possible to the VLSS observations to complement the wide-
field, medium-scale measurements provided by the field-based calibration
process. To do this, we had to first determine how bright a source needed to
be to accurately compute $\delta\mbox{TEC}$ using self-calibration-derived
phase corrections. We did this by first identifying all sources in the VLSS
catalog with peak intensities $>\\!6$ Jy beam-1. We then used each source to
perform phase-only self-calibration (i.e., we did not solve for the amplitudes
of the complex antenna gains), assuming a simple point-source model. Since we
would like to probe the smallest scales in space and time, we did this using
the smallest time interval possible. For the VLSS data, this was 10–20
seconds. We then converted these self-calibration phases to $\delta\mbox{TEC}$
measurements by unwrapping the phases within each $\sim\\!\\!20$ minute
observations and de-trending with a linear fit. This was done separately for
each polarization so that we could use the rms difference between the de-
trended $\delta\mbox{TEC}$ measurements from the two polarizations to assess
the precision of the measurements.
In Fig. 2, we have plotted $\delta\mbox{TEC}$ rms versus the observed peak
intensity (i.e., modified by the antenna response) for these sources. Above a
limit of about 15 Jy beam-1, there is a noticeable anti-correlation between
this rms and the source intensity which is what one would expect if the
$\delta\mbox{TEC}$ measurements represent real ionospheric fluctuations and
not noise. Below this limit, the data essentially form a scatter plot, and the
self-calibration phases are likely dominated by noise caused by “confusion,”
i.e., the contribution from other sources in the field of view that were
ignored in the self-calibration process. Therefore, for the analysis that will
be described below, we have only used sources with observed intensities
$>\\!15$ Jy beam-1. There are 281 sources which meet this criterion that were
observed within 477 different VLSS pointings (recall that the pointings
overlapped). These yielded 1978 separate observations of the 281 bright
sources.
In addition to the 1978, 20-minute observations of relatively bright sources,
the VLSS data also contain several five-minute observations of the extremely
bright source Cygnus A, or “CygA.” At 74 MHz, Cyg A is brighter than 17000 Jy,
making it an excellent source to use to determine corrections for instrumental
effects which are relatively stable in time. Consequently, Cyg A was observed
289 times during the VLSS campaign, each time with a duration of five minutes.
The extreme brightness of Cyg A also provides an excellent means for probing
the smallest-scale/lowest-amplitude ionospheric fluctuations observable with
the VLA (see Helmboldt et al., 2012a, b). Therefore, to increase the dynamic
range of our study of ionospheric disturbances, we also applied self-
calibration to all of the five-minute observations of Cyg A. We note that
because the Cyg A observations were shorter, the de-trending of the
$\delta\mbox{TEC}$ measurements made from these observations biased them
toward smaller-period oscillations. This makes the Cyg A data complementary to
the data produced from the 281 bright source observations described above,
which are less sensitive to smaller-scale/smaller amplitude fluctuations
because the sources used are not nearly as bright as Cyg A.
## 3 Spectral Analysis
### 3.1 Multi-source Data
The repository of source position shifts compiled during the processing of the
VLSS provides a rich database of TEC gradient time series that can be analyzed
and searched for evidence of wavelike activity. Indeed, Helmboldt and Intema
(2012) showed that by Fourier transforming such a time series in time, and
then in space, one may produce a three-dimensional (one temporal, two spatial)
power spectrum “cube” of TEC gradient fluctuations. In practice, a spectral
cube is generated for both components (north-south and east-west) of the TEC
gradient. These two cubes are then added together such that the observed power
of a wave with a TEC amplitude $A$ and spatial frequency $\xi$ will simply be
$(2\pi\xi A)^{2}$. These spectra are also normalized by the amplitude of the
impulse response function (IRF), computed using the source positions projected
to an ionospheric height of 300 km. This is denoted in the units reported for
these spectra by including a unit of IRF-1.
The three-dimensional spectral cubes can be used to map the level and
direction of any detected wavelike structures within the data. We have applied
the spectral analysis algorithm described in Helmboldt and Intema (2012) to
each roughly twenty-minute observation of each VLSS field using outputs from
the VLSSr, yielding 1862 power-spectrum cubes. Since each spectrum is based on
position shift data from multiple sources, for the remainder of this paper, we
will referred to these spectral data as “multi-source” data for convenience.
It was demonstrated by Helmboldt and Intema (2012) that these spectra have
excellent sensitivity, being able to detect fluctuations with amplitudes
$<\\!10^{-3}$ TECU. This is in part due to the improved data
processing/calibration techniques detailed in Lane et al. (2012). They
demonstrated the effect of these improvements on ionospheric analysis using
mean fluctuation spectra from the entire survey. They showed that the use of
the original VLSS as the reference catalog as well as the new RFI mitigation
techniques substantially improved the sensitivity of the power spectra to both
large and small amplitude fluctuations. Using a better reference catalog
allows one to reliably search a larger area for each calibration source,
facilitating the detection of large amplitude fluctuations. Reducing the noise
in each snapshot image by implementing new RFI-subtracting and flagging
software allowed for the detection of smaller position shifts. Specifically,
the spatial frequency at which the mean fluctuation spectrum reached the noise
“floor” was increased by a factor of two by the RFI-mitigation software,
decreasing the typical wavelength of the weakest detectable disturbances from
70 to 35 km.
### 3.2 Single-source Data
While the power spectrum cubes described above provide a wealth of information
about TEC fluctuations on a range of scales, their sensitivity is generally
limited at relatively high spatial and temporal frequencies due to the
effective smoothing applied to the data. The position shifts of the calibrator
sources give the mean TEC gradient over the span of the VLA, which in the case
of the VLSS amounts to smoothing the data with an 11-km wide circular filter.
In addition, snapshots were made using 1–2 minutes of data, resulting in
temporal smoothing.
However, the measurements of $\delta\mbox{TEC}$ made using individual bright
sources described in Section 2.3 used temporal sampling of 10–20 seconds with
mean antenna spacings $<\\!1$ km. This implies that there is potentially more
information within these data about fluctuations on scales too small to be
detected with the multi-source spectra. To help explore phenomena with smaller
amplitudes and sizes, we sought to apply the methods developed by Helmboldt et
al. (2012a, b) and refined by Helmboldt and Intema (2012) to use 74 MHz VLA
observations of a single bright source to characterize TEC fluctuations on
significantly smaller scales.
The single-source spectral analysis is similar to that of its multi-source
counterpart in that a three-dimensional Fourier transform of the TEC gradient
is used. However, in this case, the gradient is measured at each antenna for a
single source rather than toward multiple sources. Because of the Y-shape of
the VLA, the $\delta\mbox{TEC}$ measurements obtained from self-calibration
(see Section 2.3) cannot be used to directly compute the full two-dimensional
TEC gradient at each antenna. However, Helmboldt et al. (2012a) showed that
most of the structure in the TEC surface observed by the VLA can be recovered
with a second-order polynomial fit to $\delta\mbox{TEC}$ as a function of
antenna position.
Using these polynomial fits to estimate the gradient at each antenna, the same
Fourier-based procedure can be applied as was used with the multi-source
approach. The details of how this is implemented are discussed at length by
Helmboldt and Intema (2012). There are two main difference between the
spectral cubes produced from the single source data and those generated using
multi-source data. First, the spectral resolution in the spatial regime of the
single-source-based spectra is much lower given the compact size of the VLA
(11-km). Because of this, each temporal mode within each spectral cube is well
approximated by a single spatial frequency and direction. Consequently, for
each single-source spectrum, we computed a single peak power and weighted
(with the spectral power) mean spatial frequency in the north-south,
$\xi_{NS}$, and east-west, $\xi_{EW}$, directions for each temporal mode. When
combining all temporal frequencies together, this allows for a measurement of
the distribution of fluctuation strength in the $\xi_{NS}$,$\xi_{EW}$ plane
with much better spectral resolution than could normally be achieved with such
a compact array. This is because, by using a single spatial mode for each
temporal mode, we have essentially converted the relatively long temporal
baselines to spatial ones. For instance, for a TID with a speed of 100 m s-1,
a 20-minute baseline amounts to a 120-km baseline, more than ten times larger
than the B-configuration VLA.
The second difference between the single- and multi-source analyses is that
because the cosmic sources are essentially infinitely far away, the lines of
sight of the antennas toward a single source are basically parallel. The lines
of sight toward multiple sources, however, are far from parallel. Thus, the
physical separation among pierce-points for the multi-source observations is
proportional to the assumed height, whereas for the single-source data, the
pierce-point separations remain virtually the same for any assumed altitude.
This renders the multi-source data incapable of sensing disturbances in the
upper ionosphere and plasmasphere that have wavelengths less than about 1000
km. Conversely, the single-source observations are sensitive to $\sim\\!10$-km
and larger sized fluctuations, evan at plasmaspheric heights (see, e.g.,
Helmboldt and Intema, 2012).
The single-source spectral analysis was applied to all 20-minute observations
of the 281 identified bright sources and all five-minute observations of Cyg A
(see Section 2.3). In doing so, we found that for the observations conducted
in the hybrid BnA configuration (see Section 2.1), the extended, 20-km long
northern arm of the VLA posed a problem for the polynomial fit-based analysis.
In short, the polynomial fits only provide a uniform estimate of the TEC
gradient across the array when it is symmetric. Fitting a two-dimensional
polynomial to an asymmetric array can lead to biasses within the spectral
analysis performed using the results from such polynomial fits. For instance,
we found that data from this configuration produced two-dimensional
fluctuation spectra that were, on average, elongated along the north/south
axis. Because of this inherent bias, single-source BnA observations (360 for
the bright sources; 87 for Cyg A), were excluded from this analysis.
In addition to this three-dimensional spectral analysis, we also used the
single source data to yield a statistical description of any isotropic
fluctuations on the finest scales observable within the VLSS data. This was
done according to the prescription of Helmboldt et al. (2012a) for using the
self-calibration-determined $\delta\mbox{TEC}$ values to numerically compute
the projection of the TEC gradient along each VLA “arm” at each antenna. While
directional information is essentially lost within this approach, one may use
it to recover information about isotropic fluctuations on scales $<\\!1$ km.
To this end, we have computed a median power spectrum among all antennas from
each single-source observation by simply performing a Fourier transform of the
projected TEC gradient time series for each antenna and determining the median
spectral power at each temporal frequency. Using the median rather than the
mean helps to mitigate the influences of both strong wavelike disturbances and
any spurious data that might be present. These median one-dimensional spectra
therefore provide an excellent means to explore the climatological nature of
turbulent fluctuations, which is quite complementary to the wave-based
analysis provided by the multi-source spectral cubes and single-source, two-
dimensional spectral maps.
## 4 Climatology
### 4.1 Multi-source 3-D Spectra
As described above, we used the outputs from the VLSSr field-based calibration
procedure for each of the 1862, $\sim\\!\\!20$-minute observations to
construct fluctuation spectrum cubes, one temporal dimension and two spatial,
of the TEC gradient. We then binned these by local time and then within local
time bins, by day of the year, $K_{p}$ index, and $F10.7$. We used four time
of day bins, dawn (04:00–08:00), daytime (08:00–16:00), dusk (16:00–20:00),
and nighttime (20:00-04:00). We likewise used four day of the year bins,
spring (34–124), summer (125–215), autumn (216–306), and winter (307–33).
During the observations, the $K_{p}$ index varied between 0 and a little more
than 5, and we consequently designed our four $K_{p}$ bins to span the range
0–5. The values of $F10.7$ ranged from about 70 Solar Flux Units (SFU; 1 SFU
$=10^{-22}$ W m-2 Hz-1) to more than 220 SFU between 2001 and 2007 (i.e.,
roughly solar maximum to solar minimum). We therefore constructed four
logarithmically spaced bins spanning log $F10.7=1.85$–2.35.
In Fig. 3, we display the peak power over all temporal frequencies as a
function of $\xi_{NS}$ and $\xi_{EW}$ of the average power spectrum cube for
each time-of-day/time-of-year bin. Maps of the accompanying peak temporal
frequency are shown in Fig. 4. Within each panel, the number of observations
used in the average is printed. For all of the spectra, the full width at half
maximum (FWHM) of the IRF is 0.009 km-1. Within the spectral maps, there are a
variety of features that are $\gtrsim$ the size of the IRF which likely
represent real wavelike structures rather than random fluctuations. For
example, if one examines the spring nighttime panel in Fig. 3, one can see a
prominent feature in the lower left quadrant of the plot. These maps are
constructed such that north is up and east is to the right, implying that this
feature corresponds to southwestward-directed waves. There is a polar grid
plotted as dotted grey lines to help one estimate direction and wavelength. In
this case, the direction is about $45^{\circ}$ south of west and the spatial
frequency, $\xi$, is about 0.015 km-1, implying a wavelength of 67 km (i.e.,
$1/\xi$). If one examines the same region in the corresponding panel in Fig.
4, on can see that the peak temporal frequency is about 5 hr-1. Taken with the
wavelength, this gives a speed of roughly 330 km hr-1, or about 90 m s-1.
Some of the features apparent within the panels of Fig. 3 and 4 seem to be
specific to a given season and/or time of day. Some coincide with the known
climatological behavior of previously explored fluctuations. A summary of
prominent features found within these data and the single-source spectral
analysis presented below is given in Table 1, including comparisons with
previous work.
As alluded to in Section 1, southwestward-directed MSTIDs have been previously
observed to be prominent during summer nighttime in North America. Likewise,
winter daytime MSTIDs are known to be common in the same region, usually
propagating toward the southeast (e.g., Hernández-Pajares et al., 2006;
Tsugawa et al., 2007). We see direct evidence of these same phenomena within
the multi-source spectra. The summer nighttime spectrum shows a southwest-
directed feature with a wavelength of about 100 km and a period of about 17
minutes, implying a speed of 100 m s-1. This is roughly consistent with the
known properties of MSTIDs. Similarly, the winter daytime spectrum shows a
southeastward-directed feature with a peak at a wavelength of 190 km, a period
of 17 minutes, and an implied speed of 190 m s-1, which is again, consistent
with MSTIDs.
There are also many apparent features within the multi-source spectra that are
not commonly known. For instance, several show evidence of wave activity
directed toward the northeast, namely spring dawn, summer daytime and dusk,
and autumn dusk and nighttime. There are southwestward-directed waves during
spring nighttime, but they are smaller ($\sim\\!\\!50$ km) and slower
($\sim\\!\\!60$ m s-1) than the MSTIDs seen during summer nighttime. In
addition, the spring nighttime spectrum shows evidence of MSTID-like,
westward-directed waves not seen during other time periods. While medium-scale
wave activity seems to be common during the daytime, the predominant direction
appears to have a seasonal dependence with southwestward-directed waves being
more prevalent during summer daytime and southeastward-directed waves
dominating during autumn and winter. The autumn dusk and nighttime spectra are
particularly interesting as they show a plethora of features with no apparent
dominant direction or size/speed. Most of the spectra show evidence of
relatively small ($|\xi|\\!>\\!0.01$ km-1) features in the periphery of the
areas they occupy within the $\xi_{NS}$,$\xi_{EW}$ plane.
In Fig. 5, we have plotted spectra similar to those displayed in Fig. 3 but
for bins of time-of-day and $K_{p}$ index. Once can see from these that most
of the features seen in Fig. 3 do not appear to have a clear dependence on
geomagnetic activity. For instance, southeastward-directed waves are seen
during the daytime at all values of $K_{p}$. However, it does appear that the
northeastward-directed waves that appear at dawn are more prominent at low to
moderate values of $K_{p}$ as are the southwestward-directed nighttime waves
(both the $\sim\\!\\!50$ and $\sim\\!\\!100$ km sized waves).
Similar spectra are plotted again in Fig. 6, this time for bins of time-of-day
and $\mbox{log }F10.7$. Like the results for the $K_{p}$ index, most of the
features seen in Fig. 3 do not seem to have noticeable dependences on solar
activity. Again, the most notable exceptions are the dawn northeastward-
directed waves and nighttime southwestward-directed waves, both of which are
seen most prominently at the lowest levels of solar activity. The larger
southwestward-directed waves seem to actually be most prominent at the highest
solar activity levels. However, we note that all of the summer observations
were taken in 2002 near solar maximum. Indeed, the spectrum in the high
$\mbox{log }F10.7$, nighttime bin looks very similar to the summer nighttime
spectrum in Fig. 3. The fact that these waves have been seen by others to have
a strong seasonal dependence suggests that it is the time of year rather than
solar activity levels that has had the strongest influence on the appearance
of this feature in this case.
For the lowest $K_{p}$ and $\mbox{log }F10.7$ bins, the nighttime spectra show
a significant eastward-directed feature that is not prominent in the nighttime
spectra in Fig. 3. The feature is actually closer to magnetic east, which is
about $12^{\circ}$ south of due east at the VLA. The wavelength and speed of
this feature are about 150 km and 150 m s-1, respectively. Because these waves
are predominantly magnetic eastward-directed with relatively large speeds,
they may in fact be field-aligned irregularities within the plasmasphere
similar to those discovered with the VLA by Jacobson and Erickson (1992). For
instance, co-rotation within the plasmasphere will yield an observed speed for
such irregularities of 150 m s-1 for heights of about 2000 km. For these
observations, this corresponds to a McIlwain $L$-parameter of about 2.1, which
is typical for the irregularities previously discovered with the VLA
(Hoogeveen and Jacobson, 1997). These disturbances will be discussed further
in Section 5.
### 4.2 Single-source 2-D Spectra
As discussed in Section 3.2, the spectral analysis of each single-source
observation yielded a measurement of the distribution of spectral power in the
$\xi_{NS}$,$\xi_{EW}$ plane that reaches higher spatial frequencies than the
multi-source spectra. We have binned these spectral maps according to local
time, season, $K_{p}$ index and $F10.7$ in the same way as the multi-source
spectra and have displayed the results in Fig. 7–9. In each panel of each of
these figures, we have plotted the mean power over all temporal frequencies
from the corresponding mean multi-source spectral cube as blue contours.
The spectral maps shown in Fig. 7 show that at all times of day and year,
there are significant fluctuations at scales too small to detect with the
multi-source data. However, when averaged together, they seem to be largely
isotropic with a few notable exceptions. First, the map for winter daytime
seems to be especially asymmetric, showing an over-density in the northwest
quadrant. This northwest over-density is dominant on scales of about 100 km.
Fig. 8 and 9 show that the northwest over-density seen during winter daytime
is more prominent at moderate solar activity levels, but does not seem to have
a dependence on $K_{p}$.
The autumn dawn map shows a population of nearly northward-propagating
disturbances with wavelengths of about 50 km not seen in the corresponding
multi-source spectrum. There is some indication from Fig. 9 that these waves
occur more frequently during times of relatively high solar activity.
Perhaps the most striking feature is a significant group of small-scale waves
directed toward the southeast during spring dawn with wavelengths of about 40
km which represent a distinct population of waves not detectable with the
multi-source data. Fig. 8 and 9 show that these waves also appear prominently
at times of low geomagnetic and solar activity.
We also discussed in Sections 2.3 and 3.2 separate five-minute calibration
observations that were made of the extremely bright source, Cyg A, which we
also used to produce spectral maps. There were significantly fewer of these
observations (202 versus 1618 in the B configuration), and their time-of-day
and seasonal coverage is not as good. However, as also discussed in Section
3.2, the spectral maps produced from these observations are more sensitive to
smaller-scale fluctuations, making them complementary to the single-source
maps shown in Fig. 7–9. Therefore, we have displayed in Fig. 10–12 Cyg A-based
spectral maps binned by local time and season (Fig. 10), $K_{p}$ index (Fig.
11), and $F10.7$ (Fig. 12).
The Cyg A-based spectra show no signs of any significant groups of waves.
However, as expected, they typically extend to larger spatial frequencies than
their counterparts shown in Fig. 7–9. While the maps often show roughly
isotropic distributions of fluctuations, reminiscent of turbulence, many show
some evidence of “preferred” directions. For instance, the spring nighttime
map is elongated along the northwest-southeast axis with more power apparent
in the southeast direction. The same appears to be true during dawn for higher
$K_{p}$ indices, but with a much larger fraction of the total spectral power
in the southeastern quadrant. The summer dawn spectrum shows a significant
northwestern extension toward higher spatial frequencies.
The summer dawn map has an extension toward the northwest that reaches spatial
frequencies as high as about 0.04 km-1 (or, scales of 25 km). Similar features
can be seen in the dawn maps in Fig. 12 for moderate to high levels of solar
activity. There is also a prominent group of northwestward-directed waves
extending to large (0.05 km-1) spatial frequencies in the spring dusk map, but
this is based on a single observation.
### 4.3 Single-source 1-D Spectra and Turbulence
As described in Section 3.2, we have produced median TEC gradient fluctuation
spectra using the arm-based method of computing the projected TEC gradient
from all single-source data. Since the projected gradients do not rely on
polynomial fits, we have used all 1978 single-source observations for this
analysis, including those conducted in the hybrid BnA configuration. These
one-dimensional spectra probe the shape of the spectrum of isotropic of
fluctuations down to the smallest periods possible, 40 seconds. If we assume
that the population of these fluctuations does not change significantly during
each observation, the temporal frequencies can be directly related to spatial
ones using the apparent speed of the observed cosmic source at ionospheric
heights. Since turbulence is likely to be much more prominent at lower
altitudes where ion-neutral coupling is more important, we have computed the
median apparent source speed at the height of the $E$-region, 100 km, which is
roughly 10 m s-1. This implies that these spectra probe scales as small as 0.4
km.
Continuing with the assumption of “frozen” turbulence, we can characterize the
level of turbulent activity by fitting a simple power-law model to each
spectrum. Specifically, following the work of Kolmogorov (1941a, b) and
Tatarski (1961), the turbulent spectrum of phase fluctuations, which in this
case is the same as TEC fluctuations, should be proportional to $\xi^{-11/3}$.
Therefore, the TEC gradient spectrum should be $\propto\xi^{-5/3}$ and the
temporal spectra should be well fit by a spectrum $\propto\nu^{-5/3}$ for
frozen turbulence. We have displayed mean one-dimensional single-source
spectra in Fig. 13, binned by season and local time, to illustrate that this
is generally the case. For this analysis, we have excluded the five-minute
Cyg-A observations because the shorter duration yields lower temporal spectral
resolution which significantly alters the shapes of the spectra at lower
frequencies. We also found that the 20-minute, single-source observations were
more than adequate for characterizing the spectrum of turbulent fluctuations
(see Fig. 13) and that the Cyg A-based spectra did not significantly improve
or add to this analysis.
For each single-source spectrum, we have fit a simple model of
$P_{T}\nu^{-5/3}+N$, where $P_{T}$ is a parameter that characterizes the level
of turbulent activity and $N$ is a constant that models the effect of noise
within the spectrum. The fits were constrained to $\nu\\!\\!>20$ hr-1 because
below this limit, the influence of the window function used (i.e., a simple
20-minute wide boxcar) as well as relatively strong, medium-to-large scale
fluctuations on the shape of the spectrum can be significant. The fits are
plotted as red curves in Fig. 13. One can see that these turbulence model fits
provide a good approximation of the data in all cases.
To obtain a climatological picture of turbulent activity, we also fit the
turbulence model described above to the arm-based spectrum from each single-
source observation. Among these spectra, the parameter $P_{T}$ varied between
roughly 1 and 20 (mTECU km-1 IRF-1)2 and the median value for the parameter
$N$ was $10^{-4}$ (mTECU km-1 IRF-1)2. The mean value of $P_{T}$ was then
computed within bins of local time and day of the year, $K_{p}$ index, and
$\mbox{log }F10.7$. These two-dimensional distributions are displayed in the
panels of Fig. 14. During the day, the turbulent activity appears to be
highest during winter months, which is when daytime MSTID activity also peaks.
During dusk, it seems to be higher in the late winter/early spring while post-
midnight and before dawn, there seems to be some increased activity during the
summer. There also seems to be a “spike” of turbulent activity during the day
at moderately high values of $K_{p}$ ($4+$ to $5-$). Daytime turbulent
activity seems to be relatively high during moderate levels of solar activity
($F10.7\sim\\!\\!100$ SFU). Post-midnight and before dawn, the turbulent
activity also has peaks near $F10.7\\!\approx\\!120$ and 170 SFU.
## 5 Discussion
We have used a large, 74 MHz survey of the northern sky, the VLSS, to explore
the rich environment of ionospheric disturbances over the southwestern United
States and northern Mexico (see Fig. 1). Some of the results were expected
from previous analysis, namely the prominent MSTIDs during summer nighttime
and winter daytime, which propagate toward the southwest and southeast,
respectively (see, e.g., Hernández-Pajares et al., 2006; Tsugawa et al.,
2007). However, there are several features unique to this study, and we
discuss a subset of them below.
### 5.1 Muti-directional Waves
The results shown in Fig. 3 show evidence of waves moving in several different
directions on different scales. Most notable are the results for autumn dusk
and nighttime, each showing this variety within a single mean spectrum. While
several factors may influence the appearance of these waves, it seems most
likely that they are related to orographic gravity waves (see, e.g., Vadas and
Crowley, 2010). This is because beneath the region of the ionosphere probed by
the VLSS observations, the terrain is quite mountainous. The is illustrated
within the bottom panel of Fig. 1 where one can see from the displayed relief
map that the latitudes and longitudes of the ionospheric pierce-points for the
VLSS observations cover the most mountainous regions of northern Mexico, New
Mexico, and Arizona. In addition, the Rocky Mountains lie largely to the north
and west of these observations, implying that features like the southeastward-
directed, 40-km sized waves seen during spring dawn may be related to waves
generated by airflow over the Rockies.
### 5.2 Northeastward Waves
A prominent and repeating feature within the spectra shown in Fig. 3 is a
class of northeastward-directed waves. These are similar in size, and often in
strength, to MSTIDs, but are traveling in a relatively abnormal direction for
these types of disturbances. Similar waves were observed with the VLA during
nighttime observations in August, 2003 by Helmboldt et al. (2012b). The relief
map in Fig. 1 does indicate that there are mountains to the southwest of many
of the ionospheric pierce-points, implying that gravity waves may be involved.
However, these northeastward-directed waves are often relatively strong
compared to other spectral features seen in Fig. 3 (see spring dawn and summer
daytime for examples). The mountains to the southwest are not particularly
large, especially when compared to the Rockies to the northwest, implying that
there may be more at work here then vertically propagating gravity waves. The
northeastward-directed waves detected by Helmboldt et al. (2012b) were seen to
coincident with drops in $F$-region height measured with relatively nearby
ionosondes data. This is similar to what was observed over Japan by Shiokawa
et al. (2008). It is therefore plausible that $F$-region compressions are
related to the appearance of these features within the VLSS data. However, a
thorough investigation using contemporaneous ionosondes data is required to
resolve this and is currently underway.
### 5.3 Magnetic Eastward Waves
As noted in Section 3.1, there are relatively unique, eastward-directed
features in the nighttime spectra for the lowest $K_{p}$ and $\mbox{log
}F10.7$ bins shown in Fig. 5 and 6. These could be similar to the magnetic
eastward-directed waves discovered with the VLA and established by Hoogeveen
and Jacobson (1997) to be plasmaspheric waves. They could also be associated
with disturbances within plasma flow from the plasmasphere to the ionosphere
discovered with the VLA by Helmboldt and Intema (2012). The latter seems
particularly likely because (1) they are only seen during the night when such
flows are most likely to occur and (2) the spectra also show evidence of
westward-directed, presumably ionospheric waves which were also found by
Helmboldt and Intema (2012) to occur simultaneously with the plasmaspheric
disturbances. The fact that they are most prominent during periods of low
geomagnetic and solar activity may imply that a relatively undisturbed
plasmasphere is required to produce these features. A more detailed study of
this phenomenon is currently being conducted using GPS-based TEC maps, $K_{p}$
and AE indices, and VLSS data. The preliminary results suggest that these
disturbed flows may be triggered by localized depletions within the nighttime
ionosphere triggered by forcing from the lower atmosphere, likely related to
tides. The full results will be presented in a subsequent paper.
### 5.4 Turbulence
The results shown in Fig. 13 confirm what was demonstrated by Cohen and
Röttgering (2009) and (Helmboldt et al., 2012b), that the median behavior of
the spectrum of ionospheric fluctuations is turbulent. Fig. 14 shows that
while the level of turbulence does vary over time, the difference between the
maximum and minimum levels is only about a factor of five. Contrast this with
the results for wave activity displayed in Fig. 3–12 where the range in
spectral power spans at least two orders of magnitude. The level of turbulent
activity does appear to be related in some way to the occurrence of certain
wave phenomena. In particular, $P_{T}$ is relatively large when MSTIDs are
prominent in winter daytime and summer nighttime.
## 6 Conclusions
By using a large database of 74 MHz observations of cosmic sources, we have
been able to probe the climatological behavior of a variety of ionospheric
disturbances over the southwestern United States and parts of Mexico. We have
shown that small-scale, turbulent fluctuations are present nearly all of the
time with only a weak dependence on time of day, time of year, and/or the
level of geomagnetic and solar activity. We found a variety of small (roughly
40 km wavelength) and medium (100 km and larger) scale phenomena present at
different times. The sources of these disturbances may range from gravity
waves, plasmaspheric interactions, coupling between different ionospheric
layers, and plasma instabilities (e.g., the Perkins instability). Several of
these wavelike irregularities have relatively small amplitudes too weak to
detect with other methods such as GPS-based TEC measurements. In some cases,
the wave activity was coincident with increases in turbulent activity, namely
during winter daytime and summer nighttime.
We note that in general, the impact of the results presented is somewhat
hampered by the limited seasonal and time-of-day coverage of the VLSS shown in
the upper panel of Fig. 1. For instance, all of the summer observations were
conducted in the month of June. The fact that the summer nighttime MSTIDs
appear weaker than their winter daytime counterparts may be heavily influenced
by the lack of data for July and August. In addition, the only observations
conducted during the dawn hours in the summer were calibration observations of
Cyg A. We are now designing a survey with the new 330 MHz system for the VLA,
which is currently being commissioned. This new survey will be scheduled to
optimize both time-of-day and seasonal coverage. It will benefit from two new
arrays of GPS receivers operating continuously within New Mexico which have
recently increased the number of such stations from $\sim\\!\\!6$ during the
time of the VLSS to nearly 40. There will also be a new digital ionosonde
operating in nearby Kirtland Air Force Base in Albuquerque which will
potentially provide information about the height where detected disturbances
occur. Thus, we will have a contemporaneous GPS-based account of the largest
fluctuations, providing a complete and unique inventory of the seasonal
dependence of ionospheric disturbances that will expand upon the work
presented here.
###### Acknowledgements.
Basic research in astronomy at the Naval Research Laboratory is supported by
6.1 base funding. The Very Large Array is operated by The National Radio
Astronomy Observatory. The National Radio Astronomy Observatory is a facility
of the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc.
## References
* Borries et al. (2009) Borries, C., N. Jankowski, and V. Wilken (2009), Storm induced large scale TIDs observed in GPS derived TEC Ann. Geophys., 27, 1605–1612
* Cohen et al. (2007) Cohen, A. S., W. M. Lane, W. D. Cotton, N. E. Kassim, T. J. W. Lazio, R. A. Perley, J. J. Condon, and W. C. Erickson (2007), The VLA Low-Frequency Sky Survey Astron. J., 134, 1245–1262
* Condon et al. (1998) Condon, J. J., W. D. Cotton, E. W. Greisen, Q. F. Yin, R. A. Perley, G. B. Taylor, and J. J. Broderick (1998), The NRAO VLA Sky Survey, Astron. J. 115, 1693–1716
* Cohen and Röttgering (2009) Cohen, A. S. and H. J. A. Röttgering (2009), Probing Fine-scale Ionosphere Structure with the Very Large Array Radio Telescope, Astron. J., 138, 439–447
* Coker et al. (2009) Coker, C., S. E. Thonnard, K. F. Dymond, T. J. W. Lazio, J. J. Makela, and P. J. Loughmiller (2009), Simultaneous radio interferometer and optical observations of ionospheric structure at the Very Large Array Rad. Sci., 44, RS0A11
* Cornwell et al. (1999) Cornwell, T., R. Braun, and D. S. Briggs (1999), Deconvolution, in Synthesis Imaging in Radio Astronomy II, ASP Conference Ser., vol. 180, edited by G. B. Taylor, C. L. Carilli, and R. A. Perley, pp. 151–170, ASP, San Francisco, Calif.
* Cornwell and Fomalont (1999) Cornwell, T. and E. B. Fomalont (1999), Self-Calibration, in Synthesis Imaging in Radio Astronomy II, ASP Conference Ser., vol. 180, edited by G. B. Taylor, C. L. Carilli, and R. A. Perley, pp. 187–199, ASP, San Francisco, Calif.
* Cosgrove and Tsunoda (2004) Cosgrove, R. B. and R. T. Tsunoda (2004), Instability of the $E--F$ coupled nighttime midlatitude ionosphere, J. of Geophys. Res., 109, A04305
* Cosgrove (2007) Cosgrove, R. B. (2007), Generation of mesoscale $F$ layer structure and electric fields by the combined Perkins and $E_{s}$ layer instabilities, in simulation, Ann. Geophys., 25, 1579–1601
* Cotton et al. (2004) Cotton, W. D., J. J. Condon, R. A. Perley, N. Kassim, J. Lazio, A. Cohen, W. Lane, and W. C. Erickson (2004), Beyond the isoplanatic patch in the VLA Low-frequency Sky Survey, Proc. SPIE, 5489, 180
* Dymond et al. (2011) Dymond, K. F., et al. (2011), A medium-scale traveling ionospheric disturbance observed from the ground and from space, Radio Sci., 46, RS5010
* Foster et al. (2002) Foster, J. C., P. J. Erickson, A. J. Coster, J. Goldstein, and F. J. Rich (2002), Ionospheric signatures of plasmaspheric tails, Geophys. Res. Let., 29, 1623–1626
* Helmboldt et al. (2012a) Helmboldt, J. F., T. J. W. Lazio, H. Intema, and K. Dymond (2012), High-precision Measurements of Ionospheric TEC Gradients with the Very Large Array VHF System Rad. Sci., 47, RS0K02
* Helmboldt et al. (2012b) Helmboldt, J. F., T. J. W. Lazio, H. Intema, and K. Dymond (2012), A New Technique for Spectral Analysis of Ionospheric TEC Fluctuations Observed with the Very Large Array VHF System: From QP Echoes to MSTIDs Rad. Sci., 47, RS0L02
* Helmboldt and Intema (2012) Helmboldt, J. F. and H. T. Intema (2012), Very Large Array Observations of Disturbed Ion Flow from the Plasmasphere to the Nighttime Ionosphere, Rad. Sci.. 47, RS0K03
* Hernández-Pajares et al. (2006) Hernández-Parajes, M., J. M. Juan, and J. Sanz (2006), Medium-scale traveling ionospheric disturbances affecting GPS measurements: Spatial and temporal analysis, J. of Geophys. Res., 111, A07S11
* Hoogeveen and Jacobson (1997) Hoogeveen, G. W. and A. R. Jacobson (1997), Improved analysis of plasmasphere motion using the VLA radio telescope, Ann. of Geophys., 15, 236–245
* Jacobson and Erickson (1992) Jacobson, A. R. and W. C. Erickson (1992), Wavenumber-resolved Observations of Ionospheric Waves Using the Very Large Array Radiotelescope, Plant. Space. Sci., 4, 447–455
* Jacobson et al. (1995) Jacobson, A. R., R. C. Carlos, R. S. Massey, and G. Wu (1995), Observations of traveling ionospheric disturbances with a satellite-beacon radio interferometer: Seasonal and local time behavior, J. Geophys. Res., 100, 1653–1665
* Kassim et al. (2007) Kassim, N. E. T. J. W. Lazio, W. C. Erickson, R. A. Perley, W. D. Cotton, E. W. Greisen, A. S. Cohen, B. Hicks, H. R. Schmitt, and D. Katz (2007), The 74 MHz System on the Very Large Array, Astrop. J. Supp., 172, 686–719
* Kelley and Makela (2001) Kelley, M. C., and J. J. Makela (2001), Resolution of the discrepancy between experiment and theory of midlatitude $F$-region structures, Geophys. Res. Let., 28, 2589–2592
* Kolmogorov (1941a) Kolmogorov, A. N. (1941a), The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers, Dokl. Akad. Nauk SSSR, 30, 301–305
* Kolmogorov (1941b) Kolmogorov, A. N. (1941b), Dissipation of energy in the locally isotropic turbulence, Dokl. Akad. Nauk SSSR, 32, 16–18
* Lane et al. (2012) Lane, W. M., W. D. Cotton, J. F. Helmboldt, and N. E. Kassim (2012), VLSS Redux: Software Improvements applied to the Very Large Array Low-frequency Sky Survey, Rad. Sci., 47, RS0K04
* Larsen et al. (2007) Larsen, M. F., D. L. Hysell, Q. H. Zhou, S. M. Smith, J. Friedman, and R. L. Bishop (2007), Imaging coherent scatter radar, incoherent scatter radar, and optical observations of quasiperiodic structures associated with sporadic $E$ layers, J. of Geophys. Res., 112, A06321
* Perkins (1973) Perkins, F. (1973), Spread F and Ionospheric Currents, J. Geophys. Res., 78, 218–226
* Rengelink et al. (1997) Rengelink, R. B., Y. Tang, A. G. de Bruyn, G. K. Miley, M. N. Bremer, H. J. A. Roettgering, and M. A. R. Bremer (1997), The Westerbork Northern Sky Survey (WENSS), I. A 570 square degree Mini-Survey around the North Ecliptic Pole, Astron. and Astrophys. Sup., 124, 259–280
* Shiokawa et al. (2008) Shiokawa, K., Y. Otsuka, N. Nishitani, T. Ogawa, T. Tsugawa, T. Maruyama, S. E. Smirnov, V. V. Bychkov, and B. M. Shevtsov (2008), Northeastward motion of nighttime medium-scale traveling ionospheric disturbances at middle latitudes observed by an airglow imager J. of Geophys. Res., 113, A12312
* Tatarski (1961) Tatarski, V. I. (1961), Wave Propagation in a Turbulent Medium, McGraw-Hill, New York, New York ASP, San Francisco, Calif.
* Tsugawa et al. (2007) Tsugawa, T. Y., Y. Otsuka, A. J. Coster, and A. Saito (2007), Medium-scale traveling ionospheric disturbances detected with dense and wide TEC maps over North America, Geophys. Res. Lett., 34, L22101
* Yamamoto et al. (2005) Yamamoto, M., S. Fukao, R. T. Tsunoda, R. Pfaff, and H. Hayakawa (2005), SEEK-2 (Sporadic-$E$ Experiment over Kyushu 2) – Project Outline, and Significance, Ann. Geophys., 23, 2295–2305
* Yokoyama et al. (2009) Yokoyama, T., D. L. Hysell, Y. Otsuka, and M. Yamamoto (2009), Three-dimensional simulation of the coupled Perkins and $E_{s}$-layer instabilities in the nighttime midlatitude ionosphere J. of Geophys. Res., 114, A03308
* Vadas and Crowley (2010) Vadas, S. L. and G. Crowley (2010), Sources of the traveling ionospheric disturbances observed by the ionospheric TIDDBIT sounder near Wallops Island on 30 October 2007, J. Geophys. Res., 115, A07324
Table 1: Summary of Observed Phenomena Phenomenon | Season(s) | Time(s) of Day | Notes
---|---|---|---
SW-directed MSTIDs | summer | daytime–nighttime | previously detected with GPS and airglow imagers; e.g.,
| | | her06,tsu07
SE-directed MSTIDs | autumn/winter | daytime | same as above
NE-directed MSTIDs | spring/summer | dawn/daytime–dusk | isolated cases detected previously by
| | | shi08 and hel12b
multi-directional, | autumn | dusk–nighttime | likely associated with gravity waves
medium-scale waves | | |
SW- and W-directed | spring | nighttime | seen with magnetic east-directed waves at low $K_{p}$;
medium-scale waves | | | isolated case detected by helint
SE-directed 40-km- | spring | dawn | found when $K_{p}$ is low;
scale waves | | | direction close to magnetic east
turbulence | spring/summer/ | dusk/nighttime/ | expands on diurnal variation
| winter | daytime | seen by coh09
Figure 1: Upper: The local time and day of the year of each of the VLSS
observations; points are color-coded by year. Lower: The latitude and
longitude of the ionospheric pierce-point for the center of the field of view
for each VLSS observation assuming a height of 300 km. The same color-coding
by year is used as in the upper panel. A relief map from
http://www.ngdc.noaa.gov/mgg/global/global.html is also displayed for
reference.
Figure 2: The rms difference in $\delta\mbox{TEC}$ between the two
polarizations among all baselines and time steps for each observation of each
VLSS source brighter than 6 Jy versus its observed peak intensity (i.e.,
modified by the antenna response). The vertical grey line shows the chosen
limit of an intensity of $>\\!15$ Jy beam-1 for single-source data. See
Section 2.2 for more discussion.
Figure 3: From mean fluctuation spectrum cubes, derived from multi-source data
and binned by local time and season (see Section 3.1), the peak power over all
temporal frequencies as a function of north-south and east-west spatial
frequencies. Locations where the mean power over all temporal frequencies is
lower than $5\times\mbox{MAD}$ are set to zero, where MAD is the median
absolute deviation among the mean power values of all spatial frequencies. The
number of observations averaged together to make each mean spectrum is printed
in each panel.
Figure 4: From mean fluctuation spectrum cubes, derived from multi-source data
and binned by local time and season (see Section 3.1), the peak temporal
frequency as a function of north-south and east-west spatial frequencies.
Locations where the mean power over all temporal frequencies is lower than
$5\times\mbox{MAD}$ are set to zero, where MAD is the median absolute
deviation among the mean power values of all spatial frequencies. The number
of observations averaged together to make each mean spectrum is printed in
each panel.
Figure 5: The same as Fig. 3, but for spectra binned by local time and $K_{p}$
index.
Figure 6: The same as Fig. 3, but for spectra binned by local time and
$\mbox{log }F10.7$.
Figure 7: Mean maps of TEC gradient fluctuation power, derived from single-
source data and binned by local time and season (see Section 3.2). The blue
contours represent the mean power over all temporal frequencies from the
corresponding spectral cubes shown in Fig. 3 and 4. The number of observations
averaged together to make each mean map is printed in each panel.
Figure 8: The same as Fig. 7, but for spectra binned by local time and $K_{p}$
index.
Figure 9: The same as Fig. 7, but for spectra binned by local time and
$\mbox{log }F10.7$.
Figure 10: Mean maps of TEC gradient fluctuation power, derived from five-
minute observations of the bright calibration source, Cygnus A. These maps
were made by binning the observations by local time and season (see Section
3.2). The magenta contours represent the mean power over all temporal
frequencies from the corresponding spectral cubes shown in Fig. 3 and 4. The
number of observations averaged together to make each mean map is printed in
each panel.
Figure 11: The same as Fig. 10, but for spectra binned by local time and
$K_{p}$ index.
Figure 12: The same as Fig. 10, but for spectra binned by local time and
$\mbox{log }F10.7$.
Figure 13: Within bins of local time and season, the mean one-dimensional
fluctuation spectra derived from single-source data using the arm-based method
described in Section 2.2. The turbulence model described in Section 3.3 was
fit to each of these spectra and is plotted as a red curve.
Figure 14: The strength of the turbulent fluctuation power, $P_{T}$ (see
Section 3.3), derived from single-source data as a function of local time and
day of the year (upper), $K_{p}$ index (middle), and $F10.7$ (lower).
|
arxiv-papers
| 2012-09-05T16:22:30 |
2024-09-04T02:49:34.816208
|
{
"license": "Public Domain",
"authors": "J. F. Helmboldt, W. M. Lane, and W. D. Cotton",
"submitter": "Joe Helmboldt",
"url": "https://arxiv.org/abs/1209.1030"
}
|
1209.1215
|
# Sharp endpoint estimates for the $X$-ray transform and the Radon transform
in finite fields
Doowon Koh Department of Mathematics
Chungbuk National University
Cheongju city, Chungbuk-Do 361-763 Korea [email protected]
###### Abstract.
This note establishes sharp $L^{p}-L^{r}$ estimates for $X$-ray transforms and
Radon transforms in finite fields.
###### 2010 Mathematics Subject Classification:
43A32, 43A15
Key words and phrases:$k$-plane transform, $X$-ray transform, Radon transform,
finite fields.
This research was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education, Science and Technology(2012010487)
## 1\. Introduction
Let $\mathbb{F}_{q}^{d},d\geq 2,$ be a $d$-dimensional vector space over the
finite field $\mathbb{F}_{q}$ with $q$ elements. We endow $\mathbb{F}_{q}^{d}$
with a normalized counting measure $dx.$ For each $q$, we denote by $M_{q}$ a
collection of certain subsets of $\mathbb{F}_{q}^{d}.$ Recall that a
normalized surface measure $d\sigma$ on $M_{q}$ can be defined by the relation
$\int\Omega(w)d\sigma(w)=\frac{1}{|M_{q}|}\sum_{w\in M_{q}}\Omega(w)$
where $|M_{q}|$ denotes the cardinality of $M_{q}$ and $\Omega$ is a complex-
valued function on $M_{q}.$ For any complex-valued function $f$ on
$(\mathbb{F}_{q}^{d},dx)$ and $w\in M_{q}$, we consider an operator
$T_{M_{q}}$ defined by
$T_{M_{q}}f(w)=\frac{1}{|w|}\sum_{x\in w}f(x).$
We are interested in determining exponents $1\leq p,r\leq\infty$ such that the
following inequality holds:
$\|T_{M_{q}}f\|_{L^{r}(M_{q},d\sigma)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$
where the operator norm of $T_{M_{q}}$ is independent of $q$, the size of the
underlying finite field $\mathbb{F}_{q}.$ If $M_{q}$ is $\Pi_{k},$ a
collection of all $k$-planes in $\mathbb{F}_{q}^{d}$ with $1\leq k\leq d-1$,
then the operator $T_{\Pi_{k}}$ is called as the $k$-plane transform. In
particular, $T_{\Pi_{1}}$ and $T_{\Pi_{(d-1)}}$ are known as the $X$-ray
transform and the Radon transform respectively. In Euclidean space, the
complete mapping properties of $k$-plane transforms were proved by M. Christ
in [1]. Readers may refer to [9], [8], [3] for the description of $k$-plane
transforms in the Euclidean setting. In 2008, Carbery, Stones, and Wright [2]
initially studied the mapping properties of $k$-plane transforms in finite
fields. Using combinatorial arguments, they proved the following theorem.
###### Theorem 1.1.
Let $d\geq 2$ and $1\leq k\leq d-1$ be an integer. If
(1.1) $\|T_{\Pi_{k}}f\|_{L^{r}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}$
holds with $C$ independent of $|\mathbb{F}_{q}|$, then $(1/p,1/r)$ lies in the
convex hull $H$ of
$((k+1)/(d+1),1/(d+1)),(0,0),(1,1)~{}~{}\mbox{and}~{}~{}(0,1).$
Conversely, if $(1/p,1/r)$ lies in
$H\setminus\left\\{((k+1)/(d+1),1/(d+1))\right\\},$ then (1.1) holds with $C$
independent of $|\mathbb{F}_{q}|.$ Finally, if
$(1/p,1/r)=((k+1)/(d+1),1/(d+1))$, then the restricted type inequality
(1.2) $\|T_{\Pi_{k}}f\|_{L^{d+1}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{k+1},1}(\mathbb{F}_{q}^{d},dx)}$
holds with $C$ independent of $|\mathbb{F}_{q}|.$
Notice from Theorem 1.1 that if one could show that the restricted type
inequality (1.2) can be replaced by the strong type inequality, then the
mapping properties of the $k$-plane transforms in finite fields would be
completely established. Namely, our task would prove the following conjecture.
###### Conjecture 1.2.
Let $d\geq 2$ and $1\leq k\leq(d-1)$ be integers. Then, we have
$\|T_{\Pi_{k}}f\|_{L^{d+1}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{k+1}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$
where $C$ is independent of $|\mathbb{F}_{q}|.$
### 1.1. Statement of main results
In this paper we prove that Conjecture 1.2 is true for the $X$-ray transform
and the Radon transform. More precisely, we obtain the following theorem.
###### Theorem 1.3.
Let $d\geq 2$ be any integer. If $k=1$ or $k=d-1,$ then
$\|T_{\Pi_{k}}f\|_{L^{d+1}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{k+1}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$
where $C$ is independent of $|\mathbb{F}_{q}|.$
In order to prove Theorem 1.3 for the $X$-ray transform ($k=1$), we shall
adapt both the combinatorial arguments in [2] and the skills in [6] for
endpoint estimates. On the other hand, a Fourier analytic argument will be
required to prove Theorem 1.3 for the Radon transform ($k=d-1$) .
###### Remark 1.4.
After writing this paper, the author realized that our result for the $X$-ray
transform is a corollary of Theorem 1.1 in the paper [4]. This was pointed out
by R. Oberlin.
## 2\. Proof of the mapping properties of the $X$-ray transform
In this section, we restate and prove Theorem 1.3 in the case of the $X$-ray
transform. Namely, we prove the following statement which implies the sharp
boundedness of the $X$-ray transform.
###### Theorem 2.1.
Let $d\geq 2$ be any integer.
$\|T_{\Pi_{1}}f\|_{L^{d+1}(\Pi_{1},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{2}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$
where $C$ is independent of $|\mathbb{F}_{q}|.$
###### Proof.
We begin by following the argument in [6]. Without loss of generality, we may
assume that $f$ is a non-negative real-valued function and
(2.1) $\sum_{x\in\mathbb{F}_{q}^{d}}f(x)^{\frac{d+1}{2}}=1.$
Thus, it is natural to assume that $\|f\|_{\infty}\leq 1.$ Furthermore, we may
assume that $f$ is written by a step function
(2.2) $f(x)=\sum_{i=0}^{\infty}2^{-i}E_{i}(x),$
where the sets $E_{i}$ are disjoint subsets of $\mathbb{F}_{q}^{d}$, and here,
and throughout the paper, we write $E(x)$ for the characteristic function on a
set $E\subset\mathbb{F}_{q}^{d}.$ From (2.1) and (2.2), we also assume that
(2.3) $\sum_{j=0}^{\infty}2^{-\frac{(d+1)j}{2}}|E_{j}|=1\quad\mbox{and
}~{}~{}|E_{j}|\leq 2^{\frac{(d+1)j}{2}}~{}~{}\mbox{for
all}~{}~{}j=0,1,\cdots.$
Since $dx$ is the normalized counting measure on $\mathbb{F}_{q}^{d},$ the
assumption (2.1) shows that we only need to prove
(2.4) $\|T_{\Pi_{1}}f\|^{d+1}_{L^{d+1}(\Pi_{1},d\sigma)}\lesssim q^{-2d},$
where $f$ satisfies (2.2) and (2.3). Since we have assumed that $f\geq 0$, it
is clear that $T_{\Pi_{1}}f$ is also a non-negative real-valued function on
$\Pi_{1}.$ By expanding the left-hand side of (2.4) and using the facts that
$|w|=q$ for $w\in\Pi_{1}$ and $|\Pi_{1}|\sim q^{2(d-1)}$, we see that
$\|T_{\Pi_{1}}f\|^{d+1}_{L^{d+1}(\Pi_{1},d\sigma)}=\frac{1}{|\Pi_{1}|}\sum_{w\in\Pi_{1}}\left(T_{\Pi_{1}}f(w)\right)^{d+1}$
$\sim\frac{1}{q^{d+1}}\frac{1}{q^{2(d-1)}}\sum_{i_{0}=0}^{\infty}\dots\sum_{i_{d}=0}^{\infty}2^{-(i_{0}+\dots+i_{d})}\sum_{(x_{0},\dots,x_{d})\in
E_{i_{0}}\times\dots\times E_{i_{d}}}\sum_{w\in\Pi_{1}}w(x_{0})\dots w(x_{d})$
$\sim\frac{1}{q^{d+1}}\frac{1}{q^{2(d-1)}}\sum_{0=i_{0}\leq i_{1}\leq\dots\leq
i_{d}<\infty}2^{-(i_{0}+\dots+i_{d})}\sum_{(x_{0},\dots,x_{d})\in
E_{i_{0}}\times\dots\times E_{i_{d}}}\sum_{w\in\Pi_{1}}w(x_{0})\dots
w(x_{d}),$
where the last line follows from the symmetry of $i_{0},\cdots,i_{d}.$ We now
follows the argument in [2]. Notice that we can write
$\sum_{(x_{0},\dots,x_{d})\in E_{i_{0}}\times\dots\times
E_{i_{d}}}=\sum_{s=0}^{\infty}\sum_{(x_{0},\dots,x_{d})\in\Delta(s,i_{0},\dots,i_{d})},$
where $\Delta(s,i_{0},\dots,i_{d})=\\{(x_{0},\dots,x_{d})\in
E_{i_{0}}\times\dots\times E_{i_{d}}:[x_{0},\dots,x_{d}]~{}\mbox{is
a}~{}s\mbox{-plane}\\}$ and $[x_{0},\dots,x_{d}]$ denotes the smallest affine
subspace containing the elements $x_{0},\dots,x_{d}.$ In addition, observe
that if $s>1$ and $(x_{0},\dots,x_{d})\in\Delta(s,i_{0},\dots,i_{d}),$ then
the sum over $w\in\Pi_{1}$ vanishes. On the other hand, if $s=0,1,$ then the
sum over $w\in\Pi_{1}$ is same as the number of lines containing the unique
$s$-plane, that is $\sim q^{(d-1)(1-s)}.$ From these observations and (2.4),
it is enough to prove that for all $E_{i},i=0,1,\dots,$ satisfying (2.3),
(2.5) $\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}\sum_{s=0}^{1}|\Delta(s,i_{0},\dots,i_{d})|q^{-s(d-1)}\lesssim
1.$
Namely, it suffices to prove that for every $d\geq 2$ and $s=0,1$,
$\mbox{A}=\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|\Delta(s,i_{0},\dots,i_{d})|q^{-s(d-1)}\lesssim
1,$
where $\Delta(s,i_{0},\dots,i_{d})$ is defined as before, and the sets
$E_{i}$, i=0,1,…, satisfy (2.3). Suppose that $s=0.$ Since
$|\Delta(0,i_{0},\dots,i_{d})|\leq|E_{i_{0}}|$, it follows
$\mbox{A}\leq\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}|.$
Since the sum of a convergent geometric series is similar to the value of the
first term, we have the desirable conclusion for $s=0$:
$\mbox{A}\lesssim\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{-(d+1)i_{0}}\leq\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{-\frac{(d+1)i_{0}}{2}}=1,$
where the last equality is obtained from (2.3).
Next, we assume that $s=1.$ We must show that for all $E_{i},i=0,1,\dots$,
satisfying (2.3), we have
(2.6) $\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|\Delta(1,i_{0},\dots,i_{d})|q^{-(d-1)}\lesssim
1.$
We estimate the upper bound of $|\Delta(1,i_{0},\dots,i_{d})|.$ Fix
$x_{i_{0}}\in E_{i_{0}}$ which has $|E_{i_{0}}|$ choices. Notice that if
$(x_{i_{0}},\dots,x_{i_{d}})\in\Delta(1,i_{0},\dots,i_{d}),$ then all points
$x_{i_{0}},\dots,x_{i_{d}}$ must lie on a line, which is determined by at
least two different points of them . Therefore, for each $l=1,2,\dots,d,$, we
can define
$L(l)=\\{(x_{i_{0}},\dots,x_{i_{d}})\in\Delta(1,i_{0},\dots,i_{d}):[x_{i_{0}},\dots,x_{i_{l}}]~{}\mbox{is
a line, and}~{}[x_{i_{0}},\dots,x_{i_{l-1}}]~{}\mbox{is a point}\\},$
where we recall that $[\alpha_{1},\dots,\alpha_{s}]$ means the smallest affine
subspace containing all points $\alpha_{1},\dots,\alpha_{s}$ in
$\mathbb{F}_{q}^{d}.$ It is clear that
$\Delta(1,i_{0},\dots,i_{d})=\cup_{l=1}^{d}L(l),$ which implies that
(2.7) $|\Delta(1,i_{0},\dots,i_{d})|\leq\sum_{l=1}^{d}|L(l)|.$
By the definition of $L(l),l=1,\dots,d,$ it follows that for every
$l=1,\dots,d,$
(2.8) $|L(l)|\leq|E_{i_{0}}||E_{i_{l}}|q^{d-l}.$
To see this, first fix $x_{i_{0}}\in E_{i_{0}}$ which has $|E_{i_{0}}|$
choices. For each fixed $x_{i_{0}}\in E_{i_{0}},$ if
$(x_{i_{0}},\dots,x_{d})\in L(l)$, then all points
$x_{i_{1}},\dots,x_{i_{l-1}}$ are automatically chosen as $x_{i_{0}},$ and
there are at most $|E_{i_{l}}|$ choices for $x_{i_{l}}\in E_{i_{l}}.$ Since
$x_{0}$ and $x_{i_{l}}$ determine a fixed line, all points
$x_{i_{l+1}},\dots,x_{i_{d}}$ must lie on the line. Thus, there are at most
$q$ choices for each $x_{i_{l+1}},\dots,x_{i_{d}}$, because a line contains
exactly $q$ points in $\mathbb{F}_{q}^{d}.$
From (2.6), (2.7), and (2.8), it suffices to prove that for every
$l=1,\dots,d,$
$\mbox{B}=\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}||E_{i_{l}}|q^{1-l}\lesssim
1.$
Since $|E_{i_{l}}|\leq q^{d}$ and $l\geq 1,$ it is easy to see that
$|E_{i_{l}}|^{(l-1)/d}q^{1-l}\lesssim 1.$ Therefore, it follows that
$\mbox{B}\lesssim\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}||E_{i_{l}}|^{\frac{d+1-l}{d}}.$
Since $\frac{d+1-l}{d}>0$, applying (2.3) gives
$\mbox{B}\lesssim\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}|2^{\frac{(d+1-l)(d+1)i_{l}}{2d}}.$
Compute the inner summations by checking that each of them is a convergent
geometric series. It follows that
$\mbox{B}\lesssim\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{\frac{(-d^{2}-dl-l+1)i_{0}}{2d}}\leq\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{-\frac{(d+1)i_{0}}{2}}=1,$
where the last equality follows from (2.3). Thus, we complete the proof of
Theorem 2.1. ∎
###### Remark 2.2.
It seems that the similar arguments as above work for settling Conjecture 1.2,
but it may not be simple to estimate $|\Delta(s,i_{0},\dots,i_{d})|.$
## 3\. Proof of mapping properties of the Radon transform
In this section, we prove Theorem 1.3 in the case of the Radon transform.
Namely, we shall prove the following.
###### Theorem 3.1.
Let $d\geq 2$ be any integer. Then,
$\|T_{\Pi_{d-1}}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$
where $C$ is independent of $|\mathbb{F}_{q}|.$
###### Proof.
First, notice that if the dimension $d$ is two, then the statement of Theorem
3.1 follows immediately from Theorem 2.1. We therefore assume that $d\geq 3.$
As before, we may assume that $f$ is a non-negative real function and
(3.1) $\sum_{x\in\mathbb{F}_{q}^{d}}[f(x)]^{(d+1)/d}=1.$
Moreover, we may assume that the function $f$ is a step function:
(3.2) $f(x)=\sum_{i=0}^{\infty}2^{-i}E_{i}(x),$
where the sets $E_{i}$ are disjoint subsets of $\mathbb{F}_{q}^{d}.$ Notice
that (3.1) and (3.2) imply that
(3.3) $\sum_{j=0}^{\infty}2^{-\frac{(d+1)j}{d}}|E_{j}|=1\quad\mbox{and
}~{}~{}|E_{j}|\leq 2^{\frac{(d+1)j}{d}}~{}~{}\mbox{for all}~{}~{}j=0,1,\dots.$
We write $\Pi_{d-1}=H\cup\Theta$ where $H$ and $\Theta$ are defined by
$H:=\\{w\in\Pi_{d-1}:(0,\dots,0)\notin w\\}$
and
$\Theta:=\\{w\in\Pi_{d-1}:(0,\dots,0)\in w\\}.$
It is clear that $H$ and $\Theta$ are disjoint. Notice that we can identify
$H$ with $\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}$ in the sense that if
$w\in H$, then there exists a unique
$w^{\prime}\in\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}$ such that
$w=\\{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot x=1\\}.$
Thus, if $w\in H$, then we may assume that
$T_{\Pi_{d-1}}f(w)=\frac{1}{|w|}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=1}f(x)=\frac{1}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=1}f(x).$
On the other hand, for a fixed $w\in\Theta,$ there is a unique line passing
through the origin, say $L_{w}$, such that
$w=\\{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot x=0~{}\mbox{for
all}~{}w^{\prime}\in L_{w}\setminus\\{(0,\dots,0\\}\\}.$
By selecting one specific $w^{\prime}\in L_{w}\setminus\\{(0,\dots,0)\\}$ we
can identify $w\in\Theta$ with the specific point $w^{\prime}\in
L_{w}\setminus\\{(0,\dots,0)\\}.$ Throughout the paper, we denote by $S$ the
collection of the specific points each of which is chosen from
$L_{w}\setminus\\{(0,\dots,0)\\}$ for every $w\in\Theta.$ 111In the Euclidean
setting, one can consider the set $S$ as a half part of the unit sphere.
However, it is not true in general in the finite field setting. For example,
if the dimension $d$ is four and $-1\in\mathbb{F}_{q}$ is a square number,
then the line $l=\\{t(i,1,i,1):t\in\mathbb{F}_{q}\\}$ does not intersect the
set $\\{x\in\mathbb{F}_{q}^{4}:x_{1}^{2}+\cdots+x_{4}^{2}=1\\}.$ Thus, we also
assume that if $w\in\Theta,$ then
$T_{\Pi_{d-1}}f(w)=\frac{1}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=0}f(x),$
where $w^{\prime}\in S.$ Since $\Pi_{d-1}=H\cup\Theta$ and
$H\cap\Theta=\emptyset,$ the Radon transform $T_{\Pi_{d-1}}$ can be viewed as
$T_{\Pi_{d-1}}f(w)=T_{0}f(w)+T_{1}f(w)\quad\mbox{for}~{}~{}w\in\Pi_{d-1},$
where the operators $T_{0}$ and $T_{1}$ are defined as
$T_{0}f(w)=\frac{\Theta(w)}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=0}f(x)$
and
$T_{1}f(w)=\frac{H(w)}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=1}f(x).$
In order to prove Theorem 3.1, it therefore suffices to show that the
following two inequalities hold:
(3.4)
$\|T_{0}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)},$
and
(3.5)
$\|T_{1}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)},$
where the functions $f$ satisfy (3.1), (3.2), and (3.3).
### 3.1. Proof of the inequality (3.4)
Let us denote by $\chi$ the canonical additive character of $\mathbb{F}_{q}$
(see [7] or [5]). Recall that the orthogonality relation of $\chi$ holds:
$\sum_{s\in\mathbb{F}_{q}}\chi{(as)}=\left\\{\begin{array}[]{ll}0&\mbox{if}~{}~{}a\in\mathbb{F}_{q}^{*}=\mathbb{F}_{q}\setminus\\{0\\}\\\
q&\mbox{if}~{}~{}a=0.\end{array}\right.$
Using the orthogonality relation of $\chi$, we have
$T_{0}f(w)=\frac{\Theta(w)}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}}\left[\frac{1}{q}\sum_{s\in\mathbb{F}_{q}}\chi(s(w^{\prime}\cdot
x))\right]f(x)$
$=\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s=0}\chi(s(w^{\prime}\cdot
x))f(x)+\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))f(x)$
$=\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)+\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))f(x)$ $:=T_{0}^{\star}f(w)+T_{0}^{\star\star}f(w).$
Since $|T_{0}^{\star}f(w)|\leq\|f\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}$ for all
$w\in\Pi_{d-1},$ we see that
$\|T_{0}^{\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\leq\|f\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}\leq\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)},$
where we used the facts that $dx$ and $d\sigma$ are the normalized counting
measure on $\mathbb{F}_{q}^{d}$ and the normalized surface measure on
$\Pi_{d-1}$ respectively. To prove the inequality (3.4), it remains to prove
that for all functions $f$ satisfying (3.1), (3.2), and (3.3),
(3.6)
$\|T_{0}^{\star\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}=q^{-\frac{d^{2}}{d+1}},$
where the last equality follows from (3.1). We need the following lemma.
###### Lemma 3.2.
Let $d\geq 3$. Then, for every subset $E\subset\mathbb{F}_{q}^{d},$ we have
$\|T_{0}^{\star\star}E\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)}\lesssim
q^{-\frac{(d^{2}+1)}{d+1}}|E|^{\frac{d-1}{d+1}}.$
###### Proof.
Since $d\geq 3,$ we see that the statement of Lemma 3.2 follows immediately by
interpolating the following two estimates: for all indicator functions $E(x)$
on $\mathbb{F}_{q}^{d},$
(3.7) $\|T_{0}^{\star\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+1}|E|$
and
(3.8) $\|T_{0}^{\star\star}E\|_{L^{2}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+\frac{1}{2}}|E|^{\frac{1}{2}}.$
To obtain (3.7), notice that
$\|T_{0}^{\star\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}=\|T_{0}E-T_{0}^{\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}$
$\leq\|T_{0}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}+\|T_{0}^{\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}\leq
q^{-d+1}|E|+q^{-d}|E|\sim q^{-d+1}|E|.$
It remains to prove that (3.8) holds. It follows that for every set
$E\subset\mathbb{F}_{q}^{d},$
$\|T_{0}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}=\frac{1}{|\Pi_{d-1}|}\sum_{w\in\Pi_{d-1}}\left|\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))E(x)\right|^{2}$ $=\frac{1}{|\Pi_{d-1}|}\sum_{w^{\prime}\in
S}\left|\frac{1}{q^{d}}\sum_{x\in
E}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))\right|^{2}:=\frac{1}{|\Pi_{d-1}|}\sum_{w^{\prime}\in
S}\Gamma(w^{\prime}),$
where we used the fact that $S$ can be identified with $\Theta.$ Using a
change of variables, we see that for each $w^{\prime}\in
S,~{}\Gamma(w^{\prime})=\Gamma(tw^{\prime})$ for all $t\in\mathbb{F}_{q}^{*}.$
By the definition of $S,$ it therefore follows that
$\|T_{0}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\leq\frac{1}{|\Pi_{d-1}|}\frac{1}{q-1}\sum_{w^{\prime}\in\mathbb{F}_{q}^{d}}\left|\frac{1}{q^{d}}\sum_{x\in
E}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot x))\right|^{2}.$
Since $|\Pi_{d-1}|\sim q^{d}$, if we expand the square term and apply the
orthogonality relation of $\chi$ to the sum over
$w^{\prime}\in\mathbb{F}_{q}^{d}$, then we see that
$\|T_{0}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\lesssim\frac{1}{q^{3d+1}}\sum_{w^{\prime}\in\mathbb{F}_{q}^{d}}\sum_{x,x^{\prime}\in
E}\sum_{s,s^{\prime}\in\mathbb{F}_{q}^{*}}\chi(w^{\prime}\cdot(sx-s^{\prime}x^{\prime}))$
$=\frac{1}{q^{2d+1}}\sum_{x,x^{\prime}\in
E,s,s^{\prime}\in\mathbb{F}_{q}^{*}:sx=s^{\prime}x^{\prime}}1\leq\frac{|E|}{q^{2d-1}}.$
Thus, the proof of Lemma 3.2 is complete. ∎
We now prove (3.6). Since we have assumed that $f$ is considered as a step
function (3.2), it follows that
$\|T_{0}^{\star\star}f\|^{2}_{L^{d+1}(\Pi_{d-1},d\sigma)}=\|(T_{0}^{\star\star}f)(T_{0}^{\star\star}f)\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)}$
$\leq\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}2^{-i-j}\|(T_{0}^{\star\star}E_{i})(T_{0}^{\star\star}E_{j})\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)}$
$\sim\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}\|(T_{0}^{\star\star}E_{i})(T_{0}^{\star\star}E_{j})\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)},$
where the last line follows from the symmetry of $i,j.$ By Hölder’s
inequality, the inequality (3.7), and Lemma 3.2, (3.6) will follow if we prove
that
$\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}|E_{i}||E_{j}|^{\frac{d-1}{d+1}}\lesssim 1.$
This can be justified by making use of (3.3) and computing the summation over
$j$ variable:
$\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}|E_{i}||E_{j}|^{\frac{d-1}{d+1}}\leq\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}|E_{i}|\left(2^{\frac{(d+1)j}{d}}\right)^{\frac{d-1}{d+1}}\sim\sum_{i=0}^{\infty}|E_{i}|2^{-\frac{(d+1)i}{d}}=1,$
which completes the proof of (3.4).
### 3.2. Proof of the inequality (3.5)
By showing that the inequality (3.5) holds, we shall complete the proof of
Theorem 3.1. We shall take the same steps as in the previous subsection. From
the orthogonality relation of $\chi,$ it follows that for all $w\in\Pi_{d-1},$
$T_{1}f(w)=\frac{H(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)+\frac{H(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x-1))f(x):=T_{1}^{\star}(w)+T_{1}^{\star\star}(w).$
As before, it is easy to see that
$\|T_{1}^{\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\leq\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}.$
Thus, it is enough to prove that for all functions $f$ satisfying (3.1),
(3.2), and (3.3),
$\|T_{1}^{\star\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}=q^{-\frac{d^{2}}{d+1}},$
where the last equality follows from (3.1). From the same arguments as in the
proof of (3.4), our task is only to obtain Lemma 3.2 for the operator
$T_{1}^{\star\star}.$ As in the proof of Lemma 3.2, it suffices to prove the
following two equalities: for every subset $E$ of $\mathbb{F}_{q}^{d},$
(3.9) $\|T_{1}^{\star\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+1}|E|$
and
(3.10) $\|T_{1}^{\star\star}E\|_{L^{2}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+\frac{1}{2}}|E|^{\frac{1}{2}}.$
The inequality (3.9) follows immediately from the same argument as before. To
prove (3.10), we observe that
$\|T_{1}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}=\frac{1}{|\Pi_{d-1}|}\sum_{w\in\Pi_{d-1}}\left|\frac{H(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x-1))E(x)\right|^{2}$ $=\frac{1}{|\Pi_{d-1}|}\sum_{w\in
H}\left|\frac{1}{q^{d}}\sum_{x\in
E}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot x-1))\right|^{2}.$
Recall that $H\subset\Pi_{d-1}$ can be identified with
$\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}.$ Thus, if we dominate the sum
over $w\in H$ by the sum over $w^{\prime}\in\mathbb{F}_{q}^{d}$, expand the
square term, and use the orthogonality relation of $\chi$ over the variable
$w^{\prime}\in\mathbb{F}_{q}^{d},$ then it follows that
$0\leq\|T_{1}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\leq\frac{1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,s,s^{\prime}\in\mathbb{F}_{q}^{*}:sx=s^{\prime}x^{\prime}}\chi(-s+s^{\prime}).$
$=\frac{(q-1)|E|}{|\Pi_{d-1}|q^{d}}+\frac{1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,s,s^{\prime}\in\mathbb{F}_{q}^{*}:sx=s^{\prime}x^{\prime},s\neq
s^{\prime}}\chi(-s+s^{\prime})=\mbox{I}+\mbox{II}.$
Since $|\Pi_{d-1}|\sim q^{d}$, it is clear that
$\mbox{I}\sim\frac{|E|}{q^{2d-1}}.$ We claim that $\mbox{II}\leq 0.$ Indeed,
if we use a change of the variables by putting $s=t,\frac{s^{\prime}}{s}=u,$
then we see that
$\mbox{II}=\frac{1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,t,u\in\mathbb{F}_{q}^{*}:x=ux^{\prime},u\neq 0,1}\chi(-t(1-u)).$
Since $u\neq 1,$ the summation over $t\in\mathbb{F}_{q}^{*}$ is exactly $-1.$
Thus, our claim follows from the observation:
$\mbox{II}=\frac{-1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,u\in\mathbb{F}_{q}^{*}:x=ux^{\prime},u\neq 0,1}1\leq 0.$
Therefore, we conclude that
$\|T_{1}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\leq\mbox{I}+\mbox{II}\leq\mbox{I}\sim\frac{|E|}{q^{2d-1}},$
which implies that the inequality (3.10) holds . We have finished proving
Theorem 3.1. ∎
## References
* [1] M. Christ, Estimates for the $k$-plane transform, Indiana Univ. Math. J. 33 (1984), 891–910.
* [2] A. Carbery, B. Stones, and J. Wright, Averages in vector spaces over finite fields, Math. Proc. Camb. Phil. Soc. (2008), 144, 13, 13–27.
* [3] S.W. Drury, Generalizations of Riesz potentials and $L^{p}$ estimates for certain $k$-plane transforms, Illinois J. Math. 28 (1984), 495–512.
* [4] J. S. Ellenberg, R. Oberlin, and T. Tao, The Kakeya set and maximal conjectures for algebraic varieties over finite fields, Mathematika, to appear (www.arxiv.org).
* [5] H. Iwaniec, and E. Kowalski, Analytic Number Theory, Colloquium Publications 53 (2004).
* [6] A. Lewko and M. Lewko, _Endpoint restriction estimates for the paraboloid over finite fields_ , Proc. Amer. Math. Soc. 140 (2012), 2013-2028.
* [7] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge (1997).
* [8] D.M. Oberlin and E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), 641–650.
* [9] D. Solmon, A note on $k$-plane integral transforms, J. Math. Anal. Appl. 71 (1979), 351–358.
|
arxiv-papers
| 2012-09-06T08:12:48 |
2024-09-04T02:49:34.826070
|
{
"license": "Public Domain",
"authors": "Doowon Koh",
"submitter": "Doowon Koh",
"url": "https://arxiv.org/abs/1209.1215"
}
|
1209.1220
|
# Averaging operators over nondegenerate quadratic surfaces in finite fields
Doowon Koh Department of Mathematics
Chungbuk National University
Cheongju city, Chungbuk-Do 361-763 Korea [email protected]
###### Abstract.
We study mapping properties of the averaging operator related to the variety
$V=\\{x\in\mathbb{F}_{q}^{d}:Q(x)=0\\},$ where $Q(x)$ is a nondegenerate
quadratic polynomial over a finite field $\mathbb{F}_{q}$ with $q$ elements.
This paper is devoted to eliminating the logarithmic bound appearing in the
paper [5]. As a consequence, we settle down the averaging problems over the
quadratic surfaces $V$ in the case when the dimensions $d\geq 4$ are even and
$V$ contains a $d/2$-dimensional subspace.
###### 2010 Mathematics Subject Classification:
43A32, 43A15, 11T99
Key words and phrases: averaging operator, finite fields.
This research was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education, Science and Technology(2012010487)
## 1\. Introduction
Let $V\subset\mathbb{R}^{d}$ be a smooth hypersurface and $d\sigma$ a smooth,
compactly supported surface measure on $V.$ An averaging operator $A$ over $V$
is given by
$Af(x)=f\ast d\sigma(x)=\int_{V}f(x-y)d\sigma(y)$
where $f$ is a complex valued function on $\mathbb{R}^{d}.$ In this Euclidean
setting, the averaging problem is to determine the optimal range of exponents
$1\leq p,r\leq\infty$ such that
(1.1) $\|f\ast d\sigma\|_{L^{r}(\mathbb{R}^{d})}\leq
C_{p,r,d}\|f\|_{L^{p}(\mathbb{R}^{d})},~{}~{}f\in\mathcal{S}(\mathbb{R}^{d})$
where $\mathcal{S}(\mathbb{R}^{d})$ denotes the space of Schwartz functions.
When $V$ is the unit sphere $\mathbb{S}^{d-1},$ this problem is closely
related to regularity estimates of the solutions to the wave equation at time
$t=1$, and it was studied by R.S. Strichartz [12]. It is well known that
$L^{p}-L^{r}$ averaging results can be obtained by the decay estimates of the
Fourier transform of the surface measure $d\sigma$ on $V$. For instance, if
$|\widehat{d\sigma}(\xi)|=\left|\int_{V}e^{-2\pi
ix\cdot\xi}d\sigma(x)\right|\lesssim(1+|\xi|)^{-\alpha}$ for some $\alpha>0$,
then the averaging inequality (1.1) holds whenever
$1\leq p\leq
2,\quad\frac{1}{p}-\frac{1}{2}\leq\frac{1}{2}\left(\frac{\alpha}{\alpha+1}\right),~{}~{}\mbox{and}~{}~{}r=p^{\prime},$
where $p^{\prime}$ denotes the exponent conjugate to $p$ (see [6, 11]). Thus,
if $|\widehat{d\sigma}(\xi)|\lesssim(1+|\xi|)^{-(d-1)/2}$ and
$(1/p,1/r)=(d/(d+1),1/(d+1)),$ then the averaging estimate (1.1) holds. Since
$L^{1}-L^{1}$ and $L^{\infty}-L^{\infty}$ estimates are clearly possible, we
see from the interpolation theorem that if
$|\widehat{d\sigma}(\xi)|\lesssim(1+|\xi|)^{-(d-1)/2},$ then $L^{p}-L^{r}$
estimates hold whenever $(1/p,1/r)$ lies in the triangle $\Delta_{d}$ with
vertices $(0,0),(1,1),$ and $(d/(d+1),1/(d+1)).$ Moreover, it is well known
that $L^{p}-L^{r}$ estimates are impossible if $(1/p,1/r)$ lies outside of the
triangle $\Delta_{d}.$ Such analogous phenomena were also observed in the
finite field setting (see, for example, [1, 4, 5]).
On the other hand, if the optimal Fourier decay estimate of the surface
measure $d\sigma$ is given by
$|\widehat{d\sigma}(\xi)|\lesssim(1+|\xi|)^{-\alpha}\quad\mbox{for
some}~{}~{}\alpha<(d-1)/2,$
then it is in general hard to prove sharp averaging results and some technical
arguments are required to deal with the case (see [3]). In the finite field
case, this was also pointed out by the authors in [1].
As an analogue of Euclidean averaging problems, Carbery, Stones, and Wright
[1] initially introduced and studied the averaging problem in finite fields
and the author with Shen has recently investigated the averaging problem over
homogeneous varieties. In this introduction, we shortly review notation and
definitions for averaging problems in the finite field setting and readers are
referred to [5] for further information and motivation on the averaging
problem. Let $\mathbb{F}_{q}^{d}$ be a $d$-dimensional vector space over a
finite field $\mathbb{F}_{q}$ with $q$ elements. We endow the space
$\mathbb{F}_{q}^{d}$ with a normalized counting measure $``dx"$. Let
$V\subset\mathbb{F}_{q}^{d}$ be an algebraic variety. Then a normalized
surface measure $\sigma$ supported on $V$ can be defined by the relation
$\int f(x)~{}d\sigma(x)=\frac{1}{|V|}\sum_{x\in V}f(x)$
where $|V|$ denotes the cardinality of $V$ (see [9]). An averaging operator
$A$ can be defined by
$Af(x)=f\ast d\sigma(x)=\int f(x-y)~{}d\sigma(y)=\frac{1}{|V|}\sum_{y\in
V}f(x-y)$
where both $f$ and $Af$ are functions on $(\mathbb{F}_{q}^{d},dx).$ In this
setting the averaging problem over $V$ is to determine $1\leq p,r\leq\infty$
such that
(1.2) $\|Af\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\leq
C\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)},$
where the constant $C>0$ is independent of functions $f$ and $q$, the
cardinality of the underlying finite field $\mathbb{F}_{q}.$
###### Definition 1.1.
Let $1\leq p,r\leq\infty.$ We denote by $A(p\to r)\lesssim 1$ to indicate that
the averaging inequality (1.2) holds.
The main purpose of this paper is to obtain the complete $L^{p}-L^{r}$
estimates of the averaging operators over varieties determined by
nondegenerate quadratic form over $\mathbb{F}_{q}$. Let
$Q(x)\in\mathbb{F}_{q}[x_{1},\dots,x_{d}]$ be a nondegenerate quadratic form.
Define a variety
$S=\\{x\in\mathbb{F}_{q}^{d}:Q(x)=0\\}.$
We shall name this kind of varieties as a nondegenerate quadratic surface in
$\mathbb{F}_{q}^{d}.$ Since $Q(x)$ is a nondegenerate quadratic form, it can
be transformed into a diagonal form $a_{1}x_{1}^{2}+\dots+a_{d}x_{d}^{2}$ with
$a_{j}\neq 0$ by means of a linear substitution (see [8]). Therefore, we may
assume that any nondegenerate quadratic surface can be written by the form
(1.3) $S=\\{x\in\mathbb{F}_{q}^{d}:a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2}=0\\}$
where $a_{j}\in\mathbb{F}_{q}\setminus\\{0\\},j=1,\dots,d.$ The necessary
conditions for the averaging estimates over $S$ were given in [1, 5]. In fact,
$A(p\to r)\lesssim 1$ only if $(1/p,1/r)$ lies in the convex hull of
$(0,0),(0,1),(1,1),$ and $(d/(d+1),1/(d+1)).$ It is known from [5] that this
necessary conditions for $A(p\to r)\lesssim 1$ are sufficient conditions if
the dimension $d\geq 3$ is odd. On the other hand, it was observed in [5] that
if $d\geq 4$ is even and $S$ contains a subspace $H$ with $|H|=q^{d/2},$ then
$A(p\to r)\lesssim 1$ only if $(1/p,1/r)$ lies in the convex hull of
(1.4)
$(0,0),(0,1),(1,1),\left(\frac{d^{2}-2d+2}{d(d-1)},~{}\frac{1}{d-1}\right),~{}~{}\mbox{and}~{}~{}\left(\frac{d-2}{d-1},~{}\frac{d-2}{d(d-1)}\right).$
In this paper we show that (1.4) is also the sufficient conditions for $A(p\to
r)\lesssim 1$ in the specific case when the variety $S$ contains
$d/2$-dimensional subspace with $d\geq 4$ even. See Figure 1.
Figure 1. Necessary and Sufficient Conditions for $A(p\to r)\lesssim 1$ in the
case that $d\geq 4$ is even and $S$ contains a $d/2$-dimensional subspace
### 1.1. Statement of main result
###### Theorem 1.2.
Let $d\sigma$ be the normalized surface measure on the nondegenerate quadratic
surface $S\subset\mathbb{F}_{q}^{d},$ as defined in (1.3). Suppose that $d\geq
4$ is an even integer and $S$ contains a $d/2$-dimensional subspace. Then
$A(p\to r)\lesssim 1$ if and only if $(1/p,1/r)$ lies in the convex hull of
$(0,0),(0,1),(1,1),\left(\frac{d^{2}-2d+2}{d(d-1)},~{}\frac{1}{d-1}\right),~{}~{}\mbox{and}~{}~{}\left(\frac{d-2}{d-1},~{}\frac{d-2}{d(d-1)}\right).$
###### Remark 1.3.
If the dimension $d\geq 4$ is even, then the diagonal entries $a_{j}$ can be
properly chosen so that $S$ contains a $d/2$-dimensional subspace $H.$ Such an
example is essentially the following one (see Theorem 4.5.1 of [10]):
$S=\\{x\in\mathbb{F}_{q}^{d}:\sum_{k=1}^{d}(-1)^{k+1}x_{k}^{2}=0\\}$ and
$H=\\{(t_{1},t_{1},t_{2},t_{2},\dots,t_{d/2},t_{d/2})\in\mathbb{F}_{q}^{d}:t_{1},t_{2},\dots,t_{d/2}\in\mathbb{F}_{q}\\}.$
###### Remark 1.4.
From the observation (1.4), we only need to prove the “ if ” part of Theorem
1.2. Since $dx$ is the normalized counting measure on $\mathbb{F}_{q}^{d}$, it
follows from Young’s inequality that $A(p\to r)\lesssim 1$ for $1\leq r\leq
p\leq\infty.$ Thus, by duality and the interpolation theorem, it will be
enough to prove that
(1.5) $\|f\ast
d\sigma\|_{L^{d-1}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{d(d-1)/(d^{2}-2d+2)}(\mathbb{F}_{q}^{d},dx)}~{}~{}\mbox{for
all functions}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d}.$
The authors in [5] showed that this inequality holds for all characteristic
functions on subsets of $\mathbb{F}_{q}^{d}.$ Here, we improve upon their work
by obtaining the strong-type estimate.
### 1.2. Outline of this paper
In the remaining parts of this paper, we focus on providing the detail proof
of Theorem 1.2. In Section 2, we review the Fourier analysis in finite fields
and prove key lemmas which are essential in proving our main theorem. The
proof of Theorem 1.2 for even dimensions $d\geq 6$ will be completed in
Section 3. Namely, when $d\geq 6$ is any even integer, the inequality (1.5)
will be proved in Section 3. In the final section, we finish the proof of
Theorem 1.2 by proving the inequality (1.5) for $d=4.$
## 2\. key lemmas
In this section we drive key lemmas which play a crucial role in proving
Theorem 1.2. We begin by reviewing the Discrete Fourier analysis and readers
can be reffered to [5] for more information on it. Let $\mathbb{F}_{q}$ be a
finite field with $q$ elements. Throughout this paper, we assume that $q$ is a
power of odd prime so that the characteristic of $\mathbb{F}_{q}$ is greater
than two. We denote by $\chi$ a nontrivial additive character of
$\mathbb{F}_{q}.$ Recall that the orthogonality relation of the canonical
additive character $\chi$ says that
$\sum_{x\in\mathbb{F}_{q}^{d}}\chi(m\cdot
x)=\left\\{\begin{array}[]{ll}0&\mbox{if}~{}~{}m\neq(0,\dots,0)\\\
q^{d}&\mbox{if}~{}~{}m=(0,\dots,0),\end{array}\right.$
where $\mathbb{F}_{q}^{d}$ denotes the $d$-dimensional vector space over
$\mathbb{F}_{q}$ and $m\cdot x$ is the usual dot-product notation. Denote by
$(\mathbb{F}_{q}^{d},dx)$ the vector space over $\mathbb{F}_{q}$, endowed with
the normalized counting measure $``dx".$ Its dual space will be denoted by
$(\mathbb{F}_{q}^{d},dm)$ and we endow it with a counting measure $``dm".$ If
$f:(\mathbb{F}_{q}^{d},dx)\rightarrow\mathbb{C},$ then the Fourier transform
of the function $f$ is defined on $(\mathbb{F}_{q}^{d},dm)$:
$\widehat{f}(m)=\int_{\mathbb{F}_{q}^{d}}f(x)\chi(-x\cdot
m)~{}dx=\frac{1}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)\chi(-x\cdot
m)\quad\mbox{for}~{}~{}m\in\mathbb{F}_{q}^{d}.$
We also recall the Plancherel theorem:
$\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{f}(m)|^{2}=\frac{1}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{2}.$
Throughout this paper, we identify the set $E\subset\mathbb{F}_{q}^{d}$ with
the characteristic function on the set $E.$ We denote by $(d\sigma)^{\vee}$
the inverse Fourier transform of the normalized surface measure $d\sigma$ on
$S$ in (1.3) . Recall that
$(d\sigma)^{\vee}(m)=\int_{S}\chi(m\cdot
x)~{}d\sigma(x)=\frac{1}{|S|}\sum_{x\in S}\chi(m\cdot x).$
### 2.1. Gauss sums and estimates of $(d\sigma)^{\vee}$
Let $\eta$ denote the quadratic character of ${\mathbb{F}_{q}}.$ For each
$t\in{\mathbb{F}_{q}},$ the Gauss sum $G_{t}(\eta,\chi)$ is defined by
$G_{t}(\eta,\chi)=\sum_{s\in{\mathbb{F}_{q}}\setminus\\{0\\}}\eta(s)\chi(ts).$
The absolute value of the Gauss sum is given as follows (see [8, 2]):
$|G_{t}(\eta,\chi)|=\left\\{\begin{array}[]{ll}q^{\frac{1}{2}}&\mbox{if}~{}~{}t\neq
0\\\ 0&\mbox{if}~{}~{}t=0,\end{array}\right..$
It turns out that the inverse Fourier transform of $d\sigma$ can be written in
terms of the Gauss sum. The following was given in Lemma 4.1 of [5].
###### Lemma 2.1.
Let $S$ be the variety in $\mathbb{F}_{q}^{d}$ as defined in (1.3), and let
$d\sigma$ be the normalized surface measure on $S.$ If $d\geq 2$ is even, then
we have
$(d\sigma)^{\vee}(m)=\left\\{\begin{array}[]{ll}q^{d-1}|S|^{-1}+\frac{G_{1}^{d}}{|S|}(1-q^{-1})\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}m=(0,\dots,0)\\\
\frac{G_{1}^{d}}{|S|}(1-q^{-1})\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}m\neq(0,\dots,0),~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\\
-\frac{G_{1}^{d}}{q|S|}\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0.\end{array}\right.$
Here and throughout this paper we denote by $\eta$ the quadratic character of
${\mathbb{F}_{q}}$ and we write $G_{1}$ for the Gauss sum $G_{1}(\eta,\chi).$
Lemma 2.1 yields the following corollary.
###### Corollary 2.2.
Let $S$ be the variety in $\mathbb{F}_{q}^{d}$ as defined in (1.3) and let
$d\sigma$ be the normalized surface measure on $S.$ If $d\geq 4$ is even, then
we have
(2.1) $|S|=q^{d-1}+G_{1}^{d}(1-q^{-1})\eta(a_{1}\cdots a_{d})\sim q^{d-1},$
and
(2.2)
$|(d\sigma)^{\vee}(m)|\sim\left\\{\begin{array}[]{ll}q^{-\frac{(d-2)}{2}}&\mbox{if}~{}~{}m\neq(0,\dots,0),~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\\
q^{-\frac{d}{2}}&\mbox{if}~{}~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0.\end{array}\right.$
###### Proof.
By the definition of $(d\sigma)^{\vee}(0,\ldots,0)$, it is clear that
$(d\sigma)^{\vee}(0,\ldots,0)=1.$ Comparison with Lemma 2.1 shows that
$|S|=q^{d-1}+G_{1}^{d}(1-q^{-1})\eta(a_{1}\cdots a_{d}).$
Since $|G_{1}|=q^{1/2}$, it follows that $|S|\sim q^{d-1}$ for $d\geq 4$ even.
This proves (2.1). The inequality (2.2) follows immediately from Lemma 2.1
because $|G_{1}|=q^{1/2}$ and $|S|\sim q^{d-1}$ for $d\geq 4$ even. ∎
###### Remark 2.3.
It is clear from (2.2) that if $d\geq 4$ is even and $S$ is any nondegenerate
quadratic surface in $\mathbb{F}_{q}^{d},$ then
(2.3) $|(d\sigma)^{\vee}(m)|=\left|\frac{1}{|S|}\sum_{x\in S}\chi(m\cdot
x)\right|\sim\frac{1}{q^{d-1}}\left|\sum_{x\in S}\chi(m\cdot x)\right|\lesssim
q^{-\frac{(d-2)}{2}}~{}~{}\mbox{for}~{}~{}m\in\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}.$
### 2.2. Bochner-Riesz kernel
Recall that $d\sigma$ is the normalized surface measure on the nondegenerate
quadratic surface $S$ . In the finite field setting, the Bochner-Riesz kernel
$K$ is a function on $(\mathbb{F}_{q}^{d},dm)$ and it satisfies that
$K=(d\sigma)^{\vee}-\delta_{0}.$ Recall that $dm$ denotes the counting measure
on $\mathbb{F}_{q}^{d}.$ Notice that $K(m)=0$ if $m=(0,\ldots,0)$, and
$K(m)=(d\sigma)^{\vee}(m)$ otherwise. Also observe that
$d\sigma=\widehat{K}+\widehat{\delta_{0}}=\widehat{K}+1.$
Here, the last equality follows because $\delta_{0}$ is defined on the vector
space with the counting measure $dm,$ and its Fourier transform
$\widehat{\delta_{0}}$ is defined on the dual space with the normalized
counting measure $dx.$ More precisely, if $x\in(\mathbb{F}_{q}^{d},dx),$ then
$\widehat{\delta_{0}}(x)=\int_{m\in\mathbb{F}_{q}^{d}}\chi(-m\cdot
x)\delta_{0}(m)~{}dm=\sum_{m\in\mathbb{F}_{q}^{d}}\chi(-m\cdot
x)\delta_{0}(m)=1.$
Our main lemma is as follows.
###### Lemma 2.4.
Suppose that $d\geq 6$ is even. Then, for every $E\subset\mathbb{F}_{q}^{d},$
we have
(2.4)
$\|E\ast\widehat{K}\|_{L^{\frac{d-1}{2}}(\mathbb{F}_{q}^{d},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{\frac{-d^{2}+2d-3}{d-1}}|E|^{\frac{d-3}{d-1}}&\mbox{if}~{}~{}1\leq|E|\leq
q^{\frac{d-2}{2}}\\\
q^{\frac{-d^{2}+d-1}{d-1}}|E|&\mbox{if}~{}~{}q^{\frac{d-2}{2}}\leq|E|\leq
q^{\frac{d}{2}}\\\
q^{-d+1}|E|^{\frac{d-3}{d-1}}&\mbox{if}~{}~{}q^{\frac{d}{2}}\leq|E|\leq
q^{d}.\end{array}\right.$
where $K$ is the Bochner-Riesz kernel. On the other hand, for every
$E\subset\mathbb{F}_{q}^{4},$ it follows that
(2.5)
$\|E\ast\widehat{K}\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-\frac{19}{6}}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}1\leq|E|\leq
q\\\ q^{-\frac{10}{3}}|E|&\mbox{if}~{}~{}q\leq|E|\leq q^{2}\\\
q^{-3}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}q^{2}\leq|E|\leq
q^{4}.\end{array}\right.$
###### Proof.
Using the interpolation theorem, it suffices to prove that the following two
inequalities hold for all d$\geq 4$ even:
(2.6) $\|E\ast\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\lesssim
q^{-d+1}|E|$
and
(2.7)
$\|E\ast\widehat{K}\|_{L^{2}(\mathbb{F}_{q}^{d},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{\frac{-2d+1}{2}}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}1\leq|E|\leq
q^{\frac{d-2}{2}}\\\
q^{\frac{-5d+4}{4}}|E|&\mbox{if}~{}~{}q^{\frac{d-2}{2}}\leq|E|\leq
q^{\frac{d}{2}}\\\
q^{-d+1}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}q^{\frac{d}{2}}\leq|E|\leq
q^{d}.\end{array}\right.$
The estimate (2.6) can be obtained by applying Young’s inequality. In fact, we
see that
$\|E\ast\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\leq\|\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\|E\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}.$
Since $\|\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\lesssim q$ and
$\|E\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}=q^{-d}|E|,$ the inequality (2.6) is
established. To prove the inequality (2.7), first use the Plancherel theorem.
It follows that
$\|E\ast\widehat{K}\|^{2}_{L^{2}(\mathbb{F}_{q}^{d},dx)}=\|\widehat{E}K\|^{2}_{L^{2}(\mathbb{F}_{q}^{d},dm)}.$
Now, we recall that $dx$ is the normalized counting measure but $dm$ is the
counting measure. Thus, the expression above is given by
$\displaystyle\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{E}(m)|^{2}|K(m)|^{2}=$
$\displaystyle\sum_{m\neq(0,\dots,0)}|\widehat{E}(m)|^{2}|(d\sigma)^{\vee}(m)|^{2}$
$\displaystyle\sim\frac{1}{q^{d-2}}\sum_{\begin{subarray}{c}m\neq(0,\dots,0):\\\
\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\end{subarray}}|\widehat{E}(m)|^{2}+\frac{1}{q^{d}}\sum_{\begin{subarray}{c}m\neq(0,\dots,0):\\\
\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0\end{subarray}}|\widehat{E}(m)|^{2}=\mbox{I}+\mbox{II},$
where the first line and the second line follow from the definition of $K$ and
the inequality (2.2) in Corollary 2.2, respectively. Applying the Plancherel
theorem, it is clear that
(2.8)
$\mbox{II}\leq\frac{1}{q^{d}}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{E}(m)|^{2}=q^{-2d}|E|.$
In order to obtain a good upper bound of I, we shall conduct two different
estimates on $\mbox{I}.$ First, the Plancherel theorem yields
(2.9)
$\mbox{I}\leq\frac{1}{q^{d-2}}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{E}(m)|^{2}=\frac{|E|}{q^{2d-2}}.$
On the other hand, it follows that
$\mbox{I}\leq\frac{1}{q^{d-2}}\sum_{\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0}|\widehat{E}(m)|^{2}=\frac{1}{q^{3d-2}}\sum_{\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0}\sum_{x,y\in
E}\chi(-m\cdot(x-y)).$
Now, let
$S_{a}=\\{m\in\mathbb{F}_{q}^{d}:\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\}$
which is also a nondegenerate quadratic surface with $|S_{a}|\sim q^{d-1}.$
Then the expression above can be written by
$\frac{1}{q^{3d-2}}\sum_{x,y\in E:x=y}|S_{a}|+\frac{1}{q^{3d-2}}\sum_{x,y\in
E:x\neq y}\left(\sum_{m\in S_{a}}\chi(-m\cdot(x-y))\right).$
Now, we see from (2.3) that if $x\neq y$, then $\left|\sum_{m\in
S_{a}}\chi(-m\cdot(x-y))\right|\lesssim q^{\frac{d}{2}}.$ Thus, we obtain that
$\mbox{I}\lesssim q^{-2d+1}|E|+q^{\frac{-5d+4}{2}}|E|^{2}.$
Combining this with the inequality (2.9) gives
$\mbox{I}\lesssim\min\left(\frac{|E|}{q^{2d-2}},~{}q^{-2d+1}|E|+q^{\frac{-5d+4}{2}}|E|^{2}\right).$
In conjunction with the inequality (2.8), this shows that
$\|E\ast\widehat{K}\|^{2}_{L^{2}(\mathbb{F}_{q}^{d},dx)}\lesssim\min\left(\frac{|E|}{q^{2d-2}},~{}q^{-2d+1}|E|+q^{\frac{-5d+4}{2}}|E|^{2}\right)~{}+q^{-2d}|E|.$
Since $(\alpha+\beta)^{1/2}\sim\alpha^{1/2}+\beta^{1/2}$ for $\alpha,\beta\geq
0$, it also follows that
$\|E\ast\widehat{K}\|_{L^{2}(\mathbb{F}_{q}^{d},dx)}\lesssim\min\left(q^{-d+1}|E|^{\frac{1}{2}},~{}q^{\frac{-2d+1}{2}}|E|^{\frac{1}{2}}+q^{\frac{-5d+4}{4}}|E|\right)~{}+q^{-d}|E|^{\frac{1}{2}}.$
A direct computation shows that this implies the inequality (2.7). We complete
the proof of Lemma 2.4.
∎
## 3\. Proof of Theorem 1.2 for $d\geq 6$
In this section we provide the complete proof of Theorem 1.2 in the case that
$d\geq 6$ is even. The proof for $d=4$ shall be independently given in the
following section. The main reason is as follows. Lemma 2.4 shall be used to
prove Theorem 1.2. If $d\geq 6$ is even, then we have seen that the inequality
(2.4) of Lemma 2.4 follows by interpolating (2.6) and (2.7). However, if $d$
is four, then such an interpolation is too meaningless to assert that (2.4)
holds for $d=4.$ As an alternative approach, the inequality (2.5) of Lemma 2.4
shall be applied to complete the proof for $d=4.$ In this case we need more
delicate estimates.
Now we start proving Theorem 1.2 for $d\geq 6$ even. As mentioned in Remark
1.4, it is enough to prove the following statement.
###### Theorem 3.1.
Let $S$ be the variety in $\mathbb{F}_{q}^{d}$ as defined in (1.3). If $d\geq
6$ is even, then we have
$\|f\ast
d\sigma\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}~{}~{}\mbox{for
all functions}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$
where $(p,r)=\left(\frac{d^{2}-d}{d^{2}-2d+2},d-1\right).$
###### Proof.
Let $p=\frac{d^{2}-d}{d^{2}-2d+2}$ and $r=d-1.$ We aim to prove that for every
complex-valued function $f$ on $\mathbb{F}_{q}^{d},$
$\|f\ast
d\sigma\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}=\left(q^{-d}\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{p}\right)^{\frac{1}{p}}.$
As in [7] we proceed with the proof by decomposing the function $f$ to which
the operator is applied into level sets. Without loss of generality, we may
assume that $f$ is a nonnegative real-valued function and
$\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{p}=1.$
Therefore, we may also assume that
(3.1) $f=\sum_{k=0}^{\infty}2^{-k}E_{k},$
where $E_{0},E_{1},\dots$ are disjoint subsets of $\mathbb{F}_{q}^{d}.$ It
follows from these assumptions that
(3.2) $\sum_{j=0}^{\infty}2^{-pj}|E_{j}|=1,$
and hence for every $j=0,1,\dots,$
(3.3) $|E_{j}|\leq 2^{pj}.$
Recall that $d\sigma=\widehat{K}+1$ where $K$ is the Bochner-Riesz kernel. It
follows that
$\|f\ast
d\sigma\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\leq\|f\ast\widehat{K}\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}+\|f\ast
1\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}.$
Since $r>p$ and $dx$ is the normalized counting measure on
$\mathbb{F}_{q}^{d},$ it is clear from Young’s inequality that
$\|f\ast
1\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\leq\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}.$
Therefore, it suffices to prove the following inequality
$\|f\ast\widehat{K}\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}.$
Since we have assumed that $\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{p}=1$, we see
that $\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}=q^{-d/p}.$ Also observe that
$\|f\ast\widehat{K}\|^{2}_{L^{r}(\mathbb{F}_{q}^{d},dx)}=\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}.$
From these observations, our task is to show that
(3.4)
$q^{\frac{2d}{p}}\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}\lesssim
1.$
Using (3.1), we see that
$\displaystyle
q^{\frac{2d}{p}}\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}$
$\displaystyle\leq
q^{\frac{2d}{p}}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}2^{-k-j}\|(E_{k}\ast\widehat{K})(E_{j}\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}$
$\displaystyle\lesssim
q^{\frac{2d}{p}}q^{-d+1}\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-k-j}|E_{k}|\|(E_{j}\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)},$
where the last line follows by the symmetry of $k$ and $j,$ and the inequality
(2.6). Now, for each $j=0,1,2,\dots,$ we consider the following three sets:
$J_{1}=\\{j:1\leq|E_{j}|\leq q^{\frac{d-2}{2}}\\},$
$J_{2}=\\{j:q^{\frac{d-2}{2}}<|E_{j}|\leq q^{\frac{d}{2}}\\},$
and
$J_{3}=\\{j:q^{\frac{d}{2}}<|E_{j}|\leq q^{d}\\}.$
Since $r/2=(d-1)/2,$ it is clear from (2.4) in Lemma 2.4 that our goal is to
prove the following three inequalities:
(3.5)
$A_{1}:=q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+2d-3}{d-1}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{1}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{\frac{d-3}{d-1}}\lesssim 1,$ (3.6)
$A_{2}:=q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{2}}^{\infty}2^{-k-j}|E_{k}||E_{j}|\lesssim 1,$ (3.7)
$A_{3}:=q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{3}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{\frac{d-3}{d-1}}\lesssim 1.$
First, we prove that the inequality (3.5) holds. Since
$p=\frac{d^{2}-d}{d^{2}-2d+2},$ a direct computation shows that
$q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+2d-3}{d-1}}=1.$ Now recall from (3.3)
that $|E_{j}|\leq 2^{pj}$ for all $j=0,1,\dots.$ Therefore, it follows that
$A_{1}\leq\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{1}}^{\infty}2^{-k-j}|E_{k}|2^{\frac{jp(d-3)}{d-1}}\leq\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{j\left(-1+\frac{p(d-3)}{d-1}\right)}.$
Since $-1+\frac{p(d-3)}{d-1}=\frac{-d-2}{d^{2}-2d+2}<0$ and the sum over $j$
is a geometric series, we see that
$\sum_{j=k}^{\infty}2^{j\left(-1+\frac{p(d-3)}{d-1}\right)}\sim
2^{k\left(-1+\frac{p(d-3)}{d-1}\right)}.$ Thus, the inequality (3.5) is
established as follows:
$A_{1}\lesssim\sum_{k=0}^{\infty}|E_{k}|2^{k\left(-2+\frac{p(d-3)}{d-1}\right)}\leq\sum_{k=0}^{\infty}|E_{k}|2^{-pk}=1,$
where we used the simple observation that $-2+\frac{p(d-3)}{d-1}<-p,$ and then
the assumption $(\ref{as1}).$
Second, we prove that the inequality (3.6) holds. Let
$\varepsilon=\frac{2d-4}{d^{2}-d}.$ Since $d\geq 6,$ we see that
$0<\varepsilon<1.$ Write $A_{2}$ as follows:
$A_{2}=q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{2}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{1-\varepsilon}|E_{j}|^{\varepsilon}.$
Since $0<\varepsilon<1,$ we notice from (3.3) that
$|E_{j}|^{1-\varepsilon}\leq 2^{p(1-\varepsilon)j}.$ By the definition of the
set $J_{2}$, we also see that $|E_{j}|^{\varepsilon}\leq
q^{\frac{d\varepsilon}{2}}$ for all $j\in J_{2}.$ Then, we have
$\displaystyle A_{2}$ $\displaystyle\leq
q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}q^{\frac{d\varepsilon}{2}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{2}}^{\infty}2^{-k-j}|E_{k}|2^{p(1-\varepsilon)j}$ $\displaystyle\leq
q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}q^{\frac{d\varepsilon}{2}}\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{j\left(-1+p(1-\varepsilon)\right)}.$
Notice that
$q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}q^{\frac{d\varepsilon}{2}}=1,$
and the geometric series over $j$ converges to $\sim
2^{k\left(-1+p(1-\varepsilon)\right)}$ because
$-1+p(1-\varepsilon)=\frac{-d+2}{d^{2}-2d+2}<0$ for $d\geq 6.$ From this
observation and (3.2), the inequality (3.6) follows because we have
$A_{2}\lesssim\sum_{k=0}^{\infty}|E_{k}|2^{k\left(-2+p(1-\varepsilon)\right)}=\sum_{k=0}^{\infty}|E_{k}|2^{-pk}=1.$
Finally, we show that the inequality (3.7) holds. As in the proof of the
inequality (3.6), we let $0<\delta=\frac{4}{d^{2}-d}<1$ for $d\geq 6.$ The
value $A_{3}$ is written by
$A_{3}=q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{3}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{\delta+\frac{d-3}{d-1}}|E_{j}|^{-\delta}.$
Notice from (3.3) that $|E_{j}|^{\delta+\frac{d-3}{d-1}}\leq
2^{p\left(\delta+\frac{d-3}{d-1}\right)j}$ for all $j=0,1,2,\dots.$ By the
definition of $J_{3}$, it is easy to notice that $|E_{j}|^{-\delta}\leq
q^{\frac{-d\delta}{2}}$ for $j\in J_{3}.$ It therefore follows that
$\displaystyle A_{3}$ $\displaystyle\leq
q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}q^{\frac{-d\delta}{2}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{3}}^{\infty}2^{-k-j}|E_{k}|2^{p\left(\delta+\frac{d-3}{d-1}\right)j}$
$\displaystyle\leq\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{\left(-1+p\left(\delta+\frac{d-3}{d-1}\right)\right)j}$
$\displaystyle\sim\sum_{k=0}^{\infty}|E_{k}|2^{\left(-2+p\left(\delta+\frac{d-3}{d-1}\right)\right)k}=\sum_{k=0}^{\infty}|E_{k}|2^{-pk}=1,$
where we used the facts that
$q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}q^{\frac{-d\delta}{2}}=1,$
$\left(-1+p\left(\delta+\frac{d-3}{d-1}\right)\right)=\frac{-d+2}{d^{2}-2d+2}<0$
for $d\geq 6,$ and $\left(-2+p\left(\delta+\frac{d-3}{d-1}\right)\right)=-p,$
and then the assumption (3.2) for the last equality. Thus, the inequality
(3.7) holds and the proof of Theorem 3.1 is complete.∎
## 4\. Proof of Theorem 1.2 for $d=4$
As observed in Remark 1.4, it amounts to showing the following statement.
###### Theorem 4.1.
Let $S$ be the variety in $\mathbb{F}_{q}^{4}$ as defined in (1.3). Then, we
have
$\|f\ast
d\sigma\|_{L^{3}(\mathbb{F}_{q}^{4},dx)}\lesssim\|f\|_{L^{\frac{6}{5}}(\mathbb{F}_{q}^{4},dx)}~{}~{}\mbox{for
all functions}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{4}.$
###### Proof.
We will proceed by the similar ways as in the previous section. However, the
proof of the theorem will be based on (2.5), rather than (2.4) in Lemma 2.4.
We begin by recalling from (2.5) and (2.7) that
(4.1)
$\|E\ast\widehat{K}\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-\frac{19}{6}}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}1\leq|E|\leq
q\\\ q^{-\frac{10}{3}}|E|&\mbox{if}~{}~{}q\leq|E|\leq q^{2}\\\
q^{-3}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}q^{2}\leq|E|\leq
q^{4},\end{array}\right.$
and
(4.2)
$\|E\ast\widehat{K}\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-\frac{7}{2}}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}1\leq|E|\leq
q\\\ q^{-4}|E|&\mbox{if}~{}~{}q\leq|E|\leq q^{2}\\\
q^{-3}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}q^{2}\leq|E|\leq
q^{4}.\end{array}\right.$
We must show that for all complex-valued functions $f$ on
$\mathbb{F}_{q}^{4},$
$\|f\ast
d\sigma\|_{L^{3}(\mathbb{F}_{q}^{4},dx)}\lesssim\|f\|_{L^{\frac{6}{5}}(\mathbb{F}_{q}^{4},dx)}.$
As noticed in the previous section, it suffices to prove this inequality under
the following assumptions:
$\sum_{x\in\mathbb{F}_{q}^{4}}|f(x)|^{\frac{6}{5}}=1\quad\mbox{and}\quad
f=\sum_{k=0}^{\infty}2^{-k}E_{k},$
where $E_{0},E_{1},\dots$ are disjoint subsets of $\mathbb{F}_{q}^{4}.$ From
these assumptions, it is clear that
(4.3) $\sum_{j=0}^{\infty}2^{-\frac{6j}{5}}|E_{j}|=1\quad\mbox{for
all}~{}j=0,1,\dots.$
This clearly implies that
(4.4) $|E_{j}|\leq 2^{\frac{6j}{5}}\quad\mbox{for all}~{}j=0,1,\dots.$
According to (3.4), it is enough to prove that
$q^{\frac{20}{3}}\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{3}{2}}(\mathbb{F}_{q}^{4},dx)}\lesssim
1.$
Since $f=\sum_{k=0}^{\infty}2^{-k}E_{k},$ it is enough to show that
$q^{\frac{20}{3}}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}2^{-k-j}\|(E_{k}\ast\widehat{K})(E_{j}\ast\widehat{K})\|_{L^{\frac{3}{2}}(\mathbb{F}_{q}^{4},dx)}\lesssim
1.$
By the symmetry of $k$ and $j,$ and the Hölder inequality, our task is to
prove
$q^{\frac{20}{3}}\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}\lesssim
1.$
Main steps to prove this inequality are summarized as follows. By considering
the sizes of $|E_{k}|$ and $|E_{j}|$, we first decompose
$\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}$ as nine parts. Next, using the
estimates (4.1),(4.2), (4.4), (4.3), and a convergence property of a geometric
series, we show that each part of them is $\lesssim 1,$ which completes the
proof of Theorem 4.1. For the sake of completeness, we shall give full
details. To do this, let us define the following $9$ sets: for
$N=\\{0,1,\dots\\},$
$I_{1}=\\{(k,j)\in N\times N:k\leq j,~{}1\leq|E_{k}|\leq q,~{}1\leq|E_{j}|\leq
q\\},$ $I_{2}=\\{(k,j)\in N\times N:k\leq j,~{}1\leq|E_{k}|\leq
q,~{}q<|E_{j}|\leq q^{2}\\},$ $I_{3}=\\{(k,j)\in N\times N:k\leq
j,~{}1\leq|E_{k}|\leq q,~{}q^{2}<|E_{j}|\leq q^{4}\\},$ $I_{4}=\\{(k,j)\in
N\times N:k\leq j,~{}q<|E_{k}|\leq q^{2},~{}1\leq|E_{j}|\leq q\\},$
$I_{5}=\\{(k,j)\in N\times N:k\leq j,~{}q<|E_{k}|\leq q^{2},~{}q<|E_{j}|\leq
q^{2}\\},$ $I_{6}=\\{(k,j)\in N\times N:k\leq j,~{}q<|E_{k}|\leq
q^{2},~{}q^{2}<|E_{j}|\leq q^{4}\\},$ $I_{7}=\\{(k,j)\in N\times N:k\leq
j,~{}q^{2}<|E_{k}|\leq q^{4},~{}1\leq|E_{j}|\leq q\\},$ $I_{8}=\\{(k,j)\in
N\times N:k\leq j,~{}q^{2}<|E_{k}|\leq q^{4},~{}q<|E_{j}|\leq q^{2}\\},$
$I_{9}=\\{(k,j)\in N\times N:k\leq j,~{}q^{2}<|E_{k}|\leq
q^{4},~{}q^{2}<|E_{j}|\leq q^{4}\\}.$
### 4.1. Estimate of the sum over $I_{1}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{1}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle\sum_{(k,j)\in
I_{1}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}\leq\sum_{(k,j)\in
I_{1}}2^{-k-j}|E_{k}|2^{\frac{3j}{5}}\quad\mbox{since}\quad|E_{j}|^{\frac{1}{2}}\leq
2^{\frac{3j}{5}}\quad\mbox{by (\ref{size})}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}|E_{k}|2^{-k}\sum_{j=k}^{\infty}2^{-\frac{2j}{5}}\sim\sum_{k=0}^{\infty}|E_{k}|2^{-\frac{7k}{5}}\leq\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1\quad\mbox{by
(\ref{one})}.$
### 4.2. Estimate of the sum over $I_{2}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{2}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{1}{2}}\sum_{(k,j)\in
I_{2}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|\leq q^{-\frac{1}{2}}\sum_{(k,j)\in
I_{2}}2^{-j}|E_{j}|\quad\mbox{since}\quad 2^{-k}|E_{k}|^{\frac{5}{6}}\leq
1\quad\mbox{by (\ref{size})}$ $\displaystyle\leq$ $\displaystyle\sum_{(k,j)\in
I_{2}}2^{-j}|E_{j}|^{\frac{3}{4}}\quad\mbox{since}\quad|E_{j}|^{\frac{1}{4}}\leq
q^{\frac{1}{2}}\quad\mbox{for}~{}~{}(k,j)\in I_{2}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-\frac{j}{10}}\quad\mbox{by
(\ref{size})}$ $\displaystyle\sim$
$\displaystyle\sum_{k=0}^{\infty}2^{-\frac{k}{10}}\sim 1.$
### 4.3. Estimate of the sum over $I_{3}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{3}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{1}{2}}\sum_{(k,j)\in
I_{3}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}\leq
q^{\frac{1}{2}}\sum_{(k,j)\in I_{3}}2^{-j}|E_{j}|^{\frac{1}{2}}\quad\mbox{by
(\ref{size})}$ $\displaystyle=$ $\displaystyle\sum_{(k,j)\in
I_{3}}2^{-j}|E_{j}|^{\frac{3}{4}}q^{\frac{1}{2}}|E_{j}|^{-\frac{1}{4}}<\sum_{(k,j)\in
I_{3}}2^{-j}|E_{j}|^{\frac{3}{4}}\quad\mbox{since}\quad
q^{2}<|E_{j}|~{}~{}\mbox{for}~{}~{}(k,j)\in I_{3}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-\frac{j}{10}}\sim\sum_{k=0}^{\infty}2^{-\frac{k}{10}}\sim
1.$
where (4.4) was also used to obtain the last inequality .
### 4.4. Estimate of the sum over $I_{4}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{4}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{1}{6}}\sum_{(k,j)\in
I_{4}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}}\leq q^{-\frac{1}{6}}\sum_{(k,j)\in
I_{4}}2^{-k}|E_{k}|2^{-j}2^{\frac{3j}{5}}\quad\mbox{by (\ref{size})}$
$\displaystyle\leq$ $\displaystyle
q^{-\frac{1}{6}}\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{2j}{5}}\sim
q^{-\frac{1}{6}}\sum_{k=0}^{\infty}2^{-\frac{7k}{5}}|E_{k}|$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1\quad\mbox{by
(\ref{one})}.$
### 4.5. Estimate of the sum over $I_{5}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{5}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{2}{3}}\sum_{(k,j)\in
I_{5}}2^{-k-j}|E_{k}||E_{j}|\leq\sum_{(k,j)\in
I_{5}}2^{-k-j}|E_{k}||E_{j}|^{\frac{2}{3}}\quad\mbox{since}~{}~{}|E_{j}|^{\frac{1}{3}}\leq
q^{\frac{2}{3}}\quad\mbox{for}~{}~{}(k,j)\in I_{5}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1,$
where we used (4.4), the convergence of a geometric series, and (4.3) in the
last line.
### 4.6. Estimate of the sum over $I_{6}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{6}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{1}{3}}\sum_{(k,j)\in
I_{6}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}}<\sum_{(k,j)\in
I_{6}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}+\frac{1}{6}}\quad\mbox{since}~{}~{}|E_{j}|^{-\frac{1}{6}}<q^{-\frac{1}{3}}\quad\mbox{for}~{}~{}(k,j)\in
I_{6}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1,$
where (4.4), the convergence of a geometric series, and (4.3) were also
applied in the last line.
### 4.7. Estimate of the sum over $I_{7}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{7}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{1}{6}}\sum_{(k,j)\in
I_{7}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}=q^{\frac{1}{6}}\sum_{(k,j)\in
I_{7}}2^{-k-j}|E_{k}||E_{k}|^{-\frac{1}{6}}|E_{j}|^{\frac{1}{2}}$
$\displaystyle<$ $\displaystyle q^{\frac{1}{6}-\frac{1}{3}}\sum_{(k,j)\in
I_{7}}2^{-k}|E_{k}|2^{-\frac{2j}{5}}\quad\mbox{by
observing}\quad|E_{k}|^{-\frac{1}{6}}<q^{-\frac{1}{3}}\quad{for}~{}~{}(k,j)\in
I_{7}~{}~{}\mbox{and by (\ref{size})}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{2j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{7k}{5}}|E_{k}|\leq\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1.$
### 4.8. Estimate of the sum over $I_{8}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{8}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{1}{3}}\sum_{(k,j)\in
I_{8}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|=q^{-\frac{1}{3}}\sum_{(k,j)\in
I_{8}}2^{-k-j}|E_{k}||E_{j}|^{\frac{2}{3}}|E_{k}|^{-\frac{1}{6}}|E_{j}|^{\frac{1}{3}}$
$\displaystyle<$ $\displaystyle\sum_{(k,j)\in
I_{8}}2^{-k-j}|E_{k}||E_{j}|^{\frac{2}{3}}\quad\mbox{since}~{}~{}|E_{k}|^{-\frac{1}{6}}<q^{-\frac{1}{3}},\quad|E_{j}|^{\frac{1}{3}}\leq
q^{\frac{2}{3}}~{}~{}\mbox{for}~{}~{}(k,j)\in I_{8}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1.$
### 4.9. Estimate of the sum over $I_{9}$
It follows from (4.1) and (4.2) that
$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{9}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{2}{3}}\sum_{(k,j)\in
I_{9}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}=q^{\frac{2}{3}}\sum_{(k,j)\in
I_{9}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}+\frac{1}{6}}|E_{k}|^{-\frac{1}{6}}|E_{j}|^{-\frac{1}{6}}$
$\displaystyle<$ $\displaystyle\sum_{(k,j)\in
I_{9}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}+\frac{1}{6}}\quad\mbox{since}~{}~{}|E_{k}|^{-\frac{1}{6}},|E_{j}|^{-\frac{1}{6}}<q^{-\frac{1}{3}}~{}~{}\mbox{for}~{}~{}(k,j)\in
I_{9}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-j}|E_{j}|^{\frac{2}{3}}\leq\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1,$
where we also used (4.4), the convergence of a geometric series, and (4.3) in
the last line. ∎
Acknowledgment : The author would like to thank the referee for his/her
valuable comments for developing the final version of this paper.
## References
* [1] A. Carbery, B. Stones, and J. Wright, _Averages in vector spaces over finite fields,_ Math. Proc. Camb. Phil. Soc. (2008), 144, 13, 13–27.
* [2] H. Iwaniec and E. Kowalski, _Analytic Number Theory,_ Colloquium Publications, 53 (2004).
* [3] A. Iosevich and E. Sawyer, _Sharp $L^{p}-L^{r}$ estimates for a class of averaging operators,_ Ann. Inst. Fourier, Grenoble, 46, 5 (1996), 1359–1384.
* [4] D. Koh and C. Shen, _Harmonic analysis related to homogeneous varieties in three dimensional vector spaces over finite fields_ , Canad. J. Math. 64 (2012), 1036-1057.
* [5] D. Koh and C. Shen, _Extension and averaging operators for finite fields,_ Proc. Edinb. Math. Soc., To appear.
* [6] W. Littman, _$L^{p}-L^{q}$ estimates for singular integral operators,_ Proc. Symp. Pure Math., 23(1973), 479–481.
* [7] A. Lewko and M. Lewko, _Endpoint restriction estimates for the paraboloid over finite fields_ , Proc. Amer. Math. Soc. 140 (2012), 2013-2028.
* [8] R. Lidl and H. Niederreiter, _Finite fields,_ Cambridge University Press, (1997).
* [9] G. Mockenhaupt, and T. Tao, _Restriction and Kakeya phenomena for finite fields_ , Duke Math. J. 121(2004), no. 1, 35–74.
* [10] W. Scharlau, _Quadratic forms_ , Queen’s papers on pure and applied math., no. 22, Queen’s Univ., Kingston, Ontario, 1969.
* [11] E.M.Stein, _Harmonic analysis,_ Princeton University Press, 1993.
* [12] R.S. Strichartz, _Convolutions with kernels having singularities on a sphere,_ Trans. Amer. Math. Soc. 148 (1970), 461–471.
|
arxiv-papers
| 2012-09-06T08:26:02 |
2024-09-04T02:49:34.831546
|
{
"license": "Public Domain",
"authors": "Doowon Koh",
"submitter": "Doowon Koh",
"url": "https://arxiv.org/abs/1209.1220"
}
|
1209.1304
|
# Nonplanar Periodic Solutions for Spatial Restricted N+1-Body Problems
***Supported by National Natural Science Foundation of China.
Fengying Li and Shiqing Zhang and Xiaoxiao Zhao
Yangtze Center of Mathematics and College of Mathematics, Sichuan University,
Chengdu 610064, People’s Republic of China
> Abstract: We use variational minimizing methods to study spatial restricted
> N+1-body problems with a zero mass moving on the vertical axis of the moving
> plane for N equal masses. We prove that the minimizer of the Lagrangian
> action on the anti-T/2 or odd symmetric loop space must be a non-planar
> periodic solution for any $N\geq 2$.
>
> Keywords: Restricted N+1-body problems; nonplanar periodic solutions;
> variational minimizers; Jacobi’s necessary conditions.
>
> 2000 AMS Subject Classification 70F07, 34C25, 58E30
## 1 Introduction and Main Result
Spatial restricted 3-body model was studied by Sitnikov [5]. Mathlouthis [3]
etc. studied the periodic solutions for the spatial circular restricted 3-body
problems by minimax variational methods.
In this paper, we study spatial circular restricted N+1-body problems with a
zero mass moving on the vertical axis of the moving plane for N equal masses.
Suppose point masses $m_{1}=\cdots=m_{N}=1$ move centered at the center of
masses on a circular orbit. The motion for the zero mass is governed by the
gravitational forces of $m_{1},\cdots,m_{N}$. Let
$\rho_{j}=e^{\sqrt{-1}\frac{2\pi j}{N}}$ and
$q_{1}(t)=re^{\sqrt{-1}2\pi t}\rho_{1},\cdots,\
q_{j}(t)=\rho_{j}q_{1}(t),\cdots,\ q_{N}(t)=re^{\sqrt{-1}2\pi t}$ (1.1)
satisfy the Newtonian equations:
$m_{i}\ddot{q_{i}}=\frac{\partial U}{\partial q_{i}},\ \ \ \ i=1,\cdots,N,$
(1.2)
where
$U=\sum\limits_{1\leq i<j\leq N}\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}.$ (1.3)
The orbit $q(t)=(0,0,z(t))\in R^{3}$ for zero mass satisfies the following
equation
$\ddot{q}=\sum\limits_{i=1}^{N}\frac{m_{i}(q_{i}-q)}{|q_{i}-q|^{3}}.$ (1.4)
Define
$f(q)=\int_{0}^{1}\big{[}\frac{1}{2}|\dot{q}|^{2}+\sum\limits_{i=1}^{N}\frac{1}{|q-q_{i}|}\big{]}dt,\
\ \ \ q\in\Lambda_{i},$ (1.5)
then
$f(q)=\int_{0}^{1}\big{[}\frac{1}{2}|z^{\prime}|^{2}+\frac{N}{\sqrt{r^{2}+z^{2}}}\big{]}dt\triangleq
f(z),\ \ \ \ q\in\Lambda_{i},$ (1.6)
where
$\Lambda_{1}=\left\\{\begin{array}[]{c}q(t)=(0,0,z(t))|z(t)\in
W^{1,2}(R/Z,R)\\\ z(t+\frac{1}{2})=-z(t),\ q(t)\neq q_{i}(t),\ \forall t\in
R,i=1,2,\cdots,N\end{array}\right\\},$
$\Lambda_{2}=\left\\{\begin{array}[]{c}q(t)=(0,0,z(t))|z(t)\in
W^{1,2}(R/Z,R)\\\ q(-t)=-q(t)\end{array}\right\\},\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ $
$W^{1,2}(R/Z,R)=\left\\{x(t)\bigg{|}\begin{array}[]{c}x(t),\dot{x}(t)\in
L^{2}(R,R)\\\ x(t+1)=x(t)\end{array}\right\\}.$
Notice that the symmetry in $\Lambda_{1}$ is related with Italian symmetry
[1].
In this paper,our main result is the following:
Theorem 1.1 The minimizer of $f(q)$ on the closure $\overline{\Lambda}_{i}$ of
$\Lambda_{i}$(i=1,2) is a nonplanar and noncollision periodic solution.
## 2 Proof of Theorem 1.1
We define the inner product and equivalent norm of $W^{1,2}(R/Z,R)$:
$<u,v>=\int_{0}^{1}(uv+u^{\prime}\cdot v^{\prime})dt,\ \ \ \ \ \ \ \ \ \ \ \ \
\ $ (2.1) $\displaystyle\|u\|$
$\displaystyle=\Big{[}\int_{0}^{1}|u|^{2}dt\Big{]}^{\frac{1}{2}}+\Big{[}\int_{0}^{1}|u^{\prime}|^{2}dt\Big{]}^{\frac{1}{2}}$
(2.2)
$\displaystyle\cong\Big{[}\int_{0}^{1}|u^{\prime}|^{2}dt\Big{]}^{\frac{1}{2}}+|u(0)|.$
Lemma 2.1(Palais’s Symmetry Principle([4])) Let $\sigma$ be an orthogonal
representation of a finite or compact group $G$ in the real Hilbert space $H$
such that for $\forall\sigma\in G,f(\sigma\cdot x)=f(x)$, where
$f:H\rightarrow R$.
Let $S=\\{x\in H|\sigma\cdot x=x,\ \forall\sigma\in G\\}$. Then the critical
point of $f$ in $S$ is also a critical point of $f$ in $H$.
By Palais’s Symmetry Principle, we know that the critical point of $f(q)$ in
$\overline{\Lambda}_{i}$ is a noncollission periodic solution of Newtonian
equation (1.4).
In order to prove Theorem 1.1, we need
Lemma 2.2([6]) Let $X$ be a reflexive Banach space, $S$ be a weakly closed
subset of $X$, $f:S\rightarrow R\cup+\infty,\ \ f\not\equiv+\infty$ is weakly
lower semi-continuous and coercive($f(x)\rightarrow+\infty$ as
$\|x\|\rightarrow+\infty$), then $f$ attains its infimum on $S$.
Lemma 2.3(Poincare-Wirtinger Inequality) Let $q\in W^{1,2}(R/Z,R^{N})$ and
$\int_{0}^{T}q(t)dt=0$, then
$\int_{0}^{T}|\dot{q}(t)|^{2}dt\geq\Big{(}\frac{2\pi}{T}\Big{)}^{2}\int_{0}^{T}|q(t)|^{2}dt.$
(2.3)
Lemma 2.4 $f(q)$ in (1.6) attains its infimum on
$\bar{\Lambda}_{1}=\Lambda_{1}$ or $\bar{\Lambda}_{2}=\Lambda_{2}$.
Proof. By Lemma 2.2 and Lemma 2.3, it is easy to prove Lemma 2.4.
Lemma 2.5(Jacobi’s Necessary Condition[2]) If the critical point
$u=\tilde{u}(t)$ corresponds to a minimum of the functional
$\int_{a}^{b}F(t,u(t),u^{\prime}(t))dt$ and if $F_{u^{\prime}u^{\prime}}>0$
along this critical point, then the open interval $(a,b)$ contains no points
conjugate to $a$, that is, for $\forall c\in(a,b)$, the following boundary
value problem:
$\left\\{\begin{array}[]{ll}-\frac{d}{dt}(Ph^{\prime})+Qh=0,&\\\ h(a)=0,\ \
h(c)=0,\end{array}\right.$ (2.4)
has only the trivial solution $h(t)\equiv 0,\ \forall t\in(a,c)$, where
$P=\frac{1}{2}F_{u^{\prime}u^{\prime}}|_{u=\tilde{u}},\ \ \ \ \ \ \ \ \ \ \ \
\ $ (2.5) $Q=\frac{1}{2}(F_{uu}-\frac{d}{dt}F_{uu^{\prime}})|_{u=\tilde{u}}.$
(2.6)
Lemma 2.6 The radius $r$ for the moving orbit of N equal masses is
$r=\Big{(}\frac{1}{4\pi}\Big{)}^{\frac{2}{3}}\Big{[}\sum\limits_{1\leq j\leq
N-1}csc(\frac{\pi}{N}j)\Big{]}^{\frac{1}{3}}.$
Proof. By (1.1)-(1.3), we have
$\ddot{q}_{N}=\sum\limits_{j\neq N}\frac{q_{j}-q_{N}}{|q_{j}-q_{N}|^{3}},$
(2.7)
Substituting (1.1) into (2.7), we have
$-4\pi^{2}=\sum\limits_{j\neq
N}\frac{\rho_{j}-\rho_{N}}{r^{3}|\rho_{j}-\rho_{N}|^{3}}$ (2.8) $\displaystyle
4\pi^{2}r^{3}$ $\displaystyle=\sum\limits_{j\neq
N}\frac{1-\rho_{j}}{|1-\rho_{j}|^{3}}$ (2.9)
$\displaystyle=\frac{1}{4}\sum\limits_{1\leq j\leq N-1}csc(\frac{\pi}{N}j)$
Then
$r^{3}=\frac{1}{16\pi^{2}}\sum\limits_{1\leq j\leq N-1}csc(\frac{\pi}{N}j).$
(2.10)
Therefore
$r=\Big{(}\frac{1}{4\pi}\Big{)}^{\frac{2}{3}}\Big{[}\sum\limits_{1\leq j\leq
N-1}csc(\frac{\pi}{N}j)\Big{]}^{\frac{1}{3}}.$ (2.11)
Lemma 2.7([8]) $\sum\limits_{j=1}^{N-1}csc(\frac{\pi}{N}j)=\frac{4}{N}$.
For the functional (1.6), let
$F(z,z^{\prime})=\frac{1}{2}|z^{\prime}|^{2}+\frac{N}{\sqrt{r^{2}+z^{2}}}.$
Then the second variation of (1.6) in the neighborhood of $z=0$ is given by
$\int_{0}^{1}(Ph^{\prime 2}+Qh^{2})dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
(2.12)
where
$P=\frac{1}{2}F_{z^{\prime}z^{\prime}}|_{z=0}=\frac{1}{2},\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ $ (2.13)
$Q=\frac{1}{2}(F_{zz}-\frac{d}{dt}F_{zz^{\prime}})|_{z=0}=-\frac{N}{2r^{3}}.$
(2.14)
The Euler equation of (2.12) is called the Jacobi equation of the original
functional (1.6), which is
$-\frac{d}{dt}(Ph^{\prime 2})+Qh=0,\ \ \ \ \ \ \ \ \ \ \ \ \ $ (2.15)
That is,
$h^{\prime\prime}+\frac{N}{r^{3}}h=0.\ \ \ \ \ \ \ \ \ \ \ \ \ $ (2.16)
Next, we study the solution of (2.16) with initial values $h(0)=0,\
h^{\prime}(0)=1$. It is easy to get
$h(t)=\sqrt{\frac{r^{3}}{N}}\cdot sin\sqrt{\frac{N}{r^{3}}}t,$ (2.17)
which is not identically zero on $[0,\frac{1}{2}]$, but we will prove
$h(\frac{1}{2})=0$, and $h(c)=0$ for some $c\in(0,\frac{1}{2})$. Notice that
$\sqrt{\frac{N}{r^{3}}}=\sqrt{N}4\pi\Big{(}\sum\limits_{j\neq
N}csc\frac{\pi}{N}j\Big{)}^{-\frac{1}{2}}$ (2.18)
Hence
$\displaystyle\frac{1}{2}\sqrt{\frac{N}{r^{3}}}$
$\displaystyle=\sqrt{N}\Big{(}\sum\limits_{j\neq
N}csc\frac{\pi}{N}j\Big{)}^{-\frac{1}{2}}\cdot 2\pi$ (2.19)
$\displaystyle=\sqrt{N}\Big{(}\frac{4}{N}\Big{)}^{-\frac{1}{2}}\cdot 2\pi$
$\displaystyle=N\pi.$
So
$h(\frac{1}{2})=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ $ (2.20)
Given $N\geq 2$, choose $0<c=\frac{1}{2N}<\frac{1}{2}$ such that $2Nc=1$, then
$\sqrt{\frac{N}{r^{3}}}c=2N\pi c=\pi$ (2.21)
Therefore
$sin\sqrt{\frac{N}{r^{3}}}c=sin\pi=0.$ (2.22)
Hence $q(t)=(0,0,0)$ is not a local minimum for $f(q)$ on
$\bar{\Lambda}_{i}=\Lambda_{i}(i=1,2)$. So the minimizers of $f(q)$ on
$\Lambda_{i}$ are not always at the center of masses, they must oscillate
periodically on the vertical axis, that is, the minimizers are not always co-
planar, hence we get the non-planar periodic solutions.
Acknowledgements The authors sincerely thank the referee for his/her valuable
comments and helpful suggestions.
## References
* [1] Bessi. U, Coti Zelati. V, Symmetries and non-collision closed orbits for planar N-body type problems,Nonlinear Analysis TMA 16(1991),587-598.
* [2] Gelfand. I, Formin. S, Calculus of Variations, Nauka, Moscow(Russian), English edition, Prentice-Hall, Englewood Cliffs, NJ, 1965.
* [3] Mathlouthis, Periodic oribits of the restricted three-body problem, Trans. AMS 350(1998), 2265-2276.
* [4] Palais. R, The principle of symmetric criticality, CMP69(1979), 19-30.
* [5] Sitnikov, K., Existence of oscillating motions for the three-body problem, Dokl. Akad. Nauk, USSR, 133(1960), 303-306.
* [6] Struwe. M., Variational Methods, Third Edition, Springer, 2000.
* [7] Wintner. A, The Analytic Foundations of Celestial Mechanics, Princeton Univ. Press, 1941.
* [8] Yu. X, Zhang S. Q, Twisted angels for central configurations formed by two twisted regular polygons, JDE 253(2012), 2106-2122.
|
arxiv-papers
| 2012-09-06T14:53:36 |
2024-09-04T02:49:34.842552
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fengying Li and Shiqing Zhang and Xiaoxiao Zhao",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1209.1304"
}
|
1209.1370
|
# Construction of wedge-local QFT through Longo-Witten endomorphisms111This
article has been prepared for the proceedings of ICMP 12.
Yoh Tanimoto222Supported by Deutscher Akademischer Austauschdienst and in part
by Courant Research Centre “Higher Order Structures in Mathematics”.
e-mail: [email protected]
Institut für Theoretische Physik, Universtät Göttingen,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany.
###### Abstract
We review our recent construction of operator-algebraic quantum field models
with a weak localization property. Chiral components of two-dimensional
conformal fields and certain endomorphisms of their observable algebras play a
crucial role. In one case, this construction leads to a family of strictly
local (Haag-Kastler) nets.
## 1 Operator-algebraic approach to QFT
The problem of constructing interacting relativistic quantum field theory on
four-dimensional spacetime has been a long-standing open problem. On the other
hand, in lower dimensions there have been important developments in several
different approaches (e.g. Constructive QFT, form factor bootstrap). Here we
adopt the operator-algebraic approach, also known as Algebraic Quantum Field
Theory or Local Quantum Physics [9], which has recently resulted in a
construction of a large family of quantum field models in two-spacetime
dimensions [11] and a further progress [12]. Furthermore, two-dimensional
Conformal Field Theory can be successfully studied in this framework [10]. In
this contribution, we present a new method of constructing two-dimensional
operator-algebraic QFT based on chiral components of CFT [19, 1, 18].
The mathematical approach to QFT which is the closest to the notion in physics
is the Wightman axioms. A Wightman field is an operator-valued distribution
which satisfies certain properties which originate in physics, e.g. Poincaré
covariance and Einstein causality. Yet, the quantum field smeared with a
smooth function is still an unbounded operator and sometimes plagued by the
problem of domains. Instead, in algebraic QFT one considers algebras of
bounded operators.
### 1.1 Haag-Kastler nets
A Haag-Kastler net, or a Poincaré covariant net (of observables) assigns to
each open region $O\subset\mathbb{R}^{d}$ a von Neumann algebra
$\mathcal{A}(O)$ on a common Hilbert space $\mathcal{H}$ (see the book [16]
for a general account on von Neumann algebras). In addition, one assumes that
there is a continuous unitary representation $U$ of the Poincaré group on
$\mathcal{H}$ and an invariant ground state, the vacuum $\Omega$. The triple
$(\mathcal{A},U,\Omega)$ is subject to standard axioms and considered as a
model of quantum field theory [9].
If one has a Wightman field $\phi$, then one can construct the corresponding
net by defining ${\mathcal{A}}(O):=\\{e^{i\phi(f)}:{\rm supp}f\subset
O\\}^{\prime\prime}$, where ${\mathcal{M}}^{\prime}$ means the set of bounded
operators commuting with any element of ${\mathcal{M}}$. The double commutant
${\mathcal{M}}^{\prime\prime}$ is the smallest von Neumann algebra which
includes ${\mathcal{M}}$. Actually it is required that $\phi(f)$ and $\phi(g)$
have commuting spectral projections for $f,g$ with spacelike separated
support, but in many cases this is satisfied.
Conversely, when a Haag-Kastler net ${\mathcal{A}}$ is given and if certain
technical conditions are satisfied, then one can recover quantum fields. In
this way, a Haag-Kastler net is considered as the operator-algebraic
formulation of quantum field theory.
Now we are concerned with the construction problem. In any spacetime
dimension, there are the so-called (generalized) free fields, and
corresponding nets. But other examples are rare. For $d\leq 3$ the methods of
Constructive QFT have successfully constructed many examples or for $d=2$
there are plenty of examples of conformal fields, but apart from these models,
it is very hard to construct such fields or nets. In this work we address this
problem for $d=2$ with a purely von Neumann algebraic approach. An extension
to higher dimensions remains open.
### 1.2 Borchers triples
One of the difficulties in constructing Haag-Kastler nets lies in the
infiniteness of the family $\\{{\mathcal{A}}(O)\\}$ with certain compatibility
conditions. Instead, Borchers observed that for $d=2$, actually the whole net
can be recovered from the single von Neumann algebra
${\mathcal{A}}(W_{\mathrm{R}})$ associated with the (right-)wedge-shaped
regions $W_{\mathrm{R}}:=\\{a\in{\mathbb{R}}^{2}:a_{1}>|a_{0}|\\}$ and the
spacetime symmetry $U$ (under the condition called Haag-duality). Furthermore,
by the Tomita-Takesaki theory of von Neumann algebras [17], it is enough to
know the restriction of $U$ to the translation subgroup ${\mathbb{R}}^{2}$
[3].
A Borchers triple $({\mathcal{M}},T,\Omega)$ consists of a von Neumann algebra
${\mathcal{M}}$ on ${\mathcal{H}}$, a unitary representation $T$ of
${\mathbb{R}}^{2}$ with joint spectrum in $V_{+}$ and a vacuum vector $\Omega$
such that $\Omega$ is invariant under $T(a)$, ${{\rm
Ad\,}}T(a){\mathcal{M}}\subset{\mathcal{M}}$ for $a\in W_{\mathrm{R}}$ and
${\mathcal{M}}\Omega$ and ${\mathcal{M}}^{\prime}\Omega$ are dense in
${\mathcal{H}}$ (these properties are called cyclicity and separating property
of $\Omega$ for ${\mathcal{M}}$, respectively). It is easy to see that if
$({\mathcal{A}},U,\Omega)$ is a Poincaré covariant net, then
$({\mathcal{A}}(W_{\mathrm{R}}),U|_{{\mathbb{R}}^{2}},\Omega)$ is a Borchers
triple.
Conversely, starting with a Borchers triple $({\mathcal{M}},T,\Omega)$, one
can define a net as follows: in two-spacetime dimensions, any double cone can
be represented as the intersection of two-wedges
$(W_{\mathrm{R}}+a)\cap(W_{\mathrm{L}}+b)=:D_{a,b}$, where $W_{\mathrm{L}}$ is
the reflected (left-)wedge. Then one defines first von Neumann algebras
${\mathcal{A}}(D_{a,b})$ for double cones $D_{a,b}$ by
${\mathcal{A}}(D_{a,b}):={{\rm Ad\,}}T(a)({\mathcal{M}})\cap{{\rm
Ad\,}}T(b)({\mathcal{M}}^{\prime})$. For a general region $O$ one takes
${\mathcal{A}}(O):=\left(\bigcup_{D_{a,b}\subset
O}{\mathcal{A}}(D_{a,b})\right)^{\prime\prime}$. Then one can show that this
“net” ${\mathcal{A}}$ satisfies isotony and locality. Furthermore, the
representation $T$ extends to a representation $U$ of the Poincaré group which
makes ${\mathcal{A}}$ covariant and $\Omega$ is still invariant. In this way
one obtains a “net” $({\mathcal{A}},U,\Omega)$, where the only missing
property is that $\Omega$ is cyclic for ${\mathcal{A}}(O)$. In general, if one
starts with a net, goes down to the Borchers triple and back to the net as
above, this does not coincide with the original net, but such a difference is
not important when one is interested in the construction of an interacting
net.
Hence, in the operator-algebraic approach, the construction of Haag-Kastler
nets can be split into two steps: (1) to construct Borchers triples, (2) to
prove the cyclicity of $\Omega$. In the following, we carry out (1) for
massless models in Section 3 and both (1) and (2) for massive models in
Section 4.
## 2 Conformal nets and Longo-Witten endomorphisms
Here we review the operator-algebraic treatment of Conformal Field Theory,
which is the main ingredient of our construction of Borchers triples. As is
well known, two-dimensional CFT contains observables which are invariant under
right- or left-lightlike translations. They are called chiral components and
can be considered as observables defined on lightrays. They can often be
extended to $S^{1}$, the one-point compactification of a lightray. A Möbius
covariant net $(\mathcal{A}_{0},U_{0},\Omega_{0})$ on $S^{1}$ is defined
precisely like a Haag-Kastler net in $\mathbb{R}^{d}$, with the regions taken
to be intervals in $S^{1}$, and the Poincaré group is replaced by the Möbius
group $\mathrm{PSL}(2,\mathbb{R})$.
There are many examples of Möbius covariant nets: the $U(1)$-current net (free
boson), the free fermion net, the Virasoro nets, Minimal models, WZW models,
etc. However, one cannot define the notion of interaction for one-dimensional
nets (actually, it can be shown that a two-dimensional CFT does not interact
in the sense of scattering theory [20]). Rather, they are building blocks of
interacting two-dimensional QFT. Most important ones are the $U(1)$-current
and the free fermion, which admit Fock space structure. The aim of the sequel
is to demonstrate new methods to construct Borchers triples and Haag-Kastler
nets on $\mathbb{R}^{2}$ from Möbius covariant nets on $S^{1}$.
Let us now introduce the main notion in our construction. Recall that the
circle $S^{1}$ is identified with ${\mathbb{R}}\cup\\{\infty\\}$ by the
stereographic projection. A Longo-Witten endomorphism of a conformal net is an
endomorphism of ${\mathcal{A}}_{0}({\mathbb{R}}_{+})$, implemented by a
unitary $V_{0}$ which commutes with translation $T_{0}$. This notion was first
introduced in order to construct QFT with boundary [15].
The simplest example of Longo-Witten endomorphisms is the translation itself
${{\rm Ad\,}}T_{0}(t)$, $t\geq 0$. In many examples there are inner
symmetries, for which there is a unitary $V_{0}$ such that ${{\rm
Ad\,}}V_{0}({\mathcal{A}}_{0}(I))={\mathcal{A}}_{0}(I)$ and
$V_{0}\Omega_{0}=\Omega_{0}$. Then $V_{0}$ automatically commutes with
$U_{0}(g)$, especially with $T_{0}(t)$ and implements a Longo-Witten
endomorphism.
The examples above are rather of general nature. By considering a specific
net, the $U(1)$-current net ${\mathcal{A}}_{U(1)}$, Longo and Witten found a
large family of examples [15]. The $U(1)$-current net is defined on the
symmetric Fock space ${\mathcal{H}}_{0}$, on whose one-particle space the
Möbius group ${\rm PSL}(2,{\mathbb{R}})$ acts irreducibly with lowest weight
$1$. There is a operator-valued distribution $J$, the current, which generates
the net as explained above. One can promote a unitary operator $V_{1}$ on the
one-particle space ${\mathcal{H}}_{1}$ to a unitary operator $\Gamma(V_{1})$
on the full space ${\mathcal{H}}_{0}$ (second quantization). The Möbius group
representation promotes to ${\mathcal{H}}_{0}$ as well.
An inner symmetric function $\varphi$ is the boundary value of a bounded
analytic function on the upper-half plane, with $|\varphi(t)|=1$ for
$t\in{\mathbb{R}}$, and $\varphi(-t)=\overline{\varphi(t)}$.
###### Theorem 2.1 (Longo-Witten).
Let $\varphi$ be an inner symmetric function and $P_{1}$ be the generator of
the one-particle translation of the $U(1)$-current net ${\mathcal{A}}_{U(1)}$.
Then the unitary operator $\Gamma(\varphi(P_{1}))$ implements a Longo-Witten
endomorphism of ${\mathcal{A}}_{U(1)}$.
Using these operators, we construct two-dimensional models in the next
Sections.
## 3 Massless construction and scattering theory
### 3.1 Massless scattering theory
First we consider two-dimensional massless models. The reason is, as we see
below, the scattering theory is particularly simple and a general structural
result can be obtained. The following is the adaptation of the theory [4] to
Borchers triples [7].
Let $({\mathcal{M}},T,\Omega)$ be a (two-dimensional) Borchers triple. As
explained before, an element $x\in{\mathcal{M}}$ should be considered as an
observable in the wedge region $W_{\mathrm{R}}$. If one defines the following,
$x_{\pm}(h_{\mathcal{T}}):=\int h_{\mathcal{T}}(t){{\rm Ad\,}}T(t,\pm
t)(x)dt,$
where $h$ is a nonnegative smooth function with compact support with $\int
h(t)dt=1$, ${\mathcal{T}}$ is a nonzero real number and
$h_{\mathcal{T}}(t)=|{\mathcal{T}}|^{-\epsilon}h(|{\mathcal{T}}|^{-\epsilon}(t-{\mathcal{T}}))$
with some fixed number $0<\epsilon<1$. Then the strong limits $\Phi^{\rm
out}_{+}(x)=\lim_{{\mathcal{T}}\to\infty}x_{+}(h_{\mathcal{T}})$ and
$\Phi^{\rm in}_{-}(x)=\lim_{{\mathcal{T}}\to-\infty}x_{-}(h_{\mathcal{T}})$
exist. The operators $\Phi^{\rm out}_{+}(x)$ and $\Phi^{\rm in}_{-}(x)$ are
called asymptotic fields. Similarly one can define $\Phi^{\rm
out}_{-}(x^{\prime})$ and $\Phi^{\rm in}_{+}(x^{\prime})$ using an element
$x^{\prime}\in{\mathcal{M}}^{\prime}$. They have nice properties, so that
$\Phi^{\rm out}_{\pm}(\cdot)$ and $\Phi^{\rm in}_{\pm}(\cdot)$ can be
considered as operators at far-future and far-past, respectively. In
particular, $\Phi^{\rm out}_{+}(x)$ and $\Phi^{\rm out}_{-}(x^{\prime})$
commute and act like operators in tensor product. More precisely, let
${\mathcal{H}}_{\pm}$ be the space of vectors invariant under $T(t,\pm t)$.
$\Phi^{\rm out}_{\pm}(\cdot)$ can be naturally restricted to
${\mathcal{H}}_{\pm}$ respectively. Vectors $\Phi^{\rm
out}_{\pm}(\cdot)\Omega$ describe massless excitations going out in $\pm$
directions, respectively and a natural tensor product structure can be given
to the subspace generated by such operators.
If $\\{\Phi^{\rm out}_{+}(x)\Phi^{\rm
out}_{-}(x^{\prime})\Omega:x\in{\mathcal{M}},x^{\prime}\in{\mathcal{M}}^{\prime}\\}$
and $\\{\Phi^{\rm in}_{+}(x^{\prime})\Phi^{\rm
in}_{-}(x)\Omega:x\in{\mathcal{M}},x^{\prime}\in{\mathcal{M}}^{\prime}\\}$ are
total in ${\mathcal{H}}$, then we say that the Borchers triple
$({\mathcal{M}},T,\Omega)$ is asymptotically complete, or in other words, the
any vector in ${\mathcal{H}}$ can be interpreted as in- and out-scattering
states. For $\xi=\Phi^{\rm out}_{+}(x)$ and $\eta=\Phi^{\rm
out}_{-}(x^{\prime})$, let us write $\xi\overset{\rm
out}{\times}\eta=\Phi^{\rm out}_{+}(x)\Phi^{\rm out}_{-}(x^{\prime})\Omega$.
The operation $\overset{\rm out}{\times}$ naturally extends to
${\mathcal{H}}_{\pm}$. Similarly we define $\overset{\rm in}{\times}$. Then
the S-matrix defined by $S\xi\overset{\rm out}{\times}\eta=\xi\overset{\rm
in}{\times}\eta$ is a unitary operator. Note that, because of the
nondispersive nature of two-dimensional massless excitations, it is enough to
consider only such products of two operators.
Let us assume that an asymptotically complete Borchers triple
$({\mathcal{M}},T,\Omega)$ comes from a net $({\mathcal{A}},T,\Omega)$. It is
interesting to observe that the information of the S-matrix and asymptotic
fields is enough to recover the given ${\mathcal{M}}$. In fact, we have the
following simple formula [19].
###### Theorem 3.1.
Under standard assumptions (and asymptotic completeness), it holds that
${\mathcal{M}}=\\{\Phi^{\rm out}_{+}(x),{{\rm Ad\,}}S(\Phi^{\rm
out}_{-}(x^{\prime})):x\in{\mathcal{M}},x^{\prime}\in{\mathcal{M}}^{\prime}\\}^{\prime\prime}$.
As remarked before, $\Phi^{\rm out}_{\pm}(\cdot)$ are observables acting like
in tensor product. On the other hand, ${\mathcal{M}}$ is the wedge-algebra of
the interacting theory. And the net can be recovered from the Borchers triple,
namely from ${\mathcal{M}}$ and translation. Furthermore, it can be shown that
$\Phi^{\rm out}_{\pm}(\cdot)$ generate one-dimensional conformal net [20, 19].
Hence the above formula tells us how to construct interacting theory out of
free, conformal theory and S-matrix. In the next Subsection we carry out this
program.
### 3.2 Construction of massless Borchers triples
#### 3.2.1 One-parameter Longo-Witten endomorphisms
Let us take a Möbius covariant net $({\mathcal{A}}_{0},U_{0},\Omega_{0})$ on
${\mathcal{H}}_{0}$. We denote by $T_{0}$ the restriction of $U_{0}$ to the
translation subgroup as before. With its positive generator $P_{0}$, one can
write $T_{0}(t)=e^{itP_{0}}$. Only with this setting, we can already construct
nontrivial Borchers triples [19].
The full Hilbert space is now the tensor product
${\mathcal{H}}:={\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$. For a point
$(t_{+},t_{-})$ in the lightray coordinate of the Minkowski space, we define
$T(t_{+},t_{-}):=T_{0}(t_{+})\otimes T_{0}(t_{-})$. Again the simple tensor
product $\Omega:=\Omega_{0}\otimes\Omega_{0}$ plays the role of the vacuum
vector.
###### Theorem 3.2.
Let $\kappa\geq 0$ and
${\mathcal{M}}_{\kappa}:=\\{x\otimes{\mathbbm{1}},{{\rm Ad\,}}e^{i\kappa
P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}.$
Then $({\mathcal{M}}_{\kappa},T,\Omega)$ is an asymptotically complete
Borchers triple with the S-matrix $e^{i\kappa P_{0}\otimes P_{0}}$.
The proof goes as follows. The conditions on $T$ and $\Omega$ are easily
verified and it is also simple to check ${{\rm
Ad\,}}T(a)({\mathcal{M}}_{\kappa})\subset{\mathcal{M}}_{\kappa}$ for $a\in
W_{\mathrm{R}}$, since $T(a)$ and $e^{i\kappa P_{0}\otimes P_{0}}$ commute.
The cyclicity of $\Omega$ for ${\mathcal{M}}_{\kappa}$ follows immediately
from the definition of conformal nets. The only nontrivial part is the
separation by $\Omega$. In order to show this, it is enough to find another
von Neumann algebra which commutes with ${\mathcal{M}}$ and for which $\Omega$
is cyclic. Such a von Neumann algebra is given by
${\mathcal{M}}_{\kappa}^{1}:=\\{{{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}),y^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\\}^{\prime\prime}.$
Let us see that, as an example, $x\otimes{\mathbbm{1}}$ and ${{\rm
Ad\,}}e^{i\kappa P_{0}\otimes P_{0}}(x^{\prime}\otimes{\mathbbm{1}})$ commute.
Since one has $e^{i\kappa P_{0}\otimes P_{0}}=\int e^{it\kappa P_{0}}\otimes
dE_{0}(t)$, where $E_{0}$ is the spectral measure of $P_{0}$, and $e^{it\kappa
P_{0}}$ implements a Longo-Witten endomorphism, one has ${{\rm
Ad\,}}e^{i\kappa P_{0}\otimes P_{0}}(x^{\prime}\otimes{\mathbbm{1}})=\int{{\rm
Ad\,}}e^{it\kappa P_{0}\otimes P_{0}}(x^{\prime})\otimes dE_{0}(t)$ and ${{\rm
Ad\,}}e^{it\kappa P_{0}\otimes P_{0}}(x^{\prime})$ commutes with $x$. In this
way, Longo-Witten endomorphisms enter the present construction.
Note that when $\kappa=0$ then
${\mathcal{M}}_{\kappa}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$
and $S_{\kappa}={\mathbbm{1}}$, namely, the simple tensor product results in a
noninteracting triple. It has been revealed that
$({\mathcal{M}}_{\kappa},T,\Omega)$ is equivalent to the BLS deformation [5]
of $({\mathcal{M}}_{0},T,\Omega)$.
It is also possible to perform a similar construction with one-parameter inner
symmetries [19]. It is remarkable that in this case, with additional technical
conditions, one can determine the intersection of wedges. However, the
intersection turns out to be trivial in a certain sense.
#### 3.2.2 U(1)-current as building blocks
The Longo-Witten endomorphisms on the $U(1)$-current net explained above can
be used to construct Borchers triples as well. Yet the spirit is always the
same: one has only to find an S-matrix to twist the right component.
As recalled before, the Hilbert space ${\mathcal{H}}_{0}$ for the
$U(1)$-current is the symmetric Fock space. One considers first the
unsymmetrized Fock space
${\mathcal{H}}^{\Sigma}=\bigoplus_{m}{\mathcal{H}}_{1}^{\otimes m}$.
We fix an inner symmetric function $\varphi$. On ${\mathcal{H}}_{1}^{\otimes
m}$, there act $m$ commuting operators
$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
i\leq m\\}.$
Recall that functional calculus has been used in the construction by Longo and
Witten. Here we put:
* •
$P_{i,j}^{m,n}:=({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}})\otimes({\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}})$,
which acts on ${\mathcal{H}}_{1}^{\otimes m}\otimes{\mathcal{H}}_{1}^{\otimes
n}$, $1\leq i\leq m$ and $1\leq j\leq n$.
* •
$S^{m,n}_{\varphi}:=\prod_{i,j}\varphi(P_{i,j}^{m,n})$, where
$\varphi(P_{i,j}^{m,n})$ is defined by the functional calculus on
${\mathcal{H}}_{1}^{\otimes m}\otimes{\mathcal{H}}_{1}^{\otimes n}$.
* •
$S_{\varphi}:=\bigoplus_{m,n}S_{\varphi}^{m,n}=\bigoplus_{m,n}\prod_{i,j}\varphi(P^{m,n}_{i,j})$
Then one can show that $S_{\varphi}$ restricts to the symmetric Fock space and
can be decomposed into a direct integral of Longo-Witten unitaries. As in the
case of one-parameter endomorphisms, we can show the following.
###### Theorem 3.3.
Let us put
${\mathcal{M}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{{\rm
Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{U(1)}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{U(1)}({\mathbb{R}}_{+})\\}^{\prime\prime}.$
Then $({\mathcal{M}}_{\varphi},T,\Omega)$ is an asymptotically complete
Borchers triple with the S-matrix $S_{\varphi}$.
Recently we found that this construction is equivalent to the deformation of
massless free field by Lechner [13]. This will be presented elsewhere [14].
Note that the S-matrix of all these constructions preserve the particle number
in the sense of Fock space. It is also possible to construct examples which do
violate this structure as follows [1].
It is known that the $U(1)$-current net is identified as a subtheory of the
free complex fermion net ${\mathcal{F}}$ (the boson-fermion correspondence).
The free fermion net ${\mathcal{F}}$ is defined on the fermionic Fock space.
Similarly as above, by choosing an inner (this time not necessarily symmetric)
function $\varphi$, one can construct a Borchers triple with S-matrix
$S_{\varphi,{\mathcal{F}}}$ with the “free” part
${\mathcal{F}}\otimes{\mathcal{F}}$. Then $S_{\varphi,{\mathcal{F}}}$ can be
restricted to the bosonic part
${\mathcal{A}}_{U(0)}\otimes{\mathcal{A}}_{U(0)}$. Let us denote the
restriction by $S_{\varphi,{\rm r}}$
###### Theorem 3.4.
Let us put
${\mathcal{M}}_{\varphi,{\rm r}}:=\\{x\otimes{\mathbbm{1}},{{\rm
Ad\,}}S_{\varphi,{\rm r}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{U(1)}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{U(1)}({\mathbb{R}}_{+})\\}^{\prime\prime}.$
Then $({\mathcal{M}}_{\varphi,{\rm r}},T,\Omega)$ is an asymptotically
complete Borchers triple with the S-matrix $S_{\varphi,{\rm r}}$. The S-matrix
$S_{\varphi,{\rm r}}$ does not preserve $n$-particle space of the bosonic Fock
space.
So the S-matrix appears to represent particle production. However, the the
cyclicity of the vacuum for the algebras of bounded regions is not known.
Furthermore, the concept of particle is to be replaced by waves in massless
two-dimensional models [4]. Thus it is desired to construct massive models,
where these issues can be settled.
## 4 Massive construction and strict locality
It is also possible to construct massive models using Longo-Witten
endomorphisms. We only briefly sketch the construction [18].
Let $(\mathcal{A}_{\mathrm{c}},U_{\mathrm{c}},\Omega_{\mathrm{c}})$ be the
free massive complex field net. There is an action of $U(1)$ by inner
symmetry. One can take the generator $Q_{\mathrm{c}}$, such that
$V_{\mathrm{c}}(\kappa)=e^{i2\pi\kappa Q_{\mathrm{c}}},\kappa\in\mathbb{R}$.
We consider the wedge algebra
$\mathcal{M}_{\mathrm{c}}=\mathcal{A}_{\mathrm{c}}(W_{\mathrm{R}})$ and the
restriction $T_{\mathrm{c}}$ of $U_{\mathrm{c}}$ to the subgroup of
translations.
Our new massive models are constructed on the tensor product Hilbert space
$\widetilde{\mathcal{H}}_{\mathrm{c}}:=\mathcal{H}_{\mathrm{c}}\otimes\mathcal{H}_{\mathrm{c}}$.
There is a natural representation
$\widetilde{T}_{\mathrm{c}}:=T_{\mathrm{c}}\otimes T_{\mathrm{c}}$ and a
vector
$\widetilde{\Omega}_{\mathrm{c}}:=\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$.
For $\kappa\in\mathbb{R}$, we define
$\widetilde{V}_{{\mathrm{c}},\kappa}:=e^{i2\pi\kappa Q_{\mathrm{c}}\otimes
Q_{\mathrm{c}}}$.
###### Theorem 4.1.
Let us put
$\widetilde{\mathcal{M}}_{{\mathrm{c}},\kappa}:=\\{x\otimes{\mathbbm{1}},{{\rm
Ad\,}}\widetilde{V}_{{\mathrm{c}},\kappa}({\mathbbm{1}}\otimes
y):x,y\in{\mathcal{M}}_{\mathrm{c}}\\}^{\prime\prime}.$
Then
$(\widetilde{\mathcal{M}}_{{\mathrm{c}},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
is a Borchers triple.
We can show that this triple is indeed strictly local without using modular
nuclearity [6], by directly proving the wedge-split property [8, 11], hence
one can further construct nontrivial a Haag-Kastler net
$\widetilde{\mathcal{A}}_{{\mathrm{c}},\kappa}$. Since
$\widetilde{T}_{\mathrm{c}}$ is a massive representation, the scattering
theory works well. One can show that the S-matrix is nontrivial. The S-matrix
is factorizing and does not depend on the rapidity.
It should be noted that this procedure to construct nontrivial Haag-Kastler
nets can be applied to any net with either modular nuclearity or wedge-split
property which admits an action of $U(1)$ by inner symmetry. One can take the
tensor product of one of the models by Lechner [11], instead of the complex
free field. One can even repeat the procedure by taking the constructed triple
$(\widetilde{\mathcal{M}}_{{\mathrm{c}},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
as an input to construct further new nets.
We indicate another construction of Borchers triples without investigating
strict locality [18]. Here the most important observation is the following:
one considers the massive real free field (net) and takes the right-wedge
algebra $\mathcal{A}(W_{\mathrm{R}})$ and the restriction of $U$ to the
positive lightlike translations, Then it is equivalent to the $U(1)$-current
net. The negative lightlike translations (in the past direction) are redefined
as a suitable one-parameter semigroup of Longo-Witten endomorphisms
$\mathrm{Ad\,}V_{0}(s)$ with negative generator (this will be investigated in
more detail [2]). Furthermore one has a Longo-Witten endomorphism as in
Section 3.2.2 for an inner symmetric function $\varphi$. By using $\varphi$,
instead of a group action of $U(1)$, one can twist the tensor product of the
real free field net. In this way one can construct a family of Borchers
triples with two-particle S-matrix which depends on rapidity.
Models with particle production have not yet been obtained with this method.
## References
* [1] Marcel Bischoff and Yoh Tanimoto. Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II. Commun. Math. Phys., 317(3):667–695, 2013.
* [2] Marcel Bischoff and Yoh Tanimoto. Integrable QFT and Longo-Witten endomorphisms. arXiv:1305.2171, 2013.
* [3] H.-J. Borchers. The CPT-theorem in two-dimensional theories of local observables. Comm. Math. Phys., 143(2):315–332, 1992.
* [4] D. Buchholz. Collision theory for waves in two dimensions and a characterization of models with trivial $S$-matrix. Comm. Math. Phys., 45(1):1–8, 1975.
* [5] D. Buchholz, G. Lechner, and S.J. Summers. Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys., 304:95–123, 2011.
* [6] Detlev Buchholz and Gandalf Lechner. Modular nuclearity and localization. Ann. Henri Poincaré, 5(6):1065–1080, 2004.
* [7] Wojciech Dybalski and Yoh Tanimoto. Asymptotic completeness in a class of massless relativistic quantum field theories. Comm. Math. Phys., 305:427–440, 2011.
* [8] S. Doplicher and R. Longo. Standard and split inclusions of von Neumann algebras. Invent. Math., 75(3):493–536, 1984.
* [9] Rudolf Haag. Local quantum physics. Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1996. Fields, particles, algebras.
* [10] Yasuyuki Kawahigashi. Classification of operator algebraic conformal field theories in dimensions one and two. In XIVth International Congress on Mathematical Physics, pages 476–485. World Sci. Publ., Hackensack, NJ, 2005.
* [11] Gandalf Lechner. Construction of quantum field theories with factorizing $S$-matrices. Comm. Math. Phys., 277(3):821–860, 2008.
* [12] Gandalf Lechner. Deformations of operator algebras and the construction of quantum field theories. In XVIth International Congress on Mathematical Physics, pages 490–495. World Sci. Publ., Hackensack, NJ, 2010.
* [13] Gandalf Lechner. Deformations of quantum field theories and integrable models. Comm. Math. Phys., 312(1):265–302, 2011.
* [14] Gandalf Lechner, Jan Schlemmer, and Yoh Tanimoto. On the equivalence of two deformation schemes in quantum field theory. Lett. Math. Phys., 103(4):421–437, 2013.
* [15] Roberto Longo and Edward Witten. An algebraic construction of boundary quantum field theory. Commun. Math. Phys., 303:213–232, 2011.
* [16] M. Takesaki. Theory of operator algebras. I, volume 124 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5.
* [17] M. Takesaki. Theory of operator algebras. II, volume 125 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6.
* [18] Yoh Tanimoto. Construction of two-dimensional quantum field models through Longo-Witten endomorphisms. arXiv:1301.6090, 2013.
* [19] Yoh Tanimoto. Construction of wedge-local nets of observables through Longo-Witten endomorphisms. Commun. Math. Phys., 314(2):443–469, 2012.
* [20] Yoh Tanimoto. Noninteraction of waves in two-dimensional conformal field theory. Commun. Math. Phys., 314(2):419–441, 2012.
|
arxiv-papers
| 2012-09-06T19:04:20 |
2024-09-04T02:49:34.852802
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yoh Tanimoto",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1209.1370"
}
|
1209.1425
|
# The End of an Architectural Era
for Analytical Databases
Reynold S. Xin
AMPLab, UC Berkeley [email protected]
###### Abstract
Traditional enterprise warehouse solutions center around an analytical
database system that is monolithic and inflexible: data needs to be extracted,
transformed, and loaded into the rigid relational form before analysis. It
takes years of sophisticated planning to provision and deploy a warehouse;
adding new hardware resources to an existing warehouse is an equally lengthy
and daunting task.
Additionally, modern data analysis employs statistical methods that go well
beyond the typical roll-up and drill-down capabilities provided by warehouse
systems. Although it is possible to implement such methods using a combination
of SQL and UDFs [1], query engines in relational databases are ill-suited for
these.
The Hadoop ecosystem introduces a suite of tools for data analytics that
overcome some of the problems of traditional solutions. These systems,
however, forgo years of warehouse research. Memory is significantly
underutilized in Hadoop clusters, and execution engine is naive compared with
its relational counterparts.
It is time to rethink the design of data warehouse systems and take the best
from both worlds. The new generation of warehouse systems should be modular,
high performance, fault-tolerant, easy to provision, and designed to support
both SQL query processing and machine learning applications.
This paper references the Shark system developed at Berkeley as an initial
attempt [2].
Data warehouse systems should be modular, flexible, easy to provision, and
support machine learning. It’s time to rethink the system design.
## References
* [1] J. Cohen, B. Dolan, M. Dunlap, J. Hellerstein, and C. Welton. Mad skills: new analysis practices for big data. Proceedings of the VLDB Endowment, 2(2):1481–1492, 2009.
* [2] C. Engle, A. Lupher, R. Xin, M. Zaharia, M. J. Franklin, S. Shenker, and I. Stoica. Shark: fast data analysis using coarse-grained distributed memory. In Proceedings of the 2012 international conference on Management of Data, SIGMOD ’12, pages 689–692, New York, NY, USA, 2012. ACM.
|
arxiv-papers
| 2012-09-06T23:15:46 |
2024-09-04T02:49:34.858332
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Reynold S. Xin",
"submitter": "Reynold Xin",
"url": "https://arxiv.org/abs/1209.1425"
}
|
1209.1430
|
# Revising the age for the Baptistina asteroid family using WISE/NEOWISE data
Joseph R. Masiero11affiliation: Jet Propulsion Laboratory/California Institute
of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109, USA,
[email protected] , A. K. Mainzer11affiliation: Jet Propulsion
Laboratory/California Institute of Technology, 4800 Oak Grove Dr., MS 321-520,
Pasadena, CA 91109, USA, [email protected] , T. Grav22affiliation:
Planetary Science Institute, 1700 East Fort Lowell, Suite 106, Tucson, AZ
85719-2395 , J. M. Bauer11affiliation: Jet Propulsion Laboratory/California
Institute of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109,
USA, [email protected] 33affiliation: Infrared Processing and
Analysis Center, California Institute of Technology, Pasadena, CA 91125 USA ,
R. Jedicke 44affiliation: Institute for Astronomy, University of Hawaii,
Honolulu, HI 96822 USA
###### Abstract
We have used numerical routines to model the evolution of a simulated
Baptistina family to constrain its age in light of new measurements of the
diameters and albedos of family members from the Wide-field Infrared Survey
Explorer. We also investigate the effect of varying the assumed physical and
orbital parameters on the best-fitting age. We find that the physically
allowed range of assumed values for the density and thermal conductivity
induces a large uncertainty in the rate of evolution. When realistic
uncertainties in the family members’ physical parameters are taken into
account we find the best-fitting age can fall anywhere in the range of
$140-320~{}$Myr. Without more information on the physical properties of the
family members it is difficult to place a more firm constraint on Baptistina’s
age.
## 1 Introduction
The Main Belt asteroids (MBAs) offer a laboratory to study the dynamical and
collisional evolution of the inner Solar system, as well as a window into the
composition and thermal history of the protosolar disk. For nearly a century,
asteroids grouped closely in orbital element-space have been recognized as
having formed from the catastrophic disruption of a single larger parent body
(Hirayama, 1918; Zappalà et al., 1990). Through modeling of the dynamical and
the non-gravitational forces that evolve the orbits of the family members, the
time since the breakup of the parent body has been estimated. The forces and
processes that act on these small MBAs depend on the bodies’ physical
parameters, such as diameter and albedo. Previous modeling methods have used
the absolute visible magnitudes of the family members as a proxy for their
diameters (e.g. Nesvorný et al., 2005); however, this instills uncertainty in
the age determination as the derived age will depend strongly on the assumed
albedos. Assumptions about other thermophysical parameters will likewise
introduce accompanying errors on the age determination.
The chronology of asteroid family breakups is one of the few methods, along
with cratering records and petrology/radioisotope ages, for dating the history
of events in the Solar system. These collisional events in the Main Belt can
be linked to the geological record of the Earth, as well as impacts on the
terrestrial planets, other asteroids, and the Earth’s Moon (e.g. dell’Oro et
al., 2002; O’Brien & Greenberg, 2005; Farley et al., 2006; Ćuk et al., 2010;
Le Feuvre & Wieczorek, 2011). Ultimately, the goal of such analyses is to
understand the sequence of events in the Main Belt and near-Earth object (NEO)
populations that are known to have had major consequences for life on Earth
(e.g. Alvarez et al., 1980). Finally, probing the ages of the oldest families
gives us a window into the most ancient history of the Solar system, as some
family formation events may coincide or even predate the Late Heavy
Bombardment and the epoch of giant planet migration in the Solar system
(Levison et al., 2001; Tsiganis et al., 2005; Morbidelli et al., 2010a, b).
Until recently, diameter measurements were only available for a few thousand
asteroids, most of these coming from the Infrared Astronomical Satellite
(IRAS) survey (Tedesco et al., 2002). With the completion of the next-
generation all-sky thermal infrared survey by the Wide-field Infrared Survey
Explorer (WISE, Wright et al., 2010) and the identification of the small
bodies of the Solar system observed during that survey (the NEOWISE project,
Mainzer et al., 2011a) a new data set has been opened. NEOWISE allows us to
determine accurate diameters for the $>158,000$ observed Main Belt asteroids
detected during the fully cryogenic portion of the WISE mission and albedos
for the $>120,000$ that had previous optical measurements, of which more than
$33,000$ are members of previously identified asteroid families (Masiero et
al., 2011). We can use these measured diameters of family members to better
constrain the ages of asteroid families by revising predictions of their
orbital evolution, using the methods described in (Vokrouhlický et al., 2006).
However, an important consideration in any attempt to determine asteroid
family age is the error introduced in that determination by the assumed values
of physical and orbital parameters. Many physical parameters (e.g. macroscopic
density) are only poorly constrained for more than a handful of objects, yet
they play a large role in the evolution of said bodies. Similarly, the orbital
parameters of the parent body at the moment of breakup can only be assumed for
families older than a few million years (cf. Nesvorný & Bottke, 2004).
In this work we address both the uncertainty due to the assumed initial
conditions and the effect of using the newly available diameter and albedo
data from NEOWISE to the age determination of the Baptistina asteroid family,
using the work of Bottke et al. (2007) as a starting point and road map. In
Section 2 we discuss the numerical routines used to model the orbital
evolution, as well as the equations governing the thermal forces also acting
on the body. In order to test the effect of the initial conditions chosen, we
use the assumed orbital and physical parameters from Bottke et al. (2007) and
vary each independently through a range of realistic values looking for
changes in the fitted age from their best-fit value. We discuss the behavior
of the fit with respect to each of these parameters in Section 3. With these
effects quantified, we can then model the evolution of the family using the
NEOWISE diameters and albedos. We discuss the new age determination in Section
4 and its implication in Section 5.
## 2 Simulating Orbital Evolution
Under the assumption of a common location and time of origin for the members
of a family, we can simulate the evolutionary history of the orbits of family
members using a numerical integrator. For the Main Belt, the dominant force
shaping this evolution is the gravity from the major bodies of the Solar
system, in particular the Sun and Jupiter. However, non-gravitational effects
such as those arising from thermal radiation by the body can play an important
role, particularly for the smallest MBAs. We discuss these two evolutionary
forces below in the context of the software used to model them.
### 2.1 SWIFT
The dynamic evolution of minor planets due to gravitational interaction with
the Sun is simulated using the Regularized Mixed Variable Symplectic
integrator as implemented in the SWIFT code package (Levison & Duncan, 1994).
This symplectic integrator calculates the motion of a test particle by
separating its Hamiltonian into two parts: the Keplerian motion and the motion
due to gravitational interaction with other bodies, each of which can be
solved analytically. One Hamiltonian is applied for half a time step, the
other is applied for the full time step, and the first is then applied for the
remaining half-step. The Hamiltonian governing the interaction acts as an
acceleration in the particles’ velocity, a feature that is expanded on below
when non-gravitational forces are included. This method of integration ensures
that the energy of the system is conserved.
SWIFT also includes the ability to handle close-approach cases between
particles at a much higher time resolution than is used for the integration in
general. However, we have neglected this component of the routine to reduce
total run time. As cases of close-approaches/impacts with massive bodies will
remove objects from families instead of evolving them within the nominal
orbital element space, this assumption will not result in a significant
increase in the uncertainty of the family age. We note that (as discussed
below) we do include the effect of non-destructive collisions on the
reorientation of the spin states and periods of the test bodies.
Required inputs for SWIFT are the initial positions of the test particles
(assumed to be all the same and coincident with the current location of the
parent fragment), the initial diameters ($D$), and the initial velocities
relative to the parent. Each of the three velocity components were assigned
randomly up to a maximum value that is one of the tested parameters ($V_{0}$)
and scaled inversely proportionally to the diameter of the body. For this work
we used a characteristic diameter of $5~{}$km, following Vokrouhlický et al.
(2006), to allow for comparison with previous results. We compare our
simulations with the observed family using two different methods of diameter
determination (depending on the goal of the simulation, as discussed below).
For simulations that were compared to family lists generated from the
optically selected population (and thus without diameter information) we used
a single assumed albedo for the entire family and estimate diameters from the
$H$ absolute magnitude and the albedo. For comparisons to the families
identified in the WISE data (Masiero et al., 2011) we use the diameters and
albedos drawn from that work. Diameters from WISE were measured independently
of other sources of data, however the albedo measurements required a
literature $H$ magnitude and so are subject to optical observation biases and
errors. It is important to note that the family lists used in Masiero et al.
(2011) were drawn from Nesvorný et al. (2006) who determined family membership
from a sample of optically-discovered asteroids; it is expected that small,
low albedo asteroids will be underrepresented in these family lists, and that
this may alter the determination of family age. Including asteroids discovered
by WISE will begin to mitigate this problem, and this will be the subject of
future work.
### 2.2 SWIFT_RMVSY
To account for the non-gravitational forces due to thermal emission we use the
SWIFT_RMVSY modification of the SWIFT code (Broz̆, 2006). This upgrade uses
the equations derived by Vokrouhlický (1998), Vokrouhlický (1999) and
Vokrouhlický & Farinella (1999) to describe the thermal forces acting on small
Solar system objects. When the thermal force modifies the orbit of a body it
is known as the Yarkovsky effect, and it occurs when incident optical light is
absorbed by a surface and re-emitted as thermal infrared radiation in a
different direction due to the rotation of the body (see Bottke et al., 2006,
for a complete discussion). The Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP)
effect models the way thermal radiation can change the spin state of non-
spherical bodies without atmospheres (Rubincam, 2000).
To calculate these thermal forces SWIFT_RMVSY requires an input of the thermal
and physical parameters for each object: diameter, visible geometric albedo
($p_{V}$), thermal conductivity ($K$), thermal capacity ($C_{p}$), infrared
emissivity ($\epsilon$), surface density ($\rho_{s}$), bulk density ($\rho$),
rotation rate ($\omega$), and rotation pole orientation. As a starting point
for comparisons with the WISE data, we assumed values of $K=0.01~{}$W m-1 K-1,
$C_{p}=680~{}$J kg-1 K-1, $\epsilon=1$, and $\rho=\rho_{s}=2200~{}$kg m-3, and
assigned the population random rotation rates and poles, following
Vokrouhlický et al. (2006). For comparisons with literature work we use the
same values assumed there. We discuss below the effects of varying these
parameters on the best-fitting age.
As an object in the Main Belt evolves over time, it is predicted that it will
undergo small, non-disruptive impacts that can change the body’s rotation
state (both spin pole and rotation period), occurring with a characteristic
timescale depending on diameter and rotational angular momentum. We have
modified the SWIFT_RMVSY code to account for this collisional reorientation by
using the characteristic time of reorientation ($\tau_{r}$) described by
Vokrouhlický et al. (2006):
$\displaystyle\tau_{r}=B(\omega/\omega_{0})^{\beta_{1}}(D/D_{0})^{\beta_{2}}$
(1)
where $B=84.5~{}$kyr, $\beta_{1}=5/6$, $\beta_{2}=4/3$, $D_{0}=2~{}$m (i.e. a
radius of $1~{}$m, see Farinella et al., 1998), and $\omega_{0}$ corresponds
to a period of $5~{}$hr (near the peak in the debiased distribution of MBA
rotation rates, see Masiero et al., 2009). In addition to reorienting spin
poles we also allow collisions to reset the rotation rate of the body in a
random fashion.
While collisional reorientation is treated as a random event, the gradual
reorientation of the spin axis by the YORP effect is treated as a continuous
change, preferentially driving the rotation pole toward an asymptotic limit of
$0^{\circ}$ or $180^{\circ}$ (Vokrouhlický & C̆apek, 2002). We use the median
reorientation rate ($d\epsilon/dt=8.6~{}$deg/Myr) and period doubling/halving
time ($\tau_{per}=11.9~{}$Myr) derived from thermophysical simulations of test
bodies by C̆apek & Vokrouhlický (2004) for the thermal conductivity matching
our assumed value above ($K=0.01$). We note that we scale these timescales by
the rotation rate as discussed by those authors. Following Vokrouhlický et al.
(2006) we also include a multiplicative parameter $c_{YORP}$ that is applied
to both YORP parameters above ($\tau_{per}^{\prime}=\frac{11.9}{c_{YORP}}$ and
$d\epsilon/dt^{\prime}=c_{YORP}\times 8.6$) to model the uncertainty in the
age due to the weakly constrained YORP model. This parameter has been
previously found to only show a weak effect on the age determination (cf.
Vokrouhlický et al., 2006; Bottke et al., 2007) as long as it is non-zero,
though we discuss our findings further below.
### 2.3 Supercomputing Resources
Our numerical simulations make use of the supercomputing resources available
at NASA’s Jet Propulsion Laboratory. We used the Zodiac supercomputer,
comprised of 64 12-core Altix 2.66 Ghz nodes, for all simulations discussed
here. Zodiac uses a 88 terabyte Lustre parallel filesystem allowing for
improved I/O capability, especially for rapid writing to multiple files. Total
peak performance is over 19 teraflops. The range of simulations shown here
required approximately $300,000$ CPU hours of run time.
### 2.4 Integration Step Size
For all the simulations we discuss in this manuscript, we included as massive
particles Venus, Earth, Mars, Jupiter, and Saturn, in addition to the test
particles and the Sun. Uranus and Neptune are omitted as they should play a
much less significant role in the test particle evolution than Jupiter and
Saturn. As Venus has the smallest semimajor axis and perihelion of any tested
body (with the exception of MBAs ejected from the Belt into the NEO
population, which are no longer considered family members and hence are
ignored once ejected) our step size is restricted by Venus’ orbital period. It
is canonically recommended that the integration step size for a symplectic
integrator be
$\raisebox{-2.58334pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.60275pt}{$\textstyle<$}}}{{\sim}}\,$}10\%$
of the period of the innermost body (assuming a circular orbit) to prevent a
rapid accumulation of error on the total system energy (e.g. Broz̆, 2006). We
have tested the effect of step size on the simulated evolution, and show in
Figure 1 the semimajor axis of Venus as a function of time for step sizes of
10, 25, 50, and 80 days, as well as the fractional change. If the step size is
inappropriately large, we should see deviations in the the evolution of Venus
from the shortest time-step tested. For step sizes $\leq 50~{}$days we see no
significant changes in the evolution of Venus with respect to the 10-day step
simulation. For the remaining simulations in this work we use a step size of
$25~{}$days to ensure we are well within the range of acceptable step sizes,
finding it to be the best balance between integration accuracy and time
required to perform the simulations.
Figure 1: Simulated evolution of the orbit of Venus for varying integration
step sizes. For step sizes $\leq 50~{}$days there is no significant change in
the semimajor axis that would indicate an increase in error due to an
inappropriately large step size.
### 2.5 Family Membership
In order to determine the most accurate age possible for the family, the list
of family members that the simulations will be compared to must have minimal
corruption from asteroids that dynamically link to the family but are not
members. This is a particular problem for the Baptistina family, as the branch
of the family that extends to smaller semimajor axes overlaps with the much
larger and older Flora family (cf. Nesvorný et al., 2002).
Following Bottke et al. (2007) we restrict our analysis to consider only the
Baptistina family members at semimajor axes larger than the parent body. We
have accomplished this by using the Hierarchical Clustering Method (HCM,
Zappalà et al., 1990, 1994) of family identification to test a range of cutoff
velocities. We choose the highest velocity that did not link to the lower-
semimajor axis wing ($39~{}$m/s) as our cutoff for family membership,
following Bottke et al. (2007). Likewise, we have removed from our list linked
objects that are both large and distant from the parent, and thus have a high
probability of being incorrect associations. In Figure 2 we show the resultant
HCM-derived family that we use in our analysis. Objects that were rejected
from the list are shown overlaid with an ‘x’. We note that while this will
reduce uncertainty due to incorrectly identified family members, it also
decreases the sample size of WISE-measured asteroids to $360$ objects and
impacts our ability to accurately compare the models to the true distribution.
Identification of new family members and measurement of their physical
parameters will help us decrease these uncertainties.
Figure 2: Diameter vs. semimajor axis for the Baptistina family members used
in our analysis. The black lines show evenly-spaced steps of the C-parameter
(see Section 2.6) used to compare family distributions, and points overlaid
with a red ‘x’ were assumed to be background objects and were not included in
our analysis.
### 2.6 Goodness of Fit Determination
We cannot uniquely link individual test particles to observed family members
as the randomized initial conditions will not necessarily mean the evolutions
are identical. Instead we focus on the distribution of the true and test
populations to find the best matching initial conditions. To compare our
simulation to the known population, we perform a $\chi^{2}$ test of the $C$
parameter, which is defined as $C=\Delta a~{}10^{-0.2H}$ by Vokrouhlický et
al. (2006) for cases where the albedo is unknown. For tests conducted using
only objects with physical parameters measured by WISE, we define a $C_{D}$
parameter as $C_{D}=\Delta a~{}D$ (where $D$ is the diameter) that is roughly
equivalent to the $C$ parameter with a multiplicative offset. Larger objects
are predicted to have smaller drift rates from non-gravitational effects, and
so the $C$ and $C_{D}$ parameters represent lines of constant time for a given
drift strength. An important difference is that $C_{D}$ has no dependence on
the albedo of the asteroid, unlike $C$.
Figure 2 shows a series of curves indicating $C_{D}$ values from $0.025$ to
$0.2$ in steps of $0.025$ overlaid on the Baptistina family. To compare
simulations to reality, we compare the $C$ or $C_{D}$ distribution of the
simulation to the same distribution for the family. As discussed above we only
use family members at semimajor axes larger than the parent, and thus likewise
only consider simulated particles that are in that same region of semimajor
axis-space at the timestep being tested. We note that it is possible for a
particle to begin the simulation drifting inward and later through
reorientation begin moving outward. Thus it is possible for that particle to
be used for the comparison to the observed family members at some timesteps
but not others. The goodness of fit at each time step is obtained from a bin-
by-bin $\chi^{2}$ comparison of the two populations. The match to the observed
data initially improves as the test bodies disperse over time, until they
expand beyond the observed population and the $\chi^{2}$ climbs. The time at
which the minimum $\chi^{2}$ is reached is therefore the best-fit to the
present day family, and thus can be inferred to be the age of the family.
## 3 Errors Due to Assumed Physical and Orbital Parameters
The numerical simulations of the orbital evolution of family members are
deterministic in the sense that the equations of motion (both gravitational
and non-gravitational) can be described analytically. However, the specific
behavior of an individual particle depends strongly on the initial conditions
assumed for it, including the physical, orbital, and spin state parameters.
While the effect of the randomized initial conditions on the behavior of the
population should fade as the population of test particles grows (e.g. the
initial spin pole and rotation rate, the randomized collisional reorientation
of particles, etc.), other initial conditions that are singularly chosen for
the population and do not change with time may have a dramatic effect on the
overall evolution.
Before attempting to fit ages for asteroid families, we first will test our
dependence on the chosen value for each parameter. We constrain the possible
errors induced by assumptions of the thermal parameters ($K$, $\epsilon$,
$C_{p}$), orbital parameters (mean anomaly, longitude of perihelion, longitude
of the ascending node), and physical parameters ($\rho$, spin state). We
include $V_{0}$ and $c_{YORP}$ as tested parameters that are varied to find
the best-fitting age, and so will not discuss them here. Additionally, it is
beyond the scope of the work presented here to investigate the effect of
varying the equations governing the velocity distribution of the impact ejecta
(here assumed to be $V=V_{0}\frac{5km}{D}$) and collisional reorientation
(Equation 1), however these also will act as a source of uncertainty.
For the tests of the physical and orbital parameters, we follow the assumed
initial conditions and albedo for the Baptistina family from Bottke et al.
(2007) for the purpose of comparison. Once the uncertainty due to the assumed
initial conditions has been quantified, we conduct a new set of simulations
that use the measured values for the diameters and albedos in Section 4 to
update the age of the Baptistina family.
Following the best-fit values from Bottke et al. (2007) for Baptistina, we
assume a breakup velocity for the parameter tests of $V_{0}=40~{}$m s-1,
$c_{YORP}=1.0$, $K=0.01~{}$W m-1 K-1, $C_{p}=680~{}$J kg-1 K-1, $\epsilon=1$,
$\rho=\rho_{s}=1300~{}$kg m-3, and randomized rotation states. We note that
using identical initial conditions we reproduce the best-fitting age of $T\sim
160~{}$Myr for the family found by those authors. In order to compare our
results directly to previous work, we initially use the $H$ magnitudes along
with the assumed albedo used by those authors ($p_{V}=0.05$). In Section 4 we
use the WISE measured diameters and albedos.
### 3.1 Rotation State
The assumed initial rotation pole and period of a test particle will dictate
the magnitude and direction of the Yarkovsky force at the outset of the
simulation. Over the course of the evolution of the family, the YORP effect
will gradually reorient the spin axis of a test particle and slow or speed its
rotation (C̆apek & Vokrouhlický, 2004), while collisions will occasionally
abruptly randomize these values. While YORP, by driving the rotation poles to
obliquities of $0^{\circ}$ or $180^{\circ}$, will in general increase the
magnitude of the Yarkovsky effect, collisions are more likely to decrease its
strength or reverse it completely.
Figure 3: Identical simulations of the evolution of the Baptistina family
changing the initial, random spin states of the test particles. The lower plot
shows the fractional difference between the first test and the other four, for
comparison.
In Figure 3 we show five identical simulations of the Baptistina family,
allowing only the randomized spin states of the test particles to vary. The
evolution of these simulations varies in $\chi^{2}$ by $\sim 25\%$ for the
first $175~{}$Myr. After this point, when the comparisons between the
simulations and the real distribution become rapidly worse, the differences
between simulations increases however this regime is less deterministic of age
of the family. This results in an uncertainty in the specific best-fit age of
$\sim 20~{}$Myr in the case of Baptistina, however the range of likely ages
remains comparable.
### 3.2 Thermal Properties
The thermal parameters of the test particles can have a significant effect on
the evolution of the family. We therefore have tested the effect of altering
the assumed thermophysical parameters on the evolution of the test population.
In particular, we focus on varying $\epsilon$, $K$, and $C_{p}$ across ranges
typical for real-world materials around the default assumed values of
$\epsilon_{0}=1.0$, $K_{0}=0.01~{}$W m-1 K-1, and $C_{p,0}=680~{}$J kg-1 K-1.
We show in Figure 4 the evolution of the Baptistina test family for various
initial emissivity values, over the range of $0.7\leq\epsilon\leq 1.0$. We see
no significant differences between each of the cases when only thermal
emissivity is varied. Thus, our assumed value for emissivity of $\epsilon=1.0$
is valid for future tests. Likewise, in Figure 5 we show the evolution of the
test family comparing a wide range of different thermal capacities:
$250<C_{p}<2000~{}$J kg-1 K-1. We again see no significant changes at ages
less than $150~{}$Myr. Beyond this age, the simulations appear to sort roughly
corresponding to $C_{p}$, where simulations with smaller values of $C_{p}$
diverge from the real population faster than those with larger $C_{p}$. For
the purposes of finding the best fit age, an assumed value of $C_{p}=680~{}$J
kg-1 K-1 is adequate.
Figure 4: The same as Figure 3, but now testing various values of emissivity
($\epsilon$). The lower plot shows the fractional difference between the
$\epsilon=1$ case and the other tests, for comparison. Figure 5: The same as
Figure 3, but now testing various values of thermal capacity ($C_{p}$). The
lower plot shows the fractional difference between the $C_{p}=680~{}$J kg-1
K-1 case and the other tests, for comparison.
Conversely, we find that the assumed value of thermal conductivity ($K$) has a
significant impact on the strength of the thermal forces acting on the bodies,
as it is the only parameter that varies over many orders of magnitude in
realistic materials. Vokrouhlický (1998) show in their Figure 3 the relative
strength of the transverse Yarkovsky force vector as a function of the thermal
parameter $\Theta$; using $K=0.01~{}$W m-1 K-1 and the nominal assumptions for
$C_{p}$, $\epsilon$, $\rho$, and rotation rate places $\Theta$ at the peak
value for the transverse force. Changes in $K$ by half or one order of
magnitude result in a significant change in the strength of the Yarkovsky
effect. Following the thermal inertias ($\Gamma$) found by (Delbo & Tanga,
2009) for asteroids with $D<200~{}$km, we test a range of thermal inertia
values of $40<\Gamma<1200$J s-0.5 m-2 K-1 which corresponds to thermal
conductivities of $0.001<K<1$ for nominal values of density and thermal
capacity. We show the results of these simulations in Figure 6. The evolution
of the test family is significantly slower for values both larger and smaller
than $K=0.01~{}$W m-1 K-1. We note that while $K\sim 1$ is only observed for
the smallest of near-Earth asteroids that are believed to have surfaces free
of regolith and thus may not be a good analog for $D\sim 5~{}$km MBAs, the
range of $0.001<K<0.1$ is still possible for MBAs. We use $K=0.01~{}$W m-1 K-1
for future simulations, however this probably represents only a lower limit on
the family age. Determination of thermal conductivity or thermal inertia for a
number of family members will be critical to determining the true evolution of
the family.
Figure 6: The same as Figure 3, but now testing various values of thermal
conductivity ($K$).
### 3.3 Initial Orbit
In order to model the breakup of a family, we assume that all members began at
the same place in space and time, and assign them an ejection velocity that
scales inversely with their diameter (following Vokrouhlický et al., 2006),
which combines with the particle’s velocity around the sun to generate a new
orbit. As the ejection velocities typically are small compared to the motion
around the sun, this will preferentially elongate the cloud along the path of
the orbit. Although the velocity imparted on the fragments by the collision
will be randomized around a constant value, for a parent body with an
eccentric orbit the change in orbital parameters after the impact can vary
depending on the parent’s mean anomaly at the time of breakup. Nominally we
use the present day osculating orbital elements for the largest family member
as the orbit of the body prior to breakup, ensuring that the test particles
are in the same osculating system as the planets (including using the same
assumed epoch). However, we have tested the results of varying the mean
anomaly (MA), longitude of perihelion ($\varpi$), and longitude of the
ascending node ($\Omega$) on the subsequent evolution of the family.
Figure 7: The same as Figure 3, but now testing a range of Mean Anomaly values
at the time of breakup.
In Fig 7 we show a range of simulations with identical physical parameters,
$c_{YORP}$ and breakup velocity $V_{0}$, while stepping through mean anomaly
of the parent at the time of breakup. The velocity added to a test particle’s
motion upon breakup alters its initial orbit. However, the initial impulse is
more effective at changing the orbit’s aphelion when the breakup is at
perihelion than it is at changing the orbit’s perihelion when the breakup is
at aphelion. This effect is shown in Figure 7 as the offset in $\chi^{2}$ at
$T=0$, where simulations with breakups closer to perihelion have a larger
initial spread in semimajor axis and thus a lower $\chi^{2}$.
In general, after about $\sim 100~{}$Myr the differences between populations
with different initial mean anomalies are erased by the effect of Yarkovsky-
induced drifts and gravitational orbital evolution. We note that this
timescale will depend on the initial eccentricity of the parent body: parents
with low or zero eccentricity should see little difference in family member
distribution between breakups at perihelion or aphelion even at $T=0$, while
those with larger eccentricities will require more time to erase the initial
differences. This effect may thus be particularly important for high-
eccentricity families younger than $\sim 100~{}$Myr.
Figure 8: The same as Figure 3, but now testing a range of values for the
longitude of perihelion ($\varpi$) at the time of breakup. Figure 9: The same
as Figure 3, but now testing a range of values for the longitude of the
ascending node ($\Omega$) at the time of breakup.
We show in Figures 8 and 9 the results of similar simulations, testing
$\varpi$ and $\Omega$ respectively. Variations in both parameters result in no
significant change to the evolution of the population in general. While these
parameters may have an effect for other families with parents significantly
more eccentric than Baptistina, we can safely use the present-day osculating
values for all parameters for the simulations we discuss in Section 4.
### 3.4 Density
A key assumption in determining the strength of the Yarkovsky effect on the
orbit of an asteroid is the mass of the body. The Yarkovsky effect is expected
to produce a force that depends on the illuminated area of the body, but the
resultant acceleration will scale with the mass. While the SWIFT_RMVSY code
includes a parameter to allow for testing the variation in the strength of the
YORP effect due to the uncertainty in its absolute strength (the $c_{YORP}$
parameter), the Yarkovsky effect should be well quantified if the mass and
thermal parameters are known and thus does not include this scaling parameter.
With WISE we can usually derive effective diameters to within $\sim 10\%$ for
asteroids observed with good signal-to-noise (Mainzer et al., 2011b). However
the bulk density of asteroids remains poorly constrained. Likewise surface
density, which is a component of the calculation of thermal propagation in
Yarkovsky, is equally difficult to determine and is assumed here to be equal
to the bulk density.
Density measurements of meteorites can provide an upper limit to the density
we expect for different compositions of asteroid, but linking meteorites to
asteroids can be difficult, and the macro- and micro-porosity of a body (which
will strongly affect the measured bulk density) are almost impossible to
measure remotely. Conversely, asteroid masses can be obtained from their
gravitational perturbation of the other objects in the Main Belt (for the few
largest bodies), from deviations on spacecraft trajectories during fly-by (for
the handful of objects visited by spacecraft), or from the periods of
satellite bodies in orbit around the asteroid of interest (if satellites are
known to exist and the periods can be measured).
Carry (2012) provides a thorough review of the state of knowledge of asteroid
densities. They list densities for $38$ MBAs smaller than $D=200~{}$km and
with density accuracy better than $20\%$. The mean density of this group is
$\rho=2.3\pm 1.2~{}$g cm-3, however the error is inflated by the range of
compositions. Attempting to trace composition with spectral taxonomy, they
show that the range of bulk density within a given spectral taxonomic class
can still be large, due to changes in macroporosity which they attribute to
increasing compaction at larger diameters. The authors show some correlation
between density and spectral type (though even then the intrinsic scatter is
about $\sim 25\%$ in the best cases) and find similar discrepancies to the
ones seen by Mainzer et al. (2011c) when comparing taxonomy and albedo,
notably for the objects spectrally identified with the X-complex. Without an
independent measurement of the density of a significant number of family
members, age fits must be performed over the entire viable range of bulk
densities. Otherwise, improperly narrow error windows on the best-fit age will
be derived. If the family taxonomy can be linked to meteorite analogs, a
smaller window can be used, though the unknown porosity will still induce
uncertainty in the density estimate.
We show in Figure 10 simulations of evolution of the Baptistina family, in
this case only varying the density assumed for the family members over a range
of $1.0<\rho<2.8~{}$g cm-3. All other physical and orbital parameters follow
the assumptions used in Section 3.3. The rapid change in best-fitting age for
different densities is a result of the weakening of the accelerative kicks in
the orbital velocity from the Yarkovsky force (i.e. for an assumed diameter
the force will be constant, while the acceleration will be inversely
proportional to the mass and thus the density). This is shown by the
$\chi^{2}$ minimum best-fitting age $T$ following a general
$T\propto\rho^{-1}$ where
$\rho\raisebox{-2.58334pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.60275pt}{$\textstyle<$}}}{{\sim}}\,$}2$,
above which the best-fitting age increases rapidly.
Figure 10: Simulations of the evolution of the Baptistina family under varying
assumptions for the bulk and surface density of the test particles (assuming
both densities are equal).
Bottke et al. (2007) assumed that both the bulk density and surface density of
the Baptistina family members were $1.3~{}$g cm-3 from their assumption that
the spectral taxonomy was most similar to a C-type asteroid (however see Reddy
et al., 2009, 2011, for further discussion on the taxonomy of Baptistina and
its family). For our revised simulations (see Section 4.2) we adopt a bulk and
surface density of $2.2~{}$g cm-3, assuming S-type taxonomy. However testing
over the full range of probable densities ($\sim 1.6$ to $\sim 2.8$) will
result in a broadening of the best fit range.
## 4 The Age of Baptistina Incorporating WISE Results
Using the methodology developed by Vokrouhlický (1998) we revise the estimated
age of the Baptistina family by Bottke et al. (2007) by taking into account
the diameter and albedo measurements offered by NEOWISE for $\sim 1/3$ of the
known family members. One complicating factor in identifying and modeling this
family is its partial overlap in orbital element space with the much larger
and older Flora family. The albedo distinction between these two families
should enable us to use this parameter as a further restriction on family
membership, and development, testing, and analysis of this method will be
presented in a future paper. For this preliminary analysis we use a restricted
set of family members that includes only the objects that have drifted
outwards from the parent and thus are not contaminated by Flora, as discussed
above in Section 2.5.
### 4.1 New Observational Data
The diameters and albedos we use for this work are drawn from the values
derived for MBAs published in Masiero et al. (2011). The larger asteroids were
more likely to have been seen in multiple bands by WISE which allows for
fitting of the beaming parameter. Mainzer et al. (2011b) show that in cases
such as this the absolute error on diameter is $\sim 10\%$ and on albedo is
$\sim 20\%$ of the measured albedo value, however internal comparisons are
better than this limit. We note that this albedo error assumes moderate-to-low
light curve amplitudes and well characterized H and G values. In addition to
observing known objects, NEOWISE also discovered new asteroids, preferentially
with lower albedos where ground-based surveys are less sensitive. These
previously unknown objects represent a source of error in the diameter and
albedo distribution of known families as they would not be included in the
known family lists, and will tend to make the true albedo distribution darker
than the distribution seen for the previously known asteroids, most of which
were discovered by visible light surveys that are biased against detecting low
albedo objects. Future work will address the error resulting from this change
in albedo distribution.
The primary variation between the observed data and the assumed values in
Bottke et al. (2007) is the average value for the albedo of the family
measured by WISE ($p_{V}=0.21$) compared with the assumed value used
previously ($p_{V,assumed}=0.05$). The main effect of this change is to reduce
by more than a factor of two the effective size of a typical Baptistina family
member used in simulations. We note that because the albedo distribution of
the Baptistina family is fairly wide ($\pm 0.1$), the change in diameter from
assumed to measured values for each individual family member can be much
larger or smaller than the factor of two derived from applying the mean
albedos. It is therefore critical to use the actual measured diameters for
family members where available, instead of assuming a uniform albedo for all
objects. This will also remove an additional source of uncertainty that is
inherent to the $H$ magnitude measurement.
As discussed in Section 3.4, the density chosen for the test particles can
have a very large effect on the best-fit age that is determined for the
family. Bottke et al. (2007) use a density of $1.3~{}$g cm-3 appropriate for
small C-complex bodies (Carry, 2012), as Baptistina was thought to be. The
revisions in asteroid sizes and albedos from the WISE data, as well as
taxonomic classification of a larger set of Baptistina family members as
S-complex bodies (Reddy et al., 2011) drives us to assume a larger bulk
density for the objects. For our initial simulations, we assume $\rho=2.2~{}$g
cm-3, however we also test a range of densities using the updated diameters.
### 4.2 Revised Age and Error
Using a set of test particles with the same size and albedo as were measured
for the Baptistina family by WISE, we simulate their evolution over
$400~{}$Myr using for our initial conditions: present day osculating elements
for Baptistina and Venus through Saturn, $\epsilon_{0}=1.0$, $K_{0}=0.01~{}$W
m-1 K-1, $C_{p,0}=680~{}$J kg-1 K-1, and $\rho_{0}=\rho_{s,0}=2200~{}$g cm-3.
We initially test a grid of breakup velocities ($V_{0}$) and $c_{YORP}$
parameters. In Figure 11 we show $\chi^{2}$ maps of $V_{0}$ vs. age for each
of the four tested $c_{YORP}$ values. Figure 12 shows an alternate view of the
same simulations, with each map showing $c_{YORP}$ vs. age for a given
$V_{0}$. For this limited range of assumed parameters, the best fit age is
$190\pm 30~{}$Myr, with minimal dependence on $V_{0}$ and $c_{YORP}$ in the
ranges of $5<V_{0}<20~{}$m/s and $0.5<c_{YORP}<1.5$, and only a slight
preference for lower values in each case. We note that due to the albedo
assumed by Bottke et al. (2007) of $p_{V}=0.05$, the inferred diameters are
approximately a factor of two larger for the family members and thus it is not
unexpected that their best-fit $V_{0}\sim 40$ is similarly larger than the
best-fit value we find.
Figure 11: $\chi^{2}$ maps of breakup velocity $V_{0}$ vs. age for the four
tested values of $c_{YORP}$, with white shading representing the best fits and
dark shading the worst. Contours show $\chi^{2}$ levels of $6,12,18$, the
first of which defines the boundary of the region of acceptable fits. Figure
12: The same as Figure 11 but showing $c_{YORP}$ vs. age for the six tested
values of $V_{0}$.
While best-fit age has minimal dependence on $V_{0}$ and $c_{YORP}$, the
assumed value for density and thermal conductivity induce large changes in the
final age determination. We show in Figure 13 the $\chi^{2}$ map of density
vs. age for simulations using $V_{0}=10~{}$m/s and $c_{YORP}=1.0$. We note
that for $\rho=1.3~{}$g cm-3 (the value assumed by Bottke et al., 2007) the
best fit age is $\sim 80~{}$Myr which is consistent with the inverse relation
between age and the square root of the assumed albedo as specified by those
authors. For a reasonable range of assumed densities of $1.6-2.8~{}$g cm-3 we
find the best fitting age can vary from $140-320~{}$Myr. Without a better
constraint on family member density it will be difficult to more precisely
determine the age of the family.
Figure 13: The same as Figure 11 but showing density $\rho$ vs. age assuming
the best fit values of $V_{0}=10~{}$m/s and $c_{YORP}=1.0$.
Figure 14 shows a similar test, but now for a varied thermal conductivity in
the range of $0.003<K<0.03~{}$W m-1 K-1 (assuming $\rho=2.2~{}$g cm-3). Larger
values of thermal conductivity result in an increase in the best-fit age
comparable to the change caused by a larger assumed density. Like density, the
fact that thermal conductivity is relatively unconstrained sets a fundamental
limit on the accuracy of simulations of family evolution and age.
Figure 14: The same as Figure 13 but showing the log of thermal conductivity
vs. age.
We note that while our simulations can reproduce the semimajor axis
distribution of the family well for a variety of assumed parameters, there are
shortcomings to our solution. In particular, we are unable to simulate the
observed distribution of the family members in inclination-eccentricity space
for any of the range of parameters tested above. We show in Figure 15 the
inclination-eccentricity distribution for the observed Baptistina family
compared with the family simulated using $V_{0}=10~{}$m/s, $c_{YORP}=1$,
$\rho=2.2~{}$g cm-3, $K=0.01~{}$W m-1 K-1, and an age of $T=200~{}$Myr. Proper
orbital elements are calculated for the simulated particles using a frequency
modified Fourier transform (FMFT S̆idlichovský & Nesvorný, 1996) with
frequency filters described by Broz̆ (2006). The offset observed between the
two populations may indicate that the breakup had an ejection velocity
distribution that was highly anisotropic (unlike the assumed isotropic
distribution used in our simulations), that the assumed initial orbital
parameters for the parent at the time of breakup are incorrect, or that the
asteroid identified as the parent body is not the source of the breakup that
created the family. As an example we show in Figure 16 the proper orbital
elements of all objects identified as members of the Baptistina family by
Nesvorný (2010), the restricted list we use for comparisons to our simulations
(as discussed in Section 2.5), (298) Baptistina, and (1696) Nurmela: the
largest body at the center of the a-e-i distribution which has a diameter of
$D=9.9~{}$km. Future work will investigate these scenarios.
Figure 15: Proper inclination (in degrees) vs. proper eccentricity for the
observed Baptistina family members used for our analysis (black) and the
simulated model family (red) using the best-fit values of $V_{0}=10~{}$m/s,
$c_{YORP}=1$, and $T=200~{}$Myr and assumed values of $\rho=2.2~{}$g cm-3 and
$K=0.01~{}$W m-1 K-1. The cyan star indicates the location of the parent body
of the family. Figure 16: Proper orbital elements for all members of the
Baptistina collisional family in black, the restricted family list used for
comparison to our simulations in red, (298) Baptistina as the cyan star, and
(1696) Nurmela as the yellow triangle.
## 5 Conclusions
Using a symplectic integrator modified to include the effects of gravity,
Yarkovsky, and YORP, we have simulated the evolution of a synthetic Baptistina
asteroid family from breakup through $\sim 400~{}$Myr of evolution. We compare
the distribution at each timestep to the observed distribution of the
Baptistina family members to determine the age of the family. By varying all
assumed parameters, we set constraints on the effect of each parameter on the
determined age, and thus the error induced by the uncertainty in its true
value.
We find that while most physical parameters do not significantly change our
results, both the density and thermal conductivity of the surface can
drastically change the best-fit ages resulting in uncertainties greater than
$\sim 50\%$, either younger or older. While having updated values for diameter
and albedo reduces the uncertainty in the simulation and the resultant age
when compared to models conducted using only absolute magnitude, assumptions
for the other physical parameters remain a significant source of uncertainty
in the calculation.
Using the WISE-derived albedos and diameters we find a best-fitting age for
the Baptistina family of $190\pm 30~{}$Myr when we used a single assumed
density of $\rho=2.2~{}$g cm-3 and an assumed thermal conductivity of
$K=0.01~{}$W m-1 K-1. When we allow density and thermal conductivity to vary
over nominal ranges ($\pm 30\%$, and up or down by a factor of 3,
respectively) and we find that the best-fitting age can range anywhere from
$140-320~{}$Myr.
The differences between our results and the findings of Bottke et al. (2007)
are due primarily to the smaller size of the Baptistina family members that we
measure compared to their assumed values and the increase in the assumed
density. A higher assumed density will weaken the non-gravitational forces
compared to gravitational perturbation and slow the overall evolution when
strong gravitational interactions do not dominate the process. We also note
that the revised albedo and diameter measurements result in a reduction in
both the size of the pre-impact body and the number of large fragments
produced in the impact, decreasing the number available to enter the near-
Earth population.
Our simulations all assume that (298) Baptistina is the parent of the
Baptistina family and that its orbital elements at the time of breakup were
the same as today. If instead a different object is the parent of the family,
then the family age may change dramatically from the values found here. A new
suite of simulations would be required, using the updated parent, to determine
the family age. Future work will explore this possibility for the Baptistina
family.
In the end, we are unable to set a firm constraint on the age of the
Baptistina family without more information about the family’s physical
parameters ($\rho$ and $K$, specifically). However, the uncertainty in this
age determination can be greatly reduced with focused investigations of the
family members. In particular, thermophysical modeling of a selection of
Baptistina family members will allow us to better constrain the physical
parameters such as thermal conductivity, while identification and study of any
binary asteroids that may be family members will allow us decrease the
uncertainty in the density of those bodies, and by extension the family as a
whole. Future work will extend our investigation to the remaining asteroid
families observed by WISE, taking into account the caveats and concerns we
highlight here.
## Acknowledgments
We thank the referee, Bill Bottke, for his helpful and insightful comments
that resulted in a critical reanalysis of the data, greatly improving our
results and the manuscript in general. We also thank Bob McMillan for his
editing of this manuscript. J.R.M. was supported by an appointment to the NASA
Postdoctoral Program at JPL, administered by Oak Ridge Associated Universities
through a contract with NASA. Computer simulations for this research were
carried out on JPL’s Zodiac supercomputer, which is administered by the JPL
Supercomputing and Visualization Facility. The supercomputer used in this
investigation was provided by funding from the JPL Office of the Chief
Information Officer. This publication makes use of data products from the
Wide-field Infrared Survey Explorer, which is a joint project of the
University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the National
Aeronautics and Space Administration. This publication also makes use of data
products from NEOWISE, which is a project of the Jet Propulsion
Laboratory/California Institute of Technology, funded by the Planetary Science
Division of the National Aeronautics and Space Administration. This research
has made use of the NASA/IPAC Infrared Science Archive, which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.
## References
* Alvarez et al. (1980) Alvarez, L.W., Alvarez, W., Asaro, F., Michel, H.V., 1980, Science, 208, 1095.
* Bange (1998) Bange, 1998, A&A, 340, L1.
* Bottke et al. (2006) Bottke, W.F., Vokrouhlický, D., Rubincam, D.P. & Nesvorný, D., 2006, AREPS, 34, 157.
* Bottke et al. (2007) Bottke, W.F., Vokrouhlický, D. & Nesvorný, D., 2007, Nature, 449, 48.
* Broz̆ (2006) Broz̆, M., 2006, Ph.D. Thesis, http://sirrah.troja.mff.cuni.cz/$\sim$mira/mp/phdth/
* Brozović et al. (2011) Brozović, M., Benner, L.A.M., Taylor, P.A. et al., 2011, Icarus, 216, 241.
* C̆apek & Vokrouhlický (2004) C̆apek, D. & Vokrouhlický, D., 2004, Icarus, 172, 526.
* Carry (2012) Carry, B., 2012, P&SS, in press, arXiv:1203.4336v1.
* Ćuk et al. (2010) Ćuk, M., Gladman, B.J. & Stewart, S.T, 2010, Icarus, 207, 590.
* Delbo & Tanga (2009) Delbo, M. & Tanga, P., 2009, P&SS, 57, 259.
* dell’Oro et al. (2002) dell’Oro, A., Paolicchi, P., Cellino, A. & Zappalà, V., 2002, Icarus, 156, 191.
* Descamps et al. (2008) Descamps, P., Marchis, F., Pollock, J. et al., 2008, Icarus, 196, 578.
* Descamps et al. (2009a) Descamps, P., Marchis, F., Durech, J. et al., 2009a, Icarus, 203, 88.
* Descamps et al. (2009b) Descamps, P., Marchis, F., Michalowski, T. et al., 2009b, Icarus, 203, 102.
* Descamps et al. (2011) Descamps, P., Marchis, F., Berthier, J. et al., 2011, Icarus, 211, 1022.
* Farinella et al. (1998) Farinella, P., Vokrouhlický, D. & Hartmann, W.K., 1998, Icarus, 132, 378.
* Farley et al. (2006) Farley, K.A., Vokrouhlický, D., Bottke, W.F. & Nesvorný, D., 2006, Nature, 439, 295.
* Fujiwara et al. (2006) Fujiwara, A., Kawaguchi, J., Yeomans, D.K. et al., 2006, Science, 312, 1330.
* Hilton (2002) Hilton, J.L., 2002, in Asteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, Tucson, 103.
* Hirayama (1918) Hirayama, K., 1918, Icarus, 31, 185.
* Le Feuvre & Wieczorek (2011) Le Feuvre, M. & Wieczorek, M.A., 2011, Icarus, 214, 1.
* Levison & Duncan (1994) Levison, H.F. & Duncan, M.J., 1994, Icarus, 108, 18.
* Levison et al. (2001) Levison, H.F., Dones, L., Chapman, C.R., Stern, S.A., Duncan, M.J. & Zahnle, K., 2001, Icarus, 151, 286.
* Mainzer et al. (2011a) Mainzer, A.K., Bauer, J.M., Grav, T. et al., 2011a, ApJ, 731, 53.
* Mainzer et al. (2011b) Mainzer, A.K., Grav, T., Masiero, J. et al., 2011b, ApJ, 736, 100.
* Mainzer et al. (2011c) Mainzer, A.K., Grav, T., Masiero, J. et al., 2011c, ApJ, 741, 90.
* Marchis et al. (2005) Marchis, F., Descamps, P., Hestroffer, D. & Berthier, J., 2005, Nature, 436, 822.
* Masiero et al. (2009) Masiero, J., Jedicke, R., D̆urech, J., Gwyn, S., Denneau, L. & Larsen, J., 2009, Icarus, 204, 145.
* Masiero et al. (2011) Masiero, J., Mainzer, A.K., Grav, T. et al., 2011, ApJ, 741, 68.
* Merline et al. (2002) Merline, W.J., Weidenschilling, S.J., Durda, D.D., Margot, J.-L., Pravec, P. & Storrs, A., 2002, in Asteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, Tucson, 289.
* Morbidelli et al. (2010a) Morbidelli, A., Brasser, R., Nesvorny, D., Vokrouhlicky, D. & Bottke, W.F., 2010a, BAAS DPS Meeting #42, 0408.
* Morbidelli et al. (2010b) Morbidelli, A., Brasser, R., Gomes, R., Levison, H.F. & Tsiganis, K., 2010b, AJ, 140, 1391.
* Mueller et al. (2010) Mueller, M., Marchis, F., Emery, J.P., Harris, A.W., Mottola, S., Hestroffer, D., Berthier, J. & di Martino, M., 2010, Icarus, 205, 505.
* Nesvorný et al. (2002) Nesvorný, D., Morbidelli, A., Vokrouhlický, Bottke, W.F. & Broz̆, M., 2002, Icarus, 157, 155.
* Nesvorný et al. (2003) Nesvorný, D., Alvarellos, J.L.A., Dones, L. & Levison, H.F., 2003, AJ, 126, 398.
* Nesvorný & Bottke (2004) Nesvorný, D. & Bottke, W.F., 2004, Icarus, 170, 324.
* Nesvorný et al. (2005) Nesvorný, D., Jedicke, R., Whiteley, R.J. & Ivezić, Z̆, 2005, Icarus, 173, 132.
* Nesvorný et al. (2006) Nesvorný, D., Bottke, W.F., Vokrouhlický, D., Morbidelli, A. & Jedicke, R., 2006a, Asteroids, Comets, Meteors, Proc. of the 229th Symp. of the IAU, Daniela, L., Sylvio Ferraz, M., and Angel, F.J. (eds), Cambridge University Press, 289.
* Nesvorný (2010) Nesvorny, D., 2010, EAR-A-VARGBDET-5-NESVORNYFAM-V1.0, NASA Planetary Data System.
* O’Brien & Greenberg (2005) O’Brien, D.P. & Greenberg, R., 2005, Icarus, 178, 179.
* Ostro et al. (2006) Ostro, S., Margot, J.L., Benner, L.A.M. et al., 2006, Science, 314, 1276.
* Pätzold et al. (2011) Pätzold, M., Andert, T.P., Asmar, S.W. et al., 2011, Science, 334, 491.
* Petit et al. (1997) Petit, J.-M., Durda, D.D., Greenberg, R., Hurford, T.A. & Geissler, P.E., 1997, Icarus, 130, 177.
* Pitjeva (2001) Pitjeva, E.V., 2001, A&A, 371, 760.
* Reddy et al. (2009) Reddy, V., Emery, J.P., Gaffey, M.J., Bottke, W.F., Cramer, A. & Kelley, M.S., 2009, M&PS, 44, 1917.
* Reddy et al. (2011) Reddy, V., Carvano, J.M., Lazzaro, D. et al., 2011, Icarus, 216, 184.
* Rojo & Margot (2011) Rojo, P. & Margot, J.L., 2011, ApJ, 727, 69.
* Rubincam (2000) Rubincam, D.P., 2000, Icarus, 148, 2.
* Shepard et al. (2006) Shepard, M.K., Margot, J.L., Magri, C. et al., 2006, Icarus, 184, 198.
* S̆idlichovský & Nesvorný (1996) S̆idlichovský, M. & Nesvorný, D., 1996, Celestial Mechanics and Dynamical Astronomy, 65, 137.
* Tedesco et al. (2002) Tedesco, E.F., Noah, P.V., Noah, M. & Price, S.D., 2002, AJ, 123, 1056.
* Tsiganis et al. (2005) Tsiganis, K., Gomes, R., Morbidelli, A. & Levison, H.F., 2005, Nature, 435, 459.
* Veverka et al. (1997) Veverka, J., Thomas, P., Harch, A. et al., 1997, Science, 278, 2109.
* Viateau (1999) Viateau, B., 1999, A&A, 354, 725.
* Viateau & Rapaport (2001) Viateau, B. & Rapaport, M., 2001, A&A, 370, 602.
* Vokrouhlický (1998) Vokrouhlický, D., 1998, A&A, 335, 1093.
* Vokrouhlický (1999) Vokrouhlický, D., 1999, A&A, 344, 362.
* Vokrouhlický & Farinella (1999) Vokrouhlický, D. & Farinella, P., 1999, AJ, 118, 3049.
* Vokrouhlický & C̆apek (2002) Vokrouhlický, D. & C̆apek, D., 2002, Icarus, 159, 449.
* Vokrouhlický et al. (2006) Vokrouhlický, D., Broz̆, M., Bottke, W.F., Nesvorný, D. & Morbidelli, A., 2006, Icarus, 182, 118.
* Wright et al. (2010) Wright, E.L., Eisenhardt, P., Mainzer, A.K., et al., 2010, AJ, 140, 1868.
* Yeomans et al. (2000) Yeomans, D.K., Antreasian, P.G., Barriot, J.-P. et al., 2000, Science, 289, 2085.
* Zappalà et al. (1990) Zappalà, V., Cellino, A., Farinella, P. & Knezevic, Z., 1990, AJ, 100, 2030.
* Zappalà et al. (1994) Zappalà, V., Cellino, A., Farinella, P. & Milani, A., 1994, AJ, 107, 772.
|
arxiv-papers
| 2012-09-07T00:56:38 |
2024-09-04T02:49:34.862905
|
{
"license": "Public Domain",
"authors": "Joseph R. Masiero, A. K. Mainzer, T. Grav, J. M. Bauer and R. Jedicke",
"submitter": "Joseph Masiero",
"url": "https://arxiv.org/abs/1209.1430"
}
|
1209.1449
|
# Existence of Optical Vortices
Yisong Yang
Department of Mathematics
Polytechnic Institute of New York University
Brooklyn, New York 11201, USA
Ruifeng Zhang
Institute of Contemporary Mathematics
School of Mathematics
Henan University
Kaifeng, Henan 475004, PR China
###### Abstract
Optical vortices arise as phase singularities of the light fields and are of
central interest in modern optical physics. In this paper, some existence
theorems are established for stationary vortex wave solutions of a general
class of nonlinear Schrödinger equations. There are two types of results. The
first type concerns the existence of positive-radial-profile solutions which
are obtained through a constrained minimization approach. The second type
addresses the existence of saddle-point solutions through a mountain-pass-
theorem or min-max method so that the wave propagation constant may be
arbitrarily prescribed in an open interval. Furthermore some explicit
estimates for the lower bound and sign of the wave propagation constant with
respect to the light beam power and vortex winding number are also derived for
the first type solutions.
## 1 Introduction
Vortices have important applications in many areas of modern physics including
condensed matter systems, particle interactions, cosmology, and superfluids.
Research on vortices in optics also has a long history and was initiated in as
early as 1964 by Chiao, Garmire, and Townes [5] who explored some conditions
under which a light beam can produce its own waveguide and propagate without
spreading. They described such phenomenon as self-trapping, attributed it to
light propagation in materials whose dielectric coefficient increases with
field intensity in the context of high-intensity light beams such as lasers,
predicted marked optical and physical effects, and suggested the occurrence of
optical vortices. Such vortices have since then been observed in numerous
studies [2, 3, 27, 29, 32] and become a much pursued subject in optics [6, 9,
11, 19, 22, 31, 33] both theoretically and experimentally. (For a vast
literature up to 2005 and for a description of the rich features and profound
applications of optical vortices, see the nice survey article by Desyatnikov,
Kivshar, and Torner [10]. See also [16] for a more recent survey of the
subject in a broader perspective.) As waves, light propagation may be
described by a wave function. At certain spots of space, the intensity of the
waves vanishes and the phase of the waves cannot be defined. Thus, such spots
are phase singularities which were recognized in the comprehensive work of Nye
and Berry [23] as crucial characteristics of general wave motions in which
vortices are present. These phase singularities, also referred to as
dislocations or defects of waves, are the centers of vortices, around which
energy concentrates. In the context of light waves, vortices are centered
around vortexlines and light waves are twisted around the vortexlines. The
twisting arises from the phase ambiguity around a vortexline and is of a
topological nature. The twisting centers are exactly the vortex cores where
light waves cancel out leading to darkness so that light intensity measured at
any cross section vertical to a light beam axis should display concentric
ring-like patterns around the dark core. Light beams of such structure are
also vividly termed “helical beams” [4]. In optics research, a fundamental
prototype situation is when the light waves are described by a complex-valued
wave function governed by nonlinear Schrödinger equations [1, 7, 17, 18, 20,
21, 26, 28]. These theoretical studies provide a broad range of interesting
analytic problems related to the existence and properties of optical vortices
for mathematical investigation.
Our aim in the present work is to obtain some existence theorems for the
optical vortex solutions explored by Salgueiro and Kivshar in [28] through a
study of the normalized nonlinear Schrödinger equation
$\mbox{i}\frac{\partial\psi}{\partial
z}+\nabla^{2}_{\perp}\psi+(V+s|\psi|^{2})\psi=0,$ (1.1)
where $\psi$ is a complex-valued optical field propagating in the
(longitudinal) $z$-direction, $\nabla^{2}_{\perp}$ is the Laplace operator
over the (transverse) plane of coordinates $(x,y)$ which is perpendicular to
the $z$-axis, $V$ is an external potential function (cf. [18, 30]), and $s=\pm
1$ is the sign symbol indicating either a focusing or defocusing situation
[17] which is taken to be $+1$ (focusing) in [28] and will be our main focus
(the defocusing case $s=-1$ will be seen to be straightforward). The interest
of (1.1) actually goes beyond nonlinear optics. For example, it also arises in
the study of the Bose–Einstein condensates [1, 11, 15, 30] and is referred to
as the Gross–Pitaevskii equation. An important simplified situation that
allows optical vortices to present is when $V$ depends on the radial variable
only, $V=V(r),r=\sqrt{x^{2}+y^{2}}$. In this situation one may expect to find
an $n$-vortex solution of (1.1) of the form [28]
$\psi=\psi(r,\theta,z)=u(r)\mbox{e}^{\mbox{i}(n\theta+\beta z)},$ (1.2)
where $r,\theta$ are polar coordinates over ${\mathbb{R}}^{2}$, $u(r)$ is the
radial profile function which gives rise to the intensity of light waves,
integer $n\in{\mathbb{Z}}$ is the winding number or vortex charge of the
vortex solution, and $\beta\in{\mathbb{R}}$ is the wave propagation constant
[28]. This ansatz describes a vortex wave centered around the $z$-axis where
$r=0$ and propagating along the $z$-axis. Inserting (1.2) into (1.1), we
arrive at the following $n$-vortex equation
$(ru_{r})_{r}-\left(\frac{n^{2}}{r}+\beta r\right)u+r(V+su^{2})u=0,$ (1.3)
of cubic nonlinearity. The presence of the vortex core at $r=0$ requires
$u(0)=0$. As in [28] (for $n=1$), we are interested in ring-shaped vortices so
that light intensity concentrates around the vortex core which suggests that
$u(r)$ may be assumed to vanish at a sufficiently large distance.
Mathematically, this indicates that we may impose another ‘boundary’
condition, say $u(R)=0$, at a certain distance $R>0$ away from the core of the
vortex as seen in the numerical results of the work [28]. Thus, the problem of
the existence of optical vortices is reduced into a two-point boundary value
problem with undetermined parameter $\beta$ and prescribed $R$, for any given
$n\in{\mathbb{Z}}$. To tackle this problem, we shall use the methods of
calculus of variations. Our methods allow us to obtain two types of results
for the nontrivial focusing case $s=+1$. The first type of results rely on a
constrained minimization approach. The nature of minimization leads us to
obtaining positive-valued solutions in the open interval $(0,R)$ and that the
propagation constant $\beta$ arises as a Lagrange multiplier due to the
constraint. The second type of results are obtained from searching for saddle
points of the action functional associated to the problem. We will see that,
in this latter case, there is no assurance that the solutions must stay
positive-valued but the propagation constant $\beta$ arises as a prescribed
quantity.
In the next two sections, we shall concentrate on the focusing case when
$s=+1$. In Section 2, we formulate the problem of existence of optical
vortices as a constrained minimization problem, state the main existence
results regarding positive solutions, and then present the proofs. We will see
that the propagation constant $\beta$ arises as a Lagrange multiplier which is
ensured to be negative when the vortex charge $n$ is sufficiently large. We
will also derive some lower estimate for $\beta$. In Section 3, we treat
$\beta$ as a prescribed quantity and use a mountain-pass theorem approach to
establish the existence of solutions for any $R$ and vortex charge $n$. In
particular, we show that the propagation constant $\beta$ may assume any
prescribed value in an explicitly given interval. In Section 4, we briefly
discuss the defocusing case when $s=-1$.
## 2 Vortices via constrained minimization
As described in the previous section, we will be interested in ‘ring vortices’
such that (1.3) is considered over a bounded interval $(0,R)$ ($R>0$) so that
$u$ vanishes at the two endpoints of the interval. As mentioned earlier, we
will mostly concentrate on the nontrivial case, $s=+1$. Thus, our problem is a
two-point boundary value problem
$\displaystyle(ru_{r})_{r}-\frac{n^{2}}{r}u+r(V+u^{2})u$ $\displaystyle=$
$\displaystyle\beta ru,$ (2.1) $\displaystyle u(0)$ $\displaystyle=$
$\displaystyle u(R)=0,$ (2.2)
for which the parameter $\beta\in{\mathbb{R}}$ arises as an eigenvalue of the
problem.
In order to approach the problem consisting of (2.1) and (2.2), we write down
the action functional
$I(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{n^{2}}{r}u^{2}-rV(r)u^{2}-\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(2.3)
and the constraint functional
$P(u)=\int|\psi|^{2}r\,\mbox{d}r\mbox{d}\theta=2\pi\int_{0}^{R}ru^{2}\,\mbox{d}r,$
(2.4)
which measures the beam power [28] of the vortex wave. Thus, to get a solution
of (2.1)–(2.2), it suffices to prove the existence of a solution to the
following constrained minimization problem
$\min\left\\{I(u)\,|\,u\in{\cal A},P(u)=P_{0}\right\\},\quad P_{0}>0,$ (2.5)
where the admissible class $\cal A$ is defined by
${\cal A}=\left\\{u(r)\mbox{ is absolutely continuous over
}[0,R],\,u(0)=u(R)=0,\,E(u)<\infty\right\\},$ (2.6)
with
$E(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{1}{r}u^{2}+\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(2.7)
being the ‘energy’ functional, $P_{0}$ is a prescribed value for the beam
power, and $\beta$ arises as the Lagrange multiplier. Note that the finite-
energy condition $E(u)<\infty$ is imposed only to ensure that all the terms in
the indefinite action functional (2.3) stay finite.
For convenience, for a function $f$ of the variable $r$, we interchangeably
use $f_{r}$ and $f^{\prime}$ to denote its derivative. We will also need the
following decomposition and notation
$\left.\begin{array}[]{rll}V&=&V^{+}-V^{-},\quad V^{\pm}=\max\\{\pm V,0\\},\\\
&&\\\ V_{0}^{\pm}&=&\max\\{V^{\pm}(r)\,|\,r\in[0,R]\\},\\\ &&\\\
V_{0}&=&\max\\{|V(r)|\,|\,r\in[0,R]\\}=\max\\{V^{+}_{0},V^{-}_{0}\\}.\end{array}\right\\}$
(2.8)
The main results of this section may be stated as follows.
###### Theorem 2.1
. For any nonzero integer $n$ and a given continuous potential function $V(r)$
over the interval $[0,R]$ ($R>0$), consider the two-point boundary value
problem (2.1)–(2.2) governing an $n$-vortex wave solution of the nonlinear
Schrödinger equation (1.1), propagating along the $z$-axis with a propagation
constant $\beta$.
(i) The problem always has a solution pair $(u,\beta)$ with $u(r)>0$,
$r\in(0,R)$, and $\beta\in{\mathbb{R}}$, so that the associated beam power
enjoys the bound $P(u)<4\pi|n|$. In fact, such a solution may be obtained
through solving the constrained minimization problem (2.5) assuming
$P_{0}<4\pi|n|$, from which $\beta$ arises as a Lagrange multiplier.
(ii) Let $(u,\beta)$ be the solution pair of the problem (2.1)–(2.2) obtained
in part (i). Then $\beta$ has the lower bound
$\beta\geq\frac{7P_{0}}{5\pi R^{2}}-V_{0}^{-}-\frac{12}{R^{2}}(1+n^{2}[2\ln
2-1]).$ (2.9)
(iii) Let $(u,\beta)$ be the solution pair of the problem (2.1)–(2.2) obtained
in part (i). Then $\beta<0$ if $|n|$ is sufficiently large so that
$|n|>\left\\{\frac{P_{0}^{2}}{4\pi^{2}}+\max\\{r^{2}V^{+}(r)\,|\,r\in[0,R]\\}\right\\}^{\frac{1}{2}}.$
(2.10)
(iv) The problem (2.1)–(2.2) has no nontrivial small-beam-power solution
satisfying $P(u)\leq\frac{1}{2}$ if the condition
$n^{2}>r^{2}(V^{+}(r)-\beta),\quad r\in[0,R],$ (2.11)
is fulfilled. So, roughly speaking, the problem has no nontrivial small-power
$P$ and small-propagation-constant (i.e., $|\beta|$ is sufficiently small)
solution over a small interval $[0,R]$.
We now establish these results.
(i) For any function $u$ satisfying $u(0)=0$, the Schwartz inequality implies
that
$u^{2}(r)=\int^{r}_{0}2u(\rho)u_{\rho}(\rho)\,\mbox{d}\rho\leq
2\left(\int^{r}_{0}\rho
u_{\rho}^{2}(\rho)\,\mbox{d}\rho\right)^{\frac{1}{2}}\left(\int_{0}^{r}\frac{u^{2}(\rho)}{\rho}\,\mbox{d}\rho\right)^{\frac{1}{2}}.$
(2.12)
Thus, multiplying (2.12) by $ru^{2}$, integrating, and using $P(u)=P_{0}$, we
have
$\displaystyle\int_{0}^{R}ru^{4}\,\mbox{d}r$ $\displaystyle\leq$
$\displaystyle\frac{1}{\pi}P_{0}\left(\int^{R}_{0}\rho
u_{\rho}^{2}(\rho)\,\mbox{d}\rho\right)^{\frac{1}{2}}\left(\int_{0}^{R}\frac{u^{2}(\rho)}{\rho}\,\mbox{d}\rho\right)^{\frac{1}{2}}$
(2.13) $\displaystyle\leq$ $\displaystyle\varepsilon\int^{R}_{0}\rho
u_{\rho}^{2}(\rho)\,\mbox{d}\rho+\frac{1}{\varepsilon}\left(\frac{P_{0}}{2\pi}\right)^{2}\int_{0}^{R}\frac{u^{2}(\rho)}{\rho}\,\mbox{d}\rho.$
Inserting (2.13) into (2.3), we obtain
$I(u)\geq\frac{1}{2}\left(1-\frac{\varepsilon}{2}\right)\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r+\frac{1}{2}\left(n^{2}-\frac{P_{0}^{2}}{8\pi^{2}\varepsilon}\right)\int_{0}^{R}\frac{u^{2}}{r}\,\mbox{d}r-\frac{1}{4\pi}P_{0}V_{0}.$
(2.14)
In order to be able to find a suitable $\varepsilon>0$ such that in (2.14) we
have
$1-\frac{\varepsilon}{2}>0,\quad
n^{2}-\frac{P_{0}^{2}}{8\pi^{2}\varepsilon}>0,$ (2.15)
simultaneously, it suffices to assume that $P_{0}$ satisfies the condition
$P_{0}<4\pi|n|.$ (2.16)
In this situation, we can find two positive constants $C_{1},C_{2}$, depending
on $\varepsilon,n,P_{0}$ but independent of $u$, such that
$I(u)\geq
C_{1}\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r+C_{2}\int_{0}^{R}\frac{u^{2}}{r}\,\mbox{d}r-\frac{1}{4\pi}P_{0}V_{0}.$
(2.17)
Assume (2.16) and let $\\{u_{m}\\}$ be a minimizing sequence of (2.5). Then
the coercive inequality (2.17) gives us the bound
$\int_{0}^{R}r([u_{m}]_{r})^{2}\,\mbox{d}r+\int_{0}^{R}\frac{1}{r}u_{m}^{2}\,\mbox{d}r\leq
C,$ (2.18)
where $C>0$ is a constant independent of $m$.
Since both functionals $I$ and $P$ are even, we have $I(u_{m})\geq I(|u_{m}|)$
and $P(u_{m})=P(|u_{m}|)$, where we have also used the basic fact [13] that
for any function $u$ its distributional derivative must satisfy
$||u|_{r}|\leq|u_{r}|$. In other words, we see that the sequence $\\{u_{m}\\}$
may be modified so that each $u_{m}$ is nonnegative, $u_{m}\geq 0$. Thus we
may assume that the sequence $\\{u_{m}\\}$ consists of nonnegative-valued
functions. It is clear that these functions may be viewed as radially
symmetric functions over the disk
$D_{R}=\\{(x,y)\in{\mathbb{R}}^{2}\,|\,x^{2}+y^{2}\leq R^{2}\\}$ which all
vanish on $\partial D_{R}$. Moreover, since (2.18) holds, we see immediately
that $\\{u_{m}\\}$ is bounded under the radially symmetrically reduced norm
$\|\,\|$ where
$\|u\|^{2}=\int_{0}^{R}ru^{2}\,\mbox{d}r+\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r,$
(2.19)
for the standard Sobolev space $W^{1,2}_{0}(D_{R})$ since
$\int_{0}^{R}ru^{2}\,\mbox{d}r\leq
R^{2}\int_{0}^{R}\frac{1}{r}u^{2}\,\mbox{d}r.$ (2.20)
Hence we may assume without loss of generality that $\\{u_{m}\\}$ converges
weakly to an element $u\in W^{1,2}_{0}(D_{R})$ as $m\to\infty$. Applying the
compact embedding $W^{1,2}(D_{R})\to L^{p}(D_{R})$ ($p\geq 1$), we see that
$u_{m}\to u$ strongly in $L^{p}(D_{R})$ as $m\to\infty$. Of course, $u$ is
radially symmetric as well. Thus we may write it as $u=u(r)$ which satisfies
$u(R)=0$. Moreover, it is clear that for any $\varepsilon\in(0,R)$,
$\\{u_{m}\\}$ is a bounded sequence in the space $W^{1,2}(\varepsilon,R)$.
Thus, using the compact embedding $W^{1,2}(\varepsilon,R)\to
C[\varepsilon,R]$, we see that $u_{m}\to u$ as $m\to\infty$ uniformly over
$[\varepsilon,R]$. Besides, similar to (2.12), we have for any pair
$r_{1},r_{2}\in(0,R)$, $r_{1}<r_{2}$, the inequality
$\displaystyle|u_{m}^{2}(r_{2})-u_{m}^{2}(r_{1})|$ $\displaystyle\leq$
$\displaystyle
2\left(\int^{r_{2}}_{r_{1}}r([u_{m}]_{r})^{2}\,\mbox{d}r\right)^{\frac{1}{2}}\left(\int_{r_{1}}^{r_{2}}\frac{u_{m}^{2}}{r}\,\mbox{d}r\right)^{\frac{1}{2}}$
(2.21) $\displaystyle\leq$ $\displaystyle
2C^{\frac{1}{2}}\left(\int_{r_{1}}^{r_{2}}\frac{u_{m}^{2}}{r}\,\mbox{d}r\right)^{\frac{1}{2}},$
where the constant $C>0$ is as given in (2.18). Letting $m\to\infty$ in
(2.21), we arrive at
$|u^{2}(r_{2})-u^{2}(r_{1})|\leq
2C^{\frac{1}{2}}\left(\int_{r_{1}}^{r_{2}}\frac{u^{2}}{r}\,\mbox{d}r\right)^{\frac{1}{2}}.$
(2.22)
However, in view of (2.18) and Fatou’s lemma, we have
$\displaystyle\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r$ $\displaystyle\leq$
$\displaystyle\liminf_{m\to\infty}\int_{0}^{R}r([u_{m}]_{r})^{2}\,\mbox{d}r,$
(2.23) $\displaystyle\int_{0}^{R}\frac{1}{r}u^{2}\,\mbox{d}r$
$\displaystyle\leq$
$\displaystyle\liminf_{m\to\infty}\int_{0}^{R}\frac{1}{r}u_{m}^{2}\,\mbox{d}r.$
(2.24)
In particular, in view of (2.18) again, we see that $\frac{1}{r}u^{2}\in
L(0,R)$. Therefore the right-hand side of (2.22) tends to zero as
$r_{1},r_{2}\to 0$, which implies that the limit
$\eta_{0}=\lim_{r\to 0}u^{2}(r)$ (2.25)
exists. Since $\frac{1}{r}u^{2}\in L(0,R)$, we must have $\eta_{0}=0$. Hence
the boundary condition $u(0)=0$ is achieved.
Summarizing the above results, we conclude that the function $u$ obtained as
the limit of the minimizing sequence $\\{u_{m}\\}$ for the problem (2.5)
satisfies $u(0)=u(R)=0$, $u(r)\geq 0$ for all $r\in[0,R]$, $E(u)<\infty$, and
$I(u)\leq\liminf_{m\to\infty}I(u_{m}),\quad
P(u)=\lim_{m\to\infty}P(u_{m})=P_{0}.$ (2.26)
Thus, $u$ is a solution to (2.5). Consequently, there is some
$\beta\in{\mathbb{R}}$ such that $(u,\beta)$ verify the boundary value problem
(2.1)–(2.2).
If there is a point $r_{0}\in(0,R)$ such that $u(r_{0})=0$, then
$u_{r}(r_{0})=0$ since $r_{0}$ is a minimum point for the function $u(r)$.
Applying the uniqueness theorem for the initial value problem of ordinary
differential equations, we have $u(r)=0$ for all $r\in(0,R)$, which
contradicts the fact $P(u)=P_{0}>0$. This proves $u(r)>0$ for all $r\in(0,R)$.
(ii) Let $(u,\beta)$ be a solution pair just obtained. We next study the
quantity $\beta$ in (2.1). As a preparation, we establish
$\liminf_{r\to 0}\\{ru(r)|u_{r}(r)|\\}=0.$ (2.27)
Suppose otherwise that (2.27) is not valid. Then there are some
$\varepsilon_{0}>0$ and $r_{0}\in(0,R]$ such that
$ru(r)|u_{r}(r)|\geq\varepsilon_{0},\quad r\in(0,r_{0}),$ (2.28)
which leads to
$\infty=\int_{0}^{r_{0}}\frac{\varepsilon_{0}}{r}\,\mbox{d}r\leq\int_{0}^{r_{0}}u|u_{r}|\,\mbox{d}r\leq\left(\int_{0}^{r_{0}}\frac{1}{r}u^{2}\,\mbox{d}r\right)^{\frac{1}{2}}\left(\int_{0}^{r_{0}}ru_{r}^{2}\,\mbox{d}r\right)^{\frac{1}{2}},$
(2.29)
which contradicts with $E(u)<\infty$. So (2.27) is valid.
From (2.27), we can find a sequence $\\{r_{m}\\}$ so that $r_{m}\to 0$ as
$m\to\infty$ and
$\lim_{m\to\infty}\\{r_{m}u(r_{m})u_{r}(r_{m})\\}=0.$ (2.30)
Multiplying (2.1) by $u$, integrating over $[r_{m},R]$, letting $m\to\infty$,
and applying (2.30), we obtain
$\beta\int_{0}^{R}ru^{2}\,\mbox{d}r=\int_{0}^{R}(rVu^{2}+ru^{4})\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r}u^{2}+ru_{r}^{2}\right)\mbox{d}r.$
(2.31)
Let $u_{0}$ be any absolutely continuous function satisfying
$E(u_{0})<\infty$, the boundary condition (2.2), and $P(u_{0})=P_{0}$. Since
$u$ solves (2.5), we have $I(u)\leq I(u_{0})$. As a consequence of this
observation, we have the bound
$\int_{0}^{R}\left(\frac{n^{2}}{r}u^{2}+ru_{r}^{2}\right)\mbox{d}r\leq\int_{0}^{R}\left(rVu^{2}+\frac{r}{2}u^{4}\right)\mbox{d}r+2I(u_{0}).$
(2.32)
Inserting (2.32) into (2.31), we obtain
$\frac{1}{2\pi}\beta{P_{0}}\geq-2I(u_{0})+\frac{1}{2}\int_{0}^{R}ru^{4}\,\mbox{d}r.$
(2.33)
To estimate the right-hand side of (2.33), we set $R=2a$ for convenience and
define
$u_{0}(r)=\left\\{\begin{array}[]{lrl}\frac{b}{a}r,&&0\leq r\leq a,\\\ &&\\\
\frac{b}{a}(2a-r),&&a\leq r\leq 2a.\end{array}\right.$ (2.34)
Therefore, after some calculation we have
$\displaystyle P_{0}=2\pi\int_{0}^{2a}ru_{0}^{2}\,\mbox{d}r$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}a^{2}b^{2},$ (2.35)
$\displaystyle\int_{0}^{2a}r(u_{0}^{\prime}(r))^{2}\,\mbox{d}r$
$\displaystyle=$ $\displaystyle 2b^{2},$ (2.36)
$\displaystyle\int_{0}^{2a}\frac{1}{r}u_{0}^{2}\,\mbox{d}r$ $\displaystyle=$
$\displaystyle 2b^{2}(2\ln 2-1),$ (2.37)
$\displaystyle\int_{0}^{2a}ru^{4}_{0}\,\mbox{d}r$ $\displaystyle=$
$\displaystyle\frac{2}{5}a^{2}b^{4}.$ (2.38)
Using (2.35)–(2.38) and (2.8), we get
$\displaystyle I(u_{0})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left\\{r(u_{0}^{\prime}(r))^{2}+\frac{n^{2}}{r}u_{0}^{2}-\frac{r}{2}u_{0}^{4}\right\\}\,\mbox{d}r+\frac{1}{2}\int_{0}^{R}rV^{-}u^{2}\,\mbox{d}r$
(2.39) $\displaystyle\leq$ $\displaystyle b^{2}\left(1+n^{2}(2\ln
2-1)+\frac{1}{3}V_{0}^{-}a^{2}-\frac{1}{10}a^{2}b^{2}\right).$
Besides, applying the Schwartz inequality, we have
$\left(\int_{0}^{R}ru^{2}\,\mbox{d}r\right)^{2}\leq\frac{R^{2}}{2}\int_{0}^{R}ru^{4}\,\mbox{d}r.$
(2.40)
Thus, in view of (2.33), (2.39), and (2.40), we arrive at
$\frac{1}{2\pi}\beta P_{0}\geq
2b^{2}\left(\frac{7}{45}a^{2}b^{2}-\left[\frac{1}{12}V_{0}^{-}R^{2}+1+n^{2}(2\ln
2-1)\right]\right).$ (2.41)
Inserting $R=2a$ and $a^{2}b^{2}=3P_{0}/4\pi$, we obtain the lower estimate
for $\beta$:
$\beta\geq\frac{12}{R^{2}}\left(\frac{7}{60\pi}P_{0}-\left[\frac{1}{12}V_{0}^{-}R^{2}+1+n^{2}(2\ln
2-1)\right]\right).$ (2.42)
(iii) We next derive the sufficient condition stated to ensure $\beta<0$.
In fact, inserting (2.13) into (2.31), we have
$\frac{1}{2\pi}\beta
P_{0}\leq-(1-\varepsilon)\int_{0}^{R}ru^{2}_{r}\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r}-\frac{P_{0}^{2}}{4\pi^{2}\varepsilon
r}-rV\right)u^{2}\,\mbox{d}r.$ (2.43)
For convenience, we may set $\varepsilon=1$ in (2.43). Thus, whenever the
inequality
$n^{2}>\frac{1}{4\pi^{2}}P_{0}^{2}+r^{2}V^{+}(r),\quad r\in[0,R],$ (2.44)
is fulfilled, we can conclude with $\beta<0$ since $u(r)>0$ for $r\in(0,R)$.
(iv) We now consider nonexistence. For any admissible function $u$, we may
view $u$ as a radially symmetric function defined over ${\mathbb{R}}^{2}$ with
support contained in the disk
$D_{R}=\\{(x,y)\in{\mathbb{R}}^{2}\,|\,x^{2}+y^{2}\leq R^{2}\\}$. Hence,
applying the classical Gagliardo–Nirenberg inequality over ${\mathbb{R}}^{2}$,
we deduce
$\int_{0}^{R}ru^{4}\,\mbox{d}r\leq
4\pi\int_{0}^{R}ru^{2}\,\mbox{d}r\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r,$ (2.45)
From (2.31) and inserting (2.45) with $P(u)=P_{0}$, we have
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{R}(rVu^{2}+ru^{4})\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r}u^{2}+ru_{r}^{2}+\beta
ru^{2}\right)\mbox{d}r$ (2.46) $\displaystyle\leq$
$\displaystyle(2P_{0}-1)\int_{0}^{R}ru^{2}_{r}\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r^{2}}+\beta-V\right)ru^{2}\,\mbox{d}r.$
Therefore, when the conditions
$P_{0}\leq\frac{1}{2},\quad\frac{n^{2}}{r^{2}}-V(r)+\beta>0,\quad r\in(0,R],$
(2.47)
are imposed, we arrive at $u\equiv 0$, as anticipated. Thus, in this
situation, the problem consisting of (2.1)–(2.2) has no nontrivial solution.
The proof of Theorem 2.1 is complete.
## 3 Vortices as saddle points
In this section, we study the existence of optical vortices which are the
solutions of the boundary value problem (2.1)–(2.2) as the saddle points of
the action functional
$I_{\beta}(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{n^{2}}{r}u^{2}+(\beta-V(r))ru^{2}-\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(3.1)
with $|n|\geq 1$. We shall use a min-max theory approach. Suggested by the
discussion of the previous section, we introduce the function space $H$ which
is the completion of the space $X=\\{u\in C^{1}[0,R]\,|\,u(0)=u(R)=0\\}$ (the
set of differentiable functions over $[0,R]$ which vanish at the two endpoints
of the interval) equipped with the inner product
$\langle
u,v\rangle_{H}=\int_{0}^{R}\left(ru_{r}v_{r}+\frac{1}{r}uv\right)\,\mbox{d}r,\quad
u,v\in H.$ (3.2)
As seen in the discussion of the previous section, as a Hilbert space, $H$ may
be viewed as an embedded subspace of $W^{1,2}_{0}(D_{R})$ consisting of
radially symmetric functions such that any element $u\in H$ enjoys the desired
property $u(0)=0$.
In order to simplify the presentation of the study here, we assume that
$\beta$ satisfies
$\beta\geq V_{0}^{+}=\max\\{V^{+}(r)\,|\,r\in[0,R]\\}.$ (3.3)
Recall that a $C^{1}$-functional $I:H\to{\mathbb{R}}$ is said to enjoy the
Palais–Smale condition if for any sequence $\\{u_{m}\\}$ satisfying the
properties (i) $I(u_{m})\to\alpha$ as $m\to\infty$, and (ii)
$I^{\prime}(u_{m})\to 0$ as $m\to\infty$ as a sequence in the dual space of
$H$, one can extract a subsequence from $\\{u_{m}\\}$ which converges
(strongly) in $H$.
As an initial step, we have
###### Lemma 3.1
. The action functional $I_{\beta}$ given in (3.1) satisfies the Palais–Smale
condition.
Proof.It is straightforward to see that the functional (3.1) is $C^{1}$ over
$H$.
Let $\\{u_{m}\\}$ be a sequence in $H$ satisfying the properties
$\displaystyle I_{\beta}(u_{m})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left\\{r([u_{m}]_{r})^{2}+\frac{n^{2}}{r}u_{m}^{2}+\left(\beta-V(r)\right)ru_{m}^{2}-\frac{r}{2}u_{m}^{4}\right\\}\,\mbox{d}r$
(3.4) $\displaystyle\to$ $\displaystyle\alpha,\quad m\to\infty,$
$\displaystyle|I_{\beta}^{\prime}(u_{m})(v)|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{R}\left\\{r[u_{m}]_{r}v_{r}+\frac{n^{2}}{r}u_{m}v+(\beta-V(r))ru_{m}v-ru_{m}^{3}v\right\\}\,\mbox{d}r\right|$
(3.5) $\displaystyle\leq$
$\displaystyle\varepsilon_{m}\|v\|_{H},\quad\varepsilon_{m}\geq 0,\quad v\in
H,$
where $\varepsilon_{m}\to 0$ as $m\to\infty$. In (3.5), we may take $v=u_{m}$
to get
$\int_{0}^{R}ru_{m}^{4}\,\mbox{d}r\leq\int_{0}^{R}\left(r([u_{m}]_{r})^{2}+\frac{n^{2}}{r}u_{m}^{2}+(\beta-V(r))ru_{m}^{2}\right)\,\mbox{d}r+\varepsilon_{m}\|u_{m}\|_{H}.$
(3.6)
On the other hand, in view of (3.4), we may assume without loss of generality
that $I_{\beta}(u_{m})\leq\alpha+1$ for all $m=1,2,\cdots$. Hence, applying
(3.6), the assumption (3.3), and a simple interpolation inequality, we find
$\displaystyle 2(\alpha+1)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left(r([u_{m}]_{r})^{2}+\frac{n^{2}}{r}u_{m}^{2}+(\beta-V(r))ru_{m}^{2}\right)\,\mbox{d}r-\frac{1}{2}\varepsilon_{m}\|u_{m}\|_{H}$
(3.7) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\|u_{m}\|^{2}_{H}-\frac{1}{4}\varepsilon_{m}^{2},\quad
m=1,2,\cdots.$
In other words, $\\{u_{m}\\}$ is a bounded sequence in $H$. Without loss of
generality, we may assume that $\\{u_{m}\\}$ weakly converges to an element
$u\in H$ as $m\to\infty$. It is clear that $u_{m}\to u$ as $m\to\infty$
strongly in any $L^{p}(D_{R})$ or $L^{p}((0,R),r\mbox{d}r)$ ($p\geq 1$). Thus,
letting $m\to\infty$ in (3.5), we arrive at
$\int_{0}^{R}\left\\{ru_{r}v_{r}+\frac{n^{2}}{r}uv+(\beta-V(r))ruv-
ru^{3}v\right\\}\,\mbox{d}r=0,\quad\forall v\in H.$ (3.8)
Let $v=u_{m}-u$ in (3.5) and (3.8) and insert the resulting (3.8) into the
resulting (3.5). We have
$\displaystyle\left|\int_{0}^{R}\left\\{r([u_{m}-u]_{r})^{2}+\frac{n^{2}}{r}(u_{m}-u)^{2}+(\beta-V)r(u_{m}-u)^{2}-r(u_{m}^{3}-u^{3})(u_{m}-u)\right\\}\,\mbox{d}r\right|$
$\displaystyle\leq\varepsilon_{m}\|u_{m}-u\|_{H}.$ (3.9)
As a consequence of (3.9) and $\beta\geq V_{0}^{+}$, we obtain
$\|u_{m}-u\|^{2}_{H}\leq\varepsilon_{m}\|u_{m}-u\|_{H}+\int_{0}^{R}r|(u_{m}^{3}-u^{3})(u_{m}-u)|\,\mbox{d}r,\quad
m=1,2,\cdots,$ (3.10)
which immediately implies that $u_{m}\to u$ strongly in $H$ as $m\to\infty$,
as desired.
We next identify a mountain-pass structure through the following two lemmas.
###### Lemma 3.2
. There are constants $K>0$ and $C_{0}>0$ such that
$\inf\\{I_{\beta}(u)\,|\,\|u\|_{H}^{2}=K\\}\geq C_{0}.$ (3.11)
Proof.For any constant $K>0$, let $u\in H$ satisfy $\|u\|^{2}_{H}=K>0$. From
(2.12), we have
$\int_{0}^{R}ru^{4}\,\mbox{d}r\leq
4\int_{0}^{R}r\,\mbox{d}r\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r\int_{0}^{R}\frac{u^{2}}{r}\,\mbox{d}r\leq
2R^{2}K^{2}.$ (3.12)
Applying (3.12) in (3.1), we find
$I_{\beta}(u)\geq\frac{1}{2}\left(K-R^{2}K^{2}\right)\equiv f(K).$ (3.13)
However, the maximum of the function $f$ in (3.13) is attained at
$K_{0}=\frac{1}{2R^{2}}$ which gives us the value $f(K_{0})=\frac{1}{8R^{2}}$.
So, in conclusion, we have the lower bound
$I_{\beta}(u)\geq\frac{1}{8R^{2}},\quad\|u\|^{2}_{H}=\frac{1}{2R^{2}},$ (3.14)
which establishes (3.11).
###### Lemma 3.3
. For any constant $K>0$, there is an element $v\in H$ satisfying
$\|v\|_{H}^{2}>K$ and $I_{\beta}(v)<0$.
Proof.With $R=2a$, we will see that we can use the function $u_{0}$ defined in
(2.34) as a test function.
For this purpose, we first show that $u_{0}\in H$. To see this, we need to
prove that $u_{0}$ can be obtained in the limit from a sequence of functions
in $X$ under the norm of $H$. In fact, for any $0<\varepsilon<a$, we can
define
$u_{\varepsilon}(r)=\left\\{\begin{array}[]{rll}u_{0}(r),&r\in[0,a-\varepsilon)\cup(a+\varepsilon,2a],\\\
&\\\ Q_{\varepsilon}(r),&r\in[a-\varepsilon,a+\varepsilon],\end{array}\right.$
(3.15)
where $Q_{\varepsilon}(r)$ is taken to be a quadratic function satisfying
$Q_{\varepsilon}(a\pm\varepsilon)=u_{0}(a\pm\varepsilon),\quad
Q_{\varepsilon}^{\prime}(a\pm\varepsilon)=u_{0}^{\prime}(a\pm\varepsilon).$
(3.16)
Matching these conditions, we find
$Q_{\varepsilon}(r)=-\frac{b}{2a\varepsilon}(r^{2}-2ar+[a-\varepsilon]^{2}),\quad
r\in[a-\varepsilon,a+\varepsilon],$ (3.17)
which enjoys the bounds
$\frac{b}{a}(a-\varepsilon)\leq
Q_{\varepsilon}(r)\leq\frac{b}{a}\left(a-\frac{\varepsilon}{2}\right),\quad|Q^{\prime}_{\varepsilon}(r)|\leq\frac{b}{a},\quad
r\in[a-\varepsilon,a+\varepsilon].$ (3.18)
Consequently, from (3.18) we have
$\lim_{\varepsilon\to
0}\int_{a-\varepsilon}^{a+\varepsilon}\left(r[Q^{\prime}_{\varepsilon}(r)]^{2}+\frac{1}{r}Q_{\varepsilon}^{2}(r)\right)\,\mbox{d}r=0.$
(3.19)
Therefore $\\{u_{\varepsilon}\\}$ is a Cauchy sequence in $H$ as
$\varepsilon\to 0$ whose limit is clearly $u_{0}$ in view of the definition of
$\\{u_{\varepsilon}\\}$ given in (3.15). This proves $u_{0}\in H$.
Using the results (2.35)–(2.38), we have
$\displaystyle\|u_{0}\|^{2}_{H}$ $\displaystyle=$ $\displaystyle 4b^{2}\ln 2,$
(3.20) $\displaystyle I_{\beta}(u_{0})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left\\{r([u_{0}]_{r})^{2}+\frac{n^{2}}{r}u_{0}^{2}+(\beta+V^{-})ru_{0}^{2}-\frac{r}{2}u_{0}^{4}\right\\}\,\mbox{d}r$
(3.21) $\displaystyle\leq$ $\displaystyle b^{2}\left(1+n^{2}(2\ln
2-1)+\frac{1}{3}a^{2}(\beta+V^{-}_{0})-\frac{1}{10}a^{2}b^{2}\right).$
From (3.20) and (3.21), we see that for any $K>0$ we may choose $b>0$
sufficiently large to get $I_{\beta}(u_{0})<0$ and $\|u_{0}\|^{2}_{H}>K$.
Thus the lemma follows.
It is interesting to note that (3.21) implies that
$I_{\beta}(u_{0})\to-\infty$ as $b\to\infty$. In other words, the action
functional (3.1) is not bounded from below over $H$ which prevents a direct
minimization approach to the problem. Indeed, we are now prepared to obtain a
nontrivial solution of the boundary value problem (2.1)–(2.2) as a saddle
point of the functional (3.1) in the following theorem.
###### Theorem 3.4
. For any $\beta\geq V_{0}^{+}$, $|n|\geq 1$, and $R>0$, the problem
(2.1)–(2.2) has a nontrivial solution over the interval $[0,R]$. Moreover,
such a solution may be obtained from a min-max approach applied to the action
functional (3.1).
Proof.Let $I_{\beta}$ be the action functional (3.1). Then Lemma 3.1 says that
$I_{\beta}$ satisfies the Palais–Smale condition. Let $K,C_{0}>0$ be the
constant stated in Lemma 3.2. Using Lemma 3.3, we can find some $u_{0}\in H$
such that $\|u_{0}\|_{H}^{2}>K$ and $I_{\beta}(u_{0})<0$. Denote by $\Gamma$
the set of all continuous paths in $H$ that link the zero element $0$ of $H$
to $u_{0}$:
$\Gamma=\left\\{g\in C([0,1];H)\,|\,g(0)=0,g(1)=u_{0}\right\\}.$ (3.22)
Therefore there is some point $t_{g}\in(0,1)$ such that
$\|g(t_{g})\|^{2}_{H}=K$. From the classical mountain-pass theorem (cf. Evans
[12]), we know that
$c=\inf_{g\in\Gamma}\max_{t\in[0,1]}I_{\beta}(g(t))\geq C_{0},$ (3.23)
is a critical value of $I_{\beta}$. In other words, there is an element $u\in
H$ satisfying $I_{\beta}(u)=c$ which is a critical point of $I_{\beta}$. Of
course, $u$ is nontrivial. That is, $u$ cannot be the zero element of $H$.
Recall that in Theorem 2.1 the condition (2.10) is obtained to ensure the wave
propagation constant $\beta$ to assume a negative value. Although (2.9) states
a lower estimate for $\beta$, no condition has been obtained to ensure
$\beta>0$ for the solution of the constrained minimization problem (2.5).
Theorem 3.4, however, complements Theorem 2.1 in that it gives us a family of
nontrivial solutions realizing arbitrarily prescribed parameter $\beta$ in the
entire interval $[V_{0}^{+},\infty)$ for any vortex charge $|n|\geq 1$ and
$R>0$. That is, our existence result indicates that $\beta$ may take any
positive value above or at $V^{+}_{0}$.
## 4 The defocusing case when $s=-1$
If we have $s=-1$ in (1.1) instead, then the action functional (2.3) is
replaced by
$I(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{n^{2}}{r}u^{2}-rV(r)u^{2}+\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(4.1)
the difficulty with the quartic term, which was negative before, disappears,
and the constrained minimization problem (2.5) is easily solved, which gives
us a solution to the associated equation
$(ru_{r})_{r}-\frac{n^{2}}{r}u+r(V-u^{2})u=\beta ru,$ (4.2)
for some $\beta\in{\mathbb{R}}$. As before, this equation leads us to the
relation
$\int_{0}^{R}(\beta-V^{+})ru^{2}\,\mbox{d}r=-\int_{0}^{R}\left(ru_{r}^{2}+\frac{n^{2}}{r}u^{2}+rV^{-}u^{2}+ru^{4}\right)\mbox{d}r.$
(4.3)
Thus, if $\beta$ satisfies $\beta\geq V^{+}_{0}$, then $u\equiv 0$. Therefore,
regardless of the value of $R$, the problem prevents the existence of a
nontrivial solution for sufficiently large propagation constant $\beta$. This
conclusion is in sharp contrast to that in the case when $s=+1$ stated in
Theorem 3.4.
In general, the simple relation (4.3) clearly indicates that it is natural for
$\beta$ to take negative rather than positive values. For example, using the
Poincaré inequality over $D_{R}$,
$\int_{0}^{R}ru^{2}\,\mbox{d}r\leq\left(\frac{R}{r_{0}}\right)^{2}\int_{0}^{R}ru^{2}_{r}\,\mbox{d}r,$
(4.4)
where $r_{0}$ ($\approx 2.404825$) is the first positive zero of the Bessel
function $J_{0}$, and (2.20), we obtain from (4.3) the result
$\displaystyle\beta\int_{0}^{R}ru^{2}\,\mbox{d}r$ $\displaystyle\leq$
$\displaystyle
V_{0}^{+}\int_{0}^{R}ru^{2}\,\mbox{d}r-\int_{0}^{R}\left(ru_{r}^{2}+\frac{n^{2}}{r}u^{2}+rV^{-}u^{2}+ru^{4}\right)\mbox{d}r$
(4.5) $\displaystyle\leq$
$\displaystyle-\left(\left[\frac{r_{0}}{R}\right]^{2}+\left[\frac{n}{R}\right]^{2}-V_{0}^{+}\right)\int_{0}^{R}ru^{2}\,\mbox{d}r-\int_{0}^{R}ru^{4}\,\mbox{d}r.$
Consequently, we obtain
$\beta<-\left(\frac{r_{0}^{2}+n^{2}}{R^{2}}-V_{0}^{+}\right).$ (4.6)
In particular, we have $\beta\to-\infty$ as $|n|\to\infty$.
An example of the defocusing case $s=-1$ is the study carried out in [18]
where $V$ takes the form $V=pJ^{2}_{1}(br)$ given in terms of the Bessel
function $J_{1}$ and positive parameters $p,b$. Thus $V^{-}\equiv 0$. Another
example of the case $s=-1$ is in the lines of the studies [24, 30]. There,
although $V$ is not radially symmetric, it is non-positive valued,
$V^{+}\equiv 0$. Here, assuming $V$ is radial as well as non-positive, then
(4.3) indicates that $\beta<0$ is the only possibility.
Note that the ($z$-direction) angular momentum of the obtained stationary
vortex wave in view of (1.2) has the simple but elegant expression [28]
$L_{z}=\mbox{Im}\int(\psi^{*}\partial_{\theta}\psi)\,r\mbox{d}r\mbox{d}\theta=2\pi
n\int_{0}^{R}ru^{2}\,\mbox{d}r=nP.$ (4.7)
Write $\phi(x,y)=u(r)\mbox{e}^{\mbox{i}n\theta}$, where $u$ solves (1.3) and
satisfies $u(0)=0$, and $\Delta=\nabla^{2}_{\perp}$. Then $\phi$ satisfies
$\Delta\phi+(V+s|\phi|^{2})\phi=\beta\phi,$ (4.8)
away from the origin of ${\mathbb{R}}^{2}$. The condition $u(0)=0$ ensures
that the origin is a removable singularity [35] such that when $V$ is an
analytic function of $(x,y)$, so is $\phi$. Consequently, in this situation
$u$ vanishes at $r=0$ like $r^{n}$ for an $n$-vortex solution as in the
classical Ginzburg–Landau equation case [8, 14, 25].
For the focusing case $s=+1$ with a non-positive potential (cf. [34]), we have
$V^{+}\equiv 0$ and the statements of our results simplify considerably. For
example, for the solution pair $(u,\beta)$ obtained in Theorem 2.1 to have the
property $\beta<0$, it suffices that the vortex number $n$ satisfies the
condition
$|n|>\frac{P_{0}}{2\pi}.$ (4.9)
Moreover, applying Theorem 2.1 (iv), we see that there is no nontrivial
solution satisfying
$P(u)\leq\frac{1}{2}\quad\mbox{and}\quad n^{2}>-r^{2}\beta,\quad r\in[0,R].$
(4.10)
In particular, we conclude that there is no nontrivial solution with
$P(u)\leq\frac{1}{2}$ and $\beta\geq 0$. Besides, in this case Theorem 3.4
becomes an existence theory for any prescribed propagation constant $\beta\geq
0$.
## References
* [1] S. K. Adhikari, Localization of a Bose–Einstein condensate vortex in a bichromatic optical lattice, Phys. Rev. A 81 (2010) 043636.
* [2] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes, Phys. Rev. A 45 (1992) 8185–8189.
* [3] M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, Local observations of phase singularities in optical fields in waveguide structures, Phys. Rev. Lett. 85 (2000) 294–297.
* [4] A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial light beams with angular momentum, Ukrainian J. Phys. 2 (2005) 73–113.
* [5] R. Y. Chiao, E. Garmire, and C. H. Townes, Self-trapping of optical beams, Phys. Rev. Lett. 13 (1964) 479–482.
* [6] J. E. Curtis and D. G. Grier, Structure of optical vortices, Phys. Rev. Lett. 90 (2003) 133901.
* [7] T. A. Davydova and A. I. Yakimenko, Stable multi-charged localized optical vortices in cubic quintic nonlinear media, J. Optics A 97 (2004) S197–S201.
* [8] H. J. de Vega and F. A. Schaposnik, Classical vortex solution of the Abelian Higgs model, Phys. Rev. D 14 (1976) 1100–1106.
* [9] M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, Isolated optical vortex knots, Nature Phys. 6 (2010) 118–121.
* [10] A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Optical vortices and vortex solitons, Progress in Optics 47 (2005) 291–391.
* [11] Z. Dutton and J. Ruostekoski, Transfer and storage of vortex states in light and matter waves, Phys. Rev. Lett. 93 (2004) 193602.
* [12] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, 2002.
* [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin and New York, 1977.
* [14] A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkhäuser, Boston, 1980.
* [15] A. M. Kamchatnov and S. V. Korneev, Dynamics of ring dark solitons in Bose- Einstein condensates and nonlinear optics, Phys. Lett. A 374 (2010) 4625–4628.
* [16] Y. V. Kartashov, B. A. Malomed, and L. Torner, Solitons in nonlinear lattices, Rev. Mod. Phys. 83 (2011) 247–305.
* [17] Y. V. Kartashov, V. A. Vysloukh, and Lluis Torner, Rotary solitons in Bessel optical lattices, Phys. Rev. Lett. 93 (2004) 093904\.
* [18] Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Stable ring vortex solitons in Bessel optical lattices, Phys. Rev. Lett. 94 (2005) 043902.
* [19] J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, Laser beams: knotted threads of darkness, Nature 432 (2004) 165.
* [20] A. V. Mamaev, M. Saffman, and A. A. Zozulya, Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices, Phys. Rev. Lett. 76 (1996) 2262–2265.
* [21] D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Observation of discrete vortex solitons in optically-induced photonic lattices, Phys. Rev. Lett. 92 (2004) 123903.
* [22] D. Neshev, A. Nepomnyashchy, and Yu. S. Kivshar, Nonlinear Aharonov–Bohm scattering by optical vortices, Phys. Rev. Lett. 87 (2001) 043901.
* [23] J. F. Nye and M. V. Berry, Dislocations in wave trains, Proc. Roy. Soc. A 336 (1974) 165–190.
* [24] E. A. Ostrovskaya and Y. S. Kivshar, Matter-wave gap vortices in optical lattices, Phys. Rev. Lett. 93 (2004) 160405.
* [25] B. J. Plohr, The existence, regularity, and behavior of isotropic solutions of classical gauge field theories, Thesis, Princeton University, 1980\.
* [26] D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., Propagation dynamics of optical vortices, J. Optical Soc. Amer. B 14 (1997) 3054–3065.
* [27] D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., Experimental observation of fluid-like motion of optical vortices, Phys. Rev. Lett. 79 (1997) 3399–3402.
* [28] J. R. Salgueiro and Y. S. Kivshar, Switching with vortex beams in nonlinear concentric couplers, Opt. Exp. 20 (2007) 12916–12921.
* [29] J. Scheuer and M. Orenstein, Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities, Science 285 (1999) 230–233.
* [30] R. G. Scott, A. M. Martin, T. M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, Creation of solitons and vortices by Bragg reflection of Bose–Einstein condensates in an optical lattice, Phys. Rev. Lett. 90 (2003) 110404.
* [31] M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Topological charge and angular momentum of light beams carrying optical vortices, Phys. Rev. A 56 (1997) 4064–4075.
* [32] G. A. Swartzlander, Jr. and C. T. Law, Optical vortex solitons observed in Kerr nonlinear media, Phys. Rev. Lett. 69 (1992) 2503–2506.
* [33] A. Vin otte and L. Berge, Femtosecond optical vortices in air, Phys. Rev. Lett. 95 (2005) 193901.
* [34] J. Yang and Z. H. Musslimani, Fundamental and vortex solitons in a two-dimensional optical lattice, Optics Lett. 28 (2003) 2094–2096.
* [35] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, New York, 2001.
|
arxiv-papers
| 2012-09-07T05:57:43 |
2024-09-04T02:49:34.873260
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yisong Yang and Ruifeng Zhang",
"submitter": "Ruifeng Zhang",
"url": "https://arxiv.org/abs/1209.1449"
}
|
1209.1615
|
#
${{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}^{{}^{{}^{Published~{}in~{}the~{}Astrophysical~{}Journal\,,~{}Vol.~{}764\,,~{}id.~{}26\,~{}(2013)}}}}$
Tidal Friction and Tidal Lagging. Applicability Limitations of a Popular
Formula for the Tidal Torque
Michael Efroimsky
US Naval Observatory, Washington DC 20392
e-mail: michael.efroimsky @ usno.navy.mil
and
Valeri V. Makarov
US Naval Observatory, Washington DC 20392
e-mail: vvm @ usno.navy.mil
###### Abstract
Tidal torques play a key role in rotational dynamics of celestial bodies. They
govern these bodies’ tidal despinning, and also participate in the subtle
process of entrapment of these bodies into spin-orbit resonances. This makes
tidal torques directly relevant to the studies of habitability of planets and
their moons.
Our work begins with an explanation of how friction and lagging should be
built into the theory of bodily tides. Although much of this material can be
found in various publications, a short but self-consistent summary on the
topic has been lacking in the hitherto literature, and we are filling the gap.
After these preparations, we address a popular concise formula for the tidal
torque, which is often used in the literature, for planets or stars. We
explain why the derivation of this expression, offered in the paper by
Goldreich (1966; AJ 71, 1 - 7) and in the books by Kaula (1968, eqn. 4.5.29),
and Murray & Dermott (1999, eqn. 4.159), implicitly sets the time lag to be
frequency independent. Accordingly, the ensuing expression for the torque can
be applied only to bodies having a very special (and very hypothetical)
rheology which makes the time lag frequency independent, i.e, the same for all
Fourier modes in the spectrum of tide. This expression for the torque should
not be used for bodies of other rheologies. Specifically, the expression
cannot be combined with an extra assertion of the geometric lag being
constant, because at finite eccentricities the said assumption is incompatible
with the constant-time-lag condition.
## 1 Context, Motivation, and Plan
${\left.~{}~{}~{}~{}~{}~{}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\right.}^{\mbox{\small\it
The mills of God grind slowly...}}$
Usually extremely weak, tidal interactions act upon celestial bodies for
extended spans of time (up to billions of years). Over æons, tides shape
celestial bodies’ spin modes, and govern the exchange of angular momentum. The
numerous and diverse manifestations of bodily tides range from the expected
fall of Phobos to synchronous locking of the Moon, to Mercury’s capture in the
3:2 spin-orbit resonance, to bloated hot-jupiter exoplanets in tight orbits
around their host stars, to dissipational coalescence of short-period binary
stars. This makes studies of tides essential to our understanding of the
dynamical properties and evolution of stellar systems.
While the slow work of tides is responsible for circularisation, obliquity
evolution, and synchronisation of planets and moons, the wide scope of these
dynamical phenomena is not always matched by sophistication or versatility of
the tidal models employed to describe them.
### 1.1 Requirements to a consistent theory,
and the history of simpler approaches
A bona fide theory of bodily tides implies (1) decomposition of the tide into
Fourier harmonic modes and (2) endowment of each separate Fourier mode with a
phase delay and a magnitude decrease of its own. The first part of this
programme, Fourier decomposition, was accomplished in full by Kaula (1964),
though a partial sum of the Fourier series was developed yet by Darwin (1879).
The second part of this programme, the quest for an adequate frequency
dependence of the phase lags and dynamical Love numbers, is now in progress.
While earlier attempts seldom went beyond the Maxwell model, more realistic
rheologies are now coming into use. A rheology combining the Andrade model at
higher frequencies with the Maxwell model at the lowest frequencies was
investigated by Efroimsky (2012a, 2012b).111 At lower frequencies, dissipation
is predominantly viscous, and the mantle is well approximated with the Maxwell
body. Its behaviour can be represented with a viscous damper and an elastic
spring connected in series. Experiencing the same force, these elements have
their elongations summed up. This illustrates how the total strain is
comprised by a sum of viscous and elastic contributions generated by the same
stress. At higher frequencies, the strain acquires the third component, one
intended to describe inelasticity. Inelasticity is produced by defect
unpinning, a process effective at frequencies higher than a certain threshold
(about 1 yr-1, in the case of Earth’s mantle, – see Karato & Spetzler 1990).
Hence, at frequencies above the threshold, dissipation is predominantly
inelastic, and the mantle behaves as the Andrade body. A combined rheological
model written down in Efroimsky (2012a, 2012b) embraces both frequency bands.
In Makarov et al. (2012) and Efroimsky (2012a), we mistermed the Andrade creep
as anelastic. It would be more accurate to call it inelastic, which means:
irrecoverable. (The term anelastic is applied to recoverable deformations,
like the Maxwell behaviour.) The necessity for such a combined model
originates from that fact that different physical mechanisms of friction
dominate tidal dissipation at different frequencies. Several other rheological
laws were probed by Henning et al. (2009) and Nimmo et al. (2012).
Some authors tried to sidestep a Fourier decomposition by building simpler toy
models which would preserve qualitative features of the consistent tidal
theory and, ideally, would yield some reasonable quantitative estimates. Two
radically simplistic ad hoc tools known as the constant geometric lag model
(MacDonald 1964, Goldreich 1966, Murray & Dermott 1999) and the constant time
lag model (Singer 1966; Mignard 1979, 1980, 1981; Heller et al. 2011; Hut
1981) are often resorted to, and are applied to rocky moons and planets and
gas giants and stars alike. Historically, both these approaches were
introduced for the ease of analytical treatment rather than on sound physical
principles. 222 Aside from its mathematical simplicity, the constant time lag
method is sometimes motivated by its analogy with the viscously damped
harmonic oscillator. This analogy, however, appeared in the literature a
posteriori, Alexander (1973) being the earliest work known to us where this
analogy was spelled out.
### 1.2 Plan
The first of the afore-said approaches, the constant geometric lag model ,
will be addressed in this paper. Our goal is to demonstrate that the model
should be discarded, both for physical and mathematical reasons. On the one
hand, the model, is not well grounded in the physical reality, because it
assumes a constant tidal response independent of the rotation frequency
everywhere except at the 1:1 resonance where it singularly changes sign. On
the other hand, the model is genuinely contradictive in its mathematics. The
source of the inherent conflict is the popular formula for the tidal torque
(and its analogue for the tidal potential) through which the model is
implemented. It turns out that these formulae tacitly imply constancy
(frequency-independence) of the time lag, a circumstance prohibiting the
additional imposition of the constant-geometric-lag ( = frequency-independent
quality factor) condition.
Prior to executing the plan, we shall provide a comprehensive, review-style
introduction into the methods of incorporation of friction into the tidal
theory. The review will then enable us to recognise the afore-mentioned
inconsistency in the the constant geometric lag model.
In the subsequent publication (Makarov & Efroimsky 2013), we shall explore the
constant time lag model. This is an approach implying that all the tidal
strain modes should experience the same temporal delay relative to the
appropriate modes comprising the tidal stress. While the method is often
assumed 333 It can be demonstrated that the purely viscous model implies a
frequency-independent time lag at sufficiently low frequencies only. Time lag
acquires frequency dependence at frequencies higher than
$\,G\rho^{2}R^{2}/\eta\,$, where $\,G$, $\,\rho$, $\,R$, and $\,\eta\,$ are
the Newton gravity constant, mean density, radius, and the mean viscosity of
the perturbed body. This circumstance lies outside the topic of this paper,
and we shall elaborate on it elsewhere. to work in the purely viscous limit
(which, hypothetically, may be the case of stars and gaseous planets – see Hut
1981 and Eggleton et al. 1998), there is no justification for using it for
terrestrial-type bodies such as the Moon, Phobos, or any exoplanet with a
rocky or partially molten mantle. In the light of the current rheological
knowledge, the tidal response is very different, and its frequency dependence
experiences especially steep variations in the vicinity of spin-orbit
resonances. In Makarov & Efroimsky (2013), we shall demonstrate that
illegitimate application of the constant time lag model to telluric objects
leads to nonexistent phenomena like pseudosynchronous rotation – not to
mention that it squeezes the tidal-evolution timescales (Efroimsky & Lainey
2007) and alters the probabilities of capture into resonances (Makarov,
Berghea, & Efroimsky 2012).
## 2 The constant-torque model
As we mentioned above, some authors tried to circumvent a consistent but
laborious treatment, by building simpler toy models. One such attempt was
undertaken by MacDonald (1964, eqn. 20) who assumed that the dynamical tide
mimics a static tide, except for being displaced by a geometric lag angle. The
same idea underlay the study, by Goldreich (1966), of a satellite on approach
to the 1:1 spin-orbit resonance. Thus, for mathematical convenience, both
authors set the geometric lag angle to be a frequency-independent constant.
The approach is known as the constant angular lag model. This name, however,
is inexact in the sense that, within the vicinity of the 1:1 spin-orbit
resonance, the lags and torque change their signs twice over a period: when
the bulge falls behind or advances (relative to the direction towards the
perturber), the sign is positive or negative, correspondingly. So in this
discussion the term constant should be understood as frequency-independent :
both the instantaneous phase lag and the instantaneous torque
$\,\vec{\cal{T}}\,$ are independent of the tidal frequency $\,\chi\,$.
Consequently, the orbit-average (secular) tidal torque
$\,\langle\,\vec{\cal{T}}\,\rangle\,$ is also frequency-independent.
Sometimes this approach shows up in the literature under the name of MacDonald
torque (e.g., Touma & Wisdom 1994, section 2.7.1).
Unfortunately, the approach turns out to be inconsistent and should be
discarded. Physically, the constant angular lag model looks suspicious from
the beginning, because in the vicinity of the 1:1 spin-orbit resonance it
permits for abrupt switches of the torque, i.e., for situations where the
torque changes its sign, retaining the absolute value. Although the abrupt
switch can be substituted, by hand, with a continuous transition, this ad hoc
alteration still would not save the method, because it would not heal a more
fundamental defect. Mathematically, the derivation of the formula for the
tidal torque within the said model contains a subtle and often unappreciated
detail: this derivation implicitly sets the time lag $\,\Delta t\,$ to be
constant, as we shall demonstrate below. However, it can be shown that the
assertion of the time lag being frequency independent is incompatible with the
assertion of the geometric lag angle being frequency independent. In this way,
the discussed simplified approach is inherently contradictive.
Another defect of this approach is that it employs such entities as the
instantaneous phase lag and the instantaneous quality factor, the latter being
introduced as the inverse sine of the former. The so-defined instantaneous
quality factor is not guaranteed to be related to the energy damping rate in
the same manner as the regular (appropriate to a fixed frequency) quality
factor is related to the dissipation rate at that frequency (Williams &
Efroimsky 2012). Were the quality factor frequency-insensitive, this would not
be a problem. However, the latter option is excluded within the discussed
model,444 Rejection of the constant angular lag model does not, by itself,
prohibit setting the quality factor constant, at least over some limited
interval of frequencies. While realistic mantles never behave like this, such
a rheology, in principle, is not impossible. However, employment of this
rheology will not be available within a simplified model. Instead, one will
have to attribute the same value to $\,k_{l}/Q_{l}\,$ at all tidal modes, and
then will have to insert this value of $\,k_{l}/Q_{l}\,$ into all terms of the
Fourier expansion of the tidal torque (the Darwin-Kaula series). Each such
term will change its sign when the corresponding resonance gets transcended.
Up to the late 60s of the past century, there was a consensus in the
geophysical community that the seismic $\,Q\,$ of rocks should be
“substantially independent of frequency” (Knopoff 1964). This viewpoint was
later disproved by a large volume of experimental data obtained both in the
laboratory and in the field (see, e.g., Karato 2007 and references therein).
as being incompatible with the constant-$\Delta t\,$ assumption tacitly
present.
## 3 Goldreich (1966), Kaula (1968), Murray & Dermott (1999)
Numerous authors offer the following expression for the polar component of the
torque wherewith the tide-raising perturber acts on the tidally-perturbed
body:
$\displaystyle{\cal{T}}_{z}~{}=~{}\frac{3}{2}~{}G\,M_{1}^{\,2}~{}\frac{R^{5}}{r^{6}}~{}k_{2}\,\sin
2\epsilon_{g}~{}~{},$ (1)
$R$ being the radius of the disturbed body, $\,M_{1}\,$ standing for the mass
of the tide-raising perturber, $\,r\,$ denoting the instantaneous distance
between the bodies, $\,\epsilon_{g}\,$ standing for the angular lag, and the
obliquity being set nil. The subscript $\,z\,$ serves the purpose of
emphasising that the above formula furnishes the torque component orthogonal
to the equator of the tidally perturbed body.
Goldreich (1966) denotes the angular lag with $\,\Psi\,$, Kaula (1968, eqn.
4.5.29) calls it $\,\delta\,$, while Murray & Dermott (1999, eqn. 4.159) use
the letter $\,\epsilon\,$. We shall follow the latter notation, though
equipping it with the subscript $\,g\,$ which means: geometric .
Below we shall provide a detailed derivation of this formula, and shall see
that the angle standing in it is indeed the instantaneous geometric tidal lag
angle
$\displaystyle\epsilon_{g}\,\equiv~{}(\dot{\nu}-\dot{\theta})\,\Delta t~{}\,,$
(2)
i.e., the instantaneous angular separation between the direction towards the
bulge and that towards the perturber. Here $\,\Delta t\,$ is the time lag,
$\,\nu\,$ is the true anomaly of the perturber, $\,\theta\,$ is the sidereal
angle of the perturbed body, and
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\;}}$ is its spin rate. It is
important to distinguish the instantaneous geometric lag $\,\epsilon_{g}\,$
from the instantaneous phase lag
$\displaystyle\epsilon_{ph}\,\equiv~{}2\,(\dot{\nu}-\dot{\theta})\,\Delta
t~{}=~{}2~{}\epsilon_{g}$ (3)
sometimes used in the literature (Efroimsky & Williams 2009, Williams &
Efroimsky 2012).
In Section 5 of Murray & Dermott (1999, eqns. 5.2 - 5.3), the authors rewrite
the expression for the torque, employing (in fact, implying) the above
expression of the lag through the true anomaly and spin rate:
$\displaystyle\sin
2\epsilon_{g}\,=\,\sin|2\epsilon_{g}|~{}\mbox{Sgn}\left(\epsilon_{g}\right)\,=~{}-~{}\sin|2(\dot{\theta}\,-\,\dot{\nu})\,\Delta
t|~{}\mbox{Sgn}(\,\dot{\theta}\,-\,\dot{\nu}\,)\,=~{}-~{}Q_{s}^{-1}~{}\mbox{Sgn}\left(\,\dot{\eta}\,-\,\dot{\varphi}\,\right)~{}\,.~{}$
(4)
The new angles showing up in this formula are $\,\eta\equiv\theta-{\cal{M}}\,$
and $\,\varphi\equiv\nu-{\cal{M}}\,$, with $\,{\cal{M}}\,$ denoting the mean
anomaly. These angles are depicted in Figure 5.1b in Ibid. Clearly, the time
derivatives $\,\dot{\eta}=\dot{\theta}-n\,$ and
$\,\dot{\varphi}=\dot{\nu}-n\,$ are the spin rate and true anomaly rate in a
frame which is centered on the tidally perturbed body 555 In Murray & Dermott
(1999), the role of a tidally perturbed body is played by the satellite, the
planet acting as its tide-raising perturber. In a different setting, perturbed
is the planet, the star or a satellite being the perturber. and is rotating
with the mean motion $\,n\,$. Interpreting the quantity
$\,Q_{s}\,=\,1/\sin|2(\dot{\theta}\,-\,\dot{\nu})\,\Delta t|$ as the
instantaneous quality factor, the authors assume that they can set it
frequency-independent (thus making the torque frequency-independent).
This approach contains two flaws. First, as explained in Williams & Efroimsky
(2012), it is not apparently evident whether the instantaneous quality factor
introduced as the inverse sine of the instantaneous phase lag has the physical
meaning usually instilled in a tidal dissipation factor at a certain
sinusoidal mode. Second, and most important, is that in reality it is
$\,\Delta t\,$ which gets implicitly set as frequency independent in the
derivation of (1). It then becomes impossible to assume that the geometric lag
$\,\epsilon_{g}\,$ also is frequency-independent – the two assumptions are
incompatible, as we shall see shortly. Consequently, setting the factor
$\,Q_{s}\,$ to be frequency independent is no longer an option. This makes the
entire constant geometric lag model or, to be exact, its implementation with
(1), inherently contradictive. Specifically, it is illegitimate to assert that
the tidal torque is proportional to Sgn$(\dot{\nu}-\dot{\theta})\,$.
## 4 Mathematical Introduction.
The tide-raising potential $\,W\,$ created by a perturber always changes the
shape and, as a result, the potential of the perturbed body. At the point
$\vec{R}$ of the perturbed body’s surface, the potential
$\,W(\mbox{{\boldmath$\vec{R}$}},\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$
created by the perturber residing at $\,\mbox{{\boldmath$\vec{r}$}}^{*}\,$ can
be expanded into a sum of terms
$\,W_{l}(\mbox{{\boldmath$\vec{R}$}},\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$
proportional to the Legendre polynomials $\,P_{\it l}(\cos\gamma)\;$. Here
$\,\gamma\,$ is the angle between the vectors
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ and $\vec{R}$ pointing from the
perturbed body’s centre towards the perturber and the point on the perturbed
body’s surface, where the potential $\,W\,$ is measured.
Tidal distortion of the body’s geometric form renders an addition
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$ to the body’s potential in an exterior
point $\vec{r}$ . This addition turns out to be comprised of terms
$\,U_{l}(\mbox{{\boldmath$\vec{r}$}})\,$ each of which is proportional to the
term
$\,W_{l}(\mbox{{\boldmath$\vec{R}$}},\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$,
with the surface point $\vec{R}$ located exactly below (i.e., having the same
latitude and longitude as) the exterior point
$\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}\,$.
The goal of this section is to provide a squeezed introduction into this
formalism and to explain how it should be generalised from a static
configuration setting to a dynamical setting.
### 4.1 Static tides
Let a body of radius $R$ experience tides from a perturber of mass
$\,M^{*}_{1}\,$ placed at
$\,{\mbox{{\boldmath$\vec{r}$}}}^{\;*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$,
with $\,r^{*}\geq R\,$.
At a point $\,\mbox{{\boldmath$\vec{R}$}}=(R,\phi,\lambda)\,$ on the perturbed
body’s surface, the potential $\,W\,$ due to the perturber is expanded over
the Legendre polynomials $\,P_{\it l}(\cos\gamma)\;$ as 666 Summation in
formula (5) goes over $\,l\geq 2\,$. The central term ($\,l=0\,$) is regarded
as the principal, Newtonian, part of the potential generated by the perturber,
and not as a part of the perturbation $\,W\,$ caused by the finite size of the
tidally perturbed body – indeed, the $\,l=0\,$ terms bears no dependence upon
$\vec{R}$ . The reason why the $\,l=1\,$ terms falls out is more subtle and is
related to the fact that we are developing our formalism in the frame of the
tidally perturbed body, not in an inertial frame. See, e.g., Efroimsky &
Williams (2009, eqns. 5 - 11).
$\displaystyle
W(\mbox{{\boldmath$\vec{R}$}}\,,\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})$
$\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}~{}W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,~{}\mbox{{\boldmath$\vec{r}$}}^{~{}*})~{}=~{}-~{}\frac{G\;M^{*}_{1}}{r^{\,*}}~{}\sum_{{\it{l}}=2}^{\infty}\,\left(\,\frac{R}{r^{~{}*}}\,\right)^{\textstyle{{}^{\it{l}}}}\,P_{\it{l}}(\cos\gamma)~{}~{}~{}~{}$
(5) $\displaystyle=$
$\displaystyle-\,\frac{G~{}M^{*}_{1}}{r^{\,*}}\sum_{{\it{l}}=2}^{\infty}\left(\frac{R}{r^{~{}*}}\right)^{\textstyle{{}^{\it{l}}}}\sum_{m=0}^{\it
l}\frac{({\it l}-m)!}{({\it
l}+m)!}(2-\delta_{0m})P_{{\it{l}}m}(\sin\phi)P_{{\it{l}}m}(\sin\phi^{*})~{}\cos
m(\lambda-\lambda^{*})~{}~{},\quad\,\quad$
where $\,\delta_{ij}\,$ is the Kronecker delta symbol, $\,G\,$ is Newton’s
gravity constant, while $\,\gamma\,$ denotes the angular separation between
the vectors $\,{\mbox{{\boldmath$\vec{r}$}}}^{\;*}\,$ and $\vec{R}$ pointing
from the centre of the perturbed body. The longitudes $\lambda,\,\lambda^{*}$
are reckoned from a fixed meridian on the perturbed body, the latitudes
$\phi,\,\phi^{*}$ being reckoned from the equator. The integers $\,l\,$ and
$\,m\,$ are called the degree and order, accordingly. The associated Legendre
functions $\,P_{lm}(x)\,$ are referred to as the associated Legendre
polynomials when their argument is sine or cosine of some angle.
The $\,{\emph{l}}^{~{}th}\,$ term
$\,W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,~{}\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$
of the perturber’s potential introduces a distortion into the perturbed body’s
shape, assumed to be linear. Then the ensuing $\,{\emph{l}}^{~{}th}$ amendment
$\,U_{\it{l}}\,$ to the perturbed body’s potential will also be linear in
$\,W_{\it{l}}\,$. Since $\,U_{\it{l}}(\mbox{{\boldmath$\vec{r}$}})\,$ falls
off outside the body as $\,r^{-(\it{l}+1)}\,$, the overall change in the
exterior potential of the perturbed body will be:
$\displaystyle U(\mbox{{\boldmath$\vec{r}$}})~{}=~{}\sum_{{\it
l}=2}^{\infty}~{}U_{\it{l}}(\mbox{{\boldmath$\vec{r}$}})~{}=~{}\sum_{{\it
l}=2}^{\infty}~{}k_{\it l}\;\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(6)
where $\,k_{l}\,$ are the static Love numbers, $\,R\,$ is the mean equatorial
radius of the perturbed body, while
$\,\mbox{{\boldmath$\vec{R}$}}=(R,\,\phi,\,\lambda)\,$ and
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ are a surface point and
an exterior point above it, respectively, so that $\,r\geq R\,$.
Combining (6) with (5), we arrive at a useful formula for the amendment to the
potential of the tidally disturbed body:
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;\,-\,{G\;M^{*}_{1}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}\frac{({\it l}-m)!}{({\it
l}+m)!}(2-\delta_{0m})P_{{\it{l}}m}(\sin\phi)P_{{\it{l}}m}(\sin\phi^{*})\;\cos
m(\lambda-\lambda^{*})~{}~{}.~{}~{}~{}$ (7)
This is how the tidally generated change in the perturbed body’s potential is
“felt” at a point $\vec{r}$ . The change is expressed as a function of the
spherical coordinates $\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$
of this point and the spherical coordinates
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$ of
the tide-raising body. The formula may be employed when we have two exterior
bodies: if one such body, a perturber of mass $\,M^{*}_{1}\,$ located at
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ produces tides on the perturbed body,
then the other exterior body, located at $\vec{r}$ , will experience a
potential perturbation (7) due to these tides.
By changing variables from the spherical coordinates
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$ and
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ to the Keplerian
coordinates $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(\,a^{*},\,e^{*},\,{\it
i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal M}^{*}\,)\,$ and
$\,\mbox{{\boldmath$\vec{r}$}}=(\,a,\,e,\,{\it i},\,\Omega,\,\omega,\,{\cal
M}\,)\,$, one obtains a formula equivalent to (7):
$\displaystyle U(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;\sum_{{\it
l}=2}^{\infty}\;k_{\it l}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{{\it
l}+1}}}\frac{G\,M^{*}_{1}}{a^{*}}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(8) $\displaystyle~{}~{}~{}~{}\left.-\,\delta_{0m}\right)\sum_{p=0}^{\it
l}F_{{\it l}mp}({\it i}^{*})\sum_{q=-\infty}^{\infty}G_{{\it
l}pq}(e^{*})\sum_{h=0}^{\it l}F_{lmh}({\it
i})\sum_{j=-\infty}^{\infty}G_{lhj}(e)\;\cos\left(\,\left(v_{{\it
l}mpq}^{*}-m\theta^{*}\right)-\left(v_{{\it
l}mhj}-m\theta\right)\,\right)\,~{}_{\textstyle{{}_{\textstyle,}}}$
where
$\displaystyle v_{{\it l}mpq}^{*}\;\equiv\;({\it l}-2p)\,\omega^{*}\,+\,({\it
l}-2p+q){\cal M}^{*}\,+\,m\,\Omega^{*}~{}~{}~{},$ (9) $\displaystyle v_{{\it
l}mhj}\;\equiv\;({\it l}-2h)\,\omega\,+\,({\it l}-2h+j){\cal
M}\,+\,m\,\Omega~{}~{}~{},$ (10)
$\,q\,$ and $\,j\,$ being arbitrary integers, $\,p\,$ and $\,h\,$ beng
arbitrary nonnegative integers, $\,F_{lmp}({\it i})\,$ being the inclination
functions, while $\,G_{lpq}(e)$ being the eccentricity polynomials coinciding
with the Hansen coefficients
$\,X^{\textstyle{{}^{(-l-1),\,(l-2p)}}}_{\textstyle{{}_{(l-2p+q)}}}(e)\,$.
Also mind that $\,\theta^{*}\,$ is the same as $\,\theta\,$, which is the
sidereal angle of the tidally perturbed body. Following Kaula (1964), we equip
$\,\theta\,$ with an asterisk, when it shows up in expressions corresponding
to the tide-raising body. In expression (8), the terms $~{}-\,m\theta^{*}\,$
and $~{}-\,m\theta\,$ cancel one another, wherefore their presence may seem
redundant. We better keep them, though, for they will no longer cancel when
lagging comes into play.
Decomposition (8) was pioneered by Kaula (1961, 1964). However, its partial
sum, with $\,|{\it{l}}|,\,|q|,\,|j|\,\leq\,2\,$, was derived yet by Darwin
(1879). In modern notations, Darwin’s work is discussed by Ferraz-Mello,
Rodríguez & Hussmann (2008). 777 Be mindful that the convention on the meaning
of notations $\vec{r}$ and $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ in Ibid. is
opposite to ours.
For our further developments, it would be important to emphasise that formulae
(7) and (8) are equivalent to one another, because the latter is obtained from
the former simply by a change of variables.
### 4.2 Dynamical tides with no friction
Derived for a static tide, formulae (7) and (8) extend trivially to an elastic
dynamical setting where the tide adjusts instantaneously to the changing
position $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$ of the perturber.
The key point is that, to get $\,U(\mbox{{\boldmath$\vec{r}$}})\,$ at the
point $\vec{r}$ at time $\,t\,$, we insert into (7) or (8) the perturber’s
position $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$ taken at that same time
$\,t\,$, and not at an earlier time. Formulae (7) and (8) stay equivalent to
one another, and remain unchanged, except that the distances, the sidereal
angle, and the angular coordinates acquire a simultaneous time dependence: 888
The orbital parameters $\,a\,$, $\,e\,$, $\,i\,$, $\,a^{*}\,$, $\,e^{*}\,$,
$\,i^{*}\,$ acquire no time dependence, insofar as the apsidal and nodal
precession remain the only permitted variations of the orbits. $\,r\,$ becomes
$\,r(t)\,$, $\,r^{\,*}\,$ becomes $\,r^{\,*}(t)\,$; while $\,\theta\,$,
$\,\theta^{*}\,$, $\,\phi\,$, $\,\phi^{*}\,$, $\,\lambda\,$,
$\,\lambda^{*}\,$, $\,\omega\,$, $\,\omega^{*}\,$, $\,\Omega\,$,
$\,\Omega^{*}\,$, $\,{\cal{M}}\,$, $\,{\cal{M}}^{\,*}\,$ become
$\,\theta(t)\,$, $\,\theta^{*}(t)\,$, $\,\phi(t)\,$, $\,\phi^{*}(t)\,$,
$\,\lambda(t)\,$, $\,\lambda^{*}(t)\,$, $\,\omega(t)\,$, $\,\omega^{*}(t)\,$,
$\,\Omega(t)\,$, $\,\Omega^{*}(t)\,$, $\,{\cal{M}}(t)\,$,
$\,{\cal{M}}^{\,*}(t)\,$.
Thus, to obtain $\,U(\mbox{{\boldmath$\vec{r}$}}(t)\,)\,$, we take the values
of all variables at time $\,t\,$, leaving no place for any lagging. This is
possible only for an absolutely elastic, i.e., frictionless perturbed body.
### 4.3 Tidal modes and forcing frequencies
Let us now write down the modes over which the tidal disturbance of the body
gets expanded. We begin with expression (5) for the perturbing potential at a
fixed point $\,\mbox{{\boldmath$\vec{R}$}}=(R,\,\phi,\,\lambda)\,$ on the
surface of the perturbed body. Using the technique developed by Kaula (1961,
1964), we change the coordinates of the tide-raising body from
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$ to
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(\,a^{*},\,e^{*},\,{\it
i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal M}^{*}\,)\,$. However, the location
on the body’s surface, where the disturbance is observed, is still
parameterised with its spherical coordinates
$\,\mbox{{\boldmath$\vec{R}$}}=(R,\,\phi,\,\lambda)~{}$:
$\displaystyle
W(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.~{}~{}~{}-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})\left\\{\begin{array}[]{c}\cos\\\ \sin\end{array}\right\\}^{{\it
l}\,-\,m\;\;\mbox{\small even}}_{{\it l}\,-\,m\;\;\mbox{\small
odd}}\;\left(v_{{\it
l}mpq}^{*}-m(\lambda+\theta^{*})\right)~{}~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(13)
Since $\,R\,$ is the radius of the tidally perturbed body, and since the
latitude $\,\phi\,$ and the longitude $\,\lambda\,$ are reckoned from the
equator and a fixed meridian, correspondingly, then (13) is just another
expression for the perturbing potential at the fixed point
$\,(R,\,\phi,\,\lambda)\,$ of the body’s surface. In (13), the expression in
round brackets can be reshaped as
$\displaystyle v_{{\it l}mpq}^{*}-m(\lambda+\theta^{*})\,=\,\omega_{{\it
l}mpq}\,(t\,-\,t_{0})~{}-~{}m~{}\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})~{}~{},$
(14)
where
$\displaystyle\omega_{lmpq}\;\equiv\;({\it l}-2p)\;\dot{\omega}^{*}\,+\,({\it
l}-2p+q)\;n^{*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})\,~{}.~{}~{}~{}$
(15)
Here $\,n^{*}\,\equiv\,{\bf{\dot{\cal{M}}}}^{\,*}\,$ is the mean motion of the
perturber, while $\,t_{0}\,$ is the time of perigee passage wherefrom the mean
anomaly $\,{\cal{M}}^{\,*}\,$ of the perturber is reckoned.
We see from (14) that the quantities $\,\omega_{lmpq}\,$ given by (15) are the
Fourier modes over which the tidal perturbation (13) is expanded. While these
modes can be positive or negative, the physical forcing frequencies,
$\displaystyle\chi_{lmpq}~{}\equiv~{}|\,\omega_{lmpq}\,|\,~{},~{}~{}~{}$ (16)
at which the stress oscillates, are positive-definite.
Having developed formulae (8 \- 13), Kaula (1961, 1964) never stipulated 999
The linear combination standing on the right-hand side of our formula (15)
appeared in the denominator of formulae (29) and (50) in Kaula (1961).
However, Kaula did not mention that this combination is a Fourier mode of the
tide. that the Fourier modes of the tide are given by (15). Possibly, he was
not interested in the frequency dependence of the phase or time lags.
In Section 6 of his book, Lambeck (1980) explained some aspects of Kaula’s
theory. While Lambeck’s equation (6.1.13b) indicates that Lambeck could be
aware of how the Fourier modes look, he too never wrote down the formula for
the modes explicitly. Perhaps, like Kaula, Lambeck had no interest in the
frequency dependence of lags – he just introduced a time lag $\,\Delta t\,$,
which in his developments was implicitly regarded frequency independent.
While in the review by Efroimsky & Williams (2009) and in Efroimsky (2012a,
2012b) the expression for $\,\omega_{lmpq}\,$ was written down explicitly, its
derivation was omitted.
Therefore, in the literature of which we are aware, the formula for the
Fourier modes either was implied tacitly or was employed with no proof. This
was our motivation to derive it here in such detail.
To conclude, at the point
$\,\mbox{{\boldmath$\vec{R}$}}=(R,\,\phi,\,\lambda)\,$ of the surface of the
perturbed body, the perturbing potential is expressed through the tidal
Fourier modes as:
$\displaystyle
W(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})\left\\{\begin{array}[]{c}\cos\\\ \sin\end{array}\right\\}^{{\it
l}\,-\,m\;\;\mbox{\small even}}_{{\it l}\,-\,m\;\;\mbox{\small
odd}}\;\left(\,\omega_{lmpq}\,(t\,-\,t_{0})\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}\,~{},~{}~{}~{}~{}~{}~{}$
(19)
the tidal mode being given by (15). In an idealised situation, when the
extended body is frictionless and its response is instantaneous, we can employ
the static formula (6), as explained in subsection 4.2. Combining that formula
with expression (19), we see that the additional tidal potential generated by
a perfectly elastic body at the point
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ right above $\vec{R}$
will now read as:
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})~{}~{}~{}~{}$
$\displaystyle\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})~{}k_{l}~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small even}}_{{\it
l}\,-\,m\;\;\mbox{\small
odd}}\;\left(\,\omega_{lmpq}\,(t\,-\,t_{0})\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}~{}.~{}~{}~{}~{}$
(22)
It is due to the lack of friction that the $\,lmpq\,$ term of (22) is in phase
with the $\,lmpq\,$ term of (19). Below we shall see that inclusion of
friction into the picture renders a phase shift between these terms. It is
also in anticipation of the discussion of friction that we placed the Love
numbers inside the $~{}\sum_{mpq}~{}$ sum in expression (22). Mode-independent
in the perfectly elastic case, the Love numbers may acquire dependence upon
the Fourier modes $\,\omega_{lmpq}\,$, because friction may mitigate
amplitudes of distortion differently at different frequencies.
## 5 Friction and Lagging
In this section, we shall trace, step by step, how internal friction gets
included into the tidal theory. In subsection 4.2, we made an observation
that, for an absolutely elastic (frictionless) body, treatment of dynamical
tides mimics that of static tides, except that all coordinates acquire time-
dependence. Our next step will be to incorporate friction, and therefore
lagging, into the picture.
As a first step, we shall address a simplistic method implying that the
coordinates of the tide-raising body (as seen in a frame corotating with the
perturbed body) get shifted back in time by some fixed time lag $\,\Delta
t\,$. Although implementations of this method into formulae (7) and (8) look
very different, they render results which in fact are equivalent – simply
because (7) and (8) are equivalent, and because the same procedure (shift by
$\,\Delta t\,$) is performed on the quantities with asterisks in both these
formulae.
However, the difference in the mathematical form of these, equivalent, results
also prompts a more consistent way of taking care of friction. This, more
advanced, method will be implementable only in formula (8) and not in (7). The
method is the one used by Kaula (1964). Since the explanation of the method in
Ibid. was extremely concise, we shall elucidate it here in mode detail.
### 5.1 A naive way of bringing in lagging
Naively, dissipation and the ensuing lagging can be included into the picture
by assuming that the exterior body located at point $\vec{r}$ at time $\,t\,$
is subject not to the tidal potential created simultaneously by the perturber
residing at $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$, but to the potential
generated by the perturber lagging in time on its orbit 101010 Here we imply:
on its orbit as seen from the perturbed extended body. The caveat is needed,
since we are considering the physical reaction of the extended body, and thus
are interested in the location of the perturber relative to its surface, and
not to an inertial frame. It is for this reason that in equation (23) we
employ the latitudes and longitudes defined in a frame corotating with the
perturbed extended body. In an inertial frame, a shifting of the perturber
back by $\,\Delta t\,$ should then be accompanied by a shift of the
orientation of the extended body back by the same $\,\Delta t\,$; and this is
why we have $\,\theta^{*}(t-\Delta t)\,$ in equation (24). by some $\,\Delta
t\,$. Speaking loosely, the no-asterisk exterior body located at
$\,\mbox{{\boldmath$\vec{r}$}}(t)\,$ “feels” the tide given by (7) or (8), as
if the tide were generated by the asterisk perturber located on its orbit not
at $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$ but at
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t\,-\,\Delta t)\,$.
Mathematically, this implies that the tidal potential
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$ at $\,t\,$ must be calculated via (7) or
(8), by insertion of the no-asterisk coordinates taken at $\,t\,$, and the
coordinates with asterisk taken at $\,t\,-\,\Delta t\,$. The naive strategy
also implies that the Love numbers $\,k_{l}\,$ keep their static values,
though this detail is seldom spelled out.
This approach implemented, our formulae (7) and (8) will acquire the following
form:
$\displaystyle U(\,\mbox{{\boldmath$\vec{r}$}}(t)\,)\,=$ $\displaystyle-$
$\displaystyle{G\;M^{*}_{1}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r(t)^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}(t-\Delta
t)}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}\frac{\left(l-m\right)!}{({\it l}+m)!}\left(2\right.$ (23)
$\displaystyle~{}\left.-\,\delta_{0m}\right)P_{{\it{l}}m}(\,\sin\phi(t)\,)P_{{\it{l}}m}\left(\,\sin\phi^{*}\left(t-\Delta
t\right)\,\right)\;\cos m[\lambda(t)-\lambda^{*}(t-\Delta
t)]~{}\,~{}.~{}\quad~{}\quad~{}$
and
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;\sum_{l=2}^{\infty}\;k_{\it
l}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{{\it
l}+1}}}\frac{G\,M^{*}_{1}}{a^{*}}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{l}}}\sum_{m=0}^{\it
l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}\left.-\,\delta_{0m}\right)\sum_{p=0}^{l}F_{lmp}({\it
i}^{*})\sum_{q=-\infty}^{\infty}G_{lpq}(e^{*})\sum_{h=0}^{\it l}F_{lmh}({\it
i})\sum_{j=-\infty}^{\infty}G_{lhj}(e)~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\cos\left(\,\left[v_{{\it l}mpq}^{*}(t-\Delta
t)-\,m\theta^{*}(t-\Delta t)\,\right]-\left[v_{{\it
l}mhj}(t)-m\theta(t)\,\right]\,\right)\,~{}_{\textstyle{{}_{\textstyle.}}}$
(24)
Just as their static precursors (7) and (8), our dynamical formulae (23) and
(24) remain equivalent to one another. They still reflect a mere switch from
the spherical to the Keplerian coordinates, 111111 En route from (23) to
(24), one changes not only a coordinate system but also a frame of reference.
Our longitude $\,\lambda\,$ being reckoned from a meridian, the switch goes
from the corotating coordinates $\,(r,\,\phi,\,\lambda)\,$ to the Kepler
coordinates $(a,\,e,\,i,\,\Omega,\,\omega,\,{\cal{M}})$ defined in a frame
comoving but not corotating with the perturbed body. Technically, one first
substitutes $\,\lambda\,$ with $\,\tilde{\lambda}-\theta\,$, where
$\,\tilde{\lambda}=\lambda+\theta\,$ is the longitude in the comoving (not
corotating) frame. Then one can resort to the standard formulae connecting the
spherical and Kepler coordinates in the same frame. The formulae apply not to
$\,(r,\,\phi,\,\lambda)\,$ but to $\,(r,\,\phi,\,\tilde{\lambda})\,$, see
Kaula (1961). Thus, the current spin rate $\,\theta(t)\,$ pops up in (24) due
to the transition from a corotating frame to a comoving one. All said relates
equally to both the spherical and Kepler coordinates with asterisks. So the
delayed value $\,\theta^{*}(t-\Delta t)\,$, too, emerges in (24) due to the
frame switch. Recall that $\,\theta^{*}\,$ is the same spin rate as
$\,\theta\,$, except that it gets equipped with an asterisk, when it stands in
expressions corresponding to the perturber. Also recall that, within the
described approach, we model friction by simply shifting the perturber (as
seen in a frame corotating with the perturbed body) back in time by $\Delta
t$. In a frame which is comoving but not corotating, this implies not only
pulling the perturber back by $\,\Delta t\,$ but also rotating the perturbed
body back by $\,\dot{\theta}\,\Delta t\,$. Leaving the coordinates
$\,(r,\,\phi,\,{\lambda})\,$ untouched, and changing only
$\,(r^{\,*},\,\phi^{\,*},\,\lambda^{\,*})\,$ to
$\,(a^{\,*},\,e^{\,*},\,i^{\,*},\,\Omega^{\,*},\,\omega^{\,*},\,{\cal{M}}^{\,*})\,$,
one arrives at (13) and then at (19 \- 22). Applying this machinery also to
the variables with no asterisk, one ends up with (24). except that now
$\vec{r}$ is taken at time $\,t\,$, while
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ is taken at $\,t-\Delta t\,$.
The longitude reckoned from a fixed meridian on the perturbed body is
expressed through the true anomaly $\,\nu\,$, the periapse $\,\omega\,$ and
the node $\,\Omega\,$ as
$\displaystyle\lambda\,=\,-\,\theta\,+\,\Omega\,+\,\omega\,+\,\nu\,+\,O({\it
i}^{2})~{}\,.$
In neglect of the nodal and the apsidal precession, and for a small obliquity,
this results in
$\displaystyle\dot{\lambda}\,\approx\,-\,\dot{\theta}~{}+~{}\dot{\nu}\,+\,O({\it
i}^{2})~{}\,.$
In (23), the argument of cosine may now be written, in a linear approximation
over $\,\Delta t\,$, as
$\displaystyle m\,\left[\,\lambda(t)\,-\,\lambda^{*}(t-\Delta
t)\,\right]\,=\,m\,\left[\,\lambda(t)\,-\,\lambda^{*}(t)\,+\,\dot{\lambda}^{*}\,\Delta
t\,\right]\,=\,m\,\left(\,\lambda\,-\,\lambda^{*}\,\right)\,-\,m\,(\,\dot{\theta}\,-\,\dot{\nu}\,)\,\Delta
t~{}+~{}O({\it i}^{2})~{}\,.~{}\,~{}\,$ (25)
In (24), the argument of cosine may be shaped, in a linear approximation over
$\,\Delta t\,$, as
$\displaystyle\left[v_{{\it l}mpq}^{*}(t-\Delta t)-\,m\,\theta^{*}(t-\Delta
t)\,\right]-\left[v_{{\it l}mhj}(t)-m\,\theta(t)\,\right]\,~{}=$
$\displaystyle~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\left[v_{{\it
l}mpq}^{*}(t)-\,m\,\theta^{*}(t)\,\right]-\left[v_{{\it
l}mhj}(t)-m\,\theta(t)\,\right]\,~{}-~{}\epsilon_{lmpq}$
where
$\displaystyle\epsilon_{lmpq}$ $\displaystyle\equiv$
$\displaystyle\left[\dot{v}_{{\it
l}mpq}^{*}(t)-\,m\,\dot{\theta}^{*}(t)\right]\,\Delta t\,=\,\left[({\it
l}-2p)\;\dot{\omega}^{*}\,+\,({\it
l}-2p+q)\;n^{*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})\right]\,\Delta
t$ (27) $\displaystyle=$ $\displaystyle\omega_{lmpq}~{}\Delta t$
is the phase lag corresponding to the mode $\,\omega_{lmpq}\,$.
From here we observe that the above-chosen method of taking the tidal friction
into account fixes the phase lags in a very specific way: through shifting the
perturber back on its orbit by a fixed time $\,\Delta t\,$, we set the phase
lags (27) to be proportional to this $\,\Delta t\,$. It should be emphasised
once again that the shift is performed in a frame corotating with the
perturbed body. In a frame comoving but not corotating with it, the shift will
thus be accompanied by rotation of the perturbed body back by
$\,\dot{\theta}\Delta t\,$, hence the term $\,-\,\dot{\theta}\,\Delta t~{}$ in
the expression (21) for the phase lag.
Our formulae (23) and (24) can be written in another, equivalent form:
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2~{}-\;\delta_{0m}\,\right)\;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it l}\;F_{{\it l}mp}({\it
i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})~{}k_{l}~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small even}}_{{\it
l}\,-\,m\;\;\mbox{\small odd}}\;\left(\,\omega_{lmpq}\,(t\,-\,\Delta
t\,-\,t_{0})\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}~{}.~{}~{}~{}~{}$
(30)
This form is analogous to (22), except for the time lag $\,\Delta t\,$, the
same for each Fourier mode.
Expression (30) is equivalent to expression (23), and is obtained from it by a
switch from the spherical coordinates
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$ to
the Kepler elements
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(\,a^{*},\,e^{*},\,{\it
i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal M}^{*}\,)\,$, with the variables
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ kept unchanged. From
the equivalence of expressions (30) and (23), we observe that, after the same
time lag is added into all terms of (30), all the so-shifted terms add up to
the equilibrium bulge geometrically displaced in such a way as if it were a
static bulge generated by the perturber at a slightly different time. No other
rheology can make this claim because, more generally, the lag in each term in
the expansion would correspond to its own increment in $\,t\,$. This tells us
that by setting $\,\Delta t\,$ frequency-independent we impose a highly
restrictive rheological rule, obedience to which cannot be expected of
realistic mantles.
A more profound problem of this approach lies in the fact that it is
illegitimate to introduce lags, keeping at the same time the Love numbers
unchanged. Mitigation of the magnitude and lagging of the phase are
inseparably connected, though the link becomes apparent only within a
consistent approach based on the Fourier expansion of the tide and on
employment of one or another rheological law. That law will then define both
lagging in phase and reduction in magnitude.
### 5.2 A consistent way of bringing in lagging (Kaula 1964)
The above expression (27) for phase lags contains in itself an obvious hint on
how a general-type rheology should be built into the tidal theory – to that
end, one simply has to endow each mode $\,\omega_{lmpq}\,$ with a time lag
$\,\Delta t_{l}(\omega_{lmpq})\,$ of its own. Another adjustment is the mode
dependence of the Love numbers: $\,k_{l}\,=\,k_{l}(\omega_{lmpq})\,$.
Expression (24) will now become
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;\sum_{l=2}^{\infty}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{{\it
l}+1}}}\frac{G\,M^{*}_{1}}{a^{*}}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{l}}}\sum_{m=0}^{\it
l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}\left.-\,\delta_{0m}\right)\sum_{p=0}^{l}F_{lmp}({\it
i}^{*})\sum_{q=-\infty}^{\infty}G_{lpq}(e^{*})\sum_{h=0}^{\it l}F_{lmh}({\it
i})\sum_{j=-\infty}^{\infty}G_{lhj}(e)~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle
k_{\it l}(\omega_{lmpq})\;\cos\left(\,\left[v_{{\it
l}mpq}^{*}(t)-\,m\theta^{*}(t)\,\right]-\left[v_{{\it
l}mhj}(t)-m\theta(t)\,\right]\,-\,\epsilon_{l}(\omega_{lmpq})\,\right)\,~{}_{\textstyle{{}_{\textstyle,}}}~{}~{}~{}~{}$
(31)
where
$\displaystyle\epsilon_{l}(\omega_{lmpq})\,\equiv\,\omega_{lmpq}\,\Delta
t_{l}(\omega_{lmpq})\,=\,\left[\,({\it l}-2p)\;\dot{\omega}^{*}\,+\,({\it
l}-2p+q)\;n^{*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})\,\right]\,\Delta
t_{l}(\omega_{lmpq})~{}\,.~{}\,~{}\,~{}\,~{}$ (32)
In (31 \- 32), we prefer to denote the phase and time lags not as
$\,\epsilon_{lmpq}\,$ and $\,\Delta t_{lmpq}\,$, but as
$\,\epsilon_{l}(\omega_{lmpq})\,$ and $\,\Delta t_{l}(\omega_{lmpq})\,$.
Indeed, their dependence on the indices $\,mpq\,$ is solely due to the
argument $\,\omega_{lmpq}~{}$. However, we cannot strip $\,\epsilon\,$ or
$\,\Delta t\,$ of the subscript $\,l\,$, because the functional form of the
frequency dependence of the phase or time lag is different for different
$\,l\,$s. The tidal friction is not the same as the seismic friction, and the
degree $\,l\,$ affects the tidal damping rate.121212 The difference between
the tidal and seismic friction and, accordingly, the difference of dissipation
at different $\,l\,$s is unimportant in small bodies, where only the rheology
matters. However things change in large planets where self-gravitation becomes
a crucial factor in tidal friction (Efroimsky 2012a). Hence in formulae (31 \-
32) we have $\,\epsilon_{l}\,$ and $\,\Delta t_{l}\,$, and not just
$\,\epsilon\,$ or $\,\Delta t\,$.
We would also write down the expression for the tidal potential in terms of
the Keplerian elements of the perturber and the spherical coordinates of the
point where this potential is observed:
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2~{}-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)~{}\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})~{}~{}~{}~{}~{}$ $\displaystyle
k_{l}(\omega_{lmpq})~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small even}}_{{\it
l}\,-\,m\;\;\mbox{\small odd}}\;\left(\,\omega_{lmpq}\,[\,t\,-\,\Delta
t_{l}(\omega_{lmpq})\,-\,t_{0}\,]\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$
(35)
This form is analogous to (22) and (30), except for two details. First, each
mode $\,\omega_{lmpq}\,$ now has a time lag $\,\Delta t_{l}=\,\Delta
t_{l}(\omega_{lmpq})\,$ of its own. Likewise, the dynamical Love number at
each Fourier mode is a function of this mode:
$\,k_{l}=\,k_{l}(\omega_{lmpq})\,$.
The reason why we wrote down
$\,U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\,$ in
the above form is that it immediately furnishes an expression for the
geometric lag angle of an arbitrary $\,lmpq\,$ bulge:
$\displaystyle\delta_{lmpq}\,=~{}\frac{\omega_{lmpq}}{m}~{}\Delta
t_{l}(\omega_{lmpq})\,~{}.$ (36)
For example, the geometric lag angle of the principal, semidiurnal bulge is
$\,\delta_{2200}=\,\frac{\textstyle\omega_{2200}}{\textstyle 2}~{}\Delta
t_{2}\,=\,(n-\dot{\theta})\,\Delta t_{2}\,$, where the time lag is taken at
the appropriate, semidiurnal mode: $\,\Delta t_{2}=\Delta
t_{2}(\omega_{2200})\,$.
It is customary to introduce the convention that the phase and time lags and
the Love numbers are functions not of the tidal mode $\,\omega_{lmpq}\,$ but
of the positively defined frequency
$\,\chi_{lmpq}\,\equiv\,|\,\omega_{lmpq}\,|\,$. Simplifying some calculations,
this convention makes it necessary to introduce, by hand, sign factors into
the terms of the Fourier expansions of the tidal force and torque (Efroimsky
2012b).
In practical applications, most important is the special case when the
exterior body located at $\vec{r}$ coincides with the tide-raising perturber
located at $\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}\,$. In this situation, the
perturber is acting upon itself through the tide it creates on the perturbed
body. Keeping the phase lags intact, and identifying
$\,\mbox{{\boldmath$\vec{r}$}}(t)\,$ with
$\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}(t)\,$, we may be tempted to mis-assume
that the expression with no asterisk, standing in the argument of the cosine
in (31), compensates the expression with an asterisk, so the phase lag becomes
all that is left. However, this would only furnish us the secular term of the
tidal potential $\,U(\mbox{{\boldmath$\vec{r}$}})\,$ wherewith the perturber
acts upon itself through the medium of the tidally perturbed body. This term
is proportional to $\,\cos\epsilon_{lmpq}\,$. As the $\,p,\,q\,$ and
$\,h,\,j\,$ are independent pairs of integers, there also will be
contributions with $\,\\{p,\,q\\}\neq\\{h,\,j\\}\,$. These are the oscillating
components of the tidal potential $\,U(\mbox{{\boldmath$\vec{r}$}})\,$.
Although their time average is nil, they do contribute to the heat production,
while the appropriate oscillating components of the tidal torque may influence
free librations. The topic was first addressed in Efroimsky (2012b), and was
revisited by Makarov et al. (2012) who explored whether the oscillating part
of the torque can influence capture into spin-orbit resonances. It has turned
out that, naturally, the oscillating part of the torque alters the outcome of
a particular realisation of the capture scenario, but leaves the statistics
unchanged.
## 6 Calculation of the tidal torque
We would begin with emphasising a key circumstance concerning the two forms of
expansion of the tidal potential. In the absence of friction, these
expansions, (7) and (8), were equivalent. Their amended versions, (23) and
(24), remain equivalent in the presence of friction, provided the latter obeys
a special (often unphysical) restriction that the time lag $\,\Delta t\,$ is
the same at all frequencies. Beyond that threshold, the equivalence of the two
expansions failed to extend. Since expression (23) does not contain the
Fourier modes in it, it is plainly impossible to write (23) in a form that
takes into account different time lags at different frequencies. Therefore,
any calculation based on (23) will unavoidably imply the frequency-
independence of $\,\Delta t\,$ and will thus be incompatible with any other
rheological law.
### 6.1 Tidal torque, as derived from the concise expression (23)
Let us calculate the tidal torque, using (23). By employing this expression,
we automatically set the rheology to be $\,\Delta t=\,$const .
Consider an exterior body of mass $\,M_{1}\,$ located at $\vec{r}$ , which is
subject to the additional tidal potential $\,U(\mbox{{\boldmath$\vec{r}$}})\,$
of the tidally perturbed body. Then its energy in this potential will be
$\,M_{1}\,U(\mbox{{\boldmath$\vec{r}$}})\,$. When the position of the exterior
body is rendered by the spherical coordinates, the polar component of the
torque acting on it can be conveniently expressed as:
$~{}T_{z}\,=\,-\,M_{1}\;{\partial
U(\mbox{{\boldmath$\vec{r}$}})}/{\partial\lambda}~{}$. The polar torque
wherewith the exterior body acts back on the tidally perturbed body is the
negative of $\,T_{z}\,$:
$\displaystyle{\cal{T}}_{z}(\mbox{{\boldmath$\vec{r}$}})\;=\;\,M_{1}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\lambda}\,~{}.$ (37)
Here polar means: orthogonal to the perturbed body’s equator. For small
obliquities (and, therefore, small latitudes), insertion of (23) into (37)
yields
$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}(t)\,)\,=$
$\displaystyle{G\;M_{1}\;M^{*}_{1}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r(t)^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}(t-\Delta
t)}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}~{}m~{}\frac{\left(l-m\right)!}{({\it l}+m)!}\left(2\right.$ (38)
$\displaystyle~{}\left.-\,\delta_{0m}\right)P_{lm}(0)P_{lm}(0)\;\sin\left(\,m\,\left(\lambda-\lambda^{*}\right)\,-\,m\,(\dot{\theta}-\dot{\nu})\,\Delta
t\,\right)~{}+~{}O(i^{2})~{}\,~{},~{}\quad~{}\quad~{}$
where we made use of (25).
Since the integer $\,m\,$ is now entering the above expression as a
multiplier, the term with $\,m=0\,$ becomes nil. As $\,P_{21}(0)=0\,$, the
term with $\,m=1\,$ also vanishes. Hence, the term with $\,l=m=2~{}$ is
leading. Neglecting the smaller terms, we thus obtain:
$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}(t)\,)\,\approx\,\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}\;M^{*}_{1}}\,k_{2}\;\frac{R^{5}}{r(t)^{\textstyle{{}^{3}}}{r^{\;*}(t-\Delta
t)}^{\textstyle{{}^{3}}}}\;\sin\left(\,2\,\left(\lambda-\lambda^{*}\right)\,-\,\,2\,(\dot{\theta}-\dot{\nu})\,\Delta
t\,\right)~{}\,~{}.~{}\quad~{}\quad~{}$ (39)
Consider the special case when the tide-raising perturber (with the asterisk)
coincides with the other external body (with no asterisk). The perturber
creates tides on the perturbed body, and then interacts with the tides it
itself has created. Hence the perturber becomes subject to a tidal torque
$\,\vec{\bf{T}}\,$ exerted by it upon itself, through the medium of the tidal
bulge it creates on the perturbed body. Evidently, a torque
$\,\vec{\cal{T}}\,=\,-\,\vec{\bf{T}}\,$, of the same magnitude but opposite
direction, will be acting upon the perturbed body. We arrive at this torque by
setting $\,\lambda=\lambda^{*}~{}$ and $~{}M_{1}=M^{*}_{1}~{}$ in the above
expression: 131313 Be mindful that in (40) we chose to make no distinction
between $\,r(t)\,$ and $\,r(t-\Delta t)\,$. Replacement of $\,r(t-\Delta t)\,$
with $\,r(t)\,$ gives birth to an absolute error of order
$\,O(eQ^{-2}n/\chi)\,$. It is however explained in Efroimsky & Williams
(2009), that after averaging of (40) over one orbital period this error
reduces to $\,O(e^{2}Q^{-3}n^{2}/\chi^{2})\,$.
$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}\,)\,\approx\,\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin\left(\,2\,(\dot{\nu}-\dot{\theta})\,\Delta
t\,\right)~{}\,~{}.~{}\quad~{}\quad~{}$ (40)
Naturally, for the model with a frequency-independent $\,\Delta t\,$, the
quantity
$\displaystyle\epsilon_{g}\,\equiv~{}(\dot{\nu}-\dot{\theta})~{}\Delta t$ (41)
is the geometric lag, i.e., the angular separation between the planetocentric
directions towards the perturber and the bulge. Accordingly, within the said
model, the quantity
$\displaystyle\chi~{}=~{}2\,|\,\dot{\nu}-\dot{\theta}\,|$ (42)
acts as an instantaneous tidal frequency.
The quantity
$\displaystyle\epsilon_{ph}\,\equiv~{}2~{}(\dot{\nu}-\dot{\theta})~{}\Delta
t~{}=~{}2~{}\epsilon_{g}$ (43)
is commonly interpreted as an instantaneous phase lag , so the torque in
expression (40) may be written down as
$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}\,)\,\approx\,\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin\epsilon_{ph}~{}=~{}\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin
2\epsilon_{g}~{}\,~{},~{}\quad~{}\quad~{}$ (44)
which is exactly the expression (1) of our concern. In Section 2, we mentioned
several popular papers and books, including Murray & Dermott (1999, eqn.
4.159), 141414 Mind a misprint in Eqn. (4.159) of Murray & Dermott (1999): in
the denominator, $\,a^{6}$ must be changed to $\,r^{6}$. The misprint emerged
because in subsection 4.2 the distance was denoted with $\,a\,$. In formulae
(5.2 - 5.3) of Ibid. the misprint gets corrected. where this formula is
employed. Now, that we have derived this formula accurately, we see that its
validity hinges on the time lag being frequency independent.
While it is common (McDonald 1964, Goldreich 1966, Kaula 1968, Murray &
Dermott 1999) to treat
$\displaystyle
Q~{}\equiv~{}1/\,|\,\sin\left(\,2\,(\dot{\nu}-\dot{\theta})\,\Delta
t\,\right)\,|~{}=~{}1/\sin|\,\epsilon_{ph}\,|$ (45)
as an instantaneous quality factor, the validity of this interpretation of
(45) remains questionable. For a nonzero eccentricity, the instantaneous tidal
frequency (42) is varying in time. So it is not readily apparent whether the
instantaneous $\,Q\,$ is connected to the damping rate in the manner the
proper quality factor introduced at a certain frequency links to the damping
rate at that frequency. McDonald (1964) and Goldreich (1966) tried to sidestep
this difficulty by assuming that $\,Q\,$ is a frequency independent constant.
However, at finite eccentricities this assumption does not work, as it is
incompatible with the constant $\,\Delta t\,$ assumption (the latter
assumption being a necessary prerequisite to using formulae 41 \- 45, as we
saw above). For more on this see Williams & Efroimsky (2012).
### 6.2 Tidal torque, as derived from the Fourier expansion (24),
with all Fourier modes delayed by the same time lag $\,\Delta t\,$
When starting out with expression (24), it is convenient to use the formula
$\displaystyle{\cal{T}}_{z}(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;M_{1}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\theta}\;\;\;,$ (46)
instead of (37). Technically, we should first differentiate $\,U\,$ with
respect to the sidereal angle $\,\theta\,$, and then set
$\,\theta\,=\,\theta^{\,*}\,$. We should also set the orbital elements with
asterisk equal to their counterparts with no asterisk, it being understood
that the tide-raising perturber is the same as the other exterior body which
“feels” the tides on the perturbed body. The development will furnish us the
polar component of the torque with which the perturber acts upon the tidally
deformed body. For exploration of dynamics in a low-obliquity configuration,
this component is sufficient.
Insertion of the Fourier series (24) into equation (46) yields a Fourier
series for the polar component of the torque, which is presented in Efroimsky
(2012b). Here we shall not repeat this long formula, but shall only make an
important comment on it. Insofar as $\,\Delta t\,$ stays frequency independent
(i.e., has the same value for all phase lags (27) entering the expansion for
the torque), the resulting series for the torque stays fully equivalent to
(38), with $\,\lambda\,$ and $\,\lambda^{*}\,$ set equal to one another in the
latter formula.
This equivalence is ensured by the expression (24) for the potential $\,U\,$
being equivalent to the expression (23) whence formula (38) originated, and by
our agreement to keep $\,\Delta t\,$ the same for all phase lags.
### 6.3 Tidal torque, as derived from the Fourier expansion (31),
with each mode $\,\omega_{lmpq}\,$ having a time lag of its own, $\,\Delta
t_{l}(\omega_{lmpq})\,$
As soon as we abandon the assumption that the phase lags (27) contain the same
fixed $\,\Delta t\,$, i.e., as soon as we switch from the lags (27) to those
rendered by (32), we acquire an opportunity to describe a tidal torque acting
on a perturbed body of an arbitrary rheology. Indeed, as the time lags can
have an arbitrary mode-dependence, this also relates to the phase lags. Above
that, we now permit the Love numbers to be mode-dependent.
To derive the tidal torque, we now combine formula (46) not with expansion
(24) but with the expansion (31) where the time lags are, generally, all
different. The rheological emancipation, though, comes at a cost: the Fourier
decomposition for the torque, obtained through (31), with mode-dependent
$\,\Delta t_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ and
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$, will no longer be equivalent to
the concise and elegant formula (38). The customary and widely used leading-
order approximation of (38), given by (40) or by (44), will not work for an
arbitrary rheology.
If, for example, we choose to set the factors $~{}\,\frac{\textstyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})}{\textstyle
Q_{l}(\omega_{\textstyle{{}_{lmpq}}})}~{}\mbox{Sgn}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{l}\,\sin\epsilon_{l}~{}\,$
to be mode independent, then, to calculate the torque, we shall have to plug
the same value of $\,k_{l}\,\sin\epsilon_{l}\,$ into all terms of the Fourier
series for the torque (eqn. 106 from Efroimsky 2012b). However, we shall not
be able to employ the neat formula (44).
## 7 Why the $\,`\mathbf{\emph{constant angular lag}}\,$’ model is wrong
As we explained above, an accurate derivation of the popular formula (1) for
the polar component of the torque hinges on a tacit assumption that the time
lag $\,\Delta t\,$ is the same for all modes in the expansion of the tidal
potential. Through formula (41), this, mode-independent time lag is related to
the geometric lag angle $\,\epsilon_{g}\,$ present in formula (1).
Relation (41) tells us that, since $\,\Delta t\,$ is the same for all the
tidal modes involved, then the geometric lag angle $\,\epsilon_{g}\,$ cannot
be treated as a fixed constant – even if we add a caveat permitting
$\,\epsilon_{g}\,$ to switch signs when the value of $\,\dot{\nu}\,$
transcends $\,\dot{\theta}\,$. With this caveat or not, keeping
$\,|\,\epsilon_{g}\,|\,$ constant is impossible simply for the reason that
(for a nonvanishing eccentricity) the quantity $\,\dot{\nu}\,$ oscillates in
time.151515 While derivation of (1) absolutely requires $\,\Delta t\,$ to be
the same for all tidal modes, it does not require $\,\Delta t\,$ to be fixed
in time. Therefore, in theory, we can save the constant angular lag model by
tuning the time dependence of $\,\Delta t\,$ in such a special way that
$\,\epsilon_{g}\,$ in (41) stays constant in time. This however would require
the dissipative properties of the mantle to be fine-tuned, simultaneously at
all frequencies, to ensure that, first, the time lags at all frequencies
evolve but remain equal to one another and, second, that $\,\Delta
t\,(\dot{\nu}-\dot{\theta})\,$ stays constant in time. As the evolution rate
of $\,\dot{\nu}-\dot{\theta}\,$ is defined by the orbit, such fine tuning of
rheology is unrealistic. So a constant $\,\Delta t\,$ is incompatible with a
constant $\,|\,\epsilon_{g}\,|\,$.
This is the reason why the so-called constant angular lag model based on (1)
must be discarded wholesale as being inherently contradictive.
While we still retain the right to set the factors $~{}\,\frac{\textstyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})}{\textstyle
Q_{l}(\omega_{\textstyle{{}_{lmpq}}})}~{}\mbox{Sgn}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{l}\,\sin\epsilon_{l}~{}\,$
mode independent, the value of the torque resulting from this assumption has
to be calculated by insertion of these factors into the full Fourier expansion
of the torque and not into (1). We may as well use (1), but only for a
constant time lag, and not for a constant geometric lag angle.
## 8 Conclusions
We have reexamined the common formula (1) for the tidal torque, a formula
which is equivalent to the expressions given in Sections 4 and 5 of Murray &
Dermott (1999) and to the expressions offered in Goldreich (1966) and Kaula
(1968). It has turned out that an accurate derivation of this popular formula
necessarily implies a specific rheology – the assertion that the time lag
$\,\Delta t\,$ is frequency independent. As can be easily seen from (41), this
assertion is incompatible with the assertion of the geometric lag being
frequency independent. Moreover, the quantity $\,\epsilon_{g}\,$ furnished by
formula (41) can be endowed with the meaning of a geometric lag only within
the constant $\,\Delta t\,$ rheological model (and only for a small obliquity
$\,i\,$).
To conclude, whenever the analysis of bodily tides is carried out using (1),
the analysis cannot be combined with a constant geometric lag (or phase lag,
or quality factor) assumption, nor with any other assumption different from
the frequency independence of $\,\Delta t\,$. This circumstance would not, by
itself, prohibit one from considering a material for which the factors 161616
We deliberately equip the quality factors with the subscript $\,l\,$, to
emphasise that they are different from the seismic $\,Q\,$ and have different
frequency dependencies for different $\,l\,$s (Efroimsky 2012a).
$~{}\,\frac{\textstyle k_{l}(\omega_{\textstyle{{}_{lmpq}}})}{\textstyle
Q_{l}(\omega_{\textstyle{{}_{lmpq}}})}~{}\mbox{Sgn}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{l}\,\sin\epsilon_{l}~{}\,$
are insensitive to the frequency over some limited frequency band. Consistent
employment of this model will then require insertion of the same value of
$~{}\,k_{l}\,\sin\epsilon_{l}~{}\,$ into all terms of the expansion of the
torque (each term thus changing its sign as the corresponding resonance is
transcended). However, neither the quantity $\,(\dot{\nu}\,-\,\dot{\theta})\,$
nor its sign will come into play in this expression for the torque. So the
outcome will be different from the mathematically incorrect “constant angular
lag” model based on equation (1).
## Acknowledgments
The authors are grateful to Stanton Peale, who refereed the paper and whose
comments and recommendations were of great help. One of the authors (ME) is
indebted to Sylvio Ferraz Mello and James G. Williams for numerous
enlightening discussions on the theory of tides.
## References
* [1] Alexander, M. E. 1973. “The weak-friction approximation and tidal evolution in close binary systems.” Astrophysics and Space Sciences, Vol. 23, pp. 459 - 510
* [2] Darwin, G. H. 1879. “On the precession of a viscous spheroid and on the remote history of the Earth.” Philosphical Transactions of the Roy. Soc. of London, Vol. 170, pp. 447-530
* [3] Efroimsky, M., and V. Lainey. 2007. “The Physics of Bodily Tides in Terrestrial Planets, and the Appropriate Scales of Dynamical Evolution.” _Journal of Geophysical Research – Planets_ , Vol. 112, id. E12003. doi:10.1029/2007JE002908
http://arxiv.org/abs/0709.1995
* [4] Efroimsky, M., and Williams, J. G. 2009. “Tidal torques. A critical review of some techniques.” _Celestial Mechanics and Dynamical Astronomy,_ Vol. 104, pp. 257 - 289
http://arxiv.org/abs/0803.3299
* [5] Efroimsky, M. 2012 a. “Tidal dissipation compared to seismic dissipation: in small bodies, earths, and superearths.” The Astrophysical Journal, Vol. 746, id. 150
doi:10.1088/0004-637X/746/2/150 http://arxiv.org/abs/1105.3936
ERRATA: ApJ, Vol. 763, id. 150 (2013)
* [6] Efroimsky, Michael 2012 b. “Bodily tides near spin-orbit resonances.” Celestial Mechanics and Dynamical Astronomy, Vol. 112, pp. 283 - 330. Extended version: http://arxiv.org/abs/1105.6086
* [7] Eggleton, P. P.; Kiseleva, L. G.; and Hut, P. 1998. “The equilibrium tide model for tidal friction.” The Astrophysical Journal, Vol. 499, pp. 853 - 870
* [8] Ferraz-Mello, S.; Rodríguez, A.; and Hussmann, H. 2008. “Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited.” _Celestial Mechanics and Dynamical Astronomy,_ Vol. 101, pp. 171 - 201.
* [9] Goldreich, P. 1966. “Final spin states of planets and satellites.” _The Astronomical Journal_. Vol. 71, pp. 1 - 7.
* [10] Heller, R.; Leconte, J.; and Barnes, R. 2011. “Tidal obliquity evolution of potentially habitable planets.” Astronomy & Astrophysics, Vol. 528, id. A27
* [11] Henning, W.; O’Connell, R.; and Sasselov, D. 2009. “Tidally Heated Terrestrial Exoplanets: Viscoelastic Response Models.” The Astrophysical J., Vol. 707, pp. 1000 - 1015
* [12] Hut, P. 1981. “Tidal evolution in close binary systems.” Astronomy & Astrophysics, Vol. 99, pp. 126 - 140
* [13] Karato, S.-i. 2007. _Deformation of Earth Materials. An Introduction to the Rheology of Solid Earth._ Cambridge University Press, UK.
* [14] Karato, S.-i., and Spetzler, H. A. 1990. “Defect Microdynamics in Minerals and Solid-State Mechanisms of Seismic Wave Attenuation and Velocity Dispersion in the Mantle.” Reviews of Geophysics, Vol. 28, pp. 399 - 423
* [15] Kaula, W. M. 1961. “Analysis of Gravitational and Geometric Aspects of Geodetic Utilisation of Satellites.” The Geophysical Journal of the Royal Astronomical Society, Vol. 5, pp. 104 - 133
* [16] Kaula, W. M. 1964. “Tidal Dissipation by Solid Friction and the Resulting Orbital Evolution.” Reviews of Geophysics, Vol. 2, pp. 661 - 684
* [17] Kaula, W. M. 1968. An Introduction to Planetary Physics. John Wiley and Sons, NY.
* [18] Knopoff, L. 1964. “Q” Reviews of Geophysics and Space Physics, Vol. 2, pp.625 - 660
* [19] Lambeck, K. 1980. The Earth’s Variable Rotation: Geophysical Causes and Consequences. Cambridge University Press, Cambridge UK
* [20] MacDonald, G. J. F. 1964. “Tidal Friction.” Reviews of Geophysics. Vol. 2, pp. 467 - 541.
* [21] Makarov, Valeri V.; Berghea, Ciprian; and Efroimsky, Michael. 2012. “Dynamical evolution and spin-orbit resonances of potentially habitable exoplanets. The case of GJ 581d.” The Astrophysical Journal, Vol. 761, id. 83. http://arxiv.org/abs/1208.0814
ERRATUM: ApJ, Vol. 763, id. 68 (2013)
* [22] Makarov, Valeri V., and Efroimsky, Michael. 2013. “No pseudosynchronous rotation for terrestrial planets and moons.” The Astrophysical Journal, Vol. 764, id. 27
http://arxiv.org/abs/1209.1616
* [23] Mignard, F. 1979. “The Evolution of the Lunar Orbit Revisited. I.” The Moon and the Planets. Vol. 20, pp. 301 - 315.
* [24] Mignard, F. 1980. “The Evolution of the Lunar Orbit Revisited. II.” The Moon and the Planets. Vol. 23, pp. 185 - 201.
* [25] Mignard, F. 1981. “Evolution of the Martian satellites.” The Monthly Notices of the Royal Astronomical Society. Vol. 194, pp. 365 - 379.
* [26] Murray, C.D., and Dermott, S.F. 1999. Solar System Dynamics. Cambridge University Press, Cambridge UK
* [27] Nimmo, F.; Faul, U. H.; and Garnero, E. J. 2012. “Dissipation at tidal and seismic frequencies in a melt-free Moon.” Journal of Geophysical Research – Planets, Vol. 117, id. E09005
doi:10.1029/2012JE004160
* [28] Singer, S. F. 1968. “The Origin of the Moon and Geophysical Consequences.” The Geophysical Journal of the Royal Astronomical Society, Vol. 15, pp. 205 - 226
* [29] Touma, J., and Wisdom, J. 1994. “Evolution of the Earth-Moon System.” The Astronomical Journal, Vol. 108, pp. 1943 - 1961
* [30] Williams, James G., and Efroimsky, Michael: “Bodily tides near the 1:1 spin-orbit resonance. Correction to Goldreich’s dynamical model.” Celestial Mechanics and Dynamical Astronomy, Vol. 114, pp. 387 - 414
http://arxiv.org/abs/1210.2923
|
arxiv-papers
| 2012-09-07T19:13:43 |
2024-09-04T02:49:34.882216
|
{
"license": "Public Domain",
"authors": "Michael Efroimsky and Valeri V. Makarov",
"submitter": "Michael Efroimsky",
"url": "https://arxiv.org/abs/1209.1615"
}
|
1209.1616
|
#
${{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}^{{}^{{}^{Published~{}in~{}the~{}Astrophysical~{}Journal\,,~{}Vol.~{}764\,,~{}id.~{}27\,~{}~{}(2013)}}}}$
No pseudosynchronous rotation for terrestrial planets and moons
Valeri V. Makarov
US Naval Observatory, Washington DC 20392
e-mail: vvm @ usno.navy.mil
and
Michael Efroimsky
US Naval Observatory, Washington DC 20392
e-mail: michael.efroimsky @ usno.navy.mil
###### Abstract
We reexamine the popular belief that a telluric planet or satellite on an
eccentric orbit can, outside a spin-orbit resonance, be captured in a quasi-
static tidal equilibrium called pseudosynchronous rotation. The existence of
such configurations was deduced from oversimplified tidal models assuming
either a constant tidal torque or a torque linear in the tidal frequency. A
more accurate treatment requires that the torque be decomposed into the
Darwin-Kaula series over the tidal modes, and that this decomposition be
combined with a realistic choice of rheological properties of the mantle,
which we choose to be a combination of the Andrade model at ordinary
frequencies and the Maxwell model at low frequencies. This development
demonstrates that there exist no stable equilibrium states for solid planets
and moons, other than spin-orbit resonances.
## 1 Motivation
The ongoing quest for extraterrestrial life has placed exoplanets and their
properties into the forefront of scientific investigation. The trend has
provided additional momentum to the development of a broad variety of
techniques and approaches employed in the planetary sciences. For example, the
recent revival of interest in mechanics of bodily tides is partly due to the
importance of the planetary spin for the prospects of finding habitable worlds
near other stars.
Well-known examples of dynamical equilibria achieved via tidal coupling
include our Moon, which is in a 1:1 spin-orbit resonance to the Earth, and
Mercury which makes exactly three sidereal rotations over every two orbital
revolutions around the Sun. Similar behaviour is expected of the growing
number of known super-earths – especially if their composition happens to be
similar to that of the terrestrial planets of the solar system, i.e., if they
have massive solid or partially molten mantles of rocky minerals.
Unfortunately, some of the published far-reaching conclusions about specific
exoplanets are based on incomplete or ad hoc models which should never be
used for solid materials, including those with partial melt. Both these
models, introduced by Goldreich (1966) mainly for the ease of analytical
treatment, predict quasi-static pseudosynchronous rotation states, with the
planet being trapped in a slowly changing equilibrium state at a faster-than-
synchronous rotation rate and a vanishing orbit-averaged tidal torque. Except
in specific (very narrow) frequency bands, these models are incompatible with
the rheological properties of realistic mantles and crusts. Analysis based on
actual rheologies demonstrates the impossibility of pseudosynchronous rotation
for homogeneous terrestrial objects. Whether this prohibition extends to
planets and moons with internal or surface oceans remains an open issue and
needs further research.
## 2 The constant angular lag model
A consistent linear theory of bodily tides is based on Fourier decomposition
of the tide, with subsequent inclusion of the response at each separate mode.
The ensuing level of complexity has tempted many to circumvent it by
developing simpler approaches. Serving as good illustrations and reflecting
some qualitative aspects of the tidal interaction, such models are not
necessarily applicable for quantitative purposes (Efroimsky & Lainey 2007) and
should certainly be eschewed when fine features of near-resonant dynamics are
explored.
### 2.1 The essence of the method
One, often-used, toy model prescribes:
(a) to set both the Love number $\,k_{2}\,$ and geometric lag
$\,\epsilon_{g}\,$ frequency-independent;
(b) to insert their values into the popular short formula
$\displaystyle{\cal{T}}_{z}~{}=~{}\frac{3}{2}~{}G\,M_{1}^{\,2}~{}\frac{R^{5}}{r^{6}}~{}k_{2}\,\sin
2\epsilon_{g}$ (1a) for the polar component of the torque wherewith a tide-
raising perturber of mass $\,M_{1}\,$
acts on a tidally perturbed body of radius $\,R\,$ located at a distance
$\,r\,$, the obliquity $\,i\,$
assumed small; and
(c) to combine formula (1a) with the assumption that the angle
$\,\epsilon_{g}\,$ stays constant while
the tide-raising perturber stays on one side of the bulge (in the sense of the
directions
as seen from the perturbed body’s centre).
For a nonzero eccentricity $\,e\,$, and in a sufficient proximity of the 1:1
resonance, the relative orientation of the perturber and the bulge changes
twice over an orbital period. Hence, within this model, the angle
$\,\epsilon_{g}\,$ is set, by hand, to change its sign abruptly twice in a
cycle, while keeping its magnitude $\,|\,\epsilon_{g}\,|\,$ fixed. Therefore
the model can be written down as
$\displaystyle{\cal{T}}_{z}~{}=~{}\frac{3}{2}~{}G\,M_{1}^{\,2}~{}\frac{R^{5}}{r^{6}}~{}k_{2}\,\sin
2|\epsilon_{g}|~{}\,\mbox{Sgn}\,(\dot{\nu}\,-\,\dot{\theta})\,~{},$ (1b)
$\theta$ and $\dot{\theta}$ being the perturbed body’s sidereal angle and spin
rate, and $\nu$ being the true anomaly.
Historically, the method dates back to the paper by MacDonald (1964) and
therefore is often referred to as the MacDonald torque (e.g., Touma & Wisdom
1994, Section 2.7.1).
The approach is also called the constant angular lag model or the constant
tidal torque model, both names being somewhat misleading. Indeed, in the
vicinity of the 1:1 resonance the sign of the lag (and the torque) is set
positive or negative, when the bulge falls behind or advances relative to the
direction towards the perturber. So both quantities change their sign twice
over a period – a circumstance that makes the term constant inappropriate.
Furthermore, the torque depends upon the distance and is always evolving in
time unless the orbit is circular.
The abrupt switch of the sign of the torque (2.1), with its magnitude staying
unchanged, is quite a contrived assertion 111 Stated alternatively, if we
represent $\,(a/r)^{6}\,$ as a series of Fourier harmonics $\cos(j{\cal{M}})$,
$j=0,1,\ldots$, where $a$ is the semimajor axis and ${\cal{M}}$ is the mean
anomaly, we shall have to accept that the $\cos({\cal{M}})$ tidal mode
generates a positive (accelerating) torque for ${\cal{M}}\in[-\pi/2,+\pi/2]$,
abruptly switching to a negative value for ${\cal{M}}\in[\pi/2,3\pi/2]\,$.
which by itself indicates that the model is unphysical. A deeper,
mathematical, objection will be brought up in Section 2.4.
Saying goodbye to the constant-angular-lag model will not be easy, because it
has been a textbook standard for nearly half a century. Given the attractive
simplicity of the model, one will always be tempted to enquire if perhaps it
would still be producing at least qualitative results of some value. To see
that it would not, we shall have to scrutinise the principal outcomes of the
model.
The perturber’s orbit is set to lie in the equatorial plane of the perturbed
body; in other words, the obliquity is set zero. Two special situations of
interest emerge here. One is the case of exact synchronism, the other being
the case of a vanishing average tidal torque. Both settings were explored by
Goldreich (1966) whose results are explained in detail by Murray & Dermott
(1999).
### 2.2 The synchronous spin case
Suppose the tidally-perturbed body on an elliptic ($\,e\,\neq\,0\,$) orbit is
caught into the 1:1 spin-orbit resonance: $\,n=\dot{\theta}\,$. Then, as
explained in Section 5.2 of Murray & Dermott (1999), the angular motion rate
$\,\dot{\theta}\,$ exceeds $\,n\,$ over exactly one half of the orbital time
period, and falls short of $\,n\,$ during the other half of the period.
Correspondingly, the tidal torque $\,{\cal{T}}_{z}\,$ is positive
(accelerating) through the former half of the period, and is negative
(decelerating) through the latter half. The instantaneous tidal torque is
proportional to a negative power of the instantaneous distance $\,r\,$ between
the bodies. As depicted in Figure 5.3 in Ibid, when the disturbed body’s
angular motion is faster than the mean motion, the bodies are closer, so the
positive (spinning up) tidal torque is larger in absolute value than the
negative torque for the other half of the period. Thus the resultant orbit-
averaged torque $\,\langle\,{\cal{T}}_{z}\,\rangle\,$ is positive, and the net
effect is to accelerate the tidally perturbed body’s spin. (Recall that the
undisturbed body is assumed to be spherical or oblate, so the tidal torque is
the only one coming into play.)
### 2.3 The case of vanishing tidal torque
The second important application of the constant angular lag model is the
situation where the orbit-average tidal torque vanishes:
$\,\langle\,{\cal{T}}_{z}\,\rangle\,=0\,$. Vanishing of the average tidal
torque entails a dynamical equilibrium: the tidally disturbed body keeps
rotating at a steady spin rate. A calculation of this rate, borrowed from
Goldreich (1966), is presented in Murray & Dermott (1999) and is often cited
in the literature. According to that development, the equilibrium is achieved,
for a zero obliquity, at the spin rate of
$\stackrel{{\scriptstyle\mbox{\bf{\LARGE{$\,\cdot$}}}}}{{\theta}}_{\textstyle{{}_{\rm
eq}}}\,=\,n\,\left(\,1\,+\,\frac{19}{2}~{}e^{2}\,\right)~{}\,,$ (2)
$e\,$ being the eccentricity. At first glance, the result looks unassailable.
Indeed, for $\,\dot{\theta}=n\,$, the bulge is lagging behind the central line
over one half of the time period (around the periastron), so the torque
accelerates the rotation. Over the other half of the period, the torque
decelerates, being weaker due to a larger distance. So the state
$\,\dot{\theta}=n\,$ looks unstable, as the overall average torque seems to be
accelerating.
It is however well known that the Moon is not staying in this
pseudosynchronous regime (which would be 3% faster than the synchronous
rotation wherein the Moon is presently locked). Murray & Dermott (1999) point
at the lunar quadrupole moment as the reason why the Moon is not
pseudosynchronous. A deeper reason though lies in the constant geometric lag
model being genuinely flawed, and in the entire calculation leading to (2)
being invalid.
### 2.4 A major objection against the constant-angular-lag model
As well known, the generic expression for the tidal amendment to the perturbed
body’s potential is furnished by a Fourier series developed by Kaula (1964).
We term it the Darwin-Kaula expansion, as a partial sum of that series was
written down by Darwin (1879). Accordingly, the generic expression of the
tidal torque also must look as an infinite series. The series remains infinite
even if we include into it only the degree-2 terms, i.e., those proportional
to the quadrupole Love number $\,k_{2}\,$. The very fact that the expansion
for the torque can be wrapped into a short and neat form (2.1) is an indicator
of some extra, very special assumption being involved.
As was pointed out in Williams & Efroimsky (2012), such an assumption indeed
is present in the constant-angular-lag model, though this assumption is never
stipulated explicitly. The situation is elucidated in all detail in the paper
by Efroimsky & Makarov (2013) to which we refer the reader. Here we shall
provide only a brief summary.
As explained in Williams & Efroimsky (2012), the afore-presented concise
expression (2.1) for the torque is equivalent to the full Darwin-Kaula
expansion for the potential, only if the following assumptions are made:
* •
In all terms of the Darwin-Kaula series for the tidal amendment to the
potential of the perturbed body, i.e, for all Fourier tidal modes
$\,\omega_{lmpq}\,$, the time lags are endowed with the same, frequency-
independent value $\,\Delta t\,$.
* •
The obliquity is set small.
* •
Only the terms with $\,l=m=2\,$ are retained, and the Love number $\,k_{2}\,$
entering these terms is assumed frequency-independent. Here the degree $\,l\,$
and the order $\,m\,$ are the first two integers of the four-number set
$\,lmpq\,$ used to number the Fourier modes showing up in the Darwin-Kaula
expansion.
The so-processed Darwin-Kaula series for the tidal potential becomes
equivalent to a concise expression (equation 16b in Ibid.) wherefrom our
expression (2.1) for the torque ensues.
Alternatively, the above three assumptions could be applied directly to the
Darwin-Kaula expansion for the tidal torque. Once again, the outcome would be
the above expression (2.1) for the torque. This is demonstrated in Efroimsky &
Makarov (2013, equation 34).
Under the three assumptions, the instantaneous geometric lag angle turns out
to be 222 The geometric angular lag $\epsilon_{g}$ is not to be confused with
the instantaneous phase lag (or longitudinal lag)
$\displaystyle\epsilon_{ph}\,\equiv~{}2\,(\dot{\nu}-\dot{\theta})\,\Delta
t~{}=~{}2~{}\epsilon_{g}$ sometimes used in the literature (Efroimsky &
Williams 2009, Williams & Efroimsky 2012).
$\displaystyle\epsilon_{g}\,\equiv~{}(\dot{\nu}-\dot{\theta})\,\Delta t~{}\,,$
(3)
$\,\Delta t\,$ being the frequency-independent time lag, $\,\nu\,$ being the
true anomaly of the perturber, and $\,\theta\,$ being the sidereal angle of
the tidally perturbed body. From this expression, it is straightforward that
the validity of formula (2.1) is incompatible with the geometric lag being
constant. Indeed, the road to (2.1) is paved with the aforementioned three
assumptions, one of which being that of a constant $\,\Delta t\,$. As can be
observed from (3), the latter is incompatible with the geometric lag being
constant, unless the eccentricity is nil. 333 As a last resort, one can
suggest (a) to tune the time dependence of $\,\Delta t\,$ so that the lag
angle $\,\epsilon_{g}\,$ in (3) stays constant in time, and (b) to assume that
the time lags at all Fourier tidal modes are equal to the so specially
evolving $\,\Delta t\,$. This would imply that the lagging properties of the
material at all frequencies are being tuned in a fine manner, continuously and
simultaneously, so that $\,\Delta t\,$ adjusts its evolution rate, to stay
inverse to $\,\dot{\nu}-\dot{\theta}\,$ at any instant of time. The rate of
change of $\,\dot{\nu}-\dot{\theta}\,$ being defined by the orbital
parameters, existence of such a rheology in nature looks impossible.
On all these grounds, the constant-geometric-lag (constant-torque) model
should be discarded as such.
## 3 Pseudosynchronism in the constant time lag model
As distinct from the constant geometric lag approach, the constant time lag
model sets the time delay $\,\Delta t\,$ independent of the tidal mode
frequency. Pioneered by Darwin (1879), this assumption was a part of numerous
works, e.g., Hut (1981), Eggleton et al. (1998). The assumption was also the
base for one of the two models considered by Goldreich & Peale (1966, equation
23), the other model addressed in that paper being the afore-discarded
constant geometric lag method.
The constant time lag model is unique, in that it makes the full Darwin-Kaula
expansion for the tidal potential (or torque) equivalent to a much shorter and
simpler expression. In regard to the tidal potential, this is the equivalence
of the full series (16) and a simpler expression (15) in our preceding paper
Efroimsky & Makarov (2013). In application to the torque, this is the
equivalence of the appropriate full series to the simpler expression (34) in
Ibid.
If we agree to limit our approximation to the lowest degree and order,
$\,l=m=2\,$, the aforementioned simpler expressions read as
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})=~{}-~{}\frac{3}{4}~{}G\,M_{1}\,k_{2}~{}\frac{\,R^{\textstyle{{}^{5}}}\,}{\,r^{6}\,}~{}\cos(\,2\,(\dot{\nu}-\dot{\theta})\,\Delta
t\,)\,=~{}\frac{3}{4}~{}G\,M_{1}\,k_{2}~{}\frac{\,R^{\textstyle{{}^{5}}}\,}{\,r^{6}\,}~{}\cos(\,2\,|\,\dot{\nu}-\dot{\theta}\,|\,\Delta
t\,)\,~{},\,~{}\,$ (4)
for the potential, and as
$\displaystyle{\cal{T}}_{z}\,=\,\frac{\textstyle 3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin(\,2\,(\dot{\nu}-\dot{\theta})\,\Delta
t\,)~{}=~{}\frac{\textstyle 3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin(\,2\,|\,\dot{\nu}-\dot{\theta}\,|\,\Delta
t\,)~{}\,\mbox{Sgn}\,(\dot{\nu}-\dot{\theta})\,~{},\quad$ (5)
for the polar torque. A detailed derivation of (4) can be found in Williams &
Efroimsky (2012), while derivation of (5) is offered in Efroimsky & Makarov
(2013).
To average the torque, it is instrumental to insert into (5) the distance
$\,r\,$ expressed through the semimajor axis $\,a\,$, eccentricity $\,e\,$,
and true anomaly $\,\nu\,$, and to integrate over the orbital cycle. The
procedure gets simplified greatly for small lags, when one can substitute the
sine with its argument. Then the calculation (presented in detail in the
Appendix to Williams & Efroimsky 2012) renders:
$\langle\,{\cal{T}}_{z}\,\rangle~{}\propto~{}\left[\frac{~{}1~{}+~{}\frac{\textstyle
15}{\textstyle 2}~{}e^{2}\,+~{}\frac{\textstyle 45}{\textstyle
8}~{}e^{4}\,+~{}\frac{\textstyle 5}{\textstyle
16}~{}e^{6}~{}}{(1~{}-~{}e^{2})^{\textstyle{{}^{6}}}}~{}-~{}\frac{\,\dot{\theta}\,}{\,n\,}~{}\frac{~{}1~{}+~{}3~{}e^{2}\,+~{}\frac{\textstyle
3}{\textstyle
8}~{}e^{4}~{}}{(1~{}-~{}e^{2})^{\textstyle{{}^{9/2}}}}\right]~{}\,.$ (6)
This expression was obtained by Eggleton et al. (1998), though its equivalent
was present in an earlier paper by Hut (1981, equation 11). In a somewhat
disguised form, this expression can be found in a much earlier work by
Goldreich & Peale (1966, equation 24).
We find readily that the equilibrium (i.e., vanishing of the average tidal
torque) is achieved at
$\dot{\theta}_{\rm
equ}~{}=~{}n~{}\left[\,1\,+\,6\,e^{2}\,+\,\frac{3}{8}~{}e^{4}\,+\,\frac{173}{8}~{}e^{6}\,+\,O(e^{8})\,\right]~{}\,.$
(7)
Note that the pseudosynchronous rate of rotation depends only on the mean
motion and eccentricity. This enables us to solve equation (7) with respect to
$\,e\,$, for a fixed dimensionless spin rate $\,\dot{\theta}/n\,$. The outcome
will be the equilibrium eccentricity $\,e_{equ}\,$, i.e., the eccentricity
that ensures the vanishing of the average torque at a certain value of
$\,\dot{\theta}/n\,$. In Figures 1 and 2, the equilibrium eccentricity
$\,e_{equ}\,$ is depicted as a function of $\,\dot{\theta}/n\,$. For the model
leading to expression (6) for the torque, this is a monotonically rising
curve.
The curve divides the plane into two parts corresponding to the two opposite
signs of the average polar torque $\,\langle\,{\cal{T}}_{z}\,\rangle\,$. While
$\,\langle\,{\cal{T}}_{z}\,\rangle\,$ is positive (accelerating) everywhere
above the curve, is stays negative (decelerating) everywhere below the curve.
Indeed, if we fix the eccentricity and make $\,\dot{\theta}/n\,$ very large,
this will guarantee us that we get into the lower right part of the picture,
i.e., below the rising curve. In this situation, i.e., for a fixed
eccentricity and a sufficiently swift spin, the second term of the torque (6)
must be leading, wherefore the torque must be negative, i.e., despinning.
Similarly, by fixing the eccentricity and making the spin rate very small, we
ensure getting into the upper left part of the picture, and also ensure that
the first term in (6) is leading, so the torque is positive, i.e.,
accelerating the spin. Since the smoothly rising curve 444 Here and hereafter,
the symbol $\,e_{equ}(\dot{\theta}/n)\,$ implies $\,e_{equ}\,$ as a function
of the ratio $\,\dot{\theta}/n\,$. This is not a product of $\,e_{equ}\,$ and
$\,\dot{\theta}/n\,$. $~{}e_{equ}(\dot{\theta}/n)\,$ corresponds to a zero
$\,\langle\,{\cal{T}}_{z}\,\rangle\,$, it is impossible to change the sign of
$\,\langle\,{\cal{T}}_{z}\,\rangle\,$ without crossing the curve.
Figure 1: Equilibrium eccentricity (one corresponding to a vanishing average
tidal torque) depicted against the dimensionless spin rate
$\,\dot{\theta}/n\,$. Calculations were made for a tidally perturbed rotating
body with parameters of the Moon, as shown in Table LABEL:table. The
monotonically rising curve corresponds to the linear torque (constant time
lag) model. The jagged dotted line corresponds to a realistic rheology
introduced in Section 4. Both functions were computed with a step of
$\,0.01\,$ in $\,\dot{\theta}/n\,$. In both cases, the resulting curve divides
the plane into two parts corresponding to the two opposite signs of the
average polar torque $\,\langle\,{\cal{T}}_{z}\,\rangle\,$. While
$\,\langle\,{\cal{T}}_{z}\,\rangle\,$ is positive (accelerating) everywhere
above the curve, it stays negative (decelerating) everywhere below the curve.
The small arrows indicate the action of the tidal torque upon small
perturbations of the spin rate away from an equilibrium state. For the
constant time lag model, the torque is restoring, and the equilibrium is
stable. In the case of realistic rheology, though, the emerging nonzero torque
drives the rotator away from the stable spin. Figure 2: Equilibrium
eccentricities of a zero secular tidal torque acting on a tidally perturbed
super-Earth, depicted against the dimensionless spin rate
$\,\dot{\theta}/n\,$. Parameters of the super-Earth are given in Table
LABEL:table, and are consistent with those chosen for GJ581d in Makarov et al.
(2012). The monotonically rising curve represents the prediction of the
constant time lag model. The jigsaw dotted curve illustrates the prediction of
the realistic rheological model described in Section 4. The function was
computed for a grid of points at a step of $\,0.01\,$ in $\,\dot{\theta}/n\,$.
Within the constant time lag model, the function $\,e_{equ}(\dot{\theta}/n)\,$
being a smoothly rising curve explains the emergence of pseudosynchronism. To
see this, consider a point on this curve, corresponding to a certain
pseudosynchronous state. A small perturbation in $\,\dot{\theta}/n\,$ makes
the tidally perturbed body rotate either faster or slower than the
pseudosyncronous rate, and a nonzero tidal torque emerges. Illustrated by the
two counter directed short arrows on the plot in Figure 1, the tidal torque is
restoring, in that its action is opposite to the sign of perturbation. Thus,
the tidal torque will return the perturbed rotator to the equilibrium state,
i.e., to the initial position on the curve. So the equilibrium is stable. 555
Being stable, the equilibrium is quasi-static, in the following sense. As the
tidal dissipation goes on, the process of despinning continues. The argument
$\,\dot{\theta}/n\,$ slowly decreases, and so does the appropriate value of
the equilibrium eccentricity. After a perturbation in $e$ from an equilibrium
state gives birth to a torque, the torque corrects swiftly the spin rate in
such a way that the rotator returns to an equilibrium state. However, the
equilibrium state itself is evolving slowly. In the case of a two-body
problem, this evolution is always directed towards the configuration with
$\,\dot{\theta}/n\,=\,1\,$ and $\,e\,=\,0\,$. For a viscous body, this was
proven by Hut (1981). For a broader class of viscoelastic rheologies, the
proof was offered by Bambusi and Haus (2012).
The constant time lag model ignores the important contribution of rigidity
(Segatz et al. 1988) and inelasticity (Karato and Spetzler 1990) into the
tidal response of Earth-like planets. As a result, the model is incapable to
account correctly for creep. As will be discussed in the following section,
the above derivation of quasi-stable pseudosynchronism, from the linear torque
model, is inapplicable to Earth-like planets with rigid mantles. However, in
the viscous limit, this model may still be applicable to celestial bodies that
do not have appreciable rigidity or inelasticity, such as gaseous planets and
stars. Observations of binary stars, especially of short-period active stars
on eccentric orbit, hold the best prospect of proving or disproving the linear
torque model for this type of objects (Ferraz-Mello 2012, Torres et al. 2010).
The spin rate of active stars can be inferred from the characteristic periods
of photometric variations caused by the passage of large spots or groups of
spots across the visible disk of the star. The orbital period and the
eccentricity are determined from spectroscopic radial velocity measurements.
We find somewhat conflicting evidence for the existence of pseudosychronism in
binary stars. Some stars with considerable eccentricities appear to have
pseudosynchronous rotation (Hall 1986, Fekel et al. 1998), which is consistent
with the original prediction by Hut (1981). Other stars clearly rotate faster
or slower than the predicted rate (Fekel et al. 1993, Strassmeier et al.
2011). Even more puzzling, a significant number of tight binary systems have
been found on circularised orbits, albeit spinning clearly asynchronously.
This fact comes into contradiction with one of the important predictions of
the linear torque theory, the one that synchronisation (or
pseudosyncronisation) of rotation is achieved much sooner than circularisation
(P. P. Eggleton 2011, private communication). Thus the impression created by
the current body of observations is that the constant time lag model is, at
least, not universally applicable to stars. This should not come as a
surprise, because there exist theoretical indications that stars may have
magnetic rigidity (Williams 2004, 2005, 2006; Ogilvie 2008, Garaud et al.
2010). 666 Another deviation from the purely viscous model can be caused by
the so-called $\,\Lambda-$effect responsible for differential rotation (Käpylä
& Brandenburg 2008, Kichatinov 2005, Rüdiger 1989). Turbulent convection
generates an extra stress called Reynolds stress. While in a non-rotating
convection zone this stress can be described as an addition to the viscosity,
this can no longer be done when the rotation period becomes comparable to or
shorter than the convective turnover time. In that situation, a non-viscous
input, the so-called $\,\Lambda$-effect, shows up. In its presence, the stress
tensor in the stellar material will no longer be proportional to the time
derivative of the strain tensor, but will contain terms proportional directly
to velocity. Thus the purely viscous model falls apart, and a frequency-
independent time lag is no longer an option.
Finally, it should be mentioned that, contrary to a common belief, the purely
viscous model does not render a frequency-independent time lag at all
frequencies. Stated differently, the purely viscous model does not imply that
the factors $\,k_{l}(\omega_{lmpq})\,\sin\epsilon_{l}(\omega_{lmpq})\,$ are
linear functions of the tidal mode $\,\omega_{lmpq}\,$ for all values of the
mode. It can be demonstrated that this linearity takes place at low
frequencies, but gets violated at frequencies higher than
$\,G\rho^{2}R^{2}/\eta\,$, where $\,G$, $\,\rho$, $\,R$, and $\,\eta\,$ are
the Newton gravity constant, mean density, radius, and the mean viscosity of
the perturbed body. We shall address this topic elsewhere.
Table 1: Parameters of the tidal model. Name | Description | Units | Values
---|---|---|---
| | | Moon | super-Earth
| | | | ( GJ581d )
$\xi$ | moment of inertia coefficient | | 2/5 | 2/5
$R$ | radius of the perturbed body | m | $1.737\times 10^{6}$ | $1.083\times 10^{7}$
$M_{2}$ | mass of the perturbed body | kg | $7.3477\times 10^{22}$ | $4.23\times 10^{25}$
$M_{1}$ | mass of the perturbing body | kg | $5.97\times 10^{24}$ | $6.17\times 10^{29}$
$a$ | semimajor axis | m | $3.84399\times 10^{8}$ | $3.3\times 10^{10}$
$n$ | mean motion, i.e. $2\pi/P_{\rm orb}$ | yr-1 | $84$ | $34.25$
$e$ | orbital eccentricity | | 0.0549 | 0.27
$(B-A)/C$ | triaxiality | | $2.278\times 10^{-4}$ | $5\times 10^{-5}$
${G}$ | gravitational constant | m3 kg-1 yr-2 | $66468$ | $66468$
$\tau_{M}$ | Maxwell time | yr | 5 | 50
$\mu$ | unrelaxed rigidity modulus | Pa | $0.8\times 10^{11}$ | $0.8\times 10^{11}$
$\alpha$ | the Andrade parameter | | $0.2$ | $0.2$
## 4 Equilibrium torques for Earth-like planets and moons
To build a consistent theory of bodily tides, one has, first, to decompose the
tide into a Fourier series and, second, to attribute to each Fourier component
its own phase delay and magnitude decrease (the latter being expressed by the
Love number appropriate to the said Fourier mode). Development of the
decomposition technique was started by Darwin (1879) and accomplished in full
by Kaula (1964). Attribution of phase delays and Love number values to the
Fourier modes took much longer time, because of the necessity to explore
rheological properties of the mantle at various frequencies. This exploration,
by both seismological and geodetic methods, has been going on intensively
through the past dozens of years. Merger of the Darwin-Kaula decomposition
technique with the results from solid-Earth rheology is explained in Efroimsky
(2012 a). The paper relied on a combined rheological model (Andrade at higher
frequencies, Maxwell at lower frequencies), because of this model’s ability to
best match laboratory experiments and both seismic and geodetic measurements
of dissipation over a range of frequencies in the solid Earth.777 Motivation
for the combined model stems from the mantle being predominantly viscoelastic
at frequencies below some threshold, and predominantly inelastic at
frequencies above it. As explained in Karato & Spetzler (1990), dissipation
above the threshold is dominated by defect unpinning (see also Miguel et al.
2002). When the frequency descends below the threshold, the effectiveness of
this mechanism declines, because the Andrade term in the expression for the
complex compliance decreases exponentially. The response of the mantle
approaches that of the Maxwell body. So slow processes (like the postglacial
rebound) are viscoelastic. For Earth’s mantle, the threshold frequency is of
the order of $\,1$ yr${}^{-1}\,$. Its value, though, is exponentially
sensitive to the temperature and therefore may be very different for
exoplanets and exomoons. The merger of the Darwin-Kaula expansion with the
combined rheological model, worked out in Ibid., has been used to predict
spin-orbit resonances of a Mercury analogue having a constant eccentricity and
a zero obliquity (Makarov 2012) and tidal properties of super-Earths
(Efroimsky 2012 b).
### 4.1 Expression for the tidal torque
It is explained in Appendix A that the average polar component of the tidal
torque can be approximated with the following expression, provided that (1)
the obliquity is small, and (2) the perturbed body and the perturber are not
too close to one another (so only the terms with degree-2 Love number are
important) :
$\displaystyle\langle\,{\cal{T}}_{z}\rangle_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}~{}=~{}\frac{3}{2}~{}\frac{\,G\,M_{1}^{\,2}}{a}\,\left(\frac{R}{a}\right)^{5}\sum_{q=-1}^{7}\,G^{\,2}_{\textstyle{{}_{\textstyle{{}_{20\mbox{\it{q}}}}}}}(e)~{}k_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}})~{}\sin\epsilon_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}})\,+O(e^{8}\,\epsilon)+O(i^{2}\,\epsilon)~{}=~{}\quad~{}\quad\quad$
(8a)
$\displaystyle\frac{3}{2}~{}\frac{\,G\,M_{1}^{\,2}}{a}\,\left(\frac{R}{a}\right)^{5}\sum_{q=-1}^{7}\,G^{\,2}_{\textstyle{{}_{\textstyle{{}_{20\mbox{\it{q}}}}}}}(e)~{}k_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}})~{}\sin|\,\epsilon_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}})\,|\,~{}\mbox{Sgn}\,\left(\,\omega_{220q}\,\right)+O(e^{8}\,\epsilon)+O(i^{2}\,\epsilon)~{}~{}.~{}\quad~{}$
(8b)
This is the polar (orthogonal to the equator) component of the torque
wherewith the tidally-perturbed body is acted upon by the perturber. The
angular brackets denote orbital averaging, $\,G\,$ stands for the Newton
gravitational constant, $\,M_{1}\,$ signifies the mass of the perturber (the
star, if the perturbed body is its planet; or the planet, if the perturbed
body is a satellite), $\,a\,$ is the semimajor axis, while $\,R\,$ is the
radius of the tidally perturbed body. The degree-2 dynamical Love number
$\,k_{2}\,$ and the phase lag $~{}\epsilon_{2}\,$ are functions of the Fourier
tidal mode
$\displaystyle\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}}\,=~{}(2~{}+~{}q)~{}n~{}-~{}2~{}\dot{\theta}\,~{}.$
(9)
While the dynamical Love numbers
$\,k_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,$ are positive definite,
the sign of each phase lag
$\,\epsilon_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,$ coincides with
that of the Fourier mode $\,\omega_{\textstyle{{}_{220\mbox{\it{q}}}}}\,$, as
can be understood from formulae (19) and (20) in Appendix A. It is for this
reason that the products
$\,k_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,\sin\epsilon_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,$
emerging in expression (8a) are rewritten in expression (8b) as
$\,k_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,\sin|\,\epsilon_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,|~{}\,\mbox{Sgn}\,(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,$.
A generic expression for the torque implies summation over the four integer
indices $~{}lmpq~{}$ serving to number the Fourier tidal modes
$~{}\omega_{{\textstyle{{}_{lmpq}}}}~{}$ entering the spectrum – see Appendix
A below. The terms of that series depend also upon the obliquity. As the
$~{}lmpq~{}$ term contains a factor $\,(R/a)^{2l+1}\,$, it is often sufficient
to keep only the degree-2 terms ($\,l=2\,$). In this case, with an extra
assumption of small obliquity, it is enough to limit the summation to the
terms with $\,m=2\,$, $\,p=0\,$. This renders expression (8).
While the full expression for the torque implies summation over all integer
values of $\,q\,$, numerical tests demonstrate that, for eccentricities not
exceeding $\,\sim\,0.3\,$, it is enough to take into account the terms up to
$\,e^{7}\,$, inclusive. This would require summation from $\,q\,=\,-\,7\,$
through $\,q\,=\,7\,$. However, the values of the numerical factors entering
the eccentricity polynomials
$\,G_{\textstyle{{}_{\textstyle{{}_{20\mbox{\it{q}}}}}}}(e)\,$ are such that
in practice it turns out to be sufficient to include only the terms with
$\,q\,$ varying from $\,-\,1\,$ through $\,7\,$.
### 4.2 The tidal torque and the equilibrium eccentricity as functions of the
spin rate
Expression (9) makes each product $\,k_{2}\,\sin\epsilon_{2}\,$ a function of
the planetary spin rate
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta~{}}}\,$:
$\displaystyle
k_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\;\sin|\,\epsilon_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,|~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{220\mbox{\it{q}}}}}\,)~{}\quad\quad\quad\quad\quad\quad\quad\quad~{}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}$
$\displaystyle=~{}k_{2}(\,2(n-\dot{\theta})\,+\,q\,n\,)~{}~{}\sin|\,\epsilon_{2}(\,2(n-\dot{\theta})\,+\,q\,n\,)\,|~{}~{}\mbox{Sgn}\,(\,2(n-\dot{\theta})\,+\,q\,n\,)\,~{}.~{}\quad$
(10)
Consequently, the entire sum (8) can be treated as a function of
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}$. The mean motion and
eccentricity will play the role of parameters whose evolution is much slower
than that of $\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}$.
Figure 3: Angular acceleration due to the secular tidal torque (8), in the
vicinity of the 2:1 resonance. The dotted kink is the $\,q=2\,$ term which is
an odd function when centered at $\dot{\theta}/n=1+q/2=2$. The solid line
renders the total torque (8), i.e., a sum of the $q=2$ kink and the bias
comprised by the terms with $q\neq 2$. Near the resonance, the bias is a
slowly changing function of $\dot{\theta}/n$ and can be approximated with a
constant. The $q=2$ kink resides on the right slope of a more powerful $q=1$
kink which is centered at $\dot{\theta}/n=1$ and dominates the bias. So the
$q=2$ kink is shifted downward and goes through nil a tiny bit to the left of
the resonance. The figure is borrowed from our work Makarov et al. (2012)
devoted to the super-Earth GJ581d.
Each product $\,k_{2}\,\sin\epsilon_{2}\,$ is an odd function of the tidal
mode $\,\omega_{220\mbox{\it{q}}}\,$ and has the shape of a kink centered
around $\,\omega_{220\mbox{\it{q}}}=0\,$. When we employ relation (10) to
write these products as functions of the spin rate, the new functions will
still be kinks, though centered around $\,\dot{\theta}=n\left(1+{\textstyle
q}/{\textstyle 2}\right)\,$. In Figure 3, the dotted line depicts the product
888 In formula (11), the notations $~{}k_{2}(\,4\,n\,-\,2\,\dot{\theta}\,)\,$
and $~{}\epsilon_{2}(\,4\,n\,-\,2\,\dot{\theta}\,)\,$ stand for $\,k_{2}\,$
and $\,\epsilon_{2}\,$ as functions of the argument
$\,4\,n\,-\,2\,\dot{\theta}\,$. These are not products of $\,k_{2}\,$ or
$\,\epsilon_{2}\,$ by $\,(4\,n\,-\,2\,\dot{\theta})\,$. The same pertains to
formula (10).
$\displaystyle
k_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2202}}}}})~{}\sin\epsilon_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2202}}}}})\,=\,k_{2}(\,4\,n\,-\,2\,\dot{\theta}\,)~{}\sin\epsilon_{2}(\,4\,n\,-\,2\,\dot{\theta}\,)$
$\displaystyle=~{}k_{2}(\,4\,n\,-\,2\,\dot{\theta}\,)~{}~{}\sin|\,\epsilon_{2}(\,4\,n\,-\,2\,\dot{\theta}\,)\,|~{}~{}\mbox{Sgn}\,(\,4\,n\,-\,2\,\dot{\theta}\,)\,~{}.$
(11)
The kink shape of the $\,k_{2}\,\sin\epsilon_{2}\,$ products is determined by
the rheological properties of the planet and its self-gravitation (see
Appendix B for details and references). A kink transcends nil and changes sign
continuously as the spin rate goes through the appropriate resonance.
In the sum (8), the kink-shaped products stand with multipliers
$\,G^{\,2}_{\textstyle{{}_{\textstyle{{}_{20\mbox{\it{q}}}}}}}(e)\,$. So the
overall tidal torque (8), as a function of $\,\dot{\theta}~{}$, is a
superposition of many kinks having different magnitudes and centered at
different resonances (nine kinks, if we sum over
$\,q\,=\,-\,1,\,.\,.\,.\,7\,$). The ensuing curve will cross the horizontal
axis in points extremely close to the resonances
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta~{}}}=n\,\left(1+\,{\textstyle
q}/{\textstyle 2}\right)\,$, but not exactly in these resonances — like the
solid line in Figure 3.
### 4.3 The physical meaning of the kink
The physical forcing frequencies in the mantle,
$\,\chi_{\textstyle{{}_{220\mbox{\it{q}}}}}\,$, are absolute values of the
Fourier modes:
$\displaystyle\chi_{\textstyle{{}_{220\mbox{\it{q}}}}}\,=~{}|\,\omega_{\textstyle{{}_{220\mbox{\it{q}}}}}\,|\,~{}.$
(12)
The dynamical Love number is an even function of the tidal mode
$\,\omega_{\textstyle{{}_{220\mbox{\it{q}}}}}\,$, while the phase lag is an
odd function. For this reason, the product
$\,k_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,$
can be rewritten as a function of the physical frequency $\,\chi\,$,
multiplied by the sign of the appropriate Fourier mode:
$\displaystyle
k_{2}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,=~{}k_{2}(\chi_{\textstyle{{}_{220\mbox{\it{q}}}}})\,\sin|\epsilon_{2}(\chi_{\textstyle{{}_{220\mbox{\it{q}}}}})|\,{\mbox{Sgn}}\,(\omega_{\textstyle{{}_{220\mbox{\it{q}}}}})\,~{}.$
(13)
Outside the inter-peak interval (i.e., at frequencies that are not too low),
the positive definite quantity 999 This quantity is often denoted as
$\,k_{2}/Q\,$, though notation $\,k_{2}\ Q_{2}\,$ would be more appropriate.
Tidal quality factors are not identical to the seismic quality factor, the
difference becoming crucial at low frequencies.
$~{}k_{2}(\chi)\,\sin|\epsilon_{2}(\chi)|\,$ is decreasing monotonically with
increase of the frequency $\,\chi=\chi_{\textstyle{{}_{220\mbox{\it{q}}}}}\,$.
This happens for two reasons. One, intuitively obvious, is that the dynamical
Love number decreases at higher frequencies, because materials are inertial,
and it is getting harder for their shape to keep up with the varying stress as
the frequency goes up. Less obvious is the circumstance that the sine of the
phase lag (i.e., the inverse tidal quality factor), too, decreases as the
frequency grows.101010 To illustrate the decrease of the Love number, imagine
that we dip a spoon into a bowl of honey, and apply to the spoon an
oscillating force of a fixed amplitude. Naturally, the amplitude of motion of
the spoon will be larger for lower frequencies. Sadly, this simple example
will not help us to illustrate how the phase decreases with the growth of
frequency. Naively, one might expect an anti-phase response at high
frequencies, like in the case of a damped driven harmonic oscillator. This
regime would indeed be taking place, had the mantle obeyed a constant time lag
law. However, real minerals behave differently, so our parallels with a
viscously damped oscillator or a viscous liquid have their limitations.
Supported by a mighty volume of seismological, geodetic, and laboratory data,
this behaviour may look counterintuitive because this is not what one would
expect from a viscous fluid. The fact however is that at physically
interesting frequencies the mantle behaves itself not as a viscous or a
Kelvin-Vogt body but as an Andrade body dissipation wherein obeys the law
$\,\sin\epsilon\propto\chi^{\,-\,\alpha}\,$, with $\,\alpha\approx 0.14-0.4\,$
for most solids (Efroimsky 2012a, 2012b).
Finally, the steep (but still continuous) near-resonant jumps connecting the
peaks of the kink are explained by the fact that at those locations self-
gravitation “beats” rheology (Ibid.).
The kink shape of $\,k_{2}\,\sin\epsilon_{2}\,$ (generally, of
$\,k_{l}\,\sin\epsilon_{l}\,$) entails somewhat counterintuitive consequences
for the phase lag and the geometric lag angle. Consider the principal,
semidiurnal bulge. Its phase lag and the geometric lag angle are
$\displaystyle\epsilon_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2200}}}}})\,=\,\omega_{\textstyle{{}_{2200}}}~{}\Delta
t_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2200}}}}})\,=\,2\,(n\,-\,\dot{\theta})~{}\Delta
t_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2200}}}}})\,$ (14)
and
$\displaystyle\delta_{2}(\omega_{\textstyle{{}_{2200}}})~{}=~{}\frac{1}{2}~{}|\,\epsilon_{2}(\omega_{\textstyle{{}_{2200}}})\,|\,=\,|\,n\,-\,\dot{\theta}\,|~{}\Delta
t_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2200}}}}})\,~{}.$ (15)
Were a planet composed of a material with the time lag $\,\Delta
t_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2200}}}}})\,$ insensitive to the
value of the principal tidal mode
$\,\omega_{\textstyle{{}_{2200}}}\,=\,2\,|n\,-\,\dot{\theta}|\,$, the
geometric lag angle would be larger for a higher value of this frequency. This
indeed is what one would, intuitively, expect: the higher the difference
between $\,\dot{\theta}\,$ and $\,n\,$ the larger the angle. However, for a
realistic rheology, an increase of the $\,\delta_{\textstyle{{}_{2200}}}\,$
angle due to an increase in $\,2\,|n\,-\,\dot{\theta}|~{}$ will take place
only within an extremely close proximity of the the $\,1:1\,$ resonance.
Stepping beyond the kink’s peak, we shall find that an increase in
$\,2\,|n\,-\,\dot{\theta}|~{}$ will be accompanied by such a decrease in the
time lag $\,\Delta t_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{2200}}}}})\,$
that the product of these two quantities will, overall, be decreasing with
growing frequency. So both the phase lag and the geometric lag angle will
become smaller.
### 4.4 Instability of pseudosynchronous rotation. Physical interpretation
Since the mode-dependence of the products (10) follows from the rheological
properties of the mantle and from self-gravitation of the planet, these
products turn out to be functions not only of the tidal mode
$\omega_{\textstyle{{}_{220\mbox{\it{q}}}}}\,$, but also of the parameters
defining the size and rheology of the perturbed body. These parameters
(presented in Table 1) are the body’s radius $\,R\,$ and mass $\,M_{2}\,$, as
well as its unrelaxed rigidity $\,\mu\,$, Maxwell time $\,\tau_{M}\,$,
inelastic (Andrade) time $\,\tau_{A}\,$, and the Andrade parameter
$\,\alpha\,$.
For a given selection of these parameters’ values and a fixed spin rate
$\,\dot{\theta}\,$, there is a unique eccentricity $\,e_{\rm equ}\,$ at which
$~{}\langle\,{\cal{T}}_{z}\rangle_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}=0\,$.
The dependence of $\,e_{\rm equ}\,$ on the relative rate of rotation
$\,\dot{\theta}/n\,$ can be found for a grid of points by numerically
determining the roots of
$~{}\langle\,{\cal{T}}_{z}\rangle_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}~{}$
in $\,e\,$. To accomplish this, we recall that each term in Equation (8) is a
polynomial in $\,e\,$ and, therefore, so is the entire right-hand side of (8).
Computational search of the roots was carried out for two sets of parameters
listed in Table 1, one representing the Moon orbiting the Earth, and the other
a hypothetical super-Earth orbiting a solar analog. The choice of parameters
is intended to represent the range of applicability of the model. The
resulting dependencies of $\,e_{\rm equ}\,$ upon $\,\dot{\theta}/n\,$ are
presented in Figures 1 and 2.
The jigsaw shape of the found dependencies $\,e_{\rm equ}(\dot{\theta}/n)\,$
is remarkably different from the predictions of the linear torque model. The
curves for the Moon (Figure 1) and the super-Earth (Figure 2) are
monotonically descending with a rising rate of rotation, everywhere between
the spin-orbit resonances. This has profound implications for the character of
equilibrium at a zero tidal torque. Consider an arbitrary point on the
downhill portion of the curve, e.g., the one in Figure 1, from which two short
arrows of opposite direction are drawn. A perturbation in $\,\dot{\theta}\,$
away from this point, whether spinning the planet up or slowing it down, will
cause a nonzero tidal torque acting in the same direction – as indicated by
the direction of the arrows. Thereby, the tidal equilibrium achieved at
$\,e_{\rm equ}\,$ between the resonances is inherently unstable. Similarly, a
perturbation in $\,e\,$ will make the planet diverge from the curve of zero
torque rather than return to it. Hence, the states of zero tidal torque at
non-resonant spin rates are transient by nature.
In a narrow vicinity of each spin-orbit resonance, $\,e_{\rm equ}\,$ takes a
rapid upward increase. Computation of the roots of Equation 8 is numerically
difficult in these areas because of a very large gradient of the curve. The
slope of the segments at the resonances is positive; therefore, there exists a
stable equilibrium just as in the case of the linear torque model. A deviation
of the spin rate from a resonant value gives rise to a nonzero restoring
torque, as indicated by a pair of inward arrows in Figure 1 at the resonance
3:2.
To obtain a physical explanation of the unstable nature of the
pseudosynchronous rotation, recall two circumstances. First, each term of the
sum (8) has the shape of a kink. Second, each such kink, as a function of
$\,\dot{\theta}\,$, is increasing monotonically everywhere except near the
appropriate resonance, as in Figure 3. For this reason, an infinitesimal
increase in $\,\dot{\theta}/n\,$ furnishes an infinitesimal increase in the
tidal torque (not necessarily in its absolute value). This happens for an
arbitrary value of $\,e\,$ and for the values of $\,\dot{\theta}/n\,$ outside
the narrow resonances. Specifically, for $\,e=e_{equ}\,$, the torque is zero
and acquires a positive value, which leads to further spin-up. The spin-up
continues until $\,\dot{\theta}/n\,$ stumbles into a resonance. (Resonances
are depicted with near-vertical segments of the dotted curves in Figures 1 and
2.)
### 4.5 On the choice of the values for physical parameters
Figures 1 and 2 reveal, in comparison, that the structure of the equilibrium
tidal torque is similar for a wide range of values of planetary parameters.
The values employed to build these plots are shown in Table LABEL:table. The
rapid jumps of $\,e_{\rm equ}\,$ at resonances and the smooth descents between
the resonances are characteristic of small moons and large planets likewise.
The values of $\,e_{\rm equ}\,$ at resonances appear to be the same for the
Moon and the model super-Earth. The most noticeable difference is in the
amplitude of the resonance jumps, which is significantly higher for the Moon.
By experimenting with the input parameters, we found out that this amplitude
is sensitive mainly to the Maxwell time $\,\tau_{M}\,$, which differs by an
order of magnitude between our model bodies.
The choice of such a small Maxwell time for the Moon, only 5 years, is
justified by the likely presence of a high percentage of partial melt in the
lower lunar mantle. The presence of partial melt follows from the modeling
carried out by Weber et al. (2011) and also from an earlier study by Nakamura
et al. (1974). There exists data pointing at the possibility of the lunar
Maxwell time being of the order of months. 111111 The smallness of the lunar
$\,\tau_{M}\,$ ensues from the unexpected frequency dependence of the lunar
tidal $\,Q\,$ factor. According to Williams et al. (2008), the tidal $\,Q\,$
increases from $\,\sim 29\,$ at a month to $\,\sim 35\,$ at one year, a slope
incompatible with the seismic properties of rocks which are expected to have a
seismic $\,Q\,$ decreasing with increase of the period. As explained in
Efroimsky (2012a), this unexpected slope may have emerged due to the
difference between the tidal and seismic $\,Q\,$ at low frequencies. It is
possible that the frequency range in which the lunar tides were studied could
be close to or slightly left of the peak of the function
$~{}k_{2}(\omega_{2200})\,\sin|\,\epsilon_{2}(\omega_{2200})\,|~{}$. In this
case, $\,\tau_{M}\,$ of the Moon may be of the order of days. Fortunately, the
choice of the value of $\,\tau_{M}\,$ does not influence considerably the
jagged shape of the dependency $\,e_{\rm equ}(\dot{\theta}/n)\,$. The
resulting plot turns out to be similar to the jagged plot for $\,\tau_{M}=5\,$
yr depicted in Figure 1.
The choice of the parameters of the super-Earth was consistent with that made
in Makarov et al. (2012) for the planet GJ581d.
The Arrhenius law requires that planets and moons with hotter interiors have
lower viscosity of mantles and, thus, have shorter Maxwell times. So we
surmise that the secular tidal torque for such objects should be relatively
more efficient in capturing at higher spin-orbit resonances.
## 5 Resonant rotation of axisymmetric bodies
We have determined that a stable spin-orbit equilibrium is achieved at spin
rates where the value of the secular polar tidal torque is zero, while the
derivative of the equilibrium eccentricity with respect to the spin rate,
$~{}{\textstyle de_{equ}}/{\textstyle d(\dot{\theta}/n)}~{}$, is negative.
With the realistic tidal model discussed in Appendix B, this may happen only
in the vicinity of spin-orbit resonances because the derivative of the torque
is positive elsewhere. The secular torque (8), as well as the angular
acceleration caused by it, has “kinks” in the vicinity of spin-orbit
commensurabilities $\,\dot{\theta}/n\,=\,1\,+\,q/2~{}$, with an integer
$\,q\,$ of either sign. An example thereof is shown in Figure 4 for the super-
Earth model, within a segment of the spin rate around
$\,\dot{\theta}/n=5/2\,$. The kink is comprised by a local maximum below the
resonance and a minimum above the resonance.
To understand the plot in Figure 4, recall that in the vicinity of each
resonance $\,q\,^{\prime}\,$, i.e., for $\,\dot{\theta}/n\,$ being close to
$~{}1\,+\,{\textstyle q\,^{\prime}}/{\textstyle 2}~{}$, the right-hand side of
(8) can be decomposed into two parts. One part is the $\,q=q\,^{\prime}\,$
term. It is a kink-looking odd function of the tidal mode $\,\omega_{220q}\,$,
and it goes through nil at exactly $\,\omega_{220q}=0~{}$. Due to (9), this
term can also be interpreted as a function of the spin rate, antisymmetric
with respect to the resonance point $\,\dot{\theta}/n\,=\,1\,+\,{\textstyle
q\,^{\prime}}/{\textstyle 2}\,$ where this term goes through nil. The second
part, called bias, is the rest of the sum, i.e., the input of all the $\,q\neq
q\,^{\prime}\,$ Fourier modes into the values assumed by the torque in the
vicinity of the $\,q=q\,^{\prime}\,$ resonance. The bias can be negative or
positive in value, depending on the eccentricity. For not too large
eccentricities, it is usually negative. Being a very slowly changing function
within the resonance interval, it can, to a good approximation, be assumed
constant there.
Having summed up all the terms in (8), and exploring the behaviour of this sum
near $\,\dot{\theta}/n\,=\,1\,+\,{\textstyle q\,^{\prime}}/{\textstyle 2}\,$,
we see that the resulting curve does not cross the horizontal axis at
$\,\dot{\theta}/n\,=\,1\,+\,{\textstyle q\,^{\prime}}/{\textstyle 2}\,$. One
can say that the bias slightly displaces the location of zeros. The zeros are
located close to resonances but not exactly in resonances.
In Figure 4(a), the values of the overall torque (in fact, of the total
angular acceleration proportional to the torque) are defined mostly by the
$\,q=3\,$ term which, in this vicinity, looks like an antisymmetric kink.
However, the curve is shifted down due to the bias which is defined mainly by
the right slope of the $\,q=1\,$ kink located to the left. Since the right
slope of the $\,q=1\,$ kink is negative, the $\,q=3\,$ kink is shifted down.
As a result, the maximum secular torque barely rises above zero, and the curve
has two zeros located to the left of the point $\,\dot{\theta}/n\,=\,5/2\,$,
close to the maximum of the kink. The interval of the resonance is defined by
the location of the peaks: $\,\dot{\theta}/n=2.4998\,$ and $\,2.5002\,$.
Figure 4(b) is a blow-up of Figure 4(a) showing in greater detail the area
around the maximum of the kink. The root of the function within the resonance
interval is at $\,2.49985\,$ rather than exactly $\,2.5\,$. For the same
reason, the torque value is negative at $\,\dot{\theta}/n=2.5\,$.
In the framework of this model, it is reasonable to define capture in
resonance as an equilibrium state in which the body’s average rotation rate
stays between the values corresponding to these two peaks. This definition is
adequate because, as we saw above, stable equilibrium is possible only on
negative slopes of the angular acceleration (or tidal torque) as a function of
the spin rate.
Figure 4: Angular acceleration of the super-earth (Table 1) caused by the
secular tidal torque in the vicinity of the 5:2 resonance, (a) showing the
entire resonance interval, and (b) showing in more detail the same curve in
the area of the local maximum.
When a triaxial planet is captured in a 5:2 resonance, its time-averaged spin
rate is exactly $\,2.5\,n\,$. Similarly, the Moon’s spin rate, captured in
synchronous rotation, is exactly $\,1\,n\,$. However, we know that the time-
averaged tidal torque is nonzero at this spin rate. Why does not the Moon
accelerate? For a triaxial body, the nonzero secular tidal torque is
compensated by a counteracting triaxiality-caused torque, through a small tilt
of the average inertia axis with respect to the line connecting the centres of
the bodies. A nonzero time-average tilt generates a secular triaxial torque.
This mechanism of torque compensation obviously does not work for the rather
hypothetical case of axisymmetric body, which would be subject to only tidal
forces. Would the Moon be facing the Earth always with the same side if it
were completely axisymmetric? First, we have to find out if capture in spin-
orbit resonance is at all possible. The answer is yes, as long as the secular
torque changes sign in the vicinity of that resonance 121212 Triaxial bodies
can be captured in spin-orbit resonances even if the secular tidal torque is
negative everywhere in the vicinity of that resonance.. Even though the
maximum torque in Figure 4 barely rises above zero at spin rates below the
resonance, it turns out to be sufficient for capture in 5:2. Figure 5 displays
the results of a numerical integration of the evolution of spin rate for the
super-Earth model (Table 1) at $\,\tau_{M}=50$ yr with an initial spin rate of
$\,\dot{\theta}(0)=2.51\,n\,$. The planet is captured in about 8500 yr, but
the equilibrium spin rate at $\,2.49985\,n\,$ is clearly below the resonance
value. This value is consistent with the root of the secular torque within the
resonance interval, Figure 4(b). Thus, the equilibrium resonance state of an
axisymmetric body is achieved at the spin rate where the secular tidal torque
equals zero, as expected.
Figure 5: Capture of an axisymmetric super-Earth (Table 1, but with
$(B-A)/C=0$) in 5:2 resonance. Note that the ultimate equilibrium spin rate is
slightly less than $2.5\,n$.
## 6 Word of caution
As was demonstrated above, stability of pseudosynchronous spin hinges upon
rheology. Being stable for the constant time lag model, the regime is expected
to be transient for realistic mantles, insofar as their
$\,k_{2}(\chi)\,\sin\epsilon_{2}(\chi)\,$ has one pronounced peak (not to
count the opposite one at the negative value of the tidal mode) – see Figure
3. The situation will have to be re-analysed for bodies of complex structure
(with surface or internal oceans), as well as for bodies of yet unexplored
rheologies. Specifically, if it happens that somewhere in universe there exist
bodies with not one but two pronounced peaks of $\,k_{2}\,\sin\epsilon_{2}\,$
at positive frequencies, the “ditch” between these peaks may, in principle,
lead to emergence of a pseudosynchronous rotation state. Such a peak may
emerge at the boundary of two frequency bands dominated by different friction
mechanisms, i.e., when a new mechanism is “turned on” very quickly with the
increase of frequency. Although highly hypothetical, such situations should
not be written off completely.
## 7 Discussion
The shortness of Earth’s day was undoubtedly beneficial for proliferation of
biological life, making the daily temperature variation moderate. The
situation may be drastically different for the growing class of potentially
habitable super-Earth exoplanets. Due to the observational selection effect,
the spectroscopically detected super-Earths are found mostly around lower-mass
stars, whose habitable zones are inevitably narrower and closer. For such
systems, any conclusion about potential habitability of a given exoplanet
becomes especially uncertain, and the analysis becomes intricately involved
with regard to such parameters as the amount of stellar irradiation, the
chemical composition of a hypothetical atmosphere, and the internal heating.
The rate of rotation is also a crucial parameter which for now remains
unavailable from observation. The most advanced climatic simulations are based
on a certain assumption of the spin-orbit state of the planet, e.g., a tidal
synchronisation (1:1 resonance) is assumed, as in Selsis et al (2011). A
tidally synchronised planet showing the same side to its host star has this
side always exposed to plentiful irradiation, the other side staying dark.
Such planets can hardly retain an atmosphere and can hardly be habitable.
However, a spin-orbit locking into higher commensurabilities (e.g., 3:2, as in
the case of Mercury) allows the planet to rotate with respect to the host
star, and leaves the planet a possibility of sustaining a stable atmosphere
and having water in the liquid form on the surface. Three-dimensional climatic
simulations of the potentially habitable super-Earth GJ 581d, by Wordsworth et
al. (2010), were performed for a set of possible spin-orbit resonances,
including 2:1. As was later explained in Makarov et al. (2012), this resonance
is the likeliest state of GJ 581d for a wide range of rheological parameters,
assuming a terrestrial composition of its mantle. Wordsworth et al. (2010)
drew attention to the fact that a tidally synchronised atmosphere may be
short-lived because of the collapse of CO2 on the night side. A similar
phenomenon may occur on a slowly rotating planet. Both the constant angular
lag tidal model and the constant time lag model predict that oblate planets
with moderate and large eccentricities are captured in stable
pseudosynchronous rotations, in which case their spin rate only slightly
exceeds their orbital mean motion. This would make an entire class of detected
exoplanets unsuitable for biological life. In this paper we prove that the
prediction of pseudo-synchronism is germane to the above-mentioned simplified
models of tidal interactions, models inapplicable to solid planets or moons.
Super-Earth exoplanets of Earth-like composition on eccentric orbits are
likely to be captured into spin-orbit resonances higher than 1:1, but there is
no such thing as pseudo-synchronous rotation for these objects.
## Acknowledgments
The authors would like to express their thanks to the referee, Stanton Peale,
who provided several incisive comments and criticisms, and who urged the
authors to present physical interpretation of the results obtained in the
paper. One of the authors (ME) is indebted to Sylvio Ferraz Mello and James G.
Williams for numerous stimulating discussions on the topic of this work.
This research has made use of NASA’s Astrophysics Data System.
## Appendix A. The tidal torque
The additional potential $\,U\,$ of a tidally perturbed body can be expanded
into a Fourier series over the tidal modes
$\displaystyle\omega_{lmpq}\;\equiv\;({\it l}-2p)\;\dot{\omega}\,+\,({\it
l}-2p+q)\;{\bf{\dot{\cal{M}}}}\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})\,\approx\,(l-2p+q)\,n\,-\,m\,\dot{\theta}\,~{},~{}~{}~{}$
(16)
where $\,{\theta}\,$ and $\,\dot{\theta}\,$ are the sidereal angle and
rotation rate of the body, while $\,\omega\,$, $\,\Omega\,$, $\,n\,$, and
$\,{\cal{M}}\,$ are the periapse, the node, the mean motion, and the mean
anomaly of the perturber as seen from the perturbed body.
While the tidal modes $\,\omega_{\textstyle{{}_{lmpq}}}\,$ can be of either
sign, the forcing frequencies
$\displaystyle\chi_{lmpq}\,=~{}|\,\omega_{lmpq}\,|~{}\approx~{}|~{}(l-2p+q)~{}n\,-\,m~{}\dot{\theta}~{}|$
(17)
at which the strain and stress oscillate are positive-definite.
The series expansion of the additional potential $\,U\,$ was developed by
Kaula (1964), its partial sum known yet to Darwin (1879). Therefore the series
for $\,U\,$ and the resulting series for the torque are often named the
Darwin-Kaula expansions.
An accurate derivation of the expansion for the torque demonstrates that the
torque contains both a rapidly oscillating and a secular part (Efroimsky
2012a). Having a zero orbital average, the oscillating part nevertheless may
play a role in dissipation of free librations. In Makarov et al. (2012) it was
explored whether the oscillating part of the torque can influence capture into
resonances. Changing the outcome of a particular realisation of the capture
scenario, the oscillating part did not alter the statistics.
The secular part of the polar torque looks as
$\displaystyle\langle\,{\cal{T}}_{z}\rangle\,=\,2\,G\,M_{star}^{{{\,2}}}\sum_{{\it{l}}=2}^{\infty}\frac{R^{\textstyle{{}^{2l\,+\,1}}}}{a^{\textstyle{{}^{2l\,+\,2}}}}\sum_{m=0}^{l}\frac{(l-m)!}{(l+m)!}\;m\sum_{p=0}^{l}F^{\,2}_{lmp}(i)\sum^{\it\infty}_{q=-\infty}G^{\,2}_{lpq}(e)\;k_{l}(\omega_{lmpq})\;\sin\epsilon_{l}(\omega_{lmpq})\,~{},\quad$
(18)
where the angular brackets signify orbital averaging, $\,G\,$ denotes Newton’s
gravity constant, $\,a,\,i,\,e\,$ are the semimajor axis, inclination (or
obliquity), and eccentricity, while $\,F_{lmp}(i)\,$ and $\,G_{lpq}(e)\,$ are
the inclination functions and eccentricity polynomials. The Love numbers
$\,k_{\textstyle{{}_{l}}}\,$ and the phase lags
$\,\epsilon_{\textstyle{{}_{l}}}\,$ depend on the modes (16).
In the Darwin-Kaula theory, the phase lags come into being as products of the
modes $\,\omega_{lmpq}\,$ by the corresponding time delays:
$\displaystyle\epsilon_{l}(\omega_{lmpq})\,=\,\omega_{lmpq}~{}\,\Delta
t_{l}(\omega_{lmpq})\,~{},$ (19)
where, for causality reasons, the time lags $\,\Delta
t_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$ are positive-
definite. Therefore, (19) may be rewritten as
$\displaystyle\epsilon_{l}(\omega_{lmpq})\,=\,\chi_{lmpq}~{}\,\Delta
t_{l}(\omega_{lmpq})~{}\,\mbox{Sgn}\,(\,\omega_{lmpq}\,)\,~{},$ (20)
$\chi_{lmpq}\,$ being the physical forcing frequencies (17). As a result of
this, the entire expression for the polar component of the torque can be
written down as
$\displaystyle\langle\,{\cal{T}}_{z}\rangle\,=~{}~{}~{}\quad~{}\quad~{}\quad~{}\quad~{}~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad$
$\displaystyle
2\,GM_{star}^{{{\,2}}}\sum_{{\it{l}}=2}^{\infty}\frac{R^{\textstyle{{}^{2l\,+\,1}}}}{a^{\textstyle{{}^{2l\,+\,2}}}}\sum_{m=0}^{l}\frac{(l-m)!}{(l+m)!}\;m\sum_{p=0}^{l}F^{\,2}_{lmp}(i)\sum^{\it\infty}_{q=-\infty}G^{\,2}_{lpq}(e)\;k_{l}(\omega_{lmpq})~{}\sin|\,\epsilon_{l}(\omega_{lmpq})\,|\,~{}\mbox{Sgn}\,\left(\,\omega_{lmpq}\,\right)\,~{}.\quad$
(21)
Be mindful that, similar to the Love numbers, we prefer to denote the lags
with 131313 Although Kaula (1964) denoted the phase lags with
$\,\epsilon_{{\textstyle{{}_{lmpq}}}}\,$, the notation
$\,\epsilon_{l}(\omega_{{\textstyle{{}_{lmpq}}}})\,$ is more logical. It
serves to emphasise the fact that for a homogeneous near-spherical body the
functional dependence of a lag upon the tidal mode is defined by the degree
$\,l\,$ solely, while the dependence of the lag upon $\,m,\,p,\,q\,$ comes
about only due to the tidal mode $~{}\omega_{{\textstyle{{}_{lmpq}}}}~{}$
being dependent on these integers. While in the case of triaxial bodies the
functional form of the lags is parameterised by all the four integers, for
small triaxiality this complication may be ignored.
$\,\epsilon_{l}(\omega_{lmpq})\,$ and $\,\Delta t_{l}(\omega_{lmpq})\,$, and
not with $\,\epsilon_{lmpq}\,$ and $\,\Delta t_{lmpq}\,$.
When the bodies are not too close (${\textstyle R}/{\textstyle a}\ll 1$), we
drop the terms with $\,l>2\,$. For small obliquities ($i\simeq 0$), we leave
only $\,i-$independent terms (the next-order terms being quadratic in
$\,i\,$). Finally, for eccentricities not exceeding $\,\sim\,0.3\,$, only the
terms up to $\,e^{7}\,$ are important. Formally, this would imply summation
over $\,q\,=\,-\,7,\,.\,.\,.7\,$. However, the term with $\,q\,=\,-\,2\,$
vanishes identically, while those with $\,q\,=\,-\,7,\,.\,.\,.\,-\,3\,$ are
accompanied with extremely small numerical factors and can thus be dropped. So
the polar component of the torque is approximated with
$\displaystyle\langle\,{\cal{T}}_{z}\rangle_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}~{}=~{}~{}\quad~{}\quad~{}~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad$
$\displaystyle\frac{3}{2}~{}G\,M_{star}^{\,2}\,R^{5}\,a^{-6}\sum_{q=-1}^{7}\,G^{\,2}_{\textstyle{{}_{\textstyle{{}_{20\mbox{\it{q}}}}}}}(e)~{}k_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}})~{}\sin|\,\epsilon_{2}(\omega_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}})\,|\,~{}\mbox{Sgn}\,\left(\,\omega_{220q}\,\right)+O(e^{8}\,\epsilon)+O(i^{2}\,\epsilon)~{}~{},~{}\quad~{}\quad~{}$
(22)
For the first time, this expression (with a sum running over all integer
$\,q\,=\,-\,\infty,\,.\,.\,.\,\infty\,$) was written, with no proof, by
Goldreich & Peale (1966). A sketch of a proof was later suggested by
Dobrovolskis (2007).
The functional form of the dependence of the factors
$~{}k_{l}\,\sin\epsilon_{l}~{}$ upon the mode
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ is determined by the size and mass of the
body and by its rheological properties. By rheology we imply the so-called
constitutive equation of the medium, i.e., an equation interconnecting the
strain and stress. For linear deformations, such equations can be rewritten in
the frequency domain where each harmonic mode of the strain becomes expressed
algebraically through the appropriate harmonic mode of the stress. Using the
method explained in Efroimsky (2012a, 2012b), the algebraic relations can be
used to find the shape of the functions
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$
standing in the terms of the Darwin-Kaula expansion of the tidal torque.
Calculations of the shapes of these functions, presented in Ibid., are based
on a combined rheological model (Andrade at higher frequencies, Maxwell at
lower frequencies), because of this model’s ability to best match laboratory
experiments and both seismic and geodetic measurements of dissipation over a
broad range of frequencies in the solid Earth. It is reasonable to assume that
the model is applicable to the mantles of other terrestrial planets and moons.
As demonstrated in Ibid., this combined model furnishes for
$\,k_{l}~{}\sin\epsilon_{l}\,$ a kink-shaped dependence upon the Fourier mode
– as in Figure 4.
## Appendix B. Calculation of the factors
$~{}k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}$
As explained in Efroimsky (2012a, 2012b), the products
$~{}k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}$
can be expressed through the mass and radius of the planet, and the real and
imaginary parts of the complex compliance of its mantle. Thereby, the shape of
the functional dependence of $~{}k_{l}~{}\sin\epsilon_{l}~{}$ upon
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ is defined by both the self-gravitation
of the planet and its rheological properties. The functions turn out to be
odd. They go continuously through nil, changing their sign, when the argument
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ goes through nil, i.e., when a
commensurability is crossed.
These odd functions can then be written down as
$~{}k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin|\,\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,|~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{lmpq}}}\,)~{}$.
Here the product
$~{}k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin|\,\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,|~{}$
is an even function of the tidal mode and can thus be treated as a function
not of the tidal mode $\,\omega_{\textstyle{{}_{lmpq}}}\,$ but of its absolute
value
$\,\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$,
which is the actual frequency of the oscillating stress in the mantle:
$\displaystyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})$
$\displaystyle=$ $\displaystyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin|\,\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,|~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{lmpq}}}\,)$
(23) $\displaystyle=$ $\displaystyle
k_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin|\,\epsilon_{l}(\chi_{\textstyle{{}_{lmpq}}})\,|~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{lmpq}}}\,)\,~{}.$
The development in Ibid. results in the following frequency dependence:
$\displaystyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})\;\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,=\;\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{-\;A_{l}\;J\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]}{\left(\;{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\;+\;A_{l}\;J\;\right)^{2}\;+\;\left(\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\;\right)^{2}}~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{lmpq}}}\,)~{}~{}~{},~{}~{}~{}~{}~{}$
(24)
where $\,\chi\,$ is a shortened notation for the frequency
$\,\chi_{\textstyle{{}_{lmpq}}}\,$, while coefficients $\,A_{l}\,$ are given
by
$\displaystyle A_{\it
l}\,\equiv\;\frac{\textstyle{(2\,{\it{l}}^{\,2}\,+\,4\,{\it{l}}\,+\,3)\,{\mu}}}{\textstyle{{\it{l}}\,\mbox{g}\,\rho\,R}}\;=\;\frac{\textstyle{3\;(2\,{\it{l}}^{\,2}\,+\,4\,{\it{l}}\,+\,3)\,{\mu}}}{\textstyle{4\;{\it{l}}\,\pi\,G\,\rho^{2}\,R^{2}}}\;\;\;.~{}~{}~{}~{}~{}~{}~{}$
(25)
with $\,G\,$ being the Newton gravitational constant, and $\,R\,$, $\,\rho\,$,
$\,\mu\,$, and g being the radius, mean density, unrelaxed rigidity, and
surface gravity of the planet. The functions
$\displaystyle{\cal R}{\it
e}[\bar{J}(\chi)]\;=\;J\;+\;J\,(\chi\tau_{{}_{A}})^{-\alpha}\;\cos\left(\,\frac{\alpha\,\pi}{2}\,\right)\;\Gamma(\alpha\,+\,1)~{}~{}~{}~{}\quad\quad\quad\quad\quad~{}\quad\quad\quad\quad\quad\quad\quad$
(26)
and
$\displaystyle{\cal I}{\it
m}[\bar{J}(\chi)]\;=\;-\;J~{}(\chi\tau_{{}_{M}})^{-1}\;-\;J\,(\chi\tau_{{}_{A}})^{-\alpha}\;\sin\left(\,\frac{\alpha\,\pi}{2}\,\right)\;\Gamma(\alpha\,+\,1)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\quad\quad$
(27)
are the real and imaginary parts of the complex compliance $\,\bar{J}(\chi)\,$
of the mantle. Here $\,\alpha\,$ is the Andrade parameter assuming values of
about $\,0.3\,$ for solid silicates and about $\,0.14-0.2\,$ for partial
melts. In our computations, we used $\,\alpha=0.2\,$. The quantity $\,J\,$ is
the unrelaxed compliance of the mantle, which is the inverse of the mantle’s
unrelaxed rigidity $\,\mu\,$. The parameters $\,\tau_{{}_{M}}\,$ and
$\,\tau_{{}_{A}}\,$ are typical timescale characterising the mantle’s
viscoelastic and inelastic response, correspondingly.
The Maxwell time $\,\tau_{{}_{M}}\,$ is the ratio of the mantle’s viscosity
$\,\eta\,$ to its rigidity $\,\mu\,$. While for the Earth’s mantle it has a
value of about 500 years, it may be much shorter for warmer planets and moons
due to the exponential temperature-dependence of the viscosity.
The inelastic (Andrade) time $\,\tau_{{}_{A}}\,$ is expected to be of the same
order as or lower than $\,\tau_{{}_{M}}\,$, over the frequencies higher than
some threshold. For these frequencies then, the inelastic (containing
$\,\tau_{{}_{A}}\,$) terms in (26 \- 27) will be comparable to or larger than
the viscoelastic (containing $\,\tau_{{}_{M}}\,$) terms. However, at
frequencies below the threshold, inelasticity ceases to play a major role in
the internal friction, giving way to viscous friction which becomes dominant.
Therefore at very low frequencies the mantle’s behaviour approaches that of a
Maxwell body. Mathematically, this implies that below the threshold the
parameter $\,\tau_{{}_{A}}\,$ increases rapidly as the frequency goes down
(Efroimsky 2012a, 2012b). So only the first term in (26) and the first term in
(27) are important, and we arrive at the complex compliance of a Maxwell
material. The location of the frequency threshold may vary considerably for
different planets. For the Earth, it is of the order of 1 yr${}^{-1}\,$
(Karato & Spetzler 1990).
Numerical computations show that the probabilities of capture into resonances
are not very sensitive to the value of $\,\tau_{{}_{A}}\,$, nor to the
location of the threshold, nor to how quickly inelasticity yields to
viscoelasticity with the decrease of frequency. In our numerics, we treat
$\,\tau_{{}_{A}}\,$ in the same way as in Makarov et al. (2012) and Makarov
(2012). We set the threshold to be the same as for the solid Earth, 1
yr${}^{-1}\,$. We then kept $\,\tau_{{}_{A}}\,=\,\tau_{{}_{M}}\,$ over the
frequencies above the threshold. For frequencies lower than the threshold, we
set $\,\tau_{{}_{A}}\,$ to grow exponentially with the decrease of the
frequency. This way, at low frequencies the rheological model approaches the
Maxwell one exponentially. For details, see Ibid.
Writing a code, it is easier to divide both the numerator and denominator of
(24) by $\,J^{\,2}\,$:
$\displaystyle
k_{l}(\,\omega_{\textstyle{{}_{lmpq}}}\,)\;\sin\epsilon_{l}(\,\omega_{\textstyle{{}_{lmpq}}}\,)\;=\;\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{-\;A_{l}\;{\cal{I}}}{\left(\;{\cal{R}}\;+\;A_{l}\;\right)^{2}\;+\;{\cal{I}}^{\textstyle{{}^{\,2}}}}~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{lmpq}}}\,)~{}~{}~{},~{}~{}~{}~{}~{}$
(28)
where $\,{\cal R}\,$ and $\,{\cal I}\,$ are the dimensionless real and
imaginary parts of the complex compliance:
$\displaystyle{\cal
R}\;=\;1\;+\;(\chi\tau_{{}_{A}})^{-\alpha}\;\cos\left(\,\frac{\alpha\,\pi}{2}\,\right)\;\Gamma(\alpha\,+\,1)~{}~{}~{},\quad\quad\quad\quad\quad~{}\quad\quad\quad\quad\quad\quad\quad$
(29) $\displaystyle{\cal
I}\;=\;-\;(\chi\tau_{{}_{M}})^{-1}\;-\;(\chi\tau_{{}_{A}})^{-\alpha}\;\sin\left(\,\frac{\alpha\,\pi}{2}\,\right)\;\Gamma(\alpha\,+\,1)~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\quad\quad\quad\quad$
(30)
$\chi\,$ being a short notation for the physical forcing frequency
$\,\chi_{\textstyle{{}_{lmpq}}}\,\equiv\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$.
## References
* [1] Bambusi, D., & Haus, E. 2012. “Asymptotic stability of synchronous orbits for a gravitating viscoelastic sphere.” Celestial Mechanics and Dynamical Astronomy, Vol. 114, pp. 255 - 277
http://arxiv.org/abs/1012.4974
* [2] Darwin, G. H. 1879. “On the precession of a viscous spheroid and on the remote history of the Earth.” Philosophical Transactions of the Royal Society of London, Vol. 170, pp. 447 - 530
* [3] Dobrovolskis, A. 2007. “Spin states and climates of eccentric exoplanets.” Icarus, Vol. 192, pp. 1 - 23
* [4] Efroimsky, M., & Lainey, V. 2007. “The Physics of Bodily Tides in Terrestrial Planets, and the Appropriate Scales of Dynamical Evolution.” Journal of Geophysical Research – Planets, Vol. 112, id. E12003. doi:10.1029/2007JE002908
http://arxiv.org/abs/0709.1995
* [5] Efroimsky, M., & Williams, J. G. 2009. “Tidal torques. A critical review of some techniques.” Celestial Mechanics and Dynamical Astronomy, Vol. 104, pp. 257 - 289
http://arxiv.org/abs/0803.3299
* [6] Efroimsky, M. 2012 a. “Bodily tides near spin-orbit resonances.” Celestial Mechanics and Dynamical Astronomy, Vol. 112, pp. 283 - 330. Extended version: http://arxiv.org/abs/1105.6086
* [7] Efroimsky, M. 2012 b. “Tidal dissipation compared to seismic dissipation: in small bodies, earths, and superearths.” The Astrophysical Journal, Vol. 746, id. 150.
doi:10.1088/0004-637X/746/2/150 http://arxiv.org/abs/1105.3936
ERRATA: ApJ, Vol. 763, id. 150 (2013)
* [8] Efroimsky, M., and Makarov, V. V. 2013. “Tidal Friction and Tidal Lagging. Applicability Limitations of a Popular Formula for the Tidal Torque.” The Astrophysical Journal, Vol. 764, id. 26
http://arxiv.org/abs/1209.1615
* [9] Eggleton, P. P.; Kiseleva, L. G.; and Hut, P. 1998. “The equilibrium tide model for tidal friction.” The Astrophysical Journal, Vol. 499, pp. 853 - 870
* [10] Eggleton, P.P. 2011. “The Equilibrium Tide and Tidal Friction”, priv. comm., available from author upon request
* [11] Fekel, F.C.; Browning, J.C.; Henry, G.W.; Morton, M.D.; and Hall, D.S. 1993\. “Chromospherically active stars. X - Spectroscopy and photometry of HD 212280.” The Astronomical Journal, Vol. 105, pp. 2265 - 2275
* [12] Fekel, F.C., et al. 1998. “Chromospherically Active Stars. XVII. The Double-lined Binary 54 Camelopardalis (AE Lyncis).” The Astronomical Journal, Vol. 115, pp. 1153 - 1159
* [13] Ferraz-Mello, S. 2012. “Tidal synchronisation of close-in satellites and exoplanets. A rheophysical approach.” Submitted to Celestial Mechanics and Dynamical Astronomy
http://arxiv.org/abs/1204.3957
* [14] Garaud, P.; Ogilvie, G. I.; Miller, N.; and Stellmach, S. 2010. “A model of the entropy flux and Reynolds stress in turbulent convection.” Monthly Notices of the Royal Astronomical Society of London, Vol. 407, pp. 2451 - 2467
http://arxiv.org/abs/1004.3239
* [15] Goldreich, P. 1966. “Final spin states of planets and satellites.” _The Astronomical Journal_. Vol. 71, pp. 1 - 7
* [16] Goldreich, P., and Peale, S. 1966. “Spin-Orbit Coupling in the Solar System.” The Astronomical Journal, Vol. 71, pp. 425 - 438
* [17] Hall, D.S. 1986. “Pseudosynchronisation found in binaries with eccentric orbits.” The Astrophysical Journal – Letters to the Editor, Vol. 309, pp. L83 - L85
* [18] Hut, P. 1981. “Tidal evolution in close binary systems.” Astronomy & Astrophysics, Vol. 99, pp. 126 - 140
* [19] Käpylä, P. J., & Brandenburg, A. 2008. “Lambda effect from forced turbulence simulations.” The Astrophysical Journal, Vol. 488, pp. 9 - 23
* [20] Karato, S.-i., and Spetzler, H. A. 1990. “Defect Microdynamics in Minerals and Solid-State Mechanisms of Seismic Wave Attenuation and Velocity Dispersion in the Mantle.” Reviews of Geophysics, Vol. 28, pp. 399 - 423
* [21] Kaula, W. M. 1964. “Tidal Dissipation by Solid Friction and the Resulting Orbital Evolution.” Reviews of Geophysics, Vol. 2, pp. 661 - 684
* [22] Kichatinov, L. L. 2005. “The differential rotation of stars.” Physics – Uspekhi, Vol. 88, pp. 449 - 467
* [23] Makarov, V. V. 2012. “Conditions of passage and entrampent of terrestrial planets in spin-orbit resonances.” The Astrophysical Journal, Vol. 752, id. 73.
doi:10.1088/0004-637X/752/1/73 http://arxiv.org/abs/1110.2658
* [24] Makarov, V. V.; Berghea, C.; and Efroimsky, M. 2012. “Dynamical evolution and spin-orbit resonances of potentially habitable exoplanets. The case of GJ 581d.” The Astrophysical Journal, Vol. 761, id. 83.
http://arxiv.org/abs/1208.0814
ERRATUM: ApJ, Vol. 763, id. 68
* [25] MacDonald, G. J. F. 1964. “Tidal Friction.” Reviews of Geophysics. Vol. 2, pp. 467 - 541
* [26] Miguel, M.-C.; Vespignani, A.; Zaiser, M.; and Zapperi, S. 2002. “Dislocation Jamming and Andrade Creep.” _Physical Review Letters_ , Vol. 89, pp. 165501-1 - 165501-4.
* [27] Murray, C.D., and Dermott, S.F. 1999. Solar System Dynamics. Cambridge University Press, Cambridge UK
* [28] Nakamura, Y.; Latham, G.; Lammlein, D.; Ewing, M.; Duennebier, F.; and Dorman, J. 1974. “Deep lunar interior inferred from recent seismic data.” Geophysical Research Letters, Vol. 1, pp. 137 - 140.
* [29] Ogilvie, G. I. 2008. “James Clerk Maxwell and the dynamics of astrophysical discs.” Philosophical Transactions of the Royal Society A, Vol. 366, pp. 1707 - 1715
http://rsta.royalsocietypublishing.org/content/366/1871/1707.full.html
* [30] Rudiger Rüdiger, G. 1989. Differential Rotation and Stellar Convection. Gordon and Breach, NY
* [31] Segatz, M.; Spohn, T.; Ross, M.N.; and Shubert, G. 1988. “Tidal dissipation, surface heat flow, and figure of viscoelastic models of Io.” Icarus, Vol. 75, pp. 187 - 206
* [32] Selsis, F.; Wordsworth, R.D.; & Forget, F. 2011. “Thermal phase curves of nontransiting terrestrial exoplanets. I. Characterising atmospheres.” Astronomy & Astrophysics, Vol. 532, article id. A1
* [33] Strassmeier, K.G.; Carroll, T.A.; Weber, M.; Granzer, T.; Bartus, J.; Oláh, K.; and Rice, J. B. 2011. “Binary-induced magnetic activity?. Time-series echelle spectroscopy and photometry of HD 123351 = CZ CVn.” Astronomy & Astrophysics, Vol. 535, id. A98
* [34] Torres, G.; Andersen, J.; and Giménez, A. 2010. “Accurate masses and radii of normal stars: modern results and applications.” The Astronomy and Astrophysics Review, Vol. 18, pp. 67 - 126
* [35] Touma, J., and Wisdom, J. 1994. “Evolution of the Earth-Moon system.” _The Astronomical Journal._ Vol. 108, pp. 1943 - 1961.
* [36] Weber, R. C.; Lin, Pei-Ying; Garnero, E.; Williams, Q.; and Lognonné, P. 2011. “Seismic Detection of the Lunar Core.” Science, Vol. 331, Issue 6015, pp. 309 - 312.
* [37] Williams, J. G., Boggs, D. H., and Ratcliff, J. T. 2008. “Lunar Tides, Fluid Core and Core/Mantle Boundary.” The 39th Lunar and Planetary Science Conference, (Lunar and Planetary Science XXXIX), held on 10-14 March 2008 in League City, TX. LPI Contribution No. 1391., p. 1484
http://www.lpi.usra.edu/meetings/lpsc2008/pdf/1484.pdf
* [38] Williams, J. G., and Efroimsky, M. 2012. “Bodily tides near the 1:1 spin-orbit resonance. Correction to Goldreich’s dynamical model.” Celestial Mechanics and Dynamical Astronomy, Vol. 114, pp. 387 - 414
http://arxiv.org/abs/1210.2923
* [39] Williams, P. T. 2004. “Turbulent magnetohydrodynamic elasticity: Boussinesq-like approximations for steady shear.” New Astronomy, Vol. 10, p. 133-144
http://arxiv.org/abs/astro-ph/0212556
* [40] Williams, P. T. 2005 “Three routes to jet collimation by the Balbus-Hawley magnetorotational instability.” Monthly Notices of the Royal Astronomical Society of London, Vol. 361, pp. 345 - 356
http://arxiv.org/abs/astro-ph/0506184
* [41] Williams, P. T. 2006 “Turbulent Elasticity of the Solar Convective Zone and the Taylor Number Puzzle.” In: Solar MHD Theory and Observations: A High Spatial Resolution Perspective. Edited by H. Uitenbroek, J. Leibacher, and R. F. Stein. ASP Conference Series, Vol. 354, pp. 85 - 91
http://arxiv.org/abs/astro-ph/0602502 ,
http://adsabs.harvard.edu/full/2006ASPC..354…85W
* [42] Wordsworth, R.D.; Forget, F.; Selsis, F.; Madeleine, J.-B.; Millour, E.; and Eymet, V. 2010. “Is Gliese 581d habitable? Some constraints from radiative-convective climate modeling.” Astronomy & Astrophysics, Vol. 522, article id. A22
|
arxiv-papers
| 2012-09-07T19:13:51 |
2024-09-04T02:49:34.891388
|
{
"license": "Public Domain",
"authors": "Valeri V. Makarov and Michael Efroimsky",
"submitter": "Michael Efroimsky",
"url": "https://arxiv.org/abs/1209.1616"
}
|
1209.1767
|
# Perturbation analysis of $A_{T,S}^{(2)}$ on Hilbert spaces
Fapeng Du
School of Mathematical & Physical Sciences, Xuzhou Institute of Technology
Xuzhou 221008, Jiangsu Province, P.R. China
E-mail: [email protected] Yifeng Xue
Department of mathematics, East China Normal University
Shanghai 200241, P.R. China Corresponding author, E-mail:
[email protected]
###### Abstract
In this paper, we investigate the perturbation analysis of $A_{T,S}^{(2)}$
when $T,\,S$ and $A$ have some small perturbations on Hilbert spaces. We
present the conditions that make the perturbation of $A_{T,S}^{(2)}$ is
stable. The explicit representation for the perturbation of $A_{T,S}^{(2)}$
and the perturbation bounds are also obtained.
2000 Mathematics Subject Classification: 15A09, 47A55
Key words: gap, subspace, Moore–Penrose inverse, stable perturbation
## 1 Introduction
Let $X,Y$ be Banach spaces and let $B(X,Y)$ denotes the set of bounded linear
operators from $X$ to $Y$. For an operator $A\in B(X,Y)$, let $R(A)$ and
$N(A)$ denote the range and kernel of $A$, respectively. Let $T$ be a closed
subspace of $X$ and $S$ be a closed subspace of $Y$. Recall that
$A_{T,S}^{(2)}$ is the unique operator $G$ satisfying
$GAG=G,\quad R(G)=T,\quad N(G)=S.$ (1.1)
It is known that (1.1) is equivalent to the following condition:
$N(A)\cap T=\\{0\\},\quad AT\dotplus S=Y$ (1.2)
(cf. [5, 6]). It is well–known that the commonly five kinds of generalized
inverse: the Moore–Penrose inverse $A^{+}$, the weighted Moore–Penrose inverse
$A^{+}_{MN}$, the Drazin inverse $A^{D}$, the group inverse $A^{\\#}$ and the
Bott–Duffin inverse $A^{(-1)}_{(L)}$ can be reduced to a $A_{T,S}^{(2)}$ for
certain choices of $T$ and $S$.
The perturbation analysis of $A_{T,S}^{(2)}$ have been studied by several
authors (see [12, 13], [16, 17]) when $X$ and $Y$ are of finite–dimensional. A
lot of results about the error bounds have been obtained. When $X$ and $Y$ are
of infinite–dimensional Banach spaces, the perturbation analysis of
$A_{T,S}^{(2)}$ for small perturbation of $T$, $S$ and $A$ has been done in
[7].
In this paper, we assume that $X$ and $Y$ are all Hilbert spaces over the
complex field $\mathbb{C}$. Using the theory of stable perturbation of
generalized inverses established by G. Chen and Y. Xue in [2, 3], we will give
the upper bounds of $\|\bar{A}_{T^{\prime},S^{\prime}}^{(2)}\|$ and
$\|\bar{A}_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|$ respectively for
certain $T^{\prime}$, $S^{\prime}$ and $\bar{A}$. The results in this paper
improve [14, Theorem 4.4.7].
## 2 Preliminaries
Let $H$ be a complex Hilbert space. Let $V$ be a closed subspace of $H$. We
denote by $P_{V}$ the orthogonal projection of $H$ onto $V$. Let $M,\,N$ be
two closed subspaces in $H$. Set
$\delta(M,N)=\begin{cases}\sup\\{dist(x,N)\,|\,x\in
M,\,\|x\|=1\\},\quad&M\not=\\{0\\}\\\ 0\quad&M=\\{0\\}\end{cases},$
where $dist(x,N)=\inf\\{\|x-y\|\,|\,y\in N\\}$. The gap $\hat{\delta}(M,N)$ of
$M,\,N$ is given by $\hat{\delta}(M,N)=\max\\{\delta(M,N),\delta(N,M)\\}$. For
convenience, we list some properties about $\delta(M,N)$ and
$\hat{\delta}(M,N)$ which come from [9] as follows.
###### Proposition 2.1 ([9]).
Let $M,\,N$ be closed subspaces in a Hilbert space $H$.
1. $(1)$
$\delta(M,N)=0$ if and only if $M\subset N$
2. $(2)$
$\hat{\delta}(M,N)=0$ if and only if $M=N$
3. $(3)$
$\hat{\delta}(M,N)=\hat{\delta}(N,M)$
4. $(4)$
$0\leq\delta(M,N)\leq 1$, $0\leq\hat{\delta}(M,N)\leq 1$
5. $(5)$
$\hat{\delta}(M,N)=\|P_{M}-Q_{N}\|$.
Let $A\in B(X,Y)$. If there is $C\in B(Y,X)$ such that $ACA=A$ and $CAC=C$, we
call $C$ is a generalized inverse of $A$ and is denoted by $A_{GI}^{+}$. In
this case, $R(A)$ is closed in $Y$.
Recall that $A$ is Moore–Penrose invertible, if there is $B\in B(Y,X)$ such
that
$ABA=A,\ BAB=B,\ (AB)^{*}=AB,\ (BA)^{*}=BA.$ (2.1)
The operator $B$ in (2.1) is called the Moore–Penrose inverse of $A$ and is
denoted as $A^{+}$. It is well–known that $A$ is Moore–Penrose invertible iff
$R(A)$ is closed in $Y$. Thus, $A$ is Moore–Penrose invertible iff
$A_{GI}^{+}$ exists.
Let $A,\,\delta A\in B(X,Y)$ and put $\bar{A}=A+\delta A$. Recall that
$\bar{A}$ is the stable perturbation of $A$ if $R(\bar{A})\cap
R(A)^{\perp}=\\{0\\}$.
The next lemma illustrates some equivalent conditions of the stable
perturbation.
###### Lemma 2.2 ([15, 8]).
Let $A\in B(X,Y)$ with $R(A)$ closed and $\delta A\in B(X,Y)$ with
$\|A^{+}\|\|\delta A\|<1$. Put $\bar{T}=T+\delta T$.
$(A)$ The following conditions are equivalent.
1. $(1)$
$R(\bar{A})\cap R(A)^{\perp}=\\{0\\}$
2. $(2)$
$N(\bar{A})^{\perp}\cap N(A)=\\{0\\}$
3. $(3)$
$R(\bar{A})$ is closed and $\bar{A}^{+}_{GI}=A^{+}(I+\delta
AA^{+})^{-1}=(I+A^{+}\delta A)^{-1}A^{+}$
$(B)$ If $\bar{A}$ is the stable perturbation of $A$, then $R(\bar{A})$ is
closed and
$\|\bar{A}^{+}\|\leq\frac{\|A^{+}\|}{1-\|A^{+}\|\|\delta A\|},\
\|\bar{A}^{+}-A^{+}\|\leq\frac{1+\sqrt{5}}{2}\|\bar{A}^{+}\|\|A^{+}\|\|\delta
A\|.$
###### Lemma 2.3.
Let $A\in B(X,Y)$ with $R(A)$ closed. If $Z\in B(Y,X)$ satisfies the
conditions: $AZA=A$ and $ZAZ=Z$, then $A^{+}=P_{N(A)^{\perp}}ZP_{R(A)}$.
Proof. We can check that $P_{N(A)^{\perp}}ZP_{R(A)}$ satisfies the definition
of the Moore–Penrose inverse of $A$. ∎
The following result is known when $X,\,Y$ are all of finite–dimensional (cf.
[1]).
###### Lemma 2.4.
Let $A\in B(X,Y)$ and $T\subset X,S\subset Y$ be closed subspaces. If
$A_{T,S}^{(2)}$ exists, then $A_{T,S}^{(2)}=(P_{S^{\perp}}AP_{T})^{+}$ with
$R(A_{T,S}^{(2)})=T$ and $N(A_{T,S}^{(2)})=S$.
Proof. The existence of $A_{T,S}^{(2)}$ implies that $N(A)\cap T=\\{0\\}$,
$AT$ is closed and $Y=AT\dotplus S$. Let $P\colon Y\rightarrow S$ be the
idempotent operator. Since $R(P)=S$ and $R(I_{Y}-P)=AT$, it follows that
$PP_{S}=P_{S}$, $P_{S}P=P$ and $(I_{Y}-P)AT=AT$. Noting that
$\displaystyle(I_{Y}-P)(I_{Y}-P_{S})$
$\displaystyle=I_{Y}+PP_{S}-P_{S}-P=I_{Y}-P$
$\displaystyle(I_{Y}-P_{S})(I_{Y}-P)$
$\displaystyle=I_{Y}-P-P_{S}+P_{S}P=I_{Y}-P_{S},$
we have
$R(I_{Y}-P_{S})=(I_{Y}-P_{S})(R(I_{Y}-P))=(I_{Y}-P_{S})AT=P_{S^{\perp}}AT$
and hence $R(P_{S^{\bot}}AP_{T})=R(P_{S^{\bot}})=S^{\perp}$ is closed.
Let $x\in T$ and $P_{S^{\bot}}Ax=0$. Then $(I_{Y}-P)Ax=Ax,\ Ax=P_{S}Ax$ and
hence $0=PAx=PP_{S}Ax=P_{S}Ax=Ax$. Since $N(A)\cap T=\\{0\\}$, we have $x=0$
and consequently, $N(P_{S^{\bot}}AP_{T})=T^{\bot}$. Therefore,
$(P_{S^{\bot}}AP_{T})^{+}$ exists and
$\displaystyle R((P_{S^{\bot}}AP_{T})^{+})$
$\displaystyle=(N(P_{S^{\bot}}AP_{T}))^{\bot}=T$ (2.2) $\displaystyle
N((P_{S^{\bot}}AP_{T})^{+})$ $\displaystyle=(R(P_{S^{\bot}}AP_{T}))^{\bot}=S.$
(2.3)
Since
$(P_{S^{\perp}}AP_{T})^{+}P_{S^{\bot}}=(P_{S^{\bot}}AP_{T})^{+}=P_{T}(P_{S^{\bot}}AP_{T})^{+},$
by (2.2) and (2.3), it follows that
$\displaystyle(P_{S^{\bot}}AP_{T})^{+}$
$\displaystyle=(P_{S^{\bot}}AP_{T})^{+}(P_{S^{\bot}}AP_{T})(P_{S^{\bot}}AP_{T})^{+}$
$\displaystyle=(P_{S^{\bot}}AP_{T})^{+}A(P_{S^{\bot}}AP_{T})^{+}$
and so that $A_{T,S}^{(2)}=(P_{S^{\bot}}AP_{T})^{+}$.∎
###### Lemma 2.5 ([10, Theorem 11,P100]).
Let $M$ be a complemented subspace of $H$. Let $P\in B(H)$ be an idempotent
operator with $R(P)=M$. Let $M^{\prime}$ be a closed subspace of $H$
satisfying $\hat{\delta}(M,M^{\prime})<\dfrac{1}{1+\|P\|}$. Then $M^{\prime}$
is complemented, that is, $H=R(I-P)\dotplus M^{\prime}$.
## 3 main result
We begin with the key lemma as follows.
###### Lemma 3.1.
Let $A\in B(X,Y)$. Let $T\subset X$ and $S\subset Y$ be closed subspaces such
that $A_{T,S}^{(2)}$ exists. Let $T^{\prime}$ be a closed subspace of $X$ such
that $\hat{\delta}(T,T^{\prime})<\dfrac{1}{1+\|A\|\|A_{T,S}^{(2)}\|}$. Then
$\hat{\delta}(AT,AT^{\prime})\leq\frac{\|A\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})}{1-(1+\|A\|\|A_{T,S}^{(2)}\|)\hat{\delta}(T,T^{\prime})}.$
Proof. First we show
$\delta(AT,AT^{\prime})\leq\|A\|\|A_{T,S}^{(2)}\|\delta(T,T^{\prime})\leq\|A\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})$.
Let $x\in T$. Then $x=A_{T,S}^{(2)}Ax$ and $\|x\|\leq\|A_{T,S}^{(2)}\|\|Ax\|$.
For any $y\in T^{\prime}$, we have $\|Ax-Ay\|\leq\|A\|\|x-y\|$. So
$\displaystyle dist(Ax,AT^{\prime})$ $\displaystyle=\inf_{y\in
T^{\prime}}\|Ax-Ay\|\leq\|A\|\inf_{y\in T^{\prime}}\|x-y\|$
$\displaystyle=\|A\|dist(x,T^{\prime})\leq\|A\|\|x\|\delta(T,T^{\prime})$
$\displaystyle\leq\|A\|\|A_{T,S}^{(2)}\|\|Ax\|\delta(T,T^{\prime}).$
This means that
$\delta(AT,AT^{\prime})\leq\|A\|\|A_{T,S}^{(2)}\|\delta(T,T^{\prime})\leq\|A\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})$.
Next we show
$\delta(AT^{\prime},AT)\leq\frac{\|A\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})}{1-(1+\|A\|\|A_{T,S}^{(2)}\|)\hat{\delta}(T,T^{\prime})}$
when $\hat{\delta}(T,T^{\prime})<\dfrac{1}{1+\|A\|\|A_{T,S}^{(2)}\|}.$
For $x^{\prime}\in T^{\prime}$ and $x\in T$, we have
$\displaystyle\|Ax^{\prime}\|$
$\displaystyle=\|A(x^{\prime}-x+x)\|\geq\|Ax\|-\|A\|\|x^{\prime}-x\|$
$\displaystyle\geq\|A_{T,S}^{(2)}\|^{-1}\|x\|-\|A\|\|x^{\prime}-x\|$
$\displaystyle\geq\|A_{T,S}^{(2)}\|^{-1}\|x^{\prime}\|-\|A_{T,S}^{(2)}\|^{-1}\|x^{\prime}-x\|-\|A\|\|x^{\prime}-x\|$
$\displaystyle\geq\|A_{T,S}^{(2)}\|^{-1}\|x^{\prime}\|-(\|A_{T,S}^{(2)}\|^{-1}+\|A\|)\|x^{\prime}-x\|,$
Thus,
$(\|A_{T,S}^{(2)}\|^{-1}+\|A\|)\|x^{\prime}-x\|\geq\|A_{T,S}^{(2)}\|^{-1}\|x^{\prime}\|-\|Ax^{\prime}\|$
and consequently,
$\|A_{T,S}^{(2)}\|^{-1}\|x^{\prime}\|-\|Ax^{\prime}\|\leq\|x^{\prime}\|(\|A_{T,S}^{(2)}\|^{-1}+\|A\|)\delta(T^{\prime},T),$
that is,
$\|A_{T,S}^{(2)}\|\|Ax^{\prime}\|\geq\big{[}1-(1+\|A\|\|A_{T,S}^{(2)}\|)\delta(T^{\prime},T)\big{]}\|x^{\prime}\|.$
(3.1)
Therefore,
$\displaystyle dist(Ax^{\prime},AT)$
$\displaystyle\leq\|A\|dist(x^{\prime},T)\leq\|A\|\|x^{\prime}\|\delta(T^{\prime},T)$
$\displaystyle\leq\frac{\|A\|\|Ax^{\prime}\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})}{1-(1+\|A\|\|A_{T,S}^{(2)}\|)\hat{\delta}(T,T^{\prime})},$
i.e.,
$\delta(AT^{\prime},AT)\leq\dfrac{\|A\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})}{1-(1+\|A\|\|A_{T,S}^{(2)}\|)\hat{\delta}(T,T^{\prime})}$
when $\hat{\delta}(T,T^{\prime})<\dfrac{1}{1+\|A\|\|A_{T,S}^{(2)}\|}.$
The final assertion follows from above arguments. ∎
###### Proposition 3.2.
Let $A\in B(X,Y)$ and $T\subset X$, $S\subset Y$ be closed subspaces such that
$A_{T,S}^{(2)}$ exists. Let $T^{\prime}$ be a closed subspace of $X$ such that
$\hat{\delta}(T,T^{\prime})<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}$. Then
$A_{T^{\prime},S}^{(2)}$ exists and
1. $(1)$
$A_{T^{\prime},S}^{(2)}=P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}$.
2. $(2)$
$\|A_{T^{\prime},S}^{(2)}\|\leq\dfrac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}$.
3. $(3)$
$\|A_{T^{\prime},S}^{(2)}-A_{T,S}^{(2)}\|\leq\dfrac{1+\sqrt{5}}{2}\|A_{T^{\prime},S}^{(2)}\|\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime}).$
Proof. By (3.1), $N(A)\cap T^{\prime}=\\{0\\}$ when
$\hat{\delta}(T,T^{\prime})<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}$.
Let $P=AA_{T,S}^{(2)}$. Then $P$ is idempotent from $Y$ onto $AT$ along $S$.
By Lemma 3.1, we have
$\hat{\delta}(AT,AT^{\prime})\leq\frac{\|A\|\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})}{1-(1+\|A\|\|A_{T,S}^{(2)}\|)\hat{\delta}(T,T^{\prime})}<\frac{1}{1+\|A\|\|A_{T,S}^{(2)}\|}\leq\frac{1}{1+\|P\|}$
when $\hat{\delta}(T,T^{\prime})<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}.$
So $AT^{\prime}$ is complemented and $AT^{\prime}\dotplus S=Y$ by Lemma 2.5.
Therefore. $A_{T^{\prime},S}^{(2)}$ exists and
$A_{T^{\prime},S}^{(2)}=(P_{S^{\perp}}AP_{T^{\prime}})^{+}$ by Lemma 2.4.
Set $B=P_{S^{\perp}}AP_{T}$,
$\bar{B}=B+P_{S^{\perp}}A(P_{T^{\prime}}-P_{T})=P_{S^{\perp}}AP_{T^{\prime}}$.
Then $N(B^{+})=S$ and $R(\bar{B})=((N(\bar{B}^{+}))^{\perp}=S^{\perp}$. So
$R(\bar{B})\cap N(B^{+})=\\{0\\}$, that is, $\bar{B}$ is the stable
perturbation of $B$.
From Proposition 2.1 (5), we have
$\|B^{+}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T})\|\leq\|A_{T,S}^{(2)}\|\|A\|\|P_{T^{\prime}}-P_{T}\|=\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})<1.$
Hence, by Lemma 2.2 and Lemma 2.3, we have
$\displaystyle A_{T^{\prime},S}^{(2)}=\bar{B}^{+}$
$\displaystyle=P_{N(\bar{B})^{\perp}}(I+B^{+}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}B^{+}P_{R(\bar{B})}$
$\displaystyle=P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}},$
$\|A_{T^{\prime},S}^{(2)}\|\leq\dfrac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}$
and
$\displaystyle\|A_{T^{\prime},S}^{(2)}-A_{T,S}^{(2)}\|$
$\displaystyle=\|\bar{B}^{+}-B^{+}\|$
$\displaystyle\leq\frac{1+\sqrt{5}}{2}\|A_{T^{\prime},S}^{(2)}\|\|A_{T,S}^{(2)}\|\|P_{S^{\perp}}A(P_{T^{\prime}}-P_{T})\|$
$\displaystyle\leq\frac{1+\sqrt{5}}{2}\|A_{T^{\prime},S}^{(2)}\|\|A_{T,S}^{(2)}\|\|A\|\|P_{T^{\prime}}-P_{T}\|$
$\displaystyle=\frac{1+\sqrt{5}}{2}\|A_{T^{\prime},S}^{(2)}\|\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime}).$
∎
Similar to Proposition 3.2, we have
###### Proposition 3.3.
Let $A\in B(X,Y)$ and let $T\subset X$, $S\subset Y$ be closed subspaces such
that $A_{T,S}^{(2)}$ exists. Let $S^{\prime}\subset Y$ be a closed subspace
such that $\hat{\delta}(S,S^{\prime})<\dfrac{1}{2+\|A\|\|A_{T,S}^{(2)}\|}$.
Then $A_{T,S^{\prime}}^{(2)}$ exists and
1. $(1)$
$A_{T,S^{\prime}}^{(2)}=P_{T}(I_{X}+A_{T,S}^{(2)}(P_{(S^{\prime})^{\perp}}-P_{S^{\perp}})AP_{T})^{-1}A_{T,S}^{(2)}P_{(S^{\prime})^{\perp}}$.
2. $(2)$
$\|A_{T,S^{\prime}}^{(2)}\|\leq\dfrac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(S,S^{\prime})}$.
3. $(3)$
$\|A_{T,S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|\leq\dfrac{1+\sqrt{5}}{2}\|A_{T,S^{\prime}}^{(2)}\|\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(S,S^{\prime})$.
Proof. Note that $Q=I_{Y}-AA_{T,S}^{(2)}$ is an idempotent operator from $Y$
onto $S$ along $AT$ and
$\hat{\delta}(S,S^{\prime})<\frac{1}{2+\|A\|\|A_{T,S}^{(2)}\|}\leq\frac{1}{1+\|I_{Y}-Q\|}.$
So $Y=AT\dotplus S^{\prime}$ by Lemma 2.5 and hence $A_{T,S^{\prime}}^{(2)}$
exists with $A_{T,S^{\prime}}^{(2)}=(P_{{S^{\prime}}^{\perp}}AP_{T})^{+}$.
Using similar methods in the proof of Proposition 3.2, we can get the results.
∎
Now we present the main result of the paper as follows.
###### Theorem 3.4.
Let $A\in B(X,Y)$ and let $T,\,T^{\prime}\subset X$, $S,\,S^{\prime}\subset Y$
be closed subspaces such that $A_{T,S}^{(2)}$ exists and
$\max\\{\hat{\delta}(T,T^{\prime}),\hat{\delta}(S,S^{\prime})\\}<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}$.
Then $A_{T^{\prime},S^{\prime}}^{(2)}$ exists and
1. $(1)$
$A_{T^{\prime},S^{\prime}}^{(2)}=P_{T^{\prime}}\big{[}I_{X}+P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}(P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}-P_{S^{\perp}})AP_{T^{\prime}}\big{]}^{-1}\\\
\hskip 36.98866pt\times
P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}.$
2. $(2)$
$\|A_{T^{\prime},S^{\prime}}^{(2)}\|\leq\dfrac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}.$
3. $(3)$
$\|A_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|\leq\dfrac{1+\sqrt{5}}{2}\dfrac{\|A_{T,S}^{(2)}\|^{2}\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}.$
Proof. If
$\hat{\delta}(T,T^{\prime})<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}},$ then
by Proposition 3.2, $A_{T^{\prime},S}^{(2)}$ exists and
$\displaystyle A_{T^{\prime},S}^{(2)}$
$\displaystyle=P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}$
(3.2) $\displaystyle\|A_{T^{\prime},S}^{(2)}\|$
$\displaystyle\leq\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}<\|A_{T,S}^{(2)}\|(1+\|A\|\|A_{T,S}^{(2)}\|)$
(3.3)
for
$\hat{\delta}(T,T^{\prime})<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}\leq\dfrac{1}{1+\|A\|\|A_{T,S}^{(2)}\|}$.
Noting that $\|A\|\|A_{T,S}^{(2)}\|\geq\|AA_{T,S}^{(2)}\|\geq 1$ and
$(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}\geq
2+\|A\|\|A_{T,S}^{(2)}\|(1+\|A\|\|A_{T,S}^{(2)}\|)>2+\|A\|\|A_{T^{\prime},S}^{(2)}\|$
by (3.3), we have
$\hat{\delta}(S,S^{\prime})<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}<\frac{1}{2+\|A\|\|A_{T^{\prime},S}^{(2)}\|}.$
Hence $A_{T^{\prime},S^{\prime}}^{(2)}$ exists with
$\|A_{T^{\prime},S^{\prime}}^{(2)}\|\leq\dfrac{\|A_{T^{\prime},S}^{(2)}\|}{1-\|A_{T^{\prime},S}^{(2)}\|\|A\|\hat{\delta}(S,S^{\prime})}$
and
$A_{T^{\prime},S^{\prime}}^{(2)}=P_{T^{\prime}}(I_{X}+A_{T^{\prime},S}^{(2)}(P_{(S^{\prime})^{\perp}}-P_{S^{\perp}})AP_{T^{\prime}})^{-1}A_{T^{\prime},S}^{(2)}P_{(S^{\prime})^{\perp}}$
by Proposition 3.3. Thus we have
$\displaystyle A_{T^{\prime},S^{\prime}}^{(2)}$
$\displaystyle=P_{T^{\prime}}\big{[}I_{X}+P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}(P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}-P_{S^{\perp}})AP_{T^{\prime}}\big{]}^{-1}$
$\displaystyle\ \,\times
P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}$
by (3.2) and
$\displaystyle\|A_{T^{\prime},S^{\prime}}^{(2)}\|$
$\displaystyle\leq\frac{1}{1-\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}\|A\|\hat{\delta}(S,S^{\prime})}\times\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}$
$\displaystyle=\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}.$
Moreover,
$\displaystyle\|A_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|$
$\displaystyle=\|A_{T^{\prime},S^{\prime}}^{(2)}-A_{T^{\prime},S}^{(2)}+A_{T^{\prime},S}^{(2)}-A_{T,S}^{(2)}\|$
$\displaystyle\leq\|A_{T^{\prime},S^{\prime}}^{(2)}-A_{T^{\prime},S}^{(2)}\|+\|A_{T^{\prime},S}^{(2)}-A_{T,S}^{(2)}\|$
$\displaystyle\leq\frac{1+\sqrt{5}}{2}\|A_{T^{\prime},S}^{(2)}\|\|A\|(\|A_{T^{\prime},S^{\prime}}^{(2)}\|\hat{\delta}(S,S^{\prime})+\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime}))$
$\displaystyle\leq\frac{1+\sqrt{5}}{2}\frac{\|A_{T,S}^{(2)}\|\|A\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}(\|A_{T^{\prime},S^{\prime}}^{(2)}\|\hat{\delta}(S,S^{\prime})+\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime}))$
$\displaystyle\leq\frac{1+\sqrt{5}}{2}\frac{\|A_{T,S}^{(2)}\|\|A\|}{1-\|A_{T,S}^{(2)}\|\|A\|\hat{\delta}(T,T^{\prime})}$
$\displaystyle\
\,\times\bigg{(}\|A_{T,S}^{(2)}\|\hat{\delta}(T,T^{\prime})+\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}\hat{\delta}(S,S^{\prime})\bigg{)}$
$\displaystyle=\frac{1+\sqrt{5}}{2}\frac{\|A_{T,S}^{(2)}\|^{2}\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}.$
∎
###### Lemma 3.5.
Let $A,\,\bar{A}=A+E\in B(X,Y)$ and let $T\subset X$, $S\subset Y$ be closed
subspaces such that $A_{T,S}^{(2)}$ exists. Suppose that
$\|A_{T,S}^{(2)}\|\|E\|<1$. Then
$\bar{A}_{T,S}^{(2)}=(I_{X}+A_{T,S}^{(2)}E)^{-1}A_{T,S}^{(2)}=A_{T,S}^{(2)}(I_{Y}+EA_{T,S}^{(2)})^{-1}.$
and
$\|\bar{A}_{T,S}^{(2)}\|\leq\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|E\|},\quad\|\bar{A}_{T,S}^{(2)}-A_{T,S}^{(2)}\|\leq\frac{\|A_{T,S}^{(2)}\|^{2}\|E\|}{1-\|A_{T,S}^{(2)}\|\|E\|}.$
Proof. If $\|A_{T,S}^{(2)}\|\|E\|<1$, then $I_{X}+A_{T,S}^{(2)}E$ and
$I_{Y}+EA_{T,S}^{(2)}$ are invertible.
Since
$(I_{X}+A_{T,S}^{(2)}E)A_{T,S}^{(2)}=A_{T,S}^{(2)}(I_{Y}+EA_{T,S}^{(2)})$, it
follows that
$(I_{X}+A_{T,S}^{(2)}E)^{-1}A_{T,S}^{(2)}=A_{T,S}^{(2)}(I_{Y}+EA_{T,S}^{(2)})^{-1}.$
(3.4)
Put $B=(I_{X}+A_{T,S}^{(2)}E)^{-1}A_{T,S}^{(2)}$. From (3.4), we get that
$R(B)=R(A_{T,S}^{(2)})=T,\quad N(B)=N(A_{T,S}^{(2)})=S,\quad B(A+E)B=B.$
Therefore, $\bar{A}_{T,S}^{(2)}=(I_{X}+A_{T,S}^{(2)}E)^{-1}A_{T,S}^{(2)}$ and
$\|\bar{A}_{T,S}^{(2)}\|\leq\dfrac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|E\|}.$
Since
$\bar{A}_{T,S}^{(2)}-A_{T,S}^{(2)}=(I_{X}+A_{T,S}^{(2)}E)^{-1}A_{T,S}^{(2)}-A_{T,S}^{(2)}=-(I_{X}+A_{T,S}^{(2)}E)^{-1}A_{T,S}^{(2)}EA_{T,S}^{(2)},$
we have
$\|\bar{A}_{T,S}^{(2)}-A_{T,S}^{(2)}\|\leq\frac{\|A_{T,S}^{(2)}\|^{2}\|E\|}{1-\|A_{T,S}^{(2)}\|\|E\|}.$
∎
As an end of this section, we give the perturbation analysis for
$A_{T,S}^{(2)}$ when $T$, $S$ and $A$ all have small perturbation.
###### Theorem 3.6.
Let $A,\,\bar{A}=A+E\in B(X,Y)$ and let $T,\,T^{\prime}\subset X$,
$S,\,S^{\prime}\subset Y$ be closed subspaces such that $A_{T,S}^{(2)}$ exists
and
$\max\\{\hat{\delta}(T,T^{\prime}),\hat{\delta}(S,S^{\prime})\\}<\frac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}.$
If $\|A_{T,S}^{(2)}\|\|E\|<\dfrac{1}{1+\|A_{T,S}^{(2)}\|\|A\|}$, then
$\displaystyle\begin{aligned} (1)\
\bar{A}_{T^{\prime},S^{\prime}}^{(2)}&=\\{I_{X}+P_{T^{\prime}}[I_{X}+P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}\\\
&\
\,\times(P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}-P_{S^{\perp}})AP_{T^{\prime}}]^{-1}P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}\\\
&\ \,\times
P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}E\\}^{-1}P_{T^{\prime}}\\{I_{X}+P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}\\\
&\ \,\times
A_{T,S}^{(2)}(P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}-P_{S^{\perp}})AP_{T^{\prime}}\\}^{-1}\\\
&\ \,\times
P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}P_{{S^{\prime}}^{\perp}},\end{aligned}$
$\displaystyle(2)\
\|\bar{A}_{T^{\prime},S^{\prime}}^{(2)}\|\leq\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\big{[}\|E\|+\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))\big{]}},$
$\displaystyle(3)\
\|\bar{A}_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|\leq\frac{\|A_{T,S}^{(2)}\|^{2}\big{[}\|E\|+\frac{1+\sqrt{5}}{2}\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))\big{]}}{1-\|A_{T,S}^{(2)}\|\big{[}\|E\|+\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))\big{]}}.$
Proof. $A_{T^{\prime},S^{\prime}}^{(2)}$ exists with
$\|A_{T^{\prime},S^{\prime}}^{(2)}\|\leq\dfrac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}$
by Theorem 3.4 when
$\max\\{\hat{\delta}(T,T^{\prime}),\hat{\delta}(S,S^{\prime})\\}<\dfrac{1}{(1+\|A\|\|A_{T,S}^{(2)}\|)^{2}}$.
Thus
$\|A_{T^{\prime},S^{\prime}}^{(2)}\|\|E\|\leq\dfrac{\|E\|\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))}<\frac{1+\|A_{T,S}^{(2)}\|\|A\|}{1+(\|A_{T,S}^{(2)}\|\|A\|)^{2}}\leq
1,$
that is, $\|A_{T^{\prime},S^{\prime}}^{(2)}\|\|E\|<1$ by above inequalities
for $\|A_{T,S}^{(2)}\|\|A\|\geq\|A_{T,S}^{(2)}A\|\geq 1$. Consequently,
$\bar{A}_{T^{\prime},S^{\prime}}^{(2)}=(I_{X}+A_{T^{\prime},S^{\prime}}^{(2)}E)^{-1}A_{T^{\prime},S^{\prime}}^{(2)}$
by Lemma 3.5. Simple computation shows that
$\displaystyle\|\bar{A}_{T^{\prime},S^{\prime}}^{(2)}\|$
$\displaystyle\leq\frac{\|A_{T,S}^{(2)}\|}{1-\|A_{T,S}^{(2)}\|\\{\|E\|+\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))\\}},$
$\displaystyle\bar{A}_{T^{\prime},S^{\prime}}^{(2)}$
$\displaystyle=\\{I_{X}+P_{T^{\prime}}[I_{X}+P_{T^{\prime}}(I+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}(P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}-P_{S^{\perp}})$
$\displaystyle\ \,\times
AP_{T^{\prime}}]^{-1}P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}P_{(S^{\prime})^{\perp}}E\\}^{-1}$
$\displaystyle\ \,\times
P_{T^{\prime}}\\{I_{X}+P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}(P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}-P_{S^{\perp}})$
$\displaystyle\ \,\times
AP_{T^{\prime}}\\}^{-1}P_{T^{\prime}}(I_{X}+A_{T,S}^{(2)}P_{S^{\perp}}A(P_{T^{\prime}}-P_{T}))^{-1}A_{T,S}^{(2)}P_{S^{\perp}}P_{{S^{\prime}}^{\perp}}.$
Noting that
$\displaystyle\bar{A}_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}$
$\displaystyle=(I_{X}+A_{T^{\prime},S^{\prime}}^{(2)}E)^{-1}A_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}$
$\displaystyle=(I_{X}+A_{T^{\prime},S^{\prime}}^{(2)}E)^{-1}(A_{T^{\prime},S^{\prime}}^{(2)}-(I_{X}+A_{T^{\prime},S^{\prime}}^{(2)}E)A_{T,S}^{(2)})$
$\displaystyle=(I_{X}+A_{T^{\prime},S^{\prime}}^{(2)}E)^{-1}(A_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}-A_{T^{\prime},S^{\prime}}^{(2)}EA_{T,S}^{(2)}),$
we have
$\displaystyle\|\bar{A}_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|$
$\displaystyle\leq\|(I_{X}+A_{T^{\prime},S^{\prime}}^{(2)}E)^{-1}\|(\|A_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|+\|A_{T^{\prime},S^{\prime}}^{(2)}EA_{T,S}^{(2)}\|)$
$\displaystyle\leq\frac{1}{1-\|A_{T^{\prime},S^{\prime}}^{(2)}\|\|E\|}(\|A_{T^{\prime},S^{\prime}}^{(2)}-A_{T,S}^{(2)}\|+\|A_{T^{\prime},S^{\prime}}^{(2)}\|\|E\|\|A_{T,S}^{(2)}\|)$
$\displaystyle\leq\frac{\|A_{T,S}^{(2)}\|^{2}\Big{[}\|E\|+\frac{1+\sqrt{5}}{2}\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))\Big{]}}{1-\|A_{T,S}^{(2)}\|\big{[}\|E\|+\|A\|(\hat{\delta}(T,T^{\prime})+\hat{\delta}(S,S^{\prime}))\big{]}}.$
∎
Acknowledgement. The authors thank to the referee for his (or her) helpful
comments and suggestions.
## References
* [1] A. Ben-Israel and T.N.E. Greville, Generalized inverse: Theory and Applications, Springer–Verlag, New York, 2003.
* [2] G. Chen and Y. Xue, Perturbation analysis for the operator equation $Tx=b$ in Banach spaces. J. Math. Anal. Appl., 212 (1997), 107-125.
* [3] G. Chen, M. Wei and Y. Xue, Perturbation analysis of the least square solution in Hilbert spaces, Linear Alebra Appl., 244 (1996), 69-80.
* [4] Y. Chen, Iterative methods for computing the generalized inverse $A^{(2)}_{T,S}$ of a matrix $A$, Appl. Math. Comput. 75 (1996), 207-222.
* [5] D. Djordjević and Stanimirović, Splitting of operators and generalized inverses, Publ. Math. Debrecen, 59 (2001), 147–159.
* [6] D. Djordjević, Stanimirović and Y. Wei, The representation and approximations of outer generalized inverses, Acta Math. Hungar., 104(1–2) (2004), 1–26.
* [7] F. Du and Y. Xue, Perturbation analysis of $A^{(2)}_{T,S}$ on banach spaces, Electronic J. Linear Algebra, 23 (2012), 586-598.
* [8] F. Du and Y. Xue, Note on stable perturbation of bounded linear operators on Hilbert spaces, Funct. Anal. Appr. Comput., 3(2) (2011), 47–56.
* [9] T. Kato, Perturbation Theory for Linear Operators, Springer–Verlag, New York, 1984.
* [10] V. Müller, Spectral Theory of Linear Operators and spectral systems in Banach algebras. Birkhäuser Verlag AG, 2nd Edition, (2007).
* [11] Y. Wei, A characterization and representation for the generalized inverse $A^{(2)}_{T,S}$ and its applications, Linear Algebra Appl. 280 (1998), 87-96.
* [12] Y. Wei and H. Wu, On the perturbation and subproper splittings for the generalized inverse $A^{(2)}_{T,S}$ of rectangular matrix A, J. Comput. Appl. Math., 137 (2001), 317–329.
* [13] Y. Wei and H. Wu, (T–S) Splitting methods for computing the generalized inverse $A^{(2)}_{T,S}$ and rectangular systems, International Journal of Computer Mathematics, 77(2001), 401–424.
* [14] Y. Xue, Stable perturbations of operators and related topics, World Scientific, 2012.
* [15] Y. Xue and G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces, Applied Math. comput., 147 (2004), 765–772.
* [16] N. Zhang and Y. Wei, Perturbation bounds for the generalized inverse $A^{(2)}_{T,S}$ with application to constrained linear system, Appl. Math. Comput., 142 (2003), 63–78.
* [17] N. Zhang and Y. Wei, A note on the perturbation of an outer inverse, Calcolo. 45(4) (2008), 263–273.
|
arxiv-papers
| 2012-09-09T01:34:24 |
2024-09-04T02:49:34.902112
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fapeng Du and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1209.1767"
}
|
1209.1899
|
# A matrix approach for computing extensions of argumentation frameworks
Xu Yuming
$School\ of\ Mathematics,Shandong\ University,Jinan,China$ Corresponding
author. E-mail: [email protected]
Abstract
The matrices and their sub-blocks are introduced into the study of determining
various extensions in the sense of Dung’s theory of argumentation frameworks.
It is showed that each argumentation framework has its matrix representations,
and the core semantics defined by Dung can be characterized by specific sub-
blocks of the matrix. Furthermore, the elementary permutations of a matrix are
employed by which an efficient matrix approach for finding out all extensions
under a given semantics is obtained. Different from several established
approaches, such as the graph labelling algorithm, Constraint Satisfaction
Problem algorithm, the matrix approach not only put the mathematic idea into
the investigation for finding out various extensions, but also completely
achieve the goal to compute all the extensions needed.
Keywords: Argumentation framework; extension; matrix; sub-block; permutation
1\. Introduction
In recent years, the area of argumentation begins to become increasingly
central as a core study within Artificial Intelligence. A number of papers
investigated and compared the properties of different semantics which have
been proposed for argumentation frameworks (AFs, for short) as introduced by
Dung [9, 5, 4, 10, 7]. On the algorithms of abstract argumentation frameworks,
Modgil and Caminada[15] list the following questions:
1\. For a given semantics, ”global” questions concerning the existence and
construction of extensions can be addressed:
(a) Does an extension exist?
(b) Given an extension
(c) Given all extensions
2\. For a given semantics, ”local” questions concerning the existence and
construction of extensions, relative to a set $A\in\mathcal{A}$ can be
addressed:
(a) Is A contained in an extension?
(b) Is A contained in all extensions?
(c) Is A attacked by an extension?
(d) Is A attacked by all extensions?
(e) Given an extension containing A.
(f) Given all extensions containing A.
(g) Given an extension that attacks A.
(h) Given all extensions that attack A.
They reviewed two important approaches, graph labelling algorithm and argument
game algorithm, which can be used to answer a selection of the above questions
with respect to finite argumentation frameworks. Another excellent approach
was established by Amgoud and Devred which encodes argumentation frameworks as
the Constraint Satisfaction Problems[1].
Our purpose is to introduce the matrix as a tool into the research of
argumentation frameworks. In fact, there is a natural way to assign a matrix
of order $n$ for each argumentation framework with $n$ arguments. Each element
of the matrix has only two possible values: one and zero, where one represents
the attack relation and zero represents the non-attack relation between two
arguments (they can be the same one). The matrix can be thought to be a
representation of the argumentation framework, since it contains all the
information of the argumentation framework. Firstly, we define several sub-
blocks of the matrix corresponding to related extensions of an argumentation
framework, and give the matrix criterions to determine whether a conflict-free
set is a stable extension, admissible extension or complete extension. Then,
we employ the elementary permutation of matrices to handle the matrix of an
argumentation framework, and obtain a matrix approach for finding out all
extensions under a given semantics. It consists of three steps: Defining a
family of basic sets, by which we can find out all the conflict-free sets of
an argumentation framework; For each conflict-free set, turning the matrix of
the argumentation framework into a norm form through a sequence of dual
interchanges; Determining what extension the conflict-free set is according to
the corresponding criterions of related extensions.
This paper is organized as follows: Section 2 briefly presents Dung’s theory
on abstract argumentation frameworks. Section 3 introduces the matrix
representation of argumentation frameworks, as well as the concept of sub-
blocks of a matrix. Section 4 discusses the criterion of determining conflict-
free sets, and defines the concept of basic sets by which we can find out all
conflict-free sets of an argumentation framework in a programmed way. From
Section 5 to Section 7, we address the criterions of determining the core
extensions of an argumentation framework, such as stable extension, admissible
extension and complete extension. In Section 8, we mainly devote to establish
a systematic matrix approach, by which all the questions mentioned by Modgil
and Caminada can be solved. Certainly, this approach can be easily realized in
practical computing with no technical difficulty.
2\. Dung’s theory of argumentation
Argumentation is a general approach to model defeasible reasoning and
justification in Artificial Intelligence. So far, many theories of
argumentation have been established. Among them, Dung’s theory of AFs is quite
influence. In fact, it is abstract enough to manage without any assumption on
the nature of arguments and the attack relation between arguments. Let us
first recall some basic notion in Dung’s theory of AFs. We restrict them to
finite argumentation frameworks.
An argumentation framework is a pair $F=(A,R)$, where $A$ is a finite set of
arguments and $R\subset A\times A$ represents the attack-relation. For
$S\subset A$, we say that
(1) $S$ is conflict-free in $(A,R)$ if there are no $a,b\in S$ such that
$(a,b)\in R$;
(2) $a\in A$ is defeated by $S$ in $(A,R)$ if there is $b\in S$ such that
$(b,a)\in R$;
(3) $a\in A$ is defended by $S$ in $(A,R)$ if for each $b\in A$ with $(b,a)\in
R$, we
have $b$ is defeated by $S$ in $(A,R)$.
(4) $a\in A$ is acceptable with respect to $S$ if for each $b\in A$ with
$(b,a)\in R$,
there is some $c\in S$ such that $(c,b)\in R$.
The conflict-freeness, as observed by Baroni and Giacomin[2] in their study of
evaluative criteria for extension-based semantics, is viewed as a minimal
requirement to be satisfied within any computationally sensible notion of
”collection of justified arguments”. However, it is too weak a condition to be
applied as a reasonable guarantor that a set of arguments is ”collectively
acceptable”.
Semantics for AFs can be given by a function $\sigma$ which assigns each
argumentation framework $F=(A,R)$ a collection $\mathcal{S}\subset 2^{A}$ of
extensions. Here, we mainly focus on the semantic
$\sigma\in\\{s,a,p,c,g,i,ss,e\\}$ for stable, admissible, preferred, complete,
grounded, ideal, semi-stable and eager extensions, respectively.
Definition 1[16] Let $F=(A,R)$ be an argumentation framework and $S\in A$.
(1) $S$ is a stable extension of $F$, $i.e.$, $S\in s(F)$, if $S$ is conflict-
free in $F$
and each $a\in A\setminus S$ is defeated by $S$ in $F$.
(2) $S$ is an admissible extension of $F$, $i.e.$, $S\in a(F)$, if $S$ is
conflict-free
in $F$ and each $a\setminus S$ is defended by $S$ in $F$.
(3) $S$ is a preferred extension of $F$, $i.e.$, $S\in p(F)$, if $S\in a(F)$
and for
each $T\in a(F)$, we have $S\not\subset T$.
(4) $S$ is a complete extension of $F$, $i.e.$, $S\in c(F)$, if $S\in a(F)$
and for
each $a\in A$ defended by $S$ in $F$, we have $a\in S$.
(5) $S$ is a grounded extension of $F$, $i.e.$, $S\in g(F)$, if $S\in c(F)$
and for
each $T\in c(F)$, we have $T\not\subset S$.
(6) $S$ is an ideal extension of $F$, $i.e.$, $S\in i(F)$, if $S\in a(F)$,
$S\subset\cap\\{T:$
$T\in p(F)\\}$ and for each $U\in a(F)$ such that $U\subset\cap\\{T:T\in
p(F)\\}$,
we have $S\not\subset U$.
(7) $S$ is a semi-stable extension of $F$, $i.e.$, $S\in ss(F)$, if $S\in
a(F)$ and for
each $T\in a(F)$, we have $R^{+}(S)\not\subset R^{+}(T)$, where
$R^{+}(U)=U\cap\\{b:$
$(a,b)\in R,a\in U\\}$.
(8) $S$ is a eager extension of $F$, $i.e.$, $S\in e(F)$, if $S\in a(F)$,
$S\subset\cap\\{T:$
$T\in ss(F)\\}$ and for each $U\in a(F)$ such that $U\subset\cap\\{T:T\in
ss(F)\\}$,
we have $S\not\subset U$.
Note that, there are some basic properties for any argumentation framework
$F=(A,R)$ and semantic $\sigma$. If $\sigma\in\\{a,p,c,g\\}$, then we have
$\sigma(F)\neq\emptyset$. And if $\sigma\in\\{g,i,e\\}$, then $\sigma(F)$
contains exactly one extension. Furthermore, the following relations hold for
each argumentation framework $F=(A,R)$:
$s(F)\subseteq p(F)\subseteq c(F)\subseteq a(F)$.
Since every extension of an AF under the standard semantics (stable,
preferred, complete and grounded) is an admissible set, the concept of
admissible extension plays an important role in the study of AFs.
3\. The matrix of an argumentation framework
We know that the directed graph is a traditional tool in the research of AFs,
and has the feature of visualization [8, 11, 12]. It is widely used for
modeling and analyzing AFs. In this section, we shall concern another way,
that is the matrix representation of AFs. Except for the visualization, the
matrix also has the advantage of computability in analyzing the properties of
AFs and computing their various extensions.
An $m\times n$ matrix $A$ is a rectangular array of numbers, consisting of $m$
rows and $n$ columns, denoted by
$A=\left(\begin{array}[]{cccccc}a_{1,1}&a_{1,2}&.&.&.&a_{1,n}\\\
a_{2,1}&a_{2,2}&.&.&.&a_{2,n}\\\ .&.&.&.&.&.\\\
a_{m,1}&a_{m,2}&.&.&.&a_{m,n}\end{array}\right).$
The $m\times n$ numbers $a_{1,1},a_{1,2},...,a_{m,n}$ are the elements of the
matrix $A$. We often called $a_{i,j}$ the $(i,j)$th element, and write
$A=(a_{i,j})$ for short. It is important to remember that the first suffix of
$a_{i,j}$ indicates the row and the second the column of $a_{i,j}$.
An elementary operation on a matrix is an operation of one of the following
three types.
(1) The interchange of two rows (or columns).
(2) The multiplication of a row (or column) by a non-zero scalar.
(3) The addition of a multiple of one row (or column) to another row
(column).
A column matrix is an $n\times 1$ matrix, and a row matrix is an $1\times n$
matrix, denoted by
$\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ .\\\ .\\\ .\\\
x_{n}\end{array}\right),\left(\begin{array}[]{cccccc}x_{1}&x_{2}&.&.&.&x_{n}\end{array}\right)$
respectively. Matrices of both these types can be regarded as vectors and
referred to respectively as column vectors and row vectors. Usually, the $i$th
row of a matrix $A$ is denoted by $A_{i,*}$, and the $j$th column of the
matrix $A$ is denoted by $A_{*,j}$.
Definition 2 Let $A=(a_{i,j})$ be an $n\times m$ matrix, $1\leq
i_{1}<i_{2}<...,<i_{k}\leq n$ and $1\leq j_{1}<j_{2}<...,<j_{h}\leq n$. The
the matrix
$\left(\begin{array}[]{cccccc}a_{i_{1},j_{1}}&a_{i_{1},j_{2}}&.&.&.&a_{i_{1},j_{h}}\\\
a_{i_{2},j_{1}}&a_{i_{2},j_{2}}&.&.&.&a_{i_{2},j_{h}}\\\ .&.&.&.&.&.\\\
a_{i_{k},j_{1}}&a_{i_{k},j_{2}}&.&.&.&a_{i_{k},j_{h}}\end{array}\right),$
is called a $k\times h$ sub-block of the matrix $A$, and denoted by
$M_{i_{1},i_{2},...,i_{k}}^{i_{1},i_{2},...,i_{k}}$.
In particular, the matrix
$\left(\begin{array}[]{cccccc}a_{i_{1},i_{1}}&a_{i_{1},i_{2}}&.&.&.&a_{i_{1},i_{k}}\\\
a_{i_{2},i_{1}}&a_{i_{2},i_{2}}&.&.&.&a_{i_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{i_{k},i_{1}}&a_{i_{k},i_{2}}&.&.&.&a_{i_{k},i_{k}}\end{array}\right),$
is called a principal sub-block of order $k$ of the matrix $A$.
Definition 3 Let $A=(a_{i,j})$ be an $n\times m$ matrix, $1\leq
i_{1}<i_{2}<...,<i_{k}\leq n$ and $1\leq j_{1}<j_{2}<...,<j_{h}\leq n$. If in
the $n\times m$ matrix $A=(a_{i,j})$, we delete the rows and columns which
make up the sub-block $M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$,
then the remaining elements form an $(n-k)\times(m-k)$ matrix
$\left(\begin{array}[]{cccccc}a_{j_{1},i_{1}}&a_{j_{1},i_{2}}&.&.&.&a_{j_{1},i_{h}}\\\
a_{j_{2},i_{1}}&a_{j_{2},i_{2}}&.&.&.&a_{j_{2},i_{h}}\\\ .&.&.&.&.&.\\\
a_{j_{k},i_{1}}&a_{j_{k},i_{2}}&.&.&.&a_{j_{k},i_{h}}\end{array}\right),$
We call this matrix the complementary sub-block of
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$, and denote it by
$\overline{M_{i_{1},i_{2},...,i_{k}}^{i_{1},i_{2},...,i_{k}}}$.
A matrix divided by horizontal and vertical lines is called a partition
matrix. It can be represented by denoting each part array by a single matrix
symbol. For example, a $4\times 3$ matrix $A=(a_{i,j})$ can be partitioned
into the following form
$\left(\begin{array}[]{cc}A_{1,1}&A_{1,2}\\\
A_{2,1}&A_{2,2}\end{array}\right),$
where
$A_{1,1}=\left(\begin{array}[]{cc}a_{1,1}&a_{1,2}\\\
a_{2,1}&a_{2,2}\end{array}\right),A_{1,2}=\left(\begin{array}[]{cc}a_{1,3}\\\
a_{2,3}\end{array}\right),A_{2,1}=\left(\begin{array}[]{cc}a_{3,1}&a_{3,2}\\\
a_{4,1}&a_{4,2}\end{array}\right),A_{2,2}=\left(\begin{array}[]{cc}a_{3,3}\\\
a_{4,3}\end{array}\right).$
Let $F=(A,R)$ be an argumentation framework where $A$ is a finite set. It is
obvious that the notation $A=\\{a,b,...\\}$ is not convenience when the
cardinality of $A$ is too large, so we prefer to denote $A$ by
$\\{1,2,...,n\\}$, subsequently.
For the underlying finite set $A$ of $F=(A,R)$, there is no ordering in
nature. But, an ordered set can benefit us a lot in many cases. Based on this
consideration, we introduce the concept of permutation into our discussion.
A permutation of a finite set $A$ of $n$ elements is a mapping of the set onto
itself. The usual method of presenting a permutation is to write down the
elements of $A$ in a row in natural order and, under each of them to write
down its image. For convenience, we usually write down only the elements of
the image in a row without changing their original order. For example, the
following is a permutation of the set $A=\\{1,2,3,4,5\\}$
$\left(\begin{array}[]{ccccc}1&2&3&4&5\\\
3&5&2&1&4\end{array}\right)=\left(\begin{array}[]{ccccc}3&5&2&1&4\end{array}\right).$
For any the argumentation framework $F=(A,R)$ with $A=\\{1,2,...,n\\}$, let
$(i_{1},i_{2},...,i_{n})$ be a fixed permutation of $A$, then $F=(A,R)$ can be
represented by a Boolean matrix under some simple rules.
Definition 4 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$. The matrix of $F$ corresponding to the permutation
$(i_{1},i_{2},...,i_{n})$ of $A$, denoted by $M(i_{1},i_{2},...,i_{n})$, is a
Boolean matrix of order $n$, its elements are determined by the following
rules:
(1) $a_{s,t}=1$ iff $(i_{s},i_{t})\in R$;
(2) $a_{s,t}=0$ iff $(i_{s},i_{t})\notin R$.
Remark In matrix $M(i_{1},i_{2},...,i_{n})$, the elements of $r$-th row
reflect the attack relations of the argument $i_{r}$ to the other arguments,
while the elements of $r$-th column reflect the attacked relations of the
other arguments to the argument $i_{r}$. So, the argumentation framework
$F=(A,R)$ with $A=\\{1,2,...,n\\}$ has many different matrix representations,
which depend on the different selection of permutations of $A$.
Although the argumentation framework $F=(A,R)$ has many different matrix
representations, all these matrices have the same role in representing
$F=(A,R)$. In other words, they are equivalent in representing $F=(A,R)$. We
usually use the matrix $M(1,2,...,n)$ corresponding to the natural permutation
$(1,2,...,n)$ to represent the argumentation framework $F$, and denote it by
$M(F)$.
Example 5 Considering the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3\\}$ and $R=\\{(1,2),(2,3),(3,1)\\}$. By Definition 4, the matrix
of $F$ corresponding to the natural permutation $(i_{1},i_{2},i_{3})=(1,2,3)$
is
$M(F)=M(1,2,3)=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right)$
The matrix of $F$ corresponding to the permutation
$(i_{1},i_{2},i_{3})=(2,1,3)$ is
$M(2,1,3)=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right)$
In comparison with directed graph way and logical analysis way, the matrix
$M(F)$ of an argumentation framework $F$ has many excellent features. For
example, it possess a concise mathematical format and contains all the
information of $F$ by combining the arguments with attack relations in a
specific manner. Also, we can import the knowledge and method of matrix theory
to the research of AFs, or even discover some new knowledge of matrices for
the research of AFs. The more important is that a powerful soft for dealing
with the computing of matrices has been developed, by which a lot of work can
be saved.
4\. Determination of the conflict-free sets
As we have known, there is no efficient method for us to decide a conflict set
in an argumentation framework, even we can draw up the directed graph of it.
After introducing the matrix of the AF, the situation will be changed
completely. Let us see an example, firstly.
Example 6 Given an argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),(2,5),(4,3),(5,4)\\}$. Then, we can
easily to show that the family of conflict-free sets of $F$ is
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\},\\{1,3,5\\}\\}$,
by the traditional method of directed graph.
Now, we consider the matrix of $F=(A,R)$ and study its structure from the
level of sub-blocks. First, we write out the matrix of $F$ corresponding to
the natural permutation $(1,2,3,4,,5)$:
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
Secondly, we concentrate our attention on the principal sub-blocks of the
above matrix. There are five zero principal sub-blocks of order 1:
$M^{1}_{1}=\left(\begin{array}[]{c}0\end{array}\right),M^{2}_{2}=\left(\begin{array}[]{c}0\end{array}\right),M^{3}_{3}=\left(\begin{array}[]{c}0\end{array}\right),M^{4}_{4}=\left(\begin{array}[]{c}0\end{array}\right),M^{5}_{5}=\left(\begin{array}[]{c}0\end{array}\right),$
which correspond to the conflict-free sets $\\{1\\}$, $\\{2\\}$, $\\{3\\}$,
$\\{4\\}$, $\\{5\\}$, respectively. There are five zero principal sub-blocks
of order 2:
$M^{1,3}_{1,3}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{1,4}_{1,4}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{1,5}_{1,5}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{2,4}_{2,4}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),$
$M^{3,5}_{3,5}=\left(\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right).$
They exactly match with the conflict-free sets $\\{1,3\\}$, $\\{1,4\\}$,
$\\{1,5\\}$, $\\{2,4\\}$, $\\{3,5\\}$, respectively. Also, there is one zero
principal sub-block of order 3:
$M^{1,3,5}_{1,3,5}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right),$
which is followed by the conflict-free sets $\\{1,3,5\\}$.
Note that, the above sub-blocks are all principal sub-blocks which are zero in
the matrix $M(F)$, and there is a one to one correspond between the family of
all zero principal sub-blocks of $M(F)$ and the family of all conflict-free
sets of $F=(A,R)$. In fact, for any argumentation framework $F$ there exists
such corresponding relation between the family of all zero principal sub-
blocks of $M(F)$ and the family of all conflict-free sets of $F=(A,R)$.
Definition 7 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ satisfying
$1\leq i_{1}<i_{2}<...<i_{k}\leq n$. The principal sub-block
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}=\left(\begin{array}[]{cccccc}a_{i_{1},i_{1}}&a_{i_{1},i_{2}}&.&.&.&a_{i_{1},i_{k}}\\\
a_{i_{2},i_{1}}&a_{i_{2},i_{2}}&.&.&.&a_{i_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{i_{k},i_{1}}&a_{i_{k},i_{2}}&.&.&.&a_{i_{k},i_{k}}\end{array}\right)$
of order $k$ in the matrix $M(F)$ is called the $cf$-sub-block of $S$, and
denoted by $M^{cf}$ for short.
Theorem 8 Given an argumentation framework $F=(A,R)$ with $A=\\{1,2,...,n\\}$,
then $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq i_{1}<i_{2}<...<i_{k}\leq
n)$ is a conflict-free set in $F$ iff the $cf$-sub-block
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is zero.
Proof Assume that $M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}=0$, then
for arbitrary $1\leq s,t\leq k$ we have $a_{i_{s},i_{t}}=0$, $i.e.$,
$(i_{s},i_{t})\notin R$. Thus, $S=\\{i_{1},i_{2},...,i_{k}\\}$ is a conflict-
free set in $F$.
Suppose $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a conflict-free set in
$F$, then for arbitrary $1\leq s,t\leq k$ we have that $(i_{s},i_{t})\notin
R$, $i.e.$, $a_{i_{s},i_{t}}=0$. Therefore, we have
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}=0$.
Next, we shall develop a way to find out all the zero principal sub-blocks in
the matrix $M(F)$ of an argumentation framework $F$, which is corresponding to
all the conflict-free sets of $F$.
Given an argumentation framework $F=(A,R)$ with $A=\\{1,2,...,n\\}$, let
$\mathcal{S}(r)$ denote the family of all conflict-free sets in $F$ whose
cardinality are $r(1\leq r\leq n)$. Then,
$\mathcal{S}(1)=\\{\\{1\\},\\{2\\},...,\\{n\\}\\}$. Surely, $\mathcal{S}(2)$
can be easily decided according to the matrix $M(F)$ of $F=(A,R)$. In fact,
for conflict-free sets which posses more than one element we can find out them
all by the following theorem.
Theorem 9 For any argument $i\in A$ and subset $S\in\mathcal{S}(r)(1\leq r\leq
n)$, let $C(i)=\\{j\in A:a_{i,j}=0\ or\ a_{j,i}=0\\}$. If $i\notin S$ and
$S\subset C(i)$, then $S\cup\\{i\\}$ is a conflict-free set whose cardinality
is $r+1$.
Proof We need only to prove that $a_{i,k}=0$ and $a_{k,i}=0$ for each $k\in
S$. It is a direct result of the fact $S\subset C(i)$.
Remark By Theorem 9, we can find out all the families
$\mathcal{S}(1),\mathcal{S}(2),...,\mathcal{S}(n)$ in the following way:
First, determining the sets $C(1),C(2),...,C(n)$, which will be called the
basic sets of the argumentation framework $F=(A,R)$. Secondly, writing out the
family $\mathcal{S}(1)=\\{\\{1\\},\\{2\\},...,\\{n\\}\\}$ from the values of
$a_{i,i}(1\leq i\leq n)$. Then, by comparing the elements of $\mathcal{S}(1)$
with the basic sets $C(j)(1\leq j\leq n)$, we can find out all the elements of
the family $\mathcal{S}(2)$. The family $\mathcal{S}(3)$ can be decided by
comparing the elements of $\mathcal{S}(2)$ with the basic sets $C(j)(1\leq
j\leq n)$ in a similar process, and so on.
Example 10 Consider the argumentation framework $F=(A,R)$ in Example 8. The
matrix of $F$ corresponding to the natural permutation $(1,2,3,4,5)$ is
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
It is obviously that $\mathcal{S}(0)=\\{\emptyset\\}$ and
$\mathcal{S}(1)=\\{\\{1\\},\\{2\\},...,\\{5\\}\\}$. Next, we shall computer
the family $\mathcal{S}(2)$, $\mathcal{S}(3)$, $\mathcal{S}(4)$ and
$\mathcal{S}(5)$.
First, we list the basic sets $C(1)=\\{3,4,5\\}$, $C(2)=\\{4\\}$,
$C(3)=\\{1,5\\}$, $C(4)=\\{1,2\\}$ and $C(5)=\\{1,3\\}$.
Second, by the fact that $\\{3\\},\\{4\\},\\{5\\}\subset C(1)$, we have
$\\{1,3\\},\\{1,4\\},\\{1,5\\}\in\mathcal{S}(2)$. Similarly, we can conclude
that $\\{2,4\\},\\{3,5\\}\in\mathcal{S}(2)$ by the fact that $\\{4\\}\subset
C(2)$, and $\\{5\\}\subset C(3)$. Note that, $\\{1\\}\subset C(3)$ will result
in that $\\{1,3\\}\in\mathcal{S}(2)$, but it is not essential because of the
previous case. Similar situation are also arise for $\\{1\\},\\{2\\}\subset
C(4)$ and $\\{1\\},\\{3\\}\subset C(5)$. Therefore,
$\mathcal{S}(2)=\\{\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\}\\}$.
For the family $\mathcal{S}(3)$, we only need to compare $C(1)=\\{3,4,5\\}$
with the elements of $\mathcal{S}(2)$. Since $\\{3,5\\}\in\mathcal{S}(2)$ and
$\\{3,5\\}\subset C(1)$, we have $\\{1,3,5\\}\in\mathcal{S}(3)$. In fact, we
conclude that $\mathcal{S}(3)=\\{\\{1,3,5\\}\\}$.
Since there are no basic sets containing more than three elements, we claim
that $\mathcal{S}(4)$ and $\mathcal{S}(5)$ are both empty sets.
Remark In the process to computing the family $\mathcal{S}(i+1)$ from
$\mathcal{S}(i)$ and $C(j)(1\leq j\leq n)$, we need only to check the relation
between the element $S\in\mathcal{S}(i)$ and the basic set $C(j)$ satisfying
$j<max\\{k:k\in S\\}$. In fact, if $S\in\mathcal{S}(i)$ with $j\notin S$ and
there is some $k\in S$ such that $k<j$, then $S\subset C(j)$ will implies
$S\cup\\{j\\}\in\mathcal{S}(i+1)$. But this result can also be obtained when
we compare the element $(S\cup\\{j\\})\setminus\\{k\\}$ of $\mathcal{S}(i)$
and the basic set $C(k)$. Of course,
$(S\cup\\{j\\})\setminus\\{k\\}\in\mathcal{S}(i)$ and
$(S\cup\\{j\\})\setminus\\{k\\}\subset C(k)$ comes from the fact
$S\cup\\{j\\}\in\mathcal{S}(i+1)$.
5\. Determination of the stable extensions
Example 11 We continuous to study the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),(2,5),(4,3),(5,4)\\}$ in Example 10.
Since the stable extension is firstly a conflict-free set, we can look for the
stable extension from the collection
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\},\\{1,3,5\\}\\}$
of conflict-free sets. In fact, the set $S=\\{1,3,5\\}$ is the only stable
extension in $F=(A,R)$ by a short discussion.
Let us turn our attention to the matrix
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
of the $F=(A,R)$, and try to find the information contained in $M(F)$ which
insure the conflict-free set $S=\\{1,3,5\\}$ is a stable extension.
Note that, $(1,2)\in R$ implies the argument $2$ is defeated by
$S=\\{1,3,5\\}$, and $(5,4)\in R$ implies the argument $4$ is defeated by
$S=\\{1,3,5\\}$. It is exactly the two results which make the conflict-free
set $S$ to be a stable extension. By Definition 5, the conditions $(1,2)\in R$
and $(5,4)\in R$ are represented in the form of $a_{1,2}=1$ and $a_{5,4}=1$ in
the matrix $M(F)$, respectively. And, the argument $2$ is defeated by
$S=\\{1,3,5\\}$ is equivalent to the column vector
$\left(\begin{array}[]{ccc}a_{1,2}\\\ a_{3,2}\\\ a_{5,2}\end{array}\right)\neq
0,$
the argument $4$ is defeated by $S=\\{1,3,5\\}$ is equivalent to the column
vector
$\left(\begin{array}[]{ccc}a_{1,4}\\\ a_{3,4}\\\ a_{5,4}\end{array}\right)\neq
0.$
It is not difficult to see that $(1,2)\in R$ and $(5,4)\in R$ is not
sufficient for the conflict-free set $S$ to be stable. The sufficient and
necessary conditions for the conflict-free set $S$ to be a stable extension
are that, the argument $2$ is defeated by $S=\\{1,3,5\\}$ and the argument $4$
is defeated by $S=\\{1,3,5\\}$. These facts are reflected in the matrix $M(F)$
as follows:
(1) $a_{1,2}=1$ and $a_{5,4}=1$ is not the sufficient condition for the
conflict-free set $S$ to be stable,
(2) The sufficient and necessary conditions for the conflict-free set $S$ to
be a stable extension are
$\left(\begin{array}[]{ccc}a_{1,2}\\\ a_{3,2}\\\ a_{5,2}\end{array}\right)\neq
0,\ \ \ and\ \ \ \left(\begin{array}[]{ccc}a_{1,2}\\\ a_{3,2}\\\
a_{5,2}\end{array}\right)\neq 0.$
Considering the sub-block
$M^{s}=\left(\begin{array}[]{ccc}a_{1,1}&a_{1,3}&a_{1,5}\\\
a_{3,1}&a_{3,3}&a_{3,5}\\\
a_{5,1}&a_{5,3}&a_{5,5}\end{array}\right)=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$
which insures of the conflict-freeness of the set $S=\\{1,3,5\\}$, if we
combine the above two column vectors into the following matrix
$\left(\begin{array}[]{ccc}a_{1,2}&a_{1,4}\\\ a_{3,2}&a_{3,4}\\\
a_{5,2}&a_{5,4}\end{array}\right),$
then it is also a sub-block of $M(F)$ and responds to the determining of the
stableness of $S=\\{1,3,5\\}$.
The above analysis motivates us to propose the following definition, and
provide a matrix way to determine whether a conflict-free set is a stable
extension of an argumentation framework.
Definition 12 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq
i_{1}<i_{2}<...<i_{k}\leq n)$ a stable extension of $F$. The $k\times h$ sub-
block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=\left(\begin{array}[]{cccccc}a_{i_{1},j_{1}}&a_{i_{1},j_{2}}&.&.&.&a_{i_{1},j_{h}}\\\
a_{i_{2},j_{1}}&a_{i_{2},j_{2}}&.&.&.&a_{i_{2},j_{h}}\\\ .&.&.&.&.&.\\\
a_{i_{k},j_{1}}&a_{i_{k},j_{2}}&.&.&.&a_{i_{k},j_{h}}\end{array}\right)$
in the matrix $M(F)$ is called the $s$-sub-block of $S$ and denoted by $M^{s}$
for short, where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S(1\leq
j_{1}<j_{2}<...<j_{h}\leq n)$.
In other words, the elements at the intersections of rows
$i_{1},i_{2},...,i_{k}$ and columns $j_{1},j_{2},...,j_{h}$ in the matrix
$M(F)$ form the $s$-sub-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of the set $S$.
Theorem 13 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, then conflict-free set
$S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq i_{1}<i_{2}<...<i_{k}\leq n)$ is
a stable extension in $F$ iff each column vector of the $s$-sub-block $M^{s}$
of $S$ is non-zero, where $A\setminus S=\\{j_{1},j_{2},...,j_{h}\\}$ and
$1\leq j_{1}<j_{2}<...<j_{h}\leq n$.
Proof Let $S$ be a conflict-free set and $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$, then we need only to prove that every element
of $A\setminus S(1\leq t\leq h)$ is defeated by $S$ in $F$ iff all column
vectors of the $s$-sub-block $M^{s}$ of $S$ are non-zero.
Assume that every element of $A\setminus S(1\leq t\leq h)$ is defeated by $S$
in $F$. Take any column vector $M_{*,j_{t}}(1\leq t\leq h)$ of the $s$-sub-
block $M^{s}=M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$, then
$j_{t}\in A\setminus S$. By the assumption, there is some element $i_{r}\in
S(1\leq r\leq k)$ which attacks the argument $j_{t}$. It follows that
$(i_{r},j_{t})\in R$. This is reflcted by $a_{i_{r},j_{t}}=1$ in the matrix
$M(F)$, and thus the column vector $M^{s}_{*,j_{t}}$ of the $s$-sub-block
$M^{s}=M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is non-zero.
Conversely, suppose that each column vector of the $s$-sub-block $M^{s}$ of
$S$ is non-zero. Take any element $j_{t}\in A\setminus S(1\leq t\leq h)$, then
$M^{s}_{*,j_{t}}$ is a column vector of the $s$-sub-block
$M^{s}=M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$. By the
hypothesis, we know that $M_{*,j_{t}}$ is non-zero. Therefore, there is some
$i_{r}\in S(1\leq r\leq k)$ such that $a_{i_{r},j_{t}}=1$, i.e.,
$(i_{r},j_{t})\in R$. This means that the argument $i_{r}$ attacks the
argument $j_{t}$ of $S$ in $F$, and thus we claim that $j_{t}$ is defeated by
$S$ in $F$.
6\. Determination of the admissible extensions
Example 14 Let us return to the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),(2,5),(4,1),(4,3),(5,4)\\}$ in
Example 10. Since an admissible extension is necessarily a conflict-free set,
we can look for the admissible extension from the collection
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,5\\},\\{2,4\\},\\{3,5\\},\\{1,3,5\\}\\}$
of conflict-free sets. By definition, it is easy to check that $\\{1,5\\}$ and
$\\{1,3,5\\}$ are the all admissible extensions in $F=(A,R)$.
In order to find the matrix way to determine the admissible extensions, we
mainly focus our attention on the admissible extension $S=\\{1,5\\}$ which is
not stable.
Note that, $(4,1)\in R$, $(3,1)\in R$, $(5,3)\in R$ and $(5,4)\in R$ implies
that the argument $1$ is defended by $S=\\{1,5\\}$, and $(2,5)\in R$ and
$(1,2)\in R$ implies that the argument $5$ is defended by $S=\\{1,5\\}$. It
follows that $S=\\{1,5\\}$ is an admissible extensions in $F$. Interestingly,
we have anther explanation for $S=\\{1,5\\}$ to be admissible. That is the
attacker $2$ of $S$ is defeated by $S$, the attacker $3$ of $S$ is defeated by
$S$, and the attacker $4$ of $S$ is defeated by $S$.
Now, let us turn our attention to the matrix
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 1&0&0&0&0\\\
1&0&1&0&0\\\ 0&0&1&1&0\end{array}\right).$
of $F$, and try to find out the corresponding representation of the above
conditions in the matrix $M(F)$.
Firstly, $(3,1)\in R$ and $(5,3)\in R$ are reflected by $a_{3,1}=1$ and
$a_{5,3}=1$ in the matrix $M(F)$ respectively. $(2,5)\in R$ and $(1,2)\in R$
are reflected by $a_{2,5}=1$ and $a_{1,2}=1$ in the matrix $M(F)$
respectively. While, $(4,1)\in R$ and $(5,4)\in R$ are reflected by
$a_{4,1}=1$ and $a_{5,4}=1$ in the matrix $M(F)$ respectively.
Secondly, $a_{2,5}=1$, $a_{3,1}=1$ and $a_{4,1}=1$ imply the row vectors
$(a_{2,1},a_{2,5})$, $(a_{3,1},a_{3,5})$ and $(a_{4,1},a_{4,5})$ are all non-
zero. Moreover, $a_{1,2}=1$, $a_{5,3}=1$ and $a_{5,4}=1$ imply the following
column vectors are all non-zero:
$\left(\begin{array}[]{ccc}a_{1,2}\\\
a_{5,2}\end{array}\right),\left(\begin{array}[]{ccc}a_{1,3}\\\
a_{5,3}\end{array}\right),\left(\begin{array}[]{ccc}a_{1,4}\\\
a_{5,4}\end{array}\right).$
Thirdly, the attacker $2$ of $S$ is defeated by $S$ is equivalent to that when
the row vectors $(a_{2,1},a_{2,5})\neq 0$ then the column vector
$\left(\begin{array}[]{ccc}a_{1,2}\\\ a_{5,2}\end{array}\right)\neq 0.$
The attacker $3$ of $S$ is defeated by $S$ is equivalent to that when the row
vectors $(a_{3,1},a_{3,5})\neq 0$ then the column vector
$\left(\begin{array}[]{ccc}a_{1,3}\\\ a_{5,3}\end{array}\right)\neq 0,$
The attacker $4$ of $S$ is defeated by $S$ is equivalent to that when the row
vectors $(a_{4,1},a_{4,5})\neq 0$ then the column vector
$\left(\begin{array}[]{ccc}a_{1,4}\\\ a_{5,4}\end{array}\right)\neq 0.$
Finally, if we combine the row vectors $(a_{2,1},a_{2,5})$,
$(a_{3,1},a_{3,5})$ and $(a_{4,1},a_{4,5})$, then the matrix
$\left(\begin{array}[]{ccc}a_{2,1}&a_{2,5}\\\ a_{3,1}&a_{3,5}\\\
a_{4,1}&a_{4,5}\end{array}\right)$
is a sub-block of $M(F)$. While, if we combine the column vectors
$\left(\begin{array}[]{ccc}a_{1,2}\\\
a_{5,2}\end{array}\right),\left(\begin{array}[]{ccc}a_{1,3}\\\
a_{5,3}\end{array}\right)\ \ \ and\ \ \ \left(\begin{array}[]{ccc}a_{1,4}\\\
a_{5,4}\end{array}\right),$
then the matrix
$\left(\begin{array}[]{ccc}a_{1,2}&a_{1,3}&a_{1,4}\\\
a_{5,2}&a_{3,3}&a_{5,4}\end{array}\right)$
is a sub-block of $M(F)$.
The above analysis motivates us to propose the following definition, and
provide a matrix way to determine whether a conflict-free set is an admissible
extensions of an argumentation framework.
Definition 15 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq
i_{1}<i_{2}<...<i_{k}\leq n)$ is an admissible extension of $F$. The $h\times
k$ sub-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}=\left(\begin{array}[]{cccccc}a_{j_{1},i_{1}}&a_{j_{1},i_{2}}&.&.&.&a_{j_{1},i_{k}}\\\
a_{j_{2},i_{1}}&a_{j_{2},i_{2}}&.&.&.&a_{j_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{j_{h},i_{1}}&a_{j_{h},i_{2}}&.&.&.&a_{j_{h},i_{k}}\end{array}\right)$
of the matrix $M(F)$ is called the $a$-sub-block of $S$ and denoted by
$M^{a}$, where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$ and $1\leq
j_{1}<j_{2}<...<j_{h}\leq n$.
In other words, the elements at the intersection of rows
$j_{1},j_{2},...,j_{h}$ and columns $i_{1},i_{2},...,i_{k}$ in the matrix
$M(F)$ form the $a$-sub-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$.
Note that, there is a natural relation between the $a$-sub-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ and the $s$-sub-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ in matrix theory. Namely,
the $a$-sub-block $M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$
is precisely the complementary sub-block of the $s$-sub-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ in the matrix
$M(F)$.
Theorem 16 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, then a conflict-free set
$S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq i_{1}<i_{2}<...<i_{k}\leq n$ is
an admissible extension iff the column vector of the $s$-sub-block $M^{s}$ of
$S$ corresponding to the non-zero row vector of the $a$-sub-block $M^{a}$ of
$S$ is non-zero, where $A\setminus S=\\{j_{1},j_{2},...,j_{h}\\}$ and $1\leq
j_{1}<j_{2}<...<j_{h}\leq n$ .
Proof Let $S$ be a conflict-free set and $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$. We need only to prove that every $i_{r}\in
S(1\leq r\leq k)$ is defended by $S$ iff the column vector of $s$-sub-block
$M^{s}$ of $S$ corresponding to the non-zero row vector of the $a$-sub-block
$M^{a}$ of $S$ is non-zero
Assume that every $i_{r}\in S(1\leq r\leq k)$ is defended by $S$. If the row
vector $M^{a}_{t,*}(1\leq t\leq h)$ of the $a$-sub-block
$M^{a}=M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is non-zero,
then there is some $i_{r}(1\leq r\leq k)$ such that $a_{j_{t},i_{r}}=1$. It
follows that $(j_{t},i_{r})\in R$, $i.e.$, the argument $i_{r}$ is attacked by
the argument $j_{t}$. By the assumption, there is some $i_{q}\in S(1\leq q\leq
k)$ which attacks the argument $j_{t}$, i.e., $(i_{q},j_{t})\in R$. This is
reflected by $a_{i_{q},j_{t}}=1$ in the matrix $M(F)$. Obviously,
$a_{i_{q},j_{t}}$ is an element of the column vector $M^{s}_{*,t}$ of $M^{s}$.
Therefore, the column vector $M^{s}_{*,t}$ of the $s$-sub-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=M^{s}$ of $S$ is non-zero.
Conversely, suppose any column vector of the $s$-sub-block $M^{s}$ of $S$
corresponding to the non-zero row vector of the $a$-sub-block $M^{s}$ of $S$
is non-zero. Let $i_{r}\in S(1\leq r\leq k)$, which is attacked by some
$j_{t}\in A\setminus S(1\leq t\leq h)$. Then, $(j_{t},i_{r})\in R$, which is
reflected by $a_{j_{t},i_{r}}=1$ in the matrix $M(F)$. It follows that the row
vector $M^{a}_{t,*}$ of the $a$-sub-block
$M^{a}=M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is non-zero.
By the assumption, the corresponding column vector $M^{s}_{*,t}$ of the
$s$-sub-block $M^{s}=M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$
is non-zero. Therefore, there is some $i_{q}\in S(1\leq q\leq k)$ such that
$a_{i_{q},j_{t}}=1$ in the matrix $M(F)$. Correspondingly, we have that
$(i_{q},j_{t})\in R$, and thus the argument $j_{t}$ is attacked by the
argument $i_{q}\in S$. To sum up, the argument $i_{r}\in S$ is defended by $S$
in $F$.
Remark: The fact that any stable extension must be admissible is clearly
expressed by the properties of $s$-sub-blocks in the matrix $M(F)$. In other
words, the condition every column vector of the $s$-sub-block $M^{s}$ of $S$
are non-zero is stronger than that the column vector of the $s$-sub-block
$M^{s}$ of $S$ corresponding to the non-zero row vector of the $a$-sub-block
$M^{a}$ of $S$ is non-zero.
7\. Determination of the complete extensions
Example 17 Consider the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and
$R=\\{(1,4),(2,1),\\{2,3\\},(2,4),(2,5),(3,2),(4,1)\\}$. Since the admissible
extension is necessarily a conflict-free set, we can find out all admissible
extensions from the collection of conflict-free sets
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\},\\{1,3\\},\\{1,5\\},\\{3,4\\},\\{3,5\\},\\{4,5\\},\\{1,3,5\\},\\{3,4,5\\}\\}$.
By the directed graph of $F$, it is easy to check that $\\{1,3\\}$,
$\\{3,4\\}$, $\\{3,5\\}$, $\\{1,3,5\\}$ and $\\{3,4,5\\}$ are all the stable
extensions of $F$. Furthermore, one can verify that $\\{3,5\\}$ is the only
complete extension which are not stable, $\\{2\\}$, $\\{3\\}$, $\\{4\\}$ are
all the admissible extensions which are not complete.
In order to confirm that the admissible extension $S=\\{3,5\\}$ is complete,
we need to give the reasons which support that the arguments $1,2,4$ not
defended by $S=\\{3,5\\}$. There are two cases for us to deal with. One is for
the argument $2$. It is attacked by $3$ because of $(3,2)\in R$, and thus is
not defended by $S=\\{3,5\\}$. Another is for the arguments $1$ and $4$ which
has no attacker in $S$. The argument $1$ has two attackers $2$ and $4$ in
light of $(2,1),(4,1)\in R$. Since $(3,2)\in R$, the attacker $2$ of the
argument $1$ is attacked by $3$. But, $(3,4),(5,4)\notin R$ implies that the
attacker $4$ of the argument $1$ is not attacked by any elements
$S=\\{3,5\\}$. Therefore, the argument $1$ is not defended by $S$. Similar
analysis indicates that the argument $4$ is also not defended by $S$.
Note that, in the above discussion the fact that the attacker $2$ of the
argument $1$ is attacked by $3$ is not the key for which the argument $1$ is
not defended by $S$. So, we can omit the handling of this situation.
Next, we will analysis the expressions of $\\{3,5\\}$ in the matrix $M(F)$ of
$F$ (as a complete extension but not stable), and extract the matrix way to
decide that an admissible extension is complete. Let us firstly write out the
matrix of the argumentation framework $F=(A,R)$:
$M(F)=\left(\begin{array}[]{ccccc}0&0&0&1&0\\\ 1&0&1&1&1\\\ 0&1&0&0&0\\\
1&0&0&0&0\\\ 0&0&0&0&0\end{array}\right).$
For the argument $2$ of the first case above, $(3,2)\in R$ is represented in
the form $a_{3,2}=1$, which results in that the column vector
$\left(\begin{array}[]{ccc}a_{3,2}\\\ a_{5,2}\end{array}\right)\neq 0.$
For the argument $1$ of the second case above, we first note that the column
vector
$\left(\begin{array}[]{ccc}a_{3,1}\\\ a_{5,1}\end{array}\right)=0.$
Furthermore, $(4,1)\in R$ is represented in the form $a_{4,1}=1$, which
results in that the column vector
$\left(\begin{array}[]{ccc}a_{1,1}\\\ a_{2,1}\\\ a_{4,1}\end{array}\right)\neq
0.$
And, $(3,4),(5,4)\notin R$ is represented in the form $a_{3,4}=0,a_{5,4}=0$,
which results in that the column vectors
$\left(\begin{array}[]{ccc}a_{3,4}\\\ a_{5,4}\end{array}\right)=0.$
For the argument $4$ of the second case above, we can find out its
representation in the matrix $M(F)$ in a similar way.
Similar as the $s$-sub-block and the $a$-sub-block, if we combine the column
vectors
$\left(\begin{array}[]{ccc}a_{1,1}\\\ a_{2,1}\\\
a_{4,1}\end{array}\right),\left(\begin{array}[]{ccc}a_{1,2}\\\ a_{2,2}\\\
a_{4,2}\end{array}\right)\ \ \ and\ \ \ \left(\begin{array}[]{ccc}a_{1,4}\\\
a_{2,4}\\\ a_{4,4}\end{array}\right),$
then we obtain a sub-block
$\left(\begin{array}[]{ccc}a_{1,1}&a_{1,2}&a_{1,4}\\\
a_{2,1}&a_{2,2}&a_{2,4}\\\ a_{4,1}&a_{4,2}&a_{4,4}\end{array}\right).$
which is the key to determine the completeness of $S$.
To sum up, $S_{1}=\\{3,5\\}$ is a complete extension can be verified by the
following facts contained in the matrix $M(F)$:
(1) When the column vectors of the sub-block
$M^{s}=\left(\begin{array}[]{ccc}a_{3,1}&a_{3,2}&a_{3,4}\\\
a_{5,1}&a_{5,2}&a_{5,4}\end{array}\right)$
are zero, the corresponding column vectors of the sub-block
$\left(\begin{array}[]{ccc}a_{1,1}&a_{1,2}&a_{1,4}\\\
a_{2,1}&a_{2,2}&a_{2,4}\\\ a_{4,1}&a_{4,2}&a_{4,4}\end{array}\right)$
are non-zero.
(2) The column $4$ of $M^{s}$ corresponding to the row number where
$a_{4,1}=1$ is at is zero, and the column $1$ of $M^{s}$ corresponding to the
row number where $a_{1,4}=1$ is at is zero.
This motivation makes us to propose the following definition, and provide a
matrix way to determine whether a conflict-free set is a complete extension of
an argumentation framework.
Definition 18 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq
i_{1}<i_{2}<...<i_{k}\leq n)$ is a complete extension of $F$. The sub-block
$M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}=\left(\begin{array}[]{cccccc}a_{j_{1},i_{1}}&a_{j_{1},i_{2}}&.&.&.&a_{j_{1},i_{k}}\\\
a_{j_{2},i_{1}}&a_{j_{2},i_{2}}&.&.&.&a_{j_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{j_{h},i_{1}}&a_{j_{h},i_{2}}&.&.&.&a_{j_{h},i_{k}}\end{array}\right)$
of order $h$ in the matrix of $M(F)$ is called the $c$-sub-block of $S$ and
denoted by $M^{c}$ for short, where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$
and $1\leq j_{1}<j_{2}<...<j_{h}\leq n$.
In other words, the elements appearing at the intersection of rows
$j_{1},j_{2},...,j_{h}$ and the same number of columns in the matrix $M(F)$
form the $c$-sub-block $M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of
$S$.
Note that, the $c$-sub-block
$M^{c}=M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is exactly
the complementary sub-block of the $s$-sub-block
$M^{s}=M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$ of $S$, in the
matrix $M(F)$ of $F=(A,R)$.
Lemma 19 Let $F=(A,R)$ be an argumentation framework with $A=\\{1,2,...,n\\}$,
then $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq i_{1}<i_{2}<...<i_{k}\leq
n)$ is a complete extension of $F$ iff $S$ is an admissible extension and each
argument $j_{t}\in S(1\leq t\leq h)$ is not defended by $S$ in $F$.
Theorem 20 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, the admissible extension
$S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq i_{1}<i_{2}<...<i_{k}\leq n)$ is
complete in $F$ iff each column vector of $c$-sub-block $M^{c}$ of $S$
corresponding to the zero column vector of the $s$-sub-block $M^{s}$ of $S$ is
non-zero, and this column vector has at least one non-zero element such that
the column vector of the $s$-sub-block $M^{s}$ of $S$ corresponding to the row
index of it is zero, where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$ and
$1\leq j_{1}<j_{2}<...<j_{h}\leq n$.
Proof Let $S$ be an admissible extension and $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$, we need only to prove that every $j_{t}\in
S(1\leq t\leq h)$ is not defended by $S$ in $F$ iff the condition in the
theorem is hold.
Necessity. Let the column vector $M^{s}_{*,t}$ of the $s$-sub-block
$M^{i_{1},i_{2},...,i_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is zero, where
$1\leq t\leq h$. If the corresponding column vector $M^{c}_{*,t}$ in the
$c$-sub-block $M^{j_{1},j_{2},...,j_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is
zero, then $a_{i,j_{t}}=0$ for each $1\leq i\leq n$. This implies that
$(i,j_{t})\notin R$, i.e., the argument $j_{t}$ is dot attacked by $i$, for
each $1\leq i\leq n$. So, there is no argument of $A$ which attacks $j_{t}$,
and thus the argument $j_{t}\in A\setminus S$ is defended by $S$, a
contradiction with the completeness of $S$. Therefore, each column vector of
$c$-sub-block $M^{c}$ of $S$ corresponding to the zero column vector of the
$s$-sub-block $M^{s}$ of $S$ is non-zero.
Suppose the column vector $M^{c}_{*,t}$ of the $c$-sub-block $M^{c}$ of $S$
corresponding to the zero column vector $M^{s}_{*,t}$ of the $s$-sub-block
$M^{s}$ of $S$ is non-zero where $1\leq t\leq h$, but every column vector of
the $s$-sub-block $M^{s}$ of $S$ corresponding to the row index of the non-
zero element of $M^{c}_{*,t}$ is non-zero, we claim that the argument
$j_{t}\in A\setminus S$ is defended by $S$, which contradicts with the
completeness of $S$. In fact, $M^{s}_{*,t}$ indicates that
$a_{i_{r},j_{t}}=0$, i.e., the argument $i_{r}$ does not attack $j_{t}$ for
each $1\leq r\leq k$. So, the attackers of $j_{t}$ must be in the set
$A\setminus S$. Let the argument $j_{v}(1\leq v\leq h)$ be an attacker of
$j_{t}$, then $(j_{v},j_{t})\in R$, i.e., $a_{j_{v},j_{t}}=1$. By the
assumption, the column vector $M^{s}_{*,v}$ of the $s$-sub-block $M^{s}$ of
$S$ is non-zero. This implies that there is some $i_{r}(1\leq r\leq k)$ such
that $a_{i_{r},j_{v}}=1$, i.e., $(i_{r},j_{v})\in R$. Thus, the attacker
$j_{v}$ of the argument $j_{t}$ is attacked by an element $i_{r}$ of the set
$S$.
Sufficiency. Let $j_{t}\in A\setminus S(1\leq t\leq h)$, we should prove that
$j_{t}$ is not defended by $S$. If there is some argument $i_{r}\in S(1\leq
r\leq k)$ which attacks $j_{t}$, then we have done. Otherwise,
$(i_{r},j_{t})\in R$, i.e., $a_{i_{r},j_{t}}=1$ for each $1\leq r\leq k$, and
thus the column vector $M^{s}_{*,t}$ of $s$-sub-block
$M_{s}=M^{i_{1},i_{2},...,i_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is zero.
According to the assumption, the column vector $M^{c}_{*,t}$ of the $c$-sub-
block $M^{c}=M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is non-
zero, and it has one non-zero element, say $a_{j_{l},j_{t}}=1$, such that the
column vector $M^{s}_{*,l}$ of the $s$-sub-block $M^{s}$ of $S$ is zero. This
indicates that $(j_{l},j_{t})\in R$ and $(i_{r},j_{l})\notin R$ for each
$1\leq r\leq k$. Thus, the attacker $j_{l}$ of the argument $j_{t}$ does not
be attacked by any element of $S$. Therefore, the argument $j_{t}$ is not
defended by $S$.
8\. A matrix approach for computing extensions
In matrix theory, the interchange between different two rows (or columns) is
one kind of the elementary operations on a matrix, by which a matrix can be
reduce to a simply form. In this section, corresponding to the conflict-free
set $S$ found out in Section 4 we shall first turn the matrix $M(F)$ of an
argumentation framework $F$ into a norm form which is efficient enough for us
to determine whether the set $S$ is a stable (admissible, complete) extension,
by a sequence of interchanges between different two rows (or columns). Then,
combining with the results obtained in the above Sections we put foreword a
matrix approach for computing all extensions under a given semantics.
Given an argumentation framework $F=(A,R)$ with $A=\\{1,2,...,n\\}$, let
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})(1<...<k<...<l<...<n)$ be the matrix of
$F$ corresponding to the permutation $(i_{1},...,i_{k},...,i_{l},...,i_{n})$.
A dual interchange of the matrix $M(i_{1},...,i_{k},...,i_{l},...,i_{n})$
between $k$ and $l$, denoted by $k\rightleftharpoons l$, consists of two
interchanges: interchanging row $k$ and row $l$; interchanging column $k$ and
column $l$.
Lemma 21 Given an argumentation framework $F=(A,R)$ with $A=\\{1,2,...,n\\}$.
Let $M(i_{1},...,i_{k},...,i_{l},...,i_{n})(1<...<k<...<l<...<n)$ be the
matrix of $F$ corresponding to the permutation
$(i_{1},...,i_{k},...,i_{l},...,i_{n})$ of $A$, then a dual interchange
$i_{k}\rightleftharpoons i_{l}$ turns the matrix
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})$ into the matrix
$M(i_{1},...,i_{l},...,i_{k},...,i_{n})$ of $F$ corresponding to the
permutation $(i_{1},...,i_{l},...,i_{k},...,i_{n})$.
Proof Let
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})=\left(\begin{array}[]{ccccccccccccc}a_{1,1}&.&.&.&a_{1,k}&.&.&.&a_{1,l}&.&.&.&a_{1,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
a_{k,1}&.&.&.&a_{k,k}&.&.&.&a_{k,l}&.&.&.&a_{k,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
a_{l,1}&.&.&.&a_{l,k}&.&.&.&a_{l,l}&.&.&.&a_{l,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
a_{n,1}&.&.&.&a_{n,k}&.&.&.&a_{n,l}&.&.&.&a_{n,n}\end{array}\right)$
be the matrix of $F$ corresponding to the permutation
$(i_{1},...,i_{k},...,i_{l},...,i_{n})$, where $a_{s,t}=1$ if and only if
$(i_{s},i_{t})\in R(1\leq s,t\leq n)$. If we make a dual interchange
$k\rightleftharpoons l$ of the matrix
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})$, then the matrix
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})$ changes into the following matrix
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})_{(k\rightleftharpoons
l)}=\left(\begin{array}[]{ccccccccccccc}a_{1,1}&.&.&.&a_{1,l}&.&.&.&a_{1,k}&.&.&.&a_{1,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
a_{l,1}&.&.&.&a_{l,l}&.&.&.&a_{l,k}&.&.&.&a_{l,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
a_{k,1}&.&.&.&a_{k,l}&.&.&.&a_{k,k}&.&.&.&a_{k,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
a_{n,1}&.&.&.&a_{n,l}&.&.&.&a_{n,k}&.&.&.&a_{n,n}\end{array}\right)$
On the other hand, if we denote the matrix
$M(i_{1},...,i_{l},...,i_{k},...,i_{n})$ of $F$ corresponding to the
permutation $(i_{1},...,i_{l},...,i_{k},...,i_{n})$ by
$\left(\begin{array}[]{ccccccccccccc}b_{1,1}&.&.&.&b_{1,k}&.&.&.&b_{1,l}&.&.&.&b_{1,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
b_{k,1}&.&.&.&b_{k,k}&.&.&.&b_{k,l}&.&.&.&b_{k,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
b_{l,1}&.&.&.&b_{l,k}&.&.&.&b_{l,l}&.&.&.&b_{l,n}\\\
.&.&.&.&.&.&.&.&.&.&.&.&\\\
b_{n,1}&.&.&.&b_{n,k}&.&.&.&b_{n,l}&.&.&.&b_{n,n}\end{array}\right),$
then, by Definition 4, we have $b_{s,t}=1$ if and only if $(i_{s},i_{t})\in R$
when $s\neq k,l$ and $t\neq k,l$; $b_{s,k}=1$ if and only if $(i_{s},i_{l})\in
R$, $b_{s,l}=1$ if and only if $(i_{s},i_{k})\in R$, $b_{k,t}=1$ if and only
if $(i_{l},i_{t})\in R$, $b_{l,t}=1$ if and only if $(i_{k},i_{t})\in R$.
Thus, we have $b_{s,t}=a_{s,t}$ when $s\neq k,l$ and $t\neq k,l$;
$b_{s,k}=a_{s,l}$ and $b_{s,l}=a_{s,k}(1\leq s\leq n)$; $b_{k,t}=a_{l,t}$ and
$b_{l,t}=a_{k,t}(1\leq t\leq n)$. It follows that
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})_{(k\rightleftharpoons
l)}=M(i_{1},...,i_{l},...,i_{k},...,i_{n})$, and so the proof is done.
Remark By the proof of above Lemma, we can see that the dual interchange
$k\rightleftharpoons l$ can also turn the matrix
$M(i_{1},...,i_{l},...,i_{k},...,i_{n})$ corresponding to the permutation
$(i_{1},...,i_{l},...,i_{k},...,i_{n})$ into the matrix
$M(i_{1},...,i_{k},...,i_{l},...,i_{n})$ corresponding to the permutation
$(i_{1},...,i_{k},...,i_{l},...,i_{n})$. So, for any two matrices
corresponding to different permutations of $A$ we can turn one matrix into
another by a sequence of dual interchanges.
Theorem 22 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$. Let $M(F)$ be the matrix of $F$ corresponding to the
natural permutation $(1,2,...,n)$ of $A$,
$S=\\{i_{1},i_{2},...,i_{k}\\}\subset A(1\leq i_{1}<i_{2}<...<i_{k}\leq n)$,
then by a sequence of dual interchanges we can turn $M(F)$ into the matrix
$M(i_{1},i_{2},...,i_{k},j_{1},j_{2},...,j_{h})$ corresponding to the
permutation $(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$, where
$\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$ and $1\leq
j_{1}<j_{2}<...<j_{h}\leq n$. In particular, the matrix
$M(i_{1},i_{2},...,i_{k},j_{1},j_{2},...,j_{h})$ has the following partition
form
$\left(\begin{array}[]{ccc}M^{cf}&&M^{s}\\\ M^{a}&&M^{c}\end{array}\right)$,
where $M^{cf},M^{s},M^{a}$ and $M^{c}$ are the $cf$-principle sub-block,
$s$-sub-block, $M^{a}$-sub-block and $M^{c}$-sub-block of the set $S$ in the
matrix $M(F)$, respectively.
Proof Let us fist consider the argument $i_{1}$. If $i_{1}=1$, then no dual
interchange is needed, and $M(F)=M(i_{1},2,...,n)$. Otherwise, by making the
dual interchange $1\rightleftharpoons i_{1}$ of the matrix $M(F)$ we get the
matrix $M(i_{1},...,1,...)$ of $F$ corresponding to the permutation
$(i_{1},...,1,...)$, where $1$ is at the position of the natural permutation
where $i_{1}$ is at.
Secondly, we discuss the argument $i_{2}$. If $i_{1}=1$ and $i_{2}=2$, then no
dual interchange is needed, and $M(F)=M(i_{1},i_{2},2,...,n)$. If $i_{1}=1$
and $i_{2}\neq 2$, by making the dual interchange $2\rightleftharpoons i_{2}$
we turn the matrix $M(i_{1},2,...,n)$ into the matrix
$M(i_{1},i_{2},...,2,...,n)$ corresponding to the permutation
$i_{1},i_{2},...,2,...,n$. If $i_{1}=2$, then $i_{2}\neq 1$ and the dual
interchange $2\rightleftharpoons i_{2}$ turn the matrix
$M(i_{1},...,1,...)=M(i_{1},1,...)=M(2,1,...)$ into the matrix
$M(i_{1},i_{2},...,1,...)=M(2,i_{2},...,1,...)$ corresponding to the
permutation $(i_{1},i_{2},...,1,...)$, where $1$ is at the position of the
permutation $(i_{1},...,1,...)=(i_{1},1,...)$ where $i_{2}$ is at. If
$i_{1}\neq 1,2$, then $i_{2}$ is behind of $1$ in the permutation
$(i_{1},...,1,...)$ and the dual interchange $2\rightleftharpoons i_{2}$ turn
the matrix $M(i_{1},...,1,...)$ into the matrix
$M(i_{1},i_{2},...,1,...,2,...)$ corresponding to the permutation
$(i_{1},i_{2},...,1,...,2,...)$, where $2$ is at the position of the
permutation $(i_{1},...,1,...)$ where $i_{2}$ is at.
This process can be done step by step. Suppose that we have got the matrix
$M(i_{1},i_{2},...,i_{k-1},p_{1},p_{2},...,p_{n-k+1})$ where
$(p_{1},p_{2},...,p_{n-k+1})$ is a permutation of the set
$A\setminus\\{i_{1},i_{2},...,i_{k-1}\\}$, we finally handle the argument
$i_{k}$. If $i_{k}=p_{1}$, then the matrix
$M(i_{1},i_{2},...,i_{k-1},p_{1},...,p_{n-k+1})$
$=M(i_{1},i_{2},...,i_{k-1},i_{k},p_{2},...,p_{n-k+1})$. Otherwise, we can
make the dual interchange $k\rightleftharpoons i_{k}$ of the matrix
$M(i_{1},i_{2},...,i_{k-1},p_{1},p_{2},...,p_{n-k+1})$, and turn the matrix
$M(i_{1},i_{2},...,i_{k-1},p_{1},p_{2},...,p_{n-k+1})$ into the matrix
$M(i_{1},i_{2},...,i_{k},q_{1},q_{2},...,q_{n-k})$ of $F$ where
$(q_{1},q_{2},...,q_{n-k})$ is a permutation of the set
$A\setminus\\{i_{1},i_{2},...,i_{k}\\}$.
Similar as the above process for $\\{i_{1},i_{2},...,i_{k}\\}$, we can turn
the matrix $M(i_{1},i_{2},...,i_{k},q_{1},q_{2},...,q_{n-k})$ into the matrix
$M(i_{1},i_{2},...,i_{k},j_{1},j_{2},...,j_{h})$ corresponding to the
permutation $(i_{1},i_{2},...,i_{k},j_{1},j_{2},...,j_{h})$ by a sequence of
dual interchanges, where $(j_{1},j_{2},...,j_{h})$ is a permutation of the set
$\\{q_{1},q_{2},...,q_{n-k}\\}$.
Let $M(F)=M(1,2,...,n)=(a_{i,j})$, then by Definition 5 we have $a_{i,j}=1$
iff $(i,j)\in R$. Let
$M(i_{1},i_{2},...,i_{k},j_{1},j_{2},...,j_{h})=(b_{i,j})$, then
$(i_{s},i_{t})\in R$ iff $b_{s,t}=1$ where $1\leq s,t\leq k$,
$(j_{s},j_{t})\in R$ iff $b_{k+s,k+t}=1$ where $1\leq s,t\leq l$,
$(i_{s},j_{t})\in R$ iff $b_{s,k+t}=1$ where $1\leq s\leq k$ and $1\leq t\leq
l$. It follows that $b_{s,t}=a_{i_{s},i_{t}}$ where $1\leq s,t\leq k$,
$b_{k+s,k+t}=a_{j_{s},j_{t}}$ where $1\leq s,t\leq l$,
$b_{s,k+t}=a_{i_{s},j_{t}}$ where $1\leq s\leq k$ and $1\leq t\leq l$.
Therefore, we have
$M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})=\left(\begin{array}[]{ccc}M^{cf}&&M^{s}\\\
M^{a}&&M^{c}\end{array}\right)$.
where $M^{cf},M^{s},M^{a}$ and $M^{c}$ are the $cf$-principle sub-block,
$s$-sub-block, $M^{a}$-sub-block and $M^{c}$-sub-block of the set $S$ in the
matrix $M(F)$, respectively.
Corollary 23 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$. Let $M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$ be the
matrix of $F$ correspond to the permutation
$(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$ in Theorem 21, then
$S=\\{i_{1},i_{2},...,i_{k}\\}$ is a conflict-free set iff $M^{cf}$ is zero.
In other words,
$M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})=\left(\begin{array}[]{ccc}0&&M^{s}\\\
M^{a}&&M^{c}\end{array}\right)$.
Proof It follows from Theorem 8 and Theorem 22.
Corollary 24 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$. Let $M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$ be the
matrix of $F$ correspond to the permutation
$(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$ in Theorem 22, then
$S=\\{i_{1},i_{2},...,i_{k}\\}$ is a stable extension iff every column vector
of $M^{s}$ is not non-zero.
Proof It follows from Theorem 13 and Theorem 22.
Example 25 Let us continue to consider the argumentation framework $F=(A,R)$
in Example 10. The matrix of $F$ corresponding to the natural permutation
$(1,2,3,4,5)$ is
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
As have been shown in $Ex.12$, $\mathcal{S}(0)=\\{\emptyset\\}$,
$\mathcal{S}(1)=\\{\\{1\\},\\{2\\},...,\\{5\\}\\}$,
$\mathcal{S}(2)=\\{\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\}\\}$,
$\mathcal{S}(3)=\\{1,3,5\\}$, $\mathcal{S}(4)$ and $\mathcal{S}(5)$.
For $S=\\{1,3\\}$, by taking the interchange $2\rightleftharpoons 3$ the
matrix $M(F)$ is turned into the matrix
$M(1,3,2,4,5)=\left(\begin{array}[]{ccccc}0&0&1&0&0\\\ 0&0&0&0&0\\\
0&1&0&0&1\\\ 0&1&0&0&0\\\ 0&0&0&1&0\end{array}\right),$
where
$M^{cf}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{s}=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\end{array}\right),M^{a}=\left(\begin{array}[]{ccc}0&1\\\ 0&1\\\
0&0\end{array}\right),M^{c}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&1&0\end{array}\right).$
Since not every column vector of $M^{s}$ is zero, the set $S$ is not a stable
extension. Similar discussion show that other elements of $\mathcal{S}(2)$ are
also not stable extensions.
For $S^{\prime}=\\{1,3,5\\}$, by taking the interchange $3\rightleftharpoons
5$ the matrix $M(1,3,2,4,5)$ is turned into the matrix
$M(1,3,5,4,2)=\left(\begin{array}[]{ccccc}0&0&0&0&1\\\ 0&0&0&0&0\\\
0&0&0&1&0\\\ 0&1&0&0&0\\\ 0&1&1&0&0\end{array}\right).$
where
$M^{cf}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right),M^{s}=\left(\begin{array}[]{ccc}0&1\\\ 0&0\\\
1&0\end{array}\right),M^{a}=\left(\begin{array}[]{ccc}0&1&0\\\
0&1&1\end{array}\right),M^{c}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right).$
In this case, the column vectors of $M^{s}$ are all non-zero. And thus,
$S^{\prime}$ is a stable extension.
Theorem 26 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$. Let $M(F)$ be the matrix of $F$ corresponding to the
natural permutation $(1,2,...,n)$, then $S=\\{i_{1},i_{2},...,i_{k}\\}$ is a
conflict-free set iff by a sequence of dual interchanges we can turn $M(F)$
into the following partition matrix:
$\left(\begin{array}[]{ccc}O_{k,k}&O_{k,q}&S_{k,l}\\\
A_{q,k}&C_{q,q}&E_{q,l}\\\ F_{l,k}&G_{l,q}&H_{l,l}\end{array}\right)$
where each column vector of $S_{k,l}$ is non-zero, $k+q+l=n$ and $q\geq 0$.
Proof From Corollary 23, $S=\\{i_{1},i_{2},...,i_{k}\\}$ is a conflict-free
set iff by a sequence of dual interchanges $M(F)$ can be turned into the
partition matrix
$M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})=\left(\begin{array}[]{ccc}0&&M^{s}\\\
M^{a}&&M^{c}\end{array}\right)$
corresponding to the permutation $(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$,
where $M^{s},M^{a}$ and $M^{c}$ are the $s$-sub-block, $M^{a}$-sub-block and
$M^{c}$-sub-block of the set $S$ in the matrix $M(F)$, respectively.
If there is no zero column in the sub-block $M^{s}$ of
$M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$, then $q=0$ and we have done.
Otherwise, we may assume that all the zero column vectors of $M^{s}$ are the
columns $t_{1},t_{2},...,t_{q}(1\leq q\leq h)$. Certainly, they correspond to
the columns $k+t_{1}$, $k+t_{2}$, …, $k+t_{q}$ of the matrix
$M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$ respectively.
Similar as the proof of Theorem 22, after making a sequence of dual
interchanges the matrix $M(i_{1},i_{2},...,i_{k},j_{1},...,j_{h})$ shall be
turned into the matrix
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$
corresponding to the permutation
$(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$,
where $(j_{s_{1}},...,j_{s_{l}})$ is a permutation of the set
$A\setminus\\{i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}}\\}$. By now, the
elements at the intersection of first $k$ rows and first $k+q$ columns of
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$ are
zero. Therefore, the matrix
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$ has
the following partition form:
$\left(\begin{array}[]{ccc}O_{k,k}&O_{k,q}&S_{k,l}\\\
A_{q,k}&C_{q,q}&E_{q,l}\\\ F_{l,k}&G_{l,q}&H_{l,l}\end{array}\right)$
where each column vector of $S_{k,l}$ is non-zero, $k+q+l=n$ and $q\geq 1$.
Remark Since the partition matrices obtained in Corollary 23 and Theorem 26
play a central role in finding out the extensions of $F=(A,R)$, we called them
the norm form of the matrix $M(F)$ of $F=(A,R)$.
Corollary 27 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$. Let
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})=(b_{i,j})$
be the norm form of $M(F)$ corresponding to the permutation
$(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$ in
Theorem 26, then, $S=\\{i_{1},i_{2},...,i_{k}\\}$ is an admissible extension
iff $A_{q,k}=0$.
Proof Necessity. Assume that $A_{q,k}\neq 0$, then there are some $r(1\leq
r\leq k)$ and $v(1\leq u\leq q)$ such that the element at the intersection of
the $u$-th row and the $r$-th column in $A_{q,k}$ is $1$, which is at the
intersection of the $(k+u)$-th row and the $r$-th column in
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$. It
follows that $b_{k+u,r}=1$, i.e., $(j_{t_{u}},i_{r})\in R$. Thus, the argument
$i_{r}$ is attacked by $j_{s_{u}}$. But, from the zero sub-block $O_{k,q}$ we
know that $b_{w,k+u}=0$ for any $1\leq w\leq k$. So, there is no argument
$i_{w}(1\leq w\leq k)$ in $S$ which attacks $j_{t_{u}}$. And thus, the
argument $i_{r}$ is not defended by $S$, a contradiction with the hypothesis
Sufficiency. Obviously, $S$ is a conflict-free set in terms of the zero sub-
block $O_{k,k}$ in
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$.
Since $A_{q,k}=0$, we have $b_{k+u,r}=0(1\leq r\leq k,1\leq u\leq q)$, i.e.,
$(j_{t_{u}},i_{r})\notin R$. Thus, $S$ is not attacked by any argument of the
set $\\{j_{t_{1}},...,j_{t_{q}}\\}$.
Let $i_{r}(1\leq r\leq k)$ be any fixed argument of $S$, if there is some
argument $p$ which attacks $i_{r}$, then $p=j_{s_{v}}$ for some $(1\leq v\leq
l)$. Since the $v$-th column vector in $S_{k,l}$ is not zero, there is some
$w(1\leq w\leq k)$ such that $b_{w,k+q+v}=1$. It follows that
$(i_{w},j_{s_{v}})\in R$, i,e., $p=j_{s_{v}}$ is attacked by the argument
$i_{w}$ of $S$. Therefore, the conflict-free set $S$ is defended by itself,
and thus an admissible extension.
Corollary 28 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$. Let
$M(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$ be
the norm form of $M(F)$ corresponding to the permutation
$(i_{1},i_{2},...,i_{k},j_{t_{1}},...,j_{t_{q}},j_{s_{1}},...,j_{s_{l}})$ in
Theorem 26, then $S=\\{i_{1},i_{2},...,i_{k}\\}$ is a complete extension iff
$A_{q,k}=0$ and each column vector of $C_{q,q}$ is not zero.
Proof Sufficiency. First, $S$ is an admissible extension follows from
$A_{q,k}=0$. Since each column vector of $S_{k,l}$ is non-zero, for every
$v(1\leq v\leq l)$ there is some $r(1\leq r\leq k)$ such that $b_{r,k+q+v}=1$,
i.e., $(i_{r},j_{s_{v}})\in R$. Thus, the argument $j_{s_{v}}(1\leq v\leq l)$
is not defended by $S$.
Let us consider any fixed argument $j_{t_{u}}(1\leq u\leq q)$. Because each
column vector of $C_{q,q}$ is not zero, there is some $v(1\leq v\leq q)$ such
that $b_{k+v,k+u}=1$, i.e., $(j_{t_{v}},j_{t_{u}})\in R$. On the other hand,
$b_{r,k+v}=$ for every $1\leq r\leq k$ because of $O_{k,q}$. So,
$(i_{r},j_{t_{v}})\notin R(1\leq r\leq k)$. That means the argument
$j_{t_{v}}$ is not attacked by any element of $S$, and thus we conclude that
$j_{t_{u}}$ is not defended by $S$.
To sum up, $S$ is a complete extension of $F$.
Necessity. Assume that $S$ is a complete extension of $F$, then $S$ is a
conflict-free set and thus by a sequence of dual interchanges the matrix
$M(F)$ can be turn into the following the norm form
$\left(\begin{array}[]{ccc}O_{k,k}&O_{k,q}&S_{k,l}\\\
A_{q,k}&C_{q,q}&E_{q,l}\\\ F_{l,k}&G_{l,q}&H_{l,l}\end{array}\right)$
where each column vector of $S_{k,l}$ is non-zero, $k+q+l=n$ and $q\geq 1$.
Obviously, $A_{q,k}=0$ comes from the fact that $S$ is an admissible extension
of $F$. Next, we shall prove that each column vector of $C_{q,q}$ is not zero.
If there is some $u(1\leq u\leq q)$ such that the $u$-th column vector of
$C_{q,q}$ is zero, then $b_{k+v,k+u}=0$ for each $v(1\leq v\leq q)$, i.e.,
$(j_{t_{v}},j_{t_{u}})\notin R(1\leq v\leq q)$. Considering the sub-block
$O_{k,q}$, we also have $b_{r,k+u}=0$ for each $r(1\leq r\leq k)$, i.e.,
$(i_{r},j_{t_{u}})\notin R(1\leq r\leq k)$. Let $p$ be an attacker of the
argument $j_{t_{u}}$, then there must be some $w(1\leq w\leq l)$ such that
$p=j_{s_{w}}$. But, we know that each column vector of $S_{k,l}$ is non-zero,
and thus the $w$-th column vector of $S_{k,l}$ is non-zero. So, there is some
$r(1\leq r\leq k)$ such that $b_{r,k+q+w}=1$, i.e., $(i_{r},j_{s_{w}})\in R$.
That means the argument $i_{r}$ is an attacker of $p=j_{s_{w}}$. Therefore,
the argument $j_{t_{u}}$ is defended by $S$, which contradicts with the fact
that $S$ is a complete extension.
Example 29 Consider the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(1,3),(3,1),(4,5),(5,1),(5,4)\\}$. Then,
the matrix of $F$ corresponding to the natural permutation $(1,2,3,4,5)$ is
$M(F)=\left(\begin{array}[]{ccccc}0&1&1&0&0\\\ 0&0&0&0&0\\\ 1&0&0&0&0\\\
0&0&0&0&1\\\ 1&0&0&1&0\end{array}\right).$
By Theorem 9, it is easy to fin out all the conflict-free sets: $\\{1,4\\}$,
$\\{1,5\\}$, $\\{2,3\\}$ $\\{2,4\\}$ $\\{2,5\\}$ $\\{3,4\\}$ $\\{3,5\\}$
$\\{1,4,2\\}$ $\\{1,4,5\\}$ $\\{2,3,4\\}$ $\\{2,3,5\\}$.
For the set $S=\\{3,4\\}$, we make the dual interchange $1\rightleftharpoons
3$ and turn the matrix $M(F)$ into the matrix
$M(3,2,1,4,5)=\left(\begin{array}[]{ccccc}0&0&1&0&0\\\ 0&0&0&0&0\\\
1&1&0&0&0\\\ 0&0&0&0&1\\\ 0&0&1&1&0\end{array}\right).$
Furthermore, by making the dual interchange $2\rightleftharpoons 4$ on the
matrix $M(3,2,1,4,5)$ we turn the matrix $M(3,2,1,4,5)$ into the matrix
$M(3,4,1,2,5)=\left(\begin{array}[]{ccccc}0&0&1&0&0\\\ 0&0&0&0&1\\\
1&0&0&1&0\\\ 0&0&0&0&0\\\ 0&1&1&0&0\end{array}\right),$
where
$M^{s}=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&1\end{array}\right),M^{a}=\left(\begin{array}[]{ccc}1&0\\\ 0&0\\\
0&1\end{array}\right),M^{c}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\
1&0&0\end{array}\right).$
In this case, the column vectors of $M^{s}$ corresponding to the non-zero row
vectors of $M^{a}$ are non-zero. Thus, $S$ is an admissible extension
according to Theorem 16.
Next, by making the dual interchange $3\rightleftharpoons 4$ on the matrix
$M(3,4,1,2,5)$ we turn the matrix $M(3,2,1,4,5)$ into the matrix
$M(3,4,2,1,5)=\left(\begin{array}[]{ccccc}0&0&0&1&0\\\ 0&0&0&0&1\\\
0&0&0&0&0\\\ 1&0&1&0&0\\\ 0&1&0&1&0\end{array}\right),$
where
$A_{1,2}=\left(\begin{array}[]{cc}0&0\end{array}\right),S_{2,2}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),C_{1,1}=\left(\begin{array}[]{cc}0\end{array}\right),E_{1,2}=\left(\begin{array}[]{cc}0&0\end{array}\right),F_{2,2}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),$
$G_{2,1}=\left(\begin{array}[]{cc}1\\\
0\end{array}\right),H_{2,2}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right).$
In this case, $A_{2,1}=0$ and the only column vector of $C_{1,1}$ is zero.
Thus, $S$ is not a complete extension according to Corollary 28.
For the set $S^{\prime}=\\{2,3\\}$, we make the dual interchange
$1\rightleftharpoons 3$ and turn the matrix $M(F)$ into the matrix
$M(3,2,1,4,5)=\left(\begin{array}[]{ccccc}0&0&1&0&0\\\ 0&0&0&0&0\\\
1&1&0&0&0\\\ 0&0&0&0&1\\\ 0&0&1&1&0\end{array}\right).$
where
$M^{s}=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\end{array}\right),M^{a}=\left(\begin{array}[]{ccc}1&1\\\ 0&0\\\
0&0\end{array}\right),M^{c}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
1&1&0\end{array}\right).$
In this case, the column vectors of $M^{s}$ corresponding to the non-zero row
vectors of $M^{a}$ are non-zero. Thus, $S$ is an admissible extension
according to Theorem 16.
Furthermore, by making the dual interchange $3\rightleftharpoons 5$ the matrix
$M(3,2,1,4,5)$ is turned into the matrix
$M(3,2,5,4,1)=\left(\begin{array}[]{ccccc}0&0&0&0&1\\\ 0&0&0&0&0\\\
0&0&0&1&1\\\ 0&0&1&0&0\\\ 1&1&0&0&0\end{array}\right),$
where
$A_{2,2}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),S_{2,1}=\left(\begin{array}[]{cc}1\\\
0\end{array}\right),C_{2,2}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),E_{2,1}=\left(\begin{array}[]{cc}1\\\
0\end{array}\right),F_{1,2}=\left(\begin{array}[]{cc}1&1\end{array}\right),$
$G_{1,2}=\left(\begin{array}[]{cc}1&0\end{array}\right),H_{1,1}=\left(\begin{array}[]{cc}0\end{array}\right).$
In this case, $A_{2,2}=0$ and the column vectors of $C_{2,2}$ are all non-
zero. Thus, $S$ is a complete extension.
9\. Further discussion and related work
In the above Sections, we have established a matrix approach to find out all
the stable(admissible, complete) extensions of an AF. About other common
semantics of an AF not mentioned(such as preferred, grounded, ideal, semi-
stable and eager extension), we can also find out all the extensions through
the matrix approach and additional work which only concerns the comparison of
different sets. The procedure consists of two steps: Finding out all the
extensions(admissible or complete according to the need) by the matrix
approach; Comparing the related sets which are used to define the semantics,
and finding out the needed extensions in light of the definition of related
extensions. For example, if we want to find out all the preferred extensions
of an argumentation framework $F=(A,R)$, the procedure is as follows. Firstly,
finding out all the complete extensions of $F=(A,R)$ by the matrix approach.
Then, comparing all the complete extensions from the view of sets, all the
maximal sets are exactly the total preferred extensions we look for.
Now, all the ”global” questions concerning Dung’s argumentation frameworks
proposed by Modgil and Caminada[15] can be solved through the matrix approach.
For the ”local” questions, the matrix approach is still valid. We only need to
make some comparing between the related sets after finding out all the
extensions under a given semantics. If we want to decide whether a set
$A\in\mathcal{A}$ is contained in an specific extension, we only need to
compare the set $A$ with this extension, which has been found out by the
matrix approach. For other ”local” questions, we can give the similar process
based on each specific question.
There are several attempts in the literature for computing extensions of an
argumentation framework. Modgil and Caminada have developed the graph
labelling approach which was originally proposed by Pollok[17]. Argument game
approach is another efficient tool which is based on the proof theories. The
constraint satisfaction approach was built by Amgoud and Devred[1]. But, to
our knowledge no attempt was done in using the matrix to computer the
extensions of AFs.
In [15], the authors summarised the labelling approach by which the core
semantics of AFs defined by Dung and others[9, 5, 4, 10] can be found out.
But, this approach has an obvious drawback: The admissible extension found out
by graph labelling approach depend on the selection of the elements that are
illegally IN, so different selections may lead to same admissible extension.
In particular, there is no way to know whether the admissible extensions have
been found out entirely. The argument game approach mainly focus on solving
the ”local” questions, but only a selection of the ”local” questions have been
answered just as Modgil and Caminada described in [15]. The constraint
satisfaction approach possess more technical feature. It encodes an AF as a
Constraint Satisfaction Problem, and thus is able to use some powerful solvers
for computing the extensions of the argumentation framework. The problem lies
in how to find the candidate extensions which we want to verify by the
criterion established in their paper.
Our matrix approach first find out all the conflict-free sets of an AF, then
turn the matrix of the AF into a norm form with respect to a specific
semantics(stable, admissible or complete), finally select out all the
extensions according to the related criterions corresponding to different
semantics. For other semantics, such as grounded extension, preferred
extension, ideal extension, semi-stable extension and eager extension, we can
find out them from the related family of extensions by verifying whether they
satisfy the conditions(only concerning the comparison of different sets) in
their definitions. With regard to the ”local” questions, the remaining work is
only to compare the inclusion relations of deferent sets of each question,
after finding out all the needed extensions.
10\. Conclusions and perspectives
In this paper, we introduce the matrix representation $M(F)$ of an
argumentation framework $F=(A,R)$, and the $cf$-block $M^{cf}$, $s$-block
$M^{s}$, $a$-block $M^{a}$ and $c$-block $M^{c}$ with respect to a set
$S\subset A$. Several several theorems have been established in order to
determine the core extensions (stable, admissible, complete) of an
argumentation framework, by sub-blocks of the matrix $M(F)$ and the relations
between these sub-blocks. Furthermore, we propose a matrix approach finding
out all the extensions of an argumentation framework under a given semantics
(stable, admissible, complete). For other semantics, we can also compute all
the related extensions if we combine the matrix approach with some additional
work concerning the comparison of different sets.
Interestingly, the $s$-block $M^{s}$ ($a$-block $M^{a}$, $c$-block $M^{c}$) of
the set $S$ correspond to the determination for $S$ to be a stable extension
(admissible extension, complete extension respectively). And, the $c$-block of
$S$ is exactly the complementary sub-block of the $cf$-block of $S$, the
$a$-block of $S$ is exactly the complementary sub-block of the $s$-sub-block
of $S$. In addition, the dual interchanges provide us a chance to turn the
matrix $M(F)$ of the argumentation framework $F=(A,R)$ into a norm form which
can be easily employed to determine whether a conflict-free set is a specific
extension.
The prospectives are that, we can introduce or build more matrix tools into
the research of argumentation frameworks. Our future goal is to develop the
matrix approach in the related areas, such as bipolar argumentation
frameworks, fuzzy argumentation frameworks. Anther direction is to set up the
bridge from argument game to matrix operations, so as to find the possibility
of expressing the argument game by matrices[13, 14, 19].
References
## References
* [1] L. Amgoud, C. Devred, Argumentation frameworks as Constraint Satisfaction Problems, In _Proc. SUM_ , volume 6929 of _LNCS_ , 2011, 110-122. Springer.
* [2] P. Baroni, M. Giacomin, On principle-based evaluation of extension-based argumentation semantics, Artificial Intelligence 171 (2007), 675-700.
* [3] T. J. M. Bench-Capon, Paul E. Dunne, Argumentation in artificial intelligence, Artificial intelligence 171(2007)619-641
* [4] M. Caminada, Semi-stable semantics, in: Frontiers in Artificial Intelligence and its Applications, vol. 144, IOS Press, 2006, pp. 121-130.
* [5] C. Cayrol, M. C. Lagasquie-Schiex, Graduality in argumentation, J. AI Res. 23 (2005)245-297.
* [6] S. Coste-Marquis, C. Devred, Symmetric argumentation frameworks, in: Lecture Notes in Artificial Intelligence, vol. 3571, Springer-Verlag, 2005, pp. 317-328.
* [7] S. Coste-Marquis, C.Devred, P. Marquis, Prudent semantics for argumentation frameworks, in: Proc. 17th ICTAI, 2005, pp. 568-572.
* [8] Y. Dimopoulos, A. Torres, Graph theoretical structures in logic programs and default theories, Teoret. Comput. Sci. 170(1996)209-244.
* [9] P. M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and $n$-person games, Artificial Intelligence 77 (1995), 321-357.
* [10] P. M. Dung, P. Mancarella, F. Toni, A dialectic procedure for sceptical assumption-based argumentation, in: Frontiers in Artificial Intelligence and its Applications, vol. 144, IOS Press, 2006, pp. 145-156.
* [11] P. E. Dunne, Computational properties of argument systems satisfying graph-theoretic constrains, Artificial Intelligence 171 (2007), 701-729.
* [12] P. E. Dunne, T. J. M. Bench-Capon, Coherence in finite argument systems, Artificial intelligence 141(2002)187-203.
* [13] P. E. Dunne, T. J. M. Bench-Capon, Two party immediate response disputes: properties and efficiency, Artificial Intelligence 149 (2003), 221-250.
* [14] H. Jakobovits, D. Vermeir, Dialectic semantics for argumentation frameworks, in: Proc. 7th ICAIL, 1999, pp. 53-62.
* [15] S. Modgil, M. Caminada, Proof Theories and Algorithms for Abstract Argumentation Frameworks, In: Rahwan I., Simari G, editors. Argumentation in AI. Springer; 2009. p. 105-129.
* [16] E. Oikarinen, S. Woltron, Characterizing strong equivalence for argumentation frameworks, Artificial intelligence(2011), doi:10.1016/j.artint. 2011.06.003.
* [17] J. L. Pollock, Cognitive Carpentry, A Blueprint for How to Build a Person, MIT Press, Cambridge, MA, 1995.
* [18] G. Vreeswijk, Abstract argumentation system, Artificial intelligence 90(1997)225-279.
* [19] G. Vreeswijk, H. Pakken, Credulous and sceptical argument games for preferred semantics, in: Proceedings of JELIA’2000, the 7th European Workshop on Logic for Artificial Intelligence, Berlin, 2000, pp. 224-238.
|
arxiv-papers
| 2012-09-10T08:09:05 |
2024-09-04T02:49:34.909340
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xu Yuming",
"submitter": "Yuming Xu",
"url": "https://arxiv.org/abs/1209.1899"
}
|
1209.1994
|
# Non-convex penalized regression spline
Heng Peng
Department Mathematics,
The Hong Kong Baptist University
###### Abstract
Regression spline is a useful tool in nonparametric regression. However,
finding the optimal knot locations is a known difficult problem. In this
article, we introduce the Non-concave Penalized Regression Spline. This
proposal method not only produces smoothing spline with optimal convergence
rate, but also can adaptively select optimal knots simultaneously. It is
insensitive to the number of origin knots. The method’s performance in a
simulation has been studied to compare the other methods. The problem of how
to choose smoothing parameters, i.e. penalty parameters in the non-concave
regression spline is addressed.
KEY WORDS: Splines, Nonparametric regression, Non-concave penalized least
square, Power-basis, Knots selection, Additive model.
## 1 Introduction
In recently, much attention has been attracted by the penalized regression
spline. This method takes advantage of the smoothing spline and the regression
spline to simplify computation and knots selection procedures. The smoothing
spline can be regarded as a special case of penalized splines with a quadratic
roughness penalty function. Eiler and Marx (1996, 1998) used the quadratic
difference penalty function in penalized regression spline for univariate and
additive models. Mammen and van de Geer (1997) proposed to regard the total
variation of smoothing function as the penalty function and studied some
asymptotic properties for this kind penalized spline. Ruppert and Carroll
(1997, 2000) considered some properties of the $L_{1}$-penalized regression
spline. They also considered variable penalty parameters used in penalized
regression splines. Wand (1999) and Aerts, Clasekens and Wand (2002) studied
some theory of the penalized regression spline when penalties have quadratic
forms.
As introduced by Chapter 1, the trade-off between smoothness and flexibility
of the regression spline is controlled by the number and positions of knots.
Since Smith (1982) firstly used the statistical variable selection technique
to adaptively select optimal knots in fitting splines, there are a lot of
works along this direction in adaptive regression spline lecture, such as
Friedman and Silverman’s TURBO (1989), Friedman’s MARS (1991), Stone, et al.’s
POLYMARS (1997), Luo and Wahba’s HAS (1997), Ruppert’s MYOPIC (2001) etc.
Usually, this kind algorithms use stepwise procedures, such as forward or
backward. These approaches are very different from those used to establish the
traditional theory of regression spline. The gaps between the theory and
practice in the regression spline remain widely open. Though most penalized
regression spline procedures use convex function as the penalty to produce
shrinkage estimate to avoid the choice of knots, to get optimal smoothing for
the function and make the procedures insensitive to knot number and locations,
these procedures cannot avoid involving many knots, a high dimension parameter
space, just like the smoothing spline. Hence most penalized splines cannot
take the full advantage of the regression spline. In practice, the efficiency
of those penalized regression splines are still related to the optimal choice
of the number and locations of knots. Adaptive knots selection algorithms
still have to be studied for those penalized regression splines. See for
example, Ruppert and Carroll (1997), Mammen and van de Geer (1997) and Ruppert
(2002). There is little theoretical work on penalized regression splines with
various penalties. Theoretical properties for penalized splines need further
study.
In this paper, we propose a new approach, non-convex penalized regression
splines for spline fitting. It is also easily extended to multivariate
function estimation problems. Unlike the traditional penalized spline, our
penalized spline considers a non-convex function as the penalty and avoids
stepwise procedure to select knots. It can estimate smoothing function and
adaptively select knots simultaneously. It is insensitive to the number of
initial knots as long as it is large enough. This enables us to study sampling
properties of our penalized spline. In this chapter, we show that our
penalized regression spline has the optimal convergence rate empirically by
the simulation compared to other nonparametric regression methods.
To overcome the inefficiency of traditional variable selection procedures,
most attractive properties of non-concave penalty were introduced by Fan and
Li (2001). They demonstrated that penalized least squares with non-concave
penalty may produce threshold estimate. In their article, they also showed how
to select significant variables and estimate their coefficients simultaneously
via non-concave penalized likelihood, which is different to the traditional
procedures. In Chapter 2, we studied some properties of nonconcave penalized
likelihood estimate in high dimensional situations. The results may extend to
the nonparametric regression. Stone, et al.(1997) regarded the regression
spline model as an extended linear model with high dimension. Under this idea,
most variable selection methods can be modified for knots selection, so does
the non-convex penalized least-squares approach. The basic idea of our
approach just like the idea used in non-concave penalized likelihood. By
taking advantages of the non-concave penalty function, especially singularity
at origin, we get threshold estimators by non-convex penalized regression
spline with appropriate spline basis, such as the truncated power basis. By
threshold criteria, some estimated coefficient in a spline basis approximation
are shrunk to zero. This means that we can select knots automatically by the
penalized regression splines and avoid stepwise procedure. On the other hand,
the threshold estimate has the same effect as shrinkage estimate to make
trade-off between the flexibility and the smoothness. This property of the
threshold estimate has been used in the wavelets analysis, for example, by
Donoho and Johnstone (1994), Antoniadis and Fan (2001). Therefore our approach
can select knots and estimate the smoothing spline simultaneously. Eilers and
Marx (1996) claimed without proof that the penalized spline by quadratic
difference penalty is not sensitive to the number of knots if initial number
of knots is large enough. The number of knots used in smoothing spline is same
as sample size (see Green and Sliverman 1994). In these two approaches, the
trade-off between the smoothing and the flexibility is only controlled by the
regularized penalized parameters. From this insight, our new proposed approach
should be also expected to be insensitive to the initial number of knots and
the trade-off between the smoothing and flexibility is mainly controlled by
the regularized parameters. We demonstrate this property by the simulation in
Section 3. The proper penalized parameters can be selected by many methods,
such as the fivefold cross-validation proposed by Fan and Li (2001), the
generalized cross-validation method used in smoothing splines, see for example
Green and Silverman (1994), Wahba (1990) and BIC criterion (Schwarz 1978). In
this chapter, we consider two methods to select the penalized parameter for
the non-convex penalized regression spline.
Various algorithms mentioned in Chapter 2 to optimize a high-dimension
nonconcave likelihood function can be used for non-convex regression spline.
In this chapter, we apply the modified Newton-Raphson algorithm proposed by
Fan and Li (2001) to our non-convex regression spline.
In Section 2, we introduce the non-convex penalized regression spline. In
Section 3, numerical simulation results are demonstrated. In Section 4, we
give some discussions on how to our approach to multivariate regression
models.
## 2 Non-convex penalized Regression Spline
### 2.1 Penalized regression splines
Consider the nonparametric regression model as follows,
$y_{i}=f(x_{i})+\varepsilon_{i},\quad i=1,2,\ldots,n,$ (1)
where $x_{i}\in[0,1]$ are either deterministic or random design points , the
$\\{\varepsilon_{i}\\}$ are independent random error with mean zero and a
constant variance $\sigma^{2}$, $f(x)$ is a smooth regression function that
want to be estimated.
To estimate function $f(x)$, we consider spline space
$S(p,\mbox{\boldmath$t$})$ with knots
$\mbox{\boldmath$t$}=\\{0=t_{0}<t_{1}<\cdots<t_{k+1}=1\\}.$ (2)
For $p\geq 2$, $S(p,\mbox{\boldmath$t$})$ is defined as follows
$S(p,\mbox{\boldmath$t$})=\\{s(x)\in C^{p-2}[0,1]:s(x)\mbox{ is a polynomial
of order $p$ on each subinterval $[t_{i},t_{i+1}]$}\\}$
When $p=1$, $S(p,\mbox{\boldmath$t$})$ is the set of step functions with jumps
at the knots.
It is known that space $S(p,\mbox{\boldmath$t$})$ is a $k+p$ dimension linear
function space and truncated power function series
$\mbox{\boldmath$X$}_{x}=\\{1,x,x^{2},\ldots,x^{p-1},(x-t_{1})^{p-1}_{+},\ldots(x-t_{k})^{p-1}_{+}\\}$
forms a basis of $S(p,\mbox{\boldmath$t$})$ (see de Boor 1978). Thus we may
approximate $f(x)$ in model (1) by a spline with form
$f(x,\mathcal{B})=\mbox{\boldmath$X$}_{x}\mathcal{B}=\beta_{0}+\beta_{1}x+\cdots+\beta_{p-1}x^{p-1}+\sum\limits_{i=1}^{k}\beta_{p+i-1}(x-t_{i})_{+}^{p-1}.$
(3)
Hence, the nonparametric regression model (1) becomes a classic high dimension
linear regression model. Of course, we can replace the truncated power basis
by other bases of the spline space, such as the B-spline basis, in (3). Here
we like to use the truncated power basis just because deleting a knot $t_{j}$
is equivalent to setting the coefficient $\beta_{p+j-1}$ to zero. The variable
selection procedure accords with knots selection.
The penalized regression spline is defined as the minimizer
$\hat{f}(x,\mathcal{B})\in S(p,\mbox{\boldmath$t$})$ of the penalized least-
squares problem
$\min\limits_{f(x,\mbox{\boldmath$\scriptstyle\beta$})\in
S(p,\mbox{\boldmath$\scriptstyle
t$})}\sum\limits_{j=1}^{n}\\{y_{j}-f(x_{j},\mathcal{B})\\}^{2}+n\sum\limits_{j=1}^{k}p_{\lambda_{n}}(|w_{p+i-1}\beta_{p+i-1}|),$
(4)
where $p_{\lambda_{n}}(|\cdot|)$ is a penalty function and $\lambda_{n}$ is
the penalized parameter, $\\{w_{j}\\}$ are penalized weights. The latter
rescale or standardize the basis function in (3) and transform them back to
the original scale. Note that we don’t penalize the monomial terms
$1,x,\ldots,x^{p-1}$ for sake of interpretability.
### 2.2 Non-concave penalty functions
Selection of the penalty function in (4) is important for knot selection. As
discussed in Chapters 1 and 2, Fan and Li (2001) studied the non-concave
penalized likelihood for variable selection. They showed that a good penalty
function should result in an estimator with three properties: (1)
Unbiasedness, in which there is no over-penalization of large parameters to
avoid unnecessary biases; (2) sparsity, as the resulting penalized likelihood
estimator should follow a thresholding rule so that insignificant parameters
can automatically be set to zero to reduce model complexity; (3) continuity,
to avoid instability in model prediction, whereby the penalty function should
be chosen such that its corresponding penalized likelihood produces continuous
estimators of data. By the result of Fan and Li (2001), the penalty functions
satisfying sparsity and continuity must be singular at the origin. The
condition $p^{\prime}_{\lambda}(|\beta|)=0$ for large $|\beta|$ is a
sufficient condition for unbiasedness for a large true parameter.
The above three principles for penalty functions are also useful in
nonparametric regression, especially, in penalized regression splines.
Generally, the under smoothing of penalized regression spline is caused by the
excessive number of knots and this problem is attenuated by convex penalties
to produce shrinkage estimate of coefficients of the basis functions such as
the rough penalty used in smoothing splines. Here the thresholding rule
provides an attractive alternative to reduce the problem of under smoothing.
We may reduce the number of knots adaptively by a thresholding rule. On the
other hand, the properties of unbiasedness and continuity keep the smoothing
and stability of the penalized regression spline when we reduce the number of
knots.
Fan and Li (2001) proposed the Smoothly Cipped Absolute Deviation Penalty
(SCAD). It is a non-concave penalty with singular at the origin. To recall, it
is defined as follows
$p^{\prime}_{\lambda}(\theta)=\lambda\Big{\\{}I(\theta\leq\lambda)+\frac{(a\lambda-\theta)_{+}}{(a-1)\lambda}I(\theta>\lambda)\Big{\\}}$
(5)
for some $a>2$ and $\theta>0$. By Fan (1997), the simple penalized least-
squares problems
$(z-\theta)^{2}/2+p_{\lambda}(|\theta|)$ (6)
with SCAD penalty yields the solution
$\hat{\theta}=\left\\{\begin{array}[]{ll}\mathrm{sgn}(z)(|z|-\lambda)_{+},&\mbox{when}|z|\leq
2\lambda,\\\
\\{(a-1)z-\mathrm{sgn}(z)a\lambda\\}/(a-2),&\mbox{when}2\lambda<|z|\leq
a\lambda,\\\ z,&\mbox{when}|z|>a\lambda\end{array}\right.$ (7)
Hence by (6), the estimator that SCAD penalty results in has the three
properties discussed above. The discussion of other penalty functions can be
referred to Chapter 2 or Fan and Li (2001) and Antoniadis and Fan (2001). Here
we just use SCAD penalty to show the basic idea of non-convex penalized
regression spline.
### 2.3 An iterative algorithm
We may directly apply SCAD penalty in the right side of (4) to get a non-
convex penalized regression spline. However, it poses challenges to minimize
(4), which is a high-dimensional problem. Here we follow a simple iterative
algorithm proposed by Fan and Li (2001).
Suppose we have an initial value $\mathcal{B}_{0}$ that is close to the
minimizer of the right side of (4). The SCAD penalty function is singular at
the origin, and it does not have continuous first order derivatives. Thus the
first step of the algorithm is to check if the initial value of
$\beta_{j0},j=p,\ldots,p+k-1$ equal to zero. If $\beta_{j0}$ is very close to
$0$, then set $\hat{\beta}_{j}=0$. Otherwise we consider the following
quadratic approximation
$\\{p_{\lambda_{n}}(|\beta_{j}|)\\}^{\prime}=p_{\lambda_{n}}^{\prime}(|\beta_{j}|)\mathrm{sgn}(\beta_{j})\thickapprox\\{p_{\lambda_{n}}^{\prime}(|\beta_{j0}|)/|\beta_{j0}|\\}\beta_{j}$
when $\beta_{j}\neq 0$. In other words,
$p_{\lambda_{n}}(|\beta_{j}|)\thickapprox
p_{\lambda_{n}}(|\beta_{j0}|)+\frac{1}{2}\\{p_{\lambda_{n}}^{\prime}(|\beta_{j0}|)/|\beta_{j0}|\\}(\beta_{j}^{2}-\beta_{j0}^{2}),\quad\mbox{for}\quad\beta_{j}\thickapprox\beta_{j0}.$
Let $\beta_{j_{1}0},\ldots,\beta_{j_{d}0}$ be the nonzero components of
$\mathcal{B}_{0}$, In this step we also define,
$\Sigma_{\lambda_{n}}(\mathcal{B}_{0})=\mathrm{diag}\\{p_{\lambda_{n}}^{\prime}(|w_{j_{1}}\beta_{j_{1}0}|)/|w_{j_{1}}\beta_{j_{1}0}|,\ldots,p_{\lambda_{n}}^{\prime}(|w_{j_{d}}\beta_{j_{d}0}|)/|w_{j_{d}}\beta_{j_{d}0}|\\}$
$\mbox{\boldmath$X$}_{x_{i}}(\mathcal{B}_{0})=\\{1,x_{i},\ldots,x_{i}^{p-1},(x_{i}-t_{j_{1}})^{p-1}_{+},\ldots,(x_{i}-t_{j_{d}})^{p-1}_{+}\\}$
and
$\mbox{\boldmath$X$}_{n}=\\{\mbox{\boldmath$X$}_{x_{1}}^{T},\ldots,\mbox{\boldmath$X$}_{x_{n}}^{T}\\}^{T}\quad\mbox{and}\quad\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})=\\{\mbox{\boldmath$X$}_{x_{1}}^{T}(\mathcal{B}_{0}),\ldots,\mbox{\boldmath$X$}_{x_{n}}^{T}(\mathcal{B}_{0})\\}^{T}$
In the second step, we compute the ridge regression
$\mathcal{B}_{1}=\\{\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})^{T}\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})+n\Sigma_{\lambda_{n}}(\mathcal{B}_{0})\\}^{-1}\mbox{\boldmath$X$}_{n}^{T}(\mathcal{B}_{0})\mbox{\boldmath$y$}.$
The third step of the algorithm is updating $\mathcal{B}_{0}$ by
$\mathcal{B}_{1}$ and repeating the first and second steps until the the
iterative solution is numerically stable.
### 2.4 Issues on practical implementation
Fan and Li (2001) demonstrated the convergence of this algorithm by
simulation. Their test also indicates this algorithm converges quickly. A
drawback of this algorithm is that once a coefficient is shrunken to zero, it
will stay at zero. However, this drawback also significantly reduces the
computational burden.
We have claimed that the non-convex penalized regression spline is insensitive
to the number of the origin knots $k$ if it is large enough. In the following
simulation results, we can see that we only require $k$ is large enough,
generally no less than $O(n^{\frac{1}{2p+1}})$ to get optimal convergence rate
for the non-convex penalized regression spline, where $n$ is the sample size.
The knots series is defined as $t_{i}=x_{([ni/(k+1)])},i=1,\ldots,k$, where
$x_{(j)}$ is the $j$th order statistics of $x_{i}$.
In practice, we do not know the smoothness conditions of the function that we
want estimate. To guard the efficiency of our procedure to handle some spatial
inhomogeneity setting, we follows suggestion of the Friedman and Silverman
(1989) that “ A smoother should be resistant to a run of length $L$ of either
positive or negative error so long as its span in the region of the run is
large compared to $L$”. Hence we would like to adopt the formula proposed by
Friedman and Silverman (1989) to select the value of the minimum value of $k$,
$k=[n/M(n,\alpha)]+1$ (8)
where $0.05\leq\alpha\leq 0.1$ and $n\geq 15$, $M(n,\alpha)\approx
L_{\max}(\alpha)/3$ (or $L_{\max}(\alpha)/2.5$ to be conservative) denotes a
minimum span between two placed knots, and $L_{\max}(\alpha)$ is the largest
positive or negative run to be expected in $n$ binomial trials with
probability $\alpha$. By numerical approximation,
$L_{\max}(\alpha)\approx-\log_{2}\\{-(1/n)\ln(1-\alpha)\\}.$
Our procedure can also be regarded as a backward algorithm for selecting
knots. In some sense, it only reduces the flexibility of the model by deleting
unwanted knots. Hence we hope that our initial spline
$\mbox{\boldmath$X$}_{n}\mathcal{B}_{0}$ has small bias and keep it under our
procedure. When the number of knots is large enough, it is obvious that the
least-squares estimate of regression spline is a good choice. On the other
hand, in Fan and Li (2001), they require that the matrix
$n^{-1}\mbox{\boldmath$X$}_{n}^{T}\mbox{\boldmath$X$}_{n}$ is not singular. In
nonparametric regression setting,
$n^{-1}\mbox{\boldmath$X$}_{n}^{T}\mbox{\boldmath$X$}_{n}$ can be asymptotic
singular, specially when we use the truncated power basis as the basis of the
spline space, since the order of the matrix grows with $n$. In this phase the
initial estimate $\beta_{j0}$ obtained by the regression spline may has a
large variance. Thus we have to weigh the initial estimate in penalty term by
weight $w_{j}$ such that the variance of $w_{j}\beta_{j0}$ is the order of
$O(n^{-\frac{1}{2}})$, the same order as $\lambda_{n}$. Here, we take
$w_{j}=\Big{[}\big{(}\frac{1}{n}\mbox{\boldmath$X$}_{n}^{T}\mbox{\boldmath$X$}_{n}\big{)}^{-1}_{jj}\Big{]}^{-\frac{1}{2}}.$
In theory, we may able to inverse the above matrix , but in practice, we
replace the inverse operation by the generalized inverse operation. In the
classic linear model, if
$n^{-1}\mbox{\boldmath$X$}_{n}^{T}\mbox{\boldmath$X$}_{n}$ is singular, then
the least squares estimate
$\hat{\mathcal{B}}=(\mbox{\boldmath$X$}_{n}^{T}\mbox{\boldmath$X$}_{n})^{-}\mbox{\boldmath$X$}_{n}\mbox{\boldmath$y$}$
may not be a consistent estimate of $\mathcal{B}$, but
$\mbox{\boldmath$X$}_{n}\hat{\mathcal{B}}$ can still be a consistent estimable
of $\mbox{\boldmath$X$}_{n}\mathcal{B}$. Our simulation also show that though
the truncated power basis results in
$n^{-1}\mbox{\boldmath$X$}_{n}^{T}\mbox{\boldmath$X$}_{n}$ that is
asymptotically singular, this has little influence on our numerical results.
### 2.5 Selection of penalized parameters
To implement our procedure, it is more important to estimate the parameters
$a$ and $\lambda_{n}$ for the SCAD than to decide the value of $k$.
$(a,\lambda_{n})$ can be regarded as either smoothing parameters or penalty
parameters. We denote them by $\mbox{\boldmath$\theta$}=(a,\lambda_{n})$. Here
we discuss two methods of estimating $\theta$: Predictor Risk Estimation
Criterion (PREC) (often referred to as the $C_{p}$ Criterion) suggested by
Eubank (1999), Modified Generalized Cross-Validation (MGCV) proposed by Fan
and Li (2001).
Let us first consider the modified generalized cross-validation. In our
iterative algorithm, we update the estimate $\mathcal{B}$ by
$\mathcal{B}_{1}(\mbox{\boldmath$\theta$})=\\{\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})^{T}\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})+n\Sigma_{\lambda_{n}}(\mathcal{B}_{0})\\}^{-1}\mbox{\boldmath$X$}_{n}^{T}(\mathcal{B}_{0})\mbox{\boldmath$y$}.$
(9)
Thus the fitted value of $\hat{f}(x_{i})$ of $f(x_{i}),,i=1,\ldots,n$ is
$\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})\\{\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})^{T}\mbox{\boldmath$X$}_{n}(\mathcal{B}_{0})+n\Sigma_{\lambda_{n}}(\mathcal{B}_{0})\\}^{-1}\mbox{\boldmath$X$}_{n}^{T}(\mathcal{B}_{0})\mbox{\boldmath$y$},$
and the projection matrix can be defined as
$\mbox{\bf P}_{\mbox{\boldmath$\scriptstyle
X$}_{n}}\\{\hat{\mathcal{B}}(\mbox{\boldmath$\theta$})\\}=\mbox{\boldmath$X$}_{n}(\hat{\mathcal{B}})\\{\mbox{\boldmath$X$}_{n}(\hat{\mathcal{B}})^{T}\mbox{\boldmath$X$}_{n}(\hat{\mathcal{B}})+n\Sigma_{\lambda_{n}}(\hat{\mathcal{B}})\\}^{-1}\mbox{\boldmath$X$}_{n}^{T}(\hat{\mathcal{B}})$
Define the number of effective parameters in the non-convex penalized
regression spline as $e(\mbox{\boldmath$\theta$})=\mathrm{tr}[\mbox{\bf
P}_{\mbox{\boldmath$\scriptstyle
X$}_{n}}\\{\hat{\mathcal{B}}(\mbox{\boldmath$\theta$})\\}]$. Hence, the
modified generalized cross-validation statistic is
$\mathrm{MGCV}(\mbox{\boldmath$\theta$})=\frac{1}{n}\frac{\|\mbox{\boldmath$y$}-X_{n}(\hat{\mathcal{B}})\hat{\mathcal{B}}(\mbox{\boldmath$\theta$})\|^{2}}{\\{1-\gamma
e(\mbox{\boldmath$\theta$})/n\\}^{2}}$ (10)
where
$\hat{\mbox{\boldmath$\theta$}}=\mathrm{argmin}_{\mbox{\boldmath$\scriptstyle\theta$}}\\{\mathrm{MGCV}(\mbox{\boldmath$\theta$})\\}$
and $\gamma$ is the inflated factor to be specified.
The predictor risk estimation criterion is defined as follows
$\hat{\mbox{\bf
P}}\\{\hat{\mathcal{B}}(\mbox{\boldmath$\theta$})\\}=\frac{1}{n}\|\mbox{\boldmath$y$}-X_{n}(\hat{\mathcal{B}})\hat{\mathcal{B}}(\mbox{\boldmath$\theta$})\|^{2}+\frac{2\gamma\sigma^{2}}{n}e(\mbox{\boldmath$\theta$}),$
(11)
where
$\hat{\mbox{\boldmath$\theta$}}=\mathrm{argmin}_{\mbox{\boldmath$\scriptstyle\theta$}}\hat{\mbox{\bf
P}}\\{\hat{\mathcal{B}}(\mbox{\boldmath$\theta$})\\}$, and $\gamma$ is also
the inflated factor.
The inflation factor used here is due to the fact that the a lot of basis
functions used in the model are selected adaptively (see Luo and Wahba 1997
and Friedman 1991). When $\gamma=1$, the MGCV is no difference to the GCV as
suggested by Fan and Li (2001), Breiman (1995), Tibshirani (1996), and Fu
(1998). The predictor risk estimation criterion also appears to Akaike’s
information criterion, AIC (Akaike 1973). When $\gamma=\log(n)/2$, the
predictor risk estimation criterion is the Bayesian information criterion,
BIC, (Schwarz 1978). In MGCV, following discussion of Luo and Wahba (1997) and
Friedman (1991), we suggest $\gamma$ should be in $[1.2,3.5]$ to keep the
stable of the non-convex penalized regression spline. For the predictor risk
estimation criterion, our criterion is similar to the one proposed by Rao and
Wu (1989) used for the model selection in a classic regression problem. By the
strong consistent results of Rao and Wu (1989) and Bai, Rao and Wu (1999), the
value of $\gamma$ can be selected from a large range. It is only required that
$\gamma/\log\log n\to\infty$ and $\gamma=o(n)$. Thus we tend to agree that the
predictor risk estimation criterion is stable for a large range value of
$\gamma$. But conservatively, we will select the value of $\gamma$ from
$[2,5]$ or the form of $\log(n)/2$. A lot model selection criterions mentioned
in Rao and Wu (1989) and Bai, Rao and Wu (1999) can be also used here to
select the smoothing parameter $\theta$.
In fact, as shown by Fan and Li (2001), 3.7 is a good choice for the parameter
$a$ used in the SCAD. Hence, we mainly use the MGCV and PREC to select the
penalized parameter $\lambda_{n}$ for the SCAD penalized regression spline. In
the following simulation or discussion, we always set the value of $a$ as 3.7.
## 3 Simulation study
In this section, we use the following 4 examples. The first two come from Fan
and Gijbels (1995), the last two come from Donohon and Johnstone (1994).
Table 3.1: Specifications of Simulation Examples | | | | | | Sample | | | | Number of
---|---|---|---|---|---|---|---|---|---|---
Example | | f(x) | | $\sigma$ | | size (n) | | $SD(f)/\sigma$ | | replicates
1 | | $\sin(2(4x-2)+2\exp(-16x^{2})$ | | .3 | | 256 | | 2.80 | | 400
2 | | $(4x-2)+2\exp(-(16(x-0.5))^{2})$ | | .4 | | 256 | | 3.16 | | 400
3 | | $2.2(4*\sin 4\pi t-\mathrm{sgn}(t-0.3)-\mathrm{sgn}(0.72-t))$ | | 1.0 | | 2048 | | 6.54 | | 31
4 | | $22\\{t(1-t)\\}^{\frac{1}{2}}\sin\\{2\pi(1+0.05)/(t+0.05)\\}$ | | 1.0 | | 2048 | | 6.36 | | 31
### 3.1 MSE Compared to other methods
In this section, we use simulation examples to examine the performance of the
SCAD PRS and compare it with HAS (Luo and Wahba 1997), MARS (Friedman 1991),
wavelet shrinkage (SUREShrink, Donoho and Johnstone 1994), smoothing splines
SS procedures, Local PS and Globe PS (Ruppert and Carroll 2000). The
simulation results of other methods are excerpted from Luo and Wahba (1997)
and Ruppert and Carroll(2000).
Luo and Wahba (1997) used the pseudostandard normal random number generator
rnor, a Fortran subroutine from CMLIB. For SUREShrink, Luo and Wabha used the
software wavethresh, developed by Nason and Silverman (1994) in S-PLUS. They
chose the “primary resolution level” as 5 and the wavelets family “
DaubLeAsymm with filter number 8”. The Fortran routines used to compute HAS
estimates. The smoothing splines (SS) are computed by using the code GCVSPL in
Fortran by Woltring’s code, with the smoothing parameters chosen by GCV. The
code mars3.5 was used for MARS. Local PS and Globe PS were computed by using
matlab code which were programmed by Ruppert and Carroll (2000).
To compare HAS, we select 60 initial knots for our first two examples. To test
our rule for selecting the initial number of knots, we follow formula (8) and
take the value
$\left[\frac{3n}{-\log_{2}\\{(-1/n)\ln(1-0.1)\\}}\right]+1$ (12)
For the first two examples, the initial knots number is nearly 60 for $n=256$
and for the last two examples are 432 for $n=2048$. The regularization
parameter $\lambda_{n}$ is selected by MGCV with the inflation factor 2.5.
In the second, the third and the fourth example, we relax some requirement for
the tuning parameters used in our algorithm to improve the speed of our
algorithm and to observe if our algorithm is stable for these tuning
parameters. The medians of MSEs are presented in Table 3.2. The SCAD PRS fits
with median performance are shown in Figure 3.1. They accord quite well with
the true regression function.
Table 3.2: Median of MSE$\times 1000$ with different methods and the Interquartile range of MSE$\times 1000$ (in parentheses) Example | | SCAD PRS | | HAS | | SS | | SUREShrink | | MARS | | Local PS | | Global PS
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | | 5.4(3.1) | | 7(6) | | 6 (3) | | 18(4) | | 7(4) | | 5.3(3.5) | | 6.1 (2.9)
2 | | 9.3(5.3) | | 12(11) | | 10 (5) | | 42(12) | | 12 (7) | | | |
3 | | 51 (8) | | 39 (13) | | 75(5) | | 62(7) | | 150 (14) | | | |
4 | | 196 (20.3) | | 68 (15) | | 205 (11) | | 149 (13) | | | | | |
Figure 3.1: SCAD PRS with Median of MSE for Examples 1-4 (solid line for true
function, dot-dash for the estimate function)
In our simulation, we use the quadratic spline as the results are slightly
better than those of cubic spline. This phenomenon was also observed by
Ruppert and Carroll (1999) when they study their penalized regression spline
with quadratic penalty.
From Table 3.2, our procedures is the best in first two examples, and in the
last two examples, our procedure slightly outperform the smoothing spline, but
not the HAS or SUREShrink. This is due to the spatial inhomogeneity of the
simulated function, for which the HAS and SUREShrink are designed.
### 3.2 Effect of initial number of knots for SCAD PRS
In this section, we study the influence of the initial number of knots used in
SCAD penalized regression splines for the first two examples. We vary the
initial number of knots from 30 to 150 with step size 30. The median MSEs of
400 simulations are presented in Table 3.3. They show that the method is
insensitive to the number of initial knots though too many initial knots may
cause slightly over fit.
Table 3.3: Median of MSE$\times 1000$ for different initial knots and the Interquartile range of MSE$\times 1000$ (in parentheses) Initial Knots Number | | 30 | | 60 | | 90 | | 120 | | 150
---|---|---|---|---|---|---|---|---|---|---
Example 1 | | 5.5(2.9) | | 5.4(3.1) | | 5.1(2.9) | | 5.2(2.7) | | 5.8(2.7)
Example 2 | | 8.9(5.2) | | 9.3(5.3) | | 9.4(5.1) | | 9.5(5.1) | | 9.8(5.2)
### 3.3 Effect of knots selecting for SCAD PRS
Depicted in Figures 3.2 and 3.3 are the locateions of the knots finally
automatically selected by non-convex penalized regression spline. Figure 3.3
gives the frequency of every knot finally selected by SCAD PRS in the 400
simulations for Examples 1 and 2. Figures 3.3 gives the histograms for the
number of knots that the SCAD PRS finally selected in every simulation. It is
obvious that SCAD PRS used more knots than SCAD PRS of Example 1. This is
because we relax the requirement for the tuning parameters in our algorithm.
Hence, though these tuning parameters have little influence on MSE, they
affect the knots selection of SCAD PRS.
Figure 3.2: Frequencies of the initial 60 knots that are selected by SCAD PRS
Figure 3.3: The distributions of the knots selected by the SCAD PRS with 60
initial knots
### 3.4 Parameters Selection Methods
In this section, we mainly examine the performance of MGCV and PREC with
different number of initial knots. Specifically, we aim at examining the
impact of the inflation factor to the MGCV and PREC on the estimated curves.
For simplicity, we focus only on Examples 1 and 2.
From Table 3.4-3.7, the two parameter selection methods, MGCV and PREC are
efficient in selecting the penalized parameter for the non-convex penalized
regression spline. However, MGCV method is sensitive to the inflation factor.
The best value of the inflation factor for MGCV is in the interval
$[2.0,3.5]$. The PREC method is robust when the inflation factor is large.
This result is consistent with the results of Rao and Wu (1989) and Bai, Rao
and Wu (1999). The number of initial knots and the sample size may also
slightly affect the choice of the inflation factor. It needs further study.
Table 3.4: Median$\times 1000$ of MSE for different initial knots with MGCV Method and the Difference of the First and Third Quartiles of MSE (in Parentheses) for Example 1 | | | Initial Knots Number |
---|---|---|---|---
| Inflation factor $\gamma$ | | 30 | | 60 | | 90 | | 120 | | 150 |
| 1.0 | | 6.7(4.4) | | 9.0(7.7) | | 11.0(13.0) | | 14.4(18.6) | | 23.4(25.9) |
| 1.2 | | 6.2(3.8) | | 6.7(4.5) | | 6.3(5.5) | | 6.1(4.9) | | 6.6(5.5) |
| 1.5 | | 5.8(3.4) | | 5.5(3.2) | | 5.3(3.0) | | 5.4(3.0) | | 5.9(3.0) |
| 2.0 | | 5.5(3.0) | | 5.3(3.0) | | 5.2(2.8) | | 5.2(2.8) | | 5.8(2.8) |
| 2.5 | | 5.5(2.9) | | 5.4(3.1) | | 5.1(2.9) | | 5.2(2.7) | | 5.8(2.7) |
| 3.0 | | 5.6(2.9) | | 5.6(32) | | 5.2(2.9) | | 5.2(2.8) | | 52.4(14.0) |
| 3.5 | | 5.7(3.0) | | 5.8(3.3) | | 5.2(2.9) | | 5.2(2.8) | | 53.8(8.3) |
| 7.0 | | 67(42) | | 72(42) | | 32.6(6.7) | | 43.3(7.8) | | 53.8(8.3) |
| $\ln(n)/2$ | | 5.6(2.9) | | 5.5(3.0) | | 5.2(2.9) | | 5.2(2.8) | | 6.2(3.6) |
| $\ln(n)$ | | 6.2(3.4) | | 6.8(4.0) | | 32.5(7.0) | | 43.3(7.8) | | 53.8(8.3) |
| $\ln(k)/2$ | | 5.7(31) | | 5.3(3.1) | | 5.2(2.9) | | 5.2(27) | | 5.8(2.7) |
| $\ln(k)$ | | 5.7(3.0) | | 6.0(3.6) | | 5.2(2.9) | | 43.3(7.8) | | 53.8(8.3) |
Table 3.5: Median$\times 1000$ of MSE for different initial knots with PREC Method and the Difference of the First and Third Quartiles of MSE (in Parentheses) for Example 1 | | | Initial Knots Number |
---|---|---|---|---
| Inflation factor $\gamma$ | | 30 | | 60 | | 90 | | 120 | | 150 |
| 1.0 | | 6.8(4.4) | | 10.4(8.2) | | 15.7(14.1) | | 24.0(18.4) | | 34.0(17.6) |
| 1.2 | | 6.2(3.9) | | 7.6(6.2) | | 8.2(12.8) | | 9.7(19.5) | | 27.0(29.9) |
| 1.5 | | 6.0(3.7) | | 5.9(3.7) | | 5.8(4.2) | | 5.8(4.2) | | 6.6(6.6) |
| 2.0 | | 5.7(3.1) | | 5.4(3.1) | | 5.3(2.9) | | 5.4(3.0) | | 5.9(3.0) |
| 2.5 | | 5.5(3.0) | | 5.3(3.1) | | 5.2(2.8) | | 5.2(2.8) | | 5.8(2.8) |
| 3.0 | | 5.5(2.9) | | 5.5(3.1) | | 5.2(2.8) | | 5.2(2.8) | | 5.8(2.8) |
| 3.5 | | 5.5(2.9) | | 5.5(3.1) | | 5.2(2.9) | | 5.2(2.7) | | 5.8(2.7) |
| 7.0 | | 5.8(3.1) | | 6.1(3.6) | | 5.2(2.9) | | 5.2(2.8) | | 5.8(2.7) |
| $\ln(n)/2$ | | 5.6(3.0) | | 5.4(3.1) | | 5.2(2.8) | | 5.2(2.8) | | 5.8(2.8) |
| $\ln(n)$ | | 5.7(3.1) | | 5.8(3.3) | | 52(2.9) | | 5.2(2.8) | | 5.8(2.7) |
| $\ln(k)/2$ | | 5.8(3.4) | | 5.3(3.1) | | 5.2(2.8) | | 5.2(2.8) | | 5.8(2.8) |
| $\ln(k)$ | | 5.5(2.9) | | 5.6(3.2) | | 5.2(2.9) | | 5.2(2.8) | | 5.8(2.7) |
Table 3.6: Median of MSE for different initial knots with MGCV Method and the Difference of the First and Third Quartiles of MSE (in Parentheses) for Example 2 | | | Initial Knots Number |
---|---|---|---|---
| Inflation factor $\gamma$ | | 30 | | 60 | | 90 | | 120 | | 150 |
| 1.0 | | 1.08(6.7) | | 12.4(8.3) | | 14.6(14.3) | | 16.3(21.2) | | 2.64(32.1) |
| 1.2 | | 9.9(6.2) | | 10.4(6.8) | | 10.3(8.3) | | 10.8(8.9) | | 10.9(9.2) |
| 1.5 | | 9.3(5.8) | | 9.3(5.5) | | 9.1(5.5) | | 9.3(5.5) | | 9.4(5.2) |
| 2.0 | | 9.0(5.3) | | 9.1(5.0) | | 9.0(5.1) | | 9.2(5.2) | | 9.4(5.1) |
| 2.5 | | 8.9(5.2) | | 9.3(5.3) | | 9.4(5.1) | | 9.5(5.1) | | 9.8(5.2) |
| 3.0 | | 9.1(5.5) | | 9.7(5.6) | | 9.6(5.5) | | 10.1(5.6) | | 9.9(5.2) |
| 3.5 | | 9.2(5.5) | | 10.0(6.0) | | 10.2(6.2) | | 10.7(5.9) | | 10.1(5.2) |
| 7.0 | | 12.9(8.7) | | 14.5(8.8) | | 15.2(8.5) | | 13.7(6.1) | | 10.8(6.0) |
| $\ln(n)/2$ | | 9.0(5.5) | | 9.5(5.2) | | 9.5(5.3) | | 9.9(5.3) | | 9.8(5.2) |
| $\ln(n)$ | | 10.8(6.5) | | 12.8(7.6) | | 12.6(7.3) | | 13.4(6.4) | | 10.5(5.6) |
| $\ln(k)/2$ | | 9.2(5.6) | | 9.1(5.0) | | 9.2(5.1) | | 94(5.1) | | 9.8(5.2) |
| $\ln(k)$ | | 9.2(5.5) | | 10.7(6.2) | | 11.2(6.8) | | 12.4(6.3) | | 10.3(5.4) |
Table 3.7: Median$\times 1000$ of MSE for different initial knots with PREC Method and the Difference of the First and Third Quartiles of MSE (in Parentheses) for Example 2 | | | Initial Knots Number |
---|---|---|---|---
| Inflation factor $\gamma$ | | 30 | | 60 | | 90 | | 120 | | 150 |
| 1.0 | | 10.9(7.5) | | 12.6(8.8) | | 16.5(14.4) | | 21.7(20.1) | | 35.2(26.9) |
| 1.2 | | 10.2(6.5) | | 11.3(7.4) | | 12.0(13.0) | | 13.0(18.8) | | 16.8(33.5) |
| 1.5 | | 9.5(6.0) | | 9.9(6.1) | | 9.8(6.7) | | 10.0(6.6) | | 10.5(8.9) |
| 2.0 | | 9.2(5.5) | | 9.1(5.2) | | 9.0(5.4) | | 9.3(5.4) | | 9.4(5.2) |
| 2.5 | | 8.9(5.4) | | 9.2(5.2) | | 9.1(5.1) | | 9.2(5.2) | | 9.5(5.2) |
| 3.0 | | 8.9(5.3) | | 9.3(5.3) | | 9.3(5.2) | | 9.4(5.1) | | 9.8(5.2) |
| 3.5 | | 9.0(5.5) | | 9.5(5.4) | | 9.5(5.2) | | 9.7(5.3) | | 9.8(5.2) |
| 7.0 | | 10.1(5.7) | | 11.0(6.8) | | 11.1(6.6) | | 11.7(6.3) | | 10.3(5.4) |
| $\ln(n)/2$ | | 8.9(5.3) | | 9.3(5.2) | | 9.2(5.1) | | 9.3(5.1) | | 9.6(5.2) |
| $\ln(n)$ | | 9.5(5.5) | | 10.4(6.3) | | 10.4(6.1) | | 11.0(6.2) | | 10.1(5.3) |
| $\ln(k)/2$ | | 9.3(5.7) | | 9.2(5.2) | | 9.0(5.1) | | 9.2(5.3) | | 9.5(5.1) |
| $\ln(k)$ | | 9.0(5.5) | | 9.7(5.7) | | 10.1(6.1) | | 10.7(5.8) | | 10.1(5.2) |
## 4 Discussion and Extension
The non-convex penalized regression spline can be easily extended to
multivariate regression models. In Chapter 2, we have applied our non-convex
penalized regression spline to a partial linear model to reduce the modeling
bias. In section, we briefly outline how to extend our approach to
nonparametric additive model.
First, we suppose that we have $J$ predictor variables and that
$\mbox{\mbox{\boldmath$X$}}_{i}=(x_{i,1},\ldots,x_{i,J})^{T}$ is the vector of
predictor variables for the $i$th case. The additive model considered is
defined as follows:
$y_{i}=f(\mbox{\mbox{\boldmath$X$}}_{i})+\varepsilon_{i}=\mu+\Sigma_{j=1}^{J}f_{j}(x_{i,j})+\varepsilon_{i}.$
(13)
where $\mbox{E}f_{j}(x_{i,j})=0$, $1\leq j\leq J$, is imposed for
identifiability. As in univariate setting, we may use spline function
$\hat{f}_{j}(x_{i,j},\mathcal{B}_{j})$ of order $p$ with $K_{j}$ knots,
$k_{1,j},\ldots,k_{K_{j},j}$, to approximate the function $f_{j}(x_{i,j})$
subject to the constrains that
$\sum_{1}^{n}\hat{f}_{j}(x_{i,j},\mathcal{B}_{j})=0$ where
$\hat{f}_{j}(x_{i,j},\mathcal{B}_{j})$ is defined as follows,
$\hat{f}_{j}(x_{i,j},\mathcal{B}_{j})=\beta_{0,j}+\beta_{1,j}x_{i,j}+\cdots+\beta_{p-1,j}x_{i,j}^{p-1}+\sum\limits_{l=1}^{K_{j}}\beta_{p+l-1,j}(x_{i,j}-k_{l})_{+}^{p-1}.$
(14)
Therefore, we obtain an additive spline model
$\hat{f}(\mbox{\mbox{\boldmath$X$}}_{i},\mathcal{B})=\mu+\sum\limits_{j=1}^{J}\hat{f}_{j}(x_{i,j},\mathcal{B}_{j}),$
(15)
to approximate the additive model (13), where
$\mathcal{B}=(\mu,\beta_{0,1},\beta_{1,1},\ldots,\beta_{p+K_{1}-1,1},\ldots,\beta_{p+K_{J}-1,J})^{T}$
The non-convex penalized criterion is to minimize
$\sum\limits_{i=1}^{n}\\{y_{i}-\hat{f}(\mbox{\mbox{\boldmath$X$}}_{i},\mathcal{B})\\}^{2}+n\sum\limits_{j=1}^{J}\sum\limits_{k=1}^{K_{j}}p_{\lambda_{n,j}}(|w_{p+k-1,j}\beta_{p+k-1,j}|),$
(16)
subject to the constraints that
$\sum_{1}^{n}\hat{f}_{j}(x_{i,j},\mathcal{B}_{j})=0$, where
$p_{\lambda_{n,j}}(\cdot)$ is the non-concave penalty with the penalty
parameter $\lambda_{n,j}$. In practice, we replace
$\mu+\beta_{0,1}+\beta_{0,2}+\cdots+\beta_{0,J}$ in ( 15) by a single
parameter $\beta_{0}$ to release the constraints imposed on the non-concave
penalty least-squares (16). As the univariate setting, we choose the SCAD as
the non-concave penalty. Now, we have transferred the additive model to a non-
concave penalty least-squares problem. Hence it is no difference between the
minimization of (16) and that of (4). The iterative algorithm used in the
univariate setting can be still used in additive model settings.
To simplify our procedure to fit an additive spline model, we set all penalty
parameters equalling to a global penalty parameter. Then we select this globe
penalty parameter by the MGCV and the PREC as in the univariate setting. To
make our estimate for the additive model more accurately, we can also select
different value for the different penalty parameters used in (16) by the MGCV
and the PREC through some optimization algorithms in a $J$ dimension space.
## References
* [1]
* [2] Aerts, M., Claeskens, G. and Wand, M.P. (2002). Some theory for penalized spline generalized additive models, Journal statistical planning and inference, 103, 455-470.
* [3]
* [4] Agarwal, G.G. and Studden, W.J. (1980), Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minizing Splines, Ann. Statist., 8, 1307-1325.
* [5]
* [6] Blake A. (1989), Comparison of the efficiency of deterministic and stochastic algorithms for visual reconstruction, IEEE Trans. on Pattern Analysis and Machine Intelligence, 11, 2–12.
* [7]
* [8] Blake, A. and A. Zisserman (1987). Visual reconstruction, MIT Press, Cambridge.
* [9]
* [10] Eilers, P.H.C. and Marx, B.D., (1996). Flexible smoothing using B-splines and penalized likelihood (with comments and rejoinder), Statist. Sci., 11(2), 89-121.
* [11]
* [12] Fan, J. and Li, R. (2001). Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties, Journal of the American Statistical Association, 96, 1348-1360.
* [13]
* [14] Fu, W.J. (1998). Penalized regression: the bridge versus the LASSO, Journal of Computational and Graphical Statistics, 7, 397-416.
* [15]
* [16] Gilman, S. and Gilman, D. (1984). Stochastic relaxation, Gibbs distribution and Bayesian restoration of images, IEEE trans. Pattern Anal. Machine Intell., 6, 721-741.
* [17]
* [18] Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996). Markov Chain Monte Carlo in Practice, Chapman $\&$ Hall/CRC, London.
* [19]
* [20] Green, P.J. and Silverman, B.W. (1994). Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach. Chapman and Hall, London.
* [21]
* [22] Gu, C. (2002). Smoothing spline ANOVA models, Springer, New York.
* [23]
* [24] Mammen, E. and van de Geer, S. (1997), Locally Adaptive Regression Splines, Ann. Statist. 25, 387-413.
* [25]
* [26] Marx, B.D. and Eilers (1998). Direct generalized additive modeling with penalized likelihood. Comput. Statist. $\&$ Data Anal. 28, 193-209.
* [27]
* [28] Nikolova, M., Idier, J. and Mohammad-Djafari, A. (1998). Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF, IEEE Image Processing, 7, 571-585.
* [29]
* [30] Ruppert, D. and Carroll, R.J. (1997), Penalized regression splines, manuscript.
* [31]
* [32] Ruppert, D. and Carroll, R.J. (2000), Spatially-adaptive penalties for spline fitting, Australian and New Zealand Journal of Statistics, 42, 205-223.
* [33]
* [34] Smith, P.L. (1982). Curve fitting and modeling with splines using statistical variable selection methods. NASA, Langley Research Center, Hampla, VA, NASA Report 166034.
* [35]
* [36] Stone, C.J. (1985). Additive Regression and Other Nonparametric Models, Ann. Statist, 13, 689-705.
* [37]
* [38] Stone, C.J. (1990). Large-sample inference for log-spline models, Ann. Statist., 13, 689-705.
* [39]
* [40] Stone,C.J., Hansen, M., Kooperberg,C. and Truong, Y.K. (1997). Polynomial splines and their tensor products in extended linear modeling (with discussion). Ann. Statist., 25, 1371-1470.
* [41]
* [42] Tibshirani, R.J. (1996). Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society, B, 58, 267-288.
* [43]
* [44] Tibshirani, R.J. (1997). The LASSO method for variable selection in the Cox model. Statistics in Medicine, 16, 385-395.
* [45]
* [46] Wahba, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.
* [47]
* [48] Wand, M.P. (1999). On the optimal amount of smoothing in penalised spline regression, Biometrika , 86, 936-940.
* [49]
* [50] Zhou, S., Shen, X. and Wolfe, D.A. (1998), Local asymptotics for regression splines and confidence regions, Ann. Statist., 26, 1760-1782.
* [51]
|
arxiv-papers
| 2012-09-10T13:57:26 |
2024-09-04T02:49:34.918639
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Heng Peng",
"submitter": "Heng Peng",
"url": "https://arxiv.org/abs/1209.1994"
}
|
1209.2026
|
# Białynicki-Birula schemes in higher dimensional Hilbert schemes of points
Laurent Evain [email protected] Université d’Angers, Faculté des
Sciences, Département de mathématiques, 2, Boulevard Lavoisier, 49045 Angers
Cedex 01, FRANCE and Mathias Lederer [email protected] Department
of Mathematics, Cornell University, Malott Hall, Ithaca, New York 14853, USA
(Date: December 2010)
###### Abstract.
The Białynicki-Birula cells on the Hilbert scheme $H^{n}(\mathbb{A}^{d})$ are
smooth and reduced in dimension $d=2$. We prove that there is a schematic
structure in higher dimension, the Białynicki-Birula scheme, which is natural
in the sense that it represents a functor. Let
$\rho_{i}:H^{n}(\mathbb{A}^{d})\rightarrow{\rm Sym}^{n}(\mathbb{A}^{1})$ be
the Hilbert-Chow morhpism of the $i^{th}$ coordinate. We prove that a
Białynicki-Birula scheme associated with an action of a torus $T$ is
schematically included in the fiber $\rho_{i}^{-1}(0)$ if the $i^{th}$ weight
of $T$ is non positive.
###### Key words and phrases:
Hilbert scheme of points, standard sets, Białynicki-Birula cells after
Elligsrud and Strømme
The second author was supported by a Marie Curie International Outgoing
Fellowship of the EU Seventh Framework Program
## 1\. Introduction
Let $H^{n}(\mathbb{A}^{d})$ be the Hilbert scheme parametrizing zero
dimensional subschemes of length $n$ in the affine $d$-space $\mathbb{A}^{d}$
over a field $k$. This scheme is mostly called the _Hilbert scheme of points_
, sometimes also the _punctual Hilbert scheme_. There is a natural action of
the $d$-dimensional torus on $\mathbb{A}^{d}$, which induces a natural action
on $H^{n}(\mathbb{A}^{d})$. If $T$ is a general one-dimensional subtorus of
the $d$-dimensional torus, then $T$ defines the _Białynicki-Birula schemes_
$H^{BB(T,\Delta)}$ parametrizing the subschemes converging to some fixed point
$Z^{\Delta}$ under the action of $T$. The fixed point in question is a
monomial subscheme $Z^{\Delta}$.
These stratifications are preeminent in most studies of the punctual Hilbert
scheme in dimension two. For instance, they appear in the computation of the
Betti numbers (see [ES87], [ES88]), in the determination of the irreducible
components of (multi)graded Hilbert schemes (see [Eva04], [MS10]), or in the
study of the ring of symmetric functions via symmetric products of embedded
curves (see [Gro96], [Nak99]).
The Białynicki-Birula schemes in $H^{n}(\mathbb{A}^{2})$ are in fact affine
cells. In contrast, not much is known on these cells for higher dimensional
$\mathbb{A}^{d}$, and the difficulty to control and describe these cells is
probably one of the reasons why the Hilbert scheme of points is still
mysterious in higher dimension.
In dimension three or higher, the Białynicki-Birula schemes are not
irreducible, nor are they expected to be reduced. It is therefore necessary to
define them with their natural scheme structure as representing a functor.
Apart from the necessity to define them schematically, it is desirable to have
functorial descriptions of Hilbert schemes at hand, as these descriptions are
known to be both powerful and easy to handle.
In the present paper, we introduce the Białynicki-Birula functor parametrizing
families of subschemes $Z$ such that $\lim_{t\rightarrow 0,t\in
T}t.Z=Z^{\Delta}$ for some fixed monomial subscheme $Z^{\Delta}$. We will
prove (see theorem 25):
###### Theorem.
The Białynicki-Birula functor is representable by a locally closed scheme
$H^{BB(T,\Delta)}(\mathbb{A}^{d})$ of the Hilbert scheme
$H^{n}(\mathbb{A}^{d})$.
We assume that the one-dimensional subtorus $T$ of the $d$-dimensional torus
acting on $\mathbb{A}^{d}$ takes the shape
$T=\\{(t^{\xi_{1}},\dots,t^{\xi_{d}}),t\in k^{*}\\}$, for suitable weights
$\xi_{i}$.111 The action on closed points of $\mathbb{A}^{1}$ is thus given by
$t.(a_{1},\ldots,a_{d}):=(t^{\xi_{1}}a_{1},\dots,t^{\xi_{d}}a_{d})$ If
$\xi_{i}\leq 0$, then the closed points of $H^{BB(T,\Delta)}(\mathbb{A}^{d})$
correspond to subschemes $Z$ whose support is in the hyperplane $x_{i}=0$.
This follows from the the naïve observation that if $t.Z$ tends to
$Z^{\Delta}$, then the support of $Z$ tends to the support of $Z^{\Delta}$. A
much more subtle question is to ask whether this remains true at the schematic
level, when we consider the Białynicki-Birula scheme with its possibly non-
reduced structure. The answer is positive. We will prove (see theorem 28):
###### Theorem.
Let $\rho_{i}:H^{n}(\mathbb{A}^{d})\rightarrow{\rm Sym}^{n}(\mathbb{A}^{1})$
be the Hilbert-Chow morphism which associates to a subscheme $Z$ of length $n$
the unordered $n$-tuple in $k$ corresponding to the $i^{th}$ coordinate. If
$\xi_{i}\leq 0$, then $H^{BB(T,\Delta)}(\mathbb{A}^{d})$ is schematically
included in the fiber $\rho_{i}^{-1}(O)$.
For simplicity, we have considered a field $k$ in this introduction. But
throughout the paper, we shall work over a ring $k$ of arbitrary
characteristic.
Let us say a word about the proofs. The action of $T$ on $\mathbb{A}^{d}={\rm
Spec}\,k[\mathbf{x}]$ induces a partial order $<$ on the monomials of
$k[\mathbf{x}]$: the monomials are ordered according to their weight for the
$T$-action. The basic idea of the proof is that a subscheme $Z$ is in
$H^{BB(T,\Delta)}$ if, and only if, the initial ideal ${\rm
in}_{<_{\epsilon}}(I(Z))$ equals the monomial ideal $I(Z^{\Delta})$ for any
total order $<_{\epsilon}$ which is a small deformation of the partial order
$<$. It follows that the Białynicki-Birula functor is an intersection of
functors $\cap_{<_{\epsilon}}\mathcal{H}^{{\rm mon}(<_{\epsilon},\Delta)}$,
where the intersection is taken over all small deformations $<_{\epsilon}$ of
$<$, and each functor $\mathcal{H}^{{\rm mon}(<_{\epsilon},\Delta)}$ is
defined using the theory of initial ideals. We prove that each
$\mathcal{H}^{{\rm mon}(<_{\epsilon},\Delta)}$ is representable by a subscheme
$H^{{\rm mon}(<_{\epsilon},\Delta)}$ of $H^{n}(\mathbb{A}^{d})$ (see theorem
21). For avoiding the problems coming from intersecting an infinite number of
subvarieties, we realize the Białynicki-Birula functors as the intersection of
only two well-chosen functors $\mathcal{H}^{{\rm
mon}(<_{-},\Delta)}\cap\mathcal{H}^{{\rm mon}(<_{+},\Delta)}$ (see proposition
24).
When dealing with representations of functors, constructions for individual
subschemes often require uniformity lemmata when one passes to families. The
parangon of this situation is Castelnuovo-Mumford regularity, which appears in
the construction of the Hilbert scheme. Our strategy for proving
representability of the Białynicki-Birula functor is no exception; we need two
key uniformity lemmata (see lemmata 10 and 14). They are proved and explained
in section 3.
### Acknowledgements
We give many thanks to Robin Hartshorne, Diane Maclagan and Gregory G. Smith,
the organizers of the workshop Components of Hilbert Schemes held at American
Institute of Mathematics in Palo Alto, CA, in July 2010. It was there that we
first met and shared our thoughts; the exceptionally productive and at the
same time friendly atmosphere of that workshop triggered our collaboration.
Special thanks go to Bernd Sturmfels, who realized that we have many research
interests in common, and motivated us to work together. The second author
wishes to thank Allen Knutson, Jenna Rajchgot, and Mike Stillman for many
fruitful discussions.
The second author was supported by a Marie Curie International Outgoing
Fellowship of the EU Seventh Framework Program.
## 2\. Białynicki-Birula functors and $\Delta$-monic families
In this section, we introduce the Białynicki-Birula functor and reformulate
Białynicki-Birula families in terms of $\Delta$-monic ideals (see proposition
8).
In this paper, we consider schemes over a commutative ring $k$ of arbitrary
characteristic. We denote by $k[\mathbf{x}]$ the polynomial ring
$k[x_{1},\dots,x_{d}]$. Similarly, for
$e=(e_{1},\dots,e_{d})\in\mathbb{N}^{d}$, we use the multi-index notation
$\mathbf{x}^{e}:=x_{1}^{e_{1}}\dots x_{d}^{e_{d}}$. All the rings and
$k$-algebras are implicitly assumed to be noetherian.
A _standard set_ , or _staircase_ , is a subset $\Delta\subset\mathbb{N}^{d}$
whose complement $C:=\mathbb{N}^{d}\setminus\Delta$ satisfies
$C+\mathbb{N}^{d}=C$. We call the minimal generators of the
$\mathbb{N}^{d}$-module $C$ the _outer corners of $\Delta$._ All standard sets
under consideration will be of cardinality $n$, in particular, finite. The
ideal generated by the monomials $\mathbf{x}^{e},e\in C$ is denoted by
$I^{\Delta}$. We shall freely identify the monomials $\mathbf{x}^{e}$ with
their exponent $e$. In particular, the notion of a staircase of monomials
makes sense.
If $B$ is a $k$-algebra, then the tensor product $B\otimes_{k}k[\mathbf{x}]$
is just $B[\mathbf{x}]$, the ring of polynomials with coefficients in $B$.
Similarly, we write
$B[t,t^{-1},\mathbf{x}]:=B[\mathbf{x}]\otimes_{k}k[t,t^{-1}]$. Let
$\xi\in\mathbb{Z}^{d}$, and $I\subset B[\mathbf{x}]$. We denote by
$I(t,t^{-1})\subset B[t,t^{-1},\mathbf{x}]$ the ideal generated by the
elements $t.f:=\sum t^{-\xi\cdot e}c_{e}\mathbf{x}^{e}$ where $f=\sum
c_{e}\mathbf{x}^{e}\in I$. We denote by $I(t)$ the ideal
$I(t,t^{-1},\mathbf{x})\cap B[t,\mathbf{x}]$. In particular $I=I(1)$.
###### Definition 1.
We denote by $\mathcal{H}^{BB(\Delta,\xi)}(B)$, or more simply by
$\mathcal{H}^{BB(\Delta)}(B)$ when $\xi$ is obvious, the set of ideals
$I\subset B[\mathbf{x}]$ such that $\lim_{t\rightarrow 0}t.I=I^{\Delta}$,
which means:
* •
$B[t,\mathbf{x}]/I(t)$ is a locally free $B[t]$-module of rank $n=\\#\Delta$.
* •
$I(0)=I^{\Delta}$
$\mathcal{H}^{BB(\Delta)}$ is a contravariant functor from the category of
noetherian $k$-algebras to the category of sets. We call it the _Białynicki-
Birula functor_.
The first bulleted item of the definition says that ${\rm
Spec}\,B[t,\mathbf{x}]/I(t)\to{\rm Spec}\,B[t]$ is a flat family. The second
bulleted item says that its fiber over $0\in{\rm Spec}\,B[t]$ is the monomial
scheme defined by $I^{\Delta}$. The limit $\lim_{t\rightarrow 0}t.I$ is
therefore a well-defined flat limit.
Let us check that $\mathcal{H}^{BB(\Delta,\xi)}$ is indeed a functor. If
$Q:=B[t,\mathbf{x}]/I(t)$ is a locally free module of rank $\\#\Delta$ and
$B\to A$ is a ring homomorphism, then $Q\otimes_{B}A=Q\otimes_{B[t]}A[t]$ is a
locally free $A[t]$-module of the same rank. In particular, the fiber over
$0\in{\rm Spec}\,A[t]$ of the family ${\rm Spec}\,Q\otimes_{B}A\to{\rm
Spec}\,A[t]$ is a locally free $A$-module, call it $Q^{\prime}$, of the same
rank. However, the kernel of the corresponding surjection $A[\mathbf{x}]\to
Q^{\prime}$ contains the monomial ideal $I^{\Delta}$. Since the quotient
$A[\mathbf{x}]/I^{\Delta}$ is a free $A$-module of rank $\\#\Delta$, the
module $Q^{\prime}$ is in fact free of that rank; the kernel of
$A[\mathbf{x}]\to Q^{\prime}$ equals $I^{\Delta}$; and functoriality follows.
We shall now introduce the partial order on monomials in $k[\mathbf{x}]$
defined by the weight of our torus action. That order is in general not a
total order, in particular, not a monomial order. In what follows, we shall
employ techniques very similar to those in Gröbner basis theory—with the
difference that our order is not a monomial order, so the usual techniques
will have to be modified. Let us make that more precise.
###### Definition 2.
Let $\xi\in\mathbb{R}^{d}$ and $f_{\xi}\in(\mathbb{R}^{d})^{*}$ the linear
form defined by $f_{\xi}(\alpha_{1},\dots,\alpha_{d})=\sum\alpha_{i}\xi_{i}$.
We define the partial order $<_{\xi}$ on monomials in $k[\mathbf{x}]$ by
setting $\mathbf{x}^{e}<_{\xi}\mathbf{x}^{g}$ if $f_{\xi}(e)<f_{\xi}(g)$, and
letting $\mathbf{x}^{e}$ and $\mathbf{x}^{g}$ be incomparable if
$f_{\xi}(e)=f_{\xi}(g)$. Since we identify monomials and exponents, we adopt
the convention $f_{\xi}(x^{e}):=f_{\xi}(e)$.
If the weights $\xi_{i}$ are linearly independent over $\mathbb{Q}$, then
$<_{\xi}$ is a total order on monomials. Otherwise, the order is only partial.
In that case we shall refine the partial order to a total one. The refinements
we shall use may be defined either as limits or using _signed orders_.
###### Definition 3.
* •
A sequence of partial orders $<_{j}$ _converges to the total order $<$_ if for
every pair of monomials $a,b$, we have $a<b$ if, and only if, $a<_{j}b$ for
$j$ large enough.
* •
A _signed order_ on the symbols $1,\dots,d$ is a map
$(\epsilon,o):\\{1,\dots,d\\}\rightarrow\\{-1,1\\}\times\\{1,\dots,d\\}$ such
that the induced map $o:\\{1,\dots,d\\}\rightarrow\\{1,\dots,d\\}$ is a
bijection.
* •
A _sequence compatible with that signed order_ is a sequence $\xi^{j}$ in
$\mathbb{R}^{d}$ converging to $0$ such that the sign of $\xi^{j}_{k}$ is
$\epsilon(k)$ and such that the quotient $\xi^{j}_{l}/\xi^{j}_{k}$ tends to
$0$ if, and only if, $o^{-1}(k)<o^{-1}(l)$.
The connection between signed orders and convergence is as follows.
###### Proposition 4.
Let $<$ be a refinement of the order $<_{\xi}$ and $(\epsilon,o)$ a signed
order. Then the following conditions are equivalent:
* •
For all monomials $\mathbf{x}^{e}$ and $\mathbf{x}^{f}$ with
$f_{\xi}(e)=f_{\xi}(f)$, we have $\mathbf{x}^{e}<\mathbf{x}^{f}$ if, and only
if,
$(\epsilon(1)e_{o(1)},\dots,\epsilon(d)e_{o(d)})<(\epsilon(1)f_{o(1)},\dots,\epsilon(d)f_{o(d)})$
in the lexicographic order.
* •
For every sequence $\xi^{j}$ compatible with the signed order, the sequence of
orders $<_{\xi+\xi^{j}}$ converges to $<$.
###### Proof.
Easy. ∎
###### Definition 5.
A _quasi-homogeneous order of weight $\xi$_ is either a partial order
$<_{\xi}$ as above or a total order $<$ which refines $<_{\xi}$. The variable
$x_{i}$ is _positive_ with respect to $<_{\xi}$ if $\xi_{i}>0$. If the total
order $<$ refines $<_{\xi}$, then a variable $x_{i}$ with $\xi_{i}=0$ and
$\epsilon(i)=1$ in the refinement is called _positive_ too. This defines
positivity of variables for all quasi-homogeneous order of weight $\xi$.
_Negativity_ of variables is defined similarly.
When all variables are positive with respect to a total order $<$, then the
order is a monomial order and the theory of Gröbner bases is at our
disposition. In particular, there exist an algorithm for division of an
element $f$ by a monic family $f_{i}$. When all variables are negative, we are
in the setting of standard bases, in which there also exists a division
algorithm. With standard bases, the quotient is a power series. However, the
quotient is a polynomial when the family $f_{i}$ contains a power $x_{j}^{k}$
of each variable $x_{j}$. In what follows we shall consider weight vectors
$\xi$ whose signs of coordinates are neither all positive nor all negative. In
this setting, there exists a division algorithm with polynomial quotient by a
monic family $f_{i}$ containing powers of the negative variables. This leads
to the notion of a _bounded ideal_.
###### Definition 6.
Let $<$ be a quasi-homogeneous total order. An ideal $I\subset B[\mathbf{x}]$
is called _bounded by $r_{1},\dots,r_{d}$_ if for every negative (resp.
positive) variable $x_{i}$, there exists a polynomial $h_{i}\in I\cap
B[x_{i}]$ with $h_{i}=x_{i}^{r_{i}}$ (resp. ${\rm in}(h_{i})=x_{i}^{r_{i}}$).
###### Definition 7.
* •
Let $<$ be a quasi-homogeneous partial order of weight $\xi$, and $f=\sum
a_{e}\mathbf{x}^{e}\in B[\mathbf{x}]$, where $a_{e}\neq 0$ for every $e$. Let
$\mathbf{x}^{e_{1}},\dots,\mathbf{x}^{e_{l}}$ be the maximal monomials
appearing in $f$. The _initial form of $f$_ is ${\rm in}(f):=\sum
a_{e_{i}}\mathbf{x}^{e_{i}}$. This is a term when $<$ is a total order, but
may contain more than one term otherwise. We denote by ${\rm in}(I)$ the ideal
generated by the elements ${\rm in}(f)$, $f\in I$.
* •
Let $I\subset B[\mathbf{x}]$ and $m=\mathbf{x}^{e}$ be a monomial. We denote
by ${\rm in}_{m}(I)\subset B$ the ideal generated by the elements $b\in B$
such that ${\rm in}(f)=bm$ for some $f\in I$.
* •
Let $\Delta$ be a standard set of cardinality $n$. The ideal $I$ is called
$\Delta$-monic if ${\rm in}_{m}(I)=\langle 1\rangle$ if $m\notin\Delta$ and
${\rm in}_{m}(I)=0$ otherwise.
We now interpret the Białynicki-Birula functor in terms of initial ideals and
monic families with respect to some partial quasi-homogeneous order.
###### Proposition 8.
Let $\xi\in\mathbb{Z}^{d}$ and $<_{\xi}$ the associated partial order. Then
$I\in\mathcal{H}^{BB(\Delta,\xi)}(B)$ if, and only if, the following
conditions are satisfied:
* •
$I$ is $\Delta$-monic
* •
$\forall f\in I$, the initial form with respect to $<_{\xi}$ is ${\rm
in}(f)=\sum a_{e}x^{e}$, $e\notin\Delta$,
* •
$B[\mathbf{x}]/I$ is a locally free $B$-module of rank $\\#\Delta$.
###### Proof.
We only prove that the itemized conditions imply that
$I\in\mathcal{H}^{BB(\Delta,\xi)}$, the converse being easy. Let us
temporarily assume that $B[t,\mathbf{x}]/I(t)$ is a locally free
$B[t]$-module. Then the flat limit $\lim_{t\rightarrow 0}t.I$ exists; the
first condition says that the limit ideal contains $I^{\Delta}$, whereas the
second conditions says that the limit is contained in $I^{\Delta}$. For
proving that $I\in\mathcal{H}^{BB(\Delta,\xi)}(B)$, it therefore remains to
show that $P:=B[t,\mathbf{x}]/I(t)$ is a locally free $B[t]$-module.
Upon localizing $B$ (that is to say, upon replacing ${\rm Spec}\,B$ by an open
subset), we may assume that $B[\mathbf{x}]/I$ is $B$-free of rank $\\#\Delta$.
According to the second item, the monomials $\mathbf{x}^{e},e\in\Delta$ are a
basis of $B[\mathbf{x}]/I$. These monomials are therefore also a basis of the
$B[t,t^{-1}]$-module $B[\mathbf{x}]/I\otimes_{k}k[t,t^{-1}]\simeq
B[\mathbf{x},t,t^{-1}]/I(t,t^{-1})$, the latter isomorphism being given by the
torus action $t.\mathbf{x}^{e}=t^{-\xi\cdot e}\mathbf{x}^{e}$ on the
polynomial ring222 Remember that the torus takes the shape
$T=\\{(t^{\xi_{1}},\dots,t^{\xi_{d}}),t\in k^{*}\\}$ and acts by
$t.(a_{1},\ldots,a_{d}):=(t^{\xi_{1}}a_{1},\dots,t^{\xi_{d}}a_{d})$ on closed
points of $\mathbb{A}^{d}$. This corresponds to the torus action on the
polynomial ring $k[\mathbf{x}]$ which is trivial on scalars and is given by
$t.\mathbf{x}^{e}:=t^{-\xi\cdot e}\mathbf{x}^{e}$ on monomials., which induces
the torus action on $H^{n}(\mathbb{A}^{d})$. The monomials $\mathbf{x}^{e}$,
for $e\in\Delta$, remain linearly independent in the $B[t]$-module $P$. We
denote by $N:=\langle\mathbf{x}^{e}:e\in\Delta\rangle$ the free submodule of
$B[t,\mathbf{x}]/I(t)$ generated by monomials with exponents in $\Delta$, and
by $Q(t)$ the quotient
$P/N=B[t,\mathbf{x}]/(I(t)+\langle\mathbf{x}^{e}:e\in\Delta\rangle)$. Since
$I(0)=I^{\Delta}$ by hypothesis, we have $Q(0)=Q(t)/\langle t\rangle=0$. By
the above, we have $Q(t,t^{-1})=Q(t)\otimes_{B[t]}B[t,t^{-1}]=0$. Hence
$Q(t)=0$, so $P=N$. ∎
###### Remark 9.
According to the intuition from Gröbner bases, one could think that the third
condition is a consequence of the first two. This is not the case, as is shown
by the example $d=1$, $I=\langle x^{2}+x^{3}\rangle$, $\xi=-1$,
$\Delta=\\{1,x\\}$.
## 3\. Uniformity lemmata
The goal of this section is to prove the uniformity lemmata needed in our
constructions.
Uniformity considerations are a cornerstone of many constructions with
representable functors. For instance, in the construction of the Hilbert
scheme, a subscheme with Hilbert polynomial $P$ is characterized by its
equations in degree $m$ for $m$ large enough. But finding a uniform integer
$m$ suitable for all the subschemes with Hilbert polynomial $P$ requires a
uniformity lemma. In the construction of Hilbert scheme, Castelnuovo-Mumford
regularity, or the Gotzmann persistence theorem does the job. In our context,
we shall need two uniformity lemmata (see lemmata 10 and 14). Let us explain
where they are needed.
In proposition 8, we have reformulated the property
$I\in\mathcal{H}^{BB(\xi,\Delta)}(B)$ in terms of the partial order $<_{\xi}$
and of the ideal ${\rm in}_{<_{\xi}}(I)$. Later on, we shall use total orders
instead of partial orders to connect the Białynicki-Birula functors with
Gröbner basis theory. We shall slightly deform $\xi$ into $\xi_{\epsilon}$ in
such a way that $<_{\xi_{\epsilon}}$ is a total order and ${\rm
in}_{<_{\xi}}(I)={\rm in}_{<_{\xi_{\epsilon}}}(I)$. But when $<_{j}$ converges
to $<$, in the equivalence $a<b\Leftrightarrow a<_{j}b$ for $j$ large, how
large $j$ needs to be depends on the monomials $a$ and $b$. The first
uniformity lemma (lemma 10) controls the monomials appearing in a nice set of
generators of $I$ and will induce the equality ${\rm in}_{<_{\xi}}(I)={\rm
in}_{<_{\xi_{\epsilon}}}(I)$ (corollary 11).
As for the second uniformity lemma (see lemma 14), the situation is as
follows. If $Z_{0}\subset\mathbb{A}^{d}$ is a scheme of length $n$ supported
on the locus $x_{i}=0$, then $x_{i}^{n}$ vanishes on $Z$. This is not true any
more in families. The simple example $Z=V(x-\epsilon)\subset{\rm
Spec}\,k[x,\epsilon]/\epsilon^{2}$ shows a family of relative length $1$,
supported on $x=0$, schematically included in $x^{2}=0$ but not schematically
included in $x=0$. This corresponds to the fact that the functor parametrizing
families of length $n$ supported on $x_{i}=0$ is not representable by a closed
subscheme of $H^{n}(\mathbb{A}^{d})$. In contrast, we shall prove in lemma 14
that a $\Delta$-monic bounded family of relative length $n$ supported on
$x_{i}=0$ is schematically included in the locus $x_{i}^{n}=0$. This is an
ingredient to prove that the monic functors are representable by locally
closed subschemes of $H^{n}(\mathbb{A}^{d})$ (see theorem 21).
###### Lemma 10.
Let $I=\langle f_{1},\dots,f_{u}\rangle\subset B[\mathbf{x}]$ be an ideal such
that the quotient $B[\mathbf{x}]/I$ is a finite $B$-module. Let $<$ be a
quasi-homogeneous total order of weight $\xi=(\xi_{1},\dots,\xi_{d})$ with
$\sum\xi_{i}^{2}=1$. Assume moreover that one of the following two conditions
is satisfied:
1. (1)
$\xi_{i}\neq 0$ for all $i$,
2. (2)
$I$ is bounded.
Then there exists $g_{i}\in I$ such that for all $m$ and for every $a\in{\rm
in}_{m}(I)$, there exist $\lambda_{i}\in B[\mathbf{x}]$,
$b,c\in\mathbb{R}^{+}$ such that
* •
${\rm in}(\sum\lambda_{i}g_{i})=am$,
* •
the total degree of each term $\tau$ in $\sum\lambda_{i}g_{i}$ satisfies
* –
case (1): $\deg(\tau)\leq\frac{b}{\min(|\xi_{j}|)}+c$,
* –
case (2): $\deg(\tau)\leq{b}$,
* •
$b,c$ depend on $I$ and $m$, but not on the order $<$.
###### Proof.
The sequence $\langle 1,x_{i},\dots,x_{i}^{j}\rangle\subset B[\mathbf{x}]/I$,
indexed by $j$, is an increasing sequence in the noetherian module
$B[\mathbf{x}]/I$, thus it eventually becomes stationary. It follows that for
every $i$, the ideal $I$ contains an element
$h_{i}=x_{i}^{r}+\sum_{j=1}^{r-1}b_{ij}x_{i}^{j}$, $b_{ij}\in B$. We define
$g_{i}:=\begin{cases}f_{i}&\text{ for }i\leq u,\\\ h_{i-u}&\text{ for
}i=u+1,...u+d.\end{cases}$
Consider an element $f$ of $I$, which takes the shape
$f=\sum s_{i}f_{i},$
and let ${\rm in}(f)=am$, where $m$ is a monomial and $a\in B$.
Let $s_{i}=h_{1}q_{i}+r_{i}$ be the euclidean division of $s_{i}$ by $h_{1}$
with respect to the variable $x_{1}$. Then
$f=\sum r_{i}f_{i}+(\sum q_{i}f_{i})h_{1},$
where $\deg_{x_{1}}r_{i}<r$ for some natural number $r$ which is independent
of $f$. We repeat euclidean division for all other $h_{j}$, so with respect to
all other variables $x_{j}$. This yields a formula
$f=\sum t_{i}f_{i}+\sum u_{i}h_{i},$
where
$\deg_{x_{j}}t_{i}<r$
for all $i,j$ and a uniform bound $r$. We define
$D:=\max_{i,j}\\{\deg_{x_{j}}f_{i}+r-1,\deg_{x_{j}}m\\}$. Let $s$ be the index
of a positive variable. Then $\deg_{x_{s}}\sum t_{i}f_{i}\leq D$. Let $\tau$
be a term of $\sum u_{i}h_{i}$ with $\deg_{x_{s}}\tau>D\geq\deg_{x_{s}}(m)$.
Then $\tau$ is also a term of $f$, different from its initial term,
$\tau\neq{\rm in}(f)=am$. Let $f^{\prime}:=f-\frac{\tau}{x_{s}^{r}}h_{s}$.
Then ${\rm in}(f^{\prime})={\rm in}(f)$ and
$f^{\prime}=\sum t_{i}f_{i}+\sum
u_{i}^{\prime}h_{i},u_{s}^{\prime}=u_{s}-\frac{\tau}{x_{s}^{r}}h_{s},u^{\prime}_{i}=u_{i},i\neq
s.$
The term $\tau$, which appears in $\sum u_{i}h_{i}$, is gone in $\sum
u_{i}^{\prime}h_{i}$. We may repeat this construction, thus constructing
expressions $\sum u_{i}^{(l)}h_{i}$ which suppress all the terms whose degree
in some variable of positive weight is more than $D$. Finally, we obtain an
element $f^{(l)}\in I$ such that ${\rm in}(f^{(l)})=am$ and
$f^{(l)}=\sum t_{i}f_{i}+\sum v_{i}h_{i},\ \ \deg_{x_{j}}\sum v_{i}h_{i}\leq
D\text{ when }\xi_{j}>0.$
Now let $s$ be an index with $\xi_{s}<0$. In particular, $x_{s}$ is a negative
variable. Let $\tau:=c\mathbf{x}^{e}$ be a term of $\sum v_{i}h_{i}$ with
$\deg_{x_{s}}\tau>\frac{f_{\xi}(m)-dD}{\xi_{s}}+r$. In particular,
$\deg_{x_{s}}\tau>r$ since
$f_{\xi}(m)\leq\sum_{\xi_{i}>0}\xi_{i}\deg_{x_{i}}m\leq\sum_{\xi_{i}>0}D\leq
dD$, and the denominator $\xi_{s}$ is negative. Let
$q_{s}:=\frac{\tau}{x_{s}^{r}}$. Note that $f_{\xi}({\rm in}(q_{s}h_{s}))\leq
f_{\xi}({\rm
in}(q_{s}))=f_{\xi}(\tau)-r\xi_{s}<\xi_{s}\deg_{x_{s}}\tau+\sum_{\xi_{i}>0}\xi_{i}\deg_{x_{i}}\tau-r\xi_{s}\leq\frac{f_{\xi}(m)-dD}{\xi_{s}}\xi_{s}+\sum_{\xi_{i}>0}D<\frac{f_{\xi}(m)-dD}{\xi_{s}}\xi_{s}+dD=f_{\xi}(m)$.
In particular, ${\rm in}(f^{(l)})={\rm in}(f^{(l)}-h_{s}q_{s})=am$. We
therefore let $f^{(l+1)}:=f^{(l)}-h_{s}q_{s}$ and obtain have the formula
$f^{(l+1)}=\sum t_{i}f_{i}+\sum w_{i}h_{i},$
which has the virtue that the term $\tau$ does not appear any more in $\sum
w_{i}h_{i}$. Repeating the same process for all $s$ such that $\xi_{s}<0$, we
obtain an element
$f^{(l+p)}=\sum t_{i}f_{i}+\sum z_{i}h_{i}$
such that any term $\tau$ of $\sum z_{i}h_{i}$ satisfies the inequality
$\deg_{x_{s}}\tau\leq\frac{f_{\xi}(m)-dD}{\xi_{s}}+r\leq\frac{\sum\deg_{x_{i}}(m)+dD}{|\xi_{s}|}+r$
for every $s$ with $\xi_{s}<0$.
Summing things up, in the last expression, any term $\tau$ of the right hand
side satisfies
* •
$\deg_{x_{i}}\tau\leq b_{i}$ for some $b_{i}$ if $\xi_{i}>0$.
* •
$\deg_{x_{i}}\tau\leq\frac{b_{i}}{|\xi_{i}|}+c_{i}$ for some $b_{i},c_{i}$ if
$\xi_{i}<0$.
Thus we have the required expression in case (1).
It remains to cover the case when $I$ is bounded. In the above proof, the
upper bound for $\deg_{x_{i}}(\tau)$ took two different forms, depending on
the sign of $\xi_{i}$. This is necessary because the sign of $\xi_{i}$ affects
the computation of ${\rm in}(h_{i})$. However, in the bounded case, one can
take $h_{i}:=x_{i}^{r_{i}}$ for some $r_{i}$ if $\xi_{i}\leq 0$ is a negative
variable, and obtains ${\rm in}(h_{i})=x_{i}^{r_{i}}$. In other words, in the
bounded case, with this choice of $h_{i}$, the above proof for positive
variables also works for negative variables. We get the upper bound
$\deg_{x_{i}}\tau\leq b_{i}$ whatever the sign of $\xi_{i}$ is. ∎
###### Corollary 11.
Let $\Delta$ be a staircase. Let $<_{j}$ be a sequence of quasi-homogeneous
total orders converging to $<$ and $I\subset B[\mathbf{x}]$ be an ideal such
that $B[\mathbf{x}]/I$ is a finite $B$-module and ${\rm in}_{m}(I)=\langle
1\rangle$ for $m\notin\Delta$. Assume that one of the following two conditions
hold:
1. (1)
$\xi_{i}\neq 0$ for all $i$,
2. (2)
$I$ is bounded.
Then for large $j$, we have ${\rm in}_{<}(I)={\rm in}_{<_{j}}(I)$.
###### Proof.
If ${\rm in}_{<}(f)=am$, then ${\rm in}_{<_{j}}f=am$ for large $j$ by
definition of the limit order. The inclusions ${\rm in}_{m,<}(I)\subset{\rm
in}_{m,<_{j}}(I)$ and ${\rm in}_{<}(I)\subset{\rm in}_{<_{j}}(I)$ for large
$j$ then follow from the noetherian property.
Conversely, let $\xi_{j}$ and $\xi$ the normalized weights associated to
$<_{j}$ and $<$, where normalization means that
$||\xi_{j}||_{2}=||\xi||_{2}=1$. Since $<_{j}$ converges to $<$, the sequence
$\xi_{j}$ converges to $\xi$. By the previous proposition, for all $m$, there
exists $N_{m}\in\mathbb{N}$ and a finite family of monomials $m_{i}$ such that
for $j>N_{m}$, any $a\in{\rm in}_{m,<_{j}}(I)$ is equal to ${\rm
in}_{<_{j}}(f)$ for some $f=\sum b_{i}m_{i}$, $b_{i}\in B$. Let
$N^{\prime}_{m}\in\mathbb{N}$ be such that for all $n>N^{\prime}_{m}$, the
partial order $<_{j}$ and the total order $<$ define the same total order on
the finite set of monomials $m_{i}$. Then for
$n>N^{\prime\prime}_{m}:=\max(N,N^{\prime})$, we have ${\rm
in}_{m,<_{j}}(I)\subset{\rm in}_{m,<}(I)$.
When trying to find an integer large enough so that ${\rm in}_{<}(I)={\rm
in}_{<_{j}}(I)$, one a priori thinks that this integer should be at least
equal to $N_{m}$ for every monomial $m\in k[\mathbf{x}]$. But if ${\rm
in}_{m,<_{j}}(I)={\rm in}_{m,<}(I)=\langle 1\rangle$, then the same equality
is also still true if one replaces $m$ by any monomial multiple $m^{\prime}$
of $m$. In particular, it suffices to take $j>N:=\max\\{N_{m}:m\in A\\}$,
where $A$ is the union of $\Delta$ and the outside corners of $\Delta$, a
finite set. ∎
We now provide a (pseudo-)division algorithm which, in particular, tests ideal
membership. Of course such an algorithm exists in the presence of a monomial
order or with bounded families —but since in our context we need partial and
total non monomial orders, that algorithm is not good enough for us. We deform
the partial order into a total order and start a division with the total
order. Our algorithm does not necessarily terminate. Nevertheless we are able
to identify a part $R_{\Delta}$ in the remainder which stabilizes (and is
independent of the arbitrarily chosen deformed order). This division is used
to prove that most monic families are bounded (corollary 13). Here is the
precise statement.
###### Corollary 12.
Let $<$ be a quasi-homogeneous partial order of weight $\xi$, $\Delta$ a
staircase, and $I\subset B[\mathbf{x}]$ a $\Delta$-monic ideal such that the
quotient $B[\mathbf{x}]/I$ is locally free of rank $\\#\Delta$. Let
$o_{1},\dots,o_{u}$ the outside corners of $\Delta$, and $f_{1},\dots,f_{u}\in
I$ be elements with ${\rm in}_{<}(f_{i})=o_{i}$. Let $\xi^{\prime}$ a small
deformation of $\xi$ such that $<_{\xi^{\prime}}$ is a total order and ${\rm
in}_{<}(I)={\rm in}_{<_{\xi}^{\prime}}(I)$ and ${\rm in}_{<}(f_{i})={\rm
in}_{<_{\xi^{\prime}}}(f_{i})$ for all $i$. Then for all $f\in B[\mathbf{x}]$,
there exists a division
$f=\sum\lambda_{i}f_{i}+R_{\Delta}+R^{\prime}$
such that
* •
each term $\tau=b\mathbf{x}^{e}$ of $R_{\Delta}$ satisfies $e\in\Delta$,
* •
for all terms $c\mathbf{x}^{s}$ of $R^{\prime}$ and for all $m\in\Delta$,
$f_{\xi^{\prime}}(s)<f_{\xi^{\prime}}(m)$.
For every such division,
* •
$R_{\Delta}$ is independent of $\xi^{\prime}$ and of the choice of the
division,
* •
$f\in I$ if, and only if, $R_{\Delta}=0$,
* •
the map $f\mapsto R_{\Delta}$ is a homomorphism of $B$-modules
$B[\mathbf{x}]\to B[\mathbf{x}]/I$, where we identify the latter module with
$B[\Delta]:=\oplus_{e\in\Delta}B\mathbf{x}^{e}$.
###### Proof.
To construct the expected expression
$f=\sum\lambda_{i}f_{i}+R_{\Delta}+R^{\prime}$, we proceed in several steps.
At each step $j$, we have an expression $f=\sum\lambda_{ij}f_{i}+T_{j}$. For
$j=0$, we take $\lambda_{i0}:=0$, $T_{0}:=f$. We decompose
$T_{j}=R_{\Delta,j}+R^{\prime}_{j}$, with
$R_{\Delta,j}:=\sum_{m\in\Delta}c_{mj}m$ and
$R^{\prime}_{j}:=\sum_{m\notin\Delta}c_{mj}m$. The initial term ${\rm
in}_{<_{\xi^{\prime}}}(R^{\prime}_{j})$ can be written $\mu_{j}{\rm
in}_{<_{\xi^{\prime}}}(f_{i_{j}})$ for some $f_{i_{j}}$ by hypothesis. We set
$\lambda_{i_{j}(j+1)}:=\lambda_{i_{j}j}+\mu_{j}$ and
$\lambda_{i^{\prime}(j+1)}:=\lambda_{i^{\prime}j}$ for $i^{\prime}\neq i_{j}$.
Then $T_{j+1}:=f-\sum\lambda_{i(j+1)}f_{i}=T_{j}-\mu_{j}f_{i_{j}}$ decomposes,
analogously as above, into $T_{j+1}=R_{\Delta,j+1}+R^{\prime}_{j+1}$.
If, for some $j$, it happens that $R^{\prime}_{j}=0$, then we define
$R_{\Delta}:=R_{\Delta_{j}}$ and $R^{\prime}:=0$, and have constructed the
expected expression. Otherwise, there is a constant $c>0$ such that for all
$j$, $f_{\xi^{\prime}}({\rm
in}_{<_{\xi^{\prime}}}(f_{i_{j}}))>f_{\xi^{\prime}}({\rm
in}_{<_{\xi^{\prime}}}(f_{i_{j}}-{\rm in}_{<_{\xi^{\prime}}}f_{i_{j}}))+c$. If
we let $E_{j}:=\\{f_{\xi^{\prime}}(\tau):\tau\text{ a term of
}R^{\prime}_{j}\\}$, then $E_{j+1}$ is derived from $E_{j}$ by the replacement
of the maximal element $M_{j}=f_{\xi^{\prime}}({\rm
in}_{<\xi^{\prime}}(R^{\prime}_{j}))$ of $E_{j}$ by a collection
$\\{c_{1},\dots,c_{k}\\}$ with $c_{i}<M_{j}-c$ for all $i$. It follows that
for large $j$, the maximal element of $E_{j}$ is smaller than any fixed
number. In particular, $\forall m\in\Delta$, $M_{j}=f_{\xi^{\prime}}({\rm
in}_{<\xi^{\prime}}(R^{\prime}_{j}))<f_{\xi^{\prime}}(m)$ if $j$ is large
enough.
If $f\in I$, then in the expression
$f=\sum\lambda_{i}f_{i}+R_{\Delta}+R^{\prime}$, we have $R_{\Delta}=0$ since
${\rm in}_{<_{\xi^{\prime}}}(f-\sum\lambda_{i}f_{i})$ cannot lie in $\Delta$
by hypothesis. In particular, if $f=\sum\mu_{i}f_{i}+S_{\Delta}+S^{\prime}$ is
another division, we take their difference and obtain a division of $0\in I$,
which implies $R_{\Delta}=S_{\Delta}$.
It is obvious that $f\mapsto R_{\Delta}$ is a homomorphism as it is possible
to add divisions, or to multiply them with a scalar $\lambda\in B$.
Let us now consider the $B$-submodule $B[\Delta]$ of $B[\mathbf{x}]$. The
identity on $B[\Delta]$ factors as $B[\Delta]\rightarrow
B[\mathbf{x}]\rightarrow B[\Delta]$ where the first arrow is the inclusion and
the second is the morphism $R_{\Delta}$. The above implies that this
factorization induces a factorization $B[\Delta]\rightarrow
B[\mathbf{x}]/I\rightarrow B[\Delta]$. This composition is surjective between
locally free modules of the same rank, so it is an isomorphism. In particular,
we obtain that $R_{\Delta}=0$ implies $f\in I$.
To prove that in the division $f=\sum\lambda_{i}f_{i}+R_{\Delta}+R^{\prime}$,
the summand $R_{\Delta}$ is independent of $\xi^{\prime}$, it suffices to
consider two small deformations $\xi^{\prime}$ and $\xi^{\prime\prime}$ of
$\xi$ and to find a division which is valid for both $\xi^{\prime}$ and
$\xi^{\prime\prime}$. We therefore have to show that for all exponents $s$ of
terms of $R^{\prime}$ and for all $m\in\Delta$,
$f_{\xi^{\prime}}(s)<f_{\xi^{\prime}}(m)$ and
$f_{\xi^{\prime\prime}}(s)<f_{\xi^{\prime\prime}}(m)$. Choose a division
$f=\sum\lambda_{i}^{\prime\prime}f_{i}+R^{\prime\prime}_{\Delta}+R^{\prime\prime}$
which is valid for $\xi^{\prime\prime}$. Then we construct the division for
$\xi^{\prime}$ as above, except for one modification: For $j=0$, instead of
taking $\lambda_{i0}=0$, $T_{0}=f$, we set
$\lambda_{i0}:=\lambda_{i}^{\prime\prime}$,
$T_{0}:=R^{\prime\prime}_{\Delta}+R^{\prime\prime}$. The terms in
$R^{\prime}_{j}$ decrease for both $<_{\xi^{\prime}}$ and
$<_{\xi^{\prime\prime}}$, thus at each step, the expression
$f=\sum\lambda_{in}f_{ij}+R_{\Delta,j}+R_{j}$ is a division for
$\xi^{\prime\prime}$. When the process stops, we have a common division for
$\xi^{\prime}$ and $\xi^{\prime\prime}$. ∎
###### Corollary 13.
Let $\xi$ a weight vector with $\xi_{i}\neq 0$ for all $i$. If $I$ is
$\Delta$-monic and $B[\mathbf{x}]/I$ is locally free of rank $n=\\#\Delta$,
then $I$ is bounded.
###### Proof.
If $r$ is large and $x_{i}$ is a negative variable, the division of
$x_{i}^{r}$ by the collection of $f_{i}$ reads
$x_{i}^{r}=\sum 0\cdot f_{i}+(R_{\Delta}=0)+(R^{\prime}=x_{i}^{r}).$
Thus $x_{i}^{r}\in I$, as expected. If $x_{i}$ is a positive variable, the
polynomial $h_{i}$ constructed at the beginning of lemma 10 is the required
polynomial. ∎
We now come to the second uniformity lemma.
###### Lemma 14.
Let $<$ be a quasi-homogeneous total order. Let $I\subset B[\mathbf{x}]$ be a
$\Delta$-monic ideal with $B[\mathbf{x}]/I$ locally free of rank
$n=\\#\Delta$. Suppose that $I$ is bounded. Then for every negative variable
$x_{i}$, we have $x_{i}^{n}\in I$.
###### Proof.
After applying a suitable permutation, we may assume that $x_{1},\dots,x_{l}$
(resp. $x_{l+1},\dots,x_{d}$) are the negative (resp. positive) variables, and
we shall prove that $x_{1}^{n}\in I$. For every monomial $m\in
k[x_{2},\dots,x_{d}]$, there exists a unique $h(m)\in\mathbb{N}$, call it the
_height_ , such that $m^{-}:=x_{1}^{h(m)-1}m\in\Delta$ and
$m^{+}:=x_{1}^{h(m)}m\notin\Delta$. In particular, $h(m)=0$ if
$m\notin\Delta$, and all heights of all $\mathbf{x}^{e}\in
k[x_{2},\dots,x_{d}]$ sum up to $n$, the cardinality of $\Delta$.
Let $M\subset k[x_{2},\dots,x_{l}]$ be the set of monomials with exponents in
$\Delta^{\prime}:=\mathbb{N}^{l-1}\cap\Delta$. The set $M$ is finite, and we
number its elements such that $m_{1}^{+}<m_{2}^{+}<\ldots<m_{\\#M}^{+}$. For
any monomial $m\in k[x_{2},\dots,x_{d}]$, we define $H(m):=\sum_{m_{i}^{+}\leq
m^{+}}h(m_{i})$. Note that for every such $m$,
$H(m)\leq\sum_{\mathbf{x}^{e}\in k[x_{2},\dots,x_{d}]}h(\mathbf{x}^{e})=n$. In
particular, if we prove
(1) $\forall m=\mathbf{x}^{e}\in k[x_{2}\dots,x_{l}],x_{1}^{H(m)}m\in I$
then for the particular choice $m:=1$, (1) implies $x_{1}^{n}\in I$, which
concludes the proof.
Since $I$ is bounded, for each $i$ with $2\leq i\leq l$, there exists some
$r_{i}$ such that $x_{i}^{r_{i}}\in I$. Thus (1) is true if $x_{i}^{r_{i}}$
divides $m$. It follows that the set $C:=\\{m\in k[x_{2},\dots,x_{l}]:m\text{
is a monomial },x_{1}^{H(m)}m\notin I\\}$ is finite. If (1) is not true, then
$C\neq\emptyset$ and $C$ contains an element $m_{0}$ which is minimal in the
sense that $m_{0}^{+}<m^{+}$ for every $m\in C,m\neq m_{0}$. Let $f\in I$ with
${\rm in}(f)=m_{0}^{+}$. Then $f$ takes the shape
(2) $f=m_{0}^{+}+\sum_{\mathbf{x}^{e}<m_{0}^{+}}c_{e}\mathbf{x}^{e}.$
Since $I$ is bounded, we may assume by the standard division procedure that
$\mathbf{x}^{e}\in\Delta$ for every term $\mathbf{x}^{e}$ in the above
expression. We obtain
$\mathbf{x}^{e}=x_{1}^{e_{1}}\dots x_{d}^{e_{d}}\in\Delta\Rightarrow
x_{1}^{e_{1}}\dots x_{l}^{e_{l}}\in\Delta\Rightarrow
e_{1}<h(x_{2}^{e_{2}}\dots x_{l}^{e_{l}}).$
Since $x_{1}$ is a negative variable, and $x_{l+1},\dots,x_{d}$ are positive,
we get:
$(x_{2}^{e_{2}}\dots x_{l}^{e_{l}})^{+}=x_{1}^{h(x_{2}^{e_{2}}\dots
x_{l}^{e_{l}})}(x_{2}^{e_{2}}\dots
x_{l}^{e_{l}})<x_{1}^{e_{1}}x_{2}^{e_{2}}\dots
x_{l}^{e_{l}}x_{l+1}^{e_{l+1}}\dots x_{d}^{e_{d}}=\mathbf{x}^{e}$
Here is the upshot of the above: If $C\neq\emptyset$, $m_{0}\in C$ is its
minimum, and a monomial $\mathbf{x}^{e}=x_{1}^{e_{1}}\dots x_{d}^{e_{d}}$
appears in a term of $f$, then $m_{e}:=x_{2}^{e_{2}}\dots x_{l}^{e_{l}}$
satisfies $m_{e}^{+}<\mathbf{x}^{e}<m_{0}^{+}$. Thus $H(m_{e})\leq
H(m_{0})-h(m_{0})$, and by minimality of $m_{0}$, $m_{e}x_{1}^{H(m_{e})}\in
I$. It follows that the multiple $\mathbf{x}^{e}x_{1}^{H(m_{0})-h(m_{0})}$
lies in $I$. The product of the expression (2) with
$x_{1}^{H(m_{0})-h(m_{0})}$ yields
$m_{0}x_{1}^{H(m_{0})}=m_{0}^{+}x_{1}^{H(m_{0})-h(m_{0})}=fx_{1}^{H(m_{0})-h(m_{0})}-\sum
c_{e}\mathbf{x}^{e}x_{1}^{H(m_{0})-h(m_{0})}\in I,$
a contradiction. It follows that $C=\emptyset$, and (1) is true. ∎
## 4\. Base change for monic families
It is well-known that passage to the initial ideal does not commute with
arbitrary base change, but rather only with flat base change (see [BGS93]).
Thus it is difficult to define functors manipulating initial ideals through
arbitrary base changes. In contrast, we show that $\Delta$-monic families are
stable by arbitrary base change. This allows us to define a $\Delta$-monic
functor at the end of this section, after having settled the required base-
change statements.
###### Proposition 15.
Let $<$ be a quasi-homogeneous total order. Let $I\subset B[\mathbf{x}]$ be a
bounded ideal. If for all $m$, ${\rm in}_{m}(I)=0$ or ${\rm in}_{m}(I)=\langle
1\rangle$, then for every base change $B\rightarrow A$, ${\rm
in}_{m}(IA[\mathbf{x}])={\rm in}_{m}(I)A$.
###### Proof.
Obviously, ${\rm in}_{m}(I)=\langle 1\rangle$ implies ${\rm
in}_{m}(IA[\mathbf{x}])=\langle 1\rangle$. If ${\rm in}_{m}(I)=0$, we argue by
contradiction, supposing that ${\rm in}_{m}(IA[\mathbf{x}])\neq 0$. Choose
$f\in IA[\mathbf{x}]$ with ${\rm in}(f)=am$, $a\neq 0$. Let $x_{j}$ be a
negative variable. Then $x_{j}^{d_{j}}\in I$ for $d_{j}$ large. In particular,
replacing $f$ with $f-\lambda x_{j}^{e}$, we may assume that $f$ has no term
divisible by $x_{j}^{d_{j}}$. Choose an expression
(3) $f=\sum a_{i}f_{i}$
with $a_{i}\in A$ and $f_{i}\in I\subset B[\mathbf{x}]$. As above, we may
assume that none of the $f_{i}$ contains a term divisible by $x_{j}^{d_{j}}$.
Let $m^{\prime}$ be the maximal monomial appearing in the $f_{i}$. Note that
$m^{\prime}\geq m$ and, more precisely, $m^{\prime}>m$ since ${\rm
in}_{m}(I)=0$. Among all possible expressions (3), choose one with minimal
$m^{\prime}$. (Although $<$ is not necessarily a monomial order, a minimum
$m^{\prime}$ exists since we have bounded the exponent of the negative
variables). Let $J:=\\{i:{\rm in}(f_{i})=\lambda_{i}m^{\prime}\\}$ be the set
of indices of $f_{i}$ with initial monomial $m^{\prime}$. The coefficient of
$m^{\prime}$ vanishes on the right hand side of (3), thus $\sum_{i\in
J}a_{i}{\rm in}(f_{i})=\sum_{i\in J}(a_{i}\lambda_{i})m^{\prime}=0$. Thus
$\sum_{i\in J}a_{i}\lambda_{i}=0$. Let $g\in I$ with ${\rm in}(g)=m^{\prime}$
and such that $g$ has no term divisible by $x_{j}^{d_{j}}$. Let
$f^{\prime}_{i}:=f_{i}$ if $i\notin J$ and
$f^{\prime}_{i}:=f_{i}-\lambda_{i}g$ if $i\in J$. Then
$f=\sum a_{i}f^{\prime}_{i}$
and this expression contradicts the maximality of $m^{\prime}$. ∎
We remark that with the same proof, we have in fact a slightly more precise
and technical statement which will be useful later.
###### Proposition 16.
Let $<$ be a quasi-homogeneous total order. Let $I\subset B[\mathbf{x}]$ be a
bounded ideal. Let $m$ be a monomial. If for all $m^{\prime}\geq m$, either
${\rm in}_{m^{\prime}}(I)=0$ or ${\rm in}_{m^{\prime}}(I)=\langle 1\rangle$
holds, then for every base change $B\rightarrow A$, the equality ${\rm
in}_{m}(IA[\mathbf{x}])={\rm in}_{m}(I)A$ holds.
Let $X={\rm Spec}\,A$ be the affine scheme corresponding to the ring $A$. The
ideal ${\rm in}_{m}(I)\subset A$ defines a closed subscheme of $X$. If $X$ is
a scheme which is not affine, we wish to glue the local constructions we have
been working with so far. Since open immersions are flat, the following
proposition implies that gluing is possible and that for any sheaf of ideals
$\mathcal{I}\subset\mathcal{O}_{X}[x]$, there is a well-defined sheaf of
ideals ${\rm in}_{m}(\mathcal{I})\subset\mathcal{O}_{X}$ on a non-affine
scheme $X$. This allows us to speak of _bounded_ and _monic_ , resp., _ideal
sheaves_ rather than ideals.
###### Proposition 17.
Let $<$ be a quasi-homogeneous total order. Let $I\subset B[\mathbf{x}]$ be a
bounded ideal. Let $B\rightarrow A$ be a ring homomorphism which makes $A$ a
flat $B$-module. Then ${\rm in}_{m}(IA[\mathbf{x}])={\rm in}_{m}(I)A$.
###### Proof.
Theorem 3.6 of [BGS93] proves the statement in the case where $<$ is a
monomial order. The same proof also goes through in our context, provided that
we take care of the high powers of the negative variables in the same way as
we did in the proof of proposition 15. ∎
We now define the monic functor $\mathcal{H}^{{\rm mon}(<,\Delta)}$. Corollary
13 implies that monic ideals are bounded in most cases. We shall later see
that ideals arising from the Gröbner functor are always bounded (see lemma
23). It is thus natural, and not very restrictive, to add this boundedness
condition when we define functors of monic families.
###### Definition 18.
Let $<$ be a quasi-homogeneous total order and $\Delta$ a standard set. Let
$B$ be a noetherian $k$-algebra and $\mathcal{H}^{{\rm mon}(<,\Delta)}(B)$ be
the set of ideals $I\subset B[\mathbf{x}]$ such that
* •
$I$ is bounded and $\Delta$-monic
* •
$B[\mathbf{x}]/I$ is $B$ locally free of rank $\\#\Delta$
This defines a covariant functor $\mathcal{H}^{{\rm mon}(<,\Delta)}$, which we
call a _monic functor_ , from the category of noetherian $k$-algebras to the
category of sets.
## 5\. Monic functors are representable
The goal of this section is to prove that monic functors are representable
(see theorem 21)
###### Proposition 19.
Let $<$ be a quasi-homogeneous total order. Let $I\subset B[\mathbf{x}]$ be a
bounded ideal with $B[\mathbf{x}]/I$ a locally free $B$-module of rank $n$,
and let $\mathcal{I}\subset\mathcal{O}_{X}[\mathbf{x}]$ be the ideal sheaf on
$X:={\rm Spec}\,B$ defined by $I$. Let $\Delta$ be a standard set of
cardinality $n$. There exists a locally closed subscheme $Z\subset X$ such
that
* •
the restriction of $\mathcal{I}$ to $Z$ is a bounded $\Delta$-monic family,
* •
any morphism $f:{\rm Spec}\,A\rightarrow{\rm Spec}\,B$ such that
$IA[\mathbf{x}]$ is a bounded $\Delta$-monic family factors through $Z$.
The last proposition is a particular case of the following statement, which we
shall prove by induction.
###### Proposition 20.
Let $<$ be a quasi-homogeneous total order. Let $I\subset B[\mathbf{x}]$ be an
ideal bounded by $r_{1},\dots,r_{d}$ such that $B[\mathbf{x}]/I$ is a locally
free $B$-module of rank $n$, and let
$\mathcal{I}\subset\mathcal{O}_{X}[\mathbf{x}]$ be the ideal sheaf on $X:={\rm
Spec}\,B$ defined by $I$. Let $C:=\\{x^{e}:e_{i}<r_{i}\\}$ be the “hypercuboid
of monomials” of edge lengths $r_{i}$, and therefore, of cardinality
$s:=\prod_{i=1}^{d}r_{i}$. We number the monomials $m_{i}\in C$ such that
$m_{1}>m_{2}>\dots>m_{s}$. Let $r\leq s$, and fix a map
$\mu:\\{1,\dots,r\\}\rightarrow\\{0,1\\}$. Then there exists a locally closed
subscheme $Z_{r}\subset X$ (possibly empty) such that
* •
The sheaf of ideals $\mathcal{I}_{r}$, which we define as the restriction of
$\mathcal{I}$ to $Z_{r}$, is a bounded monic family with ${\rm
in}(\mathcal{I}_{r})_{m}=\langle 1\rangle$ for $m\notin C$ and ${\rm
in}(\mathcal{I}_{r})_{m_{i}}=\langle\mu(i)\rangle$ for $1\leq i\leq r$.
* •
Let $f:{\rm Spec}\,A\rightarrow{\rm Spec}\,B$ be a morphism and
$K:=IA[\mathbf{x}]$. Then ${\rm in}(K)_{m}=\langle 1\rangle$ for $m\notin C$
and ${\rm in}(K)_{m_{i}}=\langle\mu(i)\rangle$ for $1\leq i\leq r$ if, and
only if, $f$ factors through $Z_{r}$.
###### Proof.
We start with the first item, which we prove by induction on $r\geq 0$.
When $r=0$, we may take $Z_{0}=X$ since the family is bounded by
$r_{1},\dots,r_{d}$, which implies that ${\rm
in}(\mathcal{I}_{r})_{x_{i}^{r_{i}}}=\langle 1\rangle$, hence ${\rm
in}(\mathcal{I}_{r})_{m}=\langle 1\rangle$ for $m\notin C$. We may assume that
$Z_{r-1}$ is adequately defined. Let $F_{r}\subset Z_{r-1}$ the closed
subscheme defined by the sheaf of ideals
$\mathcal{I}(F_{r}):=(\mathcal{I}_{r-1})_{m_{r}}$. Let
$O_{r}:=Z_{r-1}\setminus F_{r}$. We define
$Z_{r}:=\begin{cases}F_{r}&\text{ if }\mu(r)=0\\\ O_{r}&\text{
otherwise}.\end{cases}$
By proposition 16 and the induction hypothesis, we have ${\rm
in}(\mathcal{I}_{r})_{m}=\langle 1\rangle$ for $m\notin C$ and ${\rm
in}(\mathcal{I}_{r})_{m_{i}}=\langle\mu(i)\rangle$ for $1\leq i\leq r-1$. We
have to prove that ${\rm in}(\mathcal{I}_{r})_{m_{r}}=\langle\mu(r)\rangle$.
This is a local problem, so we may assume that $Z_{r-1}\subset X$ is a closed
subscheme defined by an ideal $J_{r-1}\subset B$. If $\mu(r)=1$, the base
change $Z_{r}\hookrightarrow Z_{r-1}$ is open, thus flat. Base change
therefore shows that ${\rm in}(\mathcal{I}_{r})_{m_{r}}=\langle 1\rangle$ on a
neighborhood of any $p\in Z_{r}$.
Assume now that $\mu(r)=0$. We claim that ${\rm
in}(\mathcal{I}_{r})_{m_{r}}=0$. The problem is local, so we may assume that
both $Z_{r}$ and $Z_{r-1}$ are affine. Accordingly, we replace the sheaves
$\mathcal{I}_{r}$ and $\mathcal{I}_{r-1}$ by their respective ideals of global
sections, which we denote by $I_{r}$ and $I_{r-1}$, resp. We will argue by
contradiction, supposing that there exists some $f\in I_{r}$ with ${\rm
in}(f)=cm_{r}$, $c\neq 0$. Take some $g\in I_{r-1}$ that restricts to $f$ over
$Z_{r}$. Since $I$ is bounded by $r_{1},\dots,r_{d}$, we may assume that both
$f$ and $g$ are linear combinations of monomials in $C$, $f=\sum_{m_{i}\in
C}a_{i}m_{i}$ and $g=\sum_{m_{i}\in C}b_{i}m_{i}$, resp. Among all possible
$g$, choose one which minimizes ${\rm in}(g)$. Then ${\rm in}(g)=dm$ with
$m\geq m_{r}$. Suppose that $m>m_{r}$. Since ${\rm in}(I_{r-1})_{m}=\langle
0\rangle$ or $\langle 1\rangle$ by induction hypothesis and since $d\neq 0$,
we obtain that ${\rm in}(I_{r-1})_{m}=\langle 1\rangle$. Choose $h\in I_{r-1}$
with ${\rm in}(h)=m$. Then $g^{\prime}:=g-dh$ contradicts the minimality of
$g$. Thus $m=m_{r}$, and $d\in(I_{r-1})_{m_{r}}=I(F_{r})$ vanishes on $F_{r}$.
Since $d$ restricts to $c$ on $F_{r}$, it follows that $c=0$. This is a
contradiction, which finishes the proof of ${\rm
in}(\mathcal{I}_{r})_{m_{r}}=0$.
We now come to the second point. If $f$ factors through $Z_{r}$, then ${\rm
in}(K)_{m}=\langle 1\rangle$ for $m\notin C$ and ${\rm
in}(K)_{m_{i}}=\langle\mu(i)\rangle$ for $1\leq i\leq r$, as these properties
are inherited from $Z_{r}$ by proposition 16.
If, on the other hand, $f$ does not factor through $Z_{r}$, then we want to
prove that ${\rm in}(K)_{m_{i}}\neq\langle\mu(i)\rangle$ for some $i$, $1\leq
i\leq r$. We may assume that $f$ factors through $Z_{r-1}$, since in the
complementary case, we are done by induction.
We first consider the case $\mu(r)=0$. By the factorization property of $f$
through $Z_{r-1}$, we may assume that $B$ is the coordinate ring of the scheme
$Z_{r-1}$. We denote by $f^{\\#}:B\rightarrow A$ the morphism associated with
$f$. Since $f$ does not factor through $Z_{r}$, there exists some $g\in
I_{r-1}$ with ${\rm in}(g)=cm_{r}$, $f^{\\#}(c)\neq 0$. The pullback of $g$ to
$A[\mathbf{x}]$ shows that ${\rm in}(IA[\mathbf{x}])_{m_{r}}\neq\langle
0\rangle$, and we are done.
Now consider the case $\mu(r)=1$. Since $Z_{r}\subset Z_{r-1}$ is open in this
case, the factorization property of $f$ implies the existence of a point
$p\in{\rm Spec}\,A$ with $f(p)\in Z_{r-1}$ and $f(p)\notin
Z_{r}=Z_{r-1}\setminus F_{r}$. In particular, ${\rm in}(I\cdot
k(p)[\mathbf{x}])_{m_{r}}=0$ by proposition 16. Thus ${\rm
in}(K)_{m_{r}}\neq\langle 1\rangle$, as expected, since otherwise we would
have ${\rm in}(I\cdot k(p)[\mathbf{x}])_{m_{r}}=\langle 1\rangle$ by
proposition 16. ∎
###### Theorem 21.
Let $<$ be a quasi-homogeneous total order. Then the monic functor
$\mathcal{H}^{{\rm mon}(<,\Delta)}$ is representable by a locally closed
scheme $H^{{\rm mon}(<,\Delta)}$ of $H^{n}(\mathbb{A}^{d})$, where
$n=\\#\Delta$.
###### Proof.
Let $I\subset B[\mathbf{x}]$ be an ideal defining a flat family ${\rm
Spec}\,B[\mathbf{x}]/I\to{\rm Spec}\,B$ of relative length $n$ with $I$
bounded and $\Delta$-monic. Let $L_{i}\subset H^{n}(\mathbb{A}^{d})$ be the
closed subscheme of $H^{n}$ parametrizing the subschemes $Z$ included in the
subscheme $\\{x_{i}^{n}=0\\}\subset\mathbb{A}^{d}$. Let $L:=\cap_{x_{i}\text{
negative }}L_{i}$. Since the universal ideal of the Hilbert scheme (see
[Led11] for the construction and properties of that universal ideal) is
bounded over $L$, there is a locally closed subscheme $L_{\Delta}\subset L$
parametrizing $\Delta$-monic ideals (see proposition 19). By universal
property of the Hilbert scheme, the ideal $I$ corresponds to a unique morphism
$\phi:{\rm Spec}\,B\to H^{n}(\mathbb{A}^{d})$. Lemma 14 implies that the
morphism $\phi$ factors through $L$. Proposition 19 (the universal property of
$L_{\Delta}$) implies that $\phi$ even factors through $L_{\Delta}$.
Conversely, any morphism ${\rm Spec}\,B\rightarrow L_{\Delta}$ yields a
$\Delta$-monic bounded ideal by pullback of the universal ideal over the
Hilbert scheme. Upon defining $H^{{\rm mon}(<,\Delta)}:=L_{\Delta}$, we thus
get the required result. ∎
###### Proposition 22.
Let $<_{j}$ be a sequence of quasi-homogeneous total orders converging to
$<$.Then for $j$ large, the functors $\mathcal{H}^{\rm mon(<_{j},\Delta)}$ and
$\mathcal{H}^{\rm mon(<,\Delta)}$ are isomorphic.
###### Proof.
It suffices to prove that the representants $H^{mon(<_{j},\Delta)}$ and
$H^{mon(<,\Delta)}$ are isomorphic. Recall the subscheme $L$ introduced in the
proof of the theorem. The subschemes $H^{mon(<_{j},\Delta)}$ and
$H^{mon(<,\Delta)}$ of $L$ are constructed locally using proposition 20. By
proposition 11, both constructions coincide locally for $j>j_{O}$ on an open
set $O$. Since $L$ is quasi-compact, $H^{mon(<_{j},\Delta)}\simeq
H^{mon(<,\Delta)}$ for $j>\max_{O}j_{0}$. ∎
## 6\. Białynicki-Birula functors are representable
We have proved that our monic functors are representable by schemes. The goal
of the present section is to prove that the Białynicki-Birula functors are
representable by realizing them as an intersection of two monic functors.
Since monic functors are bounded, we first need:
###### Lemma 23.
Each ideal $I\in\mathcal{H}^{BB(\xi,\Delta)}(B)$ is bounded with respect to
any total order $<$ refining $<_{\xi}$.
###### Proof.
Up to reordering the components of $\xi$, one can assume that
$\xi_{i}>\xi_{i+1}$. We denote by $k$, $l$ the integers such that
$\xi_{i}>0\Leftrightarrow i<k$ and $\xi_{i}<0\Leftrightarrow i\geq l$.
By the definition of boundedness, we need to exhibit a family of polynomials
$h_{i}$. If $x_{i}$ is a positive variable, we construct $h_{i}$ as in the
beginning of the proof of 10.
Note that the monomials with exponents in $\Delta$ form a basis of
$B[\mathbf{x}]/I$. In particular, every monomial $\mathbf{x}^{e}$ leads to an
element $\mathbf{x}^{e}+\sum_{m\in\Delta}c_{m}\mathbf{x}^{m}$ of $I$. For
$i\geq l$, $r_{i}\gg 0$ and $\mathbf{x}^{e}=x_{i}^{r_{i}}$, we have $c_{m}=0$
since $I$ is $\Delta$-monic. Thus $h_{i}:=x_{i}^{r_{i}}\in I$.
Let $x_{i}$ be a negative variable of weight $0$. To prove that $x_{i}^{r}\in
I$ for large $r$, we shall prove that for any monomial $m\in
B[x_{l},\dots,x_{d}]$, $x_{i}^{r}m\in I$ for large $r$, and subsequently apply
that to the monomial $m=1$. If the exponent $(e_{l},\dots,e_{d})$ of
$m=x_{l}^{e_{l}}\dots x_{d}^{e_{d}}$ is large (in more concrete terms, if
$e_{i}\geq r_{i}$ for some $i$), then $m\in I$ by the above. We are therefore
left with a finite collection of $m$ for which the claim has to be checked. We
proceed by induction over $m$. Let be the $m$ minimal element of the finite
collection for which the claim has not been proved yet. By our hypothesis on
$I$, for large $r$, the monomial $x_{i}^{r}m$ lies in the limit ideal $I(0)$,
thus $I$ contains an element $f=x_{i}^{r}m+R$, where all terms of $R$ are
strictly smaller with respect to $<_{\xi}$ than $x_{i}^{r}m$. We write the
monomials appearing in $R$ as $m_{1}m_{2}m_{3}$, with $m_{1}\in
k[x_{1}\dots,x_{k-1}],m_{2}\in k[x_{k}\dots,x_{l-1}]$, $m_{3}\in
k[x_{l},\dots,x_{d}]$. Such a product satisfies
$m_{1}m_{2}m_{3}<_{\xi}x_{i}^{r}m$ only if $m_{3}<m$. By induction we know
that when multiplying by an adequate power $x_{i}^{p}$, we get
$x_{i}^{p}m_{3}\in I$, hence $x_{i}^{r+p}m=x_{i}^{p}f-x_{i}^{p}R\in I$, as
required. ∎
###### Proposition 24.
Let $\xi\in\mathbb{Z}^{d}$. Let $\delta_{j}$ be a sequence in $\mathbb{R}^{d}$
converging to $0$ such that the sequence $\xi_{j}^{+}:=\xi+\delta_{j}$
converging to $\xi$ has the property that $<_{\xi_{j}^{+}}$ converges to a
refinement $<_{+}$ of $<_{\xi}$. Let $\xi_{j}^{-}:=\xi-\delta_{j}$, and let
$<_{-}$ the limit of $<_{\xi_{j}^{-}}$. Then
$\mathcal{H}^{BB(\Delta,\xi)}=\mathcal{H}^{{\rm
mon}(<_{+},\Delta)}\cap\mathcal{H}^{{\rm mon}(<_{-},\Delta)}$.
###### Proof.
It is clear that $\mathcal{H}^{BB(\Delta,\xi)}\subset\mathcal{H}^{{\rm
mon}(<_{+},\Delta)}\cap\mathcal{H}^{{\rm mon}(<_{-},\Delta)}$ since by
proposition 8, $\mathcal{H}^{BB(\Delta,\xi)}\subset\mathcal{H}^{{\rm
mon}(<,\Delta)}$ for any refinement $<$ of $<_{\xi}$. Conversely, take
$I\in\mathcal{H}^{{\rm mon}(<_{+},\Delta)}(B)\cap\mathcal{H}^{{\rm
mon}(<_{-},\Delta)}(B)$. For proving that
$I\in\mathcal{H}^{BB(\Delta,\xi)}(B)$, we first prove that ${\rm
in}_{<_{\xi},_{m}}(I)=\langle 1\rangle$ for $m\notin\Delta$.
We argue by contradiction. Suppose that the set $C:=\\{m\notin\Delta$, ${\rm
in}_{<_{\xi},m}(I)\neq\langle 1\rangle\\}$ is non empty. Since $I$ is bounded,
$C$ is finite. Let $m^{\prime}$ be the smallest element of $C$ for the order
$<_{+}$. Since $m^{\prime}\notin\Delta$, there is an $f\in I$ with ${\rm
in}_{<_{+}}(f)=m^{\prime}$. We write $f=m^{\prime}+r+s+t$, with
$r=\sum_{f_{\xi}(e)<f_{\xi}(m^{\prime})}c_{e}\mathbf{x}^{e}$,
$s=\sum_{e\in\Delta,f_{\xi}(e)=f_{\xi}(m^{\prime}),e<_{+}m^{\prime}}c_{e}\mathbf{x}^{e}$,
$t=\sum_{e\notin\Delta,f_{\xi}(e)=f_{\xi}(m^{\prime}),e<_{+}m^{\prime}}c_{e}\mathbf{x}^{e}$.
By minimality of $m^{\prime}$, for any term $c_{e}\mathbf{x}^{e}$ in $t$,
there exists some $g_{e}\in I$ of shape
$g_{e}=\mathbf{x}^{e}+\sum_{f_{\xi}(f)<f_{\xi}(e)}d_{f}\mathbf{x}^{f}$.
Let
$h:=f-\sum_{c_{e}\mathbf{x}^{e}\text{ a term of }t}c_{e}g_{e}$
Then $h$ reads $h=m^{\prime}+r+s+u$ with
$u=\sum_{e\notin\Delta,f_{\xi}(e)=f_{\xi}(m^{\prime}),e<_{+}m^{\prime}}c_{e}(\mathbf{x}^{e}-g_{e})$.
The only terms $c_{e}\mathbf{x}^{e}$ in $h$ with
$f_{\xi}(e)=f_{\xi}(m^{\prime})$ are the terms in $s$. Note that the orders
$<_{+}$ and $<_{-}$ have been chosen such that any pair of monomials
$m,m^{\prime}$ satisfies the following:
* •
if $f_{\xi}(m)\neq f_{\xi}(m^{\prime})$, then $m<_{+}m^{\prime}\Leftrightarrow
m<_{-}m^{\prime}$
* •
if $f_{\xi}(m)=f_{\xi}(m^{\prime})$, then $m<_{+}m^{\prime}\Leftrightarrow
m^{\prime}<_{-}m$.
Thus, if $s\neq 0$, then ${\rm in}_{<_{-}}(h)$ is a term $c_{e}\mathbf{x}^{e}$
with $e\in\Delta$, contradicting the assumption that $I\in\mathcal{H}^{{\rm
mon}(<_{-},\Delta)}(B)$. We thus obtain that $s=0$ and ${\rm
in}_{<_{\xi}}(h)=m^{\prime}$, contradicting the definition of $C$.
Since $I$ is bounded, we get by definition a collection of polynomials
$h_{i}\in I\cap B[x_{i}]$ with $in(h_{i})=x_{i}^{r_{i}}$ when $x_{i}$ is
positive and $h_{i}=x_{i}^{r_{i}}$ if $x_{i}$ is negative. Then $I(t)$
contains the elements
$t^{-\xi_{i}r_{i}}(t.h_{i})=x_{i}^{r_{i}}+\sum_{r<r_{i}}\lambda_{r}(t)x_{i}^{r}$.
It follows that the quotient $B[t,\mathbf{x}]/I(t)$ is a finite $B[t]$-module.
Summing things up, ${\rm Spec}\,B[t,\mathbf{x}]/I(t)$ is a finite family over
${\rm Spec}\,B[t]$. By construction, this is a flat family of relative length
$n$ over the open set $t\neq 0$. The fiber ${\rm Spec}\,B[\mathbf{x}]/I(0)$
over $t=0$ is a quotient of ${\rm Spec}\,B[\mathbf{x}]/I^{\Delta}$ which is a
flat family of relative length $n$ over ${\rm Spec}\,B$. It follows by semi-
continuity that $B[t,\mathbf{x}]/I(t)$ is a locally free $B[t]$-module of rank
$n$ with $I(0)=I^{\Delta}$. ∎
###### Theorem 25.
The Białynicki-Birula functor $\mathcal{H}^{BB(\Delta,\xi)}$ is representable.
###### Proof.
By the above, the Białynicki-Birula functor is an intersection of two functors
$\mathcal{H}^{{\rm mon}(<_{+},\Delta)}$ and $\mathcal{H}^{{\rm
mon}(<_{-},\Delta)}$, both representable by locally closed subschemes $H^{{\rm
mon}(<_{+},\Delta)}$ and $H^{{\rm mon}(<_{-},\Delta)}$, respectively, of the
Hilbert scheme. The Białynicki-Birula functor is therefore representable by
the schematic intersection $H^{{\rm mon}(<_{+},\Delta)}\cap H^{{\rm
mon}(<_{-},\Delta)}$. ∎
Here is an application of the above to a specific choice of weight $\xi$.
###### Proposition 26.
Let $\epsilon:\\{1,\dots,d\\}\rightarrow\\{1,-1\\}$ be a sign function. Let
$\xi$ be a weight such that $|\xi_{1}|\gg|\xi_{2}|\gg\dots\gg|\xi_{d}|>0$ and
$sign(\xi_{i})=\epsilon(i)$. Let $<$ be the signed lexicographic order defined
by $(a_{1},\dots,a_{d})<(b_{1},\dots,b_{d})$ if, and only if,
$(\epsilon(1)a_{1},\dots,\epsilon(d)a_{d})<(\epsilon(1)b_{1},\dots,\epsilon(d)b_{d})$
for the usual lexicographic order. Then $H^{{\rm mon}(<,\Delta)}$ is
isomorphic to $H^{BB(\xi,\Delta)}$.
The particular weights of the proposition above are used [ES87], [ES88],
[Gro96], [Nak99], for studying the Hilbert scheme of points in the two-
dimensional case. But more generally, we have:
###### Proposition 27.
Let $\xi_{j}$ be a sequence in $\mathbb{Q}^{d}$ such that $<_{\xi_{j}}$
converges to the total quasi-homogeneous order $<$. Then for large $j$,
$\mathcal{H}^{BB(\xi_{j},\Delta)}$ is isomorphic to $\mathcal{H}^{{\rm
mon}(<,\Delta)}$.
###### Proof.
Following proposition 24, for each $\xi_{j}$, we choose a sequence
$\delta_{jk}$ such that the order $<_{\xi_{j}+\delta_{jk}}$ (resp.
$<_{\xi_{j}-\delta_{jk}}$) converges to a refinement $<_{j+}$ (resp. $<_{j-}$)
of $<_{\xi_{j}}$, and
$\mathcal{H}^{BB(\xi_{j},\Delta)}=\mathcal{H}^{\rm
mon(<_{j+},\Delta)}\cap\mathcal{H}^{\rm mon(<_{j-},\Delta)}.$
By proposition, 22, the isomorphisms
$\mathcal{H}^{\rm mon(<_{j\pm},\Delta)}\simeq\mathcal{H}^{\rm
mon(<_{\xi_{j}\pm\delta_{jk_{j}}},\Delta})$
hold for $k_{j}$ large. If we choose $k_{j}$ large enough, $\delta_{jk_{j}}$
is arbitrarily small compared to $\xi_{j}$ and we have the convergence of the
orders $\lim_{j}<_{\xi_{j}\pm\delta_{jk_{j}}}=\lim_{j}<_{\xi_{j}}=<.$ In
particular, for $j$ large,
$\mathcal{H}^{\rm
mon(<_{\xi_{j}\pm\delta_{jk_{j}}},\Delta)}\simeq\mathcal{H}^{\rm
mon(<,\Delta)}.$
The result follows from the displayed equalities. ∎
## 7\. Białynicki-Birula schemes and Hilbert-Chow morphisms
The goal of this section is to prove the following theorem:
###### Theorem 28.
If $\xi_{i}\leq 0$, then $H^{BB(\xi,\Delta)}$ is schematically included in the
fiber over the origin $\rho_{i}^{-1}(O)$, where
$\rho_{i}:H^{n}(\mathbb{A}^{d})\rightarrow{\rm Sym}^{n}(\mathbb{A}^{1})$ is
the Hilbert-Chow morphism associated with the $i^{th}$ coordinate.
To be precise, let $\rho:H^{n}(\mathbb{A}^{d})\rightarrow{\rm
Sym}^{n}(\mathbb{A}^{d})$ be the usual Hilbert-Chow morphism. The projection
$p_{i}:\mathbb{A}^{d}\rightarrow\mathbb{A}^{1}$ to the $i^{th}$-coordinate
induces a morphism $p_{i}^{n}:{\rm Sym}^{n}(\mathbb{A}^{d})\rightarrow{\rm
Sym}^{n}(\mathbb{A}^{1})$. We denote by $\rho_{i}:=p_{i}^{n}\circ\rho$ the
Hilbert-Chow morphism associated with the $i^{th}$ coordinate. We denote by
$O\in{\rm Sym}^{n}(\mathbb{A}^{1})$ the point corresponding to $n$ copies of
the origin of $\mathbb{A}^{1}$.
###### Lemma 29.
Let $I\in\mathcal{H}^{{\rm mon}(<,\Delta)}(B)$. Suppose that $x_{i}$ is a
negative variable. Let $m_{i}$ be the multiplication by $x_{i}$ in
$B[\mathbf{x}]/I$. Then there exists a basis of $B[\mathbf{x}]/I$ such that
the matrix of $m_{i}$ is strictly lower triangular.
###### Proof.
The monomials $b_{i}$ with exponents in $\Delta$ are a basis of
$k[\mathbf{x}]/I$. We order them such that $b_{1}>b_{2}\dots>b_{n}$. Since
$x_{i}$ is negative, whenever $x_{i}b_{j}\in\Delta$, then $x_{i}b_{j}=b_{l}$
for some $l>j$. If $x_{i}b_{j}\notin\Delta$, we may choose $f\in I$ with ${\rm
in}(f)=x_{i}b_{j}$ and
$f=x_{i}b_{j}+\sum_{e\in\Delta,\mathbf{x}^{e}<x_{i}b_{j}}c_{e}\mathbf{x}^{e}$.
Then in the quotient $B[\mathbf{x}]/I$, the identity
$x_{i}b_{j}=x_{i}b_{j}-f=-\sum_{e\in\Delta,\mathbf{x}^{e}<x_{i}b_{j}<b_{j}}c_{e}\mathbf{x}^{e}$
holds. ∎
###### Proof of the theorem.
According to proposition 24, it suffices to prove the inclusion $H^{{\rm
mon}(<,\Delta)}\subset\rho_{i}^{-1}(O)$ if the variable $x_{i}$ is negative
for the order $<$.
We recall the observation by Bertin [Ber10] that the Hilbert-Chow morphism is
given by the linearized determinant of Iversen. Let $I\in\mathcal{H}^{{\rm
mon}(<,\Delta)}(B)$ and $b_{1},\dots,b_{n}$ a basis of $B[\mathbf{x}]/I$. If
$P\in k[x_{i}]$, we denote by $C_{P}^{j}$ the $j^{th}$ column of the matrix
(with respect to our fixed basis) of multiplication by $P$. If
${P_{1}}\otimes\dots\otimes{P_{n}}$ is a pure tensor in $k[x_{i}]^{\otimes
n}$, we put ${\rm ld}(P_{1}\otimes\dots\otimes
P_{n}):=\det(C_{P_{1}}^{1},\dots,C_{P_{n}}^{n})$. The symmetric group $S_{n}$
acts on $k[x_{i}]^{\otimes n}$; we denote by $k[x_{i}]^{(n)}\subset
k[x_{i}]^{\otimes n}$ the invariant part. Iversen [Ive70] proved that ${\rm
ld}:k[x_{i}]^{(n)}\rightarrow B$ is a $k$-algebra homomorphism. As was
remarked by Bertin, this homomorphism corresponds to the Hilbert-Chow morphism
$\rho_{i}$. The ideal of the origin is generated by the elementary symmetric
polynomials, which have degree at least one. For proving the theorem, it
therefore suffices to show that $\det(C_{P_{1}}^{1},\dots,C_{P_{n}}^{n})=0$ if
$x_{i}$ divides some $P_{j}$. However, according to the lemma above, that
determinant is the determinant of a lower triangular matrix, and that
triangular matrix has a zero term on the diagonal if $x_{i}$ divides some
$P_{j}$. ∎
## References
* [Ber10] José Bertin, _The punctual Hilbert scheme: an introduction_ , Séminaires et Congrès 24 (2010), 1–100.
* [BGS93] Dave Bayer, André Galligo, and Mike Stillman, _Gröbner bases and extension of scalars_ , Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 198–215. MR MR1253992 (94j:13021)
* [ES87] Geir Ellingsrud and Stein Arild Strømme, _On the homology of the Hilbert scheme of points in the plane_ , Invent. Math. 87 (1987), no. 2, 343–352. MR 870732 (88c:14008)
* [ES88] by same author, _On a cell decomposition of the Hilbert scheme of points in the plane_ , Invent. Math. 91 (1988), no. 2, 365–370. MR 922805 (89f:14007)
* [Eva04] Laurent Evain, _Irreducible components of the equivariant punctual Hilbert schemes_ , Adv. Math. 185 (2004), no. 2, 328–346. MR MR2060472 (2005g:14009)
* [Gro96] I. Grojnowski, _Instantons and affine algebras. I. The Hilbert scheme and vertex operators_ , Math. Res. Lett. 3 (1996), no. 2, 275–291. MR 1386846 (97f:14041)
* [Ive70] Birger Iversen, _Linear determinants with applications to the Picard scheme of a family of algebraic curves_ , Lecture Notes in Mathematics, Vol. 174, Springer-Verlag, Berlin, 1970. MR 0292835 (45 #1917)
* [Led11] Mathias Lederer, _Gröbner strata in the Hilbert scheme of points_ , J. Commut. Algebra 3 (2011), no. 3, 349–404. MR 2826478
* [MS10] Diane Maclagan and Gregory G. Smith, _Smooth and irreducible multigraded Hilbert schemes_ , Adv. Math. 223 (2010), no. 5, 1608–1631. MR 2592504 (2011e:14009)
* [Nak99] Hiraku Nakajima, _Lectures on Hilbert schemes of points on surfaces_ , University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344 (2001b:14007)
|
arxiv-papers
| 2012-09-10T15:21:54 |
2024-09-04T02:49:34.925748
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Laurent Evain, Mathias Lederer",
"submitter": "Laurent Evain",
"url": "https://arxiv.org/abs/1209.2026"
}
|
1209.2108
|
# Anisotropic Coherence properties in a trapped quasi2D dipolar gas
Christopher Ticknor Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
###### Abstract
We consider a trapped quasi2D dipolar Bose Einstein condensate (q2D DBEC) with
a polarization tilted into the plane of motion. We show that by tilting the
polarization axis, the coherence properties are anisotropic. Such a system
will have density fluctuations whose amplitude depend on their trap location.
Additionally, interference contrast will also be anisotropic despite an
isotropic density profile. The anisotropy is related to a roton like mode that
becomes unstable and supports local collapse along the polarization axis.
###### pacs:
67.85.-d, 03.75.Kk, 03.75.Lm, 47.37.+q
Coherence properties underlie collective quantum behavior. An example of a
need to further understand coherence is in high $T_{c}$ superconductors
phaseTC , where the origin of the phase coherence leading to superconducting
behavior is still mysterious. Such solid state systems where this occurs are
complex, making the study of coherence properties a challenge. One avenue to
study coherence of quantum systems more clearly is in ultracold gases, which
have offered direct insight and unprecedented control to the study of quantum
collective behavior. For example, two macroscopic Bose Einstein condensates
(BECs) were interfered displaying phase coherence int . More refined
experiments have directly detected higher order correlations correlations
which cannot be easily accessed by electronic systems. Additionally, reduced
dimensional ultracold gases have been the focus of intense interest because
fluctuations are enhanced in such geometries. Phase coherence has been studied
at the Berezinskii Kosterlitz Thouless (BKT) transition BKT ; BKTrev where
two independent q2D ultracold gases were interfered, revealing thermally
activated phase defects are the origin of decoherence zoran . The BKT
transition is a transition where the phase coherence changes character, and
insights into controlling such coherence properties are of general importance.
In this paper we show the coherence properties of a trapped dipolar gas can be
controlled.
The progress of experiments to probe ultracold gases is remarkable. Recent
advances in imaging q2D ultracold gases have been used to detect density
fluctuations chicago ; jila ; ENS . Additionally, ultracold atomic systems in
one dimensional systems have also amazingly been used to study quantum
fluctuations directly armijo2 and other coherence properties armijo1 ; manz ;
imam ; kinoshita . For these examples of ultracold gases, the interaction
between the constituents are short range. There are ultracold dipolar systems
which have anisotropic, long range interactions, which offer an entirely new
avenue to study quantum correlations and collective behavior. Amazing
experimental progress has been made with the strongly magnetic atoms: Chromium
(Cr), Dysprosium (Dy), and Erbium (Er). All three have been Bose condensed and
displayed strong dipolar effects Cr ; Dy ; Er . The study of dipoles in
reduced geometries have begun with Cr, where it has been put into a one
dimensional lattice, forming a series of coupled q2D dipolar BECs layer .
Additionally, polar molecules have been loaded into q2D geometries to control
the rate of their chemical reaction rbk .
Correlations of ultracold dipolar gases have been studied with a perpendicular
polarization for weakly interacting gases near the roton instability klaw ;
sykes ; blair and strongly interacting gases, e.g. buchler . Using the
direction of the polarization axis as a means to control the collective
behavior and coherence properties, a q2D dipolar gas has been studied in the
many body dipolar systems at the mean field level nath ; aniso and for 2D
dipolar scattering tilt . Additional theories have investigated a tilted
polarization axis in a q2D dipolar fermi gas tilt2 .
In this work, we go beyond mean field theory and examine the correlations of a
trapped q2D DBEC at finite temperature with a partially tilted polarization
into the plane of motion. This situation leads to anisotropic coherence
properties, in which the gas maintains coherence along the direction of
polarization, but more rapidly loses coherence perpendicular to the
polarization direction. This anisotropy is evident in both the phase and
density-density coherence properties. This emerges because of a roton-like
mode in the excitation spectrum and is strongly anisotropic in character. We
consider the implications of this theory through interference experiments and
density fluctuations, both of which will exhibit anisotropic correlated
behavior. We also calculate the compressibility of the dipolar gas.
To study the finite temperature coherence properties, we employ the Hartree
Fock Bogoliubov method within the Popov approximation (HFBP) with nonlocal
interactions griffin ; CThfb . The HFB breaks the wavefunction into a
condensate and thermal component:
$\hat{\Psi}=[\sqrt{N_{0}}\phi_{0}(\vec{\rho})+\hat{\theta}(\vec{\rho})]$ where
we have replaced $\hat{a}_{0}\rightarrow\sqrt{N_{0}}$. We use the Bogoliubov
transformation:
$\hat{\theta}(\vec{\rho})=\sum_{\gamma}[u_{\gamma}(\vec{\rho})\hat{a}_{\gamma}e^{-i\omega_{\gamma}t}-v_{\gamma}^{*}(\vec{\rho})\hat{a}^{*}_{\gamma}e^{i\omega_{\gamma}t}]$
where $\hat{a}_{\gamma}$ ($\hat{a}^{*}_{\gamma}$) is the bosonic annihilation
(creation) operator for the $\gamma^{th}$ quasiparticle. The HFBP solves a
non-local, generalized Gross-Pitaevskii equation and a nonlocal Bogoliubov de
Gennes set of equations from which the eigenvalues and vectors
(quasiparticles) are self-consistently obtained CThfb .
To evaluate the interaction, we assume a dipole moment of the form:
$\vec{d}=d[\hat{x}\sin(\alpha)+\hat{z}\cos(\alpha)]$ where $\alpha$ is the
angle between $\hat{z}$ and $\hat{d}$, and we assume a single transverse
Gaussian wavefunction of width $l_{z}$ is occupied in the $z$ direction, where
$l_{i}=\sqrt{\hbar/\omega_{i}m}$ and $\omega_{i}$ is the trapping frequency.
This requires $\mu,T\ll\hbar\omega_{z}$, and this leads to a simple form of
the interaction. We use an isotropic trap ($x,y$) with $l_{z}/l_{\rho}=0.15$
($\omega_{z}/\omega_{\rho}$=44.4), and an interaction strength of
$g_{d}=0.025$ and $g=0$, where $g_{d}=d^{2}/\sqrt{2\pi}l_{z}$ and
$g=\sqrt{8\pi}a/l_{z}$ with $a$ being the 3D s-wave scattering length ($a\ll
l_{z}$). The polarization angle is set to $\alpha/\pi=0.25$, this is beyond
the critical angle. The critical angle is when the interaction is zero along
the $x$ axis, so $\sin(\alpha_{c})=1/\sqrt{3}$. For $\alpha>\alpha_{c}\sim
0.2\pi$ there is an attractive region to the interaction. This value of
$\alpha$ is picked to optimize the collapse with respect to particle number.
If it were steeper, the critical number would be less, and the depletion would
not be large enough to observe the desired behavior. In addition, if $\alpha$
were smaller then roton the mode would require many more particles to
collapse, and the collapses would be more 3D in nature. For $\alpha/\pi=0.25$,
the system is well described by the q2D formalism. For these parameters, the
trapping potential for Dy [Er] is $(\omega_{\rho},\omega_{z})/2\pi$= (12,533)
[(48,2133)] Hz. In a trapped ideal 2D gas, Bose condensation can occur trap ,
and the the critical temperature is at $T_{C}/\hbar\omega=\sqrt{6N}/\pi\sim
0.78\sqrt{N}$. This is for the pure 2D trapped gas, we only use it as a
reference temperature. Throughout this paper we work in trap units. It is
important to mention that the examples given here are weakly interacting; the
HFBP method cannot accurately handle strongly interacting dipolar gases
($\mu<10\hbar\omega_{\rho}$) CThfb . This criteria is from the deviation of
the Kohn mode, or slosh mode, from 1$\hbar\omega_{\rho}$. If the interaction
is too strong then the static thermal cloud alters the slosh mode of the
condensate; this is a numerical signature that the HFBP has broken down. This
restricted the work from entering the Thomas Fermi regime where
$l_{\rho}>\xi$, where $\xi$ is the healing length ($\sqrt{m\mu/\hbar^{2}}$).
Because of this, trapping effects are always important in this study.
Once $g_{d}$, $g$, $l_{z}$, $T$, $\alpha$ and the condensate number, $N_{0}$,
are selected, the HFBP calculation finds: the number of thermal atoms,
$\tilde{N}$; the chemical potential, $\mu$; the condensate wavefunction,
$\phi_{0}$; and the quasiparticle wavefunctions, $u_{\gamma}$ and
$v_{\gamma}$. From these wavefunctions, we construct the correlation
functions: the nonlocal thermal correlation function,
$\tilde{n}(\vec{\rho},\vec{\rho}^{\prime})=\sum_{\gamma}[u^{*}_{\gamma}(\vec{\rho})u_{\gamma}(\vec{\rho}^{\prime})+v^{*}_{\gamma}(\vec{\rho})v_{\gamma}(\vec{\rho}^{\prime})]N_{\gamma}+v^{*}_{\gamma}(\vec{\rho})v_{\gamma}(\vec{\rho}^{\prime})$
with $N_{\gamma}$ being the Bose-Einstein occupation, and the condensate
correlation function, $n_{0}(\vec{\rho},\vec{\rho}^{\prime})$, given by
$N_{0}\phi^{*}_{0}(x)\phi_{0}(x^{\prime})$. The total local density is
$n(\vec{\rho})=n_{0}(\vec{\rho})+\tilde{n}(\vec{\rho})$ where e.g.
$n_{0}(\vec{\rho})=n_{0}(\vec{\rho},\vec{\rho})$. The HFBP method uses these
correlation functions in the self-consistent calculation, and they can be
related to the more common $g_{1}$ and $g_{2}$ correlation functions phase :
$\displaystyle g_{1}(\vec{\rho};\vec{\rho}^{\prime})$
$\displaystyle={\langle\Psi^{*}(\vec{\rho})\Psi(\vec{\rho}^{\prime})\rangle\over\sqrt{n(\vec{\rho})n(\vec{\rho}^{\prime})}}={n_{0}(\vec{\rho},\vec{\rho}^{\prime})+\tilde{n}(\vec{\rho},\vec{\rho}^{\prime})\over\sqrt{n(\vec{\rho})n(\vec{\rho}^{\prime})}},$
(1) $\displaystyle g_{2}(\vec{\rho};\vec{\rho}^{\prime})$
$\displaystyle={\langle\Psi^{*}(\vec{\rho})\Psi^{*}(\vec{\rho}^{\prime})\Psi(\vec{\rho})\Psi(\vec{\rho}^{\prime})\rangle\over{n(\vec{\rho})n(\vec{\rho}^{\prime})}}$
$\displaystyle=1+g_{1}(\vec{\rho},\vec{\rho}^{\prime})^{2}-{n_{0}(\vec{\rho},\vec{\rho}^{\prime})^{2}\over
n(\vec{\rho})n(\vec{\rho}^{\prime})}.$
$g_{1}$ details phase correlations, for example it determines the fringe
contrast if a BEC were split and interfered. $g_{2}$ is the density-density
correlation function, and is related to density fluctuations and the static
structure factor zam .
We study a system with a temperature of $T/T_{C}=0.5$. At this temperature,
the condensate fraction is $N_{0}/N>0.57$ for all $N$ considered. This is well
within the region of validity for the HFBP. This temperature thermally
populates the quasiparticles, so that the anisotropy in the coherence
properties is significant. Furthermore, this temperature makes the density
more isotropic. Values of $T/T_{C}$ between $0.4$ and $0.6$ produces similar
results. Raising the temperature would lead to populating more thermal modes,
and would make correlations isotropic, reflecting the nature of the trap
($T/T_{C}\sim 1$). For lower temperatures, the correlations are determined by
the condensate’s properties alone, as the depletion and thermal population
become negligible. A pure condensate has total phase coherence, it is the
thermal fraction that leads to reduced coherence.
We now look at the correlation functions, $g_{1}(\vec{\rho};0)$ and
$g_{2}(\vec{\rho};0)$. These quantify the coherence of the system from the
origin to $\vec{\rho}$. To illustrate the anisotropic coherence of the gas, we
plot them along the polarization axis and in the perpendicular direction in
figure 1.
Figure 1: (a) The total density, (b) $g_{1}(x,0;0,0)$ (solid line) and
$g_{1}(0,y;0,0)$ (dashed), and (c) $g_{2}(x,0;0,0)$ (solid line) and
$g_{2}(0,y;0,0)$ (dashed) at $T/T_{C}$=0.50 and for $N$ =2000 (blue), 3000
(red), and 4000 (black) from top to bottom. The total density is isotropic in
the center and yet the correlation function is significantly anisotropic at
short range.
Fig. 1 (a) shows the total density and Fig. 1 (b) shows $g_{1}(\vec{\rho};0)$
along the $x$ (solid line) and $y$ (dashed) axes. The important feature is
that $g_{1}$ quickly decreases in the $y$ direction, but only steadily
decreases its value in the $x$ direction or polarization direction. We have
shown the system for three different number of total particles 2000 (blue),
3000 (red), and 4000 (black), from top to bottom at the temperature
$T/T_{C}=0.5$. Varying the number is like varying the interaction strength.
The density is isotropic near the center of the trap, even when the $g_{1}$ is
anisotropic. Near the edge of the trap the total density becomes anisotropic
and is more extended in the $x$ axis.
A recent Bragg spectroscopy experiment observed an anisotropic speed of sound
for a 3D dipolar gas bragg . A similar experiment for a q2D DBEC could be used
to probe these predicted anisotropic coherence properties. If a DBEC were
split into two equal clouds such that one is moved by $\vec{a}$ with a
relative phase of $S(\vec{\rho})$, then the total density would be
$n(\vec{\rho})=0.5(n(\vec{\rho})+n(\vec{\rho}+a))+\sqrt{n(\vec{\rho})n(\vec{\rho}+\vec{a})}g_{1}(\vec{\rho},\vec{\rho}+\vec{a})\cos[S(\vec{\rho})]$.
The fringe contrast will depend on the angle between $\vec{a}$ and $\vec{d}$
because of the anisotropy in $g_{1}(\vec{\rho},\vec{\rho}+\vec{a})$. To
maximize the difference in fringe contrast, the phase difference should be
$\pi/2$ when $\vec{a}$ is near 3/4 $l_{\rho}$. This signal is only a few
percent and would be challenging to measure.
We now move on to the density-density correlation function,
$g_{2}(\vec{\rho};0)$. In Fig. 1 (c), $g_{2}$ this is plotted along the $x$
(solid line) and $y$ (dashed) axis. The results are similar to those of
$g_{1}$, the extent of the coherence is further in the $x$ direction than the
$y$. The anisotropy is more significant in this correlation function. The fact
that the $g_{2}$ goes below one is a trap effect and is most significant when
correlating with the trap center manz ; imam . This correlation function is
important for density fluctuations, this will be discussed below.
To understand the nature of the anisotropy we look at the Bogoliubov de Gennes
excitation spectrum as a function of particle number, this is shown in Fig. 2
(a). The spectrum goes soft or an eigenvalue becomes complex implying the
system is unstable when there are more than 4850 particles. Additionally, we
show contour plots of the wavefunctions of the two modes which go soft
($u_{\gamma}$) in Fig. 2 (b,c), and the dashed black lines are $\sqrt{n_{0}}$.
The contours at 0.25, 0.5, and 0.75 of the maximum values of the individual
wavefunctions, and for $u_{\gamma}$ the shaded regions signify a region less
than zero. These quasiparticle modes start with an energy at or above
$4\hbar\omega_{\rho}$. Then as the particle number is increased, their energy
decreases. In a homogeneous q2D DBEC, the roton is a local minimum in the
excitation spectrum, which occurs near the $k_{\rho}l_{z}\sim 1$ roton1 . If
the spectrum goes soft, the system locally collapses into plane waves which
form dense stripes along the polarization axis.
The Bogoliubov de Gennes excitation spectrum has been analyzed in detail in
Ref. CThfb for a dipolar gas with $\alpha=0$ or a cylindrically symmetric
system. In that case there are degenerate modes with azimuthal symmetry $\pm
m$. In the present case this degeneracy has been broken by the tilted
polarization axis, leading to slightly different energies depending on the
excitation shape relative to the $x$ axis. This leads to band like structures
forming in Fig. 2 (a).
The mode in 2 (b) makes a significant contribution to the correlation function
for two reasons: first, it is a low energy mode, and is more thermally
populated. Second, the correlation function, $g_{i}(\vec{\rho}:0)$, has a
factor of $u_{\gamma}(0)$ and $v_{\gamma}(0)$ in it. Only a few low energy
modes have a significant contribution at the origin. The breathing mode just
below 2$\hbar\omega$ and the roton mode in 2 (b) are two low energy modes with
a maximum in the center of the trap. Other low energy modes have small or zero
amplitude in the center of the cloud because symmetry.
In the trapped case, the finite extent of the gas alters the nature of the
“roton” mode roton ; Wilson ; Wilson2 . For $\alpha=0$, the gas become 3D in
nature, $\mu\sim\hbar\omega_{z}$ blair . When the gas becomes unstable, it
locally collapses and forms regions of high density. A good demonstration of
this is given in Ref. parker . For $\alpha/\pi\sim 0.25$, the stability of the
system displays an interplay of the interaction and the trap geometry, while
maintaining $\mu\ll\hbar\omega_{z}$. The collapse is local and forms a series
density regions parallel to the polarization. In Fig. 2 (b,c), we see this
character in the quasiparticles, which have an extended amplitude along the
polarization axis and a series of nodes in the perpendicular direction. It is
this character of the quasiparticles that leads to anisotropic character to
the correlations. The nodes in the roton modes lead to destructive
interference and reduce correlation in that direction.
Figure 2: (a) The Bogoliubov de Gennes spectrum is shown as a function of
particle number for $g_{d}=0.025$ for $\alpha/\pi=0.25$. (b,c) A contour plot
of the roton quasiparticle modes ($u_{\gamma}$) which go soft for a dipolar
system with tilted polarization axis are also shown. The dashed black line is
$\sqrt{n_{0}}$ and for $u_{\gamma}$ the shaded regions are less than zero. The
contours at 0.25, 0.5, and 0.75 the maximum values of the individual
wavefunctions.
To further study the implications of the anisotropic correlations, we consider
the density fluctuations of the gas by calculating the compressibility:
$kT\kappa(\vec{\rho})=$ $\langle\int d\vec{\rho}^{\prime}\delta
n(\vec{\rho})\delta n(\vec{\rho}^{\prime})\rangle$ =$\int
d\vec{\rho}^{\prime}n(\vec{\rho})n(\vec{\rho}^{\prime})g_{2}(\vec{\rho};\vec{\rho}^{\prime})-n(\vec{\rho})(N-1)$
where $\delta
n(\vec{\rho})=\hat{\Psi}^{*}(\vec{\rho})\hat{\Psi}(\vec{\rho})-n(\vec{\rho})$
and this is the density fluctuations of the system. The compressibility and
number fluctuations have been measured in BEC experiments chicago ; jila . We
present the compressibility for a trapped dipolar gas in Fig. 3 (b) at
$T/T_{0}=0.5$ for 2000 (blue), 3000 (red), and 4000 (black) in the $x$ (solid)
and $y$ (dashed). These results show that the density fluctuations roughly
follow the anisotropic density profile. Another way to state this is: that
depending on the local value of the density, the compressibility is determined
almost uniquely.
Figure 3: (a) The total density (b) the compressibility,
$kT\kappa(\vec{\rho})$, for the dipolar gas along the $x$ (solid) and $y$
(dashed) axis for particles number 2000 (blue), 3000 (red), and 4000 (black).
In this work, we found that the correlations are anisotropic in a q2D DBEC
with a polarization axis tilted into the plane of motion. We studied the DBEC
with the HFBP at at temperature of $T/T_{C}\sim 0.5$. At this temperature,
quasiparticles are thermally populated and total density is nearly isotropic
in the central region of the trap. We computed the $g_{1}(\vec{\rho};0)$ and
$g_{2}(\vec{\rho};0)$ correlations functions, which have anisotropic character
in the center of the gas even though the density is isotropic in the central
region of the DBEC. This has measurable implications for interference
properties and density fluctuations of the gas. We found that the anisotropic
correlations occur due to roton like quasiparticle modes that can lead to
local collapse along the polarization axis.
An immediate extension of this work is to study the static structure factor
such as was recently done in Ref. blair at zero temperature. An important
difference is that the static structure will become anisotropic, i.e.
$S(k_{x},k_{y})$, and with present methods, it could be done at finite
temperature stemp . This also presents a promising method to observe the
anisotropic coherences predicted here.
Future work will be to investigate the impact of correlations at higher
temperatures and strongly interactions where they might impact the BKT
transition BKT ; BKTrev ; dBKT . This will require an improvement to the
method mora or new approach altogether. Either way, it will be intriguing to
study the impact of the anisotropic dipolar interaction on vortex correlations
near the BKT transition and the time dependence of such a gas. The present
work was not able to measure the decay of the correlation functions due to the
relatively weak interaction strength or small number the numerical method
could handle. If the next method can probe stronger interactions, we would
hope to observe a different character in the decay of the correlation
functions along and perpendicular to the polarization axis.
Acknowledgments, the author is grateful for support from the Advanced
Simulation and Computing Program (ASC) and LANL which is operated by LANS, LLC
for the NNSA of the U.S. DOE under Contract No. DE-AC52-06NA25396. The author
is grateful for discussion with A. Sykes, R. Behunin, and L. A. Collins.
## References
* (1) T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999); V. M. Loktev, R. M. Quick, S. G. Sharapov, Physics Reports, 349, 1 (2001).
* (2) M. R. Andrews et al., Science 275, 553 (1997).
* (3) For example: A. Ottl et al., Phys. Rev. Lett. 95 090404 (2005); S. S. Hodgman et al., Science 331, 1046 (2011); S. Foelling et al., Nature 434, 481 (2005).
* (4) Berezinskii, V. L. Sov. Phys. JETP 34, 610 (1972). L. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1973);
* (5) Z. Hadzibabic and J. Dalibard, Riv. Nuo. Cime., 34, 389 (2011).
* (6) Z. Hadzibabic et al., Nature 441, 1118 (2006).
* (7) T. Yefsah et al., Phys. Rev. Lett. 107, 130401 (2011).
* (8) C.L. Hung et al., Nature 470, 236 (2011); N. Gemelke et al., Nature 460, 995 (2009).
* (9) S. Tung et al., Phys. Rev. Lett. 105, 230408 (2010).
* (10) J. Armijo, Phys. Rev. Lett. 108, 225306 (2012).
* (11) S. Manz et al., Phys. Rev. A 81, 031610 (2010).
* (12) A. Imambekov et al., Phys. Rev. A 80, 033612 (2009).
* (13) T. Kinoshita, T. Wenger, and D. S. Weiss, Phys. Rev. Lett. 95, 190406 (2005).
* (14) J. Armijo et al., Phys. Rev. Lett. 105, 230402 (2010).
* (15) Th. Lahaye et al., Nature 448, 672 (2007); Th. Lahaye et al., Nature Physics 4, 218 (2008); Th. Lahaye et al., Phys. Rev. Lett 101, 080401 (2008).
* (16) K. Aikawa et al., Phys. Rev. Lett. 108, 210401 (2012).
* (17) M. Lu et al.,, Phys. Rev. Lett., 107, 109401 (2011); M. Lu et al.,, Phys. Rev. Lett., 108, 215301 (2012).
* (18) S. Meuller et al., Phys. Rev. A 84, 053601 (2011).
* (19) M. H. G. de Miranda et al., Nat. Phys. 7, 502 (2010).
* (20) M. Klawunn et al. Phys. Rev. A 84, 033612 (2012).
* (21) A. Sykes and C. Ticknor, arXiv1205.1350.
* (22) P. B. Blakie, D. Baillie, and R. N. Bisset, Phys Rev. A, 86 021604(R) (2012).
* (23) H. P. Buechler et al., Phys Rev. Lett 98, 060404 (2007);G. E. Astrakharchick et al., Phys Rev. Lett 98, 060405 (2007).
* (24) R. Nath, P. Pedri, and L. Santos, Phys. Rev. Lett. 102 050401 (2009).
* (25) C. Ticknor, R. M. Wilson, J. L. Bohn, Phys. Rev. Lett. 106, 065301 (2011).
* (26) C. Ticknor, Phys. Rev A 84, 032702 (2011).
* (27) M. M. Parish and F. M. Marchetti, Phys. Rev. Lett. 108, 145304 (2012); L. M. Sieberer and M. A. Baranov, Phys. Rev. A 84, 063633 (2012).
* (28) A. Griffin, Phys. Rev. B, 53, 9341 (1996).
* (29) C. Ticknor, Phys. Rev. A 85, 033629 (2012).
* (30) V. Bagnato and D. Kleppner, Phys. Rev. A 44, 7439 (1991).
* (31) M. Naraschewski and R. J. Glauber, Phys. Rev. A 59, 4595 (1999).
* (32) F. Zambelli et al., Phys. Rev. A 61 063608 (2000).
* (33) G. Bismut et al., arXiv 1205.6305.
* (34) L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 90, 250403 (2003).
* (35) S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Phys. Rev. Lett. 98, 030406 (2007).
* (36) R. Wilson, S. Ronen, and J.L. Bohn Phys. Rev. Lett. 104, 094501 (2010).
* (37) R. M. Wilson, S. Ronen, and J. L. Bohn, Phys. Rev. Lett. 100, 245302 (2008).
* (38) N. G. Parker et al. Phys. Rev. A 79, 013617 (2009).
* (39) A. Filinov et al. Phys. Rev. Lett. 105, 070401 (2010).
* (40) C. Mora and Y. Castin, Phys. Rev. A 67, 053615 (2003).
* (41) A. Csordas, R. Graham, and P. Szepfalusy, Phys. Rev. A 54, R2543 (1996).
|
arxiv-papers
| 2012-09-10T19:49:01 |
2024-09-04T02:49:34.934395
|
{
"license": "Public Domain",
"authors": "Christopher Ticknor",
"submitter": "Chris Ticknor",
"url": "https://arxiv.org/abs/1209.2108"
}
|
1209.2137
|
D. Lemire and L. BoytsovDecoding billions of integers per second through
vectorization
LICEF Research Center, TELUQ, Université du Québec, 5800 Saint-Denis, Montreal
(Quebec) H2S 3L5 Canada.
Natural Sciences and Engineering Research Council of Canada261437
# Decoding billions of integers per second through vectorization
D. Lemire 1 L. Boytsov 2 11affiliationmark: LICEF Research Center, TELUQ,
Montreal, QC, Canada22affiliationmark: Carnegie Mellon University,
Pittsburgh, Pennsylvania, USA
###### Abstract
In many important applications—such as search engines and relational database
systems—data is stored in the form of arrays of integers. Encoding and, most
importantly, decoding of these arrays consumes considerable CPU time.
Therefore, substantial effort has been made to reduce costs associated with
compression and decompression. In particular, researchers have exploited the
superscalar nature of modern processors and SIMD instructions. Nevertheless,
we introduce a novel vectorized scheme called SIMD-BP128⋆ that improves over
previously proposed vectorized approaches. It is nearly twice as fast as the
previously fastest schemes on desktop processors (varint-G8IU and PFOR). At
the same time, SIMD-BP128⋆ saves up to 2 bits per integer. For even better
compression, we propose another new vectorized scheme (SIMD-FastPFOR) that has
a compression ratio within 10% of a state-of-the-art scheme (Simple-8b) while
being two times faster during decoding.
###### keywords:
performance; measurement; index compression; vector processing
## 1 Introduction
Computer memory is a hierarchy of storage devices that range from slow and
inexpensive (disk or tape) to fast but expensive (registers or CPU cache). In
many situations, application performance is inhibited by access to slower
storage devices, at lower levels of the hierarchy. Previously, only disks and
tapes were considered to be slow devices. Consequently, application developers
tended to optimize only disk and/or tape I/O. Nowadays, CPUs have become so
fast that access to main memory is a limiting factor for many workloads [1, 2,
3, 4, 5]: data compression can significantly improve query performance by
reducing the main-memory bandwidth requirements.
Data compression helps to load and keep more of the data into a faster
storage. Hence, high speed compression schemes can improve the performances of
database systems [6, 7, 8] and text retrieval engines [9, 10, 11, 12, 13].
We focus on compression techniques for 32-bit integer sequences. It is best if
most of the integers are small, because we can save space by representing
small integers more compactly, i.e., using short codes. Assume, for example,
that none of the values is larger than 255. Then we can encode each integer
using one byte, thus, achieving a compression ratio of 4: an integer uses 4
bytes in the uncompressed format.
In relational database systems, column values are transformed into integer
values by dictionary coding [14, 15, 16, 17, 18]. To improve compressibility,
we may map the most frequent values to the smallest integers [19]. In text
retrieval systems, word occurrences are commonly represented by sorted lists
of integer document identifiers, also known as posting lists. These
identifiers are converted to small integer numbers through data differencing.
Other database indexes can also be stored similarly [20].
array$\to$ differential coding (e.g., $\delta_{i}=x_{i}-x_{i-1}$) $\to$
compression (e.g., SIMD-BP128) $\to\mathrm{compressed}$
(a) encoding
$\mathrm{compressed}\to$ decompression (e.g., SIMD-BP128) $\to$ differential
decoding (e.g., $x_{i}=\delta_{i}+x_{i-1}$) $\to\mathrm{array}$
(b) decoding
Figure 1: Encoding and decoding of integer arrays using differential coding
and an integer compression algorithm
A mainstream approach to data differencing is differential coding (see Fig.
1). Instead of storing the original array of sorted integers
($x_{1},x_{2},\ldots$ with $x_{i}\leq x_{i+1}$ for all $i$), we keep only the
difference between successive elements together with the initial value:
($x_{1},\delta_{2}=x_{2}-x_{1},\delta_{3}=x_{3}-x_{2},\ldots$). The
differences (or deltas) are non-negative integers that are typically much
smaller than the original integers. Therefore, they can be compressed more
efficiently. We can then reconstruct the original arrays by computing prefix
sums ($x_{j}=x_{1}+\sum_{i=2}^{j}\delta_{j}$). Differential coding is also
known as delta coding [18, 21, 22], not to be confused with Elias delta coding
(§ 2.3). A possible downside of differential coding is that random access to
an integer located at a given index may require summing up several deltas: if
needed, we can alleviate this problem by partitioning large arrays into
smaller ones.
An engineer might be tempted to compress the result using generic compression
tools such as LZO, Google Snappy, FastLZ, LZ4 or gzip. Yet this might be ill-
advised. Our fastest schemes are an order of magnitude faster than a fast
generic library like Snappy, while compressing better (see § 6.5).
Instead, it might be preferable to compress these arrays of integers using
specialized schemes based on Single-Instruction, Multiple-Data (SIMD)
operations. Stepanov et al. [12] reported that their SIMD-based varint-G8IU
algorithm outperformed the classic variable byte coding method (see § 2.4) by
300%. They also showed that use of SIMD instructions allows one to improve
performance of decoding algorithms by more than 50%.
In Table 1, we report the speeds of the fastest decoding algorithms reported
in the literature on desktop processors. These numbers cannot be directly
compared since hardware, compilers, benchmarking methodology, and data sets
differ. However, one can gather that varint-G8IU—which can be viewed as an
improvement on the Group Varint Encoding [13] (varint-GB) used by Google—is,
probably, the fastest method (except for our new schemes) in the literature.
According to our own experimental evaluation (see Tables 4, 5 and Fig. 12),
varint-G8IU is indeed one of the most efficient methods, but there are
previously published schemes that offer similar or even slightly better
performance such as PFOR [23]. We, in turn, were able to further surpass the
decoding speed of varint-G8IU by a factor of two while improving the
compression ratio.
We report our own speed in a conservative manner: (1) our timings are based on
the wall-clock time and not the commonly used CPU time, (2) our timings
incorporate all of the decoding operations including the computation of the
prefix sum whereas this is sometimes omitted by other authors [24], (3) we
report a speed of 2300 million integers per second (mis) achievable for
realistic data sets, while higher speed is possible (e.g., we report a speed
of 2500 mis on some realistic data and 2800 mis on some synthetic data).
Another observation we can make from Table 1 is that not all authors have
chosen to make explicit use of SIMD instructions. While there are has been
several variations on PFOR [23] such as NewPFD and OptPFD [10], we introduce
for the first time a variation designed to exploit the vectorization
instructions available since the introduction of the Pentium 4 and the
Streaming SIMD Extensions 2 (henceforth SSE2). Our experimental results
indicate that such vectorization is desirable: our SIMD-FastPFOR⋆ scheme
surpasses the decoding speed of PFOR by at least 30% while offering a superior
compression ratio (10%). In some instances, SIMD-FastPFOR⋆ is twice as fast as
the original PFOR.
For most schemes, the prefix sum computation is so fast as to represent 20% or
less of the running time. However, because our novel schemes are much faster,
the prefix sum can account for the majority of the running time.
Hence, we had to experiment with faster alternatives. We find that a
vectorized prefix sum using SIMD instructions can be twice as fast. Without
vectorized differential coding, we were unable to reach a speed of two billion
integers per second.
In a sense, the speed gains we have achieved are a direct application of
advanced hardware instructions to the problem of integer coding (specifically
SSE2 introduced in 2001). Nevertheless, it is instructive to show how this is
done, and to quantify the benefits that accrue.
Table 1: Recent best decoding speeds in millions of 32-bit integers per second (mis) reported by authors for integer compression on realistic data. We indicate whether the authors made explicit use of SIMD instructions. Results are not directly comparable but they illustrate the evolution of performance. | Speed | Cycles/int | Fastest scheme | Processor | SIMD
---|---|---|---|---|---
this paper | 2300 | 1.5 | SIMD-BP128⋆ | Core i7 (3.4 GHz) | SSE2
Stepanov et al. (2011) [12] | 1512 | 2.2 | varint-G8IU | Xeon (3.3 GHz) | SSSE3
Anh and Moffat (2010) [25] | 1030 | 2.3 | binary packing | Xeon (2.33 GHz) | no
Silvestri and Venturini (2010) [24] | 835 | — | VSEncoding | Xeon | no
Yan et al. (2009) [10] | 1120 | 2.4 | NewPFD | Core 2 (2.66 GHz) | no
Zhang et al. (2008) [26] | 890 | 3.6 | PFOR2008 | Pentium 4 (3.2 GHz) | no
Zukowski et al. (2006) [23, § 5] | 1024 | 2.9 | PFOR | Pentium 4 (3 GHz) | no
## 2 Related work
Some of the earliest integer compression techniques are Golomb coding [27],
Rice coding [28], as well as Elias gamma and delta coding [29]. In recent
years, several faster techniques have been added such as the Simple family,
binary packing, and patched coding. We briefly review them.
Because we work with unsigned integers, we make use of two representations:
binary and unary. In both systems numbers are represented using only two
digits: 0 and 1. The binary notation is a standard positional base-2 system
(e.g., $1\to 1$, $2\to 10$, $3\to 11$). Given a positive integer $x$, the
binary notation requires $\lceil\log_{2}(x+1)\rceil$ bits. Computers commonly
store unsigned integers in the binary notation using a fixed number of bits by
adding leading zeros: e.g., $3$ is written as $00000011$ using 8 bits. In
unary notation, we represent a number $x$ as a sequence of $x-1$ digits 0
followed by the digit 1 (e.g., $1\to 1$, $2\to 01$, $3\to 001$) [30]. If the
number $x$ can be zero, we can store $x+1$ instead.
### 2.1 Golomb and Rice coding
In Golomb coding [27], given a fixed parameter $b$ and a positive integer $v$
to be compressed, the quotient $\lfloor v/b\rfloor$ is coded in unary. The
remainder $r=v\bmod b$ is stored using the usual binary notation with no more
than $\lceil\log_{2}b\rceil$ bits. If $v$ can be zero, we can code $v+1$
instead. When $b$ is chosen to be a power of two, the resulting algorithm is
called Rice coding [28]. The parameter $b$ can be chosen optimally by assuming
some that the integers follow a known distribution [27].
Unfortunately, Golomb and Rice coding are much slower than a simple scheme
such as Variable Byte [9, 10, 31] (see § 2.4) which, itself, falls short of
our goal of decoding billions of integers per second (see § 6.4–6.5).
### 2.2 Interpolative coding
If speed is not an issue but high compression over sorted arrays is desired,
interpolative coding [32] might be appealing. In this scheme, we first store
the lowest and the highest value, $x_{1}$ and $x_{n}$, e.g., in a uncompressed
form. Then a value in-between is stored in a binary form, using the fact this
value must be in the range $(x_{1},x_{n})$. For example, if $x_{1}=16$ and
$x_{n}=31$, we know that for any value $x$ in between, the difference
$x-x_{1}$ is from 0 to 15. Hence, we can encode this difference using only 4
bits. The technique is then repeated recursively. Unfortunately, it is slower
than Golomb coding [9, 10].
### 2.3 Elias gamma and delta coding
An Elias gamma code [29, 30, 33] consists of two parts. The first part encodes
in unary notation the minimum number of bits necessary to store the positive
integer in the binary notation ($\lceil\log_{2}(x+1)\rceil$). The second part
represents the integer in binary notation less the most significant digit. If
the integer is equal to one, the second part is empty (e.g., $1\to 1$, $2\to
01\,0$, $3\to 01\,1$, $4\to 001\,00$, $5\to 001\,01$). If integers can be
zero, we can code their values incremented by one.
As numbers become large, gamma codes become inefficient. For better
compression, Elias delta codes encode the first part (the number
$\lceil\log_{2}(x+1)\rceil$) using the Elias gamma code, while the second part
is coded in the binary notation. For example, to code the number 8 using the
Elias delta code, we must first store $4=\lceil\log_{2}(8+1)\rceil$ as a gamma
code ($001\,00$) and then we can store all but the most significant bit of the
number 8 in the binary notation ($000$). The net result is $001\,00$ $000$.
However, Variable Byte is twice as fast as Elias gamma and delta coding [24].
Hence, like Golomb coding, gamma coding falls short of our objective of
compressing billions of integers per second.
#### 2.3.1 $k$-gamma
Schlegel et al. [34] proposed a version of Elias gamma coding better suited to
current processors. To ease vectorization, the data is stored in blocks of $k$
integers using the same number of bits where $k\in\\{2,4\\}$. (This approach
is similar to binary packing described in § 2.6.) As with regular gamma
coding, we use unary codes to store this number of bits though we only have
one such number for $k$ integers.
The binary part of the gamma codes are stored using the same vectorized layout
as in § 4 (known as vertical or interleaved). During decompression, we decode
integer in groups of $k$ integers. For each group we first retrieve the binary
length from a gamma code. Then, we decode group elements using a sequence of
mask-and-shift operations similar to the fast bit unpacking technique
described in § 4. This step does not require branching.
Schlegel et al. report best decoding speeds of $\approx$550 mis ($\approx$2100
MB/s) on synthetic data using an Intel Core i7-920 processor (2.67 GHz). These
results fall short of our objective to compress billions of integers per
second.
### 2.4 Variable Byte and byte-oriented encodings
Variable Byte is a popular technique [35] that is known under many names
(v-byte, variable-byte [36], var-byte, vbyte [30], varint, VInt, VB [12] or
Escaping [31]). To our knowledge, it was first described by Thiel and Heaps in
1972 [37]. Variable Byte codes the data in units of bytes: it uses the lower-
order seven bits to store the data, while the eighth bit is used as an
implicit indicator of a code length. Namely, the eighth bit is equal to 1 only
for the last byte of a sequence that encodes an integer. For example:
* •
Integers in $[0,2^{7})$ are written using one byte: The first 7 bits are used
to store the binary representation of the integer and the eighth bit is set to
1.
* •
Integers in $[2^{7},2^{14})$ are written using two bytes, the eighth bit of
the first byte is set to 0 whereas the eighth bit of the second byte is set to
1. The remaining 14 bits are used to store the binary representation of the
integer.
For a concrete example, consider the number 200. It is written as 11001000 in
the binary notation. Variable Byte would code it using 16 bits as 10000001
01001000.
When decoding, bytes are read one after the other: we discard the eighth bit
if it is zero, and we output a new integer whenever the eighth bit is one.
Though Variable Byte rarely compresses data optimally, it is reasonably
efficient. In our tests, Variable Byte encodes data three times faster than
most alternatives. Moreover, when the data is not highly compressible, it can
match the compression ratios of more parsimonious schemes.
Stepanov et al. [12] generalize Variable Byte into a family of byte-oriented
encodings. Their main characteristic is that each encoded byte contains bits
from only one integer. However, whereas Variable Byte uses one bit per byte as
descriptor, alternative schemes can use other arrangements. For example,
varint-G8IU [12] and Group Varint [13] (henceforth varint-GB) regroup all
descriptors in a single byte. Such alternative layouts make easier the
simultaneous decoding of several integers. A similar approach to placing
descriptors in a single control word was used to accelerate a variant of the
Lempel-Ziv algorithm [38].
For example, varint-GB uses a single byte to describe 4 integers, dedicating 2
bits per integer. The scheme is better explained by an example. Suppose that
we want to store the integers $2^{15}$, $2^{23}$, $2^{7}$, and $2^{31}$. In
the usual binary notation, we would use 2, 3, 1 and 4 bytes, respectively. We
can store the sequence as 2, 3, 1, 4 as 1, 2, 0, 3 if we assume that each
number is encoded using a non-zero number of bytes. Each one of these 4
integers can be written using 2 bits (as they are in {0,1,2,3}). We can pack
them into a single byte containing the bits 01,10,00, and 11. Following this
byte, we write the integer values using $2+3+1+4=10$ bytes.
Whereas varint-GB codes a fixed number of integers (4) using a single
descriptor, varint-G8IU uses a single descriptor for a group of 8 bytes, which
represent compressed integers. Each 8-byte group may store from 2 to 8
integers. A single-byte descriptor is placed immediately before this 8-byte
group. Each bit in the descriptor represents a single data byte. Whenever a
descriptor bit is set to 0, then the corresponding byte is the end of an
integer. This is symmetrical to the Variable Byte scheme described above,
where the descriptor bit value 1 denotes the last byte of an integer code.
In the example we used for varint-GB, we could only store the first 3 integers
($2^{15}$, $2^{23}$, $2^{7}$) into a single 8-byte group, because storing all
4 integers would require 10 bytes. These integers use 2, 3, and 1 bytes,
respectively, whereas the descriptor byte is equal to 11001101 (in the binary
notation). The first two bits (01) of the descriptor tell us that the first
integer uses 2 bytes. The next three bits (011) indicate that the second
integer requires 3 bytes. Because the third integer uses a single byte, the
next (sixth) bit of the descriptor would be 0. In this model, the last two
bytes cannot be used and, thus, we would set the last two bits to 1.
Figure 2: Example of simultaneous decoding of 3 integers in the scheme
varint-G8IU using the shuffle instruction. The integers $2^{15}$, $2^{23}$,
$2^{7}$ are packed into the 8-byte block with 2 bytes being unused. Byte
values are given by _hexadecimal_ numbers. The target 16-byte buffer bytes are
either copied from the source 16-byte buffer or are filled with zeros. Arrows
indicate which bytes of the source buffer are copied to the target buffer as
well as their location in the source and target buffers.
On most recent x86 processors, integers packed with varint-G8IU can be
efficiently decoded using the SSSE3 (Supplemental Streaming SIMD Extensions 3)
_shuffle_ instruction: pshufb. This assembly operation selectively copies byte
elements of a 16-element vector to specified locations of the target
16-element buffer and replaces selected elements with zeros.
The name “shuffle” is a misnomer, because certain source bytes can be omitted,
while others may be copied multiple times to a number of different locations.
The operation takes two 16 element vectors (of $16\times 8=128$ bits each):
the first vector contains the bytes to be shuffled into an output vector
whereas the second vector serves as a _shuffle mask_. Each byte in the shuffle
mask determines which value will go in the corresponding location in the
output vector. If the last bit is set (that is, if the value of the byte is
larger than 127), the target byte is zeroed. For example, if the shuffle mask
contains the byte values $127,127,\ldots,127$, then the output vector will
contain only zeros. Otherwise, the first 4 bits of the $i$th mask element
determine the index of the byte that should be copied to the target byte $i$.
For example, if the shuffle mask contains the byte values $0,1,2,\ldots,15$,
then the bytes are simply copied in their original locations.
In Fig. 2, we illustrate one step of the decoding algorithm for varint-G8IU.
We assume that the descriptor byte, which encodes the 3 numbers of bytes (2,
3, 1) required to store the 3 integers ($2^{15}$, $2^{23}$, $2^{7}$), is
already retrieved. The value of the descriptor byte was used to obtain a
proper shuffle mask for pshufb. This mask defines a hardcoded sequence of
operations that copy bytes from the source to the target buffer or fill
selected bytes of the target buffer with zeros. All these byte operations are
carried out in parallel in the following manner (byte numeration starts from
_zero_):
* •
The first integer uses 2 bytes, which are both copied to bytes 0–1 of the
target buffer Bytes 2–3 of the target buffer are zeroed.
* •
Likewise, we copy bytes 2–4 of the source buffer to bytes 4–6 of the target
buffer. Byte 7 of the target buffer is zeroed.
* •
The last integer uses only one byte 5: we copy the value of this byte to byte
8 and zero bytes 9–11.
* •
The bytes 12–15 of the target buffer are currently unused and will be filled
out by subsequent decoding steps. In the current step, we may fill them with
arbitrary values, e.g., zeros.
We do not know whether Google implemented varint-GB using SIMD instructions
[13]. However, Schlegel et al. [34] and Popov [11] described the application
of the pshufb instruction to accelerate decoding of a varint-GB scheme (which
Schlegel et al. called _4-wise null suppression_).
Stepanov et al. [12] found varint-G8IU to compress slightly better than a
SIMD-based varint-GB while being up to 20% faster. Compared to the common
Variable Byte, varint-G8IU had a slightly worse compression ratio (up to 10%),
but it is 2–3 times faster.
### 2.5 The Simple family
Whereas Variable Byte takes a fixed input length (a single integer) and
produces a variable-length output (1, 2, 3 or more bytes), at each step the
Simple family outputs a fixed number of bits, but processes a variable number
of integers, similar to varint-G8IU. However, unlike varint-G8IU, schemes from
the Simple family are not byte-oriented. Therefore, they may fare better on
highly compressible arrays (e.g., they could compress a sequence of numbers in
$\\{0,1\\}$ to $\approx$1 bit/int).
The most competitive Simple scheme on 64-bit processors is Simple-8b [25]. It
outputs 64-bit words. The first 4 bits of every 64-bit word is a selector that
indicates an encoding mode. The remaining 60 bits are employed to keep data.
Each integer is stored using the same number of bits $b$. Simple-8b has 2
schemes to encode long sequences of zeros and 14 schemes to encode positive
integers. For example:
* •
Selector values 0 or 1 represent sequences containing 240 and 120 zeros,
respectively. In this instance the 60 data bits are ignored.
* •
The selector value 2 corresponds to $b=1$. This allows us to store 60 integers
having values in {0,1}, which are packed in the data bits.
* •
The selector value 3 corresponds to $b=2$ and allows one to pack 30 integers
having values in $[0,4]$ in the data bits.
And so on (see Table 2): the larger is the value of the selector, the larger
is $b$, and the fewer integers one can fit in 60 data bits. During coding, we
try successively the selectors starting with value 0. That is, we greedily try
to fit as many integers as possible in the next 64-bit word.
Table 2: Encoding mode for Simple-8b scheme. Between 1 and 240 integers are coded with one 64-bit word. selector value | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
integers coded | 240 | 120 | 60 | 30 | 20 | 15 | 12 | 10 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1
bits per integer | 0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 10 | 12 | 15 | 20 | 30 | 60
Other schemes such as Simple-9 [9] and Simple-16 [10] use words of 32 bits.
(Simple-9 and Simple-16 can also be written as S9 and S16 [10].) While these
schemes may sometimes compress slightly better, they are generally slower.
Hence, we omitted them in our experiments. Unlike Simple-8b that can encode
integers in $[0,2^{60})$, Simple-9 and Simple-16 are restricted to integers in
$[0,2^{28})$.
While Simple-8b is not as fast as Variable Byte during encoding, it is still
faster than many alternatives. Because the decoding step can be implemented
efficiently (with little branching), we also get a good decoding speed while
achieving a better compression ratio than Variable Byte.
### 2.6 Binary Packing
Binary packing is closely related to Frame-Of-Reference (FOR) from Goldstein
et al. [39] and tuple differential coding from Ng and Ravishankar [40]. In
such techniques, arrays of values are partitioned into blocks (e.g., of 128
integers). The range of values in the blocks is first coded and then all
values in the block are written in reference to the range of values: for
example, if the values in a block are integers in the range $[1000,1127]$,
then they can be stored using 7 bits per integer
($\lceil\log_{2}(1127+1-1000)\rceil=7$) as offsets from the number 1000 stored
in the binary notation. In our approach to binary packing, we assume that
integers are small, so we only need to code a bit width $b$ per block (to
represent the range). Then, successive values are stored using $b$ bits per
integer using fast bit packing functions. Anh and Moffat called binary packing
_PackedBinary_ [25] whereas Delbru et al. [41] called their 128-integer binary
packing FOR and their 32-integer binary packing AFOR-1.
Binary packing can have a competitive compression ratio. In Appendix A, we
derive a general information-theoretic lower bound on the compression ratio of
binary packing.
### 2.7 Binary Packing with variable-length blocks
Three factors determine to the storage cost of a given block in binary
packing:
* •
the number of bits ($b$) required to store the largest integer value in the
binary notation,
* •
the block length ($B$),
* •
and a fixed per-block overhead ($\kappa$).
The total storage cost for one block is $bB+\kappa$. Binary packing uses
fixed-length blocks (e.g., $B=32$ or $B=128$).
We can vary dynamically the length of the blocks to improve the compression
ratio. This adds a small overhead to each block because we need to store not
only the corresponding bit width ($b$) but also the block length ($B$). We
then have a conventional optimization problem: we must partition the array
into blocks so that the total storage cost is minimized. The cost of each
block is still given by $bB+\kappa$, but the block length $B$ may vary from
one block to another.
The dynamic selection of block length was first proposed by Deveaux et al.
[42] who reported compression gains (15–30%). They used both a top-down and a
bottom-up heuristic.
Delbru et al. [41] also implemented two such adaptive solutions, AFOR-2 and
AFOR-3. AFOR-2 picks blocks of length 8, 16, 32 whereas AFOR-3 adds a special
processing for the case where we have successive integers. To determine the
best configuration of blocks, they pick 32 integers and try various
configurations (1 block of 32 integers, 2 blocks of 16 integers and so on).
They keep the configuration minimizing the storage cost. In effect, they apply
a greedy approach to the storage minimization problem.
Silvestri and Venturini [24] proposed two variable-length schemes, and we
selected their fastest version (henceforth VSEncoding). VSEncoding optimizes
the block length using dynamic programming over blocks of lengths 1–14, 16,
32. That is, given the integer logarithm of every integer in the array,
VSEncoding finds a partition truly minimizing the total storage cost. We
expect VSEncoding to provide a superior compression ratio compared to AFOR-2
and AFOR-3.
### 2.8 Patched coding
Binary packing might sometimes compress poorly. For example, the integers
$1,4,255,4,3,12,101$ can be stored using slightly more than 8 bits per integer
with binary packing. However, the same sequence with one large value, e.g.,
$1,4,255,4,3,12,4294967295$, is no longer so compressible: at least $32$ bits
per integer are required. Indeed, 32 bits are required for storing
$4294967295$ in the binary notation and all integers use the same bit width
under binary packing.
To alleviate this problem, Zukowski et al. [23] proposed _patching_ : we use a
small bit width $b$, but store exceptions (values greater than or equal to
$2^{b}$) in a separate location. They called this approach PFOR. (It is
sometimes written PFD [43], PFor or PForDelta when used in conjunction with
differential coding.) We begin with a partition of the input array into
subarrays that have a fixed maximal size (e.g., 32 MB). We call each such
subarray a _page_.
A single bit width is used for an entire page in PFOR. To determine the best
bit width $b$ during encoding, a sample of at most $2^{16}$ integers is
created out of the page. Then, various bit widths are tested until the best
compression ratio is achieved. In practice, to accelerate the computation, we
can construct a histogram, recording how many integers have a given integer
logarithm ($\lceil\log_{2}(x+1)\rceil$).
A page is coded in blocks of 128 integers, with a _separate_ storage array for
the exceptions. The blocks are coded using bit packing. We either pack the
integer value itself when the value is regular ($<2^{b}$), or an integer
offset pointing to the next exception in the block of 128 integers when there
is one. The offset is the difference between the index of the next exception
and the index of the current exception, minus one. For the purpose of bit
packing, we store integer values and offsets in the same array without
differentiating them. For example, consider the following array of integers in
the binary notation:
$\displaystyle 10,10,1,10,100110,10,1,11,10,100000,10,110100,\ldots$
Assume that the bit width is set to three ($b=3$), then we have exceptions at
positions $4,9,11,\ldots$, the offsets are $9-4-1=4,11-9-1=1,\ldots$ In the
binary notation we have $4\to 100$ and $1\to 1$, so we would store
$\displaystyle
10,10,1,10,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}100},10,1,11,10,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1},10,\ldots$
The bit-packed blocks are preceded by a 32-bit word containing two markers.
The first marker indicates the location of the first exception in the block of
128 integers (4 in our example), and the second marker indicates the location
of this first exception value in the array of exceptions (exception table).
Effectively, exception locations are stored using a linked list: we first read
the location of the first exception, then going to this location we find an
offset from which we retrieve the location of the next exception, and so on.
If the bit width $b$ is too small to store an offset value, that is, if the
offset is greater or equal than $2^{b}$, we have to create a _compulsory_
exception in-between. The location of the exception values themselves are
found by incrementing the location of the first exception value in the
exception table.
When there are too many exceptions, these exception tables may overflow and it
is necessary to start a new page: Zukowski et al. [23] used pages of 32 MB. In
our own experiments, we partition large arrays into arrays of at most $2^{16}$
integers (see § 6.2) so a single page is used in practice.
PFOR [23] does not compress the exception values. In an attempt to improve the
compression, Zhang et al. [26] proposed to store the exception values using
either 8, 16, or 32 bits. We implemented this approach (henceforth PFOR2008).
(See Table 3.)
#### 2.8.1 NewPFD and OptPFD
The compression ratios of PFOR and PFOR2008 are relatively modest (see § 6).
For example, we found that they fare worse than binary packing over blocks of
32 integers (BP32). To get better compression, Yang et al. [10] proposed two
new schemes called NewPFD and OptPFD. (NewPFD is sometimes called NewPFOR [44,
45] whereas OptPFD is also known as OPT-P4D [24].) Instead of using a single
bit width $b$ per page, they use a bit width per block of 128 integers. They
avoid wasteful compulsory exceptions: instead of storing exception offsets in
the bit packed blocks, they store the first $b$ bits of the exceptional
integer value. For example, given the following array
$\displaystyle 10,10,1,10,100110,10,1,11,10,100000,10,110100,\ldots$
and a bit width of 3 ($b=3$), we would pack
$\displaystyle
10,10,1,10,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}110},10,1,11,10,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0},10,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}100},\ldots$
For each block of 128 integers, the $32-b$ higher bits of the exception values
($100,100,110,\ldots$ in our example) as well as their locations (e.g.,
$4,9,11,\ldots$ ) are compressed using Simple-16. (We tried replacing
Simple-16 with Simple-8b but we found no benefit.)
Each block of 128 coded integers is preceded by a 32-bit word used to store
the bit width, the number of exceptions, and the storage requirement of the
compressed exception values in 32-bit words. NewPFD determines the bit width
$b$ by picking the smallest value of $b$ such that not more than 10% of the
integers are exceptions. OptPFD picks the value of $b$ maximizing the
compression. To accelerate the processing, the bit width is chosen among the
integer values 0–16, 20 and 32.
Table 3: Overview of the patched coding schemes: Only PFOR and PFOR2008 generate compulsory exceptions and use a single bit width $b$ per page. Only NewPFD and OptPFD store exceptions on a per block basis. We implemented all schemes with 128 integers per block and a page size of at least $2^{16}$ integers. | compulsory | bit width | exceptions | compressed exceptions
---|---|---|---|---
PFOR [23] | yes | per page | per page | no
PFOR2008 [26] | yes | per page | per page | 8, 16, 32 bits
NewPFD/OptPFD [10] | no | per block | per block | Simple-16
FastPFOR (§ 5) | no | per block | per page | binary packing
SIMD-FastPFOR (§ 5) | no | per block | per page | vectorized bin. pack.
SimplePFOR (§ 5) | no | per block | per page | Simple-8b
Ao et al. [43] also proposed a version of PFOR called ParaPFD. Though it has a
worse compression efficiency than NewPFD or PFOR, it is designed for fast
execution on graphical processing units (GPUs).
## 3 Fast differential coding and decoding
All of the schemes we consider experimentally rely on differential coding over
32-bit unsigned integers. The computation of the differences (or deltas) is
typically considered a trivial operation which accounts for only a negligible
fraction of the total decoding time. Consequently, authors do not discuss it.
But in our experience, a straightforward implementation of differential
decoding can be four times slower than the decompression of small integers.
We have implemented and evaluated two approaches to data differencing:
1. 1.
The standard form of differential coding is simple and requires merely one
subtraction per value during encoding ($\delta_{i}=x_{i}-x_{i-1}$) and one
addition per value during decoding to effectively compute the prefix sum
($x_{i}=\delta_{i}+x_{i-1}$).
2. 2.
A vectorized differential coding leaves the first four elements unmodified.
From each of the remaining elements with index $i$, we subtract the element
with the index $i-4$: $\delta_{i}=x_{i}-x_{i-4}$. In other words, the original
array ($x_{1},x_{2},\ldots$) is converted into
($x_{1},x_{2},x_{3},x_{4},\delta_{5}=x_{5}-x_{1},\delta_{6}=x_{6}-x_{2},\delta_{7}=x_{7}-x_{3},\delta_{8}=x_{8}-x_{4},\ldots$).
An advantage of this approach is that we can compute four differences using a
single SIMD operation. This operation carries out an element-wise subtraction
for two four-element _vectors_. The decoding part is symmetric and involves
the addition of the element $x_{i-4}$: $x_{i}=\delta_{i}+x_{i-4}$. Again, we
can use a single SIMD instruction to carry out four additions simultaneously.
We can get a speed of $\approx$2000 mis or 1.7 cycles/int with the standard
differential decoding (the first approach) by manually unrolling the loops.
Clearly, it is impossible to decode compressed integers at more than 2 billion
integers per second if the computation of the prefix sum itself runs at 2
billion integers per second. Hence, we implemented a vectorized version of
differential coding. Vectorized differential decoding is much faster
($\approx$5000 mis vs. $\approx$2000 mis). However, it comes at a price:
vectorized deltas are, on average, four times larger which increases the
storage cost by up to 2 bits (e.g., see Table 5).
To prevent memory bandwidth from becoming a bottleneck [1, 2, 3, 4, 5], we
prefer to compute differential coding and decoding in place. To this end, we
compute deltas in decreasing index order, starting from the largest index. For
example, given the integers 1, 4, 13, we first compute the difference between
13 and 4 which we store in last position (1, 4, 9), then we compute the
difference between 4 and 1 which we store in second position (1, 3, 9). In
contrast, the differential decoding proceeds in the increasing index order,
starting from the beginning of the array. Starting from 1, 3, 9, we first add
3 and 4 which we store in the second position (1, 4, 9), then we add 4 and 9
which we store in the last position (1, 4, 9). Further, our implementation
requires two passes: one pass to reconstruct the deltas from their compressed
format and another pass to compute the prefix sum (§ 6.2). To improve data
locality and reduce cache misses, arrays containing more than $2^{16}$
integers ($2^{16}\times 4\mathrm{\,B}=256\mathrm{\,KB}$) are broken down into
smaller arrays and each array is decompressed independently. Experiments with
synthetic data have shown that reducing cache misses by breaking down arrays
can lead to nearly a significant improvement in decoding speed for some
schemes without degrading the compression efficiency.
## 4 Fast bit unpacking
Bit packing is a process of encoding small integers in $[0,2^{b})$ using $b$
bits each: $b$ can be arbitrary and not just 8, 16, 32 or 64. Each number is
written using a string of exactly $b$ bits. Bit strings of fixed size $b$ are
concatenated together into a single bit string, which can span several 32-bit
words. If some integer is too small to use all $b$ bits, it is padded with
zeros.
struct Fields4_8 {
unsigned Int1: 4;
unsigned Int2: 4;
unsigned Int3: 4;
unsigned Int4: 4;
unsigned Int5: 4;
unsigned Int6: 4;
unsigned Int7: 4;
unsigned Int8: 4;
};
struct Fields5_8 {
unsigned Int1: 5;
unsigned Int2: 5;
unsigned Int3: 5;
unsigned Int4: 5;
unsigned Int5: 5;
unsigned Int6: 5;
unsigned Int7: 5;
unsigned Int8: 5;
};
Figure 3: Eight bit-packed integers represented as two structures in C/C++.
Integers in the left panel use 4-bit fields, while integers in the right panel
use 5-bit fields.
Languages like C and C++ support the concept of bit packing through bit
fields. An example of two C/C++ structures with bit fields is given in Fig. 3.
Each structure in this example stores 8 small integers. The structure
Fields4_8 uses 4 bits per integer ($b=4$), while the structure Fields5_8 uses
5 bits per integer ($b=5$).
Figure 4: Example of two bit-packed representations of 8 small integers. For
convenience, we indicate a starting bit number for each field (numeration
begins from zero). Integers in the left panel use 4-bit each and,
consequently, they fit into a single 32-bit word. Integers in the right panel
use 5-bit each. The complete representation uses two 32-bit words: 24-bits are
unoccupied.
Assuming that bit fields in these structures are stored compactly, i.e.,
without gaps, and the order of the bit fields is preserved, the 8 integers are
stored in the memory as shown in Fig. 4. If any bits remain unused, their
values can be arbitrary. All small integers on the left panel in Fig. 4 fit
into a single 32-bit word. However, the integers on the right panel require
two 32-bit words with 24 bits remaining unused (these bits can be arbitrary).
The field of the 7th integer crosses the 32-bit word boundary: the first 2
bits use bits 30–31 of the first words, while the remaining 3 bits occupy bits
0–2 of the second word (bits are enumerated starting from zero).
Unfortunately, language implementers are not required to ensure that the data
is fully packed. For example, the C language specification states that
_whether a bit-field that does not fit is put into the next unit or overlaps
adjacent units is implementation-defined_ [46]. Most importantly, they do not
have to provide packing and unpacking routines that are optimally fast. Hence,
we implemented bit packing and unpacking using our own procedures as proposed
by Zukowski et al. [23]. In Fig. 5, we give C/C++ implementations of such
procedures assuming that fields are laid out as depicted in Fig. 4. The
packing procedures can be implemented similarly and we omit them for
simplicity of exposition.
In some cases, we use bit packing even though some integers are larger than
$2^{b}-1$ (see § 2.8). In effect, we want to pack only the first $b$ bits of
each integer, which can be implemented by applying a bit-wise logical and
operation with the mask $2^{b}-1$ on each integer. These extra steps slow down
the bit packing (see § 6.3).
The procedure unpack4_8 decodes eight 4-bit integers. Because these integers
are tightly packed, they occupy exactly one 32-bit word. Given that this word
is already loaded in a register, each integer can be extracted using at most
four simple operations (shift, mask, store, and pointer increment). Unpacking
is efficient because it does not involve any branching.
The procedure unpack5_8 decodes eight 5-bit integers. This case is more
complicated, because the packed representation uses two words: the field for
the 7th integer crosses word boundaries. The first two (lower order) bits of
this integer are stored in the first word, while the remaining three (higher
order) bits are stored in the second word. Decoding does not involve any
branches and most integers are extracted using four simple operations.
The procedures unpack4_8 and unpack5_8 are merely examples. Separate
procedures are required for each bit width (not just 4 and 5).
void unpack4_8(const uint32_t* in,
uint32_t* out) {
*out++ = ((*in)) & 15;
*out++ = ((*in) >> 4) & 15;
*out++ = ((*in) >> 8) & 15;
*out++ = ((*in) >> 12) & 15;
*out++ = ((*in) >> 16) & 15;
*out++ = ((*in) >> 20) & 15;
*out++ = ((*in) >> 24) & 15;
*out = ((*in) >> 28);
}
void unpack5_8(const uint32_t* in,
uint32_t* out) {
*out++ = ((*in)) & 31;
*out++ = ((*in) >> 5 ) & 31;
*out++ = ((*in) >> 10) & 31;
*out++ = ((*in) >> 15) & 31;
*out++ = ((*in) >> 20) & 31;
*out++ = ((*in) >> 25) & 31;
*out = ((*in) >> 30);
++in;
*out++ |= ((*in) & 7) << 2;
*out = ((*in) >> 3) & 31;
}
Figure 5: Two procedures to unpack eight bit-packed integers. The procedure
unpack4_8 works for $b=4$ while procedure unpack5_8 works for $b=5$. In both
cases we assume that (1) integers are packed tightly, i.e., without gaps, (2)
packed representations use whole 32-bit words: values of unused bits are
undefined.
Decoding routines unpack4_8 and unpack5_8 operate on _scalar_ 32-bit values.
An effective way to improve performance of these routines involves
_vectorization_ [14, 47]. Consider listings in Fig. 5 and assume that in and
out are pointers to $m$-element vectors instead of scalars. Further, assume
that scalar operators (shifts, assignments, and bit-wise logical operations)
are vectorized. For example, a bit-wise shift is applied to all $m$ vector
elements at the same time. Then, a single call to unpack5_8 or unpack4_8
decodes $m\times 8$ rather than just eight integers.
Figure 6: Example of a vectorized bit-packed representations of 32 small
integers. For convenience, we show a starting bit number for each field
(numeration begins from zero). Integers use 5-bit each. Words in the second
row follow (i.e., have larger addresses) words of the first row. Curved lines
with arrows indicate that integers 25–28 are each split between two words.
Recent x86 processors have SIMD instructions that operate on vectors of four
32-bit integers ($m=4$) [48, 49, 50]. We can use these instructions to achieve
a better decoding speed. A sample vectorized data layout for $b=5$ is given in
Fig. 6. Integers are divided among series of four 32-bit words in a round-
robin fashion. When a series of four words overflows, the data _spills_ over
to the next series of 32-bit integers. In this example, the first 24 integers
are stored in the first four words (the first row in Fig. 6), integers 25–28
are each split between different words, and the remaining integers 29–32 are
stored in the second series of words (the second row of the Fig. 6).
These data can be processed using a vectorized version of the procedure
unpack5_8, which is obtained from unpack5_8 by replacing scalar operations
with respective SIMD instructions. With Microsoft, Intel or GNU GCC compilers,
we can almost mechanically go from the scalar procedure to the vectorized one
by replacing each C operator with the equivalent SSE2 intrinsic function:
* •
the bitwise logical and (&) becomes _mm_and_si128,
* •
the right shift ($>>$) becomes _mm_srli_epi32,
* •
and the left shift ($<<$) becomes _mm_slli_epi32.
Indeed, compare procedure unpack5_8 from Fig. 5 with procedure SIMDunpack5_8
from Fig. 7. The intrinsic functions serve the same purpose as the C operators
except that they work on vectors of 4 integers instead of single integers:
e.g., the function _mm_srli_epi32 shifts 4 integers at once. The functions
_mm_load_si128 and _mm_store_si128 load a register from memory and write the
content of a register to memory respectively; the function _mm_set1_epi32
creates a vector of 4 integers initialized with a single integer (e.g., 31
becomes 31,31,31,31).
In the beginning of the vectorized procedure the pointer in points to the
first 128-bit chunk of data displayed in row one of the Fig. 6. The first
shift-and-mask operation extracts 4 small integers at once. Then, these
integers are written to the target buffer using a _single_ 128-bit SIMD store
operation. The shift-and-mask is repeated until we extract the first 24
numbers and the first two bits of the integers 25–28. At this point the unpack
procedure increases the pointer in and loads the next 128-bit chunk into a
register. Using an additional mask operation, it extracts the remaining 3 bits
of integers 25–28. These bits are combined with already obtained first 2 bits
(for each of the integers 25–28). Finally, we store integers 25–28 and finish
processing the second 128-bit chunk by extracting numbers 29–32.
const static __m128i m7 = __mm_set1_epi32_(7U);
const static __m128i m31 = __mm_set1_epi32_(31U);
void SIMDunpack5_8(const __m128i* in, __m128i* out) {
__m128i i = __mm_load_si128_(in);
__mm_store_si128_(out++, __mm_and_si128_( i , m31));
__mm_store_si128_(out++, __mm_and_si128_( __mm_srli_epi32_(i,5) , m31));
__mm_store_si128_(out++, __mm_and_si128_( __mm_srli_epi32_(i,10) , m31));
__mm_store_si128_(out++, __mm_and_si128_( __mm_srli_epi32_(i,15) , m31));
__mm_store_si128_(out++, __mm_and_si128_( __mm_srli_epi32_(i,20) , m31));
__mm_store_si128_(out++, __mm_and_si128_( __mm_srli_epi32_(i,25) , m31));
__m128i o = __mm_srli_epi32_(i,30);
i = __mm_load_si128_(++in);
o = __mm_or_si128_(o, __mm_slli_epi32_(__mm_and_si128_(i, m7), 2));
__mm_store_si128_(out++, o);
__mm_store_si128_(out++, __mm_and_si128_( __mm_srli_epi32_(i,3) , m31));
}
Figure 7: Equivalent to the unpack5_8 procedure from Fig. 7 using SSE
intrinsic functions as illustrated by Fig. 6.
Our vectorized data layout is interleaved. That is, the first four integers
(Int 1, Int 2, Int 3, and Int 4 in Fig. 6) are packed into 4 different 32-bit
words. The first integer is immediately adjacent to the fifth integer (Int 5).
Schlegel et al. [34] called this model _vertical_. Instead we could ensure
that the integers are packed sequentially (e.g. Int 1, Int 2, and Int 3 could
be stored in the same 32-bit word). Schlegel et al. called this alternative
model _horizontal_ and it is used by Willhalm et al. [47]. In their scheme,
decoding relies on the SSSE3 shuffle operation pshufb (like varint-G8IU).
After we determine the bit width $b$ of integers in the block, one decoding
step typically includes the following operations:
1. 1.
Loading data into the source 16-byte buffer (this step may require a 16-byte
alignment).
2. 2.
Distributing 3–4 integers stored in the source buffer among four 32-bit words
of the target buffer. This step, which requires loading a shuffle mask, is
illustrated by Fig. 8 (for 5-bit integers). Note that unlike varint-G8IU, the
integers in the source buffer are not necessarily aligned by byte boundaries
(unless $b$ is 8, 16, or 32). Hence, after the shuffle operation, (1) the
integers copied the target buffer may not be aligned on boundaries of 32-bit
words, and (2) 32-bit words may contain some extra bits that do not belong to
the integers of interest.
3. 3.
Aligning integers on bit boundaries, which may require shifting several
integers to the right. Because the x86 platform currently lacks a SIMD shift
that has four different shift amounts, this step is simulated via two
operations: a SIMD multiplication by four different integers using the SSE4.1
instruction pmulld and a subsequent vectorized right shift.
4. 4.
Zeroing bits that do not belong to the integers of interest. This requires a
mask operation.
5. 5.
Storing the target buffer.
Overall, Willhalm et al. [47] require SSE4.1 for their horizontal bit packing
whereas efficient bit packing using a vertical layout only requires SSE2.
Figure 8: One step of simultaneous decoding of four 5-bit integers that are
stored in a horizontal layout (as opposed to the vertical data layout of Fig.
6). These integers are copied to four 32-bit words using the shuffle operation
pshufb. The locations in source and target buffers are indicated by arrows.
Curvy lines are used to denote integers that cross byte boundaries in the
source buffer. Hence, they are copied only partially. The boldface zero values
represent the bytes zeroed by the shuffle instruction. Note that some source
bytes are copied to multiple locations.
We compare experimentally vertical and horizontal bit packing in § 6.3.
## 5 Novel schemes: SIMD-FastPFOR, FastPFOR and SimplePFOR
Patched schemes compress arrays broken down into pages (e.g., thousands or
millions of integers). Pages themselves may be broken down into small blocks
(e.g., 128 integers). While the original patched coding scheme (PFOR) stores
exceptions on a per page basis, newer alternatives such as NewPFD and OptPFD
store exceptions on a per block basis (see Table 3). Also, PFOR picks a single
bit width for an entire page, whereas NewPFD and OptPFD may choose a separate
bit width for each block.
The net result is that NewPFD compresses better than PFOR, but PFOR is faster
than NewPFD. We would prefer a scheme that compresses as well as NewPFD but
with the speed of PFOR. For this purpose, we propose two new schemes: FastPFOR
and SimplePFOR. Instead of compressing the exceptions on a per block basis
like NewPFD and OptPFD, FastPFOR and SimplePFOR store the exceptions on a per
page basis, which is similar to PFOR. However, like NewPFD and OptPFD, they
pick a new bit width for each block.
To explain FastPFOR and SimplePFOR, we consider an example. For simplicity, we
only use 16 integers (to be encoded). In the binary notation these numbers
are:
$\displaystyle 10,10,1,10,100110,10,1,11,10,100000,10,110100,10,11,11,1$
The maximal number of bits used by an integer is 6 (e.g., because of 100000).
So we can store the data using 6 bits per value plus some overhead. However,
we might be able to do better by allowing exceptions in the spirit of patched
coding. Assume that we store the location of any exception using a byte (8
bits): in our implementation, we use blocks of 128 integers so that this is
not a wasteful choice.
We want to pick $b\leq 6$, the actual number of bits we use. That is, we store
the lowest $b$ bits of each value. If a value uses $6$ bits, then we somehow
need to store the extra $6-b$ bits as an exception. We propose to use the
difference (i.e., $6-b$) between the maximal bit width and the number of bits
allocated per truncated integer to estimate the cost of storing an exception.
This is a heuristic since we use slightly more in practice (to compress the
$6-b$ highest bits of exception values). Because we store exception locations
using 8 bits, we estimate the cost of storing each exception as $8+(6-b)=14-b$
bits. We want to choose $b$ so that $b\times 16+(14-b)\times c$ is minimized
where $c$ is the number of exceptions corresponding to the value $b$. (In our
software, we store blocks of 128 integers so that the formula would be
$b\times 128+(14-b)\times c$.)
We still need to compute the number of exceptions $c$ as a function of the bit
width $b$ in a given block of integers. For this purpose, we build a histogram
that tells us how many integers have a given bit width. In software, this can
be implemented as an array of 33 integers: one integer for each possible bit
width from 0 to 32. Creating the histogram requires the computation of the
integer logarithm ($\lceil\log_{2}(x+1)\rceil$) of every single integer to be
coded. From this histogram, we can quickly determine the value $b$ that
minimizes the expected storage simply by trying every possible value of $b$.
Looking at our data, we have 3 integers using 1 bit, 10 integers using 2 bits,
and 3 integers using 6 bits. So, if we set $b=1$, we get $c=13$ exceptions;
for $b=2$, we get $c=3$; and for $b=6$, $c=0$. The corresponding costs
($b\times 16+(14-b)\times c$) are 185, 68, and 96. So, in this case, we choose
$b=2$. We therefore have 3 exceptions (100110, 100000, 110100).
A compressed page begins with a 32-bit integer. Initially, this 32-bit integer
is left uninitialized: we come back to it later. Next, we first store the
values themselves, with the restriction that we use only $b$ lowest bits of
each value. In our example, the data corresponding to the block is
$\displaystyle
10,10,1,10,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}10},10,1,11,10,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00},10,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00},10,11,11,1.$
These truncated values are stored continuously, one block after the other (see
Fig. 9). Different blocks may use different values of $b$, but because
$128\times b$ is always divisible by $32$, the truncated values for a given
block can be stored at a memory address that is 32-bit aligned.
During the encoding of a compressed page, we write to a temporary byte array.
The byte array contains different types of information. For each block, we
store the number of bits allocated for each truncated integer (i.e., $b$) and
the maximum number of bits any actual, i.e., non-truncated, value may use. If
the maximal bit width is greater than the number of allocated bits $b$, we
store a counter $c$ indicating the number of exceptions. We also store the $c$
exception locations within the block as integers in $[0,127]$. In contrast
with schemes such as NewPFD or OptPFD, we do not attempt to compress these
numbers and simply store them using one byte each. Each value is already
represented concisely using only one byte as opposed to using a 32-bit integer
or worse.
When all integers of a page have been processed and bit-packed, the temporary
byte array is stored right after the truncated integers, preceded with a
32-bit counter indicating its size. We pad the byte array with zeros so that
the number of bytes is divisible by 4 (allowing a 32-bit memory alignment).
Then, we go back to the beginning of the compressed page, where we had left an
uninitialized 32-bit integer, and we write there the offset of the byte array
within the compressed page. This ensures that during decoding we can locate
the byte array immediately. The initial 32-bit integer and the 32-bit counter
preceding the byte array add a fixed overhead of 8 bytes per page: it is
typically negligible because it is shared by many blocks, often spanning
thousands of integers.
In our example, we write 16 truncated integers using $b=2$ bits each, for the
total of 4 bytes (32 bits). In the byte array, we store:
* •
the number of bits ($b=2$) allocated per truncated integer using one byte;
* •
the maximal bit width (6) using a byte;
* •
the number of exceptions $c=3$ (again using one byte);
* •
locations of the exceptions (4,9,11) using one byte each.
Thus, for this block alone, we use 3+4+3=10 bytes (80 bits).
Finally, we must store the highest $(6-b)=4$ bits of each exception: 1001,
1000, 1101. They are stored right after the byte array. Because the offset of
the byte array within the page is stored at the beginning of the page, and
because the byte array is stored with a header indicating its length, we can
locate the exceptions quickly during decoding. The exceptions are stored on a
per page basis in compressed form. This is in contrast to schemes such as
OptPFD and NewPFD where exceptions are stored on a per-block basis,
interleaved with the truncated values.
SimplePFOR and FastPFOR differ in how they compress high bits of the exception
values:
* •
In the SimplePFOR scheme, we collect all these values (e.g., such as 1001,
1000, 1101) in one 32-bit array and we compress them using Simple-8b. We apply
Simple-8b only once per compressed page.
* •
In the FastPFOR scheme, we store exceptions in one of 32 arrays, one for each
possible bit width (from 1 to 32). When encoding a block, the difference
between the maximal bit width and $b$ determines in which array the exceptions
are stored. Each of the 32 arrays is then bit packed using the corresponding
bit width. Arrays are padded so that their length is a multiple of 32
integers.
In our example, the 3 values corresponding to the high bits of exceptions
(1001, 1000, 1101) would be stored in the fourth array and bit-packed using 4
bits per value.
In practice, we store the 32 arrays as follows. We start with a 32-bit bitset:
each bit of the bitset corresponds to one array. The bit is set to true if the
array is not empty and to false otherwise. Then all non-empty bit-packed
arrays are stored in sequence. Each bit-packed array is preceded by a 32-bit
counter indicating its length.
In all other aspects, SimplePFOR and FastPFOR are identical.
These schemes provide effective compression even though they were designed for
speed. Indeed, suppose we could compress the highest bits of 3 exceptions of
our example (1001, 1000, 1101) using only 4 bits each. For this block alone,
we use 32 bits for the truncated data, 48 bits in the byte array plus 12 bits
for the values of the exceptions. The total would be 92 bits to store the 16
original integers, or 5.75 bits/int. This compares favorably to maximal bit
width of these integers (6). In our implementation, we use blocks of 128
integers instead of only 16 integers so that good compression is more likely.
During decoding, the exceptions are first decoded in bulk. To ensure that we
do not overwhelm the CPU cache, we process the data in pages of $2^{16}$
integers. We then unpack the integers and apply patching on a per block basis.
The exceptions locations do not need any particular decoding: they are read
byte by byte.
Though SimplePFOR and FastPFOR are similar in design to NewPFD and OptPFD, we
find that they offer better coding and decoding speed. In our tests (see § 6),
FastPFOR and SimplePFOR encode integers about twice as fast as NewPFD. It is
an indication that compressing exceptions in bulk is faster.
Data to be compressed: | …10, 10, 1, 10, 100110, 10, 1, 11, 10, 100000, 10, 110100, 10, 11, 11, 1…
---|---
Truncated data: |
($16\times 2=32$ bits) | …10, 10, 01, 10, 10, 10, 01, 11, 10, 00, 10, 00, 10, 11, 11, 01 …
Byte array: |
($6\times 8=48$ bits) | …2, 6, 3, 4, 9, 11 …
Exception data: |
(to be compressed) | …1001, 1000, 1101 …
Figure 9: Layout of a compressed page for SimplePFOR and FastPFOR schemes with
our running example. We only give numbers for a block of 16 integers: a page
contains hundreds of blocks. The beginning of each page contains the truncated
data of each block. The truncated data is then followed by a byte array
containing metadata (e.g., exception locations). At the end of the page, we
store the exceptions in compressed form.
We also designed a new scheme, SIMD-FastPFOR: it is identical to FastPFOR
except that it packs relies on vectorized bit packing for the truncated
integers and the high bits of the exception values. The compression ratio is
slightly diminished for two reasons:
* •
The 32 exception arrays are padded so that their length is a multiple of 128
integers, instead of 32 integers.
* •
We insert some padding prior to storing bit packing data so that alignment on
128-bit boundaries is preserved.
This padding adds an overhead of about 0.3–0.4 bits per integer (see Table 5).
## 6 Experiments
The goal of our experiments is to evaluate the best known integer encoding
methods. The first series of our test in § 6.4 is based on synthetic data sets
first presented by Anh and Moffat [25]: ClusterData and Uniform. They have the
benefit that they can be quickly implemented, thus helping reproducibility. We
then confirm our results in § 6.5 using large realistic data sets based on
TREC collections ClueWeb09 and GOV2.
### 6.1 Hardware
We carried out our experiments on a Linux server equipped with Intel Core i7
2600 (3.40 GHz, 8192 KB of L3 CPU cache) and 16 GB of RAM. The DDR3-1333 RAM
with dual channel has a transfer rate of $\approx$20,000 MB/s or $\approx$5300
mis. According to our tests, it can copy arrays at a rate of 2270 mis with the
C function memcpy.
### 6.2 Software
We implemented our algorithms in C++ using GNU GCC 4.7. We use the
optimization flag -O3. Because the varint-G8IU scheme requires SSSE3
instructions, we had to add the flag -mssse3. When compiling our
implementation of Willhalm et al. [47] bit unpacking, we had to use the flag
-msse4.1 since it requires SSE4 instructions. Our complete source code is
available online.111https://github.com/lemire/FastPFOR
Following Stepanov et al. [12], we compute speed based on the wall-clock in-
memory processing. Wall-clock times include the time necessary for
_differential coding and decoding_. During our tests, we do not retrieve or
store data on disk: it is impossible to decode billions of integers per second
when they are kept on disk.
Arrays containing more than $2^{16}$ integers (256 KB) are broken down into
smaller chunks. Each chunk is decoded into two passes. In the first pass, we
decompress deltas and store each delta value using a 32-bit word. In the
second pass, we carry out an in-place computation of prefix sums. As noted in
§ 3, this approach greatly improves data locality and leads to a significant
improvement in decoding speed for the fastest schemes.
Our implementation of VSEncoding, NewPFD, and OptPFD is based on software
published by Silvestri and Venturini [24]. They report that their
implementation of OptPFD was validated against an implementation provided by
the original authors [10]. We implemented varint-G8IU from Stepanov et al.
[12] as well as Simple-8b from Anh and Moffat [25]. To minimize branching, we
implemented Simple-8b using a C++ switch case that selects one of 16
functions, that is, one for each selector value. Using a function for each
selector value instead of a single function is faster because loop unrolling
eliminates branching. (Anh and Moffat [25] referred to this optimization as
_bulk unpacking_.) We also implemented the original PFOR scheme from Zukowski
et al. [23] as well as its successor PFOR2008 from Zhang et al. [26]. Zukowski
et al. made a distinction between PFOR and PFOR-Delta: we effectively use FOR-
Delta since we apply PFOR to deltas.
Reading and writing unaligned data can be as fast as reading and writing
aligned data on recent Intel processors—as long as we do not cross a 64-byte
cache line. Nevertheless, we still wish to align data on 32-bit boundaries
when using regular _binary packing_. Each block of 32 bit-packed integers
should be preceded by a descriptor that stores the bit width ($b$) of integers
in the block. The number of bits used by the block is always divisible by 32.
Hence, to keep blocks aligned on 32-bit boundaries, we group the blocks and
respective descriptors into meta-blocks each of which contains 4 successive
blocks. A meta-block is preceded by a 32-bit descriptor that combines 4 bit
widths $b$ (8 bits per width). We call this scheme BP32. We also experimented
with versions of binary packing on fewer integers (8 integers and 16
integers). Because these versions are slower, we omit them from our
experiments.
We also implemented a vectorized binary packing over blocks of 128 integers
(henceforth SIMD-BP128). Similar to regular binary packing, we want to keep
the blocks aligned on 128-bit boundaries when using vectorized binary packing.
To this end, we regroup 16 blocks into a meta-block of 2048 integers. As in
BP32, the encoded representation of a meta-block is preceded by a 128-bit
descriptor word keeping bit widths (8 bits per width).
In summary, the format of our binary packing schemes is as follows:
* •
SIMD-BP128 combines 16 blocks of 128 integers whereas BP32 combines 4 blocks
of 32 integers.
* •
SIMD-BP128 employs (vertical) vectorized bit packing whereas BP32 relies on
the regular bit packing as described in §4.
Many schemes such as BP32 and SIMD-BP128 require the computation of the
integer logarithm during encoding. If done naively, this step can take up most
of the running time: the computation of the integer logarithm is slower than a
fast operation such as a shift or an addition. We found it best to use the
_bit scan reverse_ (bsr) assembly instruction on x86 platforms (as it provides
$\lceil\log_{2}(x+1)\rceil-1$ whenever $x>0$).
For the binary packing schemes, we must determine the maximum of the integer
logarithm of the integers ($\max_{i}\lceil\log_{2}(x_{i}+1)\rceil$) during
encoding. Instead on computing one integer logarithm per integer, we carry out
a bit-wise logical or on all the integers and compute the integer logarithm of
the result. This shortcut is possible due to the equation:
$\max_{i}\lceil\log_{2}(x_{i}+1)\rceil=\lceil\log_{2}\lor_{i}(x_{i}+1)\rceil$
where $\lor$ refers to the bit-wise logical or.
Some schemes compress data in blocks of fixed length (e.g., 128 integers). We
compress the remainder using Variable Byte as in Zhang et al. [26]. In our
tests, most arrays are large compared to the block size. Thus, replacing
Variable Byte by another scheme would make no or little difference.
Speeds are reported in millions of 32-bit integers per second (mis). Stepanov
et al. report a speed of 1059 mis over the TREC GOV2 data set for their best
scheme varint-G8IU. We got a similar speed (1300 mis).
VSEncoding, FastPFOR, and SimplePFOR use buffers during compression and
decompression proportional to the size of the array. VSEncoding uses a
persistent buffer of over 256 KB. We implemented SIMD-FastPFOR, FastPFOR, and
SimplePFOR with a persistent buffer of slightly more than 64 KB. PFOR,
PFOR2008, NewPFD, and OptPFD are implemented using persistent buffers
proportional to the block size (128 integers in our tests): less than 512 KB
in persistent buffer memory are used for each scheme. Both PFOR and PFOR2008
use pages of $2^{16}$ integers or 256 KB. During compression, PFOR, PFOR2008,
SIMD-FastPFOR, FastPFOR, and SimplePFOR use a buffer to store exceptions.
These buffers are limited by the size of the pages and they are released
immediately after decoding or encoding an array.
The implementation of VSEncoding [24] uses some SSE2 instructions through
assembly during bit unpacking. Varint-G8IU makes explicit use of SSSE3
instructions through SIMD intrinsic functions whereas SIMD-FastPFOR and SIMD-
BP128 similarly use SSE2 intrinsic functions.
Though we tested vectorized differential coding with all schemes, we only
report results for schemes that make explicit use of SIMD instructions (SIMD-
FastPFOR, SIMD-BP128, and varint-G8IU). To ensure fast vector processing, we
align all initial pointers on 16-byte boundaries.
### 6.3 Computing bit packing
We implemented bit packing using hand-tuned functions as originally proposed
by Zukowski et al. [23]. Given a bit width $b$, a sequence of $K$ unsigned
32-bit integers are coded to $\lceil Kb/32\rceil$ integers. In our tests, we
used $K=32$ for the regular version, and $K=128$ for the vectorized version.
Fig. 10 illustrates the speed at which we can pack and unpack integers using
blocks of 32 integers. In some schemes, it is known that all integers are no
larger than $2^{b}-1$, while in patching schemes there are exceptions, i.e.,
integers larger than or equal to $2^{b}$. In the latter case, we enforce that
integers are smaller than $2^{b}$ through the application of a mask. This
operation slows down compression.
We can pack and unpack much faster when the number of bits is small because
less data needs to be retrieved from RAM. Also, we can pack and unpack faster
when the bit width is 4, 8, 16, 24 or 32. Packing and unpacking with bit
widths of 8 and 16 is especially fast.
The vectorized version (Fig. 10b) is roughly twice as fast as the scalar
version. We can unpack integers having a bit width of 8 or less at a rate of
$\approx$6000 mis. However, it carries the implicit constraint that integers
must be packed and unpacked in blocks of at least 128 integers. Packing is
slightly faster when the bit width is 8 or 16.
In Fig. 10b only, we report the unpacking speed when using the horizontal data
layout as described by Willhalm et al. [47] (see § 4). When the bit widths
range from 16 to 26, the vertical and horizontal techniques have the same
speed. For small ($<8$) or large ($>27$) bit widths, our approach based on a
vertical layout is preferable as it is up to 70% faster. Accordingly, all
integer coding schemes are implemented using the vertical layout.
We also experimented with the cases where we pack fewer integers ($K=8$ or
$K=16$). However, it is slower and a few bits remain unused ($\lceil
Kb/32\rceil 32-Kb$).
(a) Optimized but portable C++ (b) Vectorized with SSE2 instructions (c) Ratio
vectorized/non-vectorized
Figure 10: Wall-clock speed in millions of integers per second for bit packing
and unpacking. We use small arrays (256 KB) to minimize cache misses. When
packing integers that do not necessarily fit in $b$ bits, (as required in
patching schemes), we must apply a mask which slows down packing by as much as
30%.
### 6.4 Synthetic data sets
We used the ClusterData and the Uniform model from Anh and Moffat [25]. These
models generate sets of distinct integers that we keep in sorted order. In the
Uniform model, integers follow a uniform distribution whereas in the
ClusterData model, integer values tend to cluster. That is, we are more likely
to get long sequences of similar values. The goal of the ClusterData model is
to simulate more realistically data encountered in practice. We expect data
obtained from the ClusterData model to be more compressible.
We generated data sets of random integers in the range $[0,2^{29})$ with both
the ClusterData and the Uniform model. In the first pass, we generated
$2^{10}$ short arrays containing $2^{15}$ integers each. The average
difference between successive integers within an array is thus
$2^{29-15}=2^{14}$. We expect the compressed data to use at least 14 bits/int.
In the second pass, we generated a single long array of $2^{25}$ integers. In
this case, the average distance between successive integers is $2^{4}$: we
expect the compressed data to use at least 4 bits/int.
The results are given in Table 4 (schemes with a ⋆ by their name, e.g., SIMD-
FastPFOR⋆, use vectorized differential coding). Over short arrays, we see
little compression as expected. There is also a relatively little difference
in compression efficiency between Variable Byte and a more space-efficient
alternative such as FastPFOR. However, speed differences are large: the
decoding speed ranges from 220 mis for Variable Byte to 2500 mis for SIMD-
BP128⋆.
For long arrays, there is a greater difference between the compression
efficiencies. The schemes with the best compression ratios are SIMD-FastPFOR,
FastPFOR, SimplePFOR, Simple-8b, OptPFD. Among those, SIMD-FastPFOR is the
clear winner in terms of decoding speed. The good compression ratio of OptPFD
comes at a price: it has one of the worst encoding speeds. In fact, it is
20–50 times slower than SIMD-FastPFOR during encoding.
Though they differ significantly in implementation, FastPFOR, SimplePFOR, and
SIMD-FastPFOR have equally good compression ratios. All three schemes have
similar decoding speeds, but SIMD-FastPFOR decodes integers much faster than
FastPFOR and SimplePFOR.
In general, encoding speeds vary significantly, but binary packing schemes are
the fastest, especially when they are vectorized. Better implementations could
possibly help reduce this gap.
The version of SIMD-BP128 using vectorized differential coding (written SIMD-
BP128⋆) is always 400 mis faster during decoding than any other alternative.
Though it does not always offer the best compression ratio, it always matches
the compression ratio of Variable Byte.
The difference between using vectorized differential coding and regular
differential coding could amount to up to 2 bits per integer. Yet, typically,
the difference is less than 2 bits. For example, SIMD-BP128⋆ only uses about
one extra bit per integer when compared with SIMD-BP128. The cost of binary
packing is determined by the largest delta in a block: increasing the average
size of the deltas by a factor of 4 does not necessarily lead to a fourfold
increase in the expected largest integer (in a block of 128 deltas).
Compared to our novel schemes, performance of varint-G8IU is unimpressive.
However, variant-G8IU is about 60% faster than Variable Byte while providing a
similar compression efficiency. It is also faster than Simple-8b, though
Simple-8b has a better compression efficiency. The version with vectorized
differential coding (written varint-G8IU⋆) has poor compression over the short
arrays compared with the regular version (varint-G8IU). Otherwise, on long
arrays, varint-G8IU⋆ is significantly faster (from 1300 mis to 1600 mis) than
varint-G8IU while compressing just as well.
There is little difference between PFOR and PFOR2008 except that PFOR offers a
significantly faster encoding speed. Among all the schemes taken from the
literature, PFOR and PFOR2008 have the best decoding speed in these tests:
they use a single bit width for all blocks, determined once at the beginning
of the compression. However, they are dominated in all metrics (coding speed,
decoding speed and compression ratio) by SIMD-BP128 and SIMD-FastPFOR.
For comparison, we tested Google Snappy (version 1.0.5) as a delta compression
technique. Google Snappy is a freely available library used internally by
Google in its database engines [17]. We believe that it is competitive with
other fast generic compression libraries such as zlib or LZO. For short
ClusterData arrays, we got a decoding speed of 340 mis and almost no
compression (29 bits/int.). For long ClusterData arrays, we got a decoding
speed of 200 mis and 14 bits/int. Overall, Google Snappy has about half the
compression efficiency of SIMD-BP128⋆ while being an order of magnitude
slower.
Table 4: Coding and decoding speed in millions of integers per second over
synthetic data sets, together with number of bits per 32-bit integer. Results
are given using two significant digits. Schemes with a ⋆ by their name use
vectorized differential coding.
| coding | decoding | bits/int
---|---|---|---
SIMD-BP128⋆ | 1700 | 2500 | 17
SIMD-FastPFOR⋆ | 380 | 2000 | 16
SIMD-BP128 | 1000 | 1800 | 16
SIMD-FastPFOR | 300 | 1400 | 15
PFOR | 350 | 1200 | 18
PFOR2008 | 280 | 1200 | 18
SimplePFOR | 300 | 1100 | 15
FastPFOR | 300 | 1100 | 15
BP32 | 790 | 1100 | 15
NewPFD | 66 | 1100 | 16
varint-G8IU⋆ | 160 | 910 | 23
varint-G8IU | 150 | 860 | 18
VSEncoding | 10 | 720 | 16
Simple-8b | 260 | 690 | 16
OptPFD | 5.1 | 660 | 15
Variable Byte | 300 | 270 | 17
(a) ClusterData: Short arrays
coding | decoding | bits/int
---|---|---
1600 | 2000 | 18
360 | 1800 | 18
1100 | 1600 | 17
330 | 1400 | 16
370 | 1400 | 17
280 | 1400 | 17
330 | 1200 | 16
330 | 1200 | 16
840 | 1200 | 17
64 | 1300 | 17
140 | 650 | 25
170 | 870 | 18
10 | 690 | 18
260 | 540 | 18
4.6 | 1100 | 17
240 | 220 | 19
(b) Uniform: Short arrays
| coding | decoding | bits/int
---|---|---|---
SIMD-BP128⋆ | 1800 | 2800 | 7.0
SIMD-FastPFOR⋆ | 440 | 2400 | 6.8
SIMD-BP128 | 1100 | 1900 | 6.0
SIMD-FastPFOR | 320 | 1600 | 5.4
varint-G8IU⋆ | 270 | 1600 | 9.1
PFOR | 360 | 1300 | 6.1
PFOR2008 | 280 | 1300 | 6.1
BP32 | 840 | 1300 | 5.8
FastPFOR | 320 | 1200 | 5.4
SimplePFOR | 320 | 1200 | 5.3
varint-G8IU | 230 | 1300 | 9.0
NewPFD | 120 | 970 | 5.5
Simple-8b | 360 | 890 | 5.6
VSEncoding | 9.8 | 790 | 6.4
OptPFD | 17 | 750 | 5.4
Variable Byte | 880 | 830 | 8.1
(c) ClusterData: Long arrays
coding | decoding | bits/int
---|---|---
1900 | 2600 | 8.0
380 | 2200 | 7.6
1100 | 1800 | 7.0
340 | 1600 | 6.4
270 | 1600 | 9.0
360 | 1300 | 7.3
280 | 1300 | 7.3
810 | 1200 | 6.7
330 | 1200 | 6.3
330 | 1200 | 6.3
230 | 1300 | 9.0
110 | 1000 | 6.5
370 | 940 | 6.4
9.9 | 790 | 7.2
15 | 740 | 6.2
930 | 860 | 8.0
(d) Uniform: Long arrays
### 6.5 Realistic data sets
The posting list of a word is an array of document identifiers where the word
occurs. For more realistic data sets, we used posting lists obtained from two
TREC Web collections. Our data sets include only document identifiers, but not
positions of words in documents. For our purposes, we do not store the words
or the documents themselves, just the posting lists.
The first data set is a posting list collection extracted from the ClueWeb09
(Category B) data set [51]. The second data set is a posting list collection
built from the GOV2 data set by Silvestri and Venturini [24]. The GOV2 is a
crawl of the .gov sites, which contains 25 million HTML, text, and PDF
documents (the latter are converted to text).
This ClueWeb09 collection is a more realistic HTML collection of about 50
million crawled HTML documents, mostly in English. It represents postings for
one million most frequent words. Common stop words were excluded and different
grammar forms of words were conflated. Documents were enumerated in the order
they appear in source files, i.e., they were not reordered. Unlike GOV2, the
ClueWeb09 crawl is not limited to any specific domain. Uncompressed, the
posting lists from GOV2 and ClueWeb09 use 20 GB and 50 GB respectively.
We decomposed these data sets according to the array length, storing all
arrays of lengths $2^{K}$ to $2^{K+1}-1$ consecutively. We applied
differential coding on the arrays (integers $x_{1},x_{2},x_{3},\ldots$ are
transformed to $y_{1}=x_{1},y_{2}=x_{2}-x_{1},y_{3}=x_{3}-x_{2},\ldots$) and
computed the Shannon entropy ($-\sum_{i}p(y_{i})\log_{2}p(y_{i})$) of the
result. We estimate the probability $p(y_{i})$ of the integer value $y_{i}$ as
the number of occurrences of $y_{i}$ divided by the number of integers. As
Fig. 11 shows, longer arrays are more compressible. There are differences in
entropy values between two collections (ClueWeb09 has about two extra bits,
see Fig. 11a), but these differences are much smaller than those among
different array sizes. Fig. 11b shows the distribution of array lengths and
entropy values.
(a) Shannon entropy of the differences (deltas) (b) Data distribution
Figure 11: Description of the posting-list data sets
#### 6.5.1 Results over different array lengths
We present results per array length for selected schemes in Fig. 12. Longer
arrays are more compressible since the deltas, i.e., differences between
adjacent elements, are smaller.
We see in Figs. 12b and 12f that all schemes compress the deltas within a
factor of two of the Shannon entropy for short arrays. For long arrays
however, the compression (compared to the Shannon entropy) becomes worse for
all schemes. Yet many of them manage to remain within a factor of three of the
Shannon entropy.
Integer compression schemes are better able to compress close to the Shannon
entropy over ClueWeb09 (see Fig. 12b) than over GOV2 (see Fig. 12f). For
example, SIMD-FastPFOR, Simple-8b, and OptPFD are within a factor of two of
Shannon entropy over ClueWeb09 for all array lengths, whereas they all exceed
three times the Shannon entropy over GOV2 for the longest arrays. Similarly,
varint-G8IU, SIMD-BP128⋆, and SIMD-FastPFOR⋆ remain within a factor of six of
the Shannon entropy over ClueWeb09 but they all exceed this factor over GOV2
for long arrays. In general, it might be easier to compress data close to the
entropy when the entropy is larger.
We get poor results with varint-G8IU over the longer (and more compressible)
arrays (see Figs. 12a and 12e). We do not find this surprising because
variant-G8IU requires at least 9 bits/int. In effect, when other schemes such
as SIMD-FastPFOR and SIMD-BP128 use less than $\approx 8$ bits/int, they
surpass varint-G8IU in both compression efficiency and decoding speed.
However, when the storage exceeds 9 bits/int, Varint-G8IU is one of the
fastest methods available for these data sets. However, we also got poor
results with variant-G8IU on the ClusterData and Uniform data sets for short
(and poorly compressible) arrays in § 6.4.
We see in Figs. 12c and 12g that both SIMD-BP128 and SIMD-BP128⋆ have a
significantly better encoding speed, irrespective of the array length. The
opposite is true for OptPFD: it is much slower than the alternatives.
Examining the decoding speed as a function of array length (see Figs. 12c and
12g), we see that several schemes have a significantly worse decoding speed
over short (and poorly compressible) arrays, but the effect is most pronounced
for the new schemes we introduced (SIMD-FastPFOR, SIMD-FastPFOR⋆, SIMD-BP128,
and SIMD-BP128⋆). Meanwhile, varint-G8IU and Simple-8b have a decoding speed
that is less sensitive to the array length.
(a) Size: ClueWeb09 (bits/int) (b) Size: ClueWeb09 (relative to entropy)
(c) Encoding: ClueWeb09 (d) Decoding: ClueWeb09
(e) Size: GOV2 (bits/int) (f) Size: GOV2 (relative to entropy)
(g) Encoding: GOV2 (h) Decoding: GOV2
Figure 12: Experimental comparison of competitive schemes on ClueWeb09 and
GOV2.
#### 6.5.2 Aggregated results
Not all posting lists are equally likely to be retrieved by the search engine.
As observed by Stepanov et al. [12], it is desirable to account for different
term distributions in queries. Unfortunately, we do not know of an ideal
approach to this problem. Nevertheless, to model more closely the performance
of a major search engine, we used the AOL query log data set as a collection
of query statistics [52, 53]. It consists of about 20 million web queries
collected from 650 thousand users over three months: queries repeating within
a single user session were ignored. When possible (in about 90% of all cases),
we matched the query terms with posting lists in the ClueWeb09 data set and
obtained term frequencies (see Fig. 11b). This allowed us to estimate how
often a posting list of length between $2^{K}$ to $2^{K+1}-1$ is likely to be
retrieved for various values of $K$. This gave us a weight vector that we use
to aggregate our results.
We present aggregated results in Table 5. The results are generally similar to
what we obtained with synthetic data. The newly introduced schemes (SIMD-
BP128⋆, SIMD-FastPFOR⋆, SIMD-BP128, SIMD-FastPFOR) still offer the best
decoding speed. We find that varint-G8IU⋆ is much faster than varint-G8IU
(1500 mis vs. 1300 mis over GOV2) even though the compression ratio is the
same with a margin of 10%. PFOR and PFOR2008 offer a better compression than
varint-G8IU⋆ but at a reduced speed. However, we find that SIMD-BP128 is
preferable in every way to varint-G8IU⋆, varint-G8IU, PFOR, and PFOR2008.
Table 5: Experimental results. Coding and decoding speeds are given in
millions of 32-bit integers per second. Averages are weighted based on AOL
query logs.
| coding | decoding | bits/int
---|---|---|---
SIMD-BP128⋆ | 1600 | 2300 | 11
SIMD-FastPFOR⋆ | 330 | 1700 | 9.9
SIMD-BP128 | 1000 | 1600 | 9.5
varint-G8IU⋆ | 220 | 1400 | 12
SIMD-FastPFOR | 250 | 1200 | 8.1
PFOR2008 | 260 | 1200 | 10
PFOR | 330 | 1200 | 11
varint-G8IU | 210 | 1200 | 11
BP32 | 760 | 1100 | 8.3
SimplePFOR | 240 | 980 | 7.7
FastPFOR | 240 | 980 | 7.8
NewPFD | 100 | 890 | 8.3
VSEncoding | 11 | 740 | 7.6
Simple-8b | 280 | 730 | 7.5
OptPFD | 14 | 500 | 7.1
Variable Byte | 570 | 540 | 9.6
(a) ClueWeb09
coding | decoding | bits/int
---|---|---
1600 | 2500 | 7.6
350 | 1900 | 7.2
1000 | 1700 | 6.3
240 | 1500 | 10
290 | 1400 | 5.3
250 | 1300 | 7.9
310 | 1300 | 7.9
230 | 1300 | 9.6
790 | 1200 | 5.5
270 | 1100 | 4.8
270 | 1100 | 4.9
150 | 1000 | 5.2
11 | 810 | 5.4
340 | 780 | 4.8
23 | 710 | 4.5
730 | 680 | 8.7
(b) GOV2
For some applications, decoding speed and compression ratios are the most
important metrics. Whereas elsewhere we report the number of bits per integer
$b$, we can easily compute the compression ratio as $32/b$. We plot both
metrics for some competitive schemes (see Fig. 13). These plots suggest that
the most competitive schemes are SIMD-BP128⋆, SIMD-FastPFOR⋆, SIMD-BP128,
SIMD-FastPFOR, SimplePFOR, FastPFOR, Simple-8b, and OptPFD depending on how
much compression is desired. Fig. 13 also shows that to achieve decoding
speeds higher than 1300 mis, we must choose between SIMD-BP128, SIMD-
FastPFOR⋆, and SIMD-BP128⋆.
(a) ClueWeb09 (b) GOV2
Figure 13: Scatter plots comparing competitive schemes on decoding speed and
bits per integer weighted based on AOL query logs. We use VSE as a shorthand
for VSEncoding. For reference, Variable Byte is indicated as a red lozenge.
The novel schemes (e.g., SIMD-BP128⋆) are identified with blue markers.
Few research papers report encoding speed. Yet we find large differences: for
example, VSEncoding and OptPFD are two orders of magnitude slower during
encoding than our fastest schemes. If the compressed arrays are written to
slow disks in a batch mode, such differences might be of little practical
significance. However, for memory-based databases and network applications,
slow encoding speeds could be a concern. For example, the output of a query
might need to be compressed or we might need to index the data in real time
[54]. Our SIMD-BP128 and SIMD-BP128⋆ schemes have especially fast encoding.
Similarly to previous work [12, 24], in Table 6 we report unweighted averages.
The unweighted speed aggregates are equivalent to computing the average speed
over all arrays—irrespective of their lengths. From the distribution of
posting size logarithms in Fig. 11b, one may conclude that weighted results
should be similar to unweighted ones. These observations are supported by data
in Table 6: the decoding speeds and compression ratios for both aggregation
approaches differ by less than 15% with the weighted results presented in
Table 5.
We can compare the number of bits per integer in Table 6 with an information-
theoretic limit. Indeed, the Shannon entropy for the deltas of ClueWeb09 is
5.5 bits/int whereas it is 3.6 for GOV2. Hence, OptPFD is within 16% of the
entropy on ClueWeb09 whereas it is within 22% of the entropy on GOV2.
Meanwhile, the faster SIMD-FastPFOR is within 30% and 40% of the entropy for
ClueWeb09 and GOV2. Our fastest scheme (SIMD-BP128⋆) compresses the deltas of
GOV2 to twice the entropy. It does slightly better with ClueWeb09
($1.8\times$).
Table 6: Average speeds in millions of 32-bit integers per second and bits per
integer over all arrays of two data sets. These averages are not weighted
according to the AOL query logs.
| coding | decoding | bits/int
---|---|---|---
SIMD-BP128⋆ | 1600 | 2400 | 9.7
SIMD-FastPFOR⋆ | 340 | 1700 | 9.0
SIMD-BP128 | 1000 | 1700 | 8.7
varint-G8IU⋆ | 230 | 1400 | 12
SIMD-FastPFOR | 260 | 1300 | 7.2
PFOR2008 | 260 | 1300 | 9.6
PFOR | 330 | 1300 | 9.6
varint-G8IU | 220 | 1200 | 10
BP32 | 770 | 1100 | 7.5
SimplePFOR | 250 | 1000 | 6.9
FastPFOR | 240 | 1000 | 6.9
NewPFD | 110 | 900 | 7.4
VSEncoding | 11 | 760 | 6.9
Simple-8b | 290 | 750 | 6.7
OptPFD | 16 | 530 | 6.4
Variable Byte | 630 | 600 | 9.2
(a) ClueWeb09
coding | decoding | bits/int
---|---|---
1600 | 2500 | 7.4
350 | 1900 | 6.9
1000 | 1800 | 6.1
250 | 1500 | 10
290 | 1400 | 5.1
250 | 1300 | 7.6
310 | 1300 | 7.6
230 | 1200 | 9.6
790 | 1200 | 5.3
260 | 1100 | 4.7
270 | 1100 | 4.8
150 | 990 | 5.0
11 | 790 | 5.4
340 | 780 | 4.6
25 | 720 | 4.4
750 | 700 | 8.6
(b) GOV2
## 7 Discussion
We find that binary packing is both fast and space efficient. The vectorized
binary packing (SIMD-BP128⋆) is our fastest scheme. While it has a lesser
compression efficiency compared to Simple-8b, it is more than 3 times faster
during decoding. Moreover, in the worst case, a slower binary packing scheme
(BP32) incurred a cost of only about 1.2 bits per integer compared to the
patching scheme with the best compression ratio (OptPFD) while decoding nearly
as fast (within 10%) as the fastest patching scheme (PFOR).
Yet only few authors considered binary packing schemes or its vectorized
variants in the recent literature:
* •
Delbru et al. [41] reported good results with a binary packing scheme similar
to our BP32: in their experiments, it surpassed Simple-8b as well as a patched
scheme (PFOR2008).
* •
Anh and Moffat [9] also reported good results with a binary packing scheme: in
their tests, it decoded at least 50% faster than either Simple-8b or PFOR2008.
As a counterpart, they reported that their binary packing scheme had a poorer
compression.
* •
Schlegel et al. [34] proposed a scheme similar to SIMD-BP128. This scheme
(called $k$-gamma) uses a vertical data layout to store integers, like our
SIMD-BP128 and SIMD-FastPFOR schemes. It essentially applies binary packing to
tiny groups of integers (at most 4 elements). From our preliminary
experiments, we learned that decoding integers in small groups is not
efficient. This is also supported by results of Schlegel et al. [34]. Their
fastest decoding speed, which does not include writing back to RAM, is only
1600 mis (Core i7-920, 2.67 Ghz).
* •
Willhalm et al. [47] used a vectorized binary packing like our SIMD-BP128, but
with a horizontal data layout instead of our vertical layout. The decoding
algorithm relies on the shuffle instruction pshufb. Our experimental results
suggest that our approach based on a vertical layout might be preferable (see
Fig. 10a): our implementation of bit unpacking over a vertical layout is
sometimes between 50% to 70% faster than our reimplementation over a
horizontal layout based on the work of Willhalm et al. [47].
This performance comparison depends on the quality of our software. Yet the
speed of our reimplementation is comparable with the speed originally reported
by Willhalm et al. [47, Fig. 11]: they report a speed of $\approx$3300 mis
with a bit width of 6. In contrast, using our implementation of their
algorithms, we got a speed above 4800 mis for the same bit width and a 20%
higher clock speed on a more recent CPU architecture.
The approach described by Willhalm et al. might be more competitive on
platforms with instructions for simultaneously shifting several values by
different offsets (e.g., the vpsrld AVX2 instruction). Indeed, this must be
otherwise emulated by multiplications by powers of two followed by shifting.
Vectorized bit-packing schemes are efficient: they encode/decode integers at
speeds of 4000–8500 mis. Hence, the computation of deltas and prefix sums may
become a major bottleneck. This bottleneck can be removed through
vectorization of these operations (though at expense of poorer compression
ratios in our case). We have not encountered this approach in the literature:
perhaps, because for slower schemes the computation of the prefix sum accounts
for a small fraction of total running time. In our implementation, to ease
comparisons, we have separated differential decoding from data decompression:
an integrated approach could be up to twice as fast in some cases. Moreover,
we might be able improve the decoding speed and the compression ratios with
better vectorized algorithms. There might also be alternatives to data
differencing, which also permit vectorization, such as linear regression [43].
In our results, the original patched coding scheme (PFOR) is bested on all
three metrics (compression ratio, coding and decoding speed) by a binary
packing scheme (SIMD-BP128). Similarly, a more recent fast patching scheme
(NewPFD) is generally bested by another binary packing scheme (BP32). Indeed,
though the compression ratio of NewPFD is up to 6% better on realistic data,
NewPFD is at least 20% slower than BP32. Had we stopped our investigations
there, we might have been tempted to conclude that patched coding is not a
viable solution when decoding speed is the most important characteristic on
desktop processors. However, we designed a new vectorized patching scheme
SIMD-FastPFOR. It shows that patching remains a fruitful strategy even when
SIMD instructions are used. Indeed, it is faster than the SIMD-based
varint-G8IU while providing a much better compression ratio (by at least 35%).
In fact, on realistic data, SIMD-FastPFOR is better than BP32 on two key
metrics: decoding speed and compression ratio (see Fig. 13).
In the future, we may expect increases in the arity of SIMD operations
supported by commodity CPUs (e.g., with AVX) as well as in memory speeds
(e.g., with DDR4 SDRAM). These future improvements could make our vectorized
schemes even faster in comparison to their scalar counterparts. However, an
increase in arity means an increase in the minimum block size. Yet, when we
increase the size of the blocks in binary packing, we also make them less
space efficient in the presence of outlier values. Consider that BP32 is
significantly more space efficient than SIMD-BP128 (e.g., 5.5 bits/int vs. 6.3
bits/int on GOV2).
Thankfully, the problem of outliers in large blocks can be solved through
patching. Indeed, even though OptPFD uses the same block size as SIMD-BP128,
it offers significantly better compression (4.5 bits/int vs. 6.3 bits/int on
GOV2). Thus, patching may be more useful for future computers—capable of
processing larger vectors—than for current ones.
While our work focused on decoding speed, there is promise in directly
processing data while still in compressed form, ideally by using vectorization
[47]. We expect that conceptually simpler schemes (e.g., SIMD-BP128) might
have the advantage over relatively more sophisticated alternatives (e.g.,
SIMD-FastPFOR) for this purpose.
Many of the fastest schemes use relatively large blocks (128 integers) that
are decoded all at once. Yet not all queries require decoding the entire
array. For example, consider the computation of intersections between sorted
arrays. It is sometimes more efficient to use random access, especially when
processing arrays with vastly different lengths [55, 56, 57]. If the data is
stored in relatively large compressed blocks (e.g., 128 integers with SIMD-
BP128), the granularity of random access might be reduced (e.g., when
implementing a skip list). Hence, we may end up having to scan many more
integers than needed. However, blocks of 128 integers might not necessarily be
an impediment to good performance. Indeed, Schlegel et al. [58] were able to
accelerate the computation of intersections by a factor of 5 with
vectorization using blocks of up to 65 536 integers.
## 8 Conclusion
We have presented new schemes that are up to twice as fast as the previously
best available schemes in the literature while offering competitive
compression ratios and encoding speed. This was achieved by vectorization of
almost every step including differential decoding. To achieve both high speed
and competitive compression ratios, we introduced a new patched scheme that
stores exceptions in a way that permits a vectorization (SIMD-FastPFOR).
In the future, we might seek to generalize our results over more varied
architectures as well as to provide a greater range of tradeoffs between speed
and compression ratio. Indeed, most commodity processors support vector
processing (e.g., Intel, AMD, PowerPC, ARM). We might also want to consider
adaptive schemes that compress more aggressively when the data is more
compressible and optimize for speed otherwise. For example, one could use a
scheme such as varint-G8IU for less compressible arrays and SIMD-BP128 for the
more compressible ones. One could also use workload-aware compression:
frequently accessed arrays could be optimized for decoding speed whereas least
frequently accessed data could be optimized for high compression efficiency.
Finally, we should consider more than just 32-bit integers. For example, some
popular search engines (e.g., Sphinx [59]) support 64-bit document
identifiers. We might consider an approach similar to Schlegel et al. [58] who
decompose arrays of 32-bit integers into blocks of 16-bit integers.
Our varint-G8IU implementation is based on code by M. Caron. V. Volkov
provided better loop unrolling for differential coding. P. Bannister provided
a fast algorithm to compute the maximum of the integer logarithm of an array
of integers. We are grateful to N. Kurz for his insights and for his review of
the manuscript. We wish to thank the anonymous reviewers for their valuable
comments.
## References
* [1] Sebot J, Drach-Temam N. Memory bandwidth: The true bottleneck of SIMD multimedia performance on a superscalar processor. _Euro-Par 2001 Parallel Processing_ , _Lecture Notes in Computer Science_ , vol. 2150. Springer Berlin / Heidelberg, 2001; 439–447, 10.1007/3-540-44681-8_63.
* [2] Drepper U. What every programmer should know about memory. http://www.akkadia.org/drepper/cpumemory.pdf [Last checked August 2012.] 2007.
* [3] Manegold S, Kersten ML, Boncz P. Database architecture evolution: mammals flourished long before dinosaurs became extinct. _Proceedings of the VLDB Endowment_ Aug 2009; 2(2):1648–1653.
* [4] Zukowski M. Balancing vectorized query execution with bandwidth-optimized storage. PhD Thesis, Universiteit van Amsterdam 2009.
* [5] Harizopoulos S, Liang V, Abadi DJ, Madden S. Performance tradeoffs in read-optimized databases. _Proceedings of the 32nd international conference on Very Large Data Bases_ , VLDB ’06, VLDB Endowment, 2006; 487–498.
* [6] Westmann T, Kossmann D, Helmer S, Moerkotte G. The implementation and performance of compressed databases. _SIGMOD Record_ September 2000; 29(3):55–67, 10.1145/362084.362137∫.
* [7] Abadi D, Madden S, Ferreira M. Integrating compression and execution in column-oriented database systems. _Proceedings of the 2006 ACM SIGMOD International Conference on Management of data_ , SIGMOD ’06, ACM: New York, NY, USA, 2006; 671–682, 10.1145/1142473.1142548.
* [8] Büttcher S, Clarke CLA. Index compression is good, especially for random access. _Proceedings of the 16th ACM conference on Information and Knowledge Management_ , CIKM ’07, ACM: New York, NY, USA, 2007; 761–770, 10.1145/1321440.1321546.
* [9] Anh VN, Moffat A. Inverted index compression using word-aligned binary codes. _Information Retrieval_ 2005; 8(1):151–166, 10.1023/B:INRT.0000048490.99518.5c.
* [10] Yan H, Ding S, Suel T. Inverted index compression and query processing with optimized document ordering. _Proceedings of the 18th International Conference on World Wide Web_ , WWW ’09, ACM: New York, NY, USA, 2009; 401–410, 10.1145/1526709.1526764.
* [11] Popov P. Basic optimizations: Talk at the YaC (Yet Another Conference) held by Yandex (in Russian). http://yac2011.yandex.com/archive2010/topics/ [Last checked Sept 2012.] 2010\.
* [12] Stepanov AA, Gangolli AR, Rose DE, Ernst RJ, Oberoi PS. SIMD-based decoding of posting lists. _Proceedings of the 20th ACM International Conference on Information and Knowledge Management_ , CIKM ’11, ACM: New York, NY, USA, 2011; 317–326, 10.1145/2063576.2063627.
* [13] Dean J. Challenges in building large-scale information retrieval systems: invited talk. _Proceedings of the Second ACM International Conference on Web Search and Data Mining_ , WSDM ’09, ACM: New York, NY, USA, 2009; 1–1, 10.1145/1498759.1498761. Author’s slides: http://static.googleusercontent.com/external_content/untrusted_dlcp/research.google.com/en/us/people/jeff/WSDM09-keynote.pdf [Last checked Dec. 2012.].
* [14] Lemke C, Sattler KU, Faerber F, Zeier A. Speeding up queries in column stores: a case for compression. _Proceedings of the 12th International Conference on Data Warehousing and Knowledge Discovery_ , DaWaK’10, Springer-Verlag: Berlin, Heidelberg, 2010; 117–129, 10.1007/978-3-642-15105-7_10.
* [15] Binnig C, Hildenbrand S, Färber F. Dictionary-based order-preserving string compression for main memory column stores. _Proceedings of the 2009 ACM SIGMOD International Conference on Management of Data_ , ACM: New York, NY, USA, 2009; 283–296, 10.1145/1559845.1559877.
* [16] Poess M, Potapov D. Data compression in Oracle. _VLDB’03, Proceedings of the 29th International Conference on Very Large Data Bases_ , Morgan Kaufmann: San Francisco, CA, USA, 2003; 937–947.
* [17] Hall A, Bachmann O, Büssow R, Gänceanu S, Nunkesser M. Processing a trillion cells per mouse click. _Proceedings of the VLDB Endowment_ 2012; 5(11):1436–1446.
* [18] Raman V, Swart G. How to wring a table dry: entropy compression of relations and querying of compressed relations. _Proceedings of the 32nd International Conference on Very Large Data Bases_ , VLDB ’06, VLDB Endowment, 2006; 858–869.
* [19] Lemire D, Kaser O. Reordering columns for smaller indexes. _Information Sciences_ June 2011; 181(12):2550–2570, 10.1016/j.ins.2011.02.002.
* [20] Bjørklund TA, Grimsmo N, Gehrke J, Torbjørnsen O. Inverted indexes vs. bitmap indexes in decision support systems. _Proceedings of the 18th ACM conference on Information and Knowledge Management_ , CIKM ’09, ACM: New York, NY, USA, 2009; 1509–1512, 10.1145/1645953.1646158.
* [21] Holloway AL, Raman V, Swart G, DeWitt DJ. How to barter bits for chronons: Compression and bandwidth trade offs for database scans. _Proceedings of the 2007 ACM SIGMOD International Conference on Management of Data_ , ACM: New York, NY, USA, 2007; 389–400.
* [22] Holloway AL, DeWitt DJ. Read-optimized databases, in depth. _Proceedings of the VLDB Endowment_ 2008; 1(1):502–513.
* [23] Zukowski M, Heman S, Nes N, Boncz P. Super-scalar RAM-CPU cache compression. _Proceedings of the 22nd International Conference on Data Engineering_ , ICDE ’06, IEEE Computer Society: Washington, DC, USA, 2006; 59–71, 10.1109/ICDE.2006.150.
* [24] Silvestri F, Venturini R. VSEncoding: efficient coding and fast decoding of integer lists via dynamic programming. _Proceedings of the 19th ACM International Conference on Information and Knowledge Management_ , CIKM ’10, ACM: New York, NY, USA, 2010; 1219–1228, 10.1145/1871437.1871592.
* [25] Anh VN, Moffat A. Index compression using 64-bit words. _Software: Practice and Experience_ 2010; 40(2):131–147, 10.1002/spe.v40:2.
* [26] Zhang J, Long X, Suel T. Performance of compressed inverted list caching in search engines. _Proceedings of the 17th International Conference on World Wide Web_ , WWW ’08, ACM: New York, NY, USA, 2008; 387–396, 10.1145/1367497.1367550.
* [27] Witten IH, Moffat A, Bell TC. _Managing gigabytes (2nd ed.): compressing and indexing documents and images_. Morgan Kaufmann Publishers Inc.: San Francisco, CA, USA, 1999.
* [28] Rice R, Plaunt J. Adaptive variable-length coding for efficient compression of spacecraft television data. _IEEE Transactions on Communication Technology_ 1971; 19(6):889 –897.
* [29] Elias P. Universal codeword sets and representations of the integers. _IEEE Transactions on Information Theory_ 1975; 21(2):194–203.
* [30] Büttcher S, Clarke C, Cormack G. _Information retrieval: Implementing and evaluating search engines_. The MIT Press, 2010.
* [31] Transier F, Sanders P. Engineering basic algorithms of an in-memory text search engine. _ACM Transactions on Information Systems_ Dec 2010; 29(1):2:1–2:37, 10.1145/1877766.1877768.
* [32] Moffat A, Stuiver L. Binary interpolative coding for effective index compression. _Information Retrieval_ 2000; 3(1):25–47, 10.1023/A:1013002601898.
* [33] Walder J, Krátký M, Bača R, Platoš J, Snášel V. Fast decoding algorithms for variable-lengths codes. _Information Sciences_ Jan 2012; 183(1):66–91, 10.1016/j.ins.2011.06.019.
* [34] Schlegel B, Gemulla R, Lehner W. Fast integer compression using SIMD instructions. _Proceedings of the Sixth International Workshop on Data Management on New Hardware_ , DaMoN ’10, ACM: New York, NY, USA, 2010; 34–40, 10.1145/1869389.1869394.
* [35] Cutting D, Pedersen J. Optimization for dynamic inverted index maintenance. _Proceedings of the 13th annual international ACM SIGIR conference on Research and development in information retrieval_ , SIGIR ’90, ACM: New York, NY, USA, 1990; 405–411, 10.1145/96749.98245.
* [36] Williams H, Zobel J. Compressing integers for fast file access. _The Computer Journal_ 1999; 42(3):193–201, 10.1093/comjnl/42.3.193.
* [37] Thiel L, Heaps H. Program design for retrospective searches on large data bases. _Information Storage and Retrieval_ 1972; 8(1):1–20.
* [38] Williams R. An extremely fast ziv-lempel data compression algorithm. _Proceedings of the 1st Data Compression Conference_ , DCC ’91, 1991; 362–371.
* [39] Goldstein J, Ramakrishnan R, Shaft U. Compressing relations and indexes. _Proceedings of the Fourteenth International Conference on Data Engineering_ , ICDE ’98, IEEE Computer Society: Washington, DC, USA, 1998; 370–379.
* [40] Ng WK, Ravishankar CV. Block-oriented compression techniques for large statistical databases. _IEEE Transactions on Knowledge and Data Engineering_ Mar 1997; 9(2):314–328, 10.1109/69.591455.
* [41] Delbru R, Campinas S, Tummarello G. Searching web data: An entity retrieval and high-performance indexing model. _Web Semantics_ Jan 2012; 10:33–58, 10.1016/j.websem.2011.04.004.
* [42] Deveaux JP, Rau-Chaplin A, Zeh N. Adaptive Tuple Differential Coding. _Database and Expert Systems Applications_ , _Lecture Notes in Computer Science_ , vol. 4653. Springer Berlin / Heidelberg, 2007; 109–119, 10.1007/978-3-540-74469-6_12.
* [43] Ao N, Zhang F, Wu D, Stones DS, Wang G, Liu X, Liu J, Lin S. Efficient parallel lists intersection and index compression algorithms using graphics processing units. _Proceedings of the VLDB Endowment_ May 2011; 4(8):470–481.
* [44] Baeza-Yates R, Jonassen S. Modeling static caching in web search engines. _Advances in Information Retrieval_ , _Lecture Notes in Computer Science_ , vol. 7224. Springer Berlin / Heidelberg, 2012; 436–446, 10.1007/978-3-642-28997-2_37.
* [45] Jonassen S, Bratsberg S. Intra-query concurrent pipelined processing for distributed full-text retrieval. _Advances in Information Retrieval_ , _Lecture Notes in Computer Science_ , vol. 7224. Springer Berlin / Heidelberg, 2012; 413–425, 10.1007/978-3-642-28997-2_35.
* [46] Jones DM. _The New C Standard: A Cultural and Economic Commentary_. Addison Wesley Longman Publishing Co., Inc.: Redwood City, CA, USA, 2003.
* [47] Willhalm T, Popovici N, Boshmaf Y, Plattner H, Zeier A, Schaffner J. SIMD-scan: ultra fast in-memory table scan using on-chip vector processing units. _Proceedings of the VLDB Endowment_ Aug 2009; 2(1):385–394.
* [48] Zhou J, Ross KA. Implementing database operations using SIMD instructions. _Proceedings of the 2002 ACM SIGMOD International Conference on Management of Data_ , SIGMOD ’02, ACM: New York, NY, USA, 2002; 145–156, 10.1145/564691.564709.
* [49] Inoue H, Moriyama T, Komatsu H, Nakatani T. A high-performance sorting algorithm for multicore single-instruction multiple-data processors. _Software: Practice and Experience_ Jun 2012; 42(6):753–777, 10.1002/spe.1102.
* [50] Wassenberg J. Lossless asymmetric single instruction multiple data codec. _Software: Practice and Experience_ 2012; 42(9):1095––1106, 10.1002/spe.1109.
* [51] Boystov L. Clueweb09 posting list data set. http://boytsov.info/datasets/clueweb09gap/ [Last checked August 2012.] 2012\.
* [52] Brenes DJ, Gayo-Avello D. Stratified analysis of AOL query log. _Information Sciences_ May 2009; 179(12):1844–1858, 10.1016/j.ins.2009.01.027.
* [53] Pass G, Chowdhury A, Torgeson C. A picture of search. _Proceedings of the 1st International Conference on Scalable Information Systems_ , InfoScale ’06, ACM: New York, NY, USA, 2006, 10.1145/1146847.1146848.
* [54] Fusco F, Stoecklin MP, Vlachos M. NET-FLi: On-the-fly compression, archiving and indexing of streaming network traffic. _Proceedings of the VLDB Endowment_ 2010; 3:1382–1393.
* [55] Baeza-Yates R, Salinger A. Experimental analysis of a fast intersection algorithm for sorted sequences. _Proceedings of the 12th international conference on String Processing and Information Retrieval_ , SPIRE’05, Springer-Verlag: Berlin, Heidelberg, 2005; 13–24.
* [56] Ding B, König AC. Fast set intersection in memory. _Proceedings of the VLDB Endowment_ 2011; 4(4):255–266.
* [57] Vigna S. Quasi-succinct indices. _Proceedings of the Sixth ACM International Conference on Web Search and Data Mining_ , WSDM ’13, ACM: New York, NY, USA, 2013.
* [58] Schlegel B, Willhalm T, Lehner W. Fast sorted-set intersection using SIMD instructions. _ADMS Workshop_ , 2011.
* [59] Aksyonoff A. _Introduction to Search with Sphinx: From installation to relevance tuning_. O’Reilly Media, 2011.
## Appendix A Information-theoretic bound on binary packing
Consider arrays of $n$ distinct sorted 32-bit integers. We can compress the
deltas computed from such arrays using binary packing as described in § 2.6
(see Fig. 1). We want to prove that such an approach is reasonably efficient.
There are ${2^{32}\choose n}$ such arrays. Thus, by an information-theoretic
argument, we need at least $\log{2^{32}\choose n}$ bits to represent them. By
a well known inequality, we have that $\log{2^{32}\choose n}\geq
n\log\frac{2^{32}}{n}$. In effect, this means that we need at least
$\log\frac{2^{32}}{n}$ bits/int.
Consider binary packing over blocks of $B$ integers: e.g., for BP32 we have
$B=32$ and for SIMD-BP128 we have $B=128$. For simplicity, assume that the
array length $n$ is divisible by $B$ and that $B$ is divisible by 32. Though
our result also holds for vectorized differential coding (§ 3), assume that we
use the common version of differential coding before applying binary packing.
That is, if the original array is $x_{1},x_{2},x_{3},\ldots$ ($x_{i}>x_{i-1}$
for all $i>1$), we compress the integers
$x_{1},x_{2}-x_{1},x_{3}-x_{2},\ldots$ using binary packing.
For every block of $B$ integers, we have an overhead of $8$ bits to store the
bit width $b$. This contributes $8n/B$ bits to the total storage cost. The
storage of any given block depends also on the bit width for this block. In
turn, the bit width is bounded by the logarithm of the difference between the
largest and the smallest element in the block. If we write this difference for
block $i$ as $\Delta_{i}$, the total storage cost in bits is
$\displaystyle\frac{8n}{B}+\sum_{i=1}^{n/B}B\lceil\log(\Delta_{i})\rceil$
$\displaystyle\leq$
$\displaystyle\frac{8n}{B}+n+B\log\left(\prod_{i=1}^{n/B}\Delta_{i}\right).$
Because $\sum_{i=1}^{n/B}\Delta_{i}\leq 2^{32}$, we can show that the cost is
maximized when $\Delta_{i}=2^{32}B/n$. Thus, we have that the _total_ cost in
bits is smaller than
$\displaystyle\frac{8n}{B}+n+B\log\left(\prod_{i=1}^{n/B}\frac{2^{32}B}{n}\right)$
$\displaystyle=$
$\displaystyle\frac{8n}{B}+n+B\log\left(\frac{2^{32}B}{n}\right)^{n/B}$
$\displaystyle=$ $\displaystyle\frac{8n}{B}+n+n\log\frac{2^{32}B}{n},$
which is equivalent to $8/B+1+\log B+\log\frac{2^{32}}{n}$ bits/int. Hence, in
the worst case, binary packing is suboptimal by $8/B+1+\log B$ bits/int.
Therefore, we can show that BP32 is 2-optimal for arrays of length less than
$2^{25}$ integers: its storage cost is no more than twice the information-
theoretic limit. We also have that SIMD-BP128 is 2-optimal for arrays of
length $2^{23}$ or less.
|
arxiv-papers
| 2012-09-10T20:08:03 |
2024-09-04T02:49:34.941275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Daniel Lemire and Leonid Boytsov",
"submitter": "Daniel Lemire",
"url": "https://arxiv.org/abs/1209.2137"
}
|
1209.2178
|
arxiv-papers
| 2012-09-10T23:23:16 |
2024-09-04T02:49:34.950980
|
{
"license": "Public Domain",
"authors": "Sutanay Choudhury, Lawrence B. Holder, Abhik Ray, George Chin Jr.,\n John T. Feo",
"submitter": "Sutanay Choudhury",
"url": "https://arxiv.org/abs/1209.2178"
}
|
|
1209.2214
|
# Neutral Triple Gauge Boson production in the large extra dimensions model at
linear colliders
Sun [email protected], Zhou [email protected]
a School of Physics and Technology, University of Jinan, Jinan 250022,
Shandong Province, P.R.China
b School of Physics, Shandong University, Jinan 250100, Shandong Province,
P.R.China
###### Abstract
We consider the neutral triple gauge boson production process in the context
of large extra dimensions (LED) model including the Kaluza-Klein (KK) excited
gravitons at future linear colliders, say ILC(CLIC). We consider
$\gamma\gamma\gamma,\gamma\gamma Z,\gamma ZZ$ and $ZZZ$ production processes,
and analyse their impacts on both the total cross section and some key
distributions. These processes are important for new physics searches at
linear colliders. Our results show that KK graviton exchange has the most
significant effect on $e^{-}e^{+}\to\gamma ZZ$ among the four processes with
relatively small $M_{S}$, while it has the largest effect on
$e^{-}e^{+}\to\gamma\gamma\gamma$ with larger $M_{S}$. By using the neutral
triple gauge boson production we could set the discovery limit on the
fundamental Plank scale $M_{S}$ up to around 6-9 TeV for $\delta$ = 4 at the 3
TeV CLIC.
PACS: 12.10.-g, 13.66.Fg, 14.70.-e
## 1 Introduction
The hierarchy problem of the standard model (SM) strongly suggests new physics
at TeV scale, and the idea that there exists extra dimensions (ED) which first
proposed by Arkani-Hamed, Dimopoulos, and Dvali[1] might provide a solution to
this problem. They proposed a scenario in which the SM field is constrained to
the common 3+1 space-time dimensions (“brane”), while gravity is free to
propagate throughout a larger multidimensional space $D=\delta+4$ (“bulk”).
The picture of a massless graviton propagating in D dimensions is equal to the
picture that numerous massive Kaluza-Klein (KK) gravitons propagate in 4
dimensions. The fundamental Planck scale $M_{S}$ is related to the Plank mass
scale $M_{Pl}=G_{N}^{-1/2}=1.22\times 10^{19}~{}{\rm GeV}$ according to the
formula $M^{2}_{Pl}=8\pi M^{\delta+2}_{S}R^{\delta}$ , where $R$ and $\delta$
are the size and number of the extra dimensions, respectively. If $R$ is large
enough to make $M_{S}$ on the order of the electroweak symmetry breaking scale
($\sim 1~{}{\rm TeV}$), the hierarchy problem will be naturally solved, so
this extra dimension model is called the large extra dimension model (LED) or
the ADD model. Postulating $M_{S}$ to be 1 TeV, we get $R\sim 10^{13}~{}{\rm
cm}$ for $\delta=1$, which is obviously ruled out since it would modify
Newton’s law of gravity at solar-system distances; and we get $R\sim 1~{}{\rm
mm}$ for $\delta=2$, which is also ruled out by the torsion-balance
experiments[2]. When $\delta\geq 3$, where $R<1~{}{\rm nm}$, it is possible to
detect graviton signal at high energy colliders.
At colliders, exchange of virtual KK graviton or emission of a real KK mode
could give rise to interesting phenomenological signals at TeV scale[3, 4].
Virtual effects of KK modes could lead to the enhancement of the cross section
of pair productions in processes, for example, di-lepton, di-gauge boson
($\gamma\gamma$, $ZZ$, $W^{+}W^{-}$), dijet, $t\bar{t}$ pair, HH pair[5, 6, 7,
8, 9, 10, 11] etc. The real emission of a KK mode could lead to large missing
$E_{T}$ signals viz. mono jet, mono gauge boson[3, 4, 12, 13] etc. The CMS
Collaboration has performed a lot of search for LED on different final states
at $\sqrt{s}=7$ TeV[14, 15, 16], and they set the most stringent lower limits
to date to be $2.5~{}{\rm TeV}<M_{S}<3.8~{}{\rm TeV}$ by combining the
diphoton, dimuon and dielectron channels.
Studies for LED have been extended to three body final state processes in
recent years. Triple gauge bosons productions in the SM are important because
they involve 3-point and 4-point gauge couplings in the contributing diagrams,
which allow for restrictive tests of triple and quartic vector boson coupling.
And also they might contribute backgrounds to new physics beyond the SM.
Furthermore they are sensitive to new physics. This kind of processes have
been studied at LO[17, 18, 19] and NLO[20, 21, 22, 23, 24, 25] in the SM, and
virtual graviton exchange effects to these processes within the LED model at
LHC are studied recently[26]. Linear colliders have more advantage in testing
extra dimensions than LHC for the following reasons. First, even though the
LHC has much higher center-of-mass (c.m.s.) energy than linear colliders, the
theoretical amplitude at LHC is hampered by the unitary constraint
$\sqrt{\hat{s}}<{\rm M_{S}}$. Second, linear colliders have cleaner
environment than LHC, so it’s much easier to select the ED signals. The
capabilities of the planned International Linear Collider (ILC) and Compact
Linear Collider (CLIC) for precision Higgs studies are well documented[27,
28]. They will also provide opportunities for the search for new physics
beyond SM. So in this paper we consider the triple gauge bosons production at
ILC and CLIC $e^{-}e^{+}\to VVV$ within the LED model, where we restrict V to
be neutral gauge boson ($V=\gamma,Z$). The following four final states are the
subject of this analysis: (i) $\gamma\gamma\gamma$ (ii) $\gamma\gamma Z$ (iii)
$\gamma ZZ$ (iv) $ZZZ$. The case where $V=W^{\pm}$ is in prepare and will be
part of a different paper.
This paper is organized as follows: in section 2 we present the analytical
calculation of the processes mentioned above with a brief introduction to the
LED model, section 3 is arranged to present the numerical results of our
studies, and finally we summarize the results in the last section.
## 2 Theoretical Framework
In this section we give the analytical calculations of the process
$e^{-}e^{+}\to VVV$ with $V=\gamma,Z$ at linear colliders in the LED model. In
our calculation we use the de Donder gauge. The relevant Feynman rules
involving graviton in the LED model can be found in Ref.[4]. We denote the
process as:
$\displaystyle e^{-}(p_{1})+e^{+}(p_{2})\rightarrow
V(p_{3})+V(p_{4})+V(p_{5})$ (1)
where $p_{1}$, $p_{2}$ and $p_{3}$, $p_{4}$, $p_{5}$ represent the momenta of
the incoming and outgoing particles respectively.
Figure 1: Feynman diagrams for $e^{-}e^{+}\to VVV$ process with $V=\gamma,Z$.
(a,b) are the diagrams in the SM, and (c,d,e) are the diagrams in the LED
model.
In Fig.1 we display the Feynman diagrams for this process in both the SM and
LED model, among which (a) and (b) are SM diagrams and (c)$\sim$(e) are LED
diagrams. We have neglected the Higgs coupling to electrons because the Yukawa
coupling is proportional to the fermion mass. But the Higgs coupling to Z
bosons, which appears in $e^{-}e^{+}\to ZZZ$ process (Fig.1(b)), can’t be
neglected because of its large contribution, e.g., with $\sqrt{s}$ to be
$300-3000~{}{\rm GeV}$, the cross section for Fig.1(b) is about $26\%-9\%$ of
the total SM cross section. After considering all possible permutations, we
have 12 SM diagrams for the $ZZZ$ production process and 9 SM diagrams for the
other three processes, and we have 12 LED diagrams for the
$\gamma\gamma\gamma$ and $ZZZ$ processes and 4 LED diagrams for $\gamma\gamma
Z$ and $\gamma ZZ$ processes.
In our calculation we consider both the spin-0 and spin-2 KK mode exchange
effect. The spin-0 states only couple through the dilaton mode, which have
none contribution to $\gamma\gamma\gamma$ and $\gamma\gamma Z$ processes and
could contribute to $\gamma ZZ$ and $ZZZ$ production processes through
couplings to massive gauge bosons. However the cross sections coming from the
dilaton mode are so small that can be neglected, e.g., they are at most about
$10^{-12}$ and $10^{-8}$ times of the total cross sections for $\gamma ZZ$ and
$ZZZ$ production processes, respectively. So we focus our study on the spin-2
component of the KK states.
The couplings between gravitons and SM particles are proportional to a
constant named gravitational coupling $\kappa\equiv\sqrt{16\pi G_{N}}$, which
can be expressed in terms of the fundamental Plank scale $M_{S}$ and the size
of the compactified space R by
$\displaystyle\kappa^{2}R^{\delta}=8\pi(4\pi)^{\delta/2}\Gamma(\delta/2)M_{S}^{-(\delta+2)}$
(2)
In practical experiments, the contributions of the different Kaluza-Klein
modes have to be summed up, so the propagator is proportional to
$i/(s_{ij}-m^{2}_{\vec{n}})$, where $s_{ij}=(p_{i}+p_{j})^{2}$ and
$m_{\vec{n}}$ is the mass of the KK state $\vec{n}$. Thus, when the effects of
all the KK states are taken together, the amplitude is proportional to
$\sum\limits_{\vec{n}}\frac{i}{s_{ij}-m^{2}_{\vec{n}}+i\epsilon}=D(s)$. If
$\delta\geq 2$ this summation is formally divergent as $m_{\vec{n}}$ becomes
large. We assume that the distribution has a ultraviolet cutoff at
$m_{\vec{n}}\sim M_{S}$, where the underlying theory becomes manifest. Then
$D(s)$ can be expressed as:
$\displaystyle
D(s)=\frac{1}{\kappa^{2}}\frac{8\pi}{M_{S}^{4}}(\frac{\sqrt{s}}{M_{S}})^{\delta-2}[\pi+2iI(M_{S}/\sqrt{s})].$
(3)
The imaginary part I($\Lambda/\sqrt{s}$) is from the summation over the many
non-resonant KK states and its expression can be found in Ref.[4]. Finally the
KK graviton propagator after summing over the KK states is:
$\displaystyle\tilde{G}^{\mu\nu\alpha\beta}_{KK}=D(s)\left(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\frac{2}{D-2}\eta_{\mu\nu}\eta_{\alpha\beta}\right)$
(4)
Using the Feynman rules in the LED model and the propagator given by Eq.(4),
we can get the amplitudes for the virtual KK graviton exchange diagrams in
Fig.1. The total amplitude can be obtained by adding these LED amplitudes
together with the SM ones. The total cross section can be expressed as the
integration over the phase space of three-body final state:
$\displaystyle\sigma_{tot}$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{4}}{4|\vec{p}_{1}|\sqrt{s}}\int
d\Phi_{3}\overline{\sum}|{\cal M}_{tot}|^{2}.$ (5)
where $\overline{\sum}$ represents the summation over the spins of final
particles and the average over the spins of initial particles. The phase-space
element $d\Phi_{3}$ is defined by
$\displaystyle{d\Phi_{3}}=\delta^{(4)}\left(p_{1}+p_{2}-\sum_{i=3}^{5}p_{i}\right)\prod_{j=3}^{5}\frac{d^{3}\textbf{{p}}_{j}}{(2\pi)^{3}2E_{j}}.$
(6)
## 3 Numerical Results
### 3.1 Input parameters and kinematical cuts
We use FeynArts and FormCalc package[29, 30] to generate and reduce the
amplitudes and then implement numerical calculation. We use BASES[31] to
perform the phase space integration and CERN library to display the
distributions. The SM parameters are taken as follows[32]:
$\displaystyle m_{Z}=91.1876~{}{\rm GeV},~{}m_{W}=80.399~{}{\rm
GeV},~{}m_{e}=0.511~{}{\rm MeV},$
$\displaystyle\alpha(m^{2}_{Z})=1/127.934,~{}m_{H}=125~{}{\rm
GeV}\cite[cite]{[\@@bibref{}{SMHiggs125GeV_ATLAS, SMHiggs125GeV_CMS}{}{}]}.$
We take the cuts on final particles as:
$\displaystyle p_{T}^{\gamma,Z}\geqslant 15~{}{\rm GeV},\ \
\eta^{\gamma,Z}\leqslant 2.5,\ \ R_{\gamma\gamma}\geqslant 0.4$ (7)
where R is defined as $R=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$, with
$\Delta\phi$ and $\Delta\eta$ denoting the separation between the two
particles in azimuthal angle and pseudo-rapidity respectively.
### 3.2 Total Cross sections
In Fig.2 we present the cross sections for
$e^{-}e^{+}\rightarrow\gamma\gamma\gamma,\gamma\gamma Z,\gamma ZZ\ {\rm and}\
ZZZ$ processes as the functions of $\sqrt{s}$ for $\delta=3$ with different
values of $M_{S}$. The solid lines are the SM results. The dashed, dotted and
dot-dashed lines are corresponding to the cross sections in the LED model for
$M_{S}=3.5$ TeV, 4.5 TeV and 5.5 TeV respectively. When $\sqrt{s}$ is less
than about 1 TeV, the curves for the total cross sections including the LED
effect seem to be overlapped with that in the SM, then the LED effect becomes
significant with the increment of $\sqrt{s}$. When $M_{S}=3.5$ TeV, the
$e^{-}e^{+}\to\gamma ZZ$ process has the most significant LED effect among the
four processes considered in this paper, while the $e^{-}e^{+}\to\gamma\gamma
Z$ process has the least. When $\sqrt{s}=500~{}{\rm GeV}$, the cross sections
for $\gamma\gamma\gamma$ and $\gamma\gamma Z$ production processes are
comparable, which are 91 fb and 78 fb respectively, because they have
comparable phase-space. While the cross sections for $\gamma ZZ$ and $ZZZ$
processes are much smaller due to the less phase-space, which are 18 fb and 1
fb, respectively. With $\sqrt{s}$ increase to be 3 TeV, the cross section for
$\gamma ZZ$ are enhanced to 347 fb, which is even larger than the
$\gamma\gamma\gamma$ process (323 fb), and the cross sections for $ZZZ$
process is enhanced to 34 fb, which is comparable to the $\gamma\gamma Z$
process (36 fb). With larger $M_{S}$ value (5.5 TeV), the LED contribution to
$\gamma\gamma\gamma$ production process will exceed $\gamma ZZ$ , that’s why
$e^{-}e^{+}\to\gamma\gamma\gamma$ process puts the highest limits on $M_{S}$,
as we will see later.
Figure 2: The cross sections for the process $e^{-}e^{+}\to VVV$ with
$V=\gamma,Z$ in the SM and LED model as the function of $\sqrt{s}$ with
$M_{s}=$ 3.5, 4.5, 5.5 TeV and $\delta=3$.
In Fig.3 we present the dependence of the cross section on energy scale
$M_{S}$ with $\sqrt{s}=1$ TeV, 2 TeV and 3 TeV respectively. In each figure of
Fig.3, we present the curves for the cross sections with the extra dimension
$\delta$ value being 3, 4, 5 and 6 separately. The solid straight lines, which
are independent of $M_{S}$, are the SM results, and the dashed, dotted, dash-
dotted and dash-dot-dotted lines are the cross sections for $\delta$=3, 4, 5
and 6 respectively. It’s clear that for a given value of $\delta$, the cross
section decreases rapidly with the increment of $M_{S}$, and finally
approaches to its corresponding SM result. We can see again that the virtual
KK graviton exchange contribution decreases with the increment of the $\delta$
value. The LED effect on the cross sections with $\sqrt{s}=1$ TeV is too small
to be detected, especially for $e^{-}e^{+}\to\gamma\gamma Z$ process, which is
coincidence with Fig.2. If we got high enough c.m.s energy, say 3 TeV, the
cross sections would be very significant when $M_{S}$ is not very large. Even
with $M_{S}=6$ TeV, the cross sections are still several times of the SM ones.
Figure 3: The cross sections for the process $e^{-}e^{+}\rightarrow VVV$ in
the SM and LED model as the function of $M_{S}$ with $\sqrt{s}=$ 1, 2, 3 TeV
and $\delta=3,~{}4,~{}5,~{}6$.
### 3.3 Distributions
The distributions of the Gauge boson pair invariance mass $M_{VV}$
($VV=\gamma\gamma,~{}ZZ$) and the Gauge boson transverse momentum $p_{T}^{V}$
as well as their rapidity $y^{V}$ at the 3 TeV CILC, are shown in Fig.4-6. The
results are for $M_{S}=6.5$ TeV at the fixed value 4 for the number of extra
dimensions and obtained by taking the input parameters mentioned above.
### $e^{-}e^{+}\rightarrow\gamma\gamma\gamma$
Before selecting our event samples for triple $\gamma$ production, we order
the photons on the basis of their transverse momentum i.e.,
$p_{T}^{\gamma_{1}}\geq p_{T}^{\gamma_{2}}\geq p_{T}^{\gamma_{3}}$. For
$e^{-}e^{+}\rightarrow\gamma\gamma\gamma$, we are interested in the
$p_{T}^{\gamma}$ and $y^{\gamma}$ distribution which are displayed in the left
and right panel in Fig.4, respectively. The solid, dashed and dotted lines
refer to $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$, respectively. In high
$p_{T}^{\gamma_{1}}$ and $p_{T}^{\gamma_{2}}$ region, the LED effect dominant
the total (SM+LED) distribution, because more KK modes contribute with the
increase of $p_{T}$. Difference can be found for the $p_{T}^{\gamma_{3}}$
production, although it’s still enhanced by the LED effects, it’s low $p_{T}$
region is dominant while in high $p_{T}$ region it becomes much smaller.
Rapidity distribution of the related photon has been shown in the right panel
in Fig.4. As we can see, the rapidity distributions in the LED model show
significantly peaks around $y=0$, which implies the large contributions at
high $p_{T}^{\gamma_{1,2}}$ region. Compare with the $\gamma_{1}$ and
$\gamma_{2}$ distribution, the $y$ distribution of $\gamma_{3}$ seems much
flatter.
Figure 4: The transverse momentum($p_{T}$) and Rapidity($y$) distribution of
photons for the process $e^{-}e^{+}\rightarrow\gamma\gamma\gamma$, on the
basis of their transverse momentum $p_{T}^{\gamma_{1}}\geq
p_{T}^{\gamma_{2}}\geq p_{T}^{\gamma_{3}}$ with $M_{S}=$ 6.5TeV, $\sqrt{s}=$3
TeV and $\delta=4$. The solid, dashed and dotted lines refer to
$p_{T}^{\gamma_{1}}$($y^{\gamma_{1}}$), $p_{T}^{\gamma_{2}}$($y^{\gamma_{2}}$)
and $p_{T}^{\gamma_{3}}$($y^{\gamma_{3}}$).
### $e^{-}e^{+}\rightarrow ZZZ$
Similar to the $e^{-}e^{+}\rightarrow\gamma\gamma\gamma$ production, triple Z
bosons final particles are classified in such a way that $p_{T}^{Z_{1}}\geq
p_{T}^{Z_{2}}\geq p_{T}^{Z_{3}}$. Similar conclusion can be found for the
$e^{-}e^{+}\rightarrow ZZZ$ production. It’s not strange that the signal of
triple $\gamma$ signal is larger than the triple Z production since the three
Z bosons suppress the phase space integration extremely, so that the total
cross sections as well as the distributions become smaller as can be seen in
Fig.4 and Fig.5, the peak is around 0.0006 fb/GeV for $p_{T}^{Z_{1}}$ compared
to 0.01 fb/GeV for $p_{T}^{\gamma_{1}}$ in the high $p_{T}$ region. For the
$y$ distributions, the peaks for the $ZZZ$ production are narrower than the
$\gamma\gamma\gamma$ distributions, however, the conclusion is the same that
the rapidity distributions in the LED model show significant peaks around
$y=0$.
Figure 5: The transverse momentum($p_{T}$)and Rapidity($y$) distribution of
$Z$ bosons for the process $e^{-}e^{+}\rightarrow ZZZ$, on the basis of their
transverse momentum $p_{T}^{Z_{1}}\geq p_{T}^{Z_{2}}\geq p_{T}^{Z_{3}}$ with
$M_{S}=6.5$ TeV, $\sqrt{s}=3$ TeV and $\delta=4$. The solid, dashed and dotted
lines refer to $p_{T}^{Z_{1}}$($y^{Z_{1}}$), $p_{T}^{Z_{2}}$($y^{Z_{2}}$) and
$p_{T}^{Z_{3}}$($y^{Z_{3}}$).
### $e^{-}e^{+}\rightarrow\gamma\gamma Z$ and $e^{-}e^{+}\rightarrow\gamma
ZZ$
Now let’s see the distributions for the $e^{-}e^{+}\rightarrow\gamma\gamma Z$
and $e^{-}e^{+}\rightarrow\gamma ZZ$ productions. The photon pair decay of the
KK graviton is one of the clean decay modes, so the distribution of the
invariant mass of the photon pair ($M_{\gamma\gamma}$) is a useful observable
for $e^{-}e^{+}\to\gamma\gamma Z$ . An obvious enhancement on the tail of this
distribution makes such region of extreme interest. Typically, we find that
the KK modes dominate over SM contribution for larger values of invariant
masses (say above 1 TeV for a given set of $M_{S}$ and $\delta$ values, here
we give $M_{S}=4.5$ TeV and $\delta=4$) of photon pairs indicating the
observable nature of the signal, see the two dotted line in Fig.6. The upper
and lower ones refer to the SM predict and SM+LED effects. For the process
$e^{-}e^{+}\rightarrow\gamma ZZ$, it is similar to $\gamma\gamma Z$ production
process, and in this case, the invariant mass of Z boson pair($M_{ZZ}$) is a
useful observable. We thus display it in Fig.6, see the solid lines. These two
solid line branch at about the invariance mass 1 TeV, and the upper and lower
ones present the SM and SM+LED effects, respectively.
Figure 6: Invariant mass distribution of $M_{\gamma\gamma(ZZ)}$ for
$e^{-}e^{+}\rightarrow\gamma\gamma Z$($e^{-}e^{+}\rightarrow\gamma ZZ$) for
$M_{S}=4.5$ TeV, $\sqrt{s}=3$ TeV and $\delta=4$.
It is clear that if the deviation of the cross section from the SM prediction
is large enough, the LED effects can be found. We assume that the LED effects
can and cannot be observed, only if[35]
$\displaystyle\Delta\sigma=|\sigma_{tot}-\sigma_{SM}|\geq\frac{5\sqrt{{\cal
L}\sigma_{tot}}}{{\cal L}}\equiv 5\sigma$ (8)
and
$\displaystyle\Delta\sigma=|\sigma_{tot}-\sigma_{SM}|\leq\frac{3\sqrt{{\cal
L}\sigma_{tot}}}{{\cal L}}\equiv 3\sigma$ (9)
$\sqrt{s}$ | | 1 TeV | | 2 TeV | | 3 TeV | |
---|---|---|---|---|---|---|---|---
| | $5\sigma$ | $3\sigma$ | | $5\sigma$ | $3\sigma$ | | $5\sigma$ | $3\sigma$ | |
$e^{-}e^{+}\rightarrow\gamma\gamma\gamma$ | | 3599 | 4093 | | 6371 | 7187 | | 8906 | 10023 | |
$e^{-}e^{+}\rightarrow ZZZ$ | | 2530 | 2858 | | 4711 | 5312 | | 6730 | 7584 | |
$e^{-}e^{+}\rightarrow\gamma\gamma Z$ | | 2052 | 2289 | | 4007 | 4458 | | 5823 | 6500 | |
$e^{-}e^{+}\rightarrow\gamma ZZ$ | | 3406 | 3835 | | 6027 | 6757 | | 8412 | 9421 | |
Table 1: The discovery ($\Delta\sigma\geq 5\sigma$) and exclusion
($\Delta\sigma\leq 3\sigma$) LED model fundamental scale ($M_{S}$) values for
the $e^{-}e^{+}\to VVV$ processes at the $\sqrt{s}$ = 1, 2, 3 TeV ILC(CLIC).
${\cal L}=300~{}{\rm fb}^{-1}$, $\delta=4$.
Our final results show that by using the $\gamma\gamma\gamma$ production we
can set the discovery limit on the fundamental Plank scale $M_{S}$ up to
3.1-9.9 TeV, depending on the extra dimension $\delta\subset[3,6]$, with the
luminosity 300 fb-1 and the colliding energy 1-3 TeV. For the other three
final states $\gamma\gamma Z$ , $\gamma ZZ$ and $ZZZ$ , the limits are 1.8-6.3
TeV, 2.9-9.3 TeV and 2.2-7.4 TeV, respectively. To do a more detailed
description, in Table 1, we present the $5\sigma$ discovery and $3\sigma$
exclusion fundamental scale $M_{S}$ values at the ILC/CLIC with the luminosity
300 fb-1 for $\delta=4$ . It shows that compared to the other three channels,
$\gamma\gamma\gamma$ can set the discovery limit bounds much higher, up to 8.9
TeV. The phenomenology of the neutral triple gauge boson production at the
near future is much richer at linear colliders, though its production cannot
give compete limits as, for example, dilepton production gives, it’s still
very interesting and important.
| | ${\cal L}$ (${\rm fb}^{-1}$) |
---|---|---|---
| | $5\sigma$ | $3\sigma$
$pp\rightarrow\gamma\gamma\gamma$ | | 960 | 1500
$pp\rightarrow ZZZ$ | | 21 | 21
$pp\rightarrow\gamma\gamma Z$ | | 20 | 17
$pp\rightarrow\gamma ZZ$ | | 170 | 140
Table 2: Integrated luminosity needed at 14 TeV LHC to accomplish the
discovery and exclusion bounds at a 1 TeV linear collider, which are listed in
the first two columns in Table 1, using the neutral triple gauge boson
production processes, with $\delta=4$.
To make a comparison with Ref.[26], we repeat the $pp\to VVV$ ($V=\gamma,Z$)
process at LHC, using the same parameters and cuts with Ref.[26], and find
that our results are in good agreement with theirs. In Table 2 we list the
integrated luminosity the 14 TeV LHC needed to accomplish the discovery and
exclusion bounds at a 1 TeV LC (the first two column data listed in Table 1),
by using the corresponding $VVV$ production channels with extra dimensions
$\delta=4$. The table shows that with years of collection of data, LHC could
accomplish the discovery and exclusion limits set by a 1 TeV LC, even for the
most challenging channel $\gamma\gamma\gamma$ . While the limits set by a 2 or
3 TeV LC are much higher, and the required amounts of data for matching these
bounds are too large to be a reasonable projection for the LHC reach.
## 4 Summary and Conclusions
In a short summary, we calculate the neutral gauge boson production processes
$\gamma\gamma\gamma$, $\gamma\gamma Z$, $\gamma ZZ$ and $ZZZ$ in the SM and
LED model at ILC and CLIC. We investigate the integrated cross sections, the
distributions of some kinematic variables $M_{VV}$, $p_{T}^{V}$ and $y^{V}$.
The 5$\sigma$ discovery and 3$\sigma$ exclusion ranges for the LED parameters
$M_{S}$ are obtained and compared between different channels. It turns out
that the effects of the virtual KK graviton enhance the total cross sections
and differential distributions of kinematical observables generally. Among the
four processes we considered, $e^{-}e^{+}\to\gamma ZZ$ or
$e^{-}e^{+}\to\gamma\gamma\gamma$ process has the most significant LED effect
with relatively small or large $M_{S}$, respectively. While
$e^{-}e^{+}\to\gamma\gamma Z$ has the least contribution from LED diagrams.
With the development of linear colliders, more information related to LED
effects can be obtained experimentally through such important productions. At
the 3 TeV CLIC, it is expected that $\gamma ZZ$ production can be used to
explore a range of $M_{S}$ values up to 7.3-9.3 TeV depending on the number of
extra dimensions. Through $\gamma\gamma\gamma$ production, we can extend this
search up to 9.9 TeV, While using $\gamma\gamma Z$ and $ZZZ$ productions,
lower bounds on $M_{S}$ can be found, which are 6.3 TeV and 7.4 TeV,
respectively.
## 5 Acknowledgments
We would like to thank Prof. Zhang Ren-You for useful discussions. Project
supported by the National Natural Science Foundation of China (Grant
No.11147151, No.11205070, No.11105083, No.10947139 and No.11035003), and by
Shandong Province Natural Science Foundation (No.ZR2012AQ017).
## References
* [1] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998) [hep-ph/9803315]; N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Rev. D 59, 086004 (1999) [hep-ph/9807344].
* [2] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle and H. E. Swanson, Phys. Rev. Lett. 98, 021101 (2007) [arXiv:hep-ph/0611184].
* [3] Gian F. Giudice, Riccardo Rattazzi, James D. Wells, Nucl.Phys. B544, 3-38 (1999).
* [4] Tao Han, Joseph D. Lykken, Ren-Jie Zhang, Phys.Rev.D59:105006(1999).
* [5] J. L. Hewett, Phys. Rev. Lett. 82 (1999) 4765; Prakash Mathews, V. Ravindran, K. Sridhar and W. L. van Neerven, Nucl. Phys. B713 (2005) 333; Prakash Mathews, V. Ravindran, Nucl. Phys. B753 (2006) 1; M.C. Kumar, Prakash Mathews, V. Ravindran, Eur. Phys. J. C49 (2007) 599.
* [6] O. J. P. Eboli, Tao Han, M. B. Magro, P. G. Mercadante, Phys. Rev. D61 (2000) 094007; K.m. Cheung and G. L. Landsberg, Phys. Rev. D 62 (2000) 076003; M.C. Kumar, Prakash Mathews, V. Ravindran, Anurag Tripathi, Phys. Lett. B672 (2009) 45; Nucl. Phys. B818 (2009) 28.
* [7] M. Kober, B. Koch and M. Bleicher, Phys. Rev. D 76, 125001 (2007) [arXiv:0708.2368]; J. Gao, C. S. Li, X. Gao and J. J. Zhang, Phys. Rev. D 80, 016008 (2009) [arXiv:0903.2551]; Neelima Agarwal, V. Ravindran, V. K. Tiwari, Anurag Tripathi, Nucl. Phys. B830 (2010) 248.
* [8] Z. U. Usubov and I. A. Minashvili, Phys. Part. Nucl. Lett. 3, 153 (2006) [Pisma Fiz. Elem. Chast. Atom. Yadra 3, 24 (2006)]; K. Y. Lee, H. S. Song and J. -H. Song, Phys. Lett. B 464, 82 (1999) [hep-ph/9904355]. Neelima Agarwal, V. Ravindran, V. K. Tiwari, Anurag Tripathi, Phys. Rev. D82 (2010) 036001; B. Yu-Ming, G. Lei, L. Xiao-Zhou, M. Wen-Gan and Z. Ren-You, Phys. Rev. D 85, 016008 (2012) [arXiv:1112.4894].
* [9] Prakash Mathews, Sreerup Raychaudhuri, K. Sridhar, Phys. Lett. B450 (1999) 343; JHEP 0007 (2000) 008.
* [10] K. Y. Lee, H. S. Song, J. -H. Song and C. Yu, Phys. Rev. D 60, 093002 (1999) [hep-ph/9905227]; K. Y. Lee, S. C. Park, H. S. Song, J. -H. Song and C. Yu, Phys. Rev. D 61, 074005 (2000) [hep-ph/9910466]; hep-ph/0105326; S. C. Inan and A. A. Billur, Phys. Rev. D 84, 095002 (2011).
* [11] C. S. Kim, Kang Young Lee and Jeonghyeon Song, Phys.Rev.D64:015009(2001); A. Datta, E. Gabrielli and B. Mele, JHEP 0310, 003 (2003) [hep-ph/0303259]; Hao Sun, Ya-Jin Zhou, He Chen, Eur. Phys. J. C (2012) 72:2011.
* [12] E. A. Mirabelli, M. Perelstein and M. E. Peskin, Phys. Rev. Lett. 82, 2236 (1999) [hep-ph/9811337].
* [13] K. -m. Cheung and W. -Y. Keung, Phys. Rev. D 60, 112003 (1999) [hep-ph/9903294].
* [14] C. Collaboration [CMS Collaboration], arXiv:1204.0821 [hep-ex].
* [15] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 711, 15 (2012) [arXiv:1202.3827 [hep-ex]].
* [16] S. Chatrchyan et al. [CMS Collaboration], arXiv:1112.0688 [hep-ex].
* [17] M. Golden and S. R. Sharpe, Nucl. Phys. B 261, 217 (1985).
* [18] V. D. Barger and T. Han, Phys. Lett. B 212, 117 (1988).
* [19] Tao Han and Ron Sobey, Phys.Rev. D52 (1995) 6302-6308.
* [20] Fawzi Boudjema, Le Duc Ninh, Sun Hao, Marcus M. Weber, Phys.Rev.D81 (2010)073007; Fortsch. Phys.58 :656-659 (2010);
* [21] Su Ji-Juan, Ma Wen-Gan, Zhang Ren-You, Wang Shao-Ming, Guo Lei, Phys.Rev.D78(2008)016007;
* [22] Sun Wei, Ma Wen-Gan, Zhang Ren-You, Guo Lei, Song Mao, Phys.Lett.B680(2009)321-327;
* [23] T. Binoth, G. Ossola, C.G. Papadopoulos, R. Pittau, JHEP 0806 (2008) 082;
* [24] A. Lazopoulos, K. Melnikov, F. Petriello, Phys. Rev. D 76 (2007) 014001;
* [25] G. Bozzi, F. Campanario, V. Hankele, D. Zeppenfeld, Phys. Rev. D 81 (2010) 094030; G. Bozzi, F. Campanario, M. Rauch, D. Zeppenfeld, Phys. Rev. D 84 (2011) 074028.
* [26] M. C. Kumar, Prakash Mathews, V. Ravindran, Satyajit Seth, DESY 11-229, [arXiv:1111.7063].
* [27] G. Aarons et al. [ILC Global Design Effort and World Wide Study], International Linear Collider Reference Design Report Volume 2: Physics at the ILC, edited by A. Djouadi, J. Lykken, K. M$\ddot{o}$ig, Y. Okada, M. Oreglia and S. Yamashita, arXiv:0709.1893.
* [28] L. Linssen, A. Miyamoto, M. Stanitzki and H. Weerts (editors), Physics and Detectors at CLIC : the CLIC Conceptual Design Report, arXiv:1202.5940 [physics.ins-det].
* [29] T. Hahn, Comput.Phys.Commun. 140, 418-431 (2001).
* [30] T. Hahn, Nucl.Phys.Proc.Suppl. 89, 231-236 (2000).
* [31] S. Kawabata, Comp. Phys. Commun. 88, 309 (1995). F. Yuasa, D. Perret-Gallix, S. Kawabata, and T. Ishikawa, Nucl. Instrum. Meth. A389, 77 (1997).
* [32] Particle Data Group, K. Nakamura et al., JPG 37, 075021 (2010).
* [33] G. Aad et al. [ATLAS Collaboration],CERN-PH-EP-2012-218, [arXiv:1207.7214 [hep-ex]].
* [34] S. Chatrchyan et al. [CMS Collaboration],CMS-HIG-12-028, CERN-PH-EP-2012-220, [arXiv:1207.7235 [hep-ex]].
* [35] Sun Hao, Zhang Ren-You, Zhou Pei-Jun, Ma Wen-Gan, Jiang Yi and Han Liang, Phys.Rev. D71, 075005 (2005)
|
arxiv-papers
| 2012-09-11T03:42:25 |
2024-09-04T02:49:34.955335
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hao Sun and Ya-Jin Zhou",
"submitter": "YaJin Zhou",
"url": "https://arxiv.org/abs/1209.2214"
}
|
1209.2248
|
# An Eccentric Eclipsing Binary: CG Aur
E. Sipahi [email protected] H. A. Dal Ege University, Science
Faculty, Department of Astronomy and Space Sciences, 35100 Bornova, İzmir,
Turkey
###### Abstract
In this study, we present CG Aur’s photometric observations obtained in the
observing seasons 2011 and 2012, the first available multi-colour light
curves. Their shape indicates that the system is an Algol binary. The light
curve analyses reveal that CG Aur is a detached binary system with an
effective temperature difference between the components, approximately 1000 K.
The first estimate of the absolute dimensions of the components indicated that
the system locates on the main sequence in the HR diagram. The primary
component is slightly evolved from the ZAMS.
###### keywords:
techniques: photometric — (stars:) binaries: eclipsing — stars: early-type —
stars: individual: (CG Aur)
††journal: New Astronomy
## 1 Introduction
CG Aur ($V=11^{m}.43$) is classified as an eclipsing binary in the SIMBAD
database. The system was first discovered by Hoffleit (1935), while Kanda
(1939) gave the first light elements. The photographic light curve of the
system was obtained by Kurochkin (1951). The system was observed by Zakirov
(1995) and the colour indexes of $U-B$ and $B-V$ were given as 0m.36 and 0m.66
in this study, respectively. Presenting the $O-C$ analyses of the system, Wolf
et al. (2011) indicated that CG Aur is an interesting triple and eccentric
eclipsing system showing a slow apsidal motion with an important relativistic
contribution as well as the rapid LITE caused by a third body orbiting with a
very short period of 1.9 years. According to them, CG Aur probably belongs to
an important group of other early-type and triple eclipsing systems with a
very short third-body orbital period.
The apsidal motion in the eccentric eclipsing binaries has been used for
decades to test the models of stellar structure and evolution. CG Aur analyzed
here has some properties, which make the system an important ”astrophysical
laboratories” for studying the stellar structure and evolution. The lack of
the multi-colour light curves makes CG Aur an interesting system for including
it to our photometric programme. We followed the observations of the system
and obtained the light curves in 2011 and 2012, and we discussed the light and
colour variations. CG Aur is important due to not only being a member of the
group of the triple systems, but also having an eccentric orbit.
## 2 Observations
The observations were acquired with a thermoelectrically cooled ALTA U+42
2048$\times$2048-pixel CCD camera attached to a 40-cm Schmidt-Cassegrain MEADE
telescope at Ege University Observatory. Using exposure times of 100 s in B
filter and 40 s in V, R and I filters, the BVRI-band observations were
recorded over five nights in 2011 and four nights in 2012. Calibration images
(bias frames and twilight sky flats) were taken intermittently during
observations to correct for pixel to pixel variations on the chip. CCD
observations were reduced as follows: Bias and dark frames were subtracted
from the science frames and then corrected for the flat-fielding. These
reduced CCD images were used to obtain the differential magnitudes of the
program stars. We used GSC 1857 833 and GSC 1857 736 as a comparison and check
stars shown in Figure 1, respectively. There was no variation observed in the
brightnesses of the comparison star.
During the observations, we obtained one primary and one secondary times of
minimum light. These minima times and their errors were determined using the
method of Kwee & van Woerden (1956) and are presented in Table 1. In order to
calculate the phases of the photometric data of CG Aur, the following linear
ephemeris was used:
$JD~{}(Hel.)~{}=~{}24~{}55983.2658(6)~{}+~{}1^{d}.8048588(3)~{}\times~{}E.$
(1)
The V-light and the colour curves obtained in this study are shown in Figure
2. The shape of the light curves indicates that CG Aur is an Algol type binary
and reveals that the primary minimum, which lasts $\sim$6 hours, is deeper
than the secondary one. The mean depths of the eclipses in B, V, R, and I
filters are 0m.434, 0m.390, 0m.382, and 0m.360 in the primary minimum and
0m.168, 0m.202, 0m.218, and 0m.235 in the secondary minimum, respectively. The
$B-V$, $V-R$ and $V-I$ colour curves of the system are also displayed in
Figure 2. The system is slightly redder at the primary and bluer at the
secondary minimum which is consistent with the spectral types of the
components.
## 3 Light Curve Analysis
Photometric analysis of CG Aur was carried out using the PHOEBE V.0.31a
software (Prša & Zwitter, 2005). The software uses the version 2003 of the
Wilson-Devinney Code (Wilson & Devinney, 1971; Wilson, 1990). The BVRI light
curves were analysed simultaneously assuming the ”detached” configuration. In
the process of the computation, we initially adopted the following fixed
parameters: the mean temperature of the primary component ($T_{1}$), the
linear limb-darkening coefficients of $x_{1}$ and $x_{2}$ for various bands
(van Hamme, 1993), the gravity-darkening exponents of $g_{1}$, $g_{2}$ (Lucy,
1967) and the bolometric albedo coefficients of $A_{1}$, $A_{2}$ (Rucinski,
1969). The adjustable parameters commonly employed are the orbital inclination
($i$), the mean temperature of the secondary component ($T_{2}$), the
potentials of the components ($\Omega_{1}$ and $\Omega_{2}$) and the
monochromatic luminosity of the primary component ($L_{1}$). The third light
($L_{3}$) was used also as free parameter to check for the third light
contribution. We used the $U-B$ and $B-V$ values given by Zakirov (1995) and
determined the dereddened colours of the system as $(U-B)_{0}=0^{m}.06$,
$(B-V)_{0}=0^{m}.24$. Then, we took JHK magnitudes of the system
($J=10^{m}.261$, $H=10^{m}.036$, $K=9^{m}.955$) from the 2MASS Catalogue
(Cutri et al., 2003). Using these magnitudes, we derived dereddened colours as
a $(J-H)_{0}=0^{m}.225$ and $(H-K)_{0}=0^{m}.081$ for the system. Using the
calibrations given by Tokunaga (2000), we derived the temperature of the
primary component as 7650 K and 7475 K depending on the UBV and JHK dereddened
colours, respectively. Both of them indicate the same spectral range which is
A5-F0. We adopted the mean temperature of the primary component as 7650 K for
the light curve analyse. To find a photometric mass ratio, the solutions are
obtained for a series of fixed values of the mass ratio from $q=0.4$ to 1.0 in
increments of 0.1. The sum of the squared residuals ($\Sigma res^{2}$) for the
corresponding mass ratios are plotted in Figure 3, where the lowest value of
($\Sigma res^{2}$) was found at about $q=0.7$. The photometric elements for
the mass-ratio of 0.7 are listed in Table 2, and the corresponding light
curves are plotted in Figure 4 as continuous lines. The Roche Lobe geometry of
the system is displayed for the phase of 0.25 in Figure 5.
Although there is no available radial velocity curve of the system, we tried
to estimate the absolute parameters of the components. Considering its
spectral type, using the calibration of Tokunaga (2000), we estimate the mass
of the primary component as approximately 1.78 $M_{\odot}$, and the mass of
the secondary component is computed from the estimated mass ratio of the
system. Using Kepler’s third law, we calculate the semi-major axis ($a$), and
also the mean absolute radii of the components. We can calculate the distance
of the system by using primary and secondary component separately, via their
photometric and absolute properties. We adopted bolometric corrections from
Tokunaga (2000), while calculating the distance. Photometric properties of the
components lead to an average distance of 460 pc. All the estimated absolute
parameters are listed in Table 3. In Figure 6, we plot the components in the
$log~{}(M/M_{\odot})$-$log~{}(R/R_{\odot})$ and
$log~{}(T_{eff})$-$log~{}(L/L_{\odot})$ planes. The continuous and dotted
lines represent the ZAMS and TAMS theoretical model developed by Girardi et
al. (2000). All the tracks are taken from Girardi et al. (2000) for the stars
with $Z=0.02$. In the figure, the open circles represent the secondary
component, while the filled circles represent the primary component. Both of
the components locate in the main-sequence band, while the more massive
component is more evolved.
## 4 Summary
We have obtained the multi-colour CCD photometry for the interesting eclipsing
binary CG Aur. Based on these observations, we presented the first BVRI light
curves of the system and analysed them to find the parameters obtained from
the orbital solution, considering also the Roche configuration of the system.
The results allow us to draw the following conclusions.
$\bullet$ The physical and geometrical parameters of the components have been
derived. Orbital parameters indicate that CG Aur is a detached binary system
with a little temperature difference of approximately 1000 K.
$\bullet$ Our photometric model describes CG Aur as an Algol type eclipsing
binary in which the more massive and hotter primary component is the larger
one.
$\bullet$ In order to discuss the present evolutionary status of the
components of CG Aur, both of them were plotted on the
$log~{}(M/M_{\odot})$-$log~{}(R/R_{\odot})$ and
$log~{}(T_{eff})$-$log~{}(L/L_{\odot})$ planes. The components of the system
located inside the Main-Sequence band. The primary component is slightly
evolved from the Zero-Age Main-Sequence (ZAMS), while the secondary component
is still very close to ZAMS. CG Aur is a detached eclipsing binary and its
components are Main-Sequence dwarfs, the evolutionary states of the components
of the system could be estimated from single-star evolutionary models. We
compared the physical parameters of the system with those inferred from the
evolutionary tracks for single stars of masses in the range of 1.8 and 1.3
$M_{\odot}$, taken from Girardi et al. (2000) for solar composition. There is
a good agreement between the estimated masses and evolutionary masses for the
components of the system.
$\bullet$ The detailed $O-C$ study of the system were presented by Wolf et al.
(2011). In this study, we obtained a small amount of the third light
contribution to the total light from the light curve solution of the system
for the first time. According to our solution, the luminosity fraction of the
third body was found maximum in I filter, while the least contribution was
determined in B filter, which indicates its cool nature. The spectral type of
the third body should be late type.
$\bullet$ As said by Wolf et al. (2011), CG Aur is an interesting triple and
eccentric eclipsing system showing the slow apsidal motion with the important
relativistic contribution as well as the rapid LITE caused by a third body
orbiting with the very short period of 1.9 years. CG Aur is shown to be an
Algol-type system of special interest because of its nature. The system
belongs to the important group of other early-type and triple eclipsing
systems with a very short third-body orbital period (e.g. IM Aur, IU Aur, FZ
CMa, AO Mon) as said by Wolf et al. (2011), and also it has an eccentric
orbit. CG Aur is important due to not only being a member of the group of the
triple systems, but also having an eccentric orbit, while short period
binaries with periods less than a week generally have circular orbits. It is
recommended that more attention should be paid to this system.
$\bullet$ In future work, spectroscopic observations should be made to obtain
radial velocity curves, which will allow a better discussion of the absolute
dimensions of the components and the evolutionary status of CG Aur. New
timings of this eclipsing binary are also necessary to improve the LITE
parameters of the system.
## Acknowledgment
The author acknowledge generous allotments of observing time at the Ege
University Observatory. We also thank the referee for useful comments that
have contributed to the improvement of the paper.
## References
* Cutri et al. (2003) Cutri, R.M., Skrutskie, M.F., van Dyk, S. et al., 2003, The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC Infrared Science Archive (http://irsa.ipac.caltech.edu/applications/Gator/).
* Girardi et al. (2000) Girardi, L., Bressan, A., Bertelli, G., Chiosi, C., 2000, A&AS 141, 371
* Hoffleit (1935) Hoffleit, D., 1935, Harvard Obs. Bull., 901
* Kanda (1939) Kanda, S., 1939, Tokyo Astr. Obs. Bull. 393, 786
* Kurochkin (1951) Kurochkin, N.E., 1951, Variable Stars 8, 351
* Kwee & van Woerden (1956) Kwee, K.K., van Woerden, H., 1956, BAN 12, 327
* Lucy (1967) Lucy, L.B., 1967, Z. Astrophys, 65, 89
* Prša & Zwitter (2005) Prša, A., Zwitter, T., 2005, ApJ, 628, 426
* Rucinski (1969) Rucinski, S.M., 1969, AcA, 19, 245
* Tokunaga (2000) Tokunaga, A.T., 2000, ”Allen’s Astrophysical Quantities”, Fouth Edition, ed. A.N.Cox (Springer), p.143
* van Hamme (1993) van Hamme, W., 1993, AJ, 106, 2096
* Wilson (1990) Wilson, R.E., 1990, ApJ, 356, 613
* Wilson & Devinney (1971) Wilson, R.E., Devinney, E.J., 1971, ApJ, 166, 605
* Wolf et al. (2011) Wolf, M., Lehký, M., Šmelcer, L., Kučáková, H., Kocián, R., 2011, New Astronomy, 16, 402
* Zakirov (1995) Zakirov, M.M., Astronomy Letters, 1995, Volume 21, Issue 5, 675
Figure 1: The program star CG Aur and comparison and check stars on the sky
plane, which is in size of $23^{\prime}.27\times 23^{\prime}.27$. Figure 2: CG
Aur’s V-light and colour curves. Figure 3: The variation of the sum of
weighted squared residuals versus mass ratio. Figure 4: CG Aur’s light curves
observed in BVRI bands and the synthetic curves derived from the light curve
solutions in each band. Figure 5: CG Aur’s Roche geometry, while the system is
at the phase of 0.25. Figure 6: The places of the components of CG Aur in the
$log~{}(M/M_{\odot})$-$log~{}(R/R_{\odot})$ plane (upper panel) and
$log~{}(T_{eff})$-$log~{}(L/L_{\odot})$ plane (bottom panel). In the figures,
the filled circles represent the primary, while open circles represent the
secondary component. The continuous and dotted lines represent the ZAMS and
TAMS theoretical model developed by Girardi et al. (2000). All the tracks
taken from Girardi et al. (2000) are derived for the stars with $Z=0.02$.
Table 1: The times of minimum light for CG Aur. HJD (24 00000 +) | Sigma | Type | Filter
---|---|---|---
55983.2661 | 0.0010 | I | B
55983.2655 | 0.0006 | I | V
55983.2659 | 0.0004 | I | R
55983.2656 | 0.0004 | I | I
56002.2782 | 0.0010 | II | B
56002.2668 | 0.0018 | II | V
56002.2732 | 0.0020 | II | R
56002.2682 | 0.0008 | II | I
Table 2: The parameters obtained from the light curve analysis.
Parameter | Value | Parameter | Value
---|---|---|---
$T_{0}$ | 24 50014.6248 | $P$ (day) | 1.8048588
$q$ | 0.7 | $i$ (∘) | 87.74$\pm$0.01
$e^{1}$ | 0.124 | $w$ (∘) | 291.05$\pm$0.01
$T_{1}$ (K) | 7650 | $T_{2}$ (K) | 6600$\pm$27
$\Omega_{1}$ | 5.617$\pm$0.001 | $\Omega_{2}$ | 6.850$\pm$0.002
L1/LT $(B)$ | 0.862$\pm$0.015 | L3/LT $(B)$ | 0.0007$\pm$0.0002
L1/LT $(V)$ | 0.836$\pm$0.014 | L3/LT $(V)$ | 0.0020$\pm$0.0002
L1/LT $(R)$ | 0.814$\pm$0.014 | L3/LT $(R)$ | 0.0031$\pm$0.0002
L1/LT $(I)$ | 0.792$\pm$0.012 | L3/LT $(I)$ | 0.0080$\pm$0.0002
$g_{1}$, $g_{2}$ | 0.32, 0.32 | $A_{1}$, $A_{2}$ | 0.5, 0.5
Phase shift | 0.0146$\pm$0.0002 | $x_{1,bol}$, $x_{2,bol}$ | 0.522, 0.481
$x_{1,B}$, $x_{2,B}$ | 0.596, 0.656 | $x_{1,R}$, $x_{2,R}$ | 0.421, 0.434
$x_{1,V}$, $x_{2,V}$ | 0.518, 0.532 | $x_{1,I}$, $x_{2,I}$ | 0.329, 0.351
$<r_{1}>$ | 0.209$\pm$0.001 | $<r_{2}>$ | 0.126$\pm$0.001
* 1
Taken from Wolf et al. (2011)
Table 3: The estimated absolute parameters derived for CG Aur. Parameter | Primary | | Secondary
---|---|---|---
Mass ($M_{\odot}$) | 1.78 | | 1.25
Radius ($R_{\odot}$) | 2.09 | | 1.26
Luminosity ($L_{\odot}$) | 13.49 | | 2.72
$M_{bol}$ (mag) | 1.92 | | 3.65
$log~{}(g)$ | 4.05 | | 4.33
$d$ (pc) | | 460 |
|
arxiv-papers
| 2012-09-11T07:54:02 |
2024-09-04T02:49:34.961501
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Esin Sipahi, Hasan Ali Dal",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1209.2248"
}
|
1209.2341
|
arxiv-papers
| 2012-09-11T15:02:20 |
2024-09-04T02:49:34.968788
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.R. Balamurali, Subhabrata Mukherjee, Akshat Malu, Pushpak\n Bhattacharyya",
"submitter": "Subhabrata Mukherjee",
"url": "https://arxiv.org/abs/1209.2341"
}
|
|
1209.2352
|
arxiv-papers
| 2012-09-11T15:39:18 |
2024-09-04T02:49:34.972228
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Subhabrata Mukherjee, Pushpak Bhattacharyya",
"submitter": "Subhabrata Mukherjee",
"url": "https://arxiv.org/abs/1209.2352"
}
|
|
1209.2396
|
# The Hurst Exponent of Fermi GRBs
G. A. MacLachlan1, A. Shenoy1, E. Sonbas2,3, R. Coyne1, K. S. Dhuga1, A.
Eskandarian1, L. C. Maximon1, and W. C. Parke1
1Department of Physics, The George Washington University, Washington, D.C.
20052, USA.
2University of Adiyaman, Department of Physics, 02040, Adiyaman, Turkey.
3NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.
E-mail: [email protected] (GAM)
###### Abstract
Using a wavelet decomposition technique, we have extracted the Hurst exponent
for a sample of 46 long and 22 short Gamma-ray bursts (GRBs) detected by the
Gamma-ray Burst Monitor (GBM) aboard the Fermi satellite. This exponent is a
scaling parameter that provides a measure of long-range behavior in a time
series. The mean Hurst exponent for the short GRBs is significantly smaller
than that for the long GRBs. The separation may serve as an unbiased criterion
for distinguishing short and long GRBs.
###### keywords:
Gamma-ray bursts
## 1 Introduction
Our present understanding of complex astrophysical objects such as cataclysmic
variables (CVs), active galactic nuclei (AGNs), and gamma-ray bursts (GRBs)
comes nearly entirely from the temporal and spectral analyses of their
photoemissions (with some additional information coming from possible
associations, such as host galaxies). In this paper we consider the temporal
aspects of GRB light curves observed by the Gamma-Ray Burst Monitor (GBM)
aboard the Fermi satellite. Many studies of the temporal properties of GRB
light curves have been published, such as Nemiroff (2000); Norris et al.
(2005); Hakkila & Nemiroff (2009); Hakkila & Preece (2011); Nemiroff (2012),
from the perspective that light curves are comprised of a series of displaced
pulses and that by fitting the individual pulses and associating pulses at
various photon energies one can arrive at a holistic understanding of light
curves which in turn may be used to constrain the physics of the engines that
produce them. The main appeal of this approach is the intuitive connection
between pulses and collisions in the internal shock model. While this is a
perfectly reasonable method, issues do arise concerning the functional form to
use for pulse fitting and how to discern actual pulses from stochastic
fluctuations in the light curves. The situation is further exacerbated by the
fact that GRB light curves exhibit considerable variation in duration and in
pulse profile. We note the significant progress made in non-parametric
analyses using the Bayesian block technique (Scargle et al., 2012). In this
type of analyis the duration of a light curve is represented as a tessellated
block of time which can be partitioned into a complete array of sub-blocks in
any number of ways. An optimal partition (a partition of sub-blocks maximizing
a fitness function) is shown to exist, be unique, and computable iteratively.
The optimal partition of sub-blocks is determined, given a prior probablility
distribution for the number of blocks, by finding the model best representing
the data as sets of piece-wise constant segments or sub-blocks. This technique
shows great promise in resolving statistically significant temporal features
from noise and detector related artifacts.
An ideal complementary approach to probing light curves would be one which
handles seemingly disparate profiles on an equal footing and distills their
complex forms into a single parameter which may be used to compare one light
curve with another. One such method was pioneered by Harold Edwin Hurst
(Hurst, 1951) with a technique he invented called the rescaled range analysis
(R/S) which was later improved upon by Benoit Mandelbrot (Mandelbrot, 1968).
The eponymous parameter resulting from the rescaled range analysis is called
the Hurst exponent, $H$, and is closely related to the fractal dimension, D,
the understanding of which Mandelbrot spent much of his career developing. In
fact, fractional Brownian motion (fBm), which Mandelbrot defined in 1968, is
parametrized solely by $H$ and serves as a useful model for discussing time
series. After determining $H$ for a given time series one is in a position to
make several statements about the nature of that time series including whether
the sequence appears random or whether it is persistent or anti-persistent,
and if so, whether it exhibits long-range dependence, and over what time
scales these characteristics are operative. All of these are informative
quantitative statements, especially if the specific process generating the
time series is partially or completely unknown, in which case, these
statements are perhaps all one can really say about the process given the
available information. Some fields of research in which interesting work is
being done with Hurst exponents are financial markets, seismology,
anesthesiology, astrophysics, plasma physics and genomics.
We point out that neither the pulse fitting methods nor the Bayesian block
analysis (Scargle et al., 2012) yields information directly relatable to the
Hurst exponent as does the wavelet analysis. One approach to access the Hurst
exponent from a Bayesian block framework that seems reasonable would be an
adaptation of the Box-Counting algorithm (Feder, 1988). Such a Bayesian-Box-
Counting algorithm is outside the scope of this paper.
The estimation of the Hurst exponent and the related scaling exponent,
$\alpha$, has a history in astrophysics (Anzolin et al., 2010; Tamburini et
al., 2009; Walker & Schaefer, 2000; Fritz & Bruch, 1998) for both Cataclysmic
Variables (CVs) and GRBs. We propose that a similar determination of $H$ for
GRB light curves will be a valuable tool for categorization and we present a
separation of long and short GRBs based on $H$.
## 2 Methodology
### 2.1 Hurst Exponent and Self-Affinity
Pioneering work in self-similarity and long-range dependence was first
published in 1951 by Hurst in the study of annual Nile River levels, (Hurst,
1951). Hurst examined several decades of data to determine what should be the
minimum size of a reservoir so that it neither overflows nor runs dry due to
yearly fluctuations and made the unexpected observation that annual Nile River
levels were not independent from one another but instead exhibited a _memory_
of past events.
Figure 1: Simulated fractional Brownian motions with different values of $H$:
a) $H=0.25$, b) $H=0.50$, c) $H=0.75$.
In this analysis of time-series data we search for statistical fractals, i.e.,
fractals whose statistical characteristics are independent of time scale. Such
fractal time-series are called _self-similar_. There is another class of
statistical fractals whose scale invariance is broken but can be restored by a
multiplicative factor. These statistical fractals are called _self-affine_.
Mandelbrot (1985) defined a time-series, $X(t)$ with
$t\in\\{t_{0}\mathellipsis t_{N-1}\\}$, to be self-affine if, after a
rescaling $t\rightarrow\lambda t$ the following relation is satisfied,
$X(t)\doteq\lambda^{-H}X(\lambda t).$ (1)
The exponent, $H$, is the Hurst exponent, (Hurst, 1951) and the symbol
$\doteq$ denotes equality in distribution. The canonical example of a self-
affine time-series, also given by Mandelbrot (1968), is fractional Brownian
motion, fBm. Stationary in the context of this paper is second-order
stationarity which means the first and second moments obey the following
relations
$\displaystyle\mathbb{E}\\{X(t)\\}$ $\displaystyle=$ $\displaystyle\mu_{X}$
$\displaystyle\mathbb{E}\\{X(t_{2})X(t_{1})\\}$ $\displaystyle=$
$\displaystyle\gamma(t_{2}-t_{1})=\gamma(\tau),$ (2)
where $\mu_{X}$ is the sample-mean, $\gamma$ is the auto-covariance sequence
and $\tau\equiv t_{2}-t_{1}$ is the lag. The Hurst exponent, $H$, parametrizes
the degree of statistical self-similarity which a time-series exhibits. A
self-similar series may be sub-divided into three categories: A series with
$1/2<H<1$ is referred to as persistent or long-range dependent while a series
with $0<H<1/2$ is referred to as anti-persistent, Feder (1988). For $H=1/2$ we
have neither persistence nor anti-persistence and this corresponds to the case
of random and uncorrelated events. The Hurst exponent provides a model-
independent characterization of the data. Three examples of times series with
different values of $H$ are shown in Fig. 1. A graphical depiction of the
rescaling described by Eq. 1 for a time series with $H=0.25$ is given in Fig.
2 and for a time series with $H=0.75$ in Fig. 3.
### 2.2 Wavelet Transforms
Wavelet transformations have been shown to be a natural tool for
multiresolution analysis of non-stationary time-series (Flandrin, 1992;
Mallat, 1989). Wavelet analysis is similar to Fourier analysis in many
respects but differs in that a wavelet basis function, $\psi(t)$, is well-
localized while Fourier basis functions are global. Localization means that
outside some range the amplitudes of wavelet basis functions go to zero or are
otherwise negligibly small, Percival (2000). On the other hand, the wavelet
transform is similar to the Fourier transform because they both are expansions
into a complete orthogonal basis and resolve low-frequency, large scale
structure from high-frequency, small scale structure.
Wavelet analysis is said to be multiresolution because the time-series under
investigation is interrogated at multiple scales by a basis set of wavelets
which are rescaled and translated versions of an original wavelet commonly
referred to as the mother-wavelet, $\psi(t)$,
$\psi(t)\rightarrow\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right),$
(3)
where $a$ represents an octave or time-scale and the parameter $b$ gives the
position of the wavelet within the octave.
The continuous wavelet transform (CWT) coefficient, $C_{a,b}$, of a time-
series for some scale and position is computed as
$C_{a,b}=\frac{1}{\sqrt{a}}\int X(t)\psi_{a,b}(t)dt.$ (4)
##### Wavelet Analysis
The wavelet-transform technique for estimating self-affinity is outlined here.
By substituting the distribution relation in Eq. 1 into Eq. 4 we find
$\displaystyle C_{a,b}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{a}}\int
X(t)\psi_{a,b}(t)dt$ (5) $\displaystyle=$
$\displaystyle\frac{\lambda^{-(H+1/2)}}{\sqrt{\lambda a}}\int X(\lambda
t)\psi_{a,b}(\frac{\lambda t-\lambda b}{\lambda a})d(\lambda t)$
$\displaystyle=$ $\displaystyle\lambda^{-(H+1/2)}C_{\lambda a,\lambda b}.$
It is straightforward to see from Eq. 5 that a self-affine time-series will
have wavelet coefficients whose variance over a particular scale, $\lambda a$,
is related to the scale parameter $\lambda$ by,
$\log{\rm var}(C_{\lambda a,\lambda b})=(2H+1)\log\lambda+{\rm constant}.$ (6)
##### Fast Wavelet Transforms
Similar to the CWT, the discrete fast wavelet transform (FWT) is also a
multiresolution operation owing to the construction of the wavelets,
$\psi_{j,k}$, which form the basis of the discrete fast wavelet transform. We
employed the discrete wavelet transform because of its high degree of
computational efficiency. In order to distinguish between the CWT and its FWT
counterpart we make a slight change of notation. Just as before, the
$\psi_{j,k}$, are rescaled, translated versions of the mother wavelet, $\psi$,
$\psi_{j,k}=2^{-j/2}\psi(2^{-j}t-k).$ (7)
The coefficients of the FWT are written as
$d_{j,k}=\langle X,\psi_{j,k}\rangle,$
where $j$ and $k$ play the roles of $a$ and $b$, respectively. Moreover, the
values which $j$ and $k$ assume obey the dyadic partitioning scheme (Mallat,
1989; Addison, 2002; Percival, 2000). That is, for a time series whose number
of elements is given by $N=2^{m}$,
$0\leq j\leq m-1,$
and
$0\leq k\leq 2^{j}-1.$
Applying the dyadic partitioning scheme removes any redundant encoding of
information by the wavelet transform coefficients and guarantees orthogonality
among the wavelet basis for any change in $j$ or $k$,
$\langle\psi_{j,k},\psi_{j^{\prime},k^{\prime}}\rangle=\delta_{j,j^{\prime}}\delta_{k,k^{\prime}}.$
(8)
### 2.3 Logscale Diagrams
The average power of the light curve at time scale $j$ is expressed as
$\beta_{j}$ and may be written in terms of the variance of the FWT
coefficients as
$\beta_{j}={\rm
var}(d_{j,k})=\frac{1}{n_{j}}\sum^{n_{j}-1}_{k=0}|d_{j,k}|^{2},$ (9)
where $n_{j}$ is the number of coefficients at scale, $j$, (Abry et al., 2000,
2003). Similarly to Eq. 6, it has been shown by Flandrin (1992) that for a
series with non-stationary statistics the power-law variance of wavelet
coefficients goes like
$\log_{2}(\beta_{j})=(2H+1)j+{\rm constant},$ (10)
where $H$ is the Hurst exponent. Masry (1993) later extended this result to a
larger class of non-stationary problems with stationary increments in the low-
frequency limit and showed that fBms are a special case. A plot of Eq. 10 is
referred to as a logscale diagram. Logscale diagrams are useful for
identifying scaling regions, i.e., the range of octaves over which self-affine
scaling occurs. The slope, $\alpha$, of the scaling region is related to the
Hurst exponent through $\alpha=2H+1$.
In practice, a piecewise fitting function, $f(j;p_{i})$ is defined,
$f(j;p_{i})=\begin{cases}p_{1};1\leq j\leq p_{0}\\\ p2+p_{3}j;p_{0}\leq
j\end{cases}$
where $p_{0}$ is the value of $j$ at which the piecewise fitting function
changes definition.
### 2.4 Choice of Wavelet Basis
As in any orthogonal transformation, the basis functions to use in a wavelet
transform is a matter of strategic choice. One typically chooses a basis that
emphasizes some characteristic of interest. Commonly used families of wavelet
bases are the Coiflet, Daubechies, and Haar (Addison, 2002). We chose the Haar
wavelet basis which is the simplest of the Daubechies family.
The Haar wavelet basis was chosen from among all other possible bases because
it has the fewest number of vanishing moments and most compact support
(Addison, 2002), has a straightforward interpretation, i.e., is equivalent to
the Allan variance (Xizheng & Zhensen, 1997) and is constant over its interval
of support similar to the model assumed in the Bayesian block method (Scargle,
1998; Scargle et al., 2012).
The Haar basis is not without some defects, as noted by Kaplan & Jay Kuo
(1993) and Flandrin (1992). Namely, the Haar wavelet transformation is known
to underestimate the actual Hurst exponent and this phenomenon is a function
of the coarseness of the binning, the number of counts in the light curve, and
also of H itself. We show in Sec. 2.6 that this effect is present but smaller
than $\approx$1$\sigma$ for a set of simulated light curves and is likely to
be smaller for actual data. However, we consider that the advantages of the
Haar basis outweigh its disadvantages.
### 2.5 Minimizing Uncertainties
#### 2.5.1 Circular Permutation
Spurious artifacts due to incidental symmetries resulting from accidental
misalignment (Percival, 2000; Coifman, 1995) of light curves with wavelet
basis functions are minimized by circularly shifting the light curve against
the basis functions. Circular shifting is a form of translation invariant de-
noising (Coifman, 1995). It is possible a shift will introduce additional
artifacts by moving a different symmetry into a susceptible location. The best
approach is to circulate the signal through all possible values, or at least a
representative sampling, and then take an average over the cases which
minimizes the effect of spurious correlations.
#### 2.5.2 Reverse-Tail Concatenation
Both discrete Fourier and discrete wavelet transformations imply that the
expansion is periodic, with the longest period equal to the full time range of
the input data. This can be interpreted to mean that for a series of $N$
elements, $\\{X_{0},X_{1}\mathellipsis X_{N-1}\\}$ then $X_{0}$ is made a
surrogate for $X_{N}$ and $X_{1}$ is made a surrogate for $X_{N+1}$, and so
forth. This assumption may lead to trouble if $X_{0}$ is much different from
$X_{N-1}$. In this case, artificially large variances may be computed.
Reverse-tail concatenation minimizes this problem by making a copy of the
series which is then reversed and concatenated onto the end of the original
series resulting in a new series with a length twice that of the original.
Instead of matching boundary conditions like,
$X_{0},X_{1},\ldots,X_{N-1},X_{0},$ (11)
we match boundaries as,
$X_{0},X_{1},\ldots X_{N-1},X_{N-1},\ldots,X_{1},X_{0}.$ (12)
Note that the series length has thus artificially been increased to $2N$ by
reversing and doubling of the original series. Consequently, the wavelet
variances at the largest scale in a logscale diagram reflect this redundancy.
This is the reason that the wavelet variances at the largest scale are
excluded from least-squares fits of the scaling region.
#### 2.5.3 Poisson Operator
Photon counting statistics are considered in a bootstrapping procedure by
applying a Poisson operator, $\mathcal{P}$($\lambda_{i},X_{i}$), to every
light curve prior to analyzing. Each light curve is binned initially at 200
$\mu$-seconds and the number of counts per bin, $X_{i}$, is used as a mean
value, $\lambda_{i}$, to be supplied to a Poisson random number generator. The
value returned from $\mathcal{P}$($\lambda_{i},X_{i}$) is used to replace the
number of counts stored in $X_{i}$. The Poisson operator is applied to the
signal $X_{i}$ prior to every circular permutation. We show in Sec. 2.6 that
the Poisson operator does not affect the measured slope of logscale diagrams
above the Poisson level.
### 2.6 A Test Case: Fractional Brownian Motion
Spatial-temporal fractional Brownian motions (fBm’s) are a useful model for
studying self-similarity and long-range dependence in non-stationary time-
series, Mandelbrot (1968) and are characterized by a single parameter, $H$,
the Hurst exponent. An fBm with a particular $H$ is expressed as $B_{H}(t)$
and has the property of self-similarity over a range of scales after a
rescaling of axes,
$B_{H}(t)\doteq a^{-H}B_{H}(at),$ (13)
where $\doteq$ denotes distributional equality as in Section 2.1.
Figure 2: Graph of $B_{H}(t)$ with $H=0.25$. A box is placed around a sub-
range of $t$ (lower left hand corner). The box is zoomed into with time axis
scaled by $a$ and amplitude scaled by $a^{-H}$. This is a _self-affine_
transformation that not only makes the rescaled version qualitatively
’similar’ to the original but also preserves the variance as computed in Eq.
9. Figure 3: Graph of $B_{H}(t)$ with $H=0.75$. A box is placed around a sub-
range of $t$ (lower left hand corner). The box is zoomed into with time axis
scaled by $a$ and amplitude scaled by $a^{-H}$. This is a _self-affine_
transformation that not only makes the rescaled version qualitatively
’similar’ to the original but also preserves the variance as computed in Eq.
9.
The efficacy of the $H$ estimation procedure was tested using simulated data
in the form of fractional Brownian motion (fBm) time series. Two tests were
performed; in the first test we examine the ability of our algorithm to
determine $H$ from fBms in the presence of Poisson noise and in the second
test we examine how well we can determine $H$ at $H=0.25$, $H=0.50$, and
$H=0.75$ from noise-free fBms.
Figure 4: Panel a) shows a sample fBm pre-processed and ready to be analyzed
in black and the same light curve after applying the Poisson operator,
$\mathcal{P}$, in red. Panel b) shows the Poisson noise that has been added by
$\mathcal{P}$. In panel c) logscale diagrams illustrate the effect of Poisson
statistics on the Hurst exponent. The bare fBm is shown in black, the dressed
Poisson-type fBm is in red, and the residual Poisson noise in shown in blue.
#### 2.6.1 Test 1
The numerical computing environment MATLAB was used to produce 1000
realizations of fBms with scaling parameter $H$ randomly chosen from the range
$0.0<\alpha<1.0$ by using a uniform random number generator. Copies of the
fBms were combined with a Poisson operator as described in Sec. 2.5.3. The
fBms and the Poissonian fBms thus produced are shown in black and red
respectively in panel a) of Fig. 4. Panel b) shows the Poisson noise that has
been added by $\mathcal{P}$. The Logscale diagrams in panel c) illustrate the
effect of Poisson statistics on the Hurst exponent. The bare fBm is shown in
black, the dressed Poisson-type fBm is in red, and the residual Poisson noise
in shown in blue. The logscale diagram for the bare fBm in panel c) exhibits a
clean slope across all octaves. We see the effect of a Poisson noise operator;
it adds to the signal variance, constant across all octaves. Below some octave
the signal is completely dominated by noise but above that octave the slope of
the logscale diagrams is independent of $\mathcal{P}$. See for example the
black and red symbols for $j\geq 6$.
#### 2.6.2 Test 2
In the second test, 3000 simulated Poisson-type light curves were generated.
The simulated data were divided into three subgroups of 1000 according to $H$.
The three subgroups were $H=\\{0.25,0.50,0.75\\}$. The simulated data in each
group were analysed and an attempt was made to recover the value of the Hurst
exponent, $H$, used to generate the fBm. The Hurst exponent was estimated by a
least squares fit to the scaling portion of the logscale diagrams to determine
$\alpha$ and then $H$ is found from Eq. 10.
Results of the second test can be seen in Fig. 5 and Table 1.
Figure 5: Histograms of 3000 simulated fBm traces. Three categories of fBms
were generated with known Hurst exponents, $H=0.25$, $H=0.50$, and $H=0.75$.
These fBms were then analyzed to recover the Hurst exponent. The histograms
are the results of this analysis. Pairs of vertical lines are drawn for each
peak. The shorter of the two indicates the known $H$ used to generate the fBms
and the longer of the two indicates the $H$ extracted by our analysis. Results
are tabulated in Table 1.
The results show that the FWT analysis with the Haar wavelet basis does
underestimate the value of $H$ as discussed in Sec. 2.4 but the magnitude of
the error is not significant for our purpose.
## 3 Data Reduction
The Gamma-Ray Burst Monitor (GBM) on board Fermi observes GRBs in the energy
range 8 keV to 40 MeV. The GBM is composed of 12 thallium-activated sodium
iodide (NaI) scintillation detectors (12.7 cm in diameter by 1.27 cm thick)
that are sensitive to energies in the range of 8 keV to 1 MeV, and two bismuth
germanate (BGO) scintillation detectors (12.7 cm diameter by 12.7 cm thick)
with energy coverage between 200 keV and 40 MeV. The GBM detectors are
arranged in such a way that they provide a significant view of the sky (Meegan
et al., 2009).
In this work, we have extracted light curves for the GBM NaI detectors over
the entire energy range (8 keV - 1 MeV, also including the overflow beyond 1
MeV). Typically, the brightest three NaI detectors were chosen for the
extraction. Lightcurves for both long and short GRBs were extracted at a time
binning of 200 microseconds. The long GRBs were extracted over a duration
starting from 20 seconds before the trigger and up to about 50 seconds after
the $T_{90}$ (taken from the Fermi GBM-Burst Catalog (Paciesas et al., 2012))
for the burst without any background subtraction. For short GRBs, durations
were chosen to be 20 seconds before the trigger and 10 seconds after the
$T_{90}$. The $T_{90}$ durations were obtained from the Fermi GBM-Burst
Catalog (Paciesas et al., 2012). Summaries for the 46 long and 22 short GRBs
used in this study are tabulated in Tables 3 and 4.
Table 1: Summary of results in Fig. 5. $H$ | $H_{\mathrm{meas}}$
---|---
0.25 | 0.23$\pm$0.02
0.50 | 0.49$\pm$ 0.02
0.75 | 0.74$\pm$ 0.03
Table 2: Summary of results in Fig. 6. Type | N | STD | $\langle H\rangle$
---|---|---|---
Long | 46 | 0.18 | 0.40$\pm$0.03
Short | 22 | 0.17 | 0.23$\pm$ 0.04
Figure 6: Histogram of $H$ extracted from long and short GRBs. The result for
long GRBs is plotted as the solid blue line while the short GRB result is
plotted with the dashed red line. Note the overlap but also that the means are
displaced from one another as shown in Tab. 2 for details of plot.
## 4 Results and Discussion
We have used a technique based on wavelets to determine the Hurst exponents
for a sample of GRB prompt-emission light curves. As noted in Section 2.1, the
Hurst exponent provides a measure of correlated behavior in a time series. The
extreme values of $H$ vary from 0 to 1, and a value of 0.5 implies
uncorrelated (random) behavior. As the fBm model indicates, large $H$ values
tend to be associated with relatively smooth functions and small $H$ values
tend to favor highly jagged curves. This feature suggests that $H$ may be
useful in quantifying the variability observed in GRB prompt-emission light
curves. Plotted in Fig. 6 are the extracted $H$-exponents as histograms for
both long and short GRBs. The histograms clearly show a displacement in $H$
for the distributions of long and short GRBs, with the short GRBs indicating a
preference for small values of H (see Tab. 2). The mean displacement in $H$
raises the interesting possibility of using this feature as a way of
distinguishing between short and long GRBs. This would be in addition to the
currently employed criteria based on $T_{90}$ and spectral hardness ratios.
Interestingly, the histograms also show a significant overlap in the region of
small $H$ exponents possibly signaling similarities between the two types of
bursts in this range.
It could be argued that the sizable overlap of the distributions is
essentially a consequence of the large dispersion (in $H$) exhibited by both
short and long GRB distributions. While it is not known precisely what
processes lead to this large dispersion in $H$, we note that the dispersion
for the short GRBs is somewhat smaller than the corresponding one for long
GRBs. If the dispersion is associated with the energetics of the progenitors
of the respective systems, i.e., a merger of compact objects in the case of
short GRBs and the collapse of a rapidly rotating massive star for long GRBs,
then one might indeed expect a larger dispersion in the $H$-distribution of
long GRBs compared to the corresponding one for short GRBs based purely on the
difference in the mass range for the respective progenitors. Moreover,
additional factors such as the formation of an accretion disk, the size of the
disk, the mass of the disk, the strength of the magnetic field and the
magnitude of the accretion rate during the prompt phase, remain largely
uncertain. With the added intrinsic variability of the central engine itself,
we should not be surprised to observe a systematic difference in the extracted
Hurst exponents for long and short bursts. For completeness, we mention that
while the dispersion in $H$ is large for both distributions, the extracted
$H$-value for each individual GRB is known reasonably precisely (see Table 1).
Another way to examine the $H$-distributions is to recast the data against the
so-called minimum-time-scale parameter, MTS, extracted by MacLachlan et al.
(2013) and MacLachlan et al. (2012). Using a method based on wavelets, these
authors explored the scaling characteristics of GRBs and determined the
minimum time scale at which scaling processes dominate over random noise
processes. Furthermore, the authors have recently shown a direct connection
between the extracted MTS and the smallest pulse structures extracted by
pulse-fitting techniques. The same conclusions were confirmed independently by
Bhat (2013) using a similar technique to extract MTS by computing rescaled
Pearson variances. Furthermore, a link between pulse properties and MTS
connecting GRB prompt emission and X-ray flaring has been identified by Sonbas
et al. (2013).
In addition to this link with pulses, MTS provides an alternate scale (to
$T_{90}$) by which long and short GRBs can be separated. Shown in Fig. 7 are
the extracted $H$-exponents for both long and short GRBs versus the MTS (in
the observer frame). Short GRBs tend to cluster around small MTS values and
follow a steep trajectory in the $H$-MTS plane whereas the long GRBs are
distributed over a larger range in MTS and seem to follow a gradual power-law-
like trajectory. The behavior is a little more clear in panel (b) of Fig. 7
where the MTS is plotted on a log scale: Here the the short and long GRBs
indicate a small ($\sim 30\%$) positive correlation respectively; the combined
sample on the other hand shows a larger positive correlation ($\sim 50\%$) and
an obvious separation of the two distributions with MTS.
Other astrophysical systems for which the Hurst exponent has been extracted
includes CVs. These systems, comprising tightly-bound binaries (with periods
of the order of few hours) and a primary consisting of a compact object
(typically a white dwarf) and an accretion disk that can accommodate
significant mass transfer from the secondary may provide a benchmark for
gauging the systematics of the extracted Hurst exponents. Indeed, large
dispersions in $H$ are found for both optical and X-ray light curves of CVs.
Interestingly though, CVs apparently tend to favor large $H$-exponents i.e.,
greater than 0.5. This implies that the CV distributions are persistent as
opposed to a tendency toward antipersistence for GRBs. By their very nature,
CVs are systems that have built-in periodicity that is readily reflected, in
most cases, in the observed emissions from these systems. GRBs, on the other
hand, are transient phenomena which show very little evidence for
periodicities. It’s possible this simple difference may lead, in part at
least, to the degree of persistence or antipersistence exhibited by these
systems. Furthermore, some authors, (Fritz & Bruch, 1998; Tamburini et al.,
2009; Anzolin et al., 2010), have noted that the extracted $H$’s indicate a
sensitivity to the strength of the magnetic field of the systems under study,
and in particular, the optical and x-ray emissions from CVs exhibit different
$H$ distributions. Since the optical and x-ray emissions in CVs arise from
spatially separated regions (the optical from an extended disk and the x-ray
in the boundary layer between the inner regions of the disk and the surface of
the compact object or the polar regions in the case of a highly magnetic
system), it is tempting to surmise that such a comparison might be fruitful in
elucidating the spatial characteristics of GRB jets: Examples include the
radii and or regions that are commonly associated with the emission sites for
prompt gamma-rays (e.g., the photospheric radius in the case of a thermal
component) and the steeply declining phase of the x-ray light curves (linked
with high-latitude emission resulting from internal shell collisions). While
it is understood that GRBs and CVs are very different systems and therefore
the translation of the Hurst exponent from one system to the other is likely
to be speculative at best, it is intriguing nonetheless that a simple scaling
parameter may enable us to connect common underlying properties and processes
that ultimately produce the observed emission in these diverse systems.
Figure 7: A scatter plot of $H$ against the minimum variability time scale
from MacLachlan et al. (2013) and MacLachlan et al. (2012).
## 5 Conclusions
We have studied the temporal properties of a sample of prompt-emission light
curves for short and long-duration GRBs detected by the Fermi/GBM mission. By
using a technique based on wavelets we have extracted the Hurst exponents for
these bursts. This exponent measures the relation between variability over the
full range of available time-scales, comparing long-range with short-range
variability. The physical limits of this index are 0 and 1, where the mid-
point ($H=0.5$), is an indicator of completely uncorrelated (random) processes
that contribute to the observed time series. Often times, the $H$ is also
associated with the fractal dimension ($D$) of structures by
$D=2-H,$ (14)
and can be thought of as a measure of the degree of jaggedness of the
structures under study. In this sense the $H$ may also be indirectly linked to
the variability seen in the prompt-emission of many GRBs. Our main results are
summarized as follows:
a) The means of the $H$ distributions for the GRBs in our sample show an
offset between short and long GRBs, with the short GRBs indicating a
preference for smaller Hurst exponents compared to the long GRBs. This offset
is potentially an independent criterion for distinguishing between long and
short-duration bursts.
b) Compared to short GRBs, long-duration bursts exhibit a larger dispersion in
H. The origin of this dispersion is not known although it is possible that it
is related to the underlying energetics of the different progenitors that
produce long and short-duration bursts.
c) No distinct group or clustering is found for $H$ values corresponding to
0.5. This implies that random (or uncorrelated) processes, if present, play a
lesser role in the production of the observed prompt emission. Moreover, the
means of the $H$-distributions for both long and short GRBs indicate a
skewness toward values less than 0.5. Overall, this implies that the prompt-
emission time series exhibit antipersistence.
Finally, we note that because of the large dispersion in $H$, there exists a
significant region over which the long and short bursts overlap. This overlap
region raises the interesting possibility of exploring bursts that may possess
many more common features than would otherwise be suspected. The case for an
intermediate class of GRBs (Horvath, 1998; Gao et al., 2010) remains unsettled
and warrants further investigation.
## 6 ACKNOWLEDGEMENTS
The NASA grant NNX11AE36G provided partial support for this work and is
gratefully acknowledged. The authors (GAM and KSD) acknowledge very useful
discussions with Tilan Ukwatta.
## References
* Abry et al. (2000) Abry P., Flandrin P., Taqqu M. S., & Veitch D., 2000, Self-Similar Network Traffic and Performance Evaluation, pg 39–88, New York: Wiley, K. Park and W. Willinger
* Abry et al. (2003) Abry P., Flandrin P., Taqqu M. S., & Veitch D., 2003, Theory and Applications of Long-Range Dependence, Boston: Birkhauser, 527-556
* Anzolin et al. (2010) Anzolin G., Tamburini F., De Martino D., Bianchini A., 2002 , _A &A_, 519, A69
* Addison (2002) Addison P. S., 2002, _The Illustrated Wavelet Transform Handbook_ , IOP Publishing Ltd.
* Bhat (2013) Bhat P. N. 2013, arXiv:1301.4180v2 [astro-ph.HE].
* Coifman (1995) Coifman R. R., Donoho D. L. 1995, Springer-Verlag, 125–150
* Feder (1988) Feder J., 1988, _Fractals_ , Plenum Press
* Flandrin (1989) Flandrin P., 1989, IEEE, Transactions on Information Theory, 35, 197–199
* Flandrin (1992) Flandrin P., 1992, IEEE, Transactions on Information Theory, 38, 910–917
* Fritz & Bruch (1998) Fritz T., Bruch A., 1998, _A &A_, 332, 586-604
* Gao et al. (2010) Gao H., Lu Y., & Zhang S. N., 2010, _ApJ._ , 717, 268
* Hakkila & Nemiroff (2009) Hakkila J. & Nemiroff R. J. 2009, _ApJ._ , 705, 372
* Hakkila & Preece (2011) Hakkila J. & Preese R. 2011, _ApJ._ , 740, 104
* Horvath (1998) Horvath I., 1998, _ApJ._ , 508, 757
* Hurst (1951) Hurst H. E., 1951, Trans. Am. Sco. Civ. Eng., 116, 770
* Kaplan & Jay Kuo (1993) Kaplan L. C., Jay Kuo C. C., 1989 , IEEE, Transactions on Signal Processing, 41, 3554–3562
* MacLachlan et al. (2013) MacLachlan G. A., Shenoy A., Sonbas E., Dhuga K. S., Cobb B., Ukwatt a T. N., Morris D. C., Eskandarian A., Maximon L. C., Parke W. C., 2013, Mon. Not. R. Astron. Soc, doi: 10.1093/mnras/stt241
* MacLachlan et al. (2012) MacLachlan G. A., Shenoy A., Sonbas E., Dhuga K. S., Eskandarian A., Maximon L. C., Parke W. C., 2012, _MNRAS_ _Letters_ , 425, L32–L35
* Mallat (1989) Mallat S. G., 1989, IEEE, Transactions on Pattern Analysis and Machine Intelligence, 11, 674–693
* Mandelbrot (1968) Mandelbrot B. B. and Van Ness J. W. 1968, SIAM Review, 10, 422-437
* Mandelbrot (1985) Mandelbrot B. B, SIAM Review, vol. 10, 1968, pg 422–437
* Masry (1993) Masry E., IEEE, Transactions on Information Theory, vol. 39, 1993, pg. 260–264
* Meegan et al. (2009) Meegan C. et al. 2009, _ApJ._ , 702, 791
* Nemiroff (2000) Nemiroff R. J. et al. 2000, _ApJ._ , 544, 805
* Nemiroff (2012) Nemiroff R. J. 2012, _MNRAS_ , 419, 1650
* Norris et al. (2005) Norris J. P. et al. 2005, _ApJ._ , 627, 324
* Paciesas et al. (2012) Paciesas W. S. et al. 2012, arXiv:1201.3099v1 [astro-ph.HE].
* Percival (2000) Percival D. B. and Walden A. T. 2002, _Wavelet Methods for Time Series Analysis_ , Cambridge University Press
* Scargle (1998) Scargle J. D., 1998, _ApJ._ , 504, 405–418
* Scargle et al. (2012) Scargle J. D., Norris J. P., Jackson B., Chiang J., 2012, arXiv:1207.5578v2 [astro-ph.IM]
* Sonbas et al. (2013) Sonbas E., MacLachlan G. A., Shenoy A., Dhuga K. S., Parke W. C., 2013, _ApJ._ , 767, L28
* Tamburini et al. (2009) Tamburini F., De Martino D., & Bianchini A., 2009, _A &A_, 502, 1,1-5
* Walker & Schaefer (2000) Walker K. C., Schaefer B., 2000, _ApJ._ , 537, 264
* Xizheng & Zhensen (1997) Xizheng K., Zhensen W., 1993, Frequency Control Symposium, 1997., Proceedings of the 1997 IEEE International, 515–518
Table 3: Summary of Long GRBs. GRB Number | $H$ | $\delta H$ | $T_{90}$ [sec] | $\delta T_{90}$ [sec]
---|---|---|---|---
080723557 | 0.316 | 0.023 | 58.369 | 1.985
080723985 | 0.425 | 0.053 | 42.817 | 0.659
080724401 | 0.451 | 0.060 | 379.397 | 2.202
080804972 | 0.549 | 0.085 | 24.704 | 1.460
080806896 | 0.591 | 0.056 | 75.777 | 4.185
080807993 | 0.105 | 0.014 | 19.072 | 0.181
080810549 | 0.211 | 0.037 | 107.457 | 15.413
080816503 | 0.258 | 0.035 | 64.769 | 1.810
080817161 | 0.393 | 0.048 | 60.289 | 0.466
080825593 | 0.382 | 0.036 | 20.992 | 0.231
080906212 | 0.716 | 0.070 | 2.875 | 0.767
080916009 | 0.414 | 0.053 | 62.977 | 0.810
080925775 | 0.453 | 0.056 | 31.744 | 3.167
081009140 | 0.732 | 0.073 | 41.345 | 0.264
081101532 | 0.255 | 0.040 | 8.256 | 0.889
081125496 | 0.629 | 0.080 | 9.280 | 0.607
081129161 | 0.261 | 0.036 | 62.657 | 7.318
081215784 | 0.629 | 0.070 | 5.568 | 0.143
081221681 | 0.567 | 0.089 | 29.697 | 0.410
081222204 | 0.502 | 0.065 | 18.880 | 2.318
081224887 | 0.692 | 0.071 | 16.448 | 1.159
090102122 | 0.126 | 0.013 | 26.624 | 0.810
090131090 | 0.575 | 0.062 | 35.073 | 1.056
090202347 | 0.241 | 0.039 | 12.608 | 0.345
090323002 | 0.294 | 0.025 | 135.170 | 1.448
090328401 | 0.289 | 0.034 | 61.697 | 1.810
090411991 | 0.057 | 0.017 | 14.336 | 1.086
090424592 | 0.442 | 0.029 | 14.144 | 0.264
090425377 | 0.360 | 0.047 | 75.393 | 2.450
090516137 | 0.206 | 0.026 | 118.018 | 4.028
090516353 | 0.214 | 0.104 | 123.074 | 2.896
090528516 | 0.259 | 0.026 | 79.041 | 1.088
090618353 | 0.524 | 0.053 | 112.386 | 1.086
090620400 | 0.508 | 0.052 | 13.568 | 0.724
090626189 | 0.352 | 0.025 | 48.897 | 2.828
090718762 | 0.482 | 0.055 | 23.744 | 0.802
090809978 | 0.732 | 0.124 | 11.008 | 0.320
090810659 | 0.558 | 0.104 | 123.458 | 1.747
090829672 | 0.300 | 0.029 | 67.585 | 2.896
090831317 | 0.102 | 0.013 | 39.424 | 0.572
090902462 | 0.188 | 0.014 | 19.328 | 0.286
090926181 | 0.369 | 0.032 | 13.760 | 0.286
091003191 | 0.316 | 0.033 | 20.224 | 0.362
091127976 | 0.611 | 0.060 | 8.701 | 0.571
091208410 | 0.409 | 0.031 | 12.480 | 5.018
100414097 | 0.183 | 0.020 | 26.497 | 2.073
Table 4: Summary of Short GRBs. GRB Number | $H$ | $\delta H$ | $T_{90}$ [sec] | $\delta T_{90}$ [sec]
---|---|---|---|---
080723913 | 0.026 | 0.008 | 0.192 | 0.345
081012045 | -0.022 | -0.002 | 1.216 | 1.748
081102365 | 0.075 | 0.011 | 1.728 | 0.231
081105614 | 0.135 | 0.026 | 1.280 | 1.368
081107321 | 0.331 | 0.059 | 1.664 | 0.234
081216531 | 0.366 | 0.046 | 0.768 | 0.429
090108020 | 0.482 | 0.048 | 0.704 | 0.143
090206620 | 0.358 | 0.042 | 0.320 | 0.143
090227772 | 0.409 | 0.038 | 1.280 | 1.026
090228204 | 0.381 | 0.027 | 0.448 | 0.143
090308734 | 0.030 | 0.004 | 1.664 | 0.286
090429753 | 0.321 | 0.045 | 0.640 | 0.466
090510016 | 0.192 | 0.017 | 0.960 | 0.138
090621922 | 0.255 | 0.053 | 0.384 | 1.032
090907808 | 0.265 | 0.034 | 0.832 | 0.320
091012783 | 0.120 | 0.014 | 0.704 | 2.499
100117879 | 0.479 | 0.046 | 0.256 | 0.834
100204858 | 0.518 | 0.070 | 1.920 | 2.375
100328141 | 0.034 | 0.005 | 0.384 | 0.143
100612545 | 0.171 | 0.021 | 0.576 | 0.181
100625773 | 0.232 | 0.041 | 0.192 | 0.143
100706693 | -0.031 | -0.009 | 0.128 | 0.143
|
arxiv-papers
| 2012-09-11T18:59:24 |
2024-09-04T02:49:34.976678
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Glen MacLachlan, Ashwin Shenoy, Eda Sonbas, Rob Coyne, Kalvir Dhuga,\n Ali Eskandarian, Leonard Maximon, William Parke",
"submitter": "Glen MacLachlan",
"url": "https://arxiv.org/abs/1209.2396"
}
|
1209.2493
|
arxiv-papers
| 2012-09-12T04:33:08 |
2024-09-04T02:49:34.986018
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Subhabrata Mukherjee, Pushpak Bhattacharyya",
"submitter": "Subhabrata Mukherjee",
"url": "https://arxiv.org/abs/1209.2493"
}
|
|
1209.2495
|
arxiv-papers
| 2012-09-12T04:39:37 |
2024-09-04T02:49:34.989511
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Subhabrata Mukherjee, Akshat Malu, A.R. Balamurali, Pushpak\n Bhattacharyya",
"submitter": "Subhabrata Mukherjee",
"url": "https://arxiv.org/abs/1209.2495"
}
|
|
1209.2547
|
# On the Equivalence of Two Deformation Schemes in Quantum Field Theory
Gandalf Lechner 111Institute for Theoretical Physics, University of Leipzig,
04104 Leipzig, Germany, email: [email protected] Jan Schlemmer
222Department of Physics, University of Vienna, 1090 Vienna, Austria, email:
[email protected] GL and JS supported by FWF project P22929-N16
“Deformations of quantum field theories” Yoh Tanimoto 333Institute for
Theoretical Physics, University of Göttingen, 37077 Göttingen, Germany, email:
[email protected]. Supported by Deutscher
Akademischer Austauschdienst.
(September 11, 2012)
###### Abstract
Two recent deformation schemes for quantum field theories on two-dimensional
Minkowski space, making use of deformed field operators and Longo-Witten
endomorphisms, respectively, are shown to be equivalent.
Keywords: deformations of quantum field theories, two-dimensional models,
modular theory
MSC (2010): 81T05, 81T40
## 1 Deformations of QFTs by inner functions and their roots
In recent years, there has been a lot of interest in deformations of quantum
field theories [GL07, BS08, BLS11, DT11, LW11, LR12, MM11, Lec12, Tan12a,
BT12, Ala12, Muc12, Pla12] in the sense of specific procedures modifying
quantum field theoretic models on Minkowski space, mostly motivated by the
desire to construct new models in a non-perturbative manner. Various
constructions have been invented, relying on different methods such as smooth
group actions, non-commutative geometry, chiral conformal field theory,
boundary quantum field theory, and inverse scattering theory.
In many situations, it is possible to set up the deformation in such a way
that Poincaré covariance is completely preserved and locality partly. More
precisely, often the deformation introduces operators which are no longer
localized in arbitrarily small regions of spacetime, but rather in unbounded
regions like a Rindler wedge $W:=\\{x\in\mathbbm{R}^{d}\,:\,x_{1}>|x_{0}|\\}$.
In the operator-algebraic framework of quantum field theory [Haa96], such a
wedge-local Poincaré covariant model can be conveniently described by a so-
called Borchers triple $(\mathcal{M},U,\Omega)$ [Bor92, BLS11], consisting of
a von Neumann algebra $\mathcal{M}$ of operators localized in the wedge $W$, a
suitable representation $U$ of the translations, and an invariant (vacuum)
vector $\Omega$ (see Def. 1.2 below). Depending on the method at hand, the von
Neumann algebra $\mathcal{M}$ is generated by different objects, like deformed
field operators or twisted chiral observables.
It is the aim of this letter to show that some of the constructions on two-
dimensional Minkowski space are identical in the sense of unitary equivalence
of their associated Borchers triples. More precisely, we will show that the
deformations presented in [Tan12a], starting from a chiral field theory, are
equivalent to the deformations in terms of deformed field operators, presented
in [Lec12], for mass $m=0$ and dimension $d=1+1$ (Section 2). In the special
case of the so-called warped convolution deformation [BLS11], such an
equivalence was already observed in [Tan12a]. Here we prove that also for the
infinite family of deformations considered in [Lec12], one obtains the same
construction as in [Tan12a] in the chiral situation, where the deformation
amounts to a unitary equivalence transformation by a Longo-Witten endomorphism
[LW11] on each light ray. Furthermore, we will show that certain aspects of
the chiral construction carry over to the massive situation (Section 3).
The deformations we are interested in here take certain families of analytic
functions as input parameters, whose relations we will now clarify. We will
write $\mathbbm{H}\subset\mathbbm{C}$ for the open upper half plane,
$\mathrm{S}(0,\pi):=\\{\zeta\in\mathbbm{C}\,:\,0<\operatorname{Im}\zeta<\pi\\}$
for the strip, and $H^{\infty}(\mathbbm{H})$, $H^{\infty}(\mathrm{S}(0,\pi))$
for the Hardy spaces of bounded analytic functions on these domains. Recall
that for a function $f\in H^{\infty}(\mathbbm{H})$, the limit
$\lim_{\varepsilon\searrow 0}f(t+i\varepsilon)$ exists almost everywhere444By
“almost everywhere” (a.e.) and “almost all” we always refer to Lebesgue
measure on $\mathbbm{R}$. and defines a boundary value function in
$L^{\infty}(\mathbbm{R})$. The same holds for functions in
$H^{\infty}(\mathrm{S}(0,\pi))$ and their boundary values at $\mathbbm{R}$ and
$\mathbbm{R}+i\pi$.
###### Definition 1.1.
1. i)
A symmetric inner function is a function $\varphi\in H^{\infty}(\mathbbm{H})$
whose boundary values on the real line satisfy
$\overline{\varphi(t)}=\varphi(t)^{-1}=\varphi(-t)$ for almost all
$t\in\mathbbm{R}$.
2. ii)
A root of a symmetric inner function $\varphi$ is a function $R\in
L^{\infty}(\mathbbm{R})$ such that $\overline{R(t)}=R(t)^{-1}=R(-t)$ and
$R(t)^{2}=\varphi(t)$ for almost all $t\in\mathbbm{R}$. The family of all
roots of symmetric inner functions will be denoted $\mathcal{R}$.
3. iii)
A scattering function is a function $S\in H^{\infty}(\mathrm{S}(0,\pi))$ whose
boundary values satisfy
$\overline{S(\theta)}=S(\theta)^{-1}=S(-\theta)=S(i\pi+\theta)$ for almost all
$\theta\in\mathbbm{R}$.
Symmetric inner functions provide the input into deformations making use of
Longo-Witten endomorphisms [LW11, Tan12a, LR12, BT12], whereas scattering
functions are used in inverse scattering approaches such as [Lec03, BLM11].
For convenience, the latter are usually defined with the additional
requirement of extending continuously to the closure of $\mathrm{S}(0,\pi)$.
However, going through the construction, say in [Lec03], one realizes that
this continuity assumption is not necessary. What is required is that the
boundary conditions on $S$ hold almost everywhere, the boundary values are
regular enough to define multiplication operators on $L^{2}(\mathbbm{R})$, and
for $f\in H^{\infty}(\mathrm{S}(0,\pi))$ with Schwartz boundary values,
$[0,\pi]\ni\lambda\mapsto\int_{\mathbbm{R}}d\theta\,f(\theta+i\lambda)S(\theta+i\lambda)$
is continuous. As this is the case for any $S\in
H^{\infty}(\mathrm{S}(0,\pi))$, one can just as well work with the more
general definition of scattering function given above.
We also note that scattering functions and symmetric inner functions are in
one to one correspondence by $S(\zeta):=\varphi(\sinh\zeta)$,
$\zeta\in\mathrm{S}(0,\pi)$. As $\sinh(i\pi+\zeta)=-\sinh\zeta=\sinh(-\zeta)$,
this identification produces the required properties of the boundary values in
Def. 1.1 iii). On the other hand, $\varphi(z):=S(\sinh^{-1}z)$,
$z\in\mathbbm{H}$, is well-defined and analytic because of the crossing
symmetry $S(i\pi+\theta)=S(-\theta)$ of $S$. This identification of the strip
and the half plane via $\sinh$ is the one encountered in massive theories
[GL07]. In massless theories, also the identification
$\exp:\mathrm{S}(0,\pi)\to\mathbbm{H}$ occurs [LW11], and under this
identification, scattering functions correspond to the subset of symmetric
inner functions with the additional symmetry
$\overline{\varphi(t)}=\varphi(t^{-1})$ , see (1.20) below.
Regarding Def. 1.1 ii), we note that each symmetric inner function has
infinitely many different roots, and the family of roots $\mathcal{R}$
contains all symmetric inner functions because Def. 1.1 i) is stable under
taking squares. These roots are the input into the deformations in [Lec12,
Ala12, Pla12], where under additional regularity assumptions, they are called
deformation functions. We note that these additional requirements are only
necessary when working on the tensor algebra of test functions [Lec12], but
not when working directly on a representation space such as in [Ala12]. In
particular, the roots will not be required to be analytic, and also the
condition $R(0)=1$ [Lec12], related to fixing a root of an inner function,
will not be assumed here.
In the following, we will be concerned with deformations of free field
theories of mass $m\geq 0$ on two-dimensional Minkowski space, and now set up
some standard notation for this. We will be working on the Bose Fock space
$\displaystyle\mathcal{H}:=\Gamma(\mathcal{H}_{1})\,,\qquad\mathcal{H}_{1}:=L^{2}(\mathbbm{R},\tfrac{dp}{\omega_{m}(p)})\,,\quad\omega_{m}(p):=(m^{2}+p^{2})^{1/2}\,.$
Its Fock vacuum will be denoted $\Omega$, and we have the usual representation
$\Gamma(U_{1})$ of the proper Poincaré group as the second quantization of
$\displaystyle[U_{1}(x,\lambda)\Psi_{1}](p)$
$\displaystyle:=e^{i(x_{0}\omega_{m}(p)-x_{1}p)}\,\Psi_{1}(\lambda
p)\,,\qquad[U_{1}(j)\Psi_{1}](p):=\overline{\Psi_{1}(p)}\,,$ (1.1)
where $x=(x_{0},x_{1})\in\mathbbm{R}^{2}$ is the translation,
$\lambda\in\mathbbm{R}$ denotes the boost rapidity parameter, $\lambda
p:=-\sinh\lambda\cdot\omega_{m}(p)+\cosh\lambda\cdot p$, and $j(x)=-x$ is the
space-time reflection. We will also write $U(x):=\Gamma(U_{1}(x,0))$ for the
translations.
From an operator-algebraic point of view, a wedge-local quantum field theory
is equivalent to a Borchers triple.
###### Definition 1.2.
A Borchers triple $(\mathcal{M},U,\Omega)$ on $\mathbbm{R}^{2}$ consists of a
von Neumann algebra $\mathcal{M}\subset\mathcal{B}(\mathcal{H})$, a strongly
continuous unitary positive energy representation $U$ of the translation group
$\mathbbm{R}^{2}$ on $\mathcal{H}$, and a $U$-invariant unit vector
$\Omega\in\mathcal{H}$ such that
1. i)
$U(x)\mathcal{M}U(x)^{-1}\subset\mathcal{M}$ for any $x\in\overline{W}$,
2. ii)
$\Omega$ is cyclic and separating for $\mathcal{M}$.
Two Borchers triples $(\mathcal{M},U,\Omega)$ and
$(\tilde{\mathcal{M}},\tilde{U},\tilde{\Omega})$ will be called equivalent,
written
$(\mathcal{M},U,\Omega)\cong(\tilde{\mathcal{M}},\tilde{U},\tilde{\Omega})$,
if there exists a unitary $V$ such that
$V\mathcal{M}V^{*}=\tilde{\mathcal{M}}$, $VU(x)V^{*}=\tilde{U}(x)$ for all
$x\in\mathbbm{R}^{2}$, and $V\Omega=\tilde{\Omega}$.
Recall that by a famous theorem of Borchers [Bor92], the representation $U$
can be extended to a (anti-) unitary representation $U_{\mathcal{M}}$ of the
proper Poincaré group $\mathcal{P}_{+}$ with the help of the modular data
$J_{\mathcal{M}},\Delta_{\mathcal{M}}$ of $(\mathcal{M},\Omega)$, by
$\displaystyle
U_{\mathcal{M}}(x,\lambda):=U(x)\Delta_{\mathcal{M}}^{-\frac{i\lambda}{2\pi}}\,,\qquad
U_{\mathcal{M}}(j):=J_{\mathcal{M}}\,.$ (1.2)
As is well known, a Borchers triple gives rise to a Poincaré-covariant net of
wedge algebras [Bor92], which can under further conditions be extended to a
net of double cone algebras [BL04, Lec08]. We will not discuss the extension
question here, but rather focus on the wedge-local aspects only. Note that the
net of wedge algebras generated from a Borchers triple
$(\mathcal{M},U,\Omega)$ will transform covariantly under a representation
$U_{\mathcal{M}}$ of the Poincaré group which depends on $\mathcal{M}$.
However, in the case of two equivalent Borchers triples
$(\mathcal{M},U,\Omega)\cong(\tilde{\mathcal{M}},\tilde{U},\tilde{\Omega})$,
modular theory tells us that the modular data of $(\mathcal{M},\Omega)$ and
$(\tilde{\mathcal{M}},\tilde{\Omega})$ are related by
$VJ_{\mathcal{M}}V^{*}=J_{\tilde{\mathcal{M}}}$,
$V\Delta_{\mathcal{M}}^{it}V^{*}=\Delta_{\tilde{\mathcal{M}}}^{it}$, i.e.
equivalence of Borchers triples implies equivalence of the associated wedge-
local nets including their representations $U_{\mathcal{M}}\cong
U_{\tilde{\mathcal{M}}}$ of the proper Lorentz group.
A particular example of a Borchers triple is provided by the model of a free
scalar quantum field: Let $a(\xi)$ and $a^{\dagger}(\xi):=a(\xi)^{*}$,
$\xi\in\mathcal{H}_{1}$, denote the standard CCR annihilation and creation
operators on $\mathcal{H}$, and for $f\in\mathscr{S}(\mathbbm{R}^{2})$, let
$\displaystyle\phi_{m}(f)$
$\displaystyle:=a^{\dagger}(f^{+})+a(\overline{f^{-}})\,,\qquad
f^{\pm}(p):=\tilde{f}(\pm\omega_{m}(p),\pm p)\,,$ (1.3)
denote the free Klein-Gordon field of mass $m\geq 0$ (with $f$ restricted to
directional derivatives of test functions in the case $m=0$ because of the
well-known infrared divergence in the measure $\frac{dp}{|p|}$). With the
wedge algebra
$\displaystyle\mathcal{M}_{m}:=\\{e^{i\phi_{m}(f)}\,:\,f\in\mathscr{S}_{\mathbbm{R}}(W)\\}^{\prime\prime}\,,$
(1.4)
the Fock translations $U$ (1.1) and the Fock vacuum $\Omega$, we then have a
Borchers triple $(\mathcal{M}_{m},U,\Omega)$. In this case, the modular data
of $(\mathcal{M}_{m},\Omega)$ reproduce the Poincaré representation (1.1),
i.e. $U_{\mathcal{M}_{m}}=\Gamma(U_{1})$. For convenience of notation, we will
write
$\displaystyle
J:=J_{\mathcal{M}_{m}}=\Gamma(U_{1}(j))\,,\qquad\Delta^{it}:=\Delta^{it}_{\mathcal{M}_{m}}=\Gamma(U_{1}(0,-2\pi
t))\,.$ (1.5)
Fixing the representation $U$ of the translations and the vector $\Omega$, the
algebra $\mathcal{M}_{m}$ is however by no means the only von Neumann algebra
completing $U,\Omega$ to a Borchers triple. In the following, we will
introduce for each $R\in\mathcal{R}$ two von Neumann algebras
$\mathcal{M}_{R,m}$, $\mathcal{N}_{R}$ with this property, obtained by
(generalizations of) the deformation procedures in [Lec12] and [Tan12a],
respectively. For $R=1$, both families reduce to the undeformed situation,
i.e. $\mathcal{M}_{1,m}=\mathcal{M}_{m}$, $\mathcal{N}_{1}=\mathcal{M}_{0}$.
To define the first set of deformed wedge algebras $\mathcal{M}_{R,m}$, we
introduce a unitary-valued function
$T_{R,m}:\mathbbm{R}\to\mathcal{U}(\mathcal{H})$ [Lec12]
$\displaystyle[T_{R,m}(p)\Psi]_{n}(p_{1},\ldots,p_{n})$
$\displaystyle:=\prod_{k=1}^{n}R_{m}(p,p_{k})\cdot\Psi_{n}(p_{1},\ldots,p_{n})\,.$
(1.6)
Here the function $R_{m}\in L^{\infty}(\mathbbm{R}^{2})$ is in the case of
positive mass defined as
$\displaystyle R_{m}(p,q)$
$\displaystyle:=R\big{(}\tfrac{1}{2}(\omega_{m}(q)p-\omega_{m}(p)q)\big{)}\,,\qquad
m>0\,,$ (1.7)
where the factor $\frac{1}{2}$ is a matter of convention. Taking the limit
$m\to 0$, one observes that the argument $\frac{1}{2}(|q|p-|p|q)$ of $R$
vanishes if $p$ and $q$ have the same sign. As the root $R$ is only defined up
to equivalence in $L^{\infty}(\mathbbm{R})$, its value at $0$ is not fixed. We
therefore define
$\displaystyle R_{0}(p,q):=\left\\{\begin{array}[]{ccl}R(-pq)&;&p>0,\,q<0\\\
R(+pq)&;&p<0,\,q>0\\\
1&;&p>0,\,q>0\;\;\text{or}\;\;p<0,\,q<0\end{array}\right.\;.$ (1.11)
Note that for any mass $m\geq 0$, we have for almost all $p,q\in\mathbbm{R}$
$\displaystyle R_{m}(q,p)=R_{m}(p,q)^{-1}\,,\qquad m\geq 0\,.$ (1.12)
The assignment $a_{R}(p):=a(p)T_{R,m}(p)$ defines an operator-valued
distribution for any $R\in\mathcal{R}$, which explicitly acts on a vector
$\Psi$ of finite particle number according to, $\xi\in\mathcal{H}_{1}$,
$\displaystyle[a_{R}(\xi)\Psi]_{n}(p_{1},\ldots,p_{n})=\sqrt{n+1}\int\frac{dq}{\omega_{m}(q)}\,\overline{\xi(q)}\,\prod_{k=1}^{n}R_{m}(q,p_{k})\Psi_{n+1}(q,p_{1},\ldots,p_{n})\,.$
(1.13)
Its adjoint is denoted $a^{\dagger}_{R}(\xi):=a_{R}(\overline{\xi})^{*}$, and
the corresponding deformed field operator is
$\displaystyle\phi_{R,m}(f):=a^{\dagger}_{R}(f^{+})+a_{R}(\overline{f^{-}})\,,\qquad
f\in\mathscr{S}(\mathbbm{R}^{2})\,.$ (1.14)
As $\phi_{R,m}(f)$ is essentially self-adjoint on the subspace of finite
particle number for real $f$, one can pass to the generated von Neumann
algebra
$\displaystyle\mathcal{M}_{R,m}:=\\{e^{i\phi_{R,m}(f)}\,:\,f\in\mathscr{S}_{\mathbbm{R}}(W)\\}^{\prime\prime}\,.$
(1.15)
###### Theorem 1.3.
Let $R\in\mathcal{R}$ and $m\geq 0$. Then $(\mathcal{M}_{R,m},U,\Omega)$ is a
Borchers triple with modular data $J_{\mathcal{M}_{R,m}}=J$ and
$\Delta^{it}_{\mathcal{M}_{R,m}}=\Delta^{it}$.
For $m>0$, this has been established in [Lec12], and for $m=0$, one can use
essentially the same proofs, so that we do not have to go into details here.
In fact, as for $m=0$ the mass shell decomposes into two half-rays which are
left invariant by the Lorentz boosts, one can in this case more generally
consider three roots $R,R_{1},R_{2}\in\mathcal{R}$, with the additional
requirement $\overline{R_{k}(t)}=R_{k}(t^{-1})$ for almost all
$t\in\mathbbm{R}$, $k=1,2$, and put
$\displaystyle R_{0}(p,q):=\left\\{\begin{array}[]{ccl}R(-pq)&;&p>0,\,q<0\\\
R(+pq)&;&p<0,\,q>0\\\ R_{1}(+\tfrac{p}{q})&;&p>0,\,q>0\\\
R_{2}(-\tfrac{p}{q})&;&p<0,\,q<0\end{array}\right.\;.$ (1.20)
Also with this more general definition of $R_{0}$, the algebra
$\mathcal{M}_{R,0}$ completes $U,\Omega$ to a Borchers triple. In the
terminology of [FS93], the functions $R_{1}$, $R_{2}$ govern the left-left and
right-right “scattering” of the model, whereas $R$ determines the left-right
(wave) scattering [Bis12, Buc77, DT11]. If $R=1$, the corresponding model is
chiral – this is in particular the case for the short distance scaling limits
of the models generated by the massive wedge algebras $\mathcal{M}_{R,m}$,
$m>0$. In this context one finds $R=1$,
$R_{1}(t)^{2}=R_{2}(t)^{2}=\varphi(t-t^{-1})$ with some symmetric inner
function $\varphi$ [BLM11]. For the purposes of this letter, we will however
restrict ourselves to the case $R_{1}=R_{2}=1$ (1.11), which corresponds to
the construction in [Tan12a].
To define the second set of deformed wedge algebras $\mathcal{N}_{R}$, one
works in the massless case $m=0$, and uses the chiral structure present in
this situation. Here the Fock space, the representation $U$, the invariant
vector $\Omega$, and the wedge algebra $\mathcal{M}_{0}$ split into two
(chiral) factors. With
$\mathcal{H}_{1}^{\pm}:=L^{2}(\mathbbm{R}_{\pm},\frac{dp}{|p|})$, we have
$\mathcal{H}_{1}=\mathcal{H}_{1}^{+}\oplus\mathcal{H}_{1}^{-}$ and
$\displaystyle\mathcal{H}$
$\displaystyle\cong\mathcal{H}^{+}\otimes\mathcal{H}^{-}\,,\qquad\mathcal{H}^{\pm}:=\Gamma(\mathcal{H}_{1}^{\pm})\,,$
(1.21) $\displaystyle\Omega$
$\displaystyle\cong\Omega_{+}\otimes\Omega_{-}\,,$ (1.22) $\displaystyle U(x)$
$\displaystyle\cong U_{+}(x_{-})\otimes U_{-}(x_{+})\,,\qquad
x_{\pm}:=x_{0}\pm x_{1}\,,$ (1.23) $\displaystyle\mathcal{M}_{0}$
$\displaystyle\cong\mathcal{M}_{0,+}\otimes\mathcal{M}_{0,-}\,,$ (1.24)
where $\Omega_{\pm}$ denotes the Fock vacuum in $\mathcal{H}^{\pm}$. The
canonical unitary $V:\mathcal{H}^{+}\otimes\mathcal{H}^{-}\to\mathcal{H}$
realizing the above isomorphisms is recalled in (2.4). Note that
$\displaystyle V^{*}\Gamma(U_{1}(x,\lambda))V$
$\displaystyle=\Gamma_{+}(U_{1,+}(x_{-},\lambda))\otimes\Gamma_{-}(U_{1,-}(x_{+},\lambda))\,,$
(1.25) $\displaystyle V^{*}\Gamma(U_{1}(j))V$
$\displaystyle=\Gamma_{+}(U_{1,+}(j))\otimes\Gamma_{-}(U_{1,-}(j))\,,$ (1.26)
where $\Gamma_{\pm}$ denotes second quantization on $\mathcal{H}^{\pm}$, with
$(U_{1,\pm}(x_{\mp},\lambda)\Psi_{1})(p)=e^{\pm
ipx_{\mp}}\Psi_{1}(e^{\mp\lambda}\cdot p)$ and
$(U_{1,\pm}(j)\Psi_{1})(p)=\overline{\Psi_{1}(p)}$. For the sake of a concise
notation, we will write
$\displaystyle J_{\otimes}$
$\displaystyle:=\Gamma_{+}(U_{1,+}(j))\otimes\Gamma_{-}(U_{1,-}(j))=V^{*}JV\,,$
(1.27) $\displaystyle\Delta_{\otimes}^{it}$
$\displaystyle:=\Gamma_{+}(U_{1,+}(0,-2\pi
t))\otimes\Gamma_{-}(U_{1,-}(0,-2\pi t))=V^{*}\Delta^{it}V\,.$ (1.28)
Given $R\in\mathcal{R}$, one introduces the unitary
$S_{R}\in\mathcal{U}(\mathcal{H}^{+}\otimes\mathcal{H}^{-})$ [Tan12a],
$\displaystyle\big{[}S_{R}\Psi\big{]}_{n,n^{\prime}}(p_{1},\ldots,p_{n},q_{1},\ldots,q_{n^{\prime}})=\prod_{\begin{subarray}{c}i=1\ldots
n\\\ j=1\ldots
n^{\prime}\end{subarray}}R_{0}(p_{i},q_{j})\cdot\Psi_{n,n^{\prime}}(p_{1},\ldots,p_{n},q_{1},\ldots,q_{n^{\prime}})\,,$
(1.29)
and defines the von Neumann algebra
$\displaystyle\mathcal{N}_{R}:=(\mathcal{M}_{0,+}\otimes 1)\vee
S_{R^{2}}(1\otimes\mathcal{M}_{0,-})S_{R^{2}}^{*}\,.$ (1.30)
###### Theorem 1.4.
Let $R\in\mathcal{R}$. Then $(\mathcal{N}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})$ is a Borchers triple with modular data
$J_{\mathcal{N}_{R}}=S_{R^{2}}J_{\otimes}$ and
$\Delta^{it}_{\mathcal{N}_{R}}=\Delta_{\otimes}^{it}$.
This theorem has been proven in [Tan12a]. From (1.30), it is clear that
$\mathcal{N}_{R}$ depends on $R$ only via the symmetric inner function
$R^{2}$.
Our results can now compactly be summarized as follows ($R\in\mathcal{R}$,
$m\geq 0$):
* •
$(\mathcal{N}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})\cong(\mathcal{M}_{R,0},U,\Omega)$ (Theorem
2.4).
* •
$(\mathcal{M}_{R_{1},m},U,\Omega)\cong(\mathcal{M}_{R_{2},m},U,\Omega)$ if and
only if $R_{1}^{2}=R_{2}^{2}$ (Proposition 3.2).
As $R^{2}$ is essentially the two-particle S-matrix of the model described by
the Borchers triple $(\mathcal{M}_{R,m},U,\Omega)$, the last result amounts to
a proof of uniqueness of the solution of the inverse scattering problem in the
setting of the deformations studied here. In case of continuous $R$, such an
effect was already observed in [Ala12]. For massless nets obeying a number of
natural conditions, uniqueness of the inverse scattering problem is known once
one fixes the asymptotic algebra [Tan12a]. Furthermore, explicit examples of
S-matrices not preserving the Fock space structure are known in the massless
case [BT12].
The convenient deformation formula (1.30) is a result of the chiral structure
present in the massless case, and has no direct analogue in the massive case.
In Section 3, we will discuss why the situation is more complex in the massive
case even though (formal) relations between deformed and undeformed creation
and annihilation operators still exist.
## 2 Equivalence of the two deformations in the massless case
The aim of this section is to demonstrate the equivalence of the mass zero
Borchers triples $(\mathcal{M}_{R,0},U,\Omega)$ (1.15) and
$(\mathcal{N}_{R},U_{+}\otimes U_{-},\Omega_{+}\otimes U_{-})$ (1.30) for
arbitrary roots $R\in\mathcal{R}$. From Theorem 1.3 and Theorem 1.4, we see
that the modular groups of these von Neumann algebras coincide with the one
parameter boost groups on their respective Hilbert spaces, but their modular
conjugations differ by $S_{R^{2}}$, i.e. we have
$J_{\mathcal{N}_{R}}=S_{R^{2}}V^{*}J_{\mathcal{M}_{R,0}}V$ with the canonical
unitary $V:\mathcal{H}^{+}\otimes\mathcal{H}^{-}\to\mathcal{H}$ (2.4). We will
therefore in a first step go over to an equivalent form of $\mathcal{N}_{R}$
which has modular conjugation $V^{*}J_{\mathcal{M}_{R,0}}V$, without the
factor $S_{R^{2}}$. In general, this can be accomplished by conjugating with a
root of the “S-matrix” $J_{\mathcal{N}_{R}}V^{*}J_{\mathcal{M}_{R,0}}V$
[Wol92], and in our present situation, this amounts to considering
$\displaystyle\hat{\mathcal{N}}_{R}:=S_{R}^{*}(\mathcal{M}_{0,+}\otimes
1)S_{R}\vee S_{R}(1\otimes\mathcal{M}_{0,-})S_{R}^{*}\,.$ (2.1)
###### Lemma 2.1.
Let $R\in\mathcal{R}$. Then $(\hat{\mathcal{N}}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})$ is a Borchers triple equivalent to
$(\mathcal{N}_{R},U_{+}\otimes U_{-},\Omega_{+}\otimes\Omega_{-})$, with
modular data
$\displaystyle
J_{\hat{\mathcal{N}}_{R}}=J_{\otimes}\,,\qquad\Delta^{it}_{\hat{\mathcal{N}}_{R}}=\Delta^{it}_{\otimes}.$
(2.2)
###### Proof.
As $S_{R}$ (1.29) satisfies $S_{R}^{2}=S_{R^{2}}$, we have the unitary
equivalence of algebras $\hat{\mathcal{N}}_{R}=S_{R}^{*}\mathcal{N}_{R}S_{R}$.
The unitary $S_{R}$ clearly commutes with all translations
$U_{+}(x_{-})\otimes U_{-}(x_{+})$ and leaves $\Omega_{+}\otimes\Omega_{-}$
invariant. Hence $(\hat{\mathcal{N}}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})\cong(\mathcal{N}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})$; this also shows that
$(\hat{\mathcal{N}}_{R},U_{+}\otimes U_{-},\Omega_{+}\otimes\Omega_{-})$ is a
Borchers triple.
Regarding the modular data of
$(\hat{\mathcal{N}}_{R},\Omega_{+}\otimes\Omega_{-})$, we first have by
modular theory
$\displaystyle
J_{\hat{\mathcal{N}}_{R}}=S_{R}^{*}J_{\mathcal{N}_{R}}S_{R}\,,\qquad\Delta_{\hat{\mathcal{N}}_{R}}^{it}=S_{R}^{*}\Delta_{\mathcal{N}_{R}}^{it}S_{R}=S_{R}^{*}\Delta_{\otimes}^{it}S_{R}\,.$
(2.3)
Taking into account the action of $\Gamma_{\pm}(U_{1,\pm}(0,\lambda))$, one
sees from (1.11) and (1.29) that $S_{R}$ commutes with the Lorentz boosts. As
these coincide with the modular unitaries $\Delta^{it}_{\otimes}$, the second
equation in (2.2) follows. To establish the claimed form of the modular
conjugation, we note
$J_{\otimes}S_{R}=S_{\overline{R}}J_{\otimes}=S_{R}^{*}J_{\otimes}$ and
compute
$\displaystyle J_{\hat{\mathcal{N}}_{R}}$
$\displaystyle=S_{R}^{*}J_{\mathcal{N}_{R}}S_{R}=S_{R}^{*}S_{R^{2}}J_{\otimes}S_{R}=S_{R}^{*}S_{R}^{2}S_{R}^{*}J_{\otimes}=J_{\otimes}\,.$
This completes the proof. ∎
The equivalence between the two deformed Borchers triples with wedge algebras
$\mathcal{M}_{R,0}$ and $\hat{\mathcal{N}}_{R}$ will now be established using
the creation and annihilation operators into which the fields generating
$\mathcal{M}_{0}$ can be decomposed. Corresponding to the splitting
$\mathcal{M}_{0}=\mathcal{M}_{0,+}\otimes\mathcal{M}_{0,-}$ we have creation
and annihilation operators $a_{\pm}$, $a_{\pm}^{\dagger}$ acting on
$\mathcal{H}^{\pm}$.
In the following, we will always suppress the canonical embeddings
$\iota^{n}_{\pm}:(\mathcal{H}^{\pm}_{1})^{\otimes
n}\to\mathcal{H}_{1}^{\otimes n}$,
$(\iota^{n}_{\pm}\Psi_{n}^{\pm})(p_{1},...,p_{n}):=\Psi_{n}^{\pm}(p_{1},...,p_{n})$
for $p_{1},...,p_{n}\in\mathbbm{R}_{\pm}$ and
$(\iota^{n}_{\pm}\Psi_{n}^{\pm})(p_{1},...,p_{n}):=0$ otherwise. With these
embeddings understood,
$\mathcal{H}_{1}=\mathcal{H}_{1}^{+}\oplus\mathcal{H}_{1}^{-}$, and hence
$\mathcal{H}\cong\mathcal{H}^{+}\otimes\mathcal{H}^{-}$. This isomorphism is
given explicitly by a unitary
$V:\mathcal{H}^{+}\otimes\mathcal{H}^{-}\to\mathcal{H}\,,$
which is uniquely determined by its action on the total set [Gui72] of
“exponential vectors”
$e^{\Psi_{1}}:=\sum_{n=0}^{\infty}\frac{1}{\sqrt{n!}}\Psi_{1}^{\otimes n}$ by
$V(e^{\Psi_{1}}\otimes
e^{\Phi_{1}}):=e^{\Psi_{1}\oplus\Phi_{1}}\,,\qquad\Psi_{1}\in\mathcal{H}_{1}^{+},\;\Phi_{1}\in\mathcal{H}_{1}^{-}\,.$
(2.4)
The definitions of $\mathcal{M}_{R,0}$ and $\mathcal{N}_{R}$ make use of the
realizations of $\mathcal{H}$ as
$\Gamma(\mathcal{H}_{1}^{+}\oplus\mathcal{H}_{1}^{-})$ and
$\Gamma(\mathcal{H}_{1}^{+})\otimes\Gamma(\mathcal{H}_{1}^{-})$, respectively.
We will work on
$\mathcal{H}=\Gamma(\mathcal{H}_{1}^{+}\oplus\mathcal{H}_{1}^{-})$, and first
compute an explicit expression of $S_{R}$ on this space.
###### Lemma 2.2.
Let $R\in\mathcal{R}$ and $\hat{S}_{R}:=VS_{R}V^{*}$. Then,
$\Psi\in\mathcal{H}$,
$\displaystyle\big{[}\hat{S}_{R}\Psi\big{]}_{n}(p_{1},\ldots,p_{n})=\prod_{i,j=1}^{n}R_{0}^{+}(p_{i},p_{j})\cdot\Psi_{n}(p_{1},\ldots,p_{n})\,,$
(2.5)
where
$R_{0}^{+}(p,q)=\begin{cases}R(-pq)&\ p>0,q<0\\\ 1&\text{ else
}\end{cases}\,.$
###### Proof.
As exponential vectors form a total set in $\mathcal{H}$, it is sufficient to
compute $\hat{S}_{R}$ on $e^{\Psi_{1}\oplus\Phi_{1}}$ to verify (2.5). The
action of $V$ from (2.4) on vectors
$\Xi=\sum_{n,m=0}^{\infty}\Xi_{n,m}\in\bigoplus_{n,m=0}^{\infty}((\mathcal{H}_{1}^{+})^{\otimes_{s}n}\otimes(\mathcal{H}_{1}^{-})^{\otimes_{s}m})=\mathcal{H}^{+}\otimes\mathcal{H}^{-}$
is explicitly given by
$\big{[}V\Xi]_{n}=\sum_{k=0}^{n}\binom{n}{k}^{1/2}\operatorname{Symm}_{n}\Xi_{k,n-k}\,,$
(2.6)
where for $f:\mathbbm{R}^{n}\to\mathbbm{C}$,
$[\operatorname{Symm}_{n}f](p_{1},\ldots,p_{n}):=\frac{1}{n!}\sum_{\pi\in\textfrak{S}_{n}}f(p_{\pi(1)},\ldots,p_{\pi(n)})$
denotes total symmetrization. Combining this with (1.29), we find
$\displaystyle[$
$\displaystyle\hat{S}_{R}(e^{\Psi_{1}\oplus\Phi_{1}})]_{n}(p_{1},\ldots,p_{n})=\sum_{k=0}^{n}\binom{n}{k}^{1/2}\operatorname{Symm}_{n}(S_{R}(e^{\Psi_{1}}\otimes
e^{\Phi_{1}})_{k,n-k})(p_{1},\ldots,p_{n})$
$\displaystyle=\sum_{k=0}^{n}\frac{\binom{n}{k}^{1/2}}{n!}\sum_{\pi\in\textfrak{S}_{n}}\prod_{\begin{subarray}{c}i=1\ldots
k\\\ j=k+1\ldots
n\end{subarray}}\\!\\!R_{0}^{+}(p_{\pi(i)},p_{\pi(j)})\frac{\Psi_{1}(p_{\pi(1)})\cdots\Psi_{1}(p_{\pi(k)})\cdot\Phi_{1}(p_{\pi(k+1)})\cdots\Phi_{1}(p_{\pi(n)})}{\sqrt{k!}\sqrt{(n-k)!}}\,.$
In the second line, $R_{0}$ was replaced by $R_{0}^{+}$, which does not change
the result since the factors of $\Psi_{1}$ and $\Phi_{1}$ (explicitly writing
out the embedding $\Psi_{1}\circ\iota_{+}$ and $\Phi_{1}\circ\iota_{-}$) are
equal to zero unless $p_{\pi(i)}>0$ and $p_{\pi(j)}<0$, and
$R_{0}(p,q)=R_{0}^{+}(p,q)$ for $p>0$, $q<0$.
Next we change the range of indices $i=1,...,k$, $j=k+1,...,n$ of the product
to $i=1,...,n$, $j=1,...,n$. This does not change the result because for the
indices $i,j$ which were not present before, we have
$R_{0}^{+}(p_{\pi(i)},p_{\pi(j)})=1$ on the support of the remaining factors.
After these manipulations,
$(p_{1},...,p_{n})\mapsto\prod_{i,j=1}^{n}R_{0}^{+}(p_{\pi(i)},p_{\pi(j)})$ is
a totally symmetric function independent of $\pi$ and $k$. Thus we get
$\displaystyle[\hat{S}_{R}$
$\displaystyle(e^{\Psi_{1}\oplus\Phi_{1}})]_{n}(p_{1},\ldots,p_{n})$
$\displaystyle=\prod_{i,j=1}^{n}R_{0}^{+}(p_{i},p_{j})\sum_{k=0}^{n}\binom{n}{k}^{1/2}\frac{1}{n!}\sum_{\pi\in\textfrak{S}_{n}}\frac{\Psi_{1}(p_{\pi(1)})\cdots\Psi_{1}(p_{\pi(k)})\cdot\Phi_{1}(p_{\pi(k+1)})\cdots\Phi_{1}(p_{\pi(n)})}{\sqrt{k!}\sqrt{(n-k)!}}$
$\displaystyle=\prod_{i,j=1}^{n}R_{0}^{+}(p_{i},p_{j})\left[V(e^{\Psi_{1}}\otimes
e^{\Phi_{1}})\right]_{n}(p_{1},\ldots,p_{n})$
$\displaystyle=\prod_{i,j=1}^{n}R_{0}^{+}(p_{i},p_{j})\left[e^{\Psi_{1}\oplus\Phi_{1}}\right]_{n}(p_{1},\ldots,p_{n})\,,$
and the proof is finished. ∎
After these preparations, we can now state the precise relation between the
generators appearing in the two types of deformations.
###### Proposition 2.3.
Let $R\in\mathcal{R}$ be a root of a symmetric inner function. Then,
$\psi_{\pm}\in\mathcal{H}_{1}^{\pm}$,
$\displaystyle a_{R}(\psi_{+})=VS_{R}^{*}(a_{+}(\psi_{+})\otimes
1)S_{R}V^{*}\,,\qquad a_{R}(\psi_{-})=VS_{R}(a_{+}(\psi_{-})\otimes
1)S_{R}^{*}V^{*}\,.$ (2.7)
###### Proof.
Let $\Psi\in\mathcal{H}$ be a vector of finite particle number. Using
$a\circ\iota_{+}=V(a_{+}\otimes 1)V^{*}$, $a\circ\iota_{-}=V(1\otimes
a_{-})V^{*}$ and the corresponding relations for $a^{\dagger}$ and
$a^{\dagger}_{\pm}$, we can equivalently show
$\hat{S}_{R}^{*}a(\psi_{+})\hat{S}_{R}=a_{R}(\psi_{+})$,
$\hat{S}_{R}a(\psi_{-})\hat{S}_{R}^{*}=a_{R}(\psi_{-})$. To this end, we
compute ($p_{1},...,p_{n}\in\mathbbm{R}$):
$\displaystyle[$
$\displaystyle\hat{S}_{R}a(\psi)\hat{S}_{R}^{*}\Psi]_{n}(p_{1},\ldots,p_{n})=\prod_{i,j=1}^{n}{R_{0}^{+}}(p_{i},p_{j})\cdot\sqrt{n+1}\int\frac{dq}{\lvert
q\rvert}\,\overline{\psi(q)}\,[\hat{S}_{R}^{*}\Psi]_{n+1}(q,p_{1},\ldots,p_{n})$
$\displaystyle=\sqrt{n+1}\prod_{i,j=1}^{n}\\!\\!R_{0}^{+}(p_{i},p_{j})\int\frac{dq}{\lvert
q\rvert}\overline{\psi(q)}\,\prod_{i^{\prime},j^{\prime}=1}^{n}\\!\\!\overline{{R_{0}^{+}}(p_{i^{\prime}},p_{j^{\prime}})}\cdot\prod_{k=1}^{n}\overline{{R_{0}^{+}}(q,p_{k}){R_{0}^{+}}(p_{k},q)}\,\Psi_{n+1}(q,p_{1},..,p_{n})$
$\displaystyle=\sqrt{n+1}\>\bigg{\\{}\int_{0}^{\infty}\frac{dq}{\lvert
q\rvert}\,\overline{\psi(q)}\,\prod_{k=1}^{n}\overline{{R_{0}^{+}}(q,p_{k})}\Psi_{n+1}(q,p_{1},\ldots,p_{n})$
(2.8)
$\displaystyle\qquad\qquad\qquad\qquad+\int_{-\infty}^{0}\frac{dq}{\lvert
q\rvert}\,\overline{\psi(q)}\,\prod_{k=1}^{n}\overline{{R_{0}^{+}}(p_{k},q)}\Psi_{n+1}(q,p_{1},\ldots,p_{n})\bigg{\\}}\,,$
where in the last equality $R_{0}^{+}(p,q)=1$ unless $p>0$ and $q<0$ was used.
On the other hand, using $R_{0}(p,q)=R_{0}^{+}(p,q)$ for $p>0,q<0$, and
$R_{0}(p,q)=\overline{R_{0}^{+}(q,p)}$ for $p<0,q>0$, we find
$\displaystyle[a_{R}(\psi)\Psi]_{n}(p_{1},\ldots,p_{n})$
$\displaystyle=\sqrt{n+1}\int\frac{dq}{\lvert
q\rvert}\,\overline{\psi(p)}\,\prod_{k=1}^{n}R_{0}(q,p_{k})\Psi_{n+1}(q,p_{1},\ldots,p_{n})$
$\displaystyle=\sqrt{n+1}\bigg{\\{}\int_{0}^{\infty}\frac{dq}{\lvert
q\rvert}\,\overline{\psi(q)}\,\prod_{k=1}^{n}R_{0}^{+}(q,p_{k})\Psi_{n+1}(q,p_{1},\ldots,p_{n})$
(2.9) $\displaystyle\qquad\qquad\qquad+\int_{-\infty}^{0}\frac{dq}{\lvert
q\rvert}\,\overline{\psi(q)}\,\prod_{k=1}^{n}\overline{R_{0}^{+}(p_{k},q)}\Psi_{n+1}(q,p_{1},\ldots,p_{n})\bigg{\\}}\,,$
from which we read off $\hat{S}_{R}a(\psi)\hat{S}_{R}^{*}=a_{R}(\psi)$ for
$\text{supp}\,\psi\subset\mathbbm{R}_{-}$. For
$\text{supp}\,\psi\subset\mathbbm{R}_{+}$, the remaining integrals in (2.8)
and (2.9) agree up to complex conjugation of $R_{0}^{+}$; this is compensated
by using $\hat{S}_{R}^{*}=\hat{S}_{\overline{R}}$ , i.e. in this case we have
$\hat{S}_{R}^{*}a(\psi)\hat{S}_{R}=a_{R}(\psi)$. ∎
To obtain the equivalence of Borchers triples, recall that the massless field
$\phi_{0}$ decomposes into chiral components $\phi_{0,\pm}$, each depending on
one light ray coordinate $x_{\mp}=x_{0}\mp x_{1}$ only, namely for $f$ which
is the derivative of a function in $\mathscr{S}(\mathbbm{R}^{2})$,
$\displaystyle\phi_{0}(f)$ $\displaystyle=V\big{(}\phi_{0,+}(f_{+})\otimes
1+1\otimes\phi_{0,-}(f_{-})\big{)}V^{*}\,,\qquad\phi_{0,\pm}(f_{\pm})=a^{\dagger}_{\pm}(\tilde{f}_{\pm}|_{\mathbbm{R}_{\pm}})+a_{\pm}(\widetilde{\overline{f_{\pm}}}|_{\mathbbm{R}_{\pm}})\,,$
$\displaystyle f_{\pm}(\mp x_{\mp})$
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\int_{\mathbbm{R}}dx_{\pm}\,f\left(\tfrac{1}{2}(x_{+}+x_{-}),\tfrac{1}{2}(x_{+}-x_{-})\right)\,.$
(2.10)
The algebras in question are generated by these field operators (all of which
are essentially self-adjoint on their respective subspaces of finite particle
number) by
$\displaystyle\mathcal{M}_{0}$
$\displaystyle=\\{e^{i\phi_{0}(f)}\,:\,f\in\mathscr{S}_{\mathbbm{R}}(W)\\}^{\prime\prime}\,,$
(2.11) $\displaystyle\mathcal{M}_{0,\pm}$
$\displaystyle=\\{e^{i\phi_{0,\pm}(f_{\pm})}\,:\,f\in\mathscr{S}_{\mathbbm{R}}(W)\\}^{\prime\prime}=\\{e^{i\phi_{0,\pm}(g)}\,:\,g\in\mathscr{S}_{\mathbbm{R}}(\mathbbm{R}_{+})\\}^{\prime\prime}\,.$
(2.12)
We now come to the main result of this section.
###### Theorem 2.4.
Let $R\in\mathcal{R}$. Then
$(\mathcal{M}_{R,0},U,\Omega)\cong(\mathcal{N}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})$.
###### Proof.
The equivalence $(\mathcal{N}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})\cong(\hat{\mathcal{N}}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})$ was established already, and by
construction of $V$, we have $V(U_{+}(x_{-})\otimes U_{-}(x_{+}))V^{*}=U(x)$
and $V\Omega_{+}\otimes\Omega_{-}=\Omega$. Hence the claim follows once we
have shown $V\hat{\mathcal{N}}_{R}V^{*}=\mathcal{M}_{R,0}$.
By (2.1) and (2.12), $\hat{\mathcal{N}}_{R}$ is generated by the bounded
functions of the field operators $S_{R}^{*}(\phi_{0,+}(f_{+})\otimes 1)S_{R}$
and $S_{R}(1\otimes\phi_{0,-}(f_{-}))S_{R}^{*}$,
$f_{\pm}\in\mathscr{S}(\mathbbm{R}_{+})$. Conjugating with $V$, we have
$\displaystyle VS_{R}^{*}\big{(}\phi_{0,+}(f_{+})\otimes 1\big{)}S_{R}V^{*}$
$\displaystyle=\hat{S}_{R}^{*}\big{(}a^{\dagger}(\tilde{f}_{+}|_{\mathbbm{R}_{+}})+a(\widetilde{\overline{f_{+}}}|_{\mathbbm{R}_{+}})\big{)}\hat{S}_{R}$
$\displaystyle=a^{\dagger}_{R}(\tilde{f}_{+}|_{\mathbbm{R}_{+}})+a_{R}(\widetilde{\overline{f_{+}}}|_{\mathbbm{R}_{+}})\,,$
where in the second step, we have used Proposition 2.3, which also holds for
the creation operators by taking adjoints. Given $f_{+}$ which is the
derivative of a function in $\mathscr{S}(\mathbbm{R}_{+})$, we find $f$ which
is the derivative in $x_{+}$ direction of some function in $\mathscr{S}(W)$
such that $f_{+}$ is recovered from $f$ by (2.10) and $f_{-}=0$ (namely, one
can take the product of $f_{+}$ (which is a function of $x_{-}$) and a
function of $x_{+}$ with integral one). In this situation,
$f^{+}=\tilde{f}_{+}|_{\mathbbm{R}_{+}}$,
$\overline{f^{-}}=\widetilde{\overline{f_{+}}}|_{\mathbbm{R}_{+}}$, and thus
$\displaystyle VS_{R}^{*}\big{(}\phi_{0,+}(f_{+})\otimes
1\big{)}S_{R}V^{*}=\phi_{R,0}(f)\,.$
As all vectors of finite particle number are analytic for these field
operators, this equivalence also holds for their associated unitaries
$e^{i\phi_{R,0}(f)}$ and $e^{i\phi_{0,+}(f_{+})}$, and thus we have the
inclusion $VS_{R}^{*}(\mathcal{M}_{0,+}\otimes
1)S_{R}V^{*}\subset\mathcal{M}_{R,0}$.
Similarly, for the other light ray we obtain
$\displaystyle VS_{R}\big{(}1\otimes\phi_{0,-}(f_{-})\big{)}S_{R}^{*}V^{*}$
$\displaystyle=a^{\dagger}_{R}(\tilde{f}_{-}|_{\mathbbm{R}_{-}})+a_{R}(\widetilde{\overline{f_{-}}}|_{\mathbbm{R}_{-}})=\phi_{R,0}(f)$
for suitably chosen $f\in\mathscr{S}(W)$, and hence
$VS_{R}(1\otimes\mathcal{M}_{0,-})S_{R}^{*}V^{*}\subset\mathcal{M}_{R,0}$.
Thus $V\hat{\mathcal{N}}_{R}V^{*}\subset\mathcal{M}_{R,0}$. As $\Omega$ is
cyclic and separating for both $V\hat{\mathcal{N}}_{R}V^{*}$ and
$\mathcal{M}_{R,0}$, and their modular groups w.r.t. $\Omega$ coincide,
$\Delta_{V\hat{\mathcal{N}}_{R}V^{*}}^{it}=V\Delta_{\otimes}^{it}V^{*}=\Delta_{\mathcal{M}_{R,0}}^{it}$,
the equality of von Neumann algebras
$V\hat{\mathcal{N}}_{R}V^{*}=\mathcal{M}_{R,0}$ follows by Takesaki’s theorem
[Tak03] (see [Tan12b, Theorem A.1] for an explicit application). ∎
Recall that by construction, $\mathcal{N}_{R}$ (1.30) depends on $R$ only via
the symmetric inner function $R^{2}$, i.e.
$\mathcal{N}_{R_{1}}=\mathcal{N}_{R_{2}}$ if $R_{1}^{2}=R_{2}^{2}$. By the
equivalences
$\displaystyle(\mathcal{N}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})\cong(\hat{\mathcal{N}}_{R},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})\cong(\mathcal{M}_{R,0},U,\Omega)\,,$
this also implies
$(\mathcal{M}_{R_{1},0},U,\Omega)\cong(\mathcal{M}_{R_{2},0},U,\Omega)$ if
$R_{1}^{2}=R_{2}^{2}$.
## 3 Structure of massive deformations
The analysis in the previous section resulted in particular in two equivalence
properties of the massless deformed models: On the one hand, the two deformed
Borchers triples
$(\mathcal{M}_{R,0},U,\Omega)\cong(\mathcal{N}_{R,0},U_{+}\otimes
U_{-},\Omega_{+}\otimes\Omega_{-})$ depend only on the symmetric inner
function $\varphi=R^{2}$, i.e. choosing a different root of $\varphi$ results
in an equivalent model. On the other hand, the deformed and undeformed
(chiral) fields are unitarily equivalent. This equivalence however depends on
the light ray, and thus the triples $(\mathcal{M}_{R,0},U,\Omega)$ and
$(\mathcal{M}_{1,0},U,\Omega)$ are not equivalent for general roots
$R\in\mathcal{R}$ (the operator $S_{R^{2}}=S_{\varphi}$ appears as the
S-matrix, an invariant of Borchers triple [Tan12a, DT11]).
In this section, we show that the first property also holds in the massive
case, whereas the second one only holds in a weaker sense which is specified
below.
### Independence of the choice of root
We begin with a preparatory lemma.
###### Lemma 3.1.
Let $m\geq 0$ and $r\in\mathcal{R}$ be a root of the (trivial) symmetric inner
function $\varphi(t)=1$. Then the operator
$\displaystyle[Y_{r}\Psi]_{n}(p_{1},\ldots,p_{n}):=\prod_{1\leq i<j\leq
n}r_{m}(p_{i},p_{j})\cdot\Psi_{n}(p_{1},\ldots,p_{n})$ (3.1)
is a well-defined unitary on $\mathcal{H}$ which commutes with the
representation $U$, leaves $\Omega$ invariant, and satisfies
$\displaystyle Y_{r}\phi_{R,m}(f)Y_{r}^{*}\Psi=\phi_{r\cdot
R,m}(f)\Psi\,,\qquad
f\in\mathscr{S}(\mathbbm{R}^{2})\,,\;R\in\mathcal{R}\,,\;\Psi\in\mathcal{D}\,.$
(3.2)
###### Proof.
As $r$ is a root of $1$, it takes only the values $\pm 1$ and is in particular
real. Hence
$r_{m}(p_{j},p_{i})=\overline{r_{m}(p_{i},p_{j})}=r_{m}(p_{i},p_{j})$ is
symmetric and thus the product in (3.1) preserves the totally symmetric
subspace of $L^{2}(\mathbbm{R}^{n})$, and $Y_{r}$ defines a unitary on the
Bose Fock space $\mathcal{H}$.
It is clear that $Y_{r}$ commutes with translations and leaves $\Omega$
invariant. To establish (3.2), we first calculate for an annihilation operator
$a_{R}(\psi)$, $\psi\in\mathcal{H}_{1}$,
$\displaystyle[Y_{r}$ $\displaystyle
a_{R}(\psi)Y_{r}^{*}\Psi]_{n}(p_{1},\ldots,p_{n})$
$\displaystyle=\sqrt{n+1}\prod_{i<j}r_{m}(p_{i},p_{j})\int\frac{dq}{\omega_{m}(q)}\,\overline{\psi(q)}\,\prod_{k=1}^{n}R_{m}(q,p_{k})\cdot[Y_{r}^{*}\Psi]_{n+1}(q,p_{1},\ldots,p_{n})$
$\displaystyle=\sqrt{n+1}\prod_{i<j}|r_{m}(p_{i},p_{j})|^{2}\int\frac{dq}{\omega_{m}(q)}\,\overline{\psi(q)}\,\prod_{k=1}^{n}\Big{(}r_{m}(q,p_{k})R_{m}(q,p_{k})\Big{)}\Psi_{n+1}(q,p_{1},\ldots,p_{n})$
$\displaystyle=[a_{r\cdot R}(\psi)\Psi]_{n}(p_{1},\ldots,p_{n})\,.$
Thus $Y_{r}a_{R}(\psi)Y_{r}^{*}\Psi=a_{r\cdot R}(\psi)\Psi$, and by taking
adjoints, we also find
$Y_{r}a^{\dagger}_{R}(\psi)Y_{r}^{*}\Psi=a^{\dagger}_{r\cdot R}(\psi)\Psi$. As
$\phi_{R,m}(f)=a^{\dagger}_{R}(f^{+})+a_{R}(\overline{f^{-}})$, the claimed
equivalence (3.2) follows. ∎
With this lemma, it is now easy to show that the Borchers triple
$(\mathcal{M}_{R,m},U,\Omega)$ is independent of the choice of root up to
equivalence.
###### Proposition 3.2.
Let $R_{1},R_{2}\in\mathcal{R}$ be roots of the same symmetric inner function
$R_{1}^{2}=R_{2}^{2}$. Then
$(\mathcal{M}_{R_{1},m},U,\Omega)\cong(\mathcal{M}_{R_{2},m},U,\Omega)$,
$m\geq 0$.
###### Proof.
As $R_{1}^{2}=R_{2}^{2}$, the function $r(t):=R_{1}(t)R_{2}(t)^{-1}$ is a root
of $1$ as in Lemma 3.1, i.e. we have
$Y_{r}\phi_{R_{2},m}(f)Y_{r}^{*}=\phi_{R_{1},m}(f)$ (3.2) for any
$f\in\mathscr{S}(\mathbbm{R}^{2})$. But these field operators have the dense
subspace $\mathcal{D}$ of vectors of finite particle number as entire analytic
vectors [Lec12], and $Y_{r}\mathcal{D}=\mathcal{D}$. Hence the equivalence
(3.2) lifts to the unitaries $e^{i\phi_{R_{k},m}(f)}$, $k=1,2$,
$f\in\mathscr{S}_{\mathbbm{R}}(\mathbbm{R}^{2})$, and the von Neumann algebras
they generate, $Y_{r}\mathcal{M}_{R_{1},m}Y_{r}^{*}=\mathcal{M}_{R_{2},m}$.
Since $Y_{r}$ also commutes with $U$ and leaves $\Omega$ invariant, the
claimed equivalence of Borchers triples follows. ∎
As mentioned in Section 1, this result states that within the class of
Borchers triples considered here, the inverse scattering problem for the two-
particle S-matrix $R^{2}$ has a unique solution up to unitary equivalence. For
massless asymptotically complete nets, this uniqueness is known in the
stronger form that the wave S-matrix and the free net give an explicit formula
to construct the deformed Borchers triple [Tan12a]. It is an interesting open
problem to find its massive counterpart.
### Equivalence at fixed momentum
We now come to the discussion of equivalences between deformed and undeformed
field operators. In the massless case, this equivalence can be expressed as,
$R\in\mathcal{R}$,
$\displaystyle a_{R}(\xi)$
$\displaystyle=\begin{cases}\hat{S}_{R}^{*}a(\xi)\hat{S}_{R}&\text{supp}\,\xi\subset\mathbbm{R}_{+}\\\
\hat{S}_{R}a(\xi)\hat{S}_{R}^{*}&\text{supp}\,\xi\subset\mathbbm{R}_{-}\end{cases}\,,\qquad
m=0\,.$
For $m>0$, the Lorentz group acts transitively on the upper mass shell, so
that there is no invariant distinction between its left and right branch.
However, we still have an equivalence of the above form at sharp momentum.
Recall that for $p\in\mathbbm{R}$, the annihilator $a(p)$ is a well-defined
unbounded operator on the dense domain $\mathcal{D}_{0}\subset\mathcal{D}$ of
vectors $\Psi\in\mathcal{D}$ of finite particle number with continuous wave
functions $\Psi_{n}\in C(\mathbbm{R}^{n})$, $n\in\mathbbm{N}$.
To implement this equivalence, we define an operator-valued function
$\mathbbm{R}\ni p\mapsto\hat{S}_{R,m}(p)\in\mathcal{U}(\mathcal{H})$ by
$\displaystyle[\hat{S}_{R,m}(p)\Psi]_{n}(p_{1},\ldots,p_{n}):=\prod_{1\leq
i<j\leq n}R((p_{i}+p_{j})\wedge_{m}p)\cdot\Psi_{n}(p_{1},\ldots,p_{n})\,,$
(3.3)
where $p\wedge_{m}q:=\frac{1}{2}(\omega_{m}(q)p-\omega_{m}(p)q)$. Note that in
case the root $R$ is continuous, one has
$\hat{S}_{R,m}(p)\mathcal{D}_{0}=\mathcal{D}_{0}$.
Using $\overline{R(t)}=R(t)^{-1}=R(-t)$, the definition of $R_{m}$ (1.7), and
$(p+q)\wedge_{m}p=q\wedge_{m}p$, we then get
$\displaystyle[$
$\displaystyle\hat{S}_{R,m}(p)a(p)\hat{S}_{R,m}(p)^{*}\Psi]_{n}(p_{1},..,p_{n})$
$\displaystyle=\sqrt{n+1}\,\prod_{i<j}^{n}R((p_{i}+p_{j})\\!\wedge_{m}p)\cdot[\hat{S}_{R,m}(p)^{*}\Psi]_{n+1}(p,p_{1},..,p_{n})$
$\displaystyle=\sqrt{n+1}\,\prod_{i<j}^{n}\Big{(}R((p_{i}+p_{j})\wedge_{m}p)\overline{R((p_{i}+p_{j})\wedge_{m}p)}\Big{)}\cdot\prod_{k=1}^{n}\overline{R((p+p_{k})\wedge_{m}p)}\cdot\Psi_{n+1}(p,p_{1},...,p_{n})$
$\displaystyle=\sqrt{n+1}\,\prod_{k=1}^{n}\overline{R(p_{k}\wedge_{m}p)}\cdot\Psi_{n+1}(p,p_{1},...,p_{n})$
$\displaystyle=\sqrt{n+1}\,\prod_{k=1}^{n}R_{m}(p,p_{k})\cdot\Psi_{n+1}(p,p_{1},...,p_{n})$
$\displaystyle=[a_{R}(p)\Psi]_{n}(p_{1},...,p_{n})\,,$
where the last equality follows from comparison with (1.13). We thus have on
$\mathcal{D}_{0}$
$\displaystyle\hat{S}_{R,m}(p)a(p)\hat{S}_{R,m}(p)^{*}=a_{R,m}(p)\,.$ (3.4)
It should be noted that there is actually a big freedom in the choice of
$\hat{S}_{R,m}$ with this property, as it is only the adjoint action of
$\hat{S}_{R,m}(p)$ on $a(p)$ with the same momentum $p$ that matters in the
end. One manifestation of this freedom is the fact that $\hat{S}_{R,0}(p)$ for
$p>0$ does not agree with $\hat{S}_{R}^{*}$ (2.5), whereas their adjoint
action on $a(p)$ does. For $m>0$, another implementation of the equivalence is
$(\hat{S}_{R,m}(p)\Psi)_{n}(p_{1},\ldots p_{n})=\prod_{i<j}\overline{R({\rm
sgn}(\max(p_{j},p_{i})-p)|p_{i}\wedge_{m}p_{j}|)}\cdot\Psi_{n}(p_{1},\ldots,p_{n})\,,$
where the sign function sgn is defined with sgn$(0):=-1$. This can be checked
by a computation analogous to the previous one. Observe that if $p$ is
sufficiently large, this coincides with the root of the two-particle S-matrix
[Lec12], and for $p$ sufficiently small with its inverse. Hence this latter
implementation is analogous to the massless case, where the deformation is
given exactly by the S-matrix $\hat{S}_{R}$ and its adjoint.
By (formally) taking adjoints one gets the same relation between
$a^{\dagger}(p)$ and $a^{\dagger}_{R,m}(p)$. However, even when making this
adjoint rigorous (e.g. in the sense of quadratic forms) one cannot expect to
get an equivalence of the Fourier transform of the deformed and undeformed
field at sharp $p$. One has to keep in mind that the splitting into chiral
components is _not_ a splitting of the field according to momentum transfer
but related to a split of the one-particle Hilbert space into positive and
negative momentum parts. Thus creation and annihilation operators appear
either both with positive or both with negative momentum, so both are
transformed with $\hat{S}_{R,m}$ or $\hat{S}_{R,m}^{*}$. For the massive case
this mechanism is not available and therefore the relations between deformed
and undeformed creation and annihilation operators will not yield a
corresponding relation between the fields.
To conclude, the structure of the wedge algebra is deformed in a very
transparent manner in the chiral situation (1.30), but not for $m>0$, where
one has to rely on the use of generating fields. This observation is to some
extent in parallel with the simpler structure of the wave S-matrix in the
chiral case in comparison to the many particle S-matrix in the massive case,
and deserves further investigation.
## References
* [Ala12] S. Alazzawi. Deformations of Fermionic Quantum Field Theories and Integrable Models. _Lett. Math. Phys._ 103 (2013) 37–58
http://arxiv.org/abs/1203.2058v1
* [Bis12] M. Bischoff. Construction of Models in low-dimensional Quantum Field Theory using Operator Algebraic Methods. Ph.D. Thesis, Università di Roma “Tor Vergata” (2012)
* [BL04] D. Buchholz and G. Lechner. Modular nuclearity and localization. _Annales Henri Poincaré_ 5 (2004) 1065–1080
http://arxiv.org/abs/math-ph/0402072
* [BLM11] H. Bostelmann, G. Lechner and G. Morsella. Scaling limits of integrable quantum field theories. _Rev. Math. Phys._ 23 (2011) 1115–1156
http://arxiv.org/abs/1105.2781
* [BLS11] D. Buchholz, G. Lechner and S. J. Summers. Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories. _Commun. Math. Phys._ 304 (2011) 95–123
http://arxiv.org/abs/1005.2656
* [Bor92] H. Borchers. The CPT theorem in two-dimensional theories of local observables. _Commun. Math. Phys._ 143 (1992) 315–332
http://projecteuclid.org/euclid.cmp/1104248958
* [BS08] D. Buchholz and S. J. Summers. Warped Convolutions: A Novel Tool in the Construction of Quantum Field Theories. In E. Seiler and K. Sibold (eds.), _Quantum Field Theory and Beyond: Essays in Honor of Wolfhart Zimmermann_ , 107–121. World Scientific (2008)
http://arxiv.org/abs/0806.0349
* [BT12] M. Bischoff and Y. Tanimoto. Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II. _Commun. Math. Phys._ 317 (2013) 667–695
http://arxiv.org/abs/1111.1671v1
* [Buc77] D. Buchholz. Collision Theory for Massless Bosons. _Commun. Math. Phys._ 52 (1977) 147
http://projecteuclid.org/euclid.cmp/1103900494
* [DT11] W. Dybalski and Y. Tanimoto. Asymptotic completeness in a class of massless relativistic quantum field theories . _Commun. Math. Phys._ 305 (2011) 427–440
http://arxiv.org/abs/1006.5430
* [FS93] P. Fendley and H. Saleur. Massless integrable quantum field theories and massless scattering in 1+1 dimensions. _Technical Report USC-93-022_ (1993)
http://arxiv.org/abs/hep-th/9310058
* [GL07] H. Grosse and G. Lechner. Wedge-Local Quantum Fields and Noncommutative Minkowski Space. _JHEP_ 11 (2007) 012
http://arxiv.org/abs/0706.3992
* [Gui72] A. Guichardet. _Symmetric Hilbert Spaces and Related Topics_. Springer (1972)
* [Haa96] R. Haag. _Local Quantum Physics - Fields, Particles, Algebras_. Springer, 2 edition (1996)
* [Lec03] G. Lechner. Polarization-free quantum fields and interaction. _Lett. Math. Phys._ 64 (2003) 137–154
http://arxiv.org/abs/hep-th/0303062
* [Lec08] G. Lechner. Construction of Quantum Field Theories with Factorizing S-Matrices. _Commun. Math. Phys._ 277 (2008) 821–860
http://arxiv.org/abs/math-ph/0601022
* [Lec12] G. Lechner. Deformations of quantum field theories and integrable models. _Commun. Math. Phys._ 312 (2012) 265–302
http://arxiv.org/abs/1104.1948
* [LR12] R. Longo and K. Rehren. Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid. _Commun. Math. Phys._ 311 (2012) 769–785
http://arxiv.org/abs/1103.1141
* [LW11] R. Longo and E. Witten. An Algebraic Construction of Boundary Quantum Field Theory . _Commun. Math. Phys._ 303 (2011) 213–232
http://arxiv.org/abs/1004.0616
* [MM11] E. Morfa-Morales. Deformations of quantum field theories on de Sitter spacetime. _J. Math. Phys._ 52 (2011) 102304
http://arxiv.org/abs/1105.4856
* [Muc12] A. Much. Wedge-Local Quantum Fields on a Nonconstant Noncommutative Spacetime. _J. Math. Phys._ 53 (2012) 082303
http://arxiv.org/abs/1206.4450v1
* [Pla12] M. Plaschke. Wedge Local Deformations of Charged Fields leading to Anyonic Commutation Relations. _Preprint_ (2012)
http://arxiv.org/abs/1208.6141v1
* [Tak03] M. Takesaki. _Theory of Operator Algebras II_. Springer (2003)
* [Tan12a] Y. Tanimoto. Construction of wedge-local nets of observables through Longo-Witten endomorphisms. _Commun. Math. Phys._ 314 (2012) 443–469
http://arxiv.org/abs/1107.2629
* [Tan12b] Y. Tanimoto. Noninteraction of waves in two-dimensional conformal field theory. _Commun. Math. Phys._ 314 (2012) 419–441.
http://arxiv.org/abs/1107.2662
* [Wol92] M. Wollenberg. Notes on Perturbations of Causal Nets of Operator Algebras. _SFB 288 Preprint, N2. 36_ (1992) unpublished
|
arxiv-papers
| 2012-09-12T10:22:51 |
2024-09-04T02:49:34.997004
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Gandalf Lechner and Jan Schlemmer and Yoh Tanimoto",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1209.2547"
}
|
1209.2567
|
# Brightness variations in totally-eclipsing binary GSC 4589-2999
E. Sipahi [email protected] H. A. Dal O. Özdarcan Ege University,
Science Faculty, Department of Astronomy and Space Sciences, 35100 Bornova,
İzmir, Turkey
###### Abstract
We present multi-colour CCD photometry of GSC 4589-2999 obtained in 2008 and
2009. The observations indicate that the system is an active Algol binary.
Based on the new data, the mean brightness of the system is decreasing through
the years 2007-2009. The light curves obtained in 2008-2009 are modelled using
the Wilson-Devinney code. We also discussed the light and colour variations of
the system at different orbital phases. Evidence suggests that these
brightness and colour variations are due to the rotation of unevenly
distributed starspots on two components of the system.
###### keywords:
techniques: photometric - (stars:) binaries: eclipsing - stars: individual:
(GSC 4589-2999)
††journal: New Astronomy
## 1 Introduction
GSC 4589-2999 ($V$=10m.54) is classified as an eclipsing binary star in the
SIMBAD database. The system was discovered by Liakos and Niarchos (2007), who
provided an R band light curve. The binary has been included in our multi-
colour photometric programme. Liakos et al. (2011) presented BVRI light curves
and the radial velocities. The physical properties and the absolute parameters
of the components were given in their study.
We report the new photometric observations, and discussed the light and colour
variations of the system. The system is important not only because it joins
the group of Algol type binary systems, but also because it is a part of an
active Algol systems with total eclipse as we have showed in this study.
## 2 Observations
Observations were acquired with a thermoelectrically cooled ALTA U+42
2048$\times$2048-pixel CCD camera attached to a 40-cm Schmidt-Cassegrain MEADE
telescope at Ege University Observatory. The $BVRI$-band observations were
recorded over twelve nights in 2008 and fifteen nights in 2009. We used GSC
4589-2984 and GSC 4589-2842 as a comparison and check star, respectively. The
comparison and check stars used in all the observations are the same stars
used by Liakos et al. (2011). There was no variation observed in the
brightnesses of the comparison star.
During the observations, we obtained four primary and one secondary times of
minimum light. All the available times of minima are given in Table 1. We
combined our times of minima with the literature and obtained the new light
elements as follows:
$T{\rm(HJD)}=2455103.4986(19)+1^{d}.688646(6)\ \times\ E\ .$ (1)
All the phases in the tables and the figures are calculated with these new
light elements. The $O-C$ diagram constructed by using the light elements
given in Equation 1 indicates no variability of the period.
Table 1: The times of minimum light for GSC 4589-2999
O | Year | $(O-C)_{I}$ | $(O-C)_{II}$ | Type | Filter | REF
---|---|---|---|---|---|---
54290.4167 | 2007.51 | 0.0653 | 0.0012 | II | $RI$ | 1
54306.4601 | 2007.56 | 0.0653 | 0.0024 | I | $BVRI$ | 1
54312.3603 | 2007.57 | 0.0548 | -0.0076 | II | $BVRI$ | 1
54377.3822 | 2007.75 | 0.0587 | 0.0014 | I | $R$ | 1
54399.3355 | 2007.81 | 0.0579 | 0.0023 | I | $R$ | 1
54637.4327 | 2008.46 | 0.0373 | 0.0004 | I | $BVRI$ | 1
54642.4998 | 2008.48 | 0.0381 | 0.0016 | I | $V$ | 1
54691.4703 | 2008.61 | 0.0340 | 0.0013 | I | $V$ | 1
54752.2629 | 2008.78 | 0.0306 | 0.0027 | I | $BVRI$ | 2
54833.3082 | 2009.00 | 0.0145 | -0.0070 | I | $BVRI$ | 2
55055.3703 | 2009.61 | 0.0023 | -0.0019 | II | $I$ | 2
55076.4838 | 2009.66 | 0.0061 | 0.0036 | I | $BVRI$ | 3
55103.4982 | 2009.74 | 0.0000 | -0.0004 | I | $BVRI$ | 3
* 1
Liakos and Niarchos (2009)
* 2
Sipahi et al. (2009)
* 3
This Study
Figure 1: 2008 (left) and 2009 (right) light and colour curves obtained in
this study.
The observed V light and B-V colour curves are shown in Figure 1. The shape of
the light curves indicate that GSC 4589-2999 is an Algol type binary. The
system has a totality in the secondary eclipse, which lasts about $\sim$1.97
hours. The light curve variations which obtained in 2008 and 2009 are the
different. Both minima are asymmetrical enough in the light curves. The mean
depths of eclipses in B, V, R, and I filters are 0m.252, 0m.236, 0m.222, and
0m.204 in the primary minimum and 0m.056, 0m.061, 0m.066, and 0m.083 in the
secondary minimum, respectively. The light curves also display an asymmetrical
light variations at outside of the eclipse. The amplitudes of these variations
are $\sim$0m.052, 0m.059, 0m.062, 0m.069 in the $B$, $V$, $R$, and $I$
filters, respectively. These light variations are very similar to the
photometric variations commonly observed for RS CVn type variables, which
appear to arise from an uneven distribution of star spots over the surface of
the rotating star (Hall, 1976). The general features are not similar to those
of the light curves previously taken from Liakos et al. (2011).
## 3 Discussion and conclusions
### 3.1 The activity of both components
The previous light curves of the system have been presented and their
variations discussed by Liakos et al. (2011). They argued for a cool spot in
the secondary component. However, all the phenomena observed in the light
curves could not be explained by taking into account only one component’s spot
activity. GSC 4589-2999 has a totality in the secondary eclipse. The
brightness level at totality of the secondary minimum change in each
observation. This light variability seen in the total secondary eclipse should
be originated from the more massive primary star. Liakos et al. (2011)
analysed the light curves with the radial velocity curve and derived the
orbital parameters of the system. Using these parameters of the system in this
study, the R-band light curves observed in 2008 and 2009 were modelled by
fitting dark spots onto the surface of the primary and secondary components
with using the PHOEBE V.0.31a software (Prša & Zwitter,, 2005). The result
spot parameters are listed in Table 2. The synthetic light curves obtained
from the modelled light curves are seen in Figure 2.
Figure 2: The R-band light curves observed in 2008 (upper panel) and 2009 (bottom panel) and the synthetic curves (lines) derived from the light curve models. Table 2: The spot parameters of the components in 2008 and 2009. Year | Parameter | Primary | Secondary
---|---|---|---
| | Spot I | Spot II | Spot I | Spot II
2008 | $Co-Lat$ (∘) | 56 | 40 | 66 | -
| $Co-Long$ (∘) | 0 | 70 | 70 | -
| $R$ (∘) | 15 | 15 | 25 | -
| $T_{Spot}$/$T_{Sur}$ | 0.7 | 0.8 | 0.6 | -
2009 | $Co-Lat$ (∘) | 20 | 40 | 86 | 85
| $Co-Long$ (∘) | 350 | 45 | 290 | 95
| $R$ (∘) | 20 | 10 | 20 | 25
| $T_{Spot}$/$T_{Sur}$ | 0.6 | 0.8 | 0.6 | 0.60
### 3.2 The shape of the light curves at the bottom of the primary minima
The system shows at primary minimum either a curved bottom normal for a
partial eclipse or a flat bottom which resembles a total eclipse. In order to
study further changes at the primary minima, the observations in $R$ filter
are collected. The type of the bottom of the primary eclipse October 12, 2008
was total, later, on September 28, 2009 it was partial. When a minimum showed
asymmetry, the bottom of the eclipse shows partial shape. Both bottom types
(partial or flat) of the brightness level at the primary minima have almost
different. The shape of the minima in the light curves will be changed whether
the spots are located at minimum phases toward to observing side or not. The
brightness level of the primary minima in all filters are listed in Table 3.
The variation of the brightness level in R filter are seen in Figure 3. As
seen from Table 3 and Figure 3, the brightness level of the primary minimum is
decreasing $\sim$0m.09 during three years. It seems likely that the brightness
changes in the secondary component caused by spots is responsible for these
variations.
Figure 3: The variation of the brightness level of the primary minimum of GSC
4589-2999.
We conclude here that the system exhibits both types of bottoms of the primary
minimum and the change between two types occurs rather frequently and rapidly.
Many investigators have reported such activities on active Algol binaries.
Olson (1982a, b) discovered the variable depth, which was only a little but
considered real, of the total primary minimum in U Sge. Olson (1987) proposed
an explanation that the brightness changes could be due to the temperature
variations on the photosphere of the cool component.
### 3.3 The out-of-eclipse variation
The brightness of the system is decreasing though the years 2007-2009. The
observations of Liakos and Niarchos (2007) showed that the light variations of
out-of-eclipse had the smallest amplitude observed in 2007. In the observing
season 2008, the amplitude increased to 0m.024. These light variations are
caused by the effects of cool spot activity existing on both components of the
system.
In addition, we noticed considerable changes in the shape of the light curve
on timescales of a few weeks in our observations. The spot distribution on the
surfaces of the components are rapidly changing. The effectivity of the spots
vary from one month to the next with an amplitude of approximately 0m.07 in R
filter. A similar behaviour was found in the light variation of DK CVn by Dal
et al. (2012).
Examining carefully the light variation between the phases 0.70-0.95 in Figure
1, we revealed a rapid light variation at the same phases. One can notice that
the observations made in August 03, 13, 18 and 25 revealed that the levels of
light follow themselves in the observing season 2009. The light variations
obtained in August 30, September 1 and October 1 also follow themselves in the
general light variation. Although both these groups are corresponding to the
same phases between 0.70 and 0.95, their levels are different from each other.
Table 3: The brightness level of the primary minima of GSC 4589-2999 HJD | B | V | R | I | Type
---|---|---|---|---|---
(+24 00000) | (mag) | (mag) | (mag) | (mag) |
54296.3289 | | | -0.419 | | I
54306.4590 | | | -0.427 | | I
54377.3825 | | | -0.423 | | I
54399.3345 | | | -0.410 | | I
54752.2603 | -1.209 | -0.660 | -0.377 | -0.096 | I
54833.3150 | -1.212 | -0.656 | -0.366 | -0.106 | I
55076.4784 | -1.175 | -0.644 | -0.368 | -0.092 | I
55103.4985 | -1.153 | -0.614 | -0.347 | -0.065 | I
54290.4179 | | | -0.586 | | II
54643.3433 | | -0.860 | -0.569 | -0.266 | II
54714.2662 | -1.450 | -0.894 | -0.601 | -0.289 | II
55055.3707 | -1.411 | -0.856 | -0.574 | -0.265 | II
### 3.4 The short-term variation behaviour in light and colour curves
Apart from the long-term variations in the light curves, examining each
nightly observation day by day demonstrated that there are also some
distinctive short-term variations in even one observing season. As seen from
Figure 1, there are some distinctive relations between light and colour
curves. Comparisons of the observations phased between the same range indicate
that the brightness level of the system is rapidly changing in even one
season. In observing season 2008, the observations obtained in August 20,
November 16, and December 03 are corresponding to the same phases between 0.7
and 0.8, however their brightness levels are different from each other. In
2008, the brightness level in the $V$-band decreased from August 20 to
December 03, while the $B-V$ colour gotten bluer. Similarly, the observations
in July 28, October 12, and December 15 are corresponding to the same phase
range. The levels of $V$-band magnitude and $B-V$ colour observed in July 28
and October 12 are almost the same. However, the brightness level in the
$V$-band increased in December 15, and the $B-V$ colour gotten bluer. In the
same way, the level of the secondary minima is also changing. The observations
in June 25 and September 04 are corresponding to the same phase interval
between 0.46 and 0.68. Comparison of these observations indicates that the
level of the secondary minimum is changing. As seen in Figure 1, from June 25
to September 04, the level of the $V$-band magnitude increased, while the
$B-V$ colour gotten bluer. In observing season 2009, there are six observing
nights, in which the observations are corresponding to the same phases between
0.6 and 0.8. The observations made in August 03, 13 and 18, revealed that the
levels of both light and colour are almost the same. Similarly, the levels
observed in August 30, September 01 and October 01 are the same among
themselves. On the other hand, the mean levels obtained from these two
observing groups are clearly different from each other. The level of the
$V$-band magnitude decreased from the previous three night to the latter,
while the $B-V$ colour gotten bluer in contrast to the $V$ light. The same
phenomenon is seen in the observations made in July 23, August 04, 29 and
October 07, which all of them are corresponding to the same phases between
0.10 and 0.35. The levels of the light and colour are almost the same in July
23, August 04, 29. However, the $V$-band magnitude decreased in October 07,
while the $B-V$ colour gotten bluer.
## 4 Results
Based on the results presented in the previous sections we draw conclusions as
follows;
$\bullet$ The main features of the light curves are: a) the asymmetry in the
minima and the unequal maxima; b) the larger brightness variations at total
primary minima than at any other phase. The observed light variations of GSC
4589-2999 can be well explained by the spot activity occurring on both
components of the system.
$\bullet$ When the spots located on the components are seen at minimum phases,
the amplitude of the light variation is larger than in other phases. These
phenomenon is generally referred to a greater concentration of active regions
when the active longitude is on the hemisphere facing the companion (Lanza et
al., 1998).
$\bullet$ The out-of-eclipse brightness variation of the system were examined
in detailed. During the three years the average brightness has decreased about
0m.024. The larger brightness decreases occur at the primary eclipse. This
indicates a change in the spots on the secondary component, that may be due to
an increase in the spot coverage. The amplitude of each seasonal variation
indicates the degree of spottedness of the stellar surface. The light
variations during out-of-eclipses is highly variable, changing at times within
a few week. Similar results have found for DK CVn (Dal et al., 2012).
$\bullet$ Another important feature observed in the system is colour variation
behaviour. The $B-V$ colour is sometimes getting bluer, while the brightness
of the system is increasing. However, the $B-V$ colour is also getting bluer,
while the brightness level is decreasing. As it is well known, the long-term
photometric studies of Solar like old stars demonstrated that the bright-hot
features such as faculae provide larger excess to the total light, instead of
the cool-dark features such stellar spots. These bright-hot features are
generally located around the cool spots in the active regions on the stellar
surface. In such a case, the cool-dark spots cause general light variations
due to the rotational modulation. In addition, the bright-hot faculae cause
some additional blue excess in $B-V$ and especially $U-B$ (Wilson, 1994;
Berdyugina, 2005). If GSC 4589-2999 is a Solar like old system as offered by
Liakos et al. (2011), the bright-hot features like faculae could occur on the
stellar surface and also can cause the variations seen in the colour curves of
both 2008 and 2009.
$\bullet$ The short-term variations such as those observed from the system of
GSC 4589-2999 are generally observed in the young and rapidly rotating BY Dra
type stars (Strassmeier, 2009), such as AB Dor (Amado et al., 2001) and LO Peg
(Pandey et al., 2011) and also some low-mass close binary systems, such as DK
CVn (Dal et al., 2012). However, according to the results obtained by Liakos
et al. (2011), GSC 4589-2999 is an evolved system. In this case, the rapid
variability seen in the system can not be caused due to young age, but it
should be due to the binarity effect.
$\bullet$ It is clear that more observations will be necessary to describe the
long-term activity of GSC 4589-2999. Similar variations have also been
observed for other active Algol binaries (Olson, 1987). We conclude that GSC
4589-2999 is a new Algol type active binary with a changing light curve.
$\bullet$ GSC 4589-2999 has very high activity level. In terms of stellar
astrophysics, the chromospherically active binary systems like GSC 4589-2999
are very important to understand the stellar evolution and also evolution of
the angular momentum. Firstly, developing a model about the formation of the
close binaries, Rocha-Pinto et al. (2002) indicate some unexpected systems,
which are chromospherically young without high Li abundance, while they are
kinematically old. According to them, their components are generally rapidly
rotating due to the binarity. The initial parameters of a system are important
for such a model. The parameters of the system, which was in the main-sequence
stage, are generally assumed as the initial parameters. In this respect, the
chromospherically active binary systems, whose components are still in the
main-sequence, become very important to determine the absolute parameters such
as mass, radii, rotational velocity, and so the stellar angular momentum in
this stage. In addition, as it is seen from the literature, there are no more
chromospherically active systems, enough to determine general outline of their
absolute parameters in the main-sequence stage. Secondly, there is another
unsolved problem about the stellar absolute parameters for the
chromospherically active components of the binaries. Several studies, such as
López-Morales (2007), Casagrande et al. (2008), Morales et al. (2008, 2010),
Fernandez et al. (2009), Torres et al. (2010), and Kraus et al. (2011), have
already revealed that the radii of chromospherically active components are
found dramatically larger than expected ones. The main effect on the larger
radius seems to be the chromospheric activity. Although the general dividing
is seen for the low mass stars, there is no information for the stars from the
spectral types G and K due to fewer samples. In this respect, so many systems
like GSC 4589-2999 should be found to multiple the number of samples. In
addition, López-Morales (2007), Morales et al. (2008) and Morales et al.
(2010) mentioned that the best way to find radius of a star is the light curve
analyses of eclipsing binaries, especially having minima with totality.
Finally, it should be noted that the photometric observations of GSC 4589-2999
are continuing. Long-term photometry of the system will help paint a clearer
picture of the activity nature of this interesting active binary.
## Acknowledgments
The authors acknowledge generous allotments of observing time at the Ege
University Observatory. We also wish to thank the referee for useful comments
that have contributed to the improvement of the paper.
## References
* Amado et al. (2001) Amado, P.J., Cutispoto, G., Lanza, A.F., Rodonó, M., 2001, AB Doradus: Long and Short Term Light Variations and Spot Parameters, in _Cool Stars, Stellar Systems, and the Sun (11th Cambridge Workshop)_ , (Eds.) García López, R.J., Rebolo, R., Zapaterio Osorio, M.R., Proceedings of a meeting held at Puerto de la Cruz, Tenerife, Spain, 4 8 October 1999, vol. 223 of ASP Conference Series, pp. 895 900, Astronomical Society of the Pacific, San Francisco, U.S.A. 11, 6.1, 5
* Berdyugina (2005) Berdyugina, S.V., 2005, LRSP, 2, 8 (P.36)
* Casagrande et al. (2008) Casagrande, L., Flynn, C., and Bessell, M., 2008, MNRAS, 389, 585
* Dal et al. (2012) Dal, H.A., Sipahi, E., Özdarcan, O., 2012, PASA, 29, 54
* Fernandez et al. (2009) Fernandez, J.M., et al., 2009, ApJ, 701, 764
* Hall (1976) Hall, D.S., 1976, in ”Multiple Periodic Phenomena in Variable Stars”, IAU Colloq. No. 29, Part I, edited by W.S. Fitch (Reidel, Dordrecht)
* Kraus et al. (2011) Kraus, A.L., Tucker, R.A., Thompson, M.I., Craine, E.R., Hillenbrand, L.A., 2011, ApJ, 728, 48
* Lanza et al. (1998) Lanza, A. F., Catalano, S., Cutispoto, G., Pagano, I., Rodonó, M., 1998, A&A, 332, 541
* Liakos and Niarchos (2007) Liakos, A., Niarchos, P., 2007, IBVS, 5900, 2
* Liakos and Niarchos (2009) Liakos, A., Niarchos, P., 2009, IBVS, 5897, 1
* Liakos et al. (2011) Liakos, A., Bonfini, P., Niarchos, P., Hatzidimitriou, D., 2011, AN, 332, 602
* López-Morales (2007) López-Morales, M., 2007, ApJ, 660, 732
* Morales et al. (2008) Morales, J.C., Ribas, I., Jordi, C., 2008, A&A, 478, 507
* Morales et al. (2010) Morales, J.C., Gallardo, J., Ribas, I., Jordi, C., Baraffe, I., Chabrier, G., 2010, ApJ, 718, 502
* Olson (1982a) Olson, E.C., 1982a, ApJ, 259, 702
* Olson (1982b) Olson, E.C., 1982b, PASP, 94, 700
* Olson (1987) Olson, E.C., 1987, AJ, 94, 1043
* Pandey et al. (2011) Pandey, J.C., Medhi, B.J., Sagar, R., 2011, IAUS, 273, 455
* Prša & Zwitter, (2005) Prša, A., Zwitter, T., 2005, ApJ, 628, 426
* Rocha-Pinto et al. (2002) Rocha-Pinto, H.J., Castilho, B.V., Maciel, W.J., 2002, A&A, 384, 912
* Sipahi et al. (2009) Sipahi, E., Dal, H.A., Özdarcan, O., 2009, IBVS, 5904, 1
* Strassmeier (2009) Strassmeier, K.G., 2009, A&ARv, 17, 251
* Torres et al. (2010) Torres, G., Andersen, J., Giménez, A., 2010, A&ARv, 18, 67
* Wilson (1994) Wilson, P.R., 1994, ”Solar and stellar activity cycles”, First Edition, ed. R.F. Carswell, D.N.C. Lin and J.E. Pringle, United States of America by Cambridge University Press, New York, p.118
|
arxiv-papers
| 2012-09-12T11:27:12 |
2024-09-04T02:49:35.003950
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Sipahi, H. A. Dal, O. \\\"Ozdarcan",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1209.2567"
}
|
1209.2576
|
$\langle$reception date$\rangle$ $\langle$acception date$\rangle$
$\langle$publication date$\rangle$ Astronomical Society of JapanUsage of
pasj00.cls
Methods: data analysis — (Stars:) binaries (including multiple): close —
Stars: activity — (Stars:) starspots — stars: individual: (V369 Gem)
# V369 Gem: A New Ellipsoidal Variable System
With High Level Chromospheric Activity
Dal H. A Sipahi E Özdarcan O Department of Astronomy and Space Sciences,
University of Ege,
Bornova, 35100 İzmir, Turkey [email protected]
###### Abstract
In this study, we present BVR photometry of V369 Gem in 2006 - 2008. We
combined our V data with the data taken from the literature. We analysed all
the data together and calculated the new light elements. General features of
the light curves were discussed. The light curves were modelled with the
Fourier method and the Fourier coefficients calculated. The $\cos(2\theta)$
term was found to be dominant. The pre-whitened V light curves demonstrated
that stellar spot activity also exist. There are generally two minima in the
pre-whitened light curves. Thus, there must be two spotted areas, which are
separated about 180∘ from each other. As conclusions, we have remarked for the
first time that V369 Gem is an ellipsoidal variable with the high level
chromospheric activity.
## 1 Introduction
Many stars from the spectral types G, K and M exhibit solar like chromospheric
activity. The stellar spots on the photosphere of a star cause quasi-periodic
sinusoidal-like variations due to the star’s axial rotation. In this case, the
long-term photometric monitoring of active stars provides a powerful tool to
derive relevant parameters of stellar surface activity (Rodonó et al., 2000;
Messina & Guinan, 2003). In this study, we obtained the light curves of V369
Gem (HD 52452) in 2006-2008 years. Although V369 Gem is known as one of the
short-period non-eclipsing chromospherically active binary system discovered
so far (Messina et al., 2001), we investigate the main reasons of the light
variations seen in its light curves due to some stable properties seen in the
light curves.
In literature, for the first time, Pounds et al. (1993) detected the star as a
EUV bright source in the all sky survey of ROSAT. They listed the star as RE
J70222+255054 in the X-Ray source catalogue. Later than, Mason et al. (1995)
demonstrated that V369 Gem is the optical counterpart of the EUV bright source
RE J70222+255054. Perryman et al. (1997) catalogued the star as a suspected
variable star with a parallax of 17.0$\pm$6.4 mas in the Tycho Catalogue
(TYC1899 688 1). Messina et al. (2001) reported that V369 Gem is a triple
system. They indicated that the triple system is composed of a G4V star and
two late type G stars, which are seen as SB1. The observations of the authors
indicated that both components of the binary system exhibit photospheric spot
activity. The authors stated in their study that there was no proof of an
eclipsing in the light variation. They derived an upper value as 50∘ for
inclination of the SB1 components. It was mentioned that the system is not
young in terms of the abundance analyses of Li $\lambda$6707$\AA$ in their
spectral studies with high resolution. The $v\sin i$ value is given as
14$\pm$2 kms-1 for one of the components, while $v\sin i$ value of the other
component is higher than 60 kms-1. For the first time, photometric period of
the system was found to be 0.42304 days by Messina et al. (2001). Another
study belongs to Barway & Pandey (2004), who presented BVRI-band photometry of
the system. The authors gave the photometric periods, which they found out by
analyses of their data together with the data given by Messina et al. (2001).
The authors mentioned that the light variation of the system was due to a
stellar spot activity on the components of the system. They reported that the
amplitudes of the light curves were changing in 2000 and 2001 observing
seasons. Using the data taken from the Tycho Catalogue, Barway & Pandey (2004)
indicated that the distance of the system was 59 pc.
In this study, we suggest that there are two main effects to cause the light
variation. Examining all the light curves of the system, one of them should be
the ellipsoidal effects, which cause a stable light variation with two minima.
The second one is the chromospheric activity, which cause the short-term rapid
variation on this stable shape. In the paper, we analysed all the V-band data
of the system acquired between the years 1995 and 2009. The observational
bases of the study are presented in Section 2, then photometrically obtained
light variations and analyses are presented in Section 3, Section 4 and
Section 5. Finally, the main properties obtained in this study and the results
are discussed in Section 6.
## 2 Observations and Data
The observations were acquired with a High-Speed Three Channel Photometer
attached to the 48 cm Cassegrain telescope at Ege University Observatory. The
observations were continued in BVR-bands. The identities of programme star and
its comparisons are given in Table 1. In Table 1, the star names are given in
the first column, while J2000 coordinates are listed in second column. The V
magnitudes and B-V colours obtained in this study are listed in the following
columns, respectively.
V369 Gem, the comparison and check stars are close on the sky plane. However,
differential atmospheric extinction corrections were applied for the
observations obtained in this study. In each observing night, the observations
of the system were acquired without interruption, covering large ranges for
the airmass. In general, the observations were started before the celestial
meridian with the airmass value of 3.15, and were ended after the celestial
meridian with the airmass value of 2.25. The coefficient ($k$) of differential
atmospheric extinction correction was found to be between 0.224 and 0.390
$mag~{}airmass^{-1}$ for B-band observations. The mean value of $k$ was
computed as 0.321 $mag~{}airmass^{-1}$ for B-band observations. It was found
to be between 0.085 and 0.299 $mag~{}airmass^{-1}$ for V-band, while its mean
average is 0.221 $mag~{}airmass^{-1}$. Finally, it was found to be between
0.038 and 0.274 $mag~{}airmass^{-1}$ for R-band observations, and its mean
value was computed as 0.149 $mag~{}airmass^{-1}$. The time series analyses
also do not reveal any regular variation for these differential magnitudes.
There is no variation of differential magnitudes in the sense comparison minus
check stars. The comparison stars were observed with the standard stars also
in their vicinity and the reduced differential magnitudes, in the sense
variable minus comparison, were transformed to the standard system using
procedures outlined in Hardie (1962). The standard stars we used are listed in
the catalogue of Oja (1996). The derived transformation coefficients are given
in Equations (1, 2 and 3). Heliocentric corrections were also applied to the
times of observations. The standard deviations of observations acquired in
BVR-bands are 0m.010, 0m.007 and 0m.006, respectively.
$(V-v_{0})~{}=~{}-0.0113~{}\times~{}(b-v)_{0}~{}+~{}20.877$ (1)
$(B-V)~{}=~{}-0.9924~{}+~{}1.1592~{}\times~{}(b-v)_{0}$ (2)
$(V-R)~{}=~{}-0.0698~{}+~{}1.079~{}\times~{}(v-r)_{0}$ (3)
where the symbols $bvr$ represent the observed values, while the symbols $BVR$
represent the standard values.
In this study, we used the same comparison star with both Messina et al.
(2001) and Barway & Pandey (2004). Messina et al. (2001) given the observation
data in the standard system, while Barway & Pandey (2004) given the values of
the comparison star in their study. The comparison star brightness and colour
given by Barway & Pandey (2004) are close to the values obtained in this
study. The V data obtained by Messina et al. (2001) and Barway & Pandey (2004)
were combined to our data. In addition, we also used the data obtained from
the ASAS Database (Pojmański, 1997). The system was observed between December,
2002 and April, 2009 in the ASAS project. In Figure 1, all the V-band data
were plotted versus years.
All the V-band data were analysed with both Discrete Fourier Transform (DFT)
(Scargle, 1982) and the Phase Dispersion Minimization (PDM), which is a
statistical method (Stellingwerf, 1978). In the analyses, the period was found
to be 0d.423061. Using the method given by Kwee & van Woerden (1956) and the
data obtained in January 10, 2007 night, an epoch was determined. The new
light elements are given in Equation (4).
$JD~{}(Hel.)~{}=~{}24~{}54111.5034(4)~{}+~{}0^{d}.423061(1)~{}\times~{}E$ (4)
All the combined data were phased with the light elements given in Equation
(4). Considering that V369 Gem is an active star, the consecutive observations
were compared among themselves. The date, where the shape and the amplitude
are changed, was taken as the beginning of the new subset. Thus, all the V
data were separated into 14 subsets. Some parameters of the subsets are listed
in Table 2. In Figure 2, the BVR light curves of the system obtained in this
study are given for only two subsets, which are covered all the phase
intervals. In Set L and Set M, there are also some data obtained in this
study, but these data are not enough to cover all the phases. This is why we
did not plotted them in Figure 2. As it is seen from the figure, the shapes
and amplitudes of the light curves different especially in minima phases.
## 3 Light Curve Analysis
According to Messina et al. (2001) and Barway & Pandey (2004), the observed
light variation is caused due to the cool spots in the case of V369 Gem.
However, there are a few important characteristic features in the light
curves. First of all, there are always two minima in the light curves and one
of them is always deeper, and there is 0P.50 between them. In addition, the
phases of the minima are generally constant, though there could be some small
distortions in the general shape of the light curves time to time.
Considering these properties, we tested whether there are any eclipses in the
system by using the PHOEBE V.0.31a software (Prša & Zwitter, 2005), whose
method depends on the version 2003 of the Wilson-Devinney Code (Wilson &
Devinney, 1971; Wilson, 1990). We tried to analyse BVR-band curves with
different modes, such as the ”detached system”, ”overcontact binary not in
thermal contact”, ”semi-detached system with the primary component filling its
Roche-Lobe”, ”semi-detached system with the secondary component filling its
Roche-Lobe” and ”double contact binary” modes. Considering the minimum sum of
weighted squared residuals ($\Sigma res^{2}$) of 0.13, the initial analyses
demonstrated that a statistically acceptable result can be obtained if the
analysis is carried out in the ”overcontact binary not in thermal contact”
mode. The BVR-light curves can not be modelled by using all the other modes.
In the case of ”detached system” mode, the analysis is generally interrupted
itself due to uncomfortable configuration, while the program is working. In
the case of ”double contact binary” mode, the obtained $\Sigma res^{2}$ values
are much larger than those obtained by using the ”overcontact binary not in
thermal contact” mode. Because of these, an available solution was obtained in
just the ”overcontact binary not in thermal contact” mode.
Using spectral observations, Messina et al. (2001) estimated that the system
is G spectral type, and so the effective temperature ($T_{eff}$) of the
primary component should be 5637 $K$ according to Drilling and Landolt (2000).
Thus, the temperature of the primary component was fixed at 5637 $K$, and the
temperature of the secondary was taken as a free parameter in the analyses.
Considering the spectral type, the albedos ($A_{1}$ and $A_{2}$) and the
gravity-darkening coefficients ($g_{1}$ and $g_{2}$) of the components were
adopted for the stars with the convective envelopes (Lucy, 1967; Rucinski,
1969). The non-linear limb-darkening coefficients ($x_{1}$ and $x_{2}$) of the
components were taken from van Hamme (1993). In the analyses, the fractional
luminosity ($L_{1}$) of the primary component and the inclination ($i$) of the
system were taken as the adjustable free parameters. There is no available
spectroscopic mass ratio for the system. Because of this, we tried to find the
best photometric mass ratio of the components using the light curve analysis.
The minimum sum of weighted squared residuals ($\Sigma res^{2}$) obtained in
the analyses indicated that the mass ratio value of the system is $q=0.76\pm
0.03$. The synthetic light curves obtained in the ”overcontact binary not in
thermal contact” are shown in Figure 3, while the parameters derived from this
analysis are listed in Table 3. From the parameters found from the light curve
analysis, the 3D model of V369 Gem’s Roche geometry is shown in Figure 4 with
the geometric configurations of the system at four special phases 0.00, 0.25,
0.50 and 0.75.
According to the minimum sum of weighted squared residuals ($\Sigma res^{2}$)
of 0.13, the only solution obtained in the ”overcontact binary not in thermal
contact” is statistically acceptable one. However, it should be tested whether
it is also acceptable in the astrophysical sense. For this aim, although there
is not any available radial velocity curve, we tried to estimate the absolute
parameters of the components in order to compare them with the theoretical
models. According to Drilling and Landolt (2000), the mass of the primary
component must be 0.946 $M_{\odot}$ corresponding to its surface temperature.
Considering possible mass ratio of the system, the mass of the secondary
component was found to be 0.721 $M_{\odot}$. Using Kepler’s third law, we
calculated the semi-major axis as 2.81 $R_{\odot}$. Considering this estimated
semi-major axis, the radius of the primary component was computed as 1.06
$R_{\odot}$, while it was computed as 0.92 $R_{\odot}$ for the secondary
component. Using the estimated radii and the obtained temperatures of the
components, the luminosity of the primary component was estimated to be 1.022
$L_{\odot}$, and it was found as 0.394 $L_{\odot}$ for the secondary
component. If the obtained absolute parameters are compared with theoretical
models such as one developed for the stars with $Z=0.02$ by Girardi et al.
(2000), it is seen that the absolute parameters are generally acceptable in
the astrophysical sense.
## 4 The Ellipticity Effect
The light curve analyses demonstrated that the system do not exhibit any
eclipses. However, as seen from the Roche geometry shown in Figure 4, the
shape of the components should be distorted due to tidal effects. Thus, the
ellipticity effect due to this geometric configuration should cause the
observed light variation. As it is well known, the ellipsoidal variable stars
are non-eclipsing binary stars and especially close binaries (Morris, 1985;
Beech, 1985). The main variation seen in their light curves is due to non-
spherical shapes of the components. This like configuration of a system
creates a quasi-sinusoidal light curves. The depths of the minima in the light
curve depend on the system geometry as well as its inclination.
The one of the methods to test whether a star is an ellipsoidal variable is
the Fourier analysis (Morris, 1985; Beech, 1985). It has been also
demonstrated by Morris (1985) and Morris & Naftilan (1993), in the case of any
available radial velocity (a double lined or single lined), many parameters of
the components can be computed under some assumptions. Unfortunately, there is
no available radial velocity for V369 Gem; because of this case lots of
parameters can not be computed. As it is listed in Table 3, the inclination
angle ($i$) of the system was determined as $i=44^{\circ}.25$ from the light
curve analysis.
In this case, the stable-main variation could be caused by the ellipsoidal
effects. In order to test this, the light curves of the ellipsoidal variables
can be modelled by the Fourier analysis given by Equation (5). If the main
variation is caused due to the ellipsoidal effect, one will expect the
$\cos(2\theta)$ term must be dominant among all other terms in the results of
the Fourier analysis.
$L(\theta)=A_{0}~{}+~{}\sum_{\mbox{\scriptsize\
i=1}}^{N}~{}A_{i}~{}cos(i\theta)~{}+~{}\sum_{\mbox{\scriptsize\
i=1}}^{N}~{}B_{i}~{}sin(i\theta)$ (5)
The light curve obtained in January 10, 2007 was chosen to analyse. The
observations in this night cover homogeneously all the phases, and the minima
in the light curve are symmetric. The BVR light curves obtained in January 10,
2007 were analysed with the Fourier method. The derived Fourier fits are shown
in Figure 5, and the Fourier Coefficients are listed in Table 4. The $A$
coefficients listed in the table are the coefficients of the $\cos(i\theta)$
terms, while $B$ parameters are the coefficients of the $\sin(i\theta)$ terms
given in Equation (5). In fact, the most dominant one is $\cos(2\theta)$ term
for each of the BVR-bands. Thus, it is obvious that the main effect seen in
the light variations is the ellipticity effect. In this case, the found period
given in Equation (4) must be the orbital period of the system. As it is well
known from eclipsing binaries, the found photometric period must be the
orbital period of the system, if the main reason of the light variation is the
geometric effects of the binary. There are many systems, which are similar to
V369 Gem, such 75 Pegasi and 42 Persei are studied with this method by Martin
et al. (1990, 1991).
## 5 Stellar Spot Activity
In the Fourier Coefficients, the second dominant term is $\cos(\theta)$. Hall
(1990) discussed that the $\cos(\theta)$ term can be an indicator for the
spotted area on the surface of a star. In this case, the ellipticity effect is
not unique effect in the light variation of V369 Gem. Considering the
variation seen in Figure 2 and spectral type of the system, the second effect
should be stellar spot activity. We extracted the theoretical light curve,
which contains just the variation caused by the ellipticity effect, from all
the V-band light curves of 14 subsets. We examined all the pre-whitened light
curves together, and some parameters were calculated and given in Table 5. All
the observed and pre-whitened light curves are shown in Figure 6. The curves
seen in the first two columns of panels on the left side are observed light
curves, while the curves in the last two columns of panels on the right side
are the pre-whitened light curves. In the first two columns of the panels, the
dashed line in each panel represents the theoretical light curve, which
contains just the variation caused by the ellipticity effect. However, in the
last two columns of the panels, the dashed line in each panel represents the
theoretical light curve, which contains just the variation caused by the
stellar spots.
The variation of the mean brightness in the pre-whitened light curves are
shown in Figure 7 with the variation of the amplitudes in the light curves.
The pre-whitened V light curves were analysed by time series analyse methods.
The found periods are listed in Table 6. Excepted a few subsets, which have no
good data distribution, the periods found from each subset are close to the
period given in Equation (4). This case reveals that the components rotate
synchronously with the orbital period. The data used in analyses are not
enough to discuss about any activity cycle.
In Figure 8, we plotted minima phases ($\theta_{min}$) derived from the pre-
whitened light curves versus the years. The figure reveals two main properties
of the spot distribution on the surface. As seen from the figure, the minima
of pre-whitened V light curves are separated 0P.50 from each other. This
indicates that the spotted areas on the surface are generally inclined to be
located in two mean active longitudes. Moreover, the deeper minimum migrates
toward the earlier phases through years.
## 6 Results and Discussion
The goal in this study is determining the nature of this interesting system.
In this respect, we analysed almost all the available data of V369 Gem in the
literature. The results show some clues about its nature.
As seen in Figure 2, the light curves of the system vary from a season to the
next one. There are also some variations in the mean brightness and the shapes
of the light curves. The same cases were seen by Messina et al. (2001) and
Barway & Pandey (2004), and they explained this situation with the
chromospheric activity.
We analysed all the V data together with the time series analyses, and the
photometric period of the system was found to be 0d.423061. It is so close to
the periods found by Messina et al. (2001) and Barway & Pandey (2004).
According to the long-term photometry, another properties have come out in
this case. Although there are some small variations, the main shapes of the
light curves are usually the same. There are always two minima, and they are
almost constant according to each other. The analyses demonstrated that one of
the effects on the light curves is the ellipsoidal effect. The $\cos(2\theta)$
was found to be -0.0400$\pm$0.0004 for B, -0.0383$\pm$0.0004 for V and
-0.0358$\pm$0.0003 for R-band. These coefficients are larger than all other
coefficients in each band. According to Morris (1985) and Hall (1990), in this
case, there is an ellipsoidal effect on the light variations. Apart from the
$\cos(2\theta)$ coefficients, the second dominant coefficient is
$\cos(\theta)$. According to Hall (1990), if there was only one spotted area
on the surface of a star, it would be expected that the $\cos(\theta)$ term
must be dominant. However, Hall (1990) noticed that the $\cos(2\theta)$ term
is not be dominated due to only the ellipticity effect, but it can be also
dominated because of two spotted areas separated 180∘ from each other on the
surface of a component. However, there is a way to define the real reason of
the $\cos(2\theta)$ term dominated in the Fourier analysis. This is the
evolution and movements of the spotted areas on the surface of the star. If
the reason of the $\cos(2\theta)$ term is stellar spots separated 180∘ from
each other, the results of the Fourier analyses will change with time.
Because, the main shapes of the light curves will change. In this study, we
always found the $\cos(2\theta)$ term to be dominated for each set. This
demonstrated that the main effect on the light variation is the ellipsoidal
effect. However, one can suspect that V369 Gem may be an eclipsing binary. In
this point, the suspect can be tested by the method described by Morris (1985)
and Morris & Naftilan (1993). In the case of V369 Gem, although there is not
any available radial velocity, we determined the inclination angle ($i$) of
the system from the light curve analysis with the PHOEBE V.0.31a software, and
it was found to be 44∘.25. In this respect, it is not expected that the system
exhibits any eclipses.
In order to the find the second effect on the light variation, the ellipsoidal
effect was extracted from all the observed V light curves. A sinusoidal-like
variation is seen in all the pre-whitened light curves, and this variation
seems to be a combination of two sinusoidal waves. The consecutive light
curves demonstrated that the shapes of the pre-whitened light curves are
dramatically changing. For example, if the pre-whitened light curves of Set C,
Set D and Set E are compared, it can be seen that the level of the deeper
minimum is increasing, while the level of the first maximum seen in
$\sim$0P.40 is not changing. In contrast to the first maximum, the level of
the second maximum seen in $\sim$0P.90 is increasing and, it gets a higher
level than the first one in a few subsets. The same behaviours are seen among
the others. Although the mean brightness of the pre-whitened light curves is
increasing through years, the amplitudes are decreasing from the year 1994 to
2007. The amplitude of the pre-whitened light curve is 0m.160 in the first
light curve, while it is slowly decreasing toward the year 2007, and it is
0m.012 in 2007. However, the largest amplitude is seen in the year 2009.15, it
is as large as 0m.206. There is one minimum in some light curves, which are
generally obtained between the year 2006 and 2007. The mean brightness of the
subsets are changing. For instance, the amplitudes of the first six light
curves are changing from one to the next, though two minima always exist in
the light curves. However, the shallow minimum is beginning to disappear from
Set G, and it is disappeared in Set H. The light curve of Set H has one
minimum and completely asymmetric shape. The second minimum is started to
appear in the light curve of Set I, and the light curve of Set J has two
minima. The light curve of Set K has again one minimum and an asymmetric
shape. Although the data in both Set L and Set M do not cover all the phases
of the light curves, an asymmetry can be seen in their shapes. Finally, the
light curve of Set N has two minima. Besides the second minimum sometimes
disappearing, another dramatic variation is seen in the light curves. As seen
from Figure 8, there is a migration of the deeper minimum. The deeper minimum
migrates toward the earlier phases through years, especially the last few
years. The sinusoidal-like variations in each light curve should be caused by
dark stellar spots occurring on the surface of the system. In brief, the
behaviours like these are generally seen in chromospherically active stars
(Fekel et al., 2002).
The minima of the sinusoidal-like variation in the pre-whitened light curves
are generally separated about 0P.50 from each other. This reveals that there
are two spotted areas separated $\sim$180∘ from each others. A sudden
occurring of the deeper minimum between 0P.70-0P.80 in Set H should be caused
by changing of the active area efficiencies on the surface of the star.
Changing of the active area efficiencies exhibits itself in Figure 6. As seen
from the figure, the phases of the maxima can occur in different phases, which
are separated 0P.50 from each other. The same behaviours are generally seen in
the light curves of many close eclipsing binary systems, such as W UMa type
variables, but not limited to this type (Davidge & Milone, 1984; Kallrath &
Milone, 2009). In the W UMa systems, the chromospheric activity occurring on
the surface of the components is one of the reasons causing the phenomenon
called the O’Connell Effect.
After the extraction of the ellipsoidal effect from the light curves, the
periods found from each subset are a bit different from each other. This case
is common for active stars. These photometric periods depend on the
location(s) of the spotted area(s) on the surface of the star. In fact, if the
subsets, which have scattered data, are ignored, the periods found from each
subset seem to decrease from the year 1994 to 2009. On the other hand, the
similarity between the orbital period and the periods found from each subset
demonstrates that the components rotate nearly synchronously with the orbital
period. In fact, the rotational and revolution periods are expected to be same
for an ellipsoidal variable (Morris, 1985). In the case of V369 Gem, although
the orbital period was found to be close to the period of rotational
modulation, it is seen from Figure 8 that there should be a bit difference
between them. The minima of the pre-whitened light curves migrate toward to
the previous phases.
V369 Gem has been taken a special place among the other ellipsoidal variables.
This is because the system has chromospheric activity. In fact, there are many
systems similar to V369 Gem in the literature, such as V350 Lac, V1197 Ori,
V1764 Cyg, HD 74425 and ect (Crews et al., 1995; Lines et al., 1987; Henry &
Kaye, 1999; Hall, 1990). As it is seen from the literature, although the
ellipsoidal variables are generally from the earlier spectral types, but there
are also several ellipsoidal systems from the later spectral types.
Considering these ellipsoidal variables from the later spectral types, V369
Gem is a bit different among them due to its period. As seen from Hall (1990),
they are generally long period systems, while V369 Gem is a very short-period
one. The short period of the system indicates that V369 Gem is not a classical
ellipsoidal variable, whose shapes are just caused by the tidal effect. The
ellipsoidal shapes of V369 Gem s components should be due to not only tidal
effects but also the evolutionary status of the system. Although V369 Gem is
mentioned as a main sequence star, as seen from Figure 4, the components are
near the filling Roche-Lobes. Considering the obtained absolute parameters of
the components with the theoretical models derived for the for stars with
$Z=0.02$ by Girardi et al. (2000), both of them seem to be closer to TAMS
rather than ZAMS. Messina et al. (2001) demonstrated that very low abundance
of Li $\lambda$6707$\AA$ is obtained from this system, because of this, the
components can not be the young stars, whereas the system has a very high
level chromospheric activity. This is in agreement with our results obtained
from the light curve analysis. In fact, Rocha-Pinto et al. (2002) put
forwarded some models about the formation of W UMa type systems, indicating
some systems, which are chromospherically active without high Li abundance,
while they are kinematically old. According to Rocha-Pinto et al. (2002), the
components in these systems are also generally rapidly rotating. Messina et
al. (2001) revealed that the components of V369 Gem are also rapidly rotating
according to a star from the G spectral types. Consequently, it is possible
that the components of the system should be nearly filling their Roche-Lobes.
This also explains why the components have taken an ellipsoidal shapes.
In the future, V369 Gem should be taken into the programs of the spectroscopic
observations. A radial velocity curve of the system will be obtained, and this
supports to understand the unclear points about the system.
## 7 Acknowledgments
The authors acknowledge generous allotments of observing time at the Ege
University Observatory. We also thank the referee for useful comments that
have contributed to the improvement of the paper.
## References
* Barway & Pandey (2004) Barway, S., Pandey, S.K., 2004, IBVS, 5553
* Beech (1985) Beech, M., 1985, Ap&SS, 117, 69
* Crews et al. (1995) Crews, L.J., Hall, D.S., Henry, G.W., Lines, R.D., Lines, H.C., Fried, R.E., 1995, AJ, 109, 1346
* Davidge & Milone (1984) Davidge, T.J.and Milone, E.F., 1984, ApJ Suppl.55, 571
* Drilling and Landolt (2000) Drilling, J.S., Landolt, A.U., 2000, ”Allen s Astrophysical Quantities”, 4th ed. Edited by Arthur N. Cox. ISBN: 0-387-98746-0. Publisher: New York: AIP Press; Springer, 2000, p.381
* Fekel et al. (2002) Fekel, F.C., Henry, G.W., Eaton, J.A., Sperauskas, J., Hall, D.S., 2002, AJ, 124, 1064
* Girardi et al. (2000) Girardi, L., Bressan, A., Bertelli, G., Chiosi, C., 2000, A&AS 141, 371
* Hall (1990) Hall, D.S., 1990, AJ, 100, 554
* Hardie (1962) Hardie, R.H., 1962, in Astronomical Techniques, ed. W.A. Hiltner (Chicago: Univ.Chicago Press), 178
* Henry & Kaye (1999) Henry, G.W., Kaye, A.B., 1999, IBVS, 4684, 1
* Kallrath & Milone (2009) Kallrath, J., Milone, E.F., 2009, ”Eclipsing Binary Stars: Modeling and Analysis”, Astronomy and Astrophysics Library, Springer Dordrecht Heidelberg London and New York
* Kwee & van Woerden (1956) Kwee, K.K., van Woerden, H., 1956, BAN, 12, 327
* Lines et al. (1987) Lines, H.C., Lines, R.D., Kirkpatrick, J.D., Hall, D.S., 1987, AJ, 93, 430
* Lucy (1967) Lucy, L.B., 1967, Z. Astrophys, 65, 89
* Martin et al. (1990) Martin, B.E., Hube, D.P., Lyder, D.A., 1990, PASP, 102, 1153
* Martin et al. (1991) Martin, B.E., Hube, D.P., Brown, C., 1991, PASP, 103, 424
* Mason et al. (1995) Mason, K.O., Hassall, B.J.M., Bromage, G.E., Buckley, D.A.H., Naylor, T., O’Donoghue, D., Watson, M.G., Bertram, D., Branduardi-Raymont, G., Charles, P.A., Cooke, B., Elliott, K.H., Hawkins, M.R.S., Hodgkin, S.T., Jewell, S.J., Jomaron, C.M., Sekiguchi, K., Kellett, B.J., Lawrence, A., McHardy, I., Mittaz, J.P.D., Pike, C.D., Ponman, T.J., Schmitt, J., Voges, W., Wargau, W., Wonnacott, D., 1995, MNRAS, 274, 1194
* Messina et al. (2001) Messina, S., Cutispoto, G., Pastori, L., Rodonó, M., Tagliaferri, G., 2001, IBVS, 5014
* Messina & Guinan (2003) Messina, S., Guinan, E.F., 2003, A&A, 409, 1017
* Morris (1985) Morris, S.L., 1985, ApJ, 295, 143
* Morris & Naftilan (1993) Morris, S.L., Naftilan, S.A., 1993, ApJ, 419, 344.
* Oja (1996) Oja, T., 1996, BaltA, 5, 103
* Perryman et al. (1997) Perryman, M.A.C., Lindegren, L., Kovalevsky, J., Hoeg, E., Bastian, U., Bernacca, P.L., Crézé, M., Donati, F., Grenon, M., van Leeuwen, F., van der Marel, H., Mignard, F., Murray, C.A., Le Poole, R.S., Schrijver, H., Turon, C., Arenou, F., Froeschl, M., Petersen, C.S., 1997, A&A, 323L, 49
* Pojmański (1997) Pojmański, G., 1997, AcA, 47, 467
* Pounds et al. (1993) Pounds, K.A., Allan, D.J., Barber, C., Barstow, M.A., Bertram, D., Branduardi-Raymont, G., Brebner, G.E.C., Buckley, D., Bromage, G.E., Cole, R.E., Courtier, M., Cruise, A.M., Culhane, J.L., Denby, M., Donoghue, D.O., Dunford, E., Georgantopoulos, I., Goodall, C.V., Gondhalekar, P.M., Gourlay, J.A., Harris, A.W., Hassall, B.J.M., Hellier, C., Hodgkin, S., Jeffries, R.D., Kellett, B.J., Kent, B.J., Lieu, R., Lloyd, C., McGale, P., Mason, K.O., Matthews, L., Mittaz, J.P.D., Page, C.G., Pankiewicz, G.S., Pike, C.D., Ponman, T.J., Puchnarewicz, E.M., Pye, J.P., Quenby, J.J., Ricketts, M.J., Rosen, S.R., Sansom, A.E., Sembay, S., Sidher, S., Sims, M.R., Stewart, B.C., Sumner, T.J., Vallance, R.J., Watson, M.G., Warwick, R.S., Wells, A.A., Willingale, R., Willmore, A.P., Willoughby, G.A., Wonnacott, D., 1993, MNRAS, 260, 77
* Prša & Zwitter (2005) Prša, A., Zwitter, T., 2005, ApJ, 628, 426
* Rocha-Pinto et al. (2002) Rocha-Pinto, H.J., Castilho, B.V., Maciel, W.J., 2002, A&A, 384, 912
* Rodonó et al. (2000) Rodonó, M., Messina, S., Lanza, A.F., Cutispoto, G., Teriaca, L., 2000, A&A, 358, 624
* Rucinski (1969) Rucinski, S.M., 1969, AcA, 19, 245
* Scargle (1982) Scargle, J.D., 1982, ApJ, 263, 835
* Stellingwerf (1978) Stellingwerf, R.F., 1978, ApJ, 224, 935
* van Hamme (1993) van Hamme, W., 1993, AJ, 106, 2096
* Wilson & Devinney (1971) Wilson, R.E., Devinney, E.J., 1971, ApJ, 166, 605
* Wilson (1990) Wilson, R.E., 1990, ApJ, 356, 613
(140mm,60mm)Figure1.ps
Figure 1: All the V-band observations of V369 Gem between November, 1994 and
April, 2009.
(160mm,60mm)Figure2.ps
Figure 2: The BVR light curves of V369 Gem for different two data sets.
(175mm,60mm)Figure3.ps
Figure 3: The synthetic light curve obtained in just the ”overcontact binary
not in thermal contact” mode from the light curve analysis.
(55mm,60mm)Figure4.eps
Figure 4: The 3D model of Roche geometry and the geometric configurations at
four special phases 0.00, 0.25, 0.50 and 0.75, illustrated for V369 Gem by
using the PHOEBE V.0.31a software, using the parameters obtained in just the
”overcontact binary not in thermal contact” mode from the light curve
analysis.
(155mm,60mm)Figure5.ps
Figure 5: BVR light curves of the system obtained in January 10, 2007 (filled
circle) and the Fourier models (line).
(185mm,60mm)Figure6.ps
Figure 6: V-band light curves for each subset. The curves seen in the first
two columns of panels on the left side are observed light curves, while the
curves in the last two columns of panels on the right side are the pre-
whitened light curves. In the first two columns of the panels, the dashed line
in each panel represents the theoretical light curve derived for just the
ellipticity effect. However, in the last two columns of the panels, the dashed
line in each panel represents the theoretical light curve, which contains just
the variation caused by the stellar spots. All light curves are plotted in the
same scale to demonstrate the variations of the amplitudes and the shapes.
(140mm,60mm)Figure7.ps
Figure 7: The variations of the amplitude (a) and the mean brightness (b) of
the pre-whitened light curves for the V-band.
(140mm,60mm)Figure8.ps
Figure 8: The phases of the spot minima ($\theta_{min}$). In figure, filled
circle represents Min I, open circles represent Min II.
Table 1: Basic parameters of program stars. Stars | RA / DE (J2000) | V | B-V
---|---|---|---
| (h m s) / (∘ ′ ′′) | (mag) | mag)
V369 Gem (HD 52452) | 07 02 23.32 +25 50 45.56 | 7.931 | 0.755
HD 52071 (Comparison) | 07 00 58.15 +27 09 26.50 | 7.098 | 1.231
HD 51530 (39 Gem, Check) | 06 58 47.41 +26 04 51.89 | 6.198 | 0.488
Table 2: Data groups used in the analyses.
Data | HJD Interval | Median | Number | Source
---|---|---|---|---
Sets | (+24 00000) | Epoch | of Points | Ref.
A | 49672.6213-49772.4665 | 1995.03 | 16 | 1
B | 51592.1796-51592.1484 | 2000.13 | 28 | 2
C | 51605.1250-51627.2578 | 2000.18 | 129 | 2
D | 51963.1992-51963.1796 | 2001.14 | 93 | 2
E | 52622.7068-52729.5132 | 2003.09 | 40 | 3
F | 53007.7307-53101.4961 | 2004.13 | 21 | 3
G | 53295.8523-53412.5385 | 2004.95 | 50 | 3
H | 53647.8843-53779.5258 | 2005.94 | 45 | 3
I | 54094.2496-54111.6414 | 2007.00 | 889 | 4
J | 54383.8712-54465.7459 | 2007.88 | 16 | 3
K | 54437.3129-54442.3987 | 2007.92 | 333 | 4
L | 54477.7262-54576.4741 | 2008.06 | 209 | 3, 4
M | 54753.8753-54830.6945 | 2008.85 | 120 | 3, 4
N | 54838.6699-54941.4836 | 2009.15 | 22 | 3
1 Messina et al. (2001).
2 Barway & Pandey (2004).
3 The ASAS database, Pojmański (1997).
4 This Study.
Table 3: The parameters of the components obtained from the light curve analysis in the ”overcontact binary not in thermal contact” mode. Parameter | Value
---|---
$q$ | 0.76
$i$ (∘) | 44.25$\pm$0.12
$T_{1}$ (K) | 5637 (Fixed)
$T_{2}$ (K) | 4762$\pm$72
$\Omega_{1}$ | 3.56895
$\Omega_{2}$ | 3.56895
$L_{1}/L_{T}$ (B) | 0.763$\pm$0.019
$L_{1}/L_{T}$ (V) | 0.807$\pm$0.021
$L_{1}/L_{T}$ (R) | 0.728$\pm$0.016
$g_{1}$, $g_{2}$ | 0.32, 0.32
$A_{1}$, $A_{2}$ | 0.50, 0.50
$x_{1,bol}$, $x_{2,bol}$ | 0.646, 0.646
$x_{1,B}$, $x_{2,B}$ | 0.768, 0.768
$x_{1,V}$, $x_{2,V}$ | 0.841, 0.841
$x_{1,R}$, $x_{2,R}$ | 0.677, 0.677
$<r_{1}>$ | 0.3778$\pm$0.0001
$<r_{2}>$ | 0.3285$\pm$0.0001
$Co-Lat_{Spot}$ (∘) | 90.00
$Co-Long_{Spot}$ (∘) | 0.00
$R_{Spot}$ (∘) | 40
$T_{eff,~{}Spot}$ | 0.93
Table 4: The coefficients derived from the Fourier model. Filter | $A_{0}$ | $A_{1}$ | $A_{2}$ | $B_{1}$ | $B_{2}$
---|---|---|---|---|---
B | 0.9531$\pm$0.0003 | -0.0358$\pm$0.0004 | -0.0400$\pm$0.0004 | -0.0062$\pm$0.0004 | -0.0016$\pm$0.0004
V | 0.9494$\pm$0.0003 | -0.0344$\pm$0.0004 | -0.0383$\pm$0.0004 | 0.0006$\pm$0.0004 | -0.0026$\pm$0.0004
R | 0.9615$\pm$0.0002 | -0.0301$\pm$0.0003 | -0.0358$\pm$0.0003 | 0.0004$\pm$0.0003 | -0.0018$\pm$0.0003
Table 5: The parameters calculated from the pre-whitened light curves. Data | $\theta$ | $\theta$ | Amp. (mag) | Amp. (mag) | Mean V
---|---|---|---|---|---
Set | Min I | Min II | Min I | Min II | (mag)
A | 0.18 | 0.71 | 0.160 | 0.076 | 0.115
B | 0.20 | 0.65 | 0.076 | 0.044 | 0.100
C | 0.12 | 0.68 | 0.116 | 0.093 | 0.026
D | 0.18 | 0.61 | 0.059 | 0.016 | 0.052
E | 0.13 | 0.67 | 0.075 | 0.059 | 0.079
F | 0.13 | 0.59 | 0.049 | 0.020 | 0.062
G | 0.16 | | 0.051 | | 0.097
H | 0.72 | | 0.037 | | 0.043
I | 0.01 | 0.71 | 0.012 | 0.012 | -0.004
J | 0.07 | 0.59 | 0.032 | 0.020 | 0.003
K | 0.97 | | 0.078 | | -0.003
L | 0.02 | | 0.034 | | 0.045
M | 0.39 | | 0.043 | | -0.003
N | 0.93 | | 0.206 | | -0.044
Table 6: The period found from the pre-whitened V light curves. Set | Year | Period (day) | FAP ($\%$)
---|---|---|---
A | 1995.03 | 0.421 $\pm$ 0.001 | 4
B | 2000.13 | 0.496 $\pm$ 0.014 | 57
C | 2000.18 | 0.428 $\pm$ 0.006 | 2
D | 2001.14 | 0.421 $\pm$ 0.013 | 8
E | 2003.09 | 0.423 $\pm$ 0.001 | 8
F | 2004.13 | 0.438 $\pm$ 0.001 | 8
G | 2004.95 | 0.401 $\pm$ 0.002 | 6
H | 2005.94 | 0.423 $\pm$ 0.001 | 2
I | 2007.00 | 0.423 $\pm$ 0.001 | 2
J | 2007.88 | 0.412 $\pm$ 0.001 | 5
K | 2007.92 | 0.466 $\pm$ 0.009 | 2
L | 2008.06 | 0.402 $\pm$ 0.003 | 2
M | 2008.85 | 0.412 $\pm$ 0.001 | 2
N | 2009.15 | 0.418 $\pm$ 0.001 | 6
|
arxiv-papers
| 2012-09-12T11:44:21 |
2024-09-04T02:49:35.009544
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, E. Sipahi, O. \\\"Ozdarcan",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1209.2576"
}
|
1209.2608
|
# Hexatic phase and water-like anomalies in a two-dimensional fluid of
particles with a weakly softened core111This article appeared in _The Journal
of Chemical Physics_ 137, 104503 (2012) and may be found at
http://link.aip.org/link/?jcp/137/104503. Copyright (2012) American Institute
of Physics. This article may be downloaded for personal use only. Any other
use requires prior permission of the authors and the American Institute of
Physics.
Santi Prestipino222Email: [email protected] Università degli Studi di
Messina, Dipartimento di Fisica, Contrada Papardo, 98166 Messina, Italy Franz
Saija333Corresponding author; email: [email protected] CNR-IPCF, Viale
Ferdinando Stagno d’Alcontres 37, 98158 Messina, Italy Paolo V.
Giaquinta444Email: [email protected] Università degli Studi di Messina,
Dipartimento di Fisica, Contrada Papardo, 98166 Messina, Italy
###### Abstract
We study a two-dimensional fluid of particles interacting through a
spherically-symmetric and marginally soft two-body repulsion. This model can
exist in three different crystal phases, one of them with square symmetry and
the other two triangular. We show that, while the triangular solids first melt
into a hexatic fluid, the square solid is directly transformed on heating into
an isotropic fluid through a first-order transition, with no intermediate
tetratic phase. In the low-pressure triangular and square crystals melting is
reentrant provided the temperature is not too low, but without the necessity
of two competing nearest-neighbor distances over a range of pressures. A whole
spectrum of water-like fluid anomalies completes the picture for this model
potential.
###### pacs:
61.20.Ja, 64.70.D-, 65.20.-w, 68.60.-p
## I Introduction
When confined to two-dimensional (2D) space, condensed matter behaves
differently than in three dimensions (3D). A striking example is the physics
of the Kosterlitz-Thouless transition in superfluid films, not to mention the
phenomenon of high-temperature superconductivity in the cuprates, which is
intimately related to the motion of electrons within weakly-coupled copper-
oxide layers. Another example is provided by 2D crystals, where thermal
fluctuations are so strong as to rule out long-range translational (but not
orientational) order for non-zero temperatures, leaving it open for the
possibility of unconventional melting scenarios. The celebrated KTHNY theory
of 2D melting thereby predicts a continuous two-stage melting from a
crystalline to a hexatic phase and, subsequently, from the hexatic to an
isotropic liquid KTHNY . The intermediate hexatic phase has short-range
translational order (i.e., it is fluid-like) but quasi-long-range
orientational order, characterized by a power-law decay of bond-angle pair
correlations. Other possibilities are a standard first-order 2D-melting
transition (as in Chui’s theory Chui ) or a stronger, i.e., first-order
hexatic-to-isotropic transition, as shown by Bernard and Krauth to be the case
for hard discs Bernard . In some cases, melting is one-stage and first-order
but the hexatic phase can be accessed via a metastable route Chen ; Mejia .
The KTHNY theory has been confirmed many times in simulation and experiment,
especially for particles with long-ranged interactions, though in a few cases
with some controversy about the order of the two transition steps Murray ;
Marcus ; Zahn ; Keim ; Lin1 ; Han ; Peng ; Muto ; Lin2 ; Lee ; Qi ; Clark .
From a computational point of view, the assessment of the order of 2D melting
in practical cases can be hard, due to important finite-size effects and slow
relaxation to equilibrium. An extreme case is hard discs where the exact
nature of the melting transition could be established only recently, by
employing unprecedentedly large system sizes of the order of a million
particles in a box of fixed volume Bernard . Generally speaking, working in
the isothermal-isobaric ensemble (rather than at constant volume) can ease the
interpretation of the simulation data, since one cannot incur in any of the
finite-size artifacts that harm constant-volume simulations. Recently, we have
provided unambiguous evidence of the occurrence of continuous melting via a
hexatic phase in the 2D Gaussian-core model (GCM) Prestipino1 . This system,
taken as a prototype of the phase behavior of “empty” polymers Likos1 , has
long been known to exhibit reentrant melting (i.e., solid melting under
isothermal compression) as well as water-like anomalies in three dimensions
Stillinger ; Lang ; Prestipino2 ; Mausbach . In two dimensions, the GCM
melting becomes two-staged, with an extremely narrow hexatic region whose
properties comply with the predictions of the KTHNY theory. In particular,
besides a reentrant isotropic fluid, also a reentrant hexatic phase exists.
Recently, water-like anomalies have been reported even in the one-dimensional
GCM Speranza , where however no properly defined thermodynamic transition
occurs.
A class of systems which exhibit reentrant melting, solid polymorphism, and
other water-like anomalies is formed by particles interacting via spherically-
symmetric core-softened (CS) potentials Buldyrev . For these systems, the
strength of the repulsive component of the interaction undergoes some sort of
weakening over a range of interparticle distances. The effects of particle-
core softening on thermodynamic behavior were originally investigated by
Hemmer and Stell Hemmer , who were interested in the possibility of multiple
critical points and isostructural solid-solid transitions in simple-fluid
systems. A few years later, Young and Alder Young showed that the phase
diagram of the hard-core plus square-shoulder repulsion exhibits an anomalous
melting line of the kind observed in Cs and Ce. Later, Debenedetti Debenedetti
showed that systems of particles interacting via continuous CS potentials are
capable of contracting when heated isobarically (a behavior which goes under
the name of “density anomaly”). In the last decade or so, there has been a
renewal of interest in the phase behavior of CS potentials, which has led to
the discovery of many unusual properties such as multiple reentrant-melting
lines, polymorphism both in the fluid and solid phases, stable cluster solids
(for bounded interactions only), and a plethora of thermodynamic and dynamic
water-like anomalies Sadr ; Jagla ; Likos2 ; Kumar ; Deoliveira ; Gibson ;
Lomba ; Fomin ; Xu ; Pizio ; Pamies ; Pauschenwein ; Malescio1 . A common
feature of CS potentials is their ability to generate two distinct length
scales in the system, one related to the inner core and the other associated
with the milder component of the repulsion Buldyrev . Due to this, CS fluids
are characterized by two competing, expanded and compact, local particle
arrangements. Such an interplay has a disruptive influence on crystal order,
the fluid phase being thus recovered upon compression. This two-scale or two-
fluid property mimics the behavior of the more complex network-forming fluids,
where the looser and denser local structures arise from the incessant building
and breaking of the dynamic network generated by directional bonds. As a
result, a paradigm has been established whereby the existence of two competing
local structures is essential for the occurrence of anomalous behaviors in
simple fluids.
Unusual phase behavior has also been found in 2D systems. Scala and coworkers
Scala carried out molecular-dynamics simulations of the 2D square-shoulder
plus square-well (SSSW) potential finding water-like anomalies and identifying
two different solid phases: a triangular crystal (low-density phase) and a
square crystal (high-density phase). However, the observation of a hysteresis
loop in their simulations suggested that the liquid-solid transition is
actually first-order, thus washing out the possibility of a continuous melting
to a hexatic phase. Wilding and Magee also studied the 2D SSSW potential and
showed that the thermodynamic anomalies of the model, rather than stemming
from a metastable liquid-liquid critical point as previously surmised, were
induced by the quasicontinuous nature of the 2D freezing transition Wilding .
Almudallal and coworkers Almudallal contested this result, showing by various
free-energy techniques that all transitions in the 2D SSSW model are in fact
first-order. Malescio and Pellicane studied a 2D system of particles
interacting through a potential consisting of an impenetrable hard core plus a
square shoulder. They found a variety of stripe phases whose formation was
imputed to the existence of two characteristic length scales Malescio2 . All
such investigations suggest that the existence of two length scales would be a
prerequisite for observing water-like anomalies also in 2D.
In a series of recent papers the two-fluid picture has been overthrown and a
novel minimal scenario for the occurrence of anomalies has been established
Saija ; Prestipino3 ; Malescio3 ; Malescio4 . More specifically, it has been
shown that a weakly-softened isotropic pair repulsion, with a single
characteristic length which becomes more loosely defined in a range of
pressures, is able to give rise to an unusual phase behavior. Our purpose here
is to verify whether anything similar occurs in 2D, where in addition an
interesting interplay with hexatic order may take place. Far from being purely
academic, the present study can be relevant for many soft materials. For
instance, one monolayer of $N$-isopropylacrylamide (NIPA) microgel spheres
confined between two glass cover slips is an ideal system to study 2D melting
because the effective interparticle potential is short-ranged and repulsive,
with a temperature-tunable volume fraction Dullens ; Lin2 ; Han ; Peng ;
Alsayed . Systems like this would be natural candidates for detecting, by
video optical microscopy or light scattering, the kind of phenomena that we
are going to illustrate below.
The plan of the paper is the following. After introducing our 2D model and
method in Section II, we sketch the model phase diagram in Section III,
highlighting the unconventional structure of the fluid phase which, similarly
to other instances of CS repulsion, shows a number of water-like anomalies
without any interplay between two characteristic length scales. The melting
mechanism is studied in more detail in Section IV, where we show the existence
of a hexatic phase. Section V is finally devoted to concluding remarks.
## II Model and method
We consider a purely repulsive pair potential in two dimensions, modelled
through an exponential form which was first introduced, about four decades
ago, by Yoshida and Kamakura (YK) Yoshida :
$u(r)=\epsilon\exp\left\\{a\left(1-\frac{r}{\sigma}\right)-6\left(1-\frac{r}{\sigma}\right)^{2}\ln\left(\frac{r}{\sigma}\right)\right\\}\,,$
(2.1)
where $\epsilon$ and $\sigma$ set the energy and length scales, respectively,
and $a\geq 0$. The YK potential behaves as $r^{-6}$ for small $r$, and falls
off very rapidly for large $r$. The smaller $a$ the higher the degree of
softness of $u(r)$, i.e., the flatter the repulsive “shoulder” around
$r=\sigma$. Technically speaking Debenedetti , $u(r)$ can be regarded as soft
only for $a\lesssim 2.3$, since only in this range there is an interval of
distances where the local virial function $-ru^{\prime}(r)$ decreases when the
interparticle separation decreases. However, it is over the wider range
$a\lesssim 5.5$ that the phase diagram is expected to show a reentrant-fluid
region Prestipino3 ; Prestipino4 . We shall here focus on the case $a=3.3$
(Fig. 1), which was already shown to possess a rich anomalous phase behavior
in three dimensions which cannot simply be explained by the existence of two
distinct nearest-neighbor (NN) distances in the dense fluid Prestipino3 .
We now briefly describe the methodology of the present investigation. After
identifying the relevant crystal phases by means of total-energy calculations
at zero temperature (of the type illustrated in, e.g., Ref. Prestipino5 ), we
explore the phase diagram by Monte Carlo (MC) simulation in the $NPT$
ensemble. As is common practice, we adopt periodic boundary conditions and
employ cell linked lists in order to speed up the simulation. While systems of
about 1000 particles are suitable for investigating bulk properties and for
determining the approximate location of phase boundaries, we consider larger
systems of about 6000 particles for the search and characterization of the
hexatic phase at selected pressures. For the same $P$ values, we check the
order of the melting transition independently through thermodynamic
integration combined with exact free-energy calculations. For the fluid phase,
a dilute gas is used as a reference state, whose chemical potential is
calculated by the Widom method, whereas a low-temperature crystal is chosen as
the starting point of the simulation in the solid region of the phase diagram.
At this initial state, the excess Helmholtz free energy of the system is
computed by the Einstein-crystal method Frenkel .
## III Fluid structure and water-like anomalies
For $a=3.3$, the repulsive shoulder at $\sigma$, which is still visible in the
plot of $u(r)$ for $a\simeq 2$, has by then completely faded out, surviving
only in the form of a modest bump in the otherwise monotonously-decreasing
profile of the virial function (see Fig. 1). Yet, this almost structureless
potential shows three distinct stable crystal arrangements at $T=0$ in two
dimensions, as it follows from the calculation of the chemical potential $\mu$
as a function of the pressure $P$ for all five Bravais lattices and for the
honeycomb lattice. Upon increasing $P$, the sequence of phases is triangular-
square-triangular, the square crystal being thermodynamically stable in the
pressure range $2.27$-$4.36$, in units of $\epsilon/\sigma^{3}$ (from now on,
pressure and temperature will be given in reduced units).
Assuming that no other crystal phases come into play at nonzero temperatures,
we first sketched the overall phase diagram by the heat-until-it-melts (HUIM)
method. In practice, for selected $P$ values we run a chain of MC simulations
of the solid stable at the given $P$, for increasing $T$ values at regular
intervals of 0.0005, until we observe a clear jump in both the particle-number
density $\rho=N/V$ and total energy per particle $E/N$, which signal the
melting of the solid. The character of this transition will be addressed in
the next section. The HUIM method simply overlooks the possibility of solid
overheating, but this is usually a reasonable approximation if one is not
interested in very precise estimates of the melting temperature $T_{m}$ (a
posteriori, the typical error implied by the HUIM method in determining the
melting point of the present system was about 10%, and smaller for the square
crystal). The outcome of this analysis is reported in Fig. 2 where the most
revealing feature is the reentrance of the fluid phase as the system, already
settled in the low-density triangular solid, is further compressed at not too
low temperatures. Fluid reentrance occurs even twice in a smaller range of
temperatures above 0.020.
As we already said in the Introduction, until very recently the conventional
wisdom on the origin of the reentrant-melting phenomenon in CS systems with an
unbounded interparticle repulsion rested on a competition, with destabilizing
effects on crystal ordering, between two ways of arranging particles close to
each other, which gives rise to either an expanded or a more compact structure
in the dense fluid. This may only occur provided the pressure is strong enough
as to bring neighboring particles at distances close to $\sigma$. Clearly,
this explanation cannot work for a case like the present one, where it is hard
to maintain that there are two different length scales in the potential.
Rather, we may view the question from a solid-state perspective and argue that
a succession of reentrant-melting lines in the phase diagram is simply the
outcome of the existence of multiple solid phases at low temperature. On
increasing pressure within the range of stability of any of such solids, the
crystal strength first grows up to a maximum (and in parallel also $T_{m}$),
but then it progressively reduces on approaching the boundary of the next
stable solid at higher pressure. After all, this is exactly the rationale
behind the melting criterion discussed in Ref. Prestipino4 , which in fact is
very effective for soft repulsive interactions.
We found confirmation that the present model shows only one repulsive length
scale by computing the fluid radial distribution function (RDF) along the
isotherm at $T=0.1$, i.e., just above the maximum $T_{m}$ of the low-density
triangular solid (see Fig. 3). Looking at the position of the main RDF peak,
we see a systematic shift to lower and lower distances upon compression, with
a slight widening of the peak around $P\simeq 2$, i.e., next to the first
reentrant-melting line. We interpret this as the evidence of a unique NN
characteristic distance in the system, which becomes more loosely defined
across the region of reentrant melting. This should be contrasted with what
occurs for a more conventional CS repulsion, where the first RDF maximum is
twin-peaked, with two definite NN distances that take turns at providing the
absolute maximum for the RDF.
Aside from the existence of one repulsive length scale rather than two, many
water-like anomalies are also found in the present system, starting with a
line of $\rho$ maxima in the fluid close to the first reentrant-melting line
(see Fig. 2). This is a typical occurrence in systems with CS potentials,
where however the region of volumetric anomaly is usually more extended. We
checked that the line of the density anomaly ceases to exist just before
plunging into the solid region. Another type of anomaly is the so-called
structural anomaly, that is a non-monotonous pressure behavior of the amount
of “translational order” in the fluid, as measured through the value of minus
the pair entropy per particle ($-s_{2}$) Prestipino6 (see Fig. 2 and the left
top panel of Fig. 4). Rather than monotonously increasing with pressure at
constant temperature, the degree of spatial order reduces in the reentrant-
fluid region, as a result of a looser definition of the NN distance. A non-
sharp average separation between neighboring particles acts as a perturbing
factor for the local order, bringing about a slight decrease of $-s_{2}$ with
pressure. In the same range of pressures where $-s_{2}$ decreases, the self-
diffusion coefficient $D$, which we measure by a series of $TVN$ molecular-
dynamics runs, gets enhanced with pressure (Fig. 2 and right top panel of Fig.
4). Both lines of structural and diffusional anomalies appear to sprout out of
the point of maximum $T_{m}$ for the low-density triangular solid and roughly
terminate at the point of maximum $T_{m}$ for the square solid. This is again
similar to other CS systems, also for what concerns the crossing of the
anomaly lines (see e.g. Fig. 11 of Jabes ). Finally, we checked for just one
pressure value ($P=2$) the existence of a temperature minimum in both the
isobaric specific heat $C_{P}$ and isothermal compressibility $K_{T}$ (lower
panels of Fig. 4), similarly to what found for water at ambient pressure. Both
minima fall in the high-temperature fluid region, much far from the maximum-
density point for the same pressure.
## IV Hexatic behavior
The existence in our model of two solids with different crystal symmetry,
i.e., triangular and square, gives the opportunity to investigate the
intriguing possibility of two distinct intermediate phases (hexatic and
tetratic) between the solid and the normal fluid, with the further bonus of a
yet-to-be-observed hexatic-tetratic transition near the confluence point
between the two solids and the fluid.
In order to clarify the melting scenario for a given $P$, we run our computer
code along two simulation paths, one starting from the solid phase at
$T=0.005$ and the other from a high-temperature fluid state. We advance in
steps of $\Delta T=0.005$, equilibrating the system for long before generating
an equilibrium trajectory of $M$ sweeps (one sweep corresponding to $N$ trial
MC moves), with $M$ ranging between $5\times 10^{5}$ and $3\times 10^{6}$,
depending on how far we are from melting. Once a guess of the transition point
is made, we restart the simulation slightly ahead of it with a smaller $\Delta
T$ and/or a larger $M$ in order to better discriminate between first-order and
continuous melting. Besides $\rho$ and $E/N$, we measure two order parameters
(OP), $\psi_{T}$ and $\psi_{O}$, which are sensitive to the overall
translational and orientational triangular/square order, respectively. The
precise definition of both quantities has been given in Ref. Prestipino1 ,
with obvious modifications for the square-lattice case. Moreover, we keep
track of the OP susceptibilities $\chi_{T}$ and $\chi_{O}$, defined as $N$
times the variance of the respective OP estimator. Finally, we calculate two
orientational correlation functions (OCF) Prestipino1 , $h_{6}(r)$ and
$h_{4}(r)$, which inform on the typical size of a space region in which NN-
bond angles are strongly correlated. The KTHNY theory predicts an algebraic
$r^{-\eta(T)}$ large-distance decay of the OCF in the hexatic phase, at
variance with what occurs in an isotropic fluid where the decay is much
faster, i.e., exponential. According to the same theory, $\eta$ equals $1/4$
at the transition point between hexatic and isotropic fluid.
In Figs. 5-8, we report $\rho$ and $E/N$ for two different system sizes and
various simulation protocols, as a function of $T$ for $P=0.5,2,3$, and 5. We
clearly see that, while melting is continuous for $P=0.5,2$, and 5, it is
certainly first-order for $P=3$ (and 4, data not shown), as evidenced by the
hysteresis loops. Based on our experience with the GCM Prestipino1 , we can
conjecture that there is a narrow region of hexatic phase for $P=0.5,2$, and 5
(to be confirmed later by the analysis of OPs and OCF), whereas no definite
conclusion can be reached at this point for $P=3$, where a tetratic phase can
in principle exist even in presence of a first-order transition. A special
remark is due for $P=5$, where, in contrast with what occurs for smaller
pressures, the crystal energy decreases upon heating, which means that the
increase in kinetic energy is more than compensated for by the loss in
potential energy, whose high rate of decrease is due to a NN distance lying in
the harsh part of the potential core.
For three pressures ($0.5,2$, and 3), we plot the OPs and related
susceptibilities in Figs. 9-11. For the first two pressures, we see that
$\psi_{T}$ vanishes at a slightly smaller temperature than
$\psi_{O}\equiv\psi_{6}$, which implies that the hexatic phase is confined to
a narrow $T$ interval not wider than 0.0005 for $P=0.5$ (0.0015 for $P=2$), as
also witnessed by the maxima of the two susceptibilities occurring at slightly
different $T$ values. These temperature intervals compare well with the $T$
range of the bridging region between the solid and fluid branches in Figs. 5
and 6. We thus confirm the same phenomenology of the GCM, namely that the
width of the hexatic region increases with pressure. For $P=5$, the findings
are similar to $P=2$, with roughly the same width of the hexatic region and
comparable levels of orientational and translational order in the hot solid
(data not shown). Going to $P=3$, the picture is quite different since the two
OPs now apparently vanish at the same temperature; at that point, the
orientational susceptibility shows a spike (rather than a critical peak) which
is usually associated with a first-order transition. All evidence suggests
that standard first-order melting is a plausible explanation for $P=3$, and
the same conclusion can be made for $P=4$ (data not shown).
A more direct evidence of the existence of a bond-angle ordered fluid (or a
clue to its absence) comes from the large-distance behavior of the OCF. We
plot this function in Figs. 12-14 for various temperatures across the relevant
region, for $P=0.5,2$, and 3, respectively. It appears that, for $P=0.5$ and
2, $h_{6}(r)$ decays algebraically in a $T$ region of limited extent, roughly
corresponding to the bridging region in Figs. 5 and 6. Moreover, the decay
exponent in the hexatic region is smaller than $1/4$, becoming larger only on
transforming to the isotropic fluid, and the same is found for $P=5$ (data not
shown). On the contrary, for $P=3$ and 4, $h_{4}(r)$ switches directly from no
decay at all to an exponential damping, showing that there is no tetratic
phase in our model.
The obvious question now arises as to why, at variance with the hexatic one,
the tetratic phase is not stable in our model. This question is difficult to
answer since it involves the consideration of the delicate equilibrium between
the energy and entropy of the two competing phases, in this case the tetratic
phase and the isotropic fluid phase. A possible hint could be the level of
orientational order that is found in the two types of solid slightly before
melting. If we look at Figs. 9-11, we see that the value of
$\psi_{O}\equiv\psi_{4}$ for $P=3$ is less than a half of $\psi_{6}$ (note
that the same holds for $P=4$), and this would explain why, in the square-
crystal case, long-range orientational order does not survive (even in the
weakened form typical of a tetratic phase) the loss of quasi-long-range
positional order determined by melting.
Finally, we checked by an indipendent route the order of the melting
transition for $P=0.5,2,3$, and 4. As anticipated, for each pressure we
carried out MC simulations along two different paths, one beginning from a
cold solid and the other from a low-density fluid. Using thermodynamic
integration in combination with exact free-energy calculations at the initial
points of the paths, we were able to obtain the system chemical potential
$\mu$ along the solid and fluid branches as a function of $T$. For $P=0.5$ and
2, we did not find any crossing of the two $\mu(T)$ curves, thus confirming a
continous melting transition. Notwithstanding the care we put in keeping under
control any source of statistical error, we nevertheless found a small
discrepancy (about $0.0005\epsilon$, practically constant in a 0.01 wide $T$
range across the melting transition) between the $\mu$ levels of the solid and
the fluid, which is presumably due to the not-so-small $\Delta T$ employed
along the paths far from melting. On the contrary, for $P=3$ and 4, we
observed a clear crossing between the two $\mu(T)$ curves, respectively at
$T=0.0269$ and $T=0.0176$, which is consistent with the location of the
density and energy jumps (for $P=3$, see Fig. 7), and suggestive of a first-
order transition.
Summing up, we collected multiple evidence of a continuous melting via a
hexatic phase for the triangular crystals, while the melting transition is
certainly discontinuous and “standard” for the square crystal.
Nothwithstanding the “small” sizes of the investigated samples, we think that
our conclusions are robust since they result from many independent indicators
of the phase-transition order.
## V Conclusions
We have analyzed the phase behavior of purely-repulsive 2D particles with a
weakly-softened core. The nature of this repulsion is such as to determine a
characteristic nearest-neighbor distance in the fluid phase whose statistical
precision, expressed by the width of the main peak of the radial distribution
function, shows a non-monotonous behavior with pressure at not too high
temperatures. Notwithstanding the fact that the two-fluid paradigm of core-
softened (CS) potentials does not apply here, we anyway observe the same
phenomenology as in conventional CS systems, with solid polymorphism, multiple
reentrant-melting lines, and many other water-like anomalies. While this is
similar to the 3D case Prestipino3 , the melting transition of the present
system is different, since it is continuous and occurs via an (even reentrant)
hexatic phase for the triangular solids while being standard first-order for
the intermediate square solid (i.e., no parallel tetratic phase exists). The
hexatic behavior appears to be consistent with the KTHNY theory, as witnessed
by the value of the decay exponent of the orientational correlation function
at the hexatic-to-isotropic fluid transition temperature.
Our findings could be relevant for many real soft-matter systems. Already
nowadays, monodisperse colloidal suspensions can be engineered in such a way
as to exhibit a temperature-modulated repulsion with some amount of softness.
Probably, the day is not far when by appropriate functionalization of the
particle surface a colloidal system will be shown to exhibit the kind of
phase-transition phenomena that we have illustrated here for a rather specific
model potential.
## References
* (1) J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978); A. P. Young, Phys. Rev. B 19, 1855 (1979).
* (2) S. T. Chui, Phys. Rev. B 28, 178 (1983).
* (3) E. P. Bernard and W. Krauth, Phys. Rev. Lett. 107, 155704 (2011).
* (4) K. Chen, T. Kaplan, and M. Mostoller, Phys. Rev. Lett. 74, 4019 (1995).
* (5) S. J. Mejía-Rosales, A. Gil-Villegas, B. I. Ivlev, and J. Ruiz-Garcia, J. Phys. Chem. B 110, 22230 (2006).
* (6) C. A. Murray and D. H. Van Winkle, Phys. Rev. Lett. 58, 1200 (1987).
* (7) A. H. Marcus and S. A. Rice, Phys. Rev. Lett. 77, 2577 (1996).
* (8) K. Zahn, R. Lenke, and G. Maret, Phys. Rev. Lett. 82, 2721 (1999).
* (9) P. Keim, G. Maret, and H. H. von Grünberg, Phys Rev. E 75, 031402 (2007).
* (10) B.-J. Lin and L.-J. Chen, J. Chem. Phys 126, 034706 (2007).
* (11) Y. Han, N. Y. Ha, A. M. Alsayed, and A. G. Yodh, Phys. Rev. E 77, 041406 (2008).
* (12) Y. Peng, Z. Wang, A. M. Alsayed, A. G. Yodh, and Y. Han, Phys. Rev. Lett. 104, 205703 (2010).
* (13) S. Muto and H. Aoki, Phys. Rev. B 59, 14911 (1999).
* (14) S. Z. Lin, B. Zheng, and S. Trimper, Phys Rev. E 73, 066106 (2006).
* (15) S. I. Lee and S. J. Lee, Phys. Rev. E 78, 041504 (2008).
* (16) W.-K. Qi, Z. Wang, Y. Han, and Y. Chen, J. Chem. Phys. 133, 234508 (2010).
* (17) B. K. Clark, M. Casula, and D. M. Ceperley, Phys. Rev. Lett. 103, 055701 (2009).
* (18) S. Prestipino, F. Saija, and P. V. Giaquinta, Phys. Rev. Lett. 106, 235701 (2011).
* (19) C. N. Likos, Phys. Rep. 348, 267 (2001).
* (20) F. H. Stillinger, J. Chem. Phys. 65, 3968 (1976).
* (21) A. Lang, C. N. Likos, M. Watzlawek, and H. Löwen, J. Phys.: Condens. Matter 12, 5087 (2000).
* (22) S. Prestipino, F. Saija, and P. V. Giaquinta, Phys. Rev. E 71, 050102(R) (2005).
* (23) See e.g. P. Mausbach and H.-O. May, Fluid Phase Equilib. 249, 17 (2006).
* (24) M. C. Speranza, S. Prestipino, and P. V. Giaquinta, Mol. Phys. 109, 3001 (2011).
* (25) S. V. Buldyrev, G. Malescio, C. A. Angell, N. Giovambattista, S. Prestipino, F. Saija, H. E. Stanley, L. Xu, J. Phys.: Condens. Matter 21, 504106 (2009).
* (26) P. C. Hemmer and G. Stell, Phys. Rev. Lett. 24, 1284 (1970)
* (27) H. J. Young and B. J. Alder, Phys. Rev. Lett. 38, 1213 (1979); J. Chem. Phys. 70, 473 (1979).
* (28) P. G. Debenedetti, V. S. Raghavan, and S. S. Borick, J. Phys. Chem. 95, 4540 (1991).
* (29) M. R. Sadr-Lahijany, A. Scala, S. Buldyrev, and H. E. Stanley, Phys. Rev. Lett. 81, 4895 (1998).
* (30) E. A. Jagla, J. Chem. Phys. 111, 8980 (1999).
* (31) C. N. Likos, A. Lang, M. Watzlawek, and H. Löwen, Phys. Rev. E 63, 031206 (2001).
* (32) P. Kumar, S. V. Buldyrev, F. Sciortino, E. Zaccarelli, H. E. Stanley, Phys. Rev. E 72, 021501 (2005).
* (33) A. B. de Oliveira, P. A. Netz, T. Colla, and M. C. Barbosa, J. Chem. Phys. 124, 084505 (2006).
* (34) H. M. Gibson and N. B. Wilding, Phys. Rev. E 73, 061507 (2006).
* (35) E. Lomba, N. G. Almarza, C. Martin, C. McBridge, J. Chem. Phys. 126, 244510 (2006).
* (36) D. Yu. Fomin, N. V. Gribova, V. N. Ryzhov, S. M. Stishov, D. Frenkel, J. Chem. Phys. 129, 064512 (2008).
* (37) L. Xu, S. Buldyrev, C. A. Angell, and H. E. Stanley, J. Chem. Phys. 130, 054505 (2009).
* (38) O. Pizio, H. Dominguez, Y. Duda, and S. Sokolowski, J. Chem. Phys. 130, 174504 (2009).
* (39) J. C. Pàmies, A. Cacciuto, and D. Frenkel, J. Chem. Phys. 131, 044514 (2009).
* (40) G. J. Pauschenwein and G. Kahl, J. Chem. Phys. 129, 174107 (2008).
* (41) G. Malescio, F. Saija, and S. Prestipino, J. Chem. Phys. 129, 241101 (2008).
* (42) A. Scala, M. Reza Sadr-Lahijany, N. Giovambattista, S. V. Buldyrev, and H. E. Stanley, Phys. Rev. E 63, 041202 (2001).
* (43) N. B. Wilding and J. E Magee, Phys. Rev. E 66, 031509 (2002).
* (44) A. M. Almudallal, S. V. Buldyrev, and I. Saika-Voivod, J. Chem. Phys. 137, 034507 (2012).
* (45) G. Malescio and G. Pellicane, Nat. Mat. 2, 97 (2003).
* (46) F. Saija, S. Prestipino, and G. Malescio, Phys. Rev. E 80, 031502 (2009).
* (47) S. Prestipino, F. Saija, and G. Malescio, J. Chem. Phys. 133, 144504 (2010).
* (48) G. Malescio, S. Prestipino, and F. Saija, Mol. Phys. 109, 2837 (2011).
* (49) G. Malescio and F. Saija, J. Phys. Chem. B 115, 14091 (2011).
* (50) R. P. A. Dullens and W. K. Kegel, Phys. Rev. Lett. 92, 195702 (2004).
* (51) A. M. Alsayed, Y. Han, and A. G. Yodh, Melting and geometric frustration in temperature-sensitive colloids, A. Fernandez-Nieves et al. eds. (Wiley-VCH, 2011).
* (52) T. Yoshida and S. Kamakura, Prog. Theor. Phys. 52, 822 (1974).
* (53) S. Prestipino, J. Phys.: Condens. Matter 24, 035102 (2012).
* (54) S. Prestipino, F. Saija, and G. Malescio, Soft Matter 5, 2795 (2009).
* (55) D. Frenkel and B. Smit, Understanding molecular simulation, 2nd ed. (Academic Press, 2002).
* (56) See e.g. S. Prestipino and P. V. Giaquinta, J. Stat. Phys. 96, 135 (1999).
* (57) B. S. Jabes, M. Agarwal, and C. Chakravarty, J. Chem. Phys. 132, 234507 (2010).
Figure 1: (Color online). YK potential with $a=3.3$ (Eq. (2.1), black solid
line), its negative first-order derivative $-u^{\prime}(r)$ (dotted blue
line), and the virial function $-ru^{\prime}(r)$ (dashed red line). Figure 2:
(Color online). Phase diagram of the YK potential with $a=3.3$. Two kind of
melting points are shown: those determined by the HUIM method (blue open dots)
and others (red-filled blue dots) marking either the onset of the hexatic
phase ($P=0.5,2$, and 5) or the crossing in temperature of solid and liquid
chemical potentials ($P=3$ and 4), as discussed in Section IV. Statistical
errors are smaller than the symbols size. Another red dot is placed where the
chemical potential of the low-density triangular (T) solid takes over that of
the square (S) solid at $T=0.005$. The black solid lines are schematic
transition lines. The hexatic-isotropic fluid transition lines are not shown.
The extent of the low-pressure hexatic region can be appreciated in the inset,
which shows a magnification of the $P$ interval from 1 to 2. Of similar width
is the high-pressure hexatic region near $P=5$. Further open symbols joined by
straight lines mark the boundaries of anomaly regions: isothermal $-s_{2}$
maxima and minima (left and right green squares); isothermal $D$ minima and
maxima (left and right magenta triangles); isobaric $\rho$ maxima (red
squares). We checked that the structural and diffusional anomalies are no
longer present for $T=0.2$ and $T=0.18$, respectively; hence, we draw the
dashed lines to mean that the two branches of each anomaly are actually sewed
in from the top. Figure 3: YK potential with $a=3.3$: radial distribution
function $g(r)$ for various pressures and for $T=0.08$. Successive lines
correspond to $P=0.2,0.8,1.6,2.4,3.2,4$. The arrow marks the direction along
which the pressure increases. Figure 4: Anomalous properties of the YK
potential with $a=3.3$. Left top panel: translational-order parameter $-s2$
(in units of $k_{B}$) as a function of pressure for $T=0.08$. Right top panel:
self-diffusion coefficient $D$ (units of $\sigma(\epsilon/m)^{1/2}$, where $m$
is the particle mass) for $T=0.08$. Left bottom panel: isobaric specific heat
$C_{P}$ (units of $k_{B}$) as a function of temperature for $P=2$. Right
bottom panel: isothermal compressibility $K_{T}$ (units of
$\sigma^{2}/\epsilon$) for $P=2$. A clear minimum is seen in the plot of both
response functions. Figure 5: (Color online). YK potential with $a=3.3$ for
$P=0.5$: particle-number density (top) and total energy per particle (bottom)
for two different sizes ($N=2688$, red, green, and blue; $N=6048$, black).
Different colors denote different simulation protocols (red: $\Delta T=0.005$
and $M=5\times 10^{5}$; green: $\Delta T=0.001$ and $M=5\times 10^{5}$; blue:
$\Delta T=0.001$ and $M=2\times 10^{6}$; black: $\Delta T=0.0002$ and
$M=3\times 10^{6}$). Open dots and triangles refer to a heating path and a
cooling path, respectively. Figure 6: (Color online). YK potential with
$a=3.3$ for $P=2$: particle-number density (top) and total energy per particle
(bottom) for two different sizes ($N=2688$, red, green, and blue; $N=6048$,
black and yellow-filled black). Different colors denote different simulation
protocols (red: $\Delta T=0.005$ and $M=5\times 10^{5}$; green: $\Delta
T=0.001$ and $M=5\times 10^{5}$; blue: $\Delta T=0.001$ and $M=2\times
10^{6}$; black: $\Delta T=0.001$ and $M=3\times 10^{6}$; yellow-filled black:
$\Delta T=0.0003$ and $M=3\times 10^{6}$). Open dots and triangles refer to a
heating path and a cooling path, respectively. Figure 7: (Color online). YK
potential with $a=3.3$ for $P=3$: particle-number density (top) and total
energy per particle (bottom) for two different sizes ($N=2704$, red, green,
and blue; $N=6084$, black). Different colors denote different simulation
protocols (red: $\Delta T=0.005$ and $M=5\times 10^{5}$; green: $\Delta
T=0.001$ and $M=5\times 10^{5}$; blue: $\Delta T=0.001$ and $M=2\times
10^{6}$; black: $\Delta T=0.0001$ and $M=3\times 10^{6}$). Open dots and
triangles refer to a heating path and a cooling path, respectively. Figure 8:
(Color online). YK potential with $a=3.3$ for $P=5$: particle-number density
(top) and total energy per particle (bottom) for two different sizes
($N=2688$, red, green, and blue; $N=6048$, black). Different colors denote
different simulation protocols (red: $\Delta T=0.005$ and $M=5\times 10^{5}$;
green: $\Delta T=0.001$ and $M=5\times 10^{5}$; blue: $\Delta T=0.001$ and
$M=2\times 10^{6}$; black: $\Delta T=0.0005$ and $M=2\times 10^{6}$). Open
dots and triangles refer to a heating path and a cooling path, respectively.
Inset, total energy per particle on a wider pressure range. Figure 9: YK
potential with $a=3.3$ for $P=0.5$: order parameters and susceptibilities in
the $T$ range across the melting transition. Upper panels: the orientational
order parameter $\psi_{6}$ and its susceptibility $\chi_{6}$. Dots and
triangles mark data obtained by heating and by cooling, respectively. Lower
panels: the translational order parameter $\psi_{T}$ and its susceptibility
$\chi_{T}$ on heating. All data from different protocols and sizes are
reported, always preferring the most accurate estimate when more than one is
available. Figure 10: YK potential with $a=3.3$ for $P=2$: order parameters
and susceptibilities in the $T$ range across the melting transition. See the
caption of Fig. 9 for notation. Figure 11: YK potential with $a=3.3$ for
$P=3$: order parameters and susceptibilities in the $T$ range across the
melting transition. See the caption of Fig. 9 for notation. Figure 12: (Color
online). Orientational correlation function $h_{6}(r)$ at selected
temperatures across the hexatic region for $P=0.5$ ($N=6048$). Top: log-log
plot; bottom: log-lin plot. Upon increasing $T$ from 0.0598 to 0.0606 there is
a qualitative change in the large-distance behavior of $h_{6}(r)$, from
constant (triangular solid) to power-low decay (hexatic fluid), up to
exponential decay (isotropic fluid). Note that, consistently with the KTHNY
theory, the decay exponent $\eta$ is less than $1/4$ (which is the slope of
the dashed straight line) in the hexatic phase. The slight recovery of
correlations which is observed near the largest distance at which the OCF is
computed (roughly corresponding to half of the simulation-box length) is a
finite-size effect due to the use of periodic boundary conditions. Figure 13:
(Color online). Orientational correlation function $h_{6}(r)$ at selected
temperatures across the hexatic region for $P=2$ ($N=6048$). Top: log-log
plot; bottom: log-lin plot. Upon increasing $T$ from 0.0345 to 0.0360 there is
a qualitative change in the large-distance behavior of $h_{6}(r)$, from
constant (triangular solid) to power-low decay (hexatic fluid), up to
exponential decay (isotropic fluid). Moreover, the decay exponent $\eta$ is
less than $1/4$ (which is the slope of the dashed straight line) in the
hexatic phase. Figure 14: (Color online). Orientational correlation function
$h_{4}(r)$ at selected temperatures across the hexatic region for $P=3$
($N=6084$). Top: log-log plot; bottom: log-lin plot. At variance with the
triangular-lattice case, we assist to an abrupt change of decay mode as $T$
goes from 0.0262 to 0.0263, from constant (square solid) directly to
exponential (isotropic fluid).
|
arxiv-papers
| 2012-09-12T13:30:03 |
2024-09-04T02:49:35.016866
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santi Prestipino, Franz Saija and Paolo V. Giaquinta",
"submitter": "Paolo V. Giaquinta",
"url": "https://arxiv.org/abs/1209.2608"
}
|
1209.2664
|
# A Plan For Curating “Obsolete Data or Resources”
Michael L. Nelson
Old Dominion University Norfolk, VA USA [email protected]
###### Abstract
Our cultural discourse is increasingly carried in the web. With the initial
emergence of the web many years ago, there was a period where conventional
mediums (e.g., music, movies, books, scholarly publications) were primary and
the web was a supplementary channel. This has now changed, where the web is
often the primary channel, and other publishing mechanisms, if present at all,
supplement the web. Unfortunately, the technology for publishing information
on the web always outstrips our technology for preservation. My concern is
less that we will lose data of known importance (e.g., scientific data, census
data), but rather that we will lose data that we do not yet know is important.
In this paper I review some of the issues and, where appropriate, proposed
solutions for increasing the archivability of the web.
###### category:
H.3.7 Digital Libraries
###### keywords:
Curation, Web Archiving, Memento
## 1 Who wants “obsolete data”?
Perhaps the largest problem facing web archiving is that it remains at the
fringes of the larger web community. The most illustrative anecdote pertains
to a web archiving paper we submitted to the 2010 WWW conference. One of the
reviews stated:
> Is there (sic) any statistics to show that many or a good number of Web
> users would like to get obsolete data or resources?
This is just one reviewer, but the terminology used (“obsolete data or
resources”) succinctly captures the problem: web archiving is not widely seen
as a priority or even as in scope for a conference such as WWW. Another common
related misconception we have encountered is that the Internet Archive has
every copy of everything ever published on the web, so preservation is a
solved problem. Despite the heroic efforts of the Internet Archive, the
reality is more grim: only 16% of the resources indexed by search engines are
archived at least once in a public web archive [1].
While there are many specific challenges with regards to quality criteria,
tools, and metrics, the common thread goes back to the fact that we, the web
archiving community, have failed to articulate clear, compelling use cases and
demonstrate immediate value for web preservation. For too long web
preservation has been dominated by threats of future penalties, such as hoary
stories about file obsolescence that have not come true111David Rosenthal has
a series of convincing blog posts on this topic, see:
http://blog.dshr.org/2010/09/reinforcing-my-point.html. The lack of a
compelling use case for archives has relegated preservation to an insurance-
selling idiom, where uptake is unenthusiastic at best.
## 2 I Blame Thompson and Ritchie
The web has a poor notion of time, and it is getting worse instead of better.
An early design document for the Web addressed the problem of generic vs.
specific resources [2]. That document identified three dimensions of
genericity: time, language (e.g., English vs. French), and representation
(e.g., GIF vs. JPEG). The latter two dimensions were the basis for HTTP
content negotiation as originally defined in HTTP/1.1 [5]. Content negotiation
allowed, for example, GIF and JPEG resources to have unique URIs (i.e.,
specific resources), but to be joined together with a third, generic resource
with its own URI. When a client dereferences this generic URI, the appropriate
specific resource is selected based the client’s preferences for
representations. Content negotiation works similarly for language, but content
negotiation in the dimension of time was not part of the original HTTP core
technologies (the Memento project added content negotiation in the dimension
of time in 2009 [11]). One result of not having time as part of the core
technologies is that the web community’s concept and expectations regarding
time have not become fully mature.
I believe the reason for this underdeveloped notion of time can be traced to
the tight historical integration of HTTP and Unix, specifically the Unix
filesystem. Metadata about files in the Unix filesystem is stored in “inodes”,
and the original description of the Unix filesystem defined three notions of
time to be stored in an inode: file creation, last use, and last modification
[8]. However, at some early point the storage of the file creation time in the
inode was replaced with the last modification time of the inode itself. The
result was that we could know the last modification and access times of a
file, but the creation time, a crucial part of establishing provenance, was
lost (most URIs contain semantics, and creation time can be critical in
establishing priority). Although web resources and Unix files are logically
separate, in practice they were tightly integrated during the formative years
of the web, and so the HTTP time semantics were limited by what could be
provided by the Unix inode. For example, here is an HTTP response about a JPEG
file:
% curl -I cdn.loc.gov/images/img-head/logo-loc.png
HTTP/1.1 200 OK
Date: Sun, 19 Aug 2012 13:30:06 GMT
Server: Apache
Last-Modified: Fri, 03 Aug 2012 03:54:26 GMT
Content-Length: 1447
Connection: close
Content-Type: image/png
In the above example, the server is expressing the response was sent on August
19th, but the JPEG file itself was last modified on August 3rd. Notable by its
absence is the creation time: via the inode limitations, we cannot know when
this file was created. It might have been created on August 3rd or it might
have been created at an earlier time, and being unable to establish even this
basic level of metadata is a severe limitation for archiving and provenance.
Unfortunately, even the limited semantics of last modified are becoming less
frequent as more resources are dynamically generated. The example below is in
response for a dynamically generated home page:
% curl -I www.digitalpreservation.gov/
HTTP/1.1 200 OK
Date: Sun, 19 Aug 2012 13:30:33 GMT
Server: Apache
X-Powered-By: PHP/5.2.8
Connection: close
Content-Type: text/html
In the above example, there is the data of the response (August 19th), but
last modified times for dynamically generated representations are not defined.
Dynamically generated resources make possible the web as we know it today, but
the net result is even fewer time semantics are present in HTTP responses.
Evolving publishing technologies such as personalization, Ajax, Flash, and
streams222For example, see Anil Dash’s call to “Stop Publishing Web Pages” in
favor of streams: http://dashes.com/anil/2012/08/stop-publishing-web-
pages.html will only serve to make it more difficult to ascribe a creation
time to any particular web page.
## 3 W{h}ither Archives?
I maintain that the entire web community has a poor notion of time and are
trapped in the “perpetual now”. Because the lack of capability has shaped our
expectations, we never object when prior versions of web pages are
unavailable. We tolerate temporal inconsistency in our browsing, even 404
errors, in part because we do not know enough to expect better. Remember “lost
in hypertext” [4, 3]? That has been solved in part through better navigation
tools and design practices, but also in part due to increased familiarity with
the hypertext navigation metaphor. Now imagine if a temporal dimension was
added for each page – there would be much confusion, but eventually tools,
practices, and user awareness would prevail.
### 3.1 Archives Are Not Destinations
The most fundamental problem is that we have designed web archives as if they
are destinations in themselves. The motif of “go to the library/archive and
spend an afternoon in the stacks” has been replicated in our web archives.
Figure 1 shows the list of archived pages (or “mementos”) for cnn.com at the
Internet Archive. If you want to browse the past versions of this news site,
you go to the archive and perform a browsing session within the archive, and
then return to the live web once you are done with your journey to the past.
Figure 1: All available versions of cnn.com at the Internet Archive. This page
is not reachable from cnn.com.
In our experience, most web users do not know about the Internet Archive or
how to access it. The Memento project has demonstrated a framework for tighter
integration of the past (i.e., archived) web and the current web, but the
tools exist as add-ons for both servers and clients and have yet to reach
mainstream acceptance, which will only arrive when the archiving community can
demonstrate a “killer app” that will cause users to demand the functionality.
### 3.2 Web Archiving Is Not Social
I am not sure what an archiving killer app would look like, but there is a
good chance it will be social. People like to share links with each other via
Twitter, Facebook, Pinterest, et al. However, with the exception of Pinterest
(which makes copies of “pinned” images) this sharing is done by-reference and
not by-value, exposing it to the same link rot problems of common web pages
(for example, we found 10% of the shared links about the Egyptian Revolution
were lost after one year [9]). I am constantly surprised at the tasks that
people are willing to undertake if there is a social or gaming component
(i.e., “games with a purpose”), yet I am unaware of any such activity with a
web preservation component. Diigo (diigo.com) is a site that provides social
bookmarking services (similar to Delicious) with an archiving component, but
enthusiasm for social bookmarking seems to be less than it once was.
A web archiving application that could leverage the collection development of
Pinterest and the collaborative editing of Wikipedia and other wikis would be
a welcome development. Archive-It (archive-it.org) is nearly such an
application, but it is targeted for archiving and librarian professionals, not
as a general purpose social application. Perhaps the legal challenges333A
discussion of which is beyond the scope of this paper; for a primer see
http://1.usa.gov/QgaUZO of creating such collections would prevent the
development of such an application, but I would observe that early legal
challenges about the mechanics of HTTP and “making copies” were eventually
overcome.
### 3.3 Watchdog Archiving and Trust
Perhaps a social web archiving activity that will grow to take on a larger
role is that of distributed, citizen watchdogs of public figures and
politicians. For example, a supporter of blogger Andrew Breitbart brought down
Congressman Anthony Weiner by zealously following and archiving Weiner’s
twitter feed444See
http://en.wikipedia.org/wiki/Anthony_Weiner_sexting_scandal. Most tweets are
of arguably limited historical value, but this particular tweet and the fact
that it could not be fully redacted turned out to have significant political
and cultural implications.
In another example, consultant and commentator Richard Grenell deleted over
800 tweets after he was elevated to a senior position in the Romney campaign
in 2012555See: http://huff.to/I6dpQo. Presumably Grenell’s lesser status at
the time did not warrant a corresponding campaign to monitor and archive
Grenell’s twitter feed like there was with Weiner’s twitter feed. Grenell’s
tweets most likely do not exist outside of Twitter’s own archives (and those
they share with the Library of Congress).
And what if someone did come forward with a correspondingly damning tweet from
Grenell, how could we verify it? Aside from Weiner’s ultimate confession, was
his tweet ever verified by an independent third party? And if so, how would we
trust such a third party – where would the chain of trust terminate? Could he
not find a technologically savvy staffer to fabricate evidence that
contradicted Breitbart’s evidence (which is especially easy given the low
level of provenance regarding third-party archives)? It is easy to envision a
market for a trusted, tamper-proof archive for tweets and other social media
so a person can _deny_ that they ever released an offending tweet?
Our current approach to web archiving involves implicitly trusting the
Internet Archive and other public web archives as incorruptible. Eventually
the magnitude of scandals associated with web content will grow to the point
where less scrupulous web archives will be offered as proof. A combination of
trusted archives and citizen activism might form the basis for the first
killer app for web archiving. Instead of canvassing a neighborhood, volunteers
can canvass/archive web pages.
## 4 Wish List
This section contains a personal wish list of features that would make
archiving web pages much easier.
### 4.1 Machine-Readable Time Semantics
We have moved beyond the limitations of the Unix filesystem and its inode, so
we should increase the time semantics in our HTTP transactions. Unfortunately,
this is not the case. In the example below, when dereferencing the URI of a
specific tweet, twitter.com shows a last modified time that matches the date
the response was generated (this is true for all responses, not just this
one). More importantly, Twitter has a concept of time similar to “Memento-
Datetime”, which captures the time a page was first observed on the web (see
[7] for a discussion of how this differs from “Last-Modified”). Although this
date (June 27, 2012 in this example) is displayed in the HTML page and is
accessible to authenticated users via the Twitter API, the correct date
semantics are not presented, and the incorrect value for the last modified
time is presented instead. This phenomenon is not unique to Twitter, but
Twitter makes for a good example due to its well-known nature.
% curl -I twitter.com/machawk1/status/218015444496416768
HTTP/1.1 200 OK
Date: Mon, 20 Aug 2012 00:41:38 GMT
Content-Length: 85440
Last-Modified: Mon, 20 Aug 2012 00:41:38 GMT
Content-Type: text/html; charset=utf-8
Server: tfe
### 4.2 APIs for Archives
Talk to anyone who has built applications using archived web data and they
will have crawled and “page scraped” the archives at some point. Page scraping
puts an undue burden on the archive itself, is error prone, and doesn’t
facilitate inter-archive interaction. The Memento project defines a simple,
inter-archive HTTP access mechanism, but this is not enough. The Internet
Archive’s Wayback Machine software supports a simple API for file upload and
searching, but this API is not evolved like APIs for services like Google,
Twitter, and Facebook. If we want archives to be used in the current web
programming idiom, we have to go beyond the “afternoon in the stacks” model
(see section 3.1) and provide fully-featured APIs.
### 4.3 Impedance Matching
The Internet Archive does not have full-text search on the main Wayback
Machine. While this is a limitation, it is probably not as big a limitation as
many think, in part because it is not clear what we would do with full-text
search at this scale if we had it (cf. the discussion in section 3). The kinds
of questions that scholars wish to answer using web archives are of the form
“what role did the Tea Party play in the 2010 mid-term elections?” The kind of
access we can offer right now is “this is what cnn.com looked like November 1,
2010.” Adding full-text searching, while useful in some cases, would not
immediately help address the kinds of questions that scholars want to ask. An
example of the kind of advanced analysis that needs to be performed on web
archives is entity tracking experiments of the LAWA project [10], in which
entities (e.g., people, companies) can be tracked through time and different
URIs.
## 5 Conclusions
I expect data of known value to be successfully curated and available well
into the future. I am more concerned with our cultural record, with which we
have made a Faustian bargain of increased volume and ease of access (i.e., the
web) at the expense of permanence and provenance (i.e., paper). We are stuck
in the perpetual now and due to the initial limitations of the Unix inode
there, the notion of varying temporal access to web pages is so unexpected
that even web researchers need to be convinced of the utility.
One problem is the limited design motif for web archives: destinations that
are wholly unconnected from their live web counterparts. The related problem
is that we, as a community, have failed to envision and deliver a “killer app”
for web archiving. Perhaps it is in a watchdog role over public figures and
institutions. Or perhaps the emerging field of personal digital
preservation666See for example: http://www.personalarchiving.com/ will
energize the field and increase what are often laissez-faire user expectations
regarding archiving [6].
I would like to see a more careful approach to specifying temporal semantics
in common web services like Twitter. Similarly, I expect web archives to offer
richer APIs for accessing their content, and to eventually offer the higher-
level services, like entity tracking, that will assist scholars in using the
obsolete data or resources archives.
## 6 Acknowledgments
This work sponsored in part by the Library of Congress, NSF IIS-0643784 and
IIS-1009392.
## References
* [1] S. G. Ainsworth, A. Alsum, H. SalahEldeen, M. C. Weigle, and M. L. Nelson. How much of the web is archived? In Proceeding of the 11th annual international ACM/IEEE Joint Conference on Digital Libraries, JCDL ’11, 2011.
* [2] T. Berners-Lee. Web architecture: Generic resources. http://www.w3.org/DesignIssues/Generic.html, 1996.
* [3] J. Conklin. Hypertext: A survey and introduction. IEEE Computer, 20(9):17–41, 1987.
* [4] W. Elm and D. Woods. Getting lost: A case study in interface design. In Proceedings of the Human Factors and Ergonomics Society Annual Meeting, volume 29, pages 927–929, 1985.
* [5] R. Fielding, J. Gettys, J. Mogul, H. Frystyk, and T. Berners-Lee. Hypertex Transfer Protocol – HTTP/1.1, Internet RFC-2068, 1997.
* [6] C. Marshall, F. McCown, and M. L. Nelson. Evaluating personal archiving strategies for Internet-based in formation. In Proceedings of IS&T Archiving 2007, pages 151–156, May 2007\.
* [7] M. L. Nelson. Memento-Datetime is not Last-Modified. http://ws-dl.blogspot.com/2010/11/2010-11-05-memento-datetime-is-not-last.html, 2011\.
* [8] D. Ritchie and K. Thompson. The UNIX time-sharing system. Communications of the ACM, 17(7):365–375, 1974.
* [9] H. M. SalahEldeen and M. L. Nelson. Losing my revolution: How much social media content has been lost? In TPDL, 2012.
* [10] M. Spaniol and G. Weikum. Tracking entities in web archives: the LAWA project. In Proceedings of the 21st international conference companion on World Wide Web, WWW ’12 Companion, 2012.
* [11] H. Van de Sompel, M. L. Nelson, R. Sanderson, L. L. Balakireva, S. Ainsworth, and H. Shankar. Memento: Time Travel for the Web. Technical Report arXiv:0911.1112, 2009.
|
arxiv-papers
| 2012-09-12T16:54:45 |
2024-09-04T02:49:35.025796
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael L. Nelson",
"submitter": "Michael Nelson",
"url": "https://arxiv.org/abs/1209.2664"
}
|
1209.2697
|
$\langle$reception date$\rangle$ $\langle$acception date$\rangle$
$\langle$publication date$\rangle$ Astronomical Society of JapanUsage of
pasj00.cls
methods: data analysis — methods: statistical — stars: activity — stars: flare
— stars: individual(CR Dra)
# The Statistical Analyses of the White-Light Flares:
Two Main Results About Flare Behaviours
Dal H. A Department of Astronomy and Space Sciences, University of Ege,
Bornova, 35100 İzmir, Turkey [email protected]
###### Abstract
We present two main results, based on the models and the statistical analyses
of 1672 U-band flares. We also discuss the behaviours of the white-light
flares. In addition, the parameters of the flares detected from two years of
observations on CR Dra are presented. By comparing with the flare parameters
obtained from other UV Ceti type stars, we examine the behaviour of optical
flare processes along the spectral types. Moreover, we aimed, using large
white-light flare data,to analyse the flare time-scales in respect to some
results obtained from the X-ray observations. Using the SPSS V17.0 and the
GraphPad Prism V5.02 software, the flares detected from CR Dra were modelled
with the OPEA function and analysed with t-Test method to compare similar
flare events in other stars. In addition, using some regression calculations
in order to derive the best histograms, the time-scales of the white-light
flares were analysed. Firstly, CR Dra flares have revealed that the white-
light flares behave in a similar way as their counterparts observed in X-rays.
As seen in X-ray observations, the electron density seems to be a dominant
parameter in white-light flare process, too. Secondly, the distributions of
the flare time-scales demonstrate that the number of observed flares gets a
maximum value in some particular ratios, which are 0.5 or its multiples, and
especially positive integers. The thermal processes might be dominant for
these white-light flares, while non-thermal processes might be dominant in the
others. To reach better results for the behaviour of the white-light flare
process along the spectral types, much more stars in a wide spectral range,
from the spectral type dK5e to dM6e, must be observed in the white-light flare
patrols.
## 1 Introduction
The flare process has not been perfectly understood yet, although several
studies on this subject have been carried out since the first flare was
detected on the Sun by R. C. Carrington and R. Hodgson on September 1, 1859
(Carrington, 1859; Hodgson, 1859). The flare activity of dMe stars is modelled
based on the processes of Solar Flare Event (Gershberg, 2005; Benz & Güdel,
2010). To understand flare event of dMe stars, the levels of the flare energy
have been looked over in many studies. Gershberg & Shakhovskaya (1983)
demonstrated that there are some differences between energy levels of
different stars and star groups. They demonstrated that the flare energy
levels of the Orion association stars are located at the highest levels. The
Pleiades stars are located just below the Orion association stars. Finally,
the stars in the Galactic field are located below all of them. The levels seem
to be arranged according to the different ages. However, the differences among
the different stars and star groups might be due to the saturation level of
white-light flares detected from UV Ceti type stars. The X-ray and radio
observations indicate that some parameters of magnetic activity can reach the
saturation (Gershberg, 2005; Skumanich & McGregor, 1986; Vilhu et al., 1986;
Doyle, 1996; García-Alvarez et al., 2008; Jeffries et al., 2011). Dal & Evren
(2011a) recently revealed some clues for the saturation in the white-light
flares. They modelled the distributions of flare-equivalent durations versus
flare total duration. In the models, it is seen that flare-equivalent
durations can not be higher than a specific value and it is no matter how long
flare total duration is. According to Dal & Evren (2011a), this level defined
as the $Plateau$ parameter in the models is an indicator of the saturation
level for the white-light flares in some respects.
To understand flare event of dMe stars, the flare time-scales are as important
as emitting flare energy. There are some parameters (such as $B$, $n_{e}$ and
geometry of flaring loop) to determine the border of the time-scales in the
flare events. These parameters affect the time-scales of heating and cooling
of the regions, where the flare events occur (Temmer et al., 2001; Reeves &
Warren, 2002; Imanishi et al., 2003; Pandey & Singh, 2008). According to
Imanishi et al. (2003), the flare decay time ($\tau_{d}$) is firmly correlated
with $B$ and the electron density ($n_{e}$), while the flare rise time
($\tau_{r}$) is proportional to a larger magnetic loop length ($\ell$) and
smaller $B$ values. Reeves & Warren (2002) concluded that the observed light
curve is modelled with using multi-loop instead of a single-loop. However,
some discrepancies were observed in the decay phases and especially toward the
end of the flare light curves. In contrast to that assumed by Imanishi et al.
(2003), Pandey & Singh (2008) found that there is a sustained heating during
the decay phases of the flares. Moreover, Favata et al. (2005) demonstrated
that short flares occur in a single loop, while longer flares are generally
arcade flares and some heating is driving their decay phases. In addition,
there are some relations between the flare rise and decay times. Using the
event asymmetry index ($A_{ev}$), Temmer et al. (2001) statistically analysed
solar $H_{\alpha}$ flares, and found some relations between flare decay and
rise time-scales. Kay et al. (2003) demonstrated that there is a linear
correlation between flare rise and decay times.
In the first part of this study, the flares of CR Dra were modelled, and some
parameters were derived from the model. We compared CR Dra with other UV Ceti
type stars in respect to these parameters. In this point, CD Dra takes an
important place, because it fills a gap in the B-V range. CR Dra is a flare
star (Gershberg et al., 1999), which is classified as a double or multiple
star in the SIMBAD database. The orbital period is 4.04 yr and the semimajor
axis of the orbit is 0$\arcsec$.148. The total mass of the system is reported
as a value between 1.0 $M_{\odot}$ and 1.8 $M_{\odot}$. The distance of the
system is given between 17.4 $pc$ and 20.7 $pc$ (Tamazian et al., 2008). CR
Dra is a metal-rich stars and a member of the old disk population in the
galaxy (Stauffer & Hartmann, 1986; Veeder, 1974). The photometric magnitude
difference between the components of the system was measured as $1^{m}.8$ in
V-band by Tamazian et al. (2008). The flare activity of CR Dra was discovered
for the first time by Petit (1957). Apart from the flare activity, there are
some studies for the sinusoidal-like variations at out-of-flares, such as
Mahmoud (1991, 1993); Anderson (1979). However, it is not clear that the star
exhibits any rotational modulation. Mahmoud (1991) found that CR Dra exhibits
a sinusoidal-like variation with period of $16^{d}$ in V-band, while the
period found from B-band is $5^{d}$. The author mentioned that the amplitude
of the variation found in B-band was lower that the observational standard
deviation. On the other hand, Mahmoud (1993) found that the period of the
sinusoidal-like variation at out-of-flare is $8^{d}.776$ in V-band.
In the second part of this study, a large flare data containing 1672 U-band
flares, which were obtained in this study and collected from 42 different
studies in the literature, were analysed to test whether any relation exists
between the flare rise and decay times for the white-light flares, or not.
## 2 Observations and Data
### 2.1 Observations
Observations were acquired with the High-Speed Three-Channel Photometer
(HSTCP) attached to the 48 cm Cassegrain-type telescope at Ege University
Observatory. Observations were performed in two different manners. We used a
tracking star in the second channel of the photometer and carried out the
flare observations only in standard Johnson U-band with exposure times between
2 and 10 seconds and time resolution of 0.01 seconds. The second type of
observations was used for determining whether there was any variation out-of-
flare. We also observed CR Dra once or twice a night, when the star was close
to the celestial meridian. These observations were made with the exposure time
of 10 seconds for each band of standard Johnson BVR system. Considering the
technical properties of the HSTCP given by Meištas (2002) and following the
procedures outlined by Kirkup & Frenkel (2006), the mean average of the
standard deviations of observation times was computed as 0.08 seconds for the
U-band observations. It was found to be 0.20 seconds for the multi-band
observations. The same comparison stars were used for all observations. The
basic parameters of all the program stars (such as standard V magnitudes and
B-V colours) are given in Table 1. Although the program and comparison stars
are so close in the plane of the sky, differential extinction corrections were
applied. The extinction coefficients were obtained from observations of the
comparison stars on each night. Moreover, the comparison stars were observed
with the standard stars in their vicinity and the making use of differential
magnitudes, in the sense of variable minus comparison, were transformed to the
standard system using the procedures outlined by Hardie (1962). The standard
stars are listed in the catalogue of Landolt (1983). Heliocentric corrections
were applied to the observation times. The mean averages of the standard
deviations are $0^{m}.015$, $0^{m}.009$, $0^{m}.007$ and $0^{m}.007$ for the
observations acquired in standard Johnson UBVR bands, respectively. To compute
the standard deviations of observations, we use the standard deviations of the
reduced differential magnitudes in the sense comparisons (C1) minus check (C2)
stars for each night. There is no variation in the standard brightness
comparison stars. The differential magnitudes in the sense of comparison minus
check stars were carefully checked for each night. The comparison and check
stars were found to be constant in brightness during the period of
observations.
In general, 20 U-band flares were detected in observations of CR Dra. The star
was observed $5^{h}.38$ in 2005 and $29^{h}.10$ in 2007. All the flares were
detected in U-band observations of 2007. Apart from flares, no variation was
found in BVR bands out-of-flare activity. Some samples of detected flares are
shown in Figure 1. Several parameters, such as the flare rise time, decay
time, amplitude and equivalent duration, were computed from the light curve of
each flare. The procedure used in the calculations is similar to the methods
found in the literature (Gershberg, 1972; Moffett, 1974). The procedure schema
of calculations are shown in Figure 2 for two flares, which are seen in both
top and bottom panels of Figure 1. In calculations, we firstly separated each
flare light curve into three parts. One of them is the part indicating the
quiescent level of the brightness before the first flare on each nigh. The
brightness level without any variations (such as a flare or any oscillation)
was taken as a quiescent level of the brightness of this star. To determine
this level, we used the standard deviation of each observation point,
considering the mean average of all the observation points until this last
point. If the standard deviations of the following points get over the
$3\sigma$ level, this point was taken as the beginning of a flare. The
quiescent levels of each star were determined from all the observation points
before the first flare on each night. It must be noted that, although a
different quiescent level was found for each observing night, their levels are
almost the same considering the $3\sigma$ levels of observing condition. As
seen in Figure 2, we fitted this level with a linear function ($f_{1}(x)$ and
$g_{1}(x)$), and then, using this linear function, we computed the flare
equivalent duration, flare amplitude and all the flare time-scales (rise and
decay times). The part of the light curve above the quiescent level was also
separated into two sub-parts. First of them is the impulsive phase, in which
the flare increases. Second one is the decay phase. The impulsive and decay
phases were separated according to the maximum brightness observed in this
part. It must be noted that some flares have a few peaks. In this case, the
point of the first-highest peak was assumed as the flare maximum. To determine
the flare time-scales, we fitted the impulsive and decay phases with the
polynomial functions. In Figure 2, $f_{2}(x)$ and $g_{2}(x)$ are the fits of
the impulsive phases for the given samples, while $f_{3}(x)$ and $g_{3}(x)$
are the fits of decay phases. The best polynomial functions were chosen
according to the correlation coefficients ($r^{2}$) of fits. To determine the
beginning and end of each flare, we computed the intersection points of the
polynomial fits with the linear fit of the quiescent level and their standard
deviations. In this study, the intersection points were taken as the beginning
and end of each flare. The flare rise time ($\tau_{r}$) was taken the duration
between the beginning and the flare maximum point. For example, the duration
from the first intersection between $f_{1}(x)$ and $f_{2}(x)$ to the second
one between $f_{2}(x)$ and $f_{3}(x)$ was taken as the flare rise time
($\tau_{r}$) for this flare. In the same way, the flare decay time
($\tau_{d}$) was taken the duration between the flare maximum point and the
flare end. In Sample 1, the duration from the second intersection between
$f_{2}(x)$ and $f_{3}(x)$ to the third one between $f_{3}(x)$ and $f_{1}(x)$
was taken as the flare decay time ($\tau_{d}$). The height of the observed-
maximum points from the $f_{1}(x)$ linear fit was taken as the amplitudes of
this flare. The same procedure was used for each flare, and the Maple 12
(Monagan et al., 2008) software was performed in all calculations.
Apart from the flare time-scales and amplitudes, using Equations (1) and (2)
taken from Gershberg (1972), the equivalent durations and energies of all the
flares were computed:
$P=\int[(I_{flare}-I_{0})/I_{0}]dt$ (1)
where $I_{0}$ is the flux of the star in the observing band while in the quiet
state, and $I_{flare}$ is the intensity at the moment of flare. For instance,
the parameter $P$ is equal to about total area between the $f_{1}(x)$,
$f_{2}(x)$ and $f_{3}(x)$ for Sample 1.
$E=P\times L$ (2)
where $E$ is the energy, $P$ is the flare-equivalent duration in the observing
band, and $L$ is the intensity in the observing band while the star is in the
quiet state.
For each observed flare, the HJD of flare maximum moment, flare rise
($\tau_{r}$) and decay time ($\tau_{d}$) (s), flare total durations (s), ratio
of $\tau_{d}/\tau_{r}$, flare-equivalent duration (s), flare amplitude (mag),
and their energies (erg) were calculated. Those parameters are given in Table
2.
### 2.2 Data Used In the Analyses
The number of flares used in the analyses are 1672 from which 20 flares are
detected in the observations of CR Dra in this study. 534 flares were obtained
from other stars of this project. The remaining 1118 flares were collected
from 41 different studies in the literature. Thus, the data used in the
analyses were contained the parameters of 15 different UV Ceti stars. In Table
3, the distribution of the data taken from the literature is listed versus
references ordered by year.
In the analyses of flare activity, especially in the OPEA models, 554 flares
obtained in this project were only used. This is because the analyses need to
be comprised of parameters derived with the same method and from the flares
detected with the same optical system. Otherwise, some artificial variations
and differences can occur between the data sets. To avoid this problem,
instead of all 1672 flares, just 554 flares detected in this project, which
comprised parameters derived with the same method and the same optical system,
were used. On the other hand, all 1672 flares were used in the analyses of the
distribution of the flare numbers versus the ratio of flare decay time to rise
time. Determining and computing of the time-scales of the rise and decay
phases are almost the same in all the studies.
## 3 Analyses
### 3.1 Flare Activity and the One-Phase Exponential Association Models
The distributions of the equivalent durations in the logarithmic scale versus
flare total durations were derived in order to test whether there are any
upper limits for the distributions of the equivalent durations ($logP_{u}$) of
the flares detected in the observations of CR Dra. Although the flare energy
is generally used to analyse in the literature, the flare equivalent durations
were used to analyse in this study. This is because of the luminosity
parameter ($L$). As seen from Equations (2), the flare energy ($E$) depends on
the luminosity parameter ($L$) with flare equivalent duration ($P$). However,
the luminosity ($L$) is different for each star. Although there are small
differences among the masses of M dwarfs, the luminosities of the two M
dwarfs, whose masses are very close to each other, can be dramatically
different due to their position in the H-R diagram. This means that the
computed energies of flares are very different from each other, even if the
light variations of the flares occurring on these two stars are absolutely the
same. Because of this, the equivalent durations ($P$) were used instead of
energy ($E$) in this study. The equivalent duration parameter ($P$) depends
only on flare power.
In order to model this distribution, first of all, the best curve was
estimated with SPSS V17.0 software (Green et al., 1999) and GraphPad Prism
V5.02 software (Motulsky, 2007). The regression calculations showed that the
Exponential function is the best model. Besides, the tests done with GraphPad
Prism V5.02 software revealed that especially the One Phase Exponential
Association function (hereafter OPEA) (Motulsky, 2007; Spanier & Oldham, 1987)
given by Equation (3) is the best exponential function among all others. The
OPEA model of the distributions of the equivalent durations in the logarithmic
scale versus flare total durations was derived with using the Least-Squares
Method with GraphPad Prism V5.02 software.
$y~{}=~{}y_{0}~{}+~{}(Plateau~{}-~{}y_{0})~{}\times~{}(1~{}-~{}e^{-k~{}\times~{}x})$
(3)
There are some important parameters derived from this function, which have
some clues about the flaring loop and the condition inside the loop. One of
them is $y_{0}$, which is the lower limit of equivalent durations for observed
flares in the logarithmic scale. In contrast to $y_{0}$, the parameter of
$Plateau$ is the upper limit. The value of $y_{0}$ depends on the quality of
observations as well as flare power, while the value of $Plateau$ depends only
on power of flares. In fact, according to Equation (2), the value of $Plateau$
depends only on the energy of flares occurring on the star. The parameter $k$
in Equation (3) is a constant depending on the $x$ value, which is the total
flare time. Apart from them, one of the most important parameter is the $Half-
Life$ parameter. This one is half of the first x values, where the model
reaches the plateau values for a star. The $Half-Life$ parameter was also
computed from the derived model. The derived OPEA model is shown in top panel
of Figure 3, and the parameters of the model are listed in Table 4. In the
middle panel of Figure 3, the model derived for CR Dar is compared with the
models derived for other six UV Ceti type stars (Dal & Evren, 2011a, b). In
the bottom panel of the figure, the $Plateau$ values are compared for these
stars.
Considering the flares in the $Plateau$ phases of the OPEA model, the mean
average of maximum flare-equivalent durations was computed to test whether the
$Plateau$ value is statistically acceptable, or not. The Independent Samples
t-Test (hereafter t-Test) (Wall & Jenkins, 2003; Dawson & Trapp, 2004) was
used for the calculation. Thus, the value of $Plateau$ was tested using
another statistical method. Although the mean value computed by the t-Test is
expected to be close to the $Plateau$ value of the OPEA model, it is clear
that there can be some difference between these two values. This is because
the OPEA model depends on all the distributions of the $x$ values from the
beginning to end, while the mean value computed by the t-Test depends only on
equivalent durations of flares in the $Plateau$ phase. The results obtained
from the t-Test analyses are also listed in Table 4.
Examining the flares detected in U-band observations of CR Dra, it was
computed that the maximum flare rise time obtained from these 20 flares is
1967 s, while the maximum flare total duration is 4955 s.
### 3.2 Distribution of the Flare Numbers versus the Ratio of Flare Decay
Time to Flare Rise Time
Several parameters were computed from the light curves of all these flares
detected in this project. Then, using some statistical methods, the relations
among themselves of all the parameters were analysed with both SPSS V17.0
(Green et al., 1999) and GraphPad Prism V5.02 software (Motulsky, 2007).
During the analyses of the distribution of flare amplitudes versus the ratio
of flare decay time to rise time, a remarkable gathering was seen for some
ratios of flare decay time to rise time, which values are several times of the
values 0.5 and especially the positive integer numbers, such as 0.5, 1.0, 1.5,
2.0, 2.5, 3.0, 3.5, 4.0, 4.5, etc. These accumulations are seen in Figure 4.
As seen from the figure, the ratios of flare decay times to rise times are
especially positive integers for many flares with different amplitudes.
To examine this unusual unexpected case, first of all, the ratio of flare
decay time to rise time were computed for each flare. The values of
$\tau_{d}/\tau_{r}$ are listed for CR Dra flare in Table 2. Considering the
errors listed in Table 2, the mean error of $\tau_{d}/\tau_{r}$ values was
found to be $\pm$0.042 for the flares detected from CR Dra. However, in this
project, 554 U-band flares are detected in total. Taking all 554 flares into
account, the mean error of $\tau_{d}/\tau_{r}$ values was found to be
$\pm$0.026. As also seen from Table 2, although there are some flares, whose
$\tau_{d}/\tau_{r}$ errors are dramatically larger than the computed mean
errors, the total number of these flares is small enough to enough to be
neglected. Apart from the flare detected in this project, we also collected
1118 flares from the literature and combined them with 554 flares. The sources
of 1118 flares are listed in Table 3. It must be noted that unfortunately we
could not compute the errors of $\tau_{d}/\tau_{r}$ values for the flares
taken from the literature due to absence of errors of $\tau_{r}$ and
$\tau_{d}$ values in the sources. Secondly, the best histograms were
determined using SPSS V17.0 and GraphPad Prism V5.02 software for these data.
Table 5 shows the parameters use to produce the histograms. The analyses
indicate that the best statistically acceptable interval ratio length should
be 0.05 in the interval ratio length.
The derived numbers of the flares in intervals of 0.05 were plotted versus the
ratio of flare decay time to rise time. Then, the histogram was derived for
554 flares detected in this project. The ratio distribution is shown in Figure
5, while the results of the analysis is listed in Table 5. As seen from the
figure, the ratio is equal to 1.0 in 0.05 ratio length for 73 flares, and it
is 2.0 for 39 flares and also 3.0 for 29 flares. In brief, the ratios of flare
decay time to rise time are especially equal to 1.0 or its multiples as the
positive integers for 204 flares among 554 flares. Besides, the ratio is 0.5
for 18 flares. In total, the ratios of flare decay times to rise times are
equal to 0.5 or its multiples, such as 1.5, 2.5, 3.5, 4.5, etc. The incidence
of the flares, whose ratio of their decay times to rise times is 1.0 or
positive integers, is 36.82$\%$ over 554 flares detected in this project. It
is 15.70$\%$ for the flares, whose ratios are 0.5 or its multiples, such as
1.5, 2.5, 3.5, 4.5, etc. Considering that the positive integers are acceptable
as the multiples of 0.5, the incidence of the flares, whose ratios of their
decay times to rise times are 0.5 or its multiples, such as 1.0, 1.5, 2.0,
2.5, 3.0, 3.5, 4.0, 4.5, etc., is 52.53$\%$ over 554 flares.
In order to test whether this remarkable result obtained from the data of this
project might be occurred due to any selection effects, almost all the U-band
flare data given in the literature were collected from 41 different studies.
Thus, 1118 U-band flares were collected apart from 554 U-band flares of this
project. In fact, there are more 170 flares given in literature, but the decay
or/and rise times were not given. Because of this, the ratios of flare decay
times to rise times were not computed for these ones, and they could not be
used. Following the method mentioned above, the same examinations were done
for these 1118 U-band flares. The result is seen in Figure 6, while the
results of the analysis is listed in Table 5. According to the statistical
analyses, the incidence of the flares, whose ratio of their decay times to
rise times is 1.0 or positive integers, is 16.37$\%$ over 1118 flares
collected from the literature. It is 8.23$\%$ for the flares, whose ratios are
0.5 or its multiples, such as 1.5, 2.5, 3.5, 4.5, etc. Considering that the
positive integers are acceptable as the multiples of 0.5, the incidence of the
flares, whose ratios of their decay times to rise times are 0.5 or its
multiples, such as 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, etc., is 24.60$\%$
over these 1118 flares.
Finally, both the data obtained in this project and the data collected from
the literature were combined together in order to reach rather better result.
In brief, a data set containing 1672 U-band flares in total was obtained.
Following the same method, the distribution was derived and statistically
examined. The result of the examinations is shown in Figure 7, while the
results of the analysis is listed in Table 5. Consequently, considering 1583
U-band flares, the incidence of the flares, whose ratio of their decay times
to rise times is 1.0 or positive integers, was found as 23.09$\%$. It was
computed as 10.83$\%$ for the flares, whose ratios are 0.5 or its multiples,
such as 1.5, 2.5, 3.5, 4.5, etc. Considering that the positive integers are
acceptable as the multiples of 0.5, the incidence of the flares, whose ratios
of their decay times to rise times are 0.5 or its multiples, such as 1.0, 1.5,
2.0, 2.5, 3.0, 3.5, 4.0, 4.5, etc., is 33.91$\%$ over these 1583 flares.
## 4 Results and Discussion
### 4.1 Flare Activity and the One-Phase Exponential Association Models
The distributions of flare-equivalent durations versus flare total duration
were modelled by the OPEA function expressed by Equation (3) for CR Dra. As it
is seen from Figure 3, this function demonstrates that the equivalent
durations of the flares detected in the observations of CR Dra can not be
higher than a specific value regardless of the length of the flare total
duration. In order to test whether the $Plateau$ parameter derived from the
model is statistically acceptable, using the t-Test, the mean average of
equivalent durations was computed for the flares located in the $Plateau$
phase in the OPEA models. The mean average of equivalent durations was found
to be close to the $Plateau$ values derived from the OPEA models. The
$Plateau$ parameter derived from the OPEA model was identified as an indicator
of the saturation level for the white-light flares by Dal & Evren (2011a).
According to Dal & Evren (2011a), two stars, EV Lac and EQ Peg, have almost
different level of the $Plateau$ value. However, the observations of CR Dra
have demonstrated that the variation of the $Plateau$ value is different from
that shown by Dal & Evren (2011a). In fact, the shape of the $Plateau$ value
variation is remarkably different from the conclusions of Dal & Evren (2011a).
EV Lac and EQ Peg are not different from the others. CR Dra flares indicate
that the variation of this parameter has a trend, and both EV Lac and EQ Peg
are on the trend. The observations of CR Dra also demonstrated that white-
light flare activity could be affected by some parameters of the flare
processes as well as the flares detected in X-ray or radio observations. The
figure taken from Katsova et al. (1987) demonstrated that electron density
varies along B-V indexes (Gershberg, 2005), and its variation trend is in
contrast shape with that of the $Plateau$ values. The analyses of flares
detected in X-ray have revealed some important points of the processes.
Considering the parameters of the Solar Flare Event, the suspected parameters
could be one or several of $\nu_{A}$, $B$, $R$ or $n_{e}$. The suspected
parameters must be $B$ or/and $n_{e}$ in the atmosphere of the stars during
the flare event. According to Katsova et al. (1987) it could be $n_{e}$. On
the other hand, Doyle (1996) suggested that it can be related to some
radiative losses in the chromosphere instead of any saturation in the process.
However, Grinin (1983) demonstrated the effects of radiative losses in the
chromosphere on the white-light photometry of the flares. According to Grinin
(1983), the negative H opacity in the chromosphere causes the radiative
losses, and these are seen as pre-flare dips in the light curves of the white-
light flares.
The other parameters derived from the OPEA model are the flare time-scales.
The maximum flare duration was computed as 4955 s, while the maximum flare
rise time was computed as 1967 s from the detected flares of CR Dra. The
$Half-Life$ parameter was found to be 191.40 s from CR Dra flares. All these
time-scales were found to be in agreement with those found for other stars by
Dal & Evren (2011a, b). It is well known that the decay time-scales of the
X-ray light curves is related to the length of the loop (Haisch, 1983; van den
Oord et al., 1988; van den Oord & Barstow, 1988; Reale et al., 1997; Reale &
Micela, 1998; Reale et al., 1988). According to the results found by Favata et
al. (2005), short flare durations could be an indicator of single loop
geometry, while long flare durations could reveal arcade flares. If the same
case is valid for the white-light flares observed from UV Ceti type stars, the
lengths of the flaring loops on the surface of CR Dra are generally larger
than those on AD Leo, EV Lac, EQ Peg according to the time-scales obtained
from CR Dra. However, the general sizes of the flaring loops are almost equal
to loops occurring on V1005 Ori. In the general perspective, the decreasing of
the observed maximum time-scales (the flare rise time and the flare total
duration) of flares reveals that the lengths of the flaring loop are
decreasing toward the later spectral types among UV Ceti type stars. On the
other hand, the flare mechanism of CR Dra easily reaches the saturation in
shorter time than the stars such as AD Leo, EV Lac, EQ Peg. In this point, CR
Dra and V1005 Ori have a same nature.
It must be noted that it is a debated issue whether the white-light flares can
be affected from the parameters such as $\nu_{A}$, $B$, $R$ and $n_{e}$, which
generally affect the coronal loops and the flare emissions in the corona (Benz
& Güdel, 2010; Hawley & Fisher, 1992; Abbett et al., 1999; Hawley et al.,
2003; Allred et al., 2006). However, the results found in this study reveal
that there are some similarities between the white-light flares and their
counterparts observed in the X-ray or radio bands.
Consequently, the observations of CR Dra have demonstrated that the $Plateau$
values vary in a trend from earlier spectral types to the later. However, the
trend is not in the shape previously offered by Dal & Evren (2011a). Although
it is a debated issue, the shape of the trend in the $Plateau$ variation
versus B-V indexes is in agreement with the trend in the distribution of
$n_{e}$ versus B-V indexes. However, much more stars between the spectral type
dK5e and dM6e should be observed in order to reach more reliable result.
Especially, more stars around CR Dra should be observed.
### 4.2 Distribution of the Flares Numbers versus the Ratio of Flare Decay
Time to Flare Rise Time
The most remarkable result was found in the analyses of the distribution flare
amplitudes versus the ratios of decay times to rise times. As it can be seen
from Figure 4, although the error bars of some flares are a bit large, the
flares are gathering in the specific ratios. According to the distribution
seen in the figure, many flares seem to prefer the value of 0.5 or its
multiples, such as 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, etc., in the ratios
of decay times to rise times. To test whether these accumulations in these
ratios are statistically meaningful, the histograms were derived and analysed
to find statistically the best one. The derived best histograms are shown in
Figures 5, 6 and 7. As seen from the figures, the statistical analyses
indicated that the best width of bars is 0.05 for histograms. The mean error
is found to be $\pm$0.042 for $\tau_{d}/\tau_{r}$ of the flares detected from
CR Dra. Although the error is close to the estimated width value of bars, the
mean error is lower. In addition, when the mean error computed over all 554
flares is $\pm$0.026. In this case, the histograms are statistically
acceptable.
The histograms demonstrated some definite results. First of all, most of the
flares prefer the value of 1.0 or its multiples as positive integers. The
incidence of them over 1672 flares is 23.09$\%$. The incidence of the flares,
which prefer the value of 0.5 or its multiples, such as 1.5, 2.5, 3.5, 4.5,
etc., is 10.83$\%$. The positive integers can be taken as the multiples of the
value 0.5. In this case, the incidence of the flares, whose ratios of the
decay time to rise time are 0.5 and its multiples, reach the value of
33.91$\%$ over 1672 U-band flares. However, as it is stated in the Section
3.2, the incidences are found a bit different from the analysis of each data
set. This must be because of some small differences in the methods used to
compute the flare time-scales, such as decay or rise times. In this project,
all the parameters were computed with considering the quiescent level of the
brightness of a star. These levels were computed from the part of the
observations without any flares or any variations for each night separately.
On the other hand, in the literature, the first points of the flare beginnings
were taken as a quiescent level of the brightness of observed star in some
studies. Unfortunately, the first point of the flares does not always indicate
the quiescent level of the brightness. Taking the first points of the flare
beginnings as a quiescent level comes some difference to computed decay and
rise times. Using the data collected from different studies might cause a bit
different incidences.
In the case of many flares, preferring the ratios as 0.5 and its multiples
might be related with some parameters in the flare event process, which affect
the flare time-scales such as the flare rise or decay times. The flare decay
time ($\tau_{d}$) is firmly correlated with $B$ and $n_{e}$, while the flare
rise time ($\tau_{r}$) is proportional to a larger $\ell$ and smaller $B$
values (Temmer et al., 2001; Reeves & Warren, 2002; Imanishi et al., 2003;
Pandey & Singh, 2008). The magnetic loop length ($\ell$) also depends on
plasma electron density ($n_{e}$) and plasma temperature (Yokoyama & Shibata,
1998; Shibata & Yokoyama, 1999, 2002; Yamamoto et al., 2002). As it is seen,
the preferred ratios of the decay times to rise times is mainly related with
the electron density ($n_{e}$) in the flaring loop and quite a bit with
magnetic field strength ($B$) of the loop. In fact, with some assumptions,
Imanishi et al. (2003) has shown the relations between flare time-scales
(flare decay and rise times) and plasma electron density ($n_{e}$) and the
reconnection factor ($M_{A}$) with Equations (A5) and (A8) given by them.
It is seen from Figures 5, 6 and 7, there are two type flares. One of them is
the flares discussed above, which are shown by filled and open circles in the
figures. Other type flares are shown by small points in the figures, which are
neither the multiples of 1.0 nor the multiples of 0.5. These two types must be
the slow and fast flare types (Gurzadian, 1988). Gurzadian (1988) indicated
that thermal processes are dominant in the processes of slow flares, which are
95$\%$ of all flares observed in UV Ceti type stars. Non-thermal processes are
dominant in the processes of fast flares. In fact, the $\tau_{r}$ and
$\tau_{d}$ parameters of the fast flares can take random values according to
themselves (Gurzadian, 1988; Dal & Evren, 2010). Thus, the small points, whose
ratios are neither the multiples of 1.0 nor the multiples of 0.5, must
indicate the fast flares.
Consequently, the distribution of the number of flares versus the ratio
demonstrates that (1) the number of observed flares gets maximum in the ratio,
which are 0.5 or its multiples, and especially positive integers. As a result
of this, $\tau_{d}$/$\tau_{r}$ is usually equal to 0.5 or its multiples, and
mostly a positive integer. (2) The flare numbers are dramatically decreasing
toward the larger values of the ratio. (3) Some flares do not prefer specific
values of the ratio, thus their ratios of the decay times to rise times are
neither the multiples of 1.0 nor the multiples of 0.5. (4) The thermal
processes might be dominant in the processes of the flares preferring specific
ratios, while non-thermal processes might be dominant in the others.
## Acknowledgments
The author acknowledges the generous observing time awarded to the Ege
University Observatory. I also thank the referee for useful comments that have
contributed to the improvement of the paper.
## References
* Abbett et al. (1999) Abbett, W.P., Hawley, S.L., 1999, ApJ, 521, 906
* Allred et al. (2006) Allred, J.C., Hawley, S.L., Abbett, W.P., Carlsson, M., 2006, ApJ, 644, 484
* Anderson (1979) Anderson, C.M., 1979, PASP, 91, 202
* Antov et al. (1991) Antov, A.P., Genkov, V.V., Konstantinova-Antova, R., Kirov, N.K. 1991, IBVS, 3577, 1
* Avgoloupis et al. (1980) Avgoloupis, S., Phylactopoulos, P., Kareklidis, G., Mavridis, L.N., Varvoglis, P., 1980, IBVS, 1793, 1
* Benz & Güdel (2010) Benz, A.O., Güdel, M., 2010, ARA&A, 48, 241
* Carrington (1859) Carrington, R.C., 1859, MNRAS, 20, 13
* Contadakis et al. (1979) Contadakis, M.E., Kareklidis, G., Mavridis, L.N., Tsioumis, A.C., 1979, IBVS, 1653, 1
* Contadakis et al. (1980) Contadakis, M.E., Mahmoud, F., Mavridis, L.N., Stavridis, D., 1980, IBVS, 1784, 1
* Cristaldi & Rodonó (1970) Cristaldi, S., Rodonó, M., 1970, A&AS, 2, 223
* Cristaldi & Rodonó (1971) Cristaldi, S., Rodonó, M., 1971, IBVS, 600, 1
* Cristaldi & Rodonó (1972) Cristaldi, S., Rodonó, M., 1972, IBVS, 682, 1
* Cristaldi & Rodonó (1973a) Cristaldi, S., Rodonó, M., 1973a, IBVS, 759, 1
* Cristaldi & Rodonó (1973b) Cristaldi, S., Rodonó, M., 1973b, IBVS, 836, 1
* Dal & Evren (2010) Dal, H.A., Evren, S., 2010, AJ, 140, 483
* Dal & Evren (2011a) Dal, H.A., Evren, S., 2011a, AJ, 141, 33
* Dal & Evren (2011b) Dal, H.A., Evren, S., 2011b, PASP, 123, 659
* Dal (2011) Dal, H.A., 2011, PASA, 28, 365
* Dawson & Trapp (2004) Dawson, B., Trapp, R.G., 2004, Basic and Clinical Biostatistics (New York: McGraw-Hill), 61
* Doyle (1996) Doyle, J.G., 1996, A&A, 307, 162
* Favata et al. (2005) Favata, F., Flaccomio, E., Reale, F., Micela, G., Sciortino, S., Shang, H., Stassun, K.G., Feigelson, E.D., 2005, ApJS, 160, 469
* García-Alvarez et al. (2008) García-Alvarez, D., Drake, J.J., Kashyap, V.L., Lin, L., Ball, B., 2008, ApJ, 679, 1509
* Gershberg (1972) Gershberg, R.E., 1972, Ap&SS, 19, 75
* Gershberg (2005) Gershberg, R.E., 2005, Solar-type Activity in Main-sequence Stars (New York: Springer), 53
* Gershberg et al. (1999) Gershberg, R.E., Katsova, M.M., Lovkaya, M.N., Terebizh, A.V. and Shakhovskaya, N.I., 1999, A&AS, 139, 555
* Gershberg & Shakhovskaya (1983) Gershberg, R.E., Shakhovskaya, N.I., 1983, Astrophys. Space Sci., 95, 235
* Green et al. (1999) Green, S.B., Salkind, N.J., Akey, T. M., 1999, Using SPSS for Windows: Analyzing and Understanding Data (Upper Saddle River, NJ: Prentice Hall), 50
* Grinin (1983) Grinin, V.P., 1983, Activity in Red-dwarf Stars, Proc. Seventy-first Colloq. (Astrophys. Space Sci. Libr. 102; Dordrecht: Reidel), 613
* Gurzadian (1988) Gurzadian, G.A., 1988, ApJ, 332, 183
* Jeffries et al. (2011) Jeffries, R.D., Jackson, R.J., Briggs, K.R., Evans, P.A., Pye, J.P., 2011, MNRAS, 411, 2099
* Haisch (1983) Haisch, B.M., 1983, in Activity in Red-Dwarf Stars, eds. P.B. Byrne, M. Rodonó, Reidel, Dordrecht, p.255
* Hardie (1962) Hardie R.H., 1962, in Astronomical Techniques, ed. W. A. Hiltner (Chicago: Univ. Chicago Press), 178
* Hawley & Fisher (1992) Hawley, S.L., Fisher, G.H., 1992, ApJS, 78, 565
* Hawley et al. (2003) Hawley, S.L., Allred, J.C., Johns-Krull, C.M., Fisher, G.H., Abbett, W.P., Alekseev, I., Avgoloupis, S.I., Deustua, S.E., Gunn, A., Seiradakis, J.H., Sirk, M.M., Valenti, J.A., 2003, ApJ, 597, 535
* Herr & Brcich (1969) Herr, R.B., Brcich, J.A., 1969, IBVS, 329, 1
* Herr & Caputo (1988) Herr, R.B., Caputo, F.M., 1988, IBVS, 3243, 1
* Herr & Charache (1988) Herr, R.B., Charache, D.H., 1988, IBVS, 3270, 1
* Herr & Frank (1983) Herr, R.B., Frank, J.D., 1983, IBVS, 2426, 1
* Herr & Opie (1987) Herr, R.B., Opie, D.B., 1987, IBVS, 3069, 1
* Hodgson (1859) Hodgson, R., 1859, MNRAS, 20, 15
* Ichimura & Shimizu (1978) Ichimura, K., Shimizu, Y., 1978, IBVS, 1416, 1
* Imanishi et al. (2003) Imanishi, K., Nakajima, H., Tsujimoto, M., Koyama, K., Tsuboi, Y., 2003, PASJ, 55, 653
* Kapoor et al. (1973) Kapoor, R.C., Sanwal, B.B., Sinvhal, S.D., 1973, IBVS, 810, 1
* Kapoor & Sinvhal (1974) Kapoor, R.C., Sinvhal, S.D., 1974, IBVS, 901, 1
* Katsova et al. (1987) Katsova, M.M., Badalyan, O.G., and Livshits, M.A., 1987, Astron. Zh., 64, 1243 (= Sov. Astron., 31, 652)
* Kay et al. (2003) Kay, H.R.M., Culhane, J.L., Harra, L.K., Matthews, S.A., 2003, AdSpR, 32, 1051
* Kirkup & Frenkel (2006) Kirkup, L., Frenkel, R.B., 2006, An Introduction to Uncertainty in Measurement, Cambridge University Press
* Landolt (1983) Landolt, A.U., 1983, AJ, 88, 439
* Leto et al. (1997) Leto, G., Pagano, I., Buemi, C.S., Rodonó, M., 1997, A&A, 327, 1114
* MacConnell (1968) MacConnell, D.J., 1968, ApJ, 153, 313
* Mahmoud (1991) Mahmoud, F.M., 1991, Ap&SS, 186, 113
* Mahmoud (1993) Mahmoud, F.M., 1993, Ap&SS, 209, 237
* Meištas (2002) Meištas, E.G., 2002, High-Sped Three-Channel Photometer (HSTCP) User’s Guide, To Molétai version (Vilnius, Astronomical Observatory of Vilnius University)
* Monagan et al. (2008) Monagan, M.B., Geddes, K.O., Heal, K.M., Labahn, G., Vorkoetter, S.M., McCarron, J., DeMarco, P., 2008, Maple Introductory Programming Guide (Maplesoft, a division of Waterloo Maple Inc., Canada)
* Moffett (1974) Moffett, T.J., 1974, ApJS, 29, 1
* Moffett (1975) Moffett, T.J., 1975, IBVS, 997, 1
* Motulsky (2007) Motulsky, H., 2007, GraphPad Prism 5: Statistics Guide (San Diego, CA: GraphPad Software Inc. Press), 94
* Nicastro (1975) Nicastro, A.J., 1975, IBVS, 1045, 1
* Osawa et al. (1973) Osawa, K., Ichimura, K., Shimizu, Y., Okada, T., Okida, K., Yutani, M., Koyano, H., 1973, IBVS, 790, 1
* Osawa et al. (1974a) Osawa, K., Ichimura, K., Okada, T., Okida, K., Yutani, M., Koyano, H., 1974a, IBVS, 876, 1
* Osawa et al. (1974b) Osawa, K., Ichimura, K., Shimizu, Y., Koyano, H., 1974b, IBVS, 906, 1
* Pandey & Singh (2008) Pandey, J.C., Singh, K.P., 2008, MNRAS, 387, 1627
* Panov et al. (1983) Panov, K.P., Asteriadis, G., Mavridis, L.N., 1983, IBVS, 2358, 1
* Panov et al. (1982b) Panov, K.P., Grigorova, M., Tsintsarova, A., 1982b, IBVS, 2220, 1
* Panov et al. (1982a) Panov, K.P., Pamukchiev, I., Christov, P., Asteriadis, G., Mavridis, L. N., 1982a, IBVS, 2128, 1
* Panov et al. (1985) Panov, K.P., Piirola, V., Korhonen, T., 1985, IBVS, 2826, 1
* Panov et al. (1988) Panov, K.P., Piirola, V., Korhonen, T., 1988, A&AS, 75, 53
* Panov & Tsvetkov (1981) Panov, K.P., Tsvetkov, M.K., 1981, IBVS, 1971, 1
* Panov et al. (2000) Panov, K., Goranova, Yu., Genkov, V., 2000, IBVS, 4917, 1
* Petit (1957) Petit, M., 1957, SvA, 1783
* Pettersen et al. (1984) Pettersen, B.R., Coleman, L.A., Evans, D.S., 1984, ApJ, 282, 214
* Pettersen et al. (1986) Pettersen, B.R., Panov, K.P., Ivanova, M.S., Sandmann, W.H., 1986, A&AS, 66, 235
* Pettersen & Sundland (1991) Pettersen, B.R., Sundland, S.R., 1991, A&AS, 87, 303
* Reale et al. (1997) Reale, F., Betta, R., Peres, G., Serio, S., McTiernan, J., 1997, A&A, 325, 782
* Reale & Micela (1998) Reale, F., Micela, G., 1998, A&A, 334, 1028
* Reale et al. (1988) Reale, F., Peres, G., Serio, S., Rosner, R., Schimitt, J.H.M.M., 1988, ApJ, 328, 256
* Reeves & Warren (2002) Reeves, K.K., Warren, H.P., 2002, ApJ, 578, 590
* Rodonó (1973) Rodonó, M., 1973, IBVS, 802, 1
* Sanwal (1975) Sanwal, B.B., 1975, IBVS, 998, 1
* Shibata & Yokoyama (1999) Shibata, K., Yokoyama, T., 1999, ApJ, 526, L49
* Shibata & Yokoyama (2002) Shibata, K., Yokoyama, T., 2002, ApJ, 577, 422
* Skumanich & McGregor (1986) Skumanich, A., McGregor, K., 1986, Adv. Space Phys., 6, 151
* Spanier & Oldham (1987) Spanier, J., Oldham, K.B., 1987, An Atlas of Function (Washington, DC: Hemisphere Publishing Corporation Press), 233
* Stauffer & Hartmann (1986) Stauffer, J.R., Hartmann, L.W., 1986, ApJS, 61, 531.
* Tamazian et al. (2008) Tamazian, V.S., Docobo, J.A., Balega, Y.Y., Melikian, N.D., Maximov, A.F., Malogolovets, E.V., 2008, AJ, 136, 974
* Temmer et al. (2001) Temmer, M., Veronig, A., Hanslmeier, A., Otruba, W., and Messerotti, M., 2001, A&A 375, 1049
* Tovmassian et al. (1997) Tovmassian, H.M., Recillas, E., Cardona, O., Zalinian, V.P., 1997, RMxAA, 33, 107
* Tsvetkov et al. (1986b) Tsvetkov, M.K., Antov, A.P., Tsvetkova, A.G., 1986b, IBVS, 2972, 1
* Tsvetkov et al. (1986a) Tsvetkov, M.K., Tsvetkova, K.P., Melikian, N.D., 1986a, IBVS, 2954, 1
* van den Oord & Barstow (1988) van den Oord, G.H.J., Barstow, M.A., 1988, A&A, 207, 89
* van den Oord et al. (1996) van den Oord, G.H.J., Doyle, J.G., Rodonó, M., Gary, D.E., Henry, G.W., Byrne, P.B., Linsky, J.L., Haisch, B.M., Pagano, I., Leto, G., 1996, A&A, 310, 908
* van den Oord et al. (1988) van den Oord, G.H.J., Mewe, R., Brinkman, A.C., 1988, A&A, 205, 181
* Veeder (1974) Veeder, G.J., 1974, AJ, 79, 702V
* Vilhu et al. (1986) Vilhu, O., Neff, J.E., Walter, F.M., 1986, in ESA SP-263, New Insight in Astrophysics, ed. E. J. Rolfe (Noordwijk: ESA), 113
* Wall & Jenkins (2003) Wall, J.W. & Jenkins, C.R., 2003, In Practical Statistics For Astronomers, Cambridge University Press, p.79
* Yamamoto et al. (2002) Yamamoto, T.T., Shiota, D., Sakajiri, T., Akiyama, S., Isobe, H., Shibata, K., 2002, ApJ, 579, L45
* Yokoyama & Shibata (1998) Yokoyama, T., Shibata, K., 1998, ApJ, 494, L113
(155mm,60mm)Figure1.ps
Figure 1: Three samples for the observed flare light curves in U-band. The
flare shown in top panel was observed in April 18, 2007, while flares shown in
middle and bottom panels were detected in May 13, 2007. In the figures, filled
circle represents observations, while dashed line represents the quiescent
level of the brightness determined for each night.
(160mm,60mm)Figure2.ps
Figure 2: Two samples for computing of the flare parameters. In both figures,
the filled-black circles represent observations in the quiescent level of the
brightness. The blue-squares represent the impulsive phase of the flares,
while the red-diamonds represent the main (decay) phases of the flares. The
black lines ($f_{1}(x)$ and $g_{1}(x)$) represent the linear fits of the
observations in the quiescent level, while the blue lines ($f_{2}(x)$ and
$g_{2}(x)$) represent the linear fits of the observations in the impulsive
phase. The red lines ($f_{3}(x)$ and $g_{3}(x)$) represent the polynomial fits
of the decay phases of the flares.
(170mm,60mm)Figure3.ps
Figure 3: Comparison of CD Dra’s The OPEA with other stars. Top panel: The
OPEA model derived from the distributions of flare-equivalent duration on a
logarithmic scale vs. flare total duration for CR Dra. Filled circles
represent equivalent durations computed from flares detected from CR Dra. The
line represents the model identified with Equation (3) computed using the
least-squares method. The dotted lines represent 95$\%$ confidence intervals
for the model. Middle panel: The OPEA model (solid line) of CR Dra is compared
with the models (dotted line) derived from observations on other 6 stars
reported by Dal & Evren (2011a, b). Bottom panel: The $Plateau$ parameters vs.
the B-V index of the CR Dra and other 6 stars taken from Dal & Evren (2011a,
b) are shown. In the panel, open symbols represent the parameters taken from
Dal & Evren (2011a, b), while filled circle represents the parameter of CR
Dra.
(160mm,60mm)Figure4.ps
Figure 4: A sample for the distribution of the flare amplitudes versus the
ratio of flare decay time to flare rise time. Instead of any amplitude
variation in a regular trend, there is a remarkable gathering in some
particular ratios of flare decay time to rise time.
(155mm,60mm)Figure5.ps
Figure 5: The distribution of the number of the flares derived in intervals of
0.05 ratio length versus the ratio of flare decay time to rise time. Top
panel: the distribution is shown for ratios between 0.00 and 58.35. All the
flares seen in this figure were obtained in this project. The value of 58.35
is the largest ratio obtained from the flares detected in this work. Bottom
panel: the distribution is shown for the interval between 0.00 and 10.00 for
clarity. In both panels, the filled circles represent the flares, whose ratios
of flare decay times to rise times are equal to 1.0 or its multiples as the
positive integers. The open circles represent the flares, whose ratio is 0.5
or its multiples, such as 1.5, 2.5, 3.5, 4.5, etc. The small points represent
the flares with other ratios, which are neither the multiples of 1.0 nor the
multiples of 0.5.
(155mm,60mm)Figure6.ps
Figure 6: The distribution of the number of the flares derived in intervals of
0.05 ratio length versus the ratio of flare decay time to rise time. Top
panel: the distribution is shown for all ratios from 0.00 to 206.40. All the
flares seen in this figure were collected from the literature. The value of
206.40 is the largest ratio obtained from these flares. Bottom panel: the
distribution is shown for the interval between 0.00 and 10.00 for clarity. In
both panels, all the symbols used in this figure are the same with the symbols
used in Figure 4.
(155mm,60mm)Figure7.ps
Figure 7: The distribution of the number of the flares derived in intervals of
0.05 ratio length versus the ratio of flare decay time to rise time. Top
panel: the distribution is shown for all ratios from 0.00 to 206.40. The data
plotted in the figure are a combination of data from this work and those taken
from the literature. The value of 2006.40 is seen to be the largest ratio
obtained from all the flares. Bottom panel: the distribution is shown for the
interval between 0.00 and 10.00 for clarity. In both panels, all the symbols
used in this figure are the same with the symbols used in Figure 4.
Table 1: Basic parameters for the targets studied, comparison (C1) and check (C2) stars. Program Stars | V (mag) | B-V (mag)
---|---|---
CR Dra | 9.997 | 1.370
C1: HD 238545 | 10.195 | 0.582
C2: HD 238552 | 10.123 | 1.021
Table 2: All the parameters computed for each flare detected in flare patrols of CR Dra in this study. All the flares were obtained in the U-band. HJD of Flare | Rise | Decay | Total | | Equivalent | Flare | Flare
---|---|---|---|---|---|---|---
Maximum | Time | Time | Duration | $\tau_{d}/\tau_{r}$ | Duration | Amplitude | Energy
(+24 00000) | (s) | (s) | (s) | | (s) | (mag) | (erg)
54208.47638 | 60$\pm$0.5 | 96$\pm$0.6 | 156$\pm$0.8 | 1.60$\pm$0.03 | 20.3403$\pm$0.0007 | 0.188$\pm$0.003 | $\sim$ 9.5898 E+30
54208.48038 | 110$\pm$1.1 | 2090$\pm$2.1 | 2200$\pm$2.4 | 19.00$\pm$0.2 | 183.0304$\pm$0.0024 | 0.518$\pm$0.008 | $\sim$ 2.4743 E+32
54209.53320 | 305$\pm$1.0 | 394$\pm$2.3 | 699$\pm$2.5 | 1.29$\pm$0.06 | 75.2661$\pm$0.0006 | 0.270$\pm$0.004 | $\sim$ 3.5485 E+31
54210.53322 | 204$\pm$0.5 | 341$\pm$1.1 | 545$\pm$1.2 | 1.67$\pm$0.01 | 42.2318$\pm$0.0005 | 0.255$\pm$0.004 | $\sim$ 1.9911 E+31
54210.54189 | 48$\pm$0.6 | 288$\pm$1.5 | 336$\pm$1.6 | 6.00$\pm$0.10 | 33.3097$\pm$0.0012 | 0.299$\pm$0.004 | $\sim$ 1.5704 E+31
54211.32701 | 48$\pm$0.6 | 33$\pm$0.4 | 81$\pm$0.7 | 0.69$\pm$0.02 | 11.6477$\pm$0.0004 | 0.329$\pm$0.005 | $\sim$ 5.4915 E+30
54211.48255 | 120$\pm$0.4 | 1245$\pm$0.6 | 1365$\pm$0.8 | 10.38$\pm$0.04 | 112.4808$\pm$0.0009 | 0.496$\pm$0.007 | $\sim$ 1.0080 E+32
54227.37541 | 15$\pm$0.2 | 75$\pm$0.9 | 90$\pm$0.9 | 5.00$\pm$0.12 | 14.5103$\pm$0.0005 | 0.484$\pm$0.007 | $\sim$ 6.8411 E+30
54227.49538 | 15$\pm$0.2 | 15$\pm$0.2 | 30$\pm$0.3 | 1.00$\pm$0.02 | 5.0416$\pm$0.0002 | 0.422$\pm$0.006 | $\sim$ 2.6598 E+30
54227.50538 | 1044$\pm$1.5 | 1260$\pm$2.1 | 2304$\pm$2.6 | 1.21$\pm$0.01 | 133.2189$\pm$0.0013 | 0.258$\pm$0.004 | $\sim$ 8.1931 E+31
54234.38692 | 364$\pm$0.4 | 741$\pm$0.4 | 1106$\pm$0.5 | 2.03$\pm$0.01 | 27.8203$\pm$0.0010 | 0.277$\pm$0.004 | $\sim$ 1.3116 E+31
54234.50690 | 373$\pm$0.6 | 787$\pm$0.9 | 1160$\pm$1.1 | 2.11$\pm$0.01 | 164.2169$\pm$0.0027 | 0.488$\pm$0.007 | $\sim$ 7.7422 E+31
54235.47382 | 1967$\pm$1.1 | 2988$\pm$1.3 | 4955$\pm$1.7 | 1.52$\pm$0.01 | 145.4049$\pm$0.0031 | 0.503$\pm$0.008 | $\sim$ 1.3288 E+32
54276.37291 | 45$\pm$0.5 | 15$\pm$0.2 | 60$\pm$0.6 | 0.33$\pm$0.01 | 8.5775$\pm$0.0003 | 0.291$\pm$0.004 | $\sim$ 4.0440 E+30
54276.39774 | 90$\pm$1.1 | 210$\pm$2.5 | 300$\pm$2.7 | 2.33$\pm$0.06 | 51.0005$\pm$0.0018 | 0.342$\pm$0.005 | $\sim$ 2.4045 E+31
54276.40208 | 15$\pm$0.2 | 60$\pm$0.7 | 75$\pm$0.7 | 4.00$\pm$0.10 | 10.8760$\pm$0.0004 | 0.230$\pm$0.003 | $\sim$ 5.1277 E+30
54276.40399 | 105$\pm$0.3 | 225$\pm$0.3 | 330$\pm$0.4 | 2.14$\pm$0.01 | 50.4296$\pm$0.0018 | 0.247$\pm$0.004 | $\sim$ 2.3776 E+31
54276.41007 | 120$\pm$0.2 | 255$\pm$0.4 | 375$\pm$0.4 | 2.13$\pm$0.01 | 65.9255$\pm$0.0013 | 0.256$\pm$0.004 | $\sim$ 3.1082 E+31
54276.41927 | 30$\pm$0.2 | 30$\pm$0.2 | 60$\pm$0.2 | 1.00$\pm$0.01 | 12.2865$\pm$0.0004 | 0.465$\pm$0.007 | $\sim$ 5.7927 E+30
54276.43185 | 30$\pm$0.2 | 15$\pm$0.2 | 45$\pm$0.2 | 0.50$\pm$0.01 | 6.6471$\pm$0.0002 | 0.441$\pm$0.007 | $\sim$ 3.1339 E+30
Table 3: Summary of the reference list and the data used in the analyses.
Reference | Observed | Spectral | Number | Reference | Observed | Spectral | Number
---|---|---|---|---|---|---|---
of Data | Star | Type∗ | of Flare | of Data | Star | Type∗ | of Flare
MacConnell (1968) | AD Leo | M4.5Ve | 21 | Panov & Tsvetkov (1981) | EV Lac | M4.5V | 11
MacConnell (1968) | V1054 Oph | M3.5Ve | 11 | Panov et al. (1982a) | EV Lac | M4.5V | 7
Herr & Brcich (1969) | DO Cep | M4.0V | 10 | Panov et al. (1982b) | AD Leo | M4.5Ve | 22
Cristaldi & Rodonó (1970) | AD Leo | M4.5Ve | 2 | Osawa et al. (1973) | AD Leo | M4.5Ve | 14
Cristaldi & Rodonó (1971) | EV Lac | M4.5V | 3 | Panov et al. (1983) | EV Lac | M4.5V | 12
Cristaldi & Rodonó (1972) | AD Leo | M4.5Ve | 1 | Herr & Frank (1983) | AD Leo | M4.5Ve | 6
Cristaldi & Rodonó (1973a) | EV Lac | M4.5V | 3 | Pettersen et al. (1984) | AD Leo | M4.5Ve | 82
Rodonó (1973) | EV Lac | M4.5V | 4 | Panov et al. (1985) | EV Lac | M4.5V | 20
Kapoor et al. (1973) | AD Leo | M4.5Ve | 2 | Pettersen et al. (1986) | AD Leo | M4.5Ve | 99
Cristaldi & Rodonó (1973b) | EV Lac | M4.5V | 1 | Tsvetkov et al. (1986a) | EV Lac | M4.5V | 5
Moffett (1974) | Wolf 424 | M5 | 11 | Tsvetkov et al. (1986b) | EV Lac | M4.5V | 17
Moffett (1974) | UV Cet | M6.0V | 52 | Herr & Opie (1987) | AD Leo | M4.5Ve | 14
Moffett (1974) | EV Lac | M4.5V | 21 | Herr & Caputo (1988) | DO Cep | M4.0V | 5
Moffett (1974) | EQ Peg | M3.5 | 42 | Herr & Charache (1988) | AD Leo | M4.5Ve | 6
Moffett (1974)) | YZ CMi | M4.5V | 51 | Panov et al. (1988) | EV Lac | M4.5V | 22
Moffett (1974) | YY Gem | dM1e | 14 | Pettersen & Sundland (1991) | V577 Mon | M4.5V | 72
Moffett (1974) | AD Leo | M4.5Ve | 8 | Antov et al. (1991) | AD Leo | M4.5Ve | 4
Moffett (1974) | CN Leo | M6.0V | 102 | van den Oord et al. (1996) | YZ CMi | M4.5V | 8
Osawa et al. (1974a) | YZ CMi | M4.5V | 7 | Leto et al. (1997) | EV Lac | M4.5V | 193
Kapoor & Sinvhal (1974) | AD Leo | M4.5Ve | 4 | Tovmassian et al. (1997) | EV Lac | M4.5V | 20
Osawa et al. (1974b) | AD Leo | M4.5Ve | 4 | Panov et al. (2000) | YZ CMi | M4.5V | 7
Moffett (1975) | AD Leo | M4.5Ve | 27 | Dal & Evren (2010) | EV Lac | M4.5V | 98
Sanwal (1975) | YZ CMi | M4.5V | 1 | Dal & Evren (2010) | AD Leo | M4.5Ve | 110
Nicastro (1975) | DO Cep | M4.0V | 22 | Dal & Evren (2010) | V1054 Oph | M3.5Ve | 40
Ichimura & Shimizu (1978) | YZ CMi | M4.5V | 7 | Dal & Evren (2010) | EQ Peg | M3.5 | 73
Contadakis et al. (1979) | EV Lac | M4.5V | 23 | Dal & Evren (2011a) | V1005 Ori | M0Ve | 41
Contadakis et al. (1980) | EV Lac | M4.5V | 13 | Dal & Evren (2011b) | V1285 Aql | M3.0V | 83
Avgoloupis et al. (1980) | EV Lac | M4.5V | 5 | Dal (2011) | DO Cep | M4.0V | 89
∗ All the spectral types were taken from the SIMBAD database.
Table 4: The parameters derived from the OPEA model and the results of the t-Test analyses. Parameter | Value $\pm$Error
---|---
Parameters of The OPEA Function:
$Plateau$ = | 2.146$\pm$0.065
$y_{0}$ = | 0.673$\pm$0.088
$k$ = | 0.003621$\pm$0.000584
$Span$ = | 1.473$\pm$0.097
$Half-Life$ = | 191.40
$r^{2}$ of the model = | 0.94
The t-Test Analysis:
Mean = | 2.163$\pm$0.037
Std. Deviation = | 0.082
Table 5: The values of the best histograms, which were estimated by SPSS V17.0 and Prism V5.02 software. Parameters | In This Project | Literature | Combined
---|---|---|---
| (Figure 5) | (Figure 6) | (Figure 7)
Total Number of Flares = | 554 | 1118 | 1672
Minimum Value of $\tau_{d}$/$\tau_{r}$ = | 0.07 | 0.09 | 0.07
25$\%$ Percentile = | 1.00 | 2.02 | 1.51
Median Value of $\tau_{d}$/$\tau_{r}$ = | 2.00 | 4.79 | 3.50
75$\%$ Percentile = | 4.31 | 11.13 | 8.81
Maximum Value of $\tau_{d}$/$\tau_{r}$ = | 58.33 | 206.39 | 206.39
Mean Value of $\tau_{d}$/$\tau_{r}$ = | 4.16 | 9.35 | 7.63
Std. Deviation = | 6.03 | 14.01 | 12.21
Std. Error = | 0.26 | 0.42 | 0.30
Lower 95$\%$ CI of mean = | 3.66 | 8.53 | 7.05
Upper 95$\%$ CI of mean = | 4.66 | 10.18 | 8.22
|
arxiv-papers
| 2012-09-12T19:33:00 |
2024-09-04T02:49:35.032266
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1209.2697"
}
|
1209.2872
|
# Artificial ferroelectricity due to anomalous Hall effect in magnetic tunnel
junctions
A. Vedyayev [email protected] Department of Physics, Moscow Lomonosov State
University, Moscow 119991, Russia SPINTEC, UMR 8191 CEA-INAC/CNRS/UJF-
Grenoble 1/Grenoble-INP, 38054 Grenoble, France N. Ryzhanova Department of
Physics, Moscow Lomonosov State University, Moscow 119991, Russia SPINTEC,
UMR 8191 CEA-INAC/CNRS/UJF-Grenoble 1/Grenoble-INP, 38054 Grenoble, France N.
Strelkov Department of Physics, Moscow Lomonosov State University, Moscow
119991, Russia SPINTEC, UMR 8191 CEA-INAC/CNRS/UJF-Grenoble 1/Grenoble-INP,
38054 Grenoble, France B. Dieny [email protected] SPINTEC, UMR 8191 CEA-
INAC/CNRS/UJF-Grenoble 1/Grenoble-INP, 38054 Grenoble, France
###### Abstract
We theoretically investigated Anomalous Hall Effect (AHE) and Spin Hall Effect
(SHE) transversally to the insulating spacer O, in magnetic tunnel junctions
of the form F/O/F where F are ferromagnetic layers and O represents a tunnel
barrier. We considered the case of purely ballistic (quantum mechanical)
transport, taking into account the assymetric scattering due to spin-orbit
interaction in the tunnel barrier. AHE and SHE in the considered case have a
surface nature due to proximity effect. Their amplitude is in first order of
the scattering potential. This contrasts with ferromagnetic metals wherein
these effect are in second (side-jump scattering) and third (skew scattering)
order on these potentials. The value of AHE voltage in insulating spacer may
be much larger than in metallic ferromagnetic electrodes. For the antiparallel
orientation of the magnetizations in the two F-electrodes, a spontaneous Hall
voltage exists even at zero applied voltage. Therefore an insulating spacer
sandwiched between two ferromagnetic layers can be considered as exhibiting a
spontaneous ferroelectricity.
Hall effect, anomalous Hall effect, spin Hall effect, spintronics,
ferroelectrics
The Anomalous Hall Effect (AHE) in ferromagnetic metals and Spin Hall Effect
(SHE) in nonmagnetic materials have attracted a renewed interest in the last
decades. One can notice that AHE and SHE have the same origin, namely spin-
orbit interaction in the presence of magnetic ordering for AHE and without
magnetic ordering for SHE. Detailed analyses of the mechanisms responsible for
these two effects may be found in reviews Wölfle and Muttalib (2006); Nagaosa
(2006); Nagaosa et al. (2010). These mechanisms are divided into two groups:
intrinsic ones and extrinsic ones. The former appear in pure metals and have
topological nature, closely connected with Berry curvature. Extrinsic
mechanisms are due to asymmetric electron scattering on defects in presence of
spin-orbit interaction. Two main types of scattering are considered: skew
scattering Smit (1955, 1973) and side-jump scattering Berger (1970). Most of
theoretical papers on AHE and SHE considered the case of infinite homogeneous
samples. References Ryzhanova et al. (1998); Granovsky et al. (1994) also
investigated AHE for multilayers and for highly inhomogeneous media.
Figure 1: Schematic illustration of MTJ. F - ferromagnetic layers, O -
insulating spacer. Arrows denote the direction of magnetizations in
electrodes, for parallel (P) and antiparallel (AP) orientations. Inset
schematically illustrates the dependence of density of states ($\nu$ \- in
arbitrary units) of spin up tunnelling electrons on the distance from the
interface for P and AP orientations.
Let’s consider a magnetic tunnel junctions (i.e. a sandwich of two
ferromagnetic layers separated by a dielectric spacer, MTJ – Magnetic Tunnel
Junction) (Fig.1) submitted to a bias voltage applied between the two
F-electrodes supposed to be made of the same ferromagnetic material. In this
study, we are primarily interested by the Hall voltage which may appear
between the opposite sides of the tunnel barrier due to the Hall current
inside the spacer in presence of spin-orbit scattering on impurities. We will
show that these Hall and spin Hall current do exist and that moreover, for the
antiparallel orientation of the magnetizations in the two ferromagnetic
layers, a spontaneous transverse Hall voltage exists, even in the absence of
any applied bias voltage.
The Hall currents were calculated using Keldysh formalism Keldysh (1965). The
electrons were described as forming a free electron gas submitted to $s$-$d$
exchange interaction. As an example, the Green functions for the considered
system (Fig.1) and for $z$-projection of electron’s spin antiparallel to the
magnetization in the left electrode are:
$G^{\uparrow}_{\mathrm{AP},x>x^{\prime}}(r,r^{\prime})=\frac{1}{N}\sum_{\varkappa}-\frac{1}{2q\mathfrak{D}}{\mathrm{e}}^{i\varkappa_{i}(y-y^{\prime})}{\mathrm{e}}^{i\varkappa_{z}(z-z^{\prime})}\\\
\times\Bigl{(}{\mathrm{e}}^{q(x-x_{2})}(q+ik_{2})+{\mathrm{e}}^{-q(x-x_{2})}(q-ik_{2})\Bigr{)}\\\
\times\Bigl{(}{\mathrm{e}}^{q(x^{\prime}-x_{1})}(q-ik_{1})+{\mathrm{e}}^{-q(x^{\prime}-x_{2})}(q+ik_{1})\Bigr{)},$
(1)
$G^{\uparrow}_{\mathrm{AP},x<x^{\prime}}(r,r^{\prime})=\frac{1}{N}\sum_{\varkappa}-\frac{1}{2q\mathfrak{D}}{\mathrm{e}}^{i\varkappa_{i}(y-y^{\prime})}{\mathrm{e}}^{i\varkappa_{z}(z-z^{\prime})}\\\
\times\Bigl{(}{\mathrm{e}}^{q(x-x_{1})}(q-ik_{1})+{\mathrm{e}}^{-q(x-x_{2})}(q+ik_{1})\Bigr{)}\\\
\times\Bigl{(}{\mathrm{e}}^{q(x^{\prime}-x_{2})}(q+ik_{2})+{\mathrm{e}}^{-q(x^{\prime}-x_{2})}(q-ik_{2})\Bigr{)},$
(2)
where:
$\mathfrak{D}=\left({\mathrm{e}}^{qb}(q-ik_{1})(q-ik_{2})-{\mathrm{e}}^{-qb}(q+ik_{1})(q+ik_{2})\right),$
$q=\sqrt{\frac{2m}{\hbar^{2}}(U-E)+\varkappa^{2}},$ (3)
$k_{1,2}=\sqrt{\frac{2m}{\hbar^{2}}(E\pm J_{sd})-\varkappa^{2}}.$
In (1) and (2) “AP” means the antiparallel orientation of magnetizations in
the two ferromagnetic electrodes and $x_{1}$, $x_{2}$ – are the
ferromagnetic/insulator interfaces coordinates. In (3), $U$ is the barrier
height, $E$ is the electron energy, $J_{sd}$ is $s$-$d$ exchange energy. For
the opposite direction of spin, all projection changes in (1) and (2) are
straightforward. From (1) and (2), it follows that for the considered system,
a finite density of states exists in the energy gap within the barrier due to
proximity effect, which decreases exponentially with the distance from F/O
interfaces (Fig.1). In other words, a quasi-two-dimensional electron gas
exists inside the barrier near the interfaces. Similarly to three dimensional
topological insulator, this electron gas can give birth to charge and spin
currents Hasan and Kane (2010).
Figure 2: Schematic illustration of AHE and SHE in MTJ due to spin-orbit
scattering on impurities. $\otimes$ and $\odot$ denote the direction of
magnetizations and electrons spins. The thickness of lines are proportional to
Hall currents for the given projection of spin.
Evidently the mechanisms of creation of these surface states are different in
the two cases. Let’s suppose now that the tunnelling electrons experience
scattering on impurities within the barrier with spin-orbit interaction. This
asymmetric scattering deviates the electrons in the direction perpendicular to
the tunnel current and to the projection of their spin. So if the current is
spin-polarized, a Hall voltage appears transversally to the tunnel barrier.
Quite interestingly, in antiparallel magnetic configuration of the MTJ, this
AHE appears spontaneously even in the absence of bias voltage applied across
the tunnel barrier. In addition, as illustrated in Fig.2, if the two
ferromagnetic materials were assumed to have different spin-polarization, a
spontaneous spin unbalance (spin Hall effect) would also appear between the
two transversal sides of the tunnel barrier at zero bias voltage. This would
even be true if one of the ferromagnetic electrode was replaced by a non-
magnetic metallic electrode.
To investigate this effect we added into the free electron Hamiltonian, the
impurity potential including spin-orbit interaction and calculated the induced
perturbation to the wave functions:
$\psi=\psi_{0}(r)+\int G(r,r^{\prime})V_{so}(r^{\prime})\,d^{3}r^{\prime}=\\\
\psi_{0}(r)+\int\delta(r^{\prime}-r_{i})(a_{0}^{5}\lambda_{0})\,d^{3}r^{\prime}\\\
\times\left[G(r,r^{\prime})i\sigma_{z}\left(\frac{\overleftarrow{\partial}}{\partial
x^{\prime}}\frac{\overrightarrow{\partial}}{\partial
y^{\prime}}-\frac{\overleftarrow{\partial}}{\partial
y^{\prime}}\frac{\overrightarrow{\partial}}{\partial
x^{\prime}}\right)\psi_{0}(r^{\prime})\right].$ (4)
In (4) $\lambda_{0}$ represents the intensity of spin-orbit interaction,
$a_{0}$ – lattice parameter, $r_{i}$ – position of the impurity, $\sigma_{z}$
– $z$-component of Pauli matrix. Zero order wave function for the left-to-
right and right-to-left tunnelling electrons are correspondently:
$\psi^{\uparrow}_{\mathrm{AP},l}=\frac{2\sqrt{k_{1}}}{\mathfrak{D}}\\\
\times\left({\mathrm{e}}^{q(x-x_{2})}(q+ik_{2})+{\mathrm{e}}^{-q(x-x_{2})}(q-ik_{2})\right),$
(5) $\psi^{\uparrow}_{\mathrm{AP},r}=\frac{2\sqrt{k_{2}}}{\mathfrak{D}}\\\
\times\left({\mathrm{e}}^{q(x-x_{1})}(q-ik_{1})+{\mathrm{e}}^{-q(x-x_{1})}(q+ik_{1})\right).$
(6)
Now it is easy to calculate the Hall current in ballistic regime in the first
order on spin-orbit interaction:
$j^{\sigma}_{H}=\frac{e}{2\pi\hbar}\int\frac{f(E)}{(2\pi)^{2}}\,dE\\\
\times\int i\sigma_{z}\left(\psi_{l}^{\sigma}\frac{\partial}{\partial
y}\psi_{l}^{\sigma*}-\psi_{l}^{\sigma*}\frac{\partial}{\partial
y}\psi_{l}^{\sigma}\right)^{(1)}\,d\varkappa_{y}d\varkappa_{z}\\\
+\frac{e}{2\pi\hbar}\int\frac{f(E+eV)}{(2\pi)^{2}}\,dE\\\ \times\int
i\sigma_{z}\left(\psi_{r}^{\sigma}\frac{\partial}{\partial
y}\psi_{r}^{\sigma*}-\psi_{r}^{\sigma*}\frac{\partial}{\partial
y}\psi_{r}^{\sigma}\right)^{(1)}\,d\varkappa_{y}d\varkappa_{z},$ (7)
where $f(E)$ – Fermi distribution for the left electrode, $f(E+eV)$ – the same
for the right one, $V$ – applied voltage. Subscript “$(\dots)^{(1)}$” denotes
the first order terms on spin-orbit interaction in the expression in brackets.
Substituting (4), (5) and (6) into (7) and averaging on the position of
impurities $r_{i}$ yields the following expressions for the spin-up all
current originating respectively from left ($l$) and right ($r$) electrodes in
AP configuration:
$j^{\uparrow}_{\mathrm{AP},l}\sim\int
d\varkappa_{y}d\varkappa_{z}dE\frac{4\lambda_{0}\varkappa_{y}^{2}k_{1}}{\left|\mathfrak{D}\right|^{4}}f(E)\\\
\times\Biggl{[}\left({\mathrm{e}}^{2q(x-x_{2})}{\mathrm{e}}^{-2qb}-{\mathrm{e}}^{-2q(x-x_{2})}{\mathrm{e}}^{2qb}\right)(q^{2}+k_{2}^{2})^{2}\\\
+2\left({\mathrm{e}}^{2qb}-{\mathrm{e}}^{-2qb}\right)(q^{2}-k_{2}^{2})(k_{1}^{2}-k_{2}^{2})\\\
+\left({\mathrm{e}}^{2q(x-x_{1})}-{\mathrm{e}}^{-2q(x-x_{1})}\right)\\\
\times(q^{2}+k_{2}^{2})\left((q^{2}+k_{1}^{2})+(k_{1}^{2}-k_{2}^{2})\right)\\\
-2\left({\mathrm{e}}^{2q(x-x_{2})}-{\mathrm{e}}^{-2q(x-x_{2})}\right)\\\
\times\left((q^{2}-k_{1}k_{2})^{2}-q^{2}(k_{1}+k_{2})^{2}\right)\Biggr{]},$
(8) $j^{\uparrow}_{\mathrm{AP},r}\sim\int
d\varkappa_{y}d\varkappa_{z}dE\frac{4\lambda_{0}\varkappa_{y}^{2}k_{2}}{\left|\mathfrak{D}\right|^{4}}f(E+eV)\\\
\times\Biggl{[}\left({\mathrm{e}}^{2q(x-x_{1})}{\mathrm{e}}^{2qb}-{\mathrm{e}}^{-2q(x-x_{1})}{\mathrm{e}}^{-2qb}\right)(q^{2}+k_{1}^{2})^{2}\\\
+2\left({\mathrm{e}}^{2qb}-{\mathrm{e}}^{-2qb}\right)(q^{2}-k_{1}^{2})(k_{1}^{2}-k_{2}^{2})\\\
+\left({\mathrm{e}}^{2q(x-x_{2})}-{\mathrm{e}}^{-2q(x-x_{2})}\right)\\\
\times(q^{2}+k_{1}^{2})\left((q^{2}+k_{2}^{2})-(k_{1}^{2}-k_{2}^{2})\right)\\\
-2\left({\mathrm{e}}^{2q(x-x_{1})}-{\mathrm{e}}^{-2q(x-x_{1})}\right)\\\
\times\left((q^{2}-k_{1}k_{2})^{2}-q^{2}(k_{1}+k_{2})^{2}\right)\Biggr{]}.$
(9)
From (8) and (9), it follows that the Hall current exponentially depends on
the coordinate $x$ and reaches its maximum near the “left” interface for the
“left” electrons and at the “right” interface for “right” electrons. This
emphasizes the surface nature of the considered Hall effect.
It’s interesting to note that the present Hall effect in MTJ barrier appears
at first order on the scattering potential, whereas for infinite ferromagnetic
metals the Hall effect is in third order on the scattering potential for skew
scattering and in second order for side jump mechanism Wölfle and Muttalib
(2006); Nagaosa (2006); Nagaosa et al. (2010). This difference is due to the
strong inhomogeneity of the considered system in $x$-direction. The other
remarkable difference already pointed out is that this Hall effect
spontaneously exists even at zero bias voltage in MTJ.
Next the obtained expressions for Hall currents and spin Hall currents were
averaged over the coordinate $x$ and integration over momentum
$\vec{\varkappa}$ and energy $E$ yields in the limit ${\mathrm{e}}^{-2qb}\ll
1$, in parallel configuration of the MTJ:
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{AHE}}^{\mathrm{P,skew}}=\frac{4}{15\pi}\,\frac{e^{2}}{2\pi\hbar}\,\frac{\tilde{\lambda}c}{U^{2}b}\left(E_{F}^{\uparrow
2}k_{F}^{\uparrow}-E_{F}^{\downarrow 2}k_{F}^{\downarrow}\right)V,$
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{SHE}}^{\mathrm{P,skew}}=\frac{4}{15\pi}\,\frac{e^{2}}{2\pi\hbar}\,\frac{\tilde{\lambda}c}{U^{2}b}\left(E_{F}^{\uparrow
2}k_{F}^{\uparrow}+E_{F}^{\downarrow 2}k_{F}^{\downarrow}\right)V,$
and in antiparallel configuration:
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{AHE}}^{\mathrm{AP,skew}}=\frac{8}{105\pi}\,\frac{e}{2\pi\hbar}\,\frac{\tilde{\lambda}c}{U^{2}b}\Biggl{(}E_{F}^{\uparrow
3}k_{F}^{\uparrow}-E_{F}^{\downarrow 3}k_{F}^{\downarrow}\\\
+\left(E_{F}^{\uparrow}+eV\right)^{3}\sqrt{k_{F}^{\uparrow
2}+eV}-\left(E_{F}^{\downarrow}+eV\right)^{3}\sqrt{k_{F}^{\downarrow
2}+eV}\Biggr{)},$ (10)
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{SHE}}^{\mathrm{AP,skew}}=\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{SHE}}^{\mathrm{P,skew}},$
where $\tilde{\lambda}=2ma_{0}^{2}\lambda_{0}/\hbar^{2}$ – dimensionless
constant of spin-orbit interaction, $c$ – atomic concentration of impurities.
One may notice that in contrast to the tunnelling current through the tunnel
barrier, the expressions of the Hall and spin Hall currents do not contain the
small parameter ${\mathrm{e}}^{-2qb}$. Instead, the averaged Hall voltage
decreases inversely proportional to the barrier thickness. Its amplitude is
proportional to the small parameter $\lambda_{0}$ related to the intensity of
the spin-orbit interaction. The absence of ${\mathrm{e}}^{-2qb}$ in the
expression for $j_{H}$ further indicates that this predicted Hall and spin
Hall effects have a surface nature in contrast to the tunnelling current.
Up to now, the case of “skew” scattering was considered. In addition to this
scattering mechanism, another contribution to Hall and spin Hall currents
originates from another term in the operator of quantum mechanical velocity,
proportional to spin-orbit interaction:
$\hat{v}=\frac{d}{dt}\vec{r}=-i[\vec{r}\times\vec{H}]=\frac{\hbar\vec{k}}{m}+\lambda[\vec{\sigma}\times\vec{\nabla}V(\vec{r})],$
(11)
where $V(\vec{r})$ – potential of impurity, $\lambda$ – spin-orbit constant.
This additional contribution to the Hall current is equivalent to a “side
jump” mechanism Wölfle and Muttalib (2006). In the present case it is written
in final form as:
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{AHE}}^{\mathrm{P,sj}}=\frac{2}{3\pi}\,\frac{e^{2}}{2\pi\hbar}\,\frac{\tilde{\lambda}c}{Ub}\left(E_{F}^{\uparrow}k_{F}^{\uparrow}-E_{F}^{\downarrow}k_{F}^{\downarrow}\right)V,$
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{SHE}}^{\mathrm{P,sj}}=\frac{2}{3\pi}\,\frac{e^{2}}{2\pi\hbar}\,\frac{\tilde{\lambda}c}{Ub}\left(E_{F}^{\uparrow}k_{F}^{\uparrow}+E_{F}^{\downarrow}k_{F}^{\downarrow}\right)V,$
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{AHE}}^{\mathrm{AP,sj}}=\frac{4}{15\pi}\,\frac{e}{2\pi\hbar}\,\frac{\tilde{\lambda}c}{Ub}\Bigl{(}E_{F}^{\uparrow
2}k_{F}^{\uparrow}-E_{F}^{\downarrow 2}k_{F}^{\downarrow}\\\
+\left(E_{F}^{\uparrow 2}+eV\right)^{2}\sqrt{k_{F}^{\uparrow
2}+eV}-\left(E_{F}^{\downarrow 2}+eV\right)^{2}\sqrt{k_{F}^{\downarrow
2}+eV}\Bigr{)},$ (12)
$\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{SHE}}^{\mathrm{AP,sj}}=\left<j_{l+r}^{\uparrow+\downarrow}\right>_{\mathrm{SHE}}^{\mathrm{P,sj}}$
First of all, we note that both contributions into the Hall and spin Hall
currents are proportional to the concentration of impurities. This contrasts
to the usual Hall conductivity in ferromagnetic metals which is inversely
proportional to this concentration for the skew scattering and does not depend
on concentration for the side jump mechanism. However in the present case,
Hall current in metallic ferromagnetic electrodes is proportional to the
current in this electrode, itself proportional to the small parameter
${\mathrm{e}}^{-2qb}$. Therefore, for thick enough insulating spacer, Hall and
spin Hall effects inside the spacer may become much larger than the
corresponding effects within the ferromagnetic electrodes.
To find the Hall voltage $V_{H}$, we divided the expressions for Hall current
by conductance in $y$-direction:
$G=\frac{e^{2}}{2\pi\hbar}\frac{1}{b}\sqrt{\frac{2m}{\hbar^{2}}U}\\\
\times\left[\left(1-\sqrt{1-\frac{E_{F}^{\uparrow}}{U}}\right)+\left(1-\sqrt{1-\frac{E_{F}^{\downarrow}}{U}}\right)\right].$
(13)
Estimated value of $V_{H}$ is $(10^{-5}$ to $10^{-3})V$, for $\tilde{\lambda}$
in interval $(10^{-2}$ to $10^{-1})$ and $c$ in interval $(0.01$ to $0.1)$.
The most interesting conclusion is the existence of a Hall voltage for AP-
configuration even in absence of any applied voltage. It means that an
insulating spacer sandwiched between two ferromagnetic electrodes in AP-
configuration exhibits a spontaneous electric polarization i.e. a spontaneous
transverse ferroelectricity due to proximity effect. The latter results from
the asymmetric scattering on spin-orbit impurities of tunnelling electrons
penetrating into the insulating barrier from the ferromagnetic electrodes.
To experimentally measure this effect, one possibility would be to make
electrically isolated metallic islands aside of the tunnel barrier. These
islands would get charged by electrostatic influence with the charges arising
on the side walls of the tunnel barrier. Measuring the voltage between these
islands and the MTJ electrodes in parallel and antiparallel magnetic
configuration with an electrometer could allow to detect and measure this new
phenomenon of spontaneous transverse ferroelectricity in MTJ.
Acknowledgements: This work was partly funded by the European Commission
through the ERC HYMAGINE grant no ERC 246942.
## References
* Wölfle and Muttalib (2006) P. Wölfle and K. Muttalib, Annalen der Physik 15, 508 (2006).
* Nagaosa (2006) N. Nagaosa, Journal of the Physical Society of Japan 75, 042001 (2006).
* Nagaosa et al. (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010), URL http://link.aps.org/doi/10.1103/RevModPhys.82.1539.
* Smit (1955) J. Smit, Physica 21, 877 (1955).
* Smit (1973) J. Smit, Phys. Rev. B 8, 2349 (1973), URL http://link.aps.org/doi/10.1103/PhysRevB.8.2349.
* Berger (1970) L. Berger, Phys. Rev. B 2, 4559 (1970), URL http://link.aps.org/doi/10.1103/PhysRevB.2.4559.
* Ryzhanova et al. (1998) N. Ryzhanova, A. Vedyayev, A. Crépieux, and C. Lacroix, Phys. Rev. B 57, 2943 (1998), URL http://link.aps.org/doi/10.1103/PhysRevB.57.2943.
* Granovsky et al. (1994) A. Granovsky, A. Vedyayev, and F. Brouers, JMMM 136, 229 (1994).
* Keldysh (1965) L. V. Keldysh, JETP 20, 1018 (1965).
* Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010), URL http://link.aps.org/doi/10.1103/RevModPhys.82.3045.
|
arxiv-papers
| 2012-09-13T12:38:40 |
2024-09-04T02:49:35.048017
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Vedyayev and N. Ryzhanova and N. Strelkov and B. Dieny",
"submitter": "Anatoly Vedyayev",
"url": "https://arxiv.org/abs/1209.2872"
}
|
1209.2960
|
# Higher dimensional charged $f(R)$ black holes
Ahmad Sheykhi [email protected] Center for Excellence in Astronomy and
Astrophysics (CEAA-RIAAM) Maragha, P. O. Box 55134-441, Iran
Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454,
Iran
###### Abstract
We construct a new class of higher dimensional black hole solutions of $f(R)$
theory coupled to a nonlinear Maxwell field. In deriving these solutions the
traceless property of the energy-momentum tensor of the matter filed plays a
crucial role. In $n$-dimensional spacetime the energy-momentum tensor of
conformally invariant Maxwell field is traceless provided we take $n=4p$,
where $p$ is the power of conformally invariant Maxwell lagrangian. These
black hole solutions are similar to higher dimensional Reissner-Nordstrom AdS
black holes but only exist for dimensions which are multiples of four. We
calculate the conserved and thermodynamic quantities of these black holes and
check the validity of the first law of black hole thermodynamics by computing
a Smarr-type formula for the total mass of the solutions. Finally, we study
the local stability of the solutions and find that there is indeed a phase
transition for higher dimensional $f(R)$ black holes with conformally
invariant Maxwell source.
PACS numbers: 04.70. Bw, 04.70. Dy.
## I Introduction
A large volume of observational evidences indicate that our universe is now
experiencing a phase of accelerated expansion. There are two main approaches
for explanation of this cosmic acceleration. The first approach which try to
explain the problem in the framework of general relativity, requires the
existence of a strange type of energy called “dark energy” whose gravity is
repulsive and consist an un-clustered component through the universe. The
second approach is to modify general relativity by adding higher powers of the
scalar curvature $R$, the Riemann and Ricci tensors, or their derivatives in
the lagrangian formulation. Among the latter attempts are Lovelock gravity,
braneworld cosmology, scalar-tensor theories like Brans-Dicke one and also
$f(R)$ theories. A fairly comprehensive review on dark energy models can be
seen in Cop . $f(R)$ theories were proved to be able to mimic the whole
cosmological history, from inflation to the actual accelerated expansion era
Odin ; Capo (see also Feli for a recent review on $f(R)$ theories). Diverse
applications of $f(R)$ theories on gravitation and cosmology have been also
widely studied Feli , as well as multiple ways to observationally and
experimentally distinguish them from general relativity.
When one considers $f(R)$ theory as a modification of general relativity, it
is quite natural to ask about black hole existence and its features in this
theory. It is expected that some signatures of black holes in $f(R)$ theory
may differ from the expected physical results in Einstein’s gravity.
Therefore, the investigation on $f(R)$ black holes is of particular interest.
Some attempts have been done to construct black hole solutions in $f(R)$
theories (see Cong ; Seb ; Ade ; Hendi1 and references therein). In general,
in the presence of a matter field, the field equations of $f(R)$ gravity are
complicated and it is not easy to find exact analytical solutions. However, if
one considers the traceless energy-momentum tensor for the matter field, one
can extract exact analytical solutions from $R+f(R)$ theory coupled to a
matter field Tae . Since the energy-momentum tensor of Maxwell and Yang-Mills
fields are traceless only in four dimensions, therefore black hole solutions
from $R+f(R)$ theory coupled to the matter field were derived only in four
dimensions Tae . The studies were also generalized to the four dimensional
charged rotating black holes Alex , charged rotating black string shey1 and
magnetic string solutions shey2 in $R+f(R)$-Maxwell theory. However, since
the standard Maxwell energy-momentum tensor is not traceless in higher
dimensions, they failed to derive higher dimensional black hole/string
solutions from $R+f(R)$ gravity coupled to standard Maxwell field. A natural
question then arises: Is there an extension of Maxwell action in arbitrary
dimensions that is traceless and hence possesses the conformal invariance? The
answer is positive and the conformally invariant Maxwell action was presented
as Hass1 ,
$\displaystyle S_{m}=-\int{d^{n}x\sqrt{-g}(F_{\mu\nu}F^{\mu\nu})^{p}},$ (1)
where $p$ is a positive integer, i.e., $p\in\mathbb{N}$. The associated
energy-momentum tensor of the above conformally invariant Maxwell action is
given by
$T_{\mu\nu}=2\left(pF_{\mu\eta}F_{\nu}^{\text{
}\eta}F^{p-1}-\frac{1}{4}g_{\mu\nu}F^{p}\right),$ (2)
where $F=F_{\alpha\beta}F^{\alpha\beta}$ is the Maxwell invariant. One can
easily check that the above energy momentum tensor is traceless for $n=4p$.
The theory of conformally invariant Maxwell field is considerably richer than
that of the linear standard Maxwell field and in the special case $(p=1)$ it
recovers the Maxwell action. It is worthwhile to investigate the effects of
exponent $p$ on the behavior of the solutions and the laws of black hole
mechanics. The motivation is to take advantage of the conformal symmetry to
construct the analogues of the four dimensional Reissner-Nordstrom (RN)
solutions, in higher dimensions. Recently, the studies on the black object
solutions with a nonlinear Maxwell source in Einstein Hass1 ; Hass2 ; Hendi2
and Gauss-Bonnet Mis gravity have got a lot of attentions.
In this work we would like to extend the investigation on the conformally
invariant Maxwell field to $f(R)$ gravity. We will consider the action (1) as
the matter source of the field equations in $R+f(R)$ theory with constant
curvature scalar. Our purpose is to find the analogues of the four-dimensional
charged black hole solutions of $R+f(R)$-Maxwell theory Tae in higher
dimensional spacetime. In contrast to the higher dimensional black holes of
Einstein gravity with a conformally invariant Maxwell source presented in
Hass1 which has vanishing scalar curvature $R=0$, the spacetime we construct
here in $R+f(R)$ gravity coupled to a nonlinear Maxwell field has a constant
scalar curvature $R=R_{0}$. Our solutions also differ from higher dimensional
RN solutions in that the electric charge term in the metric coefficient goes
as $r^{-(n-2)}$ while in the standard RN case is $r^{-2(n-3)}$. Also, the
electric field in higher dimensions does not depend on $n$ and goes as
electric field in four dimensions.
This paper is outlined as follows. In Sec. II, we construct exact spherically
symmetric black hole solutions of $R+f(R)$ theory coupled to a nonlinear
Maxwell field in $n=4p$ dimensions and investigate their properties. In Sec.
III, we obtain the conserved and thermodynamic quantities of the solutions and
verify the validity of the first law of black hole thermodynamics. We also
study local stability of the solutions in this section. We summarize our
results in Sec. IV.
## II Field Equations and solutions
We consider the action of $R+f(R)$ gravity in $n$-dimensional spacetime
coupled to a conformally invariant Maxwell field
$\displaystyle
S=\int_{\mathcal{M}}d^{n}x\sqrt{-g}\left[R+f(R)-(F_{\mu\nu}F^{\mu\nu})^{p}\
\right],$ (3)
where ${R}$ is the scalar curvature, $f(R)$ is an arbitrary function of scalar
curvature, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor and $A_{\mu}$ is the electromagnetic potential.
The equations of motion can be obtained by varying action (3) with respect to
the gravitational field $g_{\mu\nu}$ and the gauge field $A_{\mu}$,
${R}_{\mu\nu}\left(1+f^{\prime}(R)\right)-\frac{1}{2}g_{\mu\nu}(R+f(R))+\left(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu}\right)f^{\prime}(R)=T_{\mu\nu},$
(4) $\partial_{\mu}(\sqrt{-g}F^{\mu\nu}F^{p-1})=0,$ (5)
where $F=F_{\alpha\beta}F^{\alpha\beta}$ is the Maxwell invariant and the
“prime” denotes differentiation with respect to $R$. In order to obtain the
constant curvature black hole solution in $f(R)$ gravity theory coupled to a
matter field, the trace of stress-energy tensor $T_{\mu\nu}$ should be zero
Tae . Hence, two candidates for the matter field in four dimensions are
Maxwell and Yang-Mills fields. Since the assumption of traceless energy-
momentum tensor is essential for deriving exact black hole solutions in $f(R)$
gravity coupled to the matter field, therefore the solutions exist only for
$n=4p$ dimensions. Assuming the constant scalar curvature $R=R_{0}=const.$,
then the trace of Eq. (4) yields
${R}_{0}\left(1+f^{\prime}(R_{0})\right)-\frac{n}{2}(R_{0}+f(R_{0}))=0.$ (6)
Solving the above equation for $R_{0}$, gives
${R}_{0}=\frac{nf(R_{0})}{2f^{\prime}(R_{0})+2-n}\equiv\frac{2n}{n-2}{\Lambda_{\rm
f}}<0.$ (7)
Substituting the above relation into Eq. (4), we obtain the following equation
for Ricci tensor
${R}_{\mu\nu}(1+f^{\prime}(R_{0}))-\frac{g_{\mu\nu}}{n}R_{0}\left(1+f^{\prime}(R_{0})\right)=T_{\mu\nu}.$
(8)
We are looking for the $n$-dimensional static spherically symmetric solutions.
Motivated by the metric of higher dimensional charged black holes in Einstein
gravity, we assume the metric has the following form
$d{s}^{2}=-N(r)dt^{2}+\frac{dr^{2}}{N(r)}+{r^{2}}d\Omega_{n-2}^{2},$ (9)
where $d\Omega_{n-2}^{2}$ denotes the metric of an unit $(n-2)$-sphere and
$N(r)$ is a functions of $r$ which should be determined. We are seeking for a
purely radial electric solution which means that the only non-vanishing
component of the Maxwell tensor is $F_{tr}$. In this case, the Maxwell
equations (5) can be integrated immediately, where, for the spherically
symmetric spacetime (9), all the components of ${F}_{\mu\nu}$ are zero except
${F}_{tr}$:
${F}_{tr}=\frac{q}{r^{\frac{n-2}{2p-1}}},$ (10)
where $q$ is an integration constant. Substituting $n=4p$ in the above
relation, the Maxwell field becomes
${F}_{tr}=\frac{q}{r^{2}}.$ (11)
It is important to note that the electric field in higher dimensions does not
depend on $n$ and its value coincides with the RN solution in four dimensions.
Using metric (9) and the Maxwell field (11), one can show that Eq. (8) has a
solution of the form
$N(r)=1-\frac{2m}{r^{n-3}}+\frac{q^{2}}{r^{n-2}}\times\frac{(-2q^{2})^{(n-4)/4}}{\left(1+f^{\prime}(R_{0})\right)}-\frac{R_{0}}{n(n-1)}{r}^{2},$
(12)
where $m$ is an integration constant which is related to the mass of the
solution. In four dimension ($n=4$) the solution recovers the result of Tae .
In order to have a real solution we should restrict ourself to the dimensions
which are multiples of four, i.e., $n=4,8,12,....$, which means that $p$
should be only positive integer, as we mentioned already.
Next we study the physical properties of the solutions. To do this, we first
look for the curvature singularities. A simple calculation shows that the
Kretschmann scalar $R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$ diverges
at $r=0$, it is finite for $r\neq 0$ and is proportional to $R_{0}^{2}$ as
$r\rightarrow\infty$. Therefore, there is a curvature singularity located at
$r=0$. As one can see from Eq. (12), the solution is ill-defined for
$f^{\prime}(R_{0})=-1$. Let us consider the cases with $f^{\prime}(R_{0})>-1$
and $f^{\prime}(R_{0})<-1$ separately. In the first case where
$f^{\prime}(R_{0})>-1$, we have a black hole solution (see Fig. 1). Indeed,
for $1+f^{\prime}(R_{0})>0$ and $R_{0}<0$ black hole can have two inner and
outer horizons, an extreme black hole or a naked singularity provided the
parameters of the solutions are chosen suitably (see Fig. 2). In the latter
case where $f^{\prime}(R_{0})<-1$, we encounter a cosmological horizon for
$R_{0}<0$. Indeed, in this case the signature of the spacetime changes and the
conserved quantities such as mass become negative, as we will see in the next
section, thus this is not a physical case and we rule it out from our
consideration.
It is apparent that the spacetime described by solution (12) is asymptotically
AdS provided we define $R_{0}=-n(n-1)/l^{2}$. However, the solution presented
here differ from the standard higher dimensional Reissner-Nordstrom AdS
(RNAdS) solutions since the electric charge term in the metric coefficient
goes as $r^{-(n-2)}$ while in the standard RNAdS case is $r^{-2(n-3)}$. In
four dimensions, with the following replacement
$\displaystyle\frac{\alpha q^{2}}{\left(1+f^{\prime}(R_{0})\right)}\rightarrow
Q^{2}$ (13) $\displaystyle{R_{0}}\rightarrow 4\Lambda$ (14)
the solution reduces to standard RNAdS black holes for $\Lambda=-3/l^{2}$.
Figure 1: The function $N(r)$ versus $r$ for $m=2$, $n=4$, $q=1$ and
$R_{0}=-12$. $f^{\prime}(R_{0})=-0.5$ (bold line) and $f^{\prime}(R_{0})=-2$
(dashed line).
Figure 2: The function $N(r)$ versus $r$ for $m=2$, $q=1$, $n=4$ and
$R_{0}=-12$. $f^{\prime}(R_{0})=-0.2$ (bold line), $f^{\prime}(R_{0})=-0.54$
(continuous line) and $f^{\prime}(R_{0})=-0.8$ (dashed line).
We use the subtraction method of Brown and York (BY) BY to calculate the
quasilocal mass of the charged $f(R)$ black hole. Such a procedure causes the
resulting physical quantities to depend on the choice of reference background.
In order to use the BY method one should write the metric in the following
form
$ds^{2}=-W(r)dt^{2}+\frac{d{r}^{2}}{V({r})}+{r}^{2}d\Omega_{n-2}^{2}.$ (15)
Since metric (9) has the above form, it is sufficient to choose the background
metric to be the metric (15) with
$W_{0}({r})=V_{0}({r})=N_{0}(r)=1+\frac{r^{2}}{l^{2}}$ (16)
Where we have defined $R_{0}=-n(n-1)/l^{2}$ which show that the solutions are
asymptotically AdS as we mentioned. It is well-known that the Ricci scalar for
AdS spacetime should have this value (see e.g. Weinberg ). To compute the
conserved mass of the spacetime, we choose a timelike Killing vector field
$\xi$ on the boundary surface ${\cal B}$ of the spacetime (15). Then the
quasilocal conserved mass can be written as
${\cal M}=\frac{1}{8\pi}\int_{{\cal
B}}d^{n-2}x\sqrt{\sigma}\left\\{\left(K_{ab}-Kh_{ab}\right)-\left(K_{ab}^{0}-K^{0}h_{ab}^{0}\right)\right\\}n^{a}\xi^{b},$
(17)
where $\sigma$ is the determinant of the metric of the boundary ${\cal B}$,
$K_{ab}^{0}$ is the extrinsic curvature of the background metric and $n^{a}$
is the timelike unit normal vector to the boundary ${\cal B}$. In the context
of counterterm method, the limit in which the boundary ${\cal B}$ becomes
infinite (${\cal B}_{\infty}$) is taken, and the counterterm prescription
ensures that the action and conserved charges are finite. Thus, we obtain the
mass through the use of the above subtraction method of BY as
${M}=\frac{(n-2)\Omega_{n-2}}{8\pi}m\left[1+f^{\prime}(R_{0})\right],$ (18)
where $\Omega_{n-2}$ is the volume of the unit $(n-2)$-sphere. In the limiting
case ($f^{\prime}(R_{0})=0$), this expression for the mass reduces to the mass
of the $n$-dimensional AdS black hole.
## III Thermodynamics of Charged $f(R)$ Black holes
In this section we are going to explore thermodynamics of higher dimensional
charged $f(R)$ black holes. The Hawking temperature of the black holes can be
easily obtained by requiring the absence of conical singularity at the horizon
in the Euclidean sector of the black hole solutions. One obtains the
associated temperature with the outer event horizon $r=r_{+}$ as
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left(\frac{dN(r)}{dr}\right)_{r=r_{+}}=\frac{[1+f^{\prime}(R_{0})]\left(2r^{2}(n-1)+2l^{2}(n-3)\right)+(-2q^{2})^{n/4}r^{2-n}l^{2}}{\pi
l^{2}r[1+f^{\prime}(R_{0})]}.$ (19)
where we have used equation $N(r_{+})=0$ for omitting the mass parameter $m$
from temperature expression. Next, we calculate the entropy of the black hole.
Let us first give a brief discussion regarding the entropy of the black hole
in $f(R)$ gravity. To this aim, we follow the arguments presented in Brevik .
If one use the Noether charge method for evaluating the entropy associated
with black hole solutions in $f(R)$ theory with constant curvature, one finds
Cong
${S}=\frac{A}{4G}f^{\prime}(R_{0}),$ (20)
where $A=4\pi r_{+}^{2}$ is the horizon area. As a result, in $f(R)$ gravity,
the entropy does not obey the area law and one obtains a modification of the
area law. Inspired by the above argument, for the $n$-dimensional charged
black hole solutions in $R+f(R)$ gravity, we find the entropy as
${S}=\frac{r_{+}^{n-2}\Omega_{n-2}}{4}\left[1+f^{\prime}(R_{0})\right].$ (21)
The charge of conformally invariant $f(R)$ black holes can be found by
calculating the flux of the electric field at infinity, yielding
${Q}=\frac{n(-2)^{(n-4)/4}q^{(n-2)/2}\Omega_{n-2}}{16\pi\sqrt{1+f^{\prime}(R_{0})}}.$
(22)
The electric potential $\Phi$, measured at infinity with respect to the
horizon, is defined by
$\Phi=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(23)
where $\chi=\partial_{t}$ is the null generator of the horizon. We find
$\Phi=\frac{q}{r_{+}}\sqrt{1+f^{\prime}(R_{0})}.$ (24)
Then, we investigate the validity of the first law of thermodynamics for
higher dimensional charged $f(R)$ black hole. For this purpose, we obtain a
Smarr-type formula, namely the mass $M$ as a function of extensive quantities
$S$, and $Q$. Using the expression for the mass, the entropy and the charge
given in Eqs. (18), (21) and (22) and the fact that $N(r_{+})=0$, we find
$\displaystyle M(S,Q)$ $\displaystyle=$ $\displaystyle\frac{n-2}{16\pi
l^{2}}\left(\frac{4S}{1+f^{\prime}(R_{0})}\right)^{-1/(n-2)}\left[[1+f^{\prime}(R_{0})]\left(\frac{4S}{1+f^{\prime}(R_{0})}\right)\times\right.$
(25)
$\displaystyle\left.\left(l^{2}+\left(\frac{4S}{1+f^{\prime}(R_{0})}\right)^{2/(n-2)}\right)+l^{2}q^{2}(-2q^{2})^{(n-4)/4}\right].$
where $q$ is a function of $Q$ according to Eq. (22). One may then regard the
parameters $S$, and $Q$ as a complete set of extensive parameters for the mass
$M(S,Q)$ and define the intensive parameters conjugate to $S$ and $Q$. These
quantities are the temperature and the electric potential
$T=\left(\frac{\partial M}{\partial S}\right)_{Q},\ \
\Phi=\left(\frac{\partial M}{\partial Q}\right)_{S}.$ (26)
Numerical calculations show that the intensive quantities calculated by Eq.
(26) coincide with Eqs. (19) and (24). Thus, the thermodynamics quantities we
obtained in this section satisfy the first law of black hole thermodynamics
$dM=TdS+\Phi d{Q}.$ (27)
In this way we constructed all conserved and thermodynamic quantities of
charged $f(R)$ black holes and verified the validity of the first law of
thermodynamics on the event horizon.
Figure 3: The function $(\partial^{2}M/\partial S^{2})_{Q}$ versus $q$ for
$l=1$, $n=4$, $r_{+}=0.8$. $f^{\prime}(R_{0})=-0.5$ (bold line),
$f^{\prime}(R_{0})=0$ (continuous line) and $f^{\prime}(R_{0})=0.5$ (dashed
line).
Figure 4: The function $(\partial^{2}M/\partial S^{2})_{Q}$ versus $r_{+}$ for
$l=1$, $f^{\prime}(R_{0})=1$ and $n=4$. $q=0.4$ (bold line), $q=1$ (continuous
line) and $q=2$ (dashed line).
Figure 5: The function $(\partial^{2}M/\partial S^{2})_{Q}$ versus $q$ for
$l=1$, $r_{+}=0.8$ and $f^{\prime}(R_{0})=1$. $n=4$ (bold line), $n=8$
(continuous line) and $n=12$ (dashed line).
Figure 6: The function $(\partial^{2}M/\partial S^{2})_{Q}$ versus $q$ for
$l=1$, $n=8$ and $r_{+}=0.8$. $f^{\prime}(R_{0})=-0.5$ (bold line),
$f^{\prime}(R_{0})=0$ (continuous line) and $f^{\prime}(R_{0})=0.5$ (dashed
line).
Figure 7: The function $(\partial^{2}M/\partial S^{2})_{Q}$ versus $q$ for
$l=1$, $n=12$ and $r_{+}=0.8$. $f^{\prime}(R_{0})=-0.5$ (bold line),
$f^{\prime}(R_{0})=0$ (continuous line) and $f^{\prime}(R_{0})=0.5$ (dashed
line).
Finally, we study the local stability of the charged $R+f(R)$ black holes in
the presence of conformally invariant Maxwell source. In the canonical
ensemble, the positivity of the heat capacity $C_{Q}=T/(\partial^{2}M/\partial
S^{2})_{Q}$ and therefore the positivity of $(\partial^{2}M/\partial
S^{2})_{Q}$ is sufficient to ensure the local stability. We have shown the
behavior of $(\partial^{2}M/\partial S^{2})_{Q}$ as a function $q$ and $r_{+}$
for different value of $n$, $q$ and $f^{\prime}(R_{0})$ in figures 3-7 where
we have taken $R_{0}=-n(n-1)/l^{2}$. From figures 3 and 4 we see that the
system is always thermally stable in four dimensions for different value of
the parameters. However, in higher dimensions the system has an unstable phase
as one can see from Fig. 5. As an example, we find that in $8$-dimensions
black holes are always unstable and there is no phase transition (see Fig. 6),
while in $12$-dimensions the system has a transition from unstable phase to
stable phase (see Fig. 7).
## IV Summary and Discussion
In general the field equations of $f(R)$ gravity coupled to a matter field are
complicated and it is not easy to construct exact analytical solutions.
Recently, it was shown Tae that by assuming a traceless energy-momentum
tensor for the matter field as well as the constant curvature scalar, one can
extract some analytical black hole solutions in $R+f(R)$ theory coupled to a
matter field. Two examples for the traceless $T_{\mu\nu}$ in four dimensions
are Maxwell and Yang-Mills fields Tae . However, the energy-momentum tensor of
Maxwell field is not traceless in higher dimensions. Seeking for a traceless
energy-momentum tensor in arbitrary dimensions, the authors of Hass1 found a
conformally invariant nonlinear Maxwell action which its energy-momentum
tensor is traceless in $n=4p$ dimensions where $p$ is a positive integer. They
also studied the black hole solutions in Einstein gravity with nonlinear
Maxwell field Hass1 .
In this paper we obtained a class of higher dimensional black holes from
$R+f(R)$ gravity with conformally invariant Maxwell source. The two key
assumptions in finding these solutions are: (i) the constant scalar curvature
$R=R_{0}$ and (ii) the traceless energy momentum tensor. These solutions are
similar to higher dimensional RNAdS black holes with appropriate replacement
of the parameters, but only exist in dimensions which are multiples of four.
Besides, the solutions presented here differ from higher dimensional RNAdS
black holes in two features. First, the electric charge term in the metric
coefficient goes as $r^{-(n-2)}$ while in the standard RNAdS case is
$r^{-2(n-3)}$. Second, the electric field in higher dimensions does not depend
on $n$ and goes as electric field in four dimensional RNAdS black holes. Our
solutions also differ from the higher dimensional black holes of Einstein
gravity with a conformally invariant Maxwell source Hass1 in that they have
vanishing scalar curvature $R=0$, while the obtained solutions here in
$R+f(R)$ gravity coupled to a nonlinear Maxwell field have a constant
curvature scalar $R=R_{0}$. In addition, the conserved and thermodynamic
quantities computed here depend on function $f^{\prime}(R_{0})$ and differ
completely from those of Einstein theory in AdS spaces. Clearly the presence
of the general function $f^{\prime}(R_{0})$ changes the physical values of
conserved and thermodynamic quantities. Furthermore, unlike Einstein gravity,
for the black hole solutions obtained here in $f(R)$ gravity, the entropy does
not obey the area law.
After studying the physical properties of the solutions, we computed the mass,
charge, electric potential and temperature of the black holes. We also found
the entropy expression which does not obey the area law for the $f(R)$ black
holes. We obtained a Smarr-type formula for the mass, and verified that the
conserved and thermodynamics quantities satisfy the first law of black hole
thermodynamics. We also studied the phase behavior of the higher dimensional
charged $f(R)$ black holes. We found that the system is always thermally
stable in four dimensions, while in higher dimensions, there is a phase
transition in the presence of the conformally invariant Maxwell field in
$R+f(R)$ theory with constant curvature.
Finally, we would like to mention that the $n$-dimensional charged $f(R)$
black hole solutions obtained here are static. Thus, it would be interesting
if one can construct charged rotating black brane/hole solutions from $R+f(R)$
theory in the presence of conformally invariant Maxwell source. These issues
are now under investigation and will be appeared elsewhere.
###### Acknowledgements.
This work has been supported financially by Center for Excellence in Astronomy
and Astrophysics (CEAA-RIAAM), Maragha, under research project number 1/2718.
## References
* (1) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006);
M. Li, X. D. Li, S. Wang, Y. Wang, Commun. Theor. Phys. 56, (2011) 525.
* (2) S. Nojiri and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003);
S. Nojiri and S. D. Odintsov, Phys. Lett. B 576, 5 (2003);
S. Nojiri and S. D. Odintsov, Phys. Rev. D 74, 086005 (2006);
S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007);
S. Nojiri and S. D. Odintsov, J. Phys. Conf. Ser. 66, 012005 (2007);
S. Nojiri and S. D. Odintsov, Phys. Rept. 505, 59 (2011);
S. Capozziello, S. Nojiri, S. D. Odintsov and A. Troisi, Phys. Lett. B 639,
135 (2006);
S. Nojiri and S. D. Odintsov, Phys. Rev. D 78, 046006 (2008).
* (3) S. Capozziello, Int. J. Mod. Phys. D 11, 483, (2002);
S. Capozziello, V. F. Cardone, and A. Troisi, J. Cosmol. Astropart. Phys. 08,
001 (2006);
S. Capozziello, V. F. Cardone, and A. Troisi, Mon. Not. R. Astron. Soc. 375,
1423 (2007);
K. Atazadeh, M. Farhoudi and H. R. Sepangi, Phys. Lett. B 660, 275 (2008);
C. Corda, Astropart. Phys. 34, 587 (2011).
* (4) A. De Felice and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010);
S. Capozziello and M. De Laurentis, arXiv:1108.6266, to appear in Physics
Reports.
* (5) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, JCAP 0502, 010 (2005).
* (6) L. Sebastiani and S. Zerbini, arXiv:1012.5230.
* (7) A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. D 80, 124011 (2009);
A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto. Phys Rev. D 83, 029903 (E)
(2011).
* (8) S. H. Hendi, Phys. Lett. B 690, 220 (2010);
S. H. Hendi, et. al., Gen Relativ Gravit 44 (2012) 835;
S. H. Hendi, and D. Momeni, Eur. Phys. J. C 71 (2011) 1823.
* (9) T. Moon, Y. S. Myung and E. J. Son, arXiv:1101.1153, to appear in Gen Relativ Grav.
* (10) A. Larranaga, arXiv:1108.6325;
J. A. R. Cembranos, et al., arXiv:1109.4519
* (11) A. Sheykhi and S. Salarpour, Submitted to Phys. Scrip.
* (12) A. Sheykhi, Submitted to Gen. Relativ Grav.
* (13) M. Hassaine and C. Martinez, Phys. Rev. D 75 (2007) 027502\.
* (14) M. Hassaine and C. Martinez, Class. Quantum Gravit. 25 (2008) 195023;
H. A. Gonzalez, M. Hassaine and C. Martinez, Phys. Rev. D 80, 104008 (2009);
H. Maeda, M. Hassaine and C. Martinez, Phys. Rev. D 79 (2009) 044012.
* (15) S. H. Hendi and B. Eslam Panah, Phys. Lett. B 684 (2010) 77;
S. H. Hendi, Eur. Phys. J. C 69 (2010) 281;
S. H. Hendi, Phys. Rev. D 82 (2010) 064040;
S. H. Hendi, Phys. Lett. B 677 (2009) 123.
* (16) O. Miskovic, R. Olea, Phys. Rev. D 83 (2001) 024011;
O. Miskovic, R. Olea, Phys. Rev. D 83 (2011) 064017\.
* (17) J. Brown and J. York, Phys. Rev. D 47, 1407 (1993);
J.D. Brown, J. Creighton, and R. B. Mann, Phys. Rev. D 50, 6394 (1994).
* (18) S. Weinberg, Gravitation and cosmology, (John Wiley 1972, Chapter 15).
* (19) I. Brevik, S. Nojiri, S.D. Odintsov and L. Vanzo, Phys. Rev. D 70, 043520 (2004).
|
arxiv-papers
| 2012-09-13T16:56:36 |
2024-09-04T02:49:35.053712
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1209.2960"
}
|
1209.2969
|
# The Extended Solar Cycle Tracked High into the Corona.
S.J. Tappin111National Solar Observatory, Sacramento Peak, Sunspot, NM, USA,
email: [email protected], R.C. Altrock222Air Force Research Laboratory, Space
Weather Center of Excellence, National Solar Observatory, Sacramento Peak,
Sunspot, NM, USA, email: [email protected]
###### Abstract
We present observations of the extended solar cycle activity in white-light
coronagraphs, and compare them with the more familiar features seen in the Fe
xiv green-line corona. We show that the coronal activity zones seen in the
emission corona can be tracked high into the corona. The peak latitude of the
activity, which occurs near solar maximum, is found to be very similar at all
heights. But we find that the equatorward drift of the activity zones is
faster at greater heights, and that during the declining phase of the solar
cycle, the lower branch of activity (that associated with the current cycle)
disappears at about 3$R_{\odot}$. This implies that that during the declining
phase of the cycle, the solar wind detected near Earth is likely to be
dominated by the next cycle. The so-called “rush to the poles” is also seen in
the higher corona. In the higher corona it is found to start at a similar time
but at lower latitudes than in the green-line corona. The structure is found
to be similar to that of the equatorward drift.
## 1 Introduction
The existence of an “extended solar cycle” in which features related to the
next cycle appear at high latitudes around or soon after the maximum of the
current cycle has been known on the solar surface and in the low corona for
many years (e.g. Labonte and Howard, 1982; Wilson et al., 1988; Altrock,
1997). Typically this activity is observed in the form of ephemeral active
regions, torsional waves and coronal streamers observed in the Fe xiv green
line. Having appeared near solar maximum, these structures then drift
equatorwards mirroring the “butterfly” pattern of sunspots and active regions
at lower latitudes in the current cycle. Following solar minimum, the sunspots
and active regions reappear at the same latitudes as this descending coronal
and rotational structure. At this time a zone of coronal activity detaches
from the main zone and moves rapidly poleward, joining the next extended cycle
around solar maximum, this is described as the “rush to the poles”
(Waldemeier, 1964), (see also Altrock (2011) for examples).
To date, studies of the extended cycle and of the rush to the poles have been
confined to the solar surface and the low corona. But now that observations
from the LASCO coronagraphs on SOHO (Brueckner et al., 1995) are available for
more than a complete solar cycle it is possible to carry out an investigation
of how these structures propagate into the higher corona.
In this paper we combine the Fe xiv green-line and white-light coronagraph
observations to trace the evolution of the zones of activity in latitude as a
function of height above the solar limb and phase of the solar cycle.
## 2 The Observations
In this paper we make use of five coronagraph datasets to track activity in
the corona from 1.15$R_{\odot}$ to 20$R_{\odot}$ (N.B. in this paper, all
heights are measured from Sun centre).
### 2.1 NSO Green Line
The Fe xiv 530.3 nm green line coronagraph observations made at the National
Solar Observatory at Sacramento Peak (NSO/SP), have been discussed in some
detail by Altrock (1997). In this study we have used those data from 1986 to
the end of 2011. In summary: On each day with sufficiently good observing
conditions a scan of the corona is made at a distance of 1.15$R_{\odot}$ from
disk centre with an angular resolution of 3∘ in position angle, and a scan
width of 1.1 arcmin.
### 2.2 MLSO Polarised White Light (Mk3 and Mk4 Coronameters)
The Mauna Loa Solar Observatory (MLSO) has been making white-light polarised
brightness images since 1980. The Mk3 coronameter (Fisher et al., 1981) was
operated from 1980 to 1999, and the Mk4 (Elmore et al., 2003) from 1998 to the
present. For both instruments a linear detector array is scanned around the
corona to build up an image in position angle and radial distance. For the Mk3
instrument a few scans were made each day (during the interval considered in
this paper there are typically two or three images on the MLSO data download
site) while for the Mk4 scans are taken every three minutes through the 5-h
observing day. These scans are published at an angular resolution of 0.5∘ in
position angle by 10 arcsec in the radial direction for Mk3, and 5 arcsec for
Mk4. Both instruments cover similar regions of the corona, 1.12$R_{\odot}$ to
2.44$R_{\odot}$ for Mk3 and 1.12$R_{\odot}$ to 2.86$R_{\odot}$ for Mk4. For
this study we have used the daily average images for all available days from
1986 to 2011 (for 2009 and 2010, where many daily average images are missing,
we have downloaded the full dataset and generated daily average images from
the individual scans). During the interval of overlap we use the Mk4 image if
both are available. At the time of writing the most recent data available were
for 19 September 2011.
### 2.3 LASCO White Light (C2 and C3)
The LASCO coronagraphs on SOHO (Brueckner et al., 1995), have been making
regular images of the corona since late 1995. For this study we use the C2
coronagraph which covers the range from about 2$R_{\odot}$ to 6$R_{\odot}$,
and the C3 coronagraph which covers the range from about 4$R_{\odot}$ to
30$R_{\odot}$. For C2 the images are taken using an orange filter with a
cadence of about 20 min for most of the mission. C3 has a cadence of around 30
min and uses a clear filter for its primary synoptic sequence.At the time of
writing the most recent data available were for 1 April 2011.
## 3 Data Processing
The first step in the data processing was to reduce the data to a common
coordinate system and resolution. For this we adopted the resolution of the
NSO Fe xiv dataset, which has the lowest resolution in position angle of the
five instruments.
For the MLSO data which were already in polar coordinates, all that needed to
be done was to average together position angle bins to reduce the resolution
from 0.5∘ to 3∘, and average radially over an annulus approximately 70arcsec
wide ($\pm 35$arcsec from the nominal radius).
Unlike MLSO where daily (i.e. 5 h) average images are available, LASCO data
are only available as individual frames. Therefore to improve the signal to
noise and also to make any time-smearing effects comparable to those in the
MLSO data, we first read a full day of images and made exposure corrections by
the method described by Tappin, Simnett and Lyons (1999). We then generated
four 6-hour average images for each day.
Since LASCO uses a camera with a rectangular CCD rather than a rotating linear
detector, the raw images are in cartesian coordinates. We therefore needed to
convert the image to polar coordinates. To do this, the field of view was
divided into radial bins of width equal to four pixels in the original data
(47.6 arcsec for C2 and 224 arcsec for C3) and into 3∘ sectors in position
angle. For each resulting cell, the pixels whose centres lie in the cell were
determined and the count rates in those pixels were averaged.
Observatory | Heights
---|---
NSO/SP | 1.15
MLSO (both) | 1.15, 1.20, 1.25, 1.30, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.2, 2.35
LASCO (C2) | 2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0
LASCO (C3) | 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 12.0, 14.0, 16.0, 18.0, 20.0
Table 1: The heights of the scans used in this study. All distances are in
units of $R_{\odot}$ from the centre of the Sun. The heights which are shown
in Figure 2 are indicated by bold font.
We then selected a number of radii at which to extract circumferential
profiles covering the range from 1.15 to 2.35$R_{\odot}$ for the MLSO data,
2.5 to 4.0$R_{\odot}$ for LASCO C2 and 5 to 20$R_{\odot}$ for LASCO C3. The
full list of heights used is shown in Table 1. It should be noted that prior
to July 1997, the majority of LASCO images were not full-field. As a result
the C2 scans above about 3.5$R_{\odot}$ and C3 scans above about 18$R_{\odot}$
have a lower signal-to-noise over the poles prior to that date.
Because the different position angles in the LASCO images were recorded with
different parts of the detector, we found that variations in the detector dark
current resulted in certain latitudes at which persistent maxima and minima
occurred. Since these instrumental features make it difficult to see the
maxima due to solar wind structures, it was necessary to find a way to remove
them. We have found that an effective method is to divide the data into one
year blocks so as to have a long enough base, but also to handle the
variations in the CCD properties with time. For each year, we determine the
fifth percentile of data at each location in the scans, and subtract the
resulting value from all the scans for that year. This is chosen in preference
to using an annual minimum as there are occasional corrupt images which are
likely to dominate the minimum.
In 2004, the Z-axis drive of SOHO’s high-gain antenna failed. Since that time
the spacecraft has performed a 180∘ roll manoeuvre four times a year to keep
Earth in the antenna beam. This was corrected after the CCD pattern removal to
align the scans correctly.
For all the datasets we then followed Altrock (2003) and defined a significant
maximum as one in which the brightness at a position angle was a local
maximum, and also its immediate neighbours exceeded the next pair of position
angles (i.e. for position angle index $i$ to be considered a significant
maximum then
$b_{i-2}<b_{i-1}<b_{i}>b_{i+1}>b_{i+2}$ (1)
must be satisfied; $b_{i}$ denotes the observed intensity at position angle
index $i$). As well as excluding noise, this criterion also eliminates most
maxima due to stars and planets (even Venus in LASCO C3 does not extend over
15∘). This method does not distinguish bright and faint maxima, but rather
gives us a measure of where streamers are present. To adjust the observation
dates of the two limbs to central meridian, maxima over the East limb were
assigned a date seven days later than the date of observation, while those
over the West limb were given a date seven days earlier. We then convert the
position angles of the maxima to latitude (on the assumption that the features
lie above the limb). For LASCO we excluded the quadrant containing the
occulter support pylon as this region is vignetted by the pylon thus
compromising our ability to detect maxima. The maxima for all four (or three
for LASCO) quadrants were then combined to give a list of maxima against
absolute latitude. We consider that the gain in signal to noise thus obtained
outweighs any loss of information about North-South asymmetries (c.f. Altrock,
2011). All the quantitative fits presented in this paper are made by fitting
to this list of maxima.
For the purposes of displaying the data we then counted the number of maxima
that were detected in each latitude bin for each Bartels (27 day) rotation,
and divided by the number of scans or images in the rotation to give a
probability of a maximum at that latitude. To facilitate visualisation we
smooth the counts of maxima using a nine rotation running mean, which was
chosen as giving the best apparent balance between signal to noise and
smearing.
## 4 Results
Figure 1: The evolution of the zones of activity from 1986 to 2012\. The
panels show (from top to bottom) the NSO green line data at 1.15$R_{\odot}$,
the MLSO white-light data at 1.15$R_{\odot}$, the MLSO data at
2.35$R_{\odot}$, and the smoothed sunspot number. A single contour of the NSO
green line map is shown in yellow over each panel to facilitate comparison.
N.B. all heights are measured from the centre of the Sun.
In Figure 1, we show the NSO green line activity map and two of those for the
MLSO white light polarised brightness for the interval 1986 to 2011 (solar
cycles 22 and 23). These maps show the number of maxima as a function of
latitude and date. We remind the reader that this does not show the amplitude
of the maxima, only the number. All three maps show a similar structure with
two overlapping cycles drifting towards the equator, and also the rush to the
poles. The most important feature to notice here is that the NSO map at
1.15$R_{\odot}$ and the MLSO map at the same height match very closely
although the latter is considerably noisier. Since the K-coronal intensity is
a function of density only, while the green line emission is a function of
both density and temperature, the close match shows that the activity zones
must be primarily density rather than temperature features. The MLSO map at
2.35$R_{\odot}$, shows a similar structure to the maps at 1.15$R_{\odot}$, but
as can be seen by comparing it with the overlaid contour of the green line
activity it is very clear that the movement of the activity towards the
equator during the declining phase of both cycles is much faster than at
1.15$R_{\odot}$.
For a more quantitative analysis of the variation of the latitude of the
activity zones with height, we concentrate on the interval for which we also
have LASCO data, namely 1996-2010 which corresponds to solar cycle 23. A
selection of these datasets are shown in Figure 2.
Figure 2: The evolution of the zones of activity for selected heights in the
corona, from 1996 to 2011 (Fexiv in green MLSO in blue, LASCO C2 in red, LASCO
C3 in grey, respectively). For each height, a single contour from the NSO
green line map (yellow) is superposed to facilitate comparison. Also
superposed on each map is an estimate of the gradient of the drift towards the
equator (black and white dashed lines), and of the “rush to the poles” (black
and white dotted lines, black on white for cycle 23, white on black for cycle
24). The rush to the poles for cycle 24 is only shown for those heights at
which it can be clearly discerned. The persistent maxima seen below 20∘
latitude in the LASCO C3 plots, the final four panels with heights of
6$R_{\odot}$ and above, is an effect of the F-corona which we have not been
able to eliminate. The date of solar maximum is indicated by the cyan and
black dashed line overlayed on the NSO map (upper left panel).
It should be noted that the constant band of “activity” below 20∘ latitude in
LASCO C3 is due to the F-corona which dominates at these altitudes. Even
though the CCD irregularity removal does also reduce this contribution, the
small changes in apparent width and inclination of the F-corona as the Earth
orbits the Sun still produce a dominant contribution to the number of detected
maxima. Unfortunately this means that we cannot track the activity bands below
about 20∘ latitude in C3.
Figure 3: The process of fitting the trends to the drift. The example here is
for LASCO C2 at 3.0$R_{\odot}$. (a) The activity map and (b) The raw maxima
(N.B. in the figure a random number between -1 and +1 is added to the latitude
of each point to separate the individual points and allow the reader to see
the variation of point density more easily). In each case the region of the
equatorward drift is outlined with a medium weight box, and the fit is shown
with a heavy line. The fit is generated by a linear regression to all the
points within the region-of-interest box.
For the purposes of this study, we have considered only the interval from 2001
to the start of 2008, where (a) the equatorward drift is approximately linear
in time and (b) it can be clearly seen in all the instruments at all heights
(we therefore do not consider the abrupt acceleration of the drift from about
the start of 2009, which is also the time at which the cycle 24 rush to the
poles separated from the equatorward drift). To quantify the trend in the
equatorward drift we firstly define manually a region which contains the
activity band of the drift, this is done simply by examining the activity maps
and drawing a box around the band of activity. We then perform a linear fit to
all the recorded maxima within that region to obtain a slope and intercept.
The chosen region and the comparison of the smoothed map and the individual
maxima is shown in Figure 3. This method was chosen in preference to fitting
to the maxima at each time (as was done by Altrock (2003) for seven-rotation
averages) since it is less sensitive to spurious maxima. The fits thus derived
are indicated in Figure 2 as dashed black and white lines.
It is clear from the activity zone maps in Figure 2 that at all heights, the
coronal activity zones reach a maximum latitude of around 75∘ near solar
maximum in 2000 and then the latitude of the activity zones moves toward the
equator through the declining phase of the cycle. However the drift to the
equator is clearly much faster at higher altitudes, and by about 4$R_{\odot}$
there is little if any overlap between the cycles. To show this more clearly,
we overlay all the fitted trends from Figure 2 in Figure 4a.
Figure 4: (a) The trends of selected equatorward drifts, and the rushes to the
poles as overlayed on Figure 2. The heights are indicated to the left of the
tracks for the cycle 23 rush to the poles, for the two trends at
1.15$R_{\odot}$, the solid green lines are the NSO green-line data and the
dotted blue lines are the MLSO data. Note that the rush of cycle 24 could not
be discerned for the MLSO data at 1.15$R_{\odot}$. (b) The variation of
equatorward drift rate with height as determined from the slope estimates in
Figure 2. (c) The variation of latitude of the start of the equatorward drift
at 2000.0 with height.
We plot the gradients of the estimates of the drift to the equator in Figure
4b, along with the latitude at the start of 2000 in Figure 4c. This shows
clearly that close to the Sun, the drift rate is around 3∘/year, but that this
increases to about 7∘/year by 3-4$R_{\odot}$, above that height the gradient
is constant within the errors of measurement. There is no clear trend in the
latitude at the start of 2000, with a value close to 75∘ at all heights. We do
not consider that the slight fluctuations are significant.
We note that there are a number of epochs during which high activity is seen
at all heights. Most notable of these features is the enhanced activity in
late 2003 and early 2004. This combined with the smooth change of latitude
with height provides good evidence that we are observing the same structures
at all heights.
The other main component of the extended cycle, the so-called “rush to the
poles” which begins soon after solar minimum and reaches its highest latitude
near to solar maximum, is also clearly visible at all heights. Historically
the rush to the poles has been tracked from about 45∘ latitude at solar
minimum up to high latitudes at maximum (Waldemeier, 1964; Altrock, 1997).
However we see that higher in the corona the rush starts at about the same
time as it does in the green-line measurements, but it begins at much lower
latitudes. The trend that is observed at the start of cycle 23 is that the
rush started at all altitudes early in 1997 when it branched away from the
previous equatorward drift, and intersected the start of the next equatorward
drift at the start of 2000. As with the equatorward drift, we have indicated
our estimates of the rush to the poles during the rising phase of cycle 23
with dotted black and white lines in Figure 2. The start of the rush to the
poles of cycle 24 is also apparent in the green line and the LASCO C2 data for
2009 and 2010.
## 5 Discussion
The most obvious trend that we see as we track the zones of activity to higher
altitudes is that the drift towards the equator becomes more rapid as we move
higher into the corona.
Could this be an observational artifact caused by changes in the Thomson-
scattering weighting functions? For this to be the case, it would be necessary
that the Thomson weighting functions become narrower (in angle as seen from
the Sun) as the closest approach of the line of sight to the Sun moves further
from the Sun. In fact the reverse is the case as the electron density falls
off faster than $1/r^{2}$ close to the Sun (e.g. Allen, 1973). In addition,
the MLSO data (heights below 2.35$R_{\odot}$) use polarised light which has a
much narrower weighting function than unpolarised (e.g. Howard and Tappin,
2009). The emission-line weighting is generally narrower than the Thomson
function at the same distance since the dependence of the emission is
approximately proportional to $N_{\mathrm{e}}^{1.7}$ and peaks at a
temperature of about $1.8\times 10^{6}$K (Mason, 1975; Guhathakurta et al.,
1992). We have verified this by some simple simulations of radial streamers.
These show that such streamers can never appear at a lower projected latitude
than their footpoint, and that except at very high footpoint latitudes (above
about 60∘) they predominantly appear close to the footpoint latitude. If (as
seems probable) we are looking at a band of activity reaching up to a
particular latitude, then a wider scattering function will increase the
sensitivity of the observations to structures well-away from the sky plane.
Since a radial streamer away from the sky plane will have an apparent latitude
greater than its true latitude this would tend to make that activity appear to
extend closer to the poles, which is the reverse of the trend that we see,
thus confirming that it cannot be an artifact of the different observing
techniques.
It is well-known that flows from coronal holes diverge (e.g. Whang, 1983;
Falceta-Gonçalves and Jatenco-Pereira, 2005), and also that during the
declining and minimum phase of the solar cycle the solar wind is well-
represented as a two-part system with low-latitude slow wind from the streamer
belt and high-latitude fast wind coming from the polar coronal holes (e.g.
Phillips et al., 1995). The behaviour of the equatorward drift may then be
understood if we assume that the upper zone of activity lies on the boundary
between the low latitude corona and the polar coronal holes (most probably
streamers overlying the polar crown filaments). That boundary moves
equatorward as the polar holes become larger during the decline of the solar
cycle. The divergence of the flow from the holes also squeezes the streamers
towards the equator at higher altitudes. When the polar hole is smaller and
weaker near solar maximum, we expect that not only will the boundary be closer
to the poles, but that there will also be less divergence. While the exact
degree of divergence will depend on the speeds and densities of the flows in
the coronal holes and the streamer belt, the degree of equatorward deflection
which we see is consistent with that found the in the MHD simulations of Whang
(1983).
Figure 5: Annual averages of the activity distribution in the K-corona for the
declining phase of cycle 23. The green line activity profile is plotted to the
left of each year. The approximate latitudes of the two bands of activity are
marked by the arrows on the green line profiles. LASCO C3 data below 20∘
latitude are omitted as these reflect the F-corona rather than the K-corona.
For 2009, the low altitude data from MLSO are noisy as there were many data
gaps during that year.
To better visualise the shape of the activity zones in latitude and radius, we
made annual averages of the activity levels as a function of radius and
latitude for the declining phase of cycle 23, these are shown in Figure 5. In
this format the image is a representation of the average path of the streamers
from close to the Sun, into the high corona. The deflection of the activity
zones towards the equator with increasing height in the low corona is very
evident, as is their more radial nature in the higher corona. It is also very
clear that the amount of deflection increases as we approach solar minimum.
We also see that the feature corresponding to the low-latitude “current” cycle
activity zone in the low corona cannot be traced above about 3$R_{\odot}$
after 2003. It is unclear whether the streamers overlying the main activity
belt simply do not extend into the high corona or whether they deflect
polewards and merge with the high-latitude zone. The latter is hinted by some
years, notably 2006 and 2007, however the apparent band joining the low-
latitude activity to the high-latitude is in the MLSO data which have
significantly poorer quality than either the NSO green line data or the LASCO
data.
Figure 6: Schematic illustration of two possible interpretations of the
streamer structure in the late declining phase of the solar cycle. (a) The
streamers over the main activity belt are closed loops and do not extend above
a few $R_{\odot}$, (b) The activity belt streamers merge with the high-
latitude streamers.
We thus have two possible interpretations of the long-term structure of the
coronal activity during the declining phase of the cycle. These are
illustrated schematically in Figure 6. If the apparent connection of the two
bands at about 2$R_{\odot}$ is an artifact, then the structure is as shown in
Figure 6(a), where the streamers over the main activity zone are closed loops
which do not extend above a few solar radii. If on the other hand the
connection is real, then the structure must be as in Figure 6(b), and the
streamers from the activity belt merge with those originating at higher
latitudes. It is of interest to compare this with the interpretation of a
coronal ray observed during a solar minimum eclipse shown in Figure 136 of
Shklovskii (1965).
Irrespective of which interpretation of the topology is correct, it is evident
that at higher altitudes ($>3R_{\odot}$) the streamer belt is predominantly
rooted in the higher-latitude activity band closer to the Sun. Hence we must
expect that the slow solar wind seen by in-situ observations near Earth during
the declining phase of the solar cycle, and especially after the lower edge of
the equatorward drift reaches the equator (around 2007 for cycle 23), will
include material characteristic of the next solar cycle.
We conjecture that the high-latitude streamers are those originating over the
polar crown filaments, while the low-latitude ones originate above active
regions. If this is the case then it is to be expected that as activity
declines to solar minimum the importance of streamers from the active regions
will decrease relative to those from the polar crown. However a full
understanding of this would require a detailed study of the rising phase of
the solar cycle, including photospheric and chromospheric observations to show
how the polar-crown region transitions to the activity-belt region.
As with the equatorward drift, we find that the highest latitudes of the rush
to the poles is similar at all heights. However at the start of the rush, it
is at much lower latitudes at higher altitudes. It branches away from the
equatorward drift at similar times at all altitudes. This may be explained in
the same way as the latitudinal variation of the equatorward drift with
altitude. That is, the polar coronal hole shrinks during the rise in activity,
and so the boundary between the polar hole region and the polar crown region
moves poleward and the flows from the polar hole become weaker. Thus as the
rush move poleward its equatorward deflection decreases.
It is also evident from Figure 4a that for those heights at which we were able
to determine the rush to the poles for cycle 24, it was taking place more
slowly than at the corresponding heights in cycle 23. However the
uncertainties in the determination of the rush for cycle 24 are large, and it
is not clear that this is actually saying anything other than that the time
between the maximum of cycle 23 and that of cycle 24 is longer than the
interval in cycles 22 and 23.
## 6 Conclusions
We have shown that the activity zones of the extended solar cycle seen in the
emission corona at 1.15$R_{\odot}$ can be traced far out into the corona in
the Thomson-scattered light of the K-corona. We find that the activity zones
are deflected towards the equator at greater heights in the corona. We also
see that as the activity moves toward the equator through the declining phase
of the cycle, the amount of deflection increases. This is consistent with the
expansion of the flows from the polar coronal holes. This deflection also seen
in the rush to the poles during the rising phase of the cycle.
During the declining phase of the solar cycle, the activity zones above a few
solar radii are dominated by structure which connect to the high-latitude
branch in the green-line activity, which is related to the upcoming solar
cycle. It therefore seems inevitable that at least in the late declining phase
of the solar cycle the low-latitude slow solar wind is more characteristic of
the upcoming cycle than of the current. This opens up the possibility of
determining some characteristics of the next cycle from in-situ measurements
made well before the start of that cycle. In the case of the current cycle, it
appears that cycle 24 should dominate from about 2007.
#### acknowledgements
The Mauna Loa coronameter data are provided courtesy of the Mauna Loa Solar
Observatory, operated by the High Altitude Observatory, as part of the
National Center for Atmospheric Research (NCAR). NCAR is supported by the
National Science Foundation. LASCO was built by a consortium of the Naval
Research Laboratory (Washington, USA), the Max-Planck-Institut für Aeronomie
(Lindau, Germany) the Laboratoire d’Astronomie Spatiale (Marseille, France)
and the University of Birmingham (Birmingham, UK) and is a part of the SOHO
mission which is a collaborative mission of ESA and NASA. Partial support for
NSO (including the work of SJT and the operation of the coronagraph) is
provided by the US Air Force under a Memorandum of Agreement.
## References
* Allen (1973) Allen, C. W.: 1973, Astrophysical Quantities, Athlone Press, London, p176.
* Altrock (1997) Altrock, R. C.: 1997, Solar Phys. 170, 411.
* Altrock (2003) Altrock, R. C.: 2003, Solar Phys. 216, 343.
* Altrock (2011) Altrock, R. C.: 2011, Solar Phys. 274, 251.
* Brueckner et al. (1995) Brueckner, G. E., Howard, R. A., Koomen, M. J., Korendyke, C. M., Michels, D. J., Moses, J. D., et al.: 1995, Solar Phys. 162, 357.
* Elmore et al. (2003) Elmore, D. F., Burkepile, J. T., Darnell, J. A., Lecinski, A. R., Stanger, A. L.: 2003, In: Fineschi, S. (ed), Polarimetry in Astronomy, Proc. SPIE 4843 66\.
* Falceta-Gonçalves and Jatenco-Pereira (2005) Falceta-Gonçalves, D. and Jatenco-Pereira, V.: 2005, In Gouveia Dal, E. M., Pino, G. L., Lazarian, A. (eds.), Magnetic Fields in the Universe: From Laboratory and Star to Primordial Structures, AIP Conf. Proc. 784 591\.
* Fisher et al. (1981) Fisher, R. R., Lee, R. H., MacQueen, R. M., Poland, A. I.: 1981, Appl. Opt. 20, 6.
* Guhathakurta et al. (1992) Guhathakurta, M., Rottman, G. J., Fisher, R. R., Orrall, F. Q., Altrock, R. C.: 1992, Astrophys. J. 388, 633.
* Howard and Tappin (2009) Howard, T. A., Tappin, S. J.: 2009, Space Sci. Rev. 147, 31.
* Labonte and Howard (1982) Labonte, B. J., Howard, R.: 1982, Solar Phys. 75, 161.
* Mason (1975) Mason, H. E.: 1975, Mon. Not. Roy. Astron. Soc. 170 651\.
* Phillips et al. (1995) Phillips, J. L., Bame, S. J., Barnes, A., Barraclough, B. L., Feldman, W. C., Goldstein, B. E., et al.: 1995, Geophys. Res. Lett. 22, 3301.
* Shklovskii (1965) Shklovskii, I. S.: 1965, Physics of the Solar Corona, 2nd edition, Pergamon Press, Oxford, p404.
* Tappin, Simnett and Lyons (1999) Tappin, S. J., Simnett, G. M., Lyons, M. A.: 1999, Astron. Astrophys. 350, 302.
* Waldemeier (1964) Waldemeier, M.: 1964, Z. Astrophys. 59, 205.
* Whang (1983) Whang, Y. C.: 1983, Solar Phys. 88, 343.
* Wilson et al. (1988) Wilson, P. R., Altrock, R. C., Harvey, K. L., Martin, S. F., Snodgrass, H. B.: 1988, Nature 333, 748.
|
arxiv-papers
| 2012-09-13T17:30:30 |
2024-09-04T02:49:35.061789
|
{
"license": "Public Domain",
"authors": "S.J. Tappin (1) and R.C. Altrock (2) ((1) National Solar Observatory,\n USA (2) Air Force Research Laboratory, USA)",
"submitter": "Richard Altrock",
"url": "https://arxiv.org/abs/1209.2969"
}
|
1209.2987
|
1NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771,
USA
2Universities Space Research Association, 10211 Wincopin Circle, Columbia, MD
21044,USA
3National Institute of Standard and Technology, 325 Broadway, Boulder, CO
80305, USA
∗Corresponding author: [email protected]
# Infrared dielectric properties of low-stress silicon nitride
Giuseppe Cataldo,1,2,∗ James A. Beall,3 Hsiao-Mei Cho,3
Brendan McAndrew,1 Michael D. Niemack,3 and Edward J. Wollack1
###### Abstract
Silicon nitride thin films play an important role in the realization of
sensors, filters, and high-performance circuits. Estimates of the dielectric
function in the far- and mid-infrared regime are derived from the observed
transmittance spectra for a commonly employed low-stress silicon nitride
formulation. The experimental, modeling, and numerical methods used to extract
the dielectric parameters with an accuracy of approximately 4% are presented.
310.3840, 310.6188, 310.6860.
The physical properties of silicon nitride thin films, namely low tensile
stress, low thermal/electrical conductance, and its overall compatibility with
other common materials, have facilitated its use in the micro-fabrication of
structures requiring mechanical support, thermal isolation, and low-loss
microwave signal propagation (e.g., [1, 2, 3, 4]). Silicon nitride films are
amorphous, highly absorbing in the mid-infrared [5], and their general
properties are functions of composition [6, 7]. Here, the optical properties
are studied in detail for a membrane with parameters commonly employed in
micro-fabrication.
The silicon nitride optical test films were prepared by a LP-CVD (Low-Pressure
Chemical-Vapor-Deposition) process optimized for low tensile stress and
refractive index [8]. The 5:1 SiH2Cl2/ NH3 gas ratio employed results in a
tensile stress $<100$ MPa and optical index greater than $\sim$ 2 [9]. The
test structure is shown schematically in Fig. 1 (inset). Double-side-polished
silicon (75-mm-diameter, 500-$\mu$m-thick) wafers [10] were used as a
mechanically robust handling structure for the SiNx membranes. A 150-nm
thermal oxide was grown on the silicon wafers by wet oxidation at 950∘C for 31
minutes. This layer was subsequently used as an etch stop to protect the
nitride during definition of the silicon handling wafer geometry. A low-stress
SiNx layer was then deposited by LP-CVD (e.g., deposition parameters for
2-$\mu$m film are 835∘C for 9.7 hours with pressure 33 Pa and 12 sccm NH3, 59
sccm SiH2Cl2). The wafers were then patterned with a resist mask and SiNx/SiO2
windows formed by deep reactive ion etching which removed all the silicon
under the window area. The residual thermal oxide was removed with HF vapor
etch leaving a set of uniform SiNx membranes each with a 10-mm diameter
aperture individually suspended from the silicon handling frame.
The optical tests were performed on SiNx samples having membrane thicknesses
of 0.5 and 2.3 $\mu$m with a uncertainty of 3%. Fabry-Perot resonators were
made by stacking multiple samples with silicon standoff frames between
adjacent samples to explore the long-wavelength response of the material in
greater detail. The silicon standoffs allowed a vent path for evacuation of
air between the nitride membranes. All optical measurements were performed in
vacuum with a residual pressure less than 100 Pa.
The samples were characterized with a Bruker 125 high-resolution Fourier
Transform Spectrometer (FTS) and were measured in transmission at the focal
plane of an $f/6$ beam. A number of different sources, beam splitters, and
detector configurations were used in combination to provide measurements over
the reported spectral range. The single-layer SiNx sample transmission was
measured over an extended range from 15 to 10000 cm-1. The mercury lamp and a
multilayer Mylar beam splitter were used to access frequencies below 600 cm-1.
Additional mid-infrared spectral data up to 2400 cm-1 were acquired using a
ceramic glow bar source, Ge-coated KBr beam splitter, and room-temperature
DTGS detector. The remaining near-infrared data up to 10000 cm-1 were taken
with a W filament source, Si on CaF2 beam splitter, and a liquid-nitrogen-
cooled InSb detector (Fig. 1). Far-infrared data between 15 and 95 cm-1 were
taken using a mercury arc lamp source and a liquid-helium-cooled 4.2-K
bolometer. Mylar beam splitters of 50-, 75- and 125-$\mu$m thicknesses and a
multilayer Mylar beam splitter were used during separate scans (Fig. 2). The
resultant transmission data were merged into a single spectra using a signal-
to-noise weighting for subsequent parameter extraction.
The dielectric response is represented as a function of frequency, $\omega$,
by the classical Maxwell-Helmholtz-Drude dispersion model [11]:
$\overset{\hat{}}{\varepsilon_{r}}(\omega)=\overset{\hat{}}{\varepsilon}_{\infty}+\sum^{M}_{j=1}\frac{\Delta\overset{\hat{}}{\varepsilon}_{j}\cdot\omega^{2}_{\mbox{{\tiny\it
T}}_{j}}}{\omega^{2}_{\mbox{{\tiny\it
T}}_{j}}-\omega^{2}-i\omega\Gamma^{\prime}_{j}(\omega)}$ (1)
where $M$ is the number of oscillators and
$\overset{\hat{}}{\varepsilon_{r}}=\varepsilon^{\prime}_{r}+i\varepsilon^{\prime\prime}_{r}$
is a complex function of $(5M+2)$ degrees of freedom, which are as follows:
the contribution to the relative permittivity
$\overset{\hat{}}{\varepsilon}_{\infty}=\overset{\hat{}}{\varepsilon}_{M+1}$
of higher lying transitions, the difference in relative complex dielectric
constant between adjacent oscillators
$\Delta\overset{\hat{}}{\varepsilon}_{j}=\overset{\hat{}}{\varepsilon}_{j}-\overset{\hat{}}{\varepsilon}_{j+1}$
which serves as a measure of the oscillator strength, the oscillator resonance
frequency $\omega_{\mbox{{\tiny\it T}}_{j}}$, and the effective Lorentzian
damping coefficient $\Gamma^{\prime}_{j}$, for $j=1,...,M$. The following
functional form is used to specify the damping:
$\Gamma^{\prime}_{j}(\omega)=\Gamma_{j}\exp{\left[-\alpha_{j}\left(\frac{\omega^{2}_{\mbox{{\tiny\it
T}}_{j}}-\omega^{2}}{\omega\Gamma_{j}}\right)^{2}\right]}$ (2)
where $\alpha_{j}$ allows interpolation between Lorentzian $(\alpha_{j}=0)$
and Gaussian wings $(\alpha_{j}>0)$ similar to the approach in [12]. The form
indicated above enables a more accurate representation of relatively strong
oscillator features.
[width=0.47]fig1_T_5osc_Final.pdf
Figure 1: (Color online) Room-temperature transmission of a silicon nitride
sample 0.5 $\mu$m thick: measured (grey), model (black dotted), and residual
(red). The shaded band’s width delimits the estimated 3$\sigma$ measurement
uncertainty. A 30 GHz (1 cm-1) resolution is employed for the measurement. The
insert depicts the geometry of the SiNx membrane and micro-machined silicon
frame.
The impedance contrast between free space and the thin-film sample forms a
Fabry-Perot resonator. The observed transmission can be modeled [13] as a
function of the dielectric response (Eq. 1), thickness, and wavenumber. The
dielectric parameters were solved by means of a non-linear least-squares fit
of the transmission equation to the laboratory FTS data. Specifically, a
sequential quadratic programming (SQP) method with computation of the Jacobian
and Hessian matrices [14, 15] was implemented. The merit function,
$\chi^{2}$,was used in a constrained minimization over frequency as follows:
$\underset{\mbox{\footnotesize DOF}}{\mbox{\footnotesize
min}}\chi^{2}=\underset{\mbox{\footnotesize DOF}}{\mbox{\footnotesize
min}}\sum^{N}_{k=1}{\left[T(\overset{\hat{}}{\varepsilon_{r}}(\omega),h)-T_{\mbox{\tiny{FTS}}_{k}}\right]^{2}}$
(3)
where $N$ is the number of data points, $T$ the modeled transmittance,
$T_{\mbox{\tiny FTS}}$ the measured transmittance data, and $h$ is the
measured sample thickness. We are guided by the Kramers-Kronig relations in
defining constraints for a passive material:
$|\overset{\hat{}}{\varepsilon}_{j}|>|\overset{\hat{}}{\varepsilon}_{j+1}|$,
$\varepsilon^{\prime\prime}_{j}>0$ and
$\overset{\hat{}}{\varepsilon_{r}}(0)=\overset{\hat{}}{\varepsilon}_{1}$ [16].
For accurate parameter determination the sample should have uniform thickness,
be adequately transparent to achieve high signal-to-noise, and have diffuse
scattering as a sub-dominate process. The method requires an a posteriori
numerical verification for Kramers-Kronig consistency. In the example
presented here, a numerical Hilbert transform [17] of
$\varepsilon_{r}^{\prime\prime}(\omega)$ reproduces
$\varepsilon_{r}^{\prime}(\omega)$ to within 2% (Fig. 3). An alternative
method employing reflectivity and phase allows a priori Kramers-Kronig
consistent results [18]. However, given the details of the thin-film samples
and available instrumentation, this approach was not implemented.
[width=0.45]fig2_3-layer_sample_0-3THz.pdf
Figure 2: (Color online) Measured (solid grey) and model (black dotted)
transmission for a 3-layer stack of silicon nitride samples 2.3 $\mu$m in
thickness with 998-$\mu$m intermembrane delays which complements the data
shown in Fig. 1. The sample response in the far-infrared was acquired with a
resolution of 3 GHz (0.1 cm-1).
Figure 1 illustrates the measured and modeled results obtained from the
analysis of a 0.5-$\mu$m-thick sample. The peak residual in the transmittance
is less than 3% and the $3\sigma=0.023$ uncertainty band indicated corresponds
to the 99.7% confidence level. The standard deviation adopted for the measured
data, $\sigma$, was estimated assuming the errors as a function of frequency
are uniform and have a reduced $\chi^{2}$ equal to unity. An additional
uncertainty in the FTS normalization influences the dielectric response
function at the 1% level. In addition to the channel spectra, the observed
spectrum shows two predominant features at 12 THz and 25 THz. Simulations with
$M=2$ oscillators lead to a peak residual on transmission of 5% and do not
enable recovery of the resonance at 25 THz. Using 5 oscillators satisfactorily
recovers the observed transmittance and reduces the peak residual by a factor
of 4.4. When the resonator’s quality factor, $Q_{\mbox{{\scriptsize\it
eff}}_{j}}=\omega_{j}/\Gamma^{\prime}_{j}$, is greater than 5, the data were
not reproducible by either a pure Lorentzian oscillator or Eq. (4.6) in [12].
In these regions, the peak transmission residuals were decreased by a factor
$\sim$ 2 through the use of Eq. (2).
In Fig. 3 the values of the real and imaginary components of the dielectric
function are illustrated as a function of frequency. The uncertainty in
$\overset{\hat{}}{\varepsilon_{r}}$ was propagated and computed as described
in [19]. Table 1 contains a summary of the best fit parameters for 5
oscillators, which can be used to reproduce the data shown in Fig. 3.
Table 1: Fit parameter summary $j$ | $\varepsilon^{\prime}_{j}$ | $\varepsilon^{\prime\prime}_{j}$ | $\omega_{\mbox{{\tiny\it T}}_{j}}/2\pi$ | $\Gamma_{j}/2\pi$ | $\alpha_{j}$
---|---|---|---|---|---
$[-]$ | $[-]$ | $[-]$ | $[$THz$]$ | $[$THz$]$ | $[-]$
1 | 7.582 | 0 | 13.913 | 5.810 | 0.0001
2 | 6.754 | 0.3759 | 15.053 | 6.436 | 0.3427
3 | 6.601 | 0.0041 | 24.521 | 2.751 | 0.0006
4 | 5.430 | 0.1179 | 26.440 | 3.482 | 0.0002
5 | 4.601 | 0.2073 | 31.724 | 5.948 | 0.0080
6 | 4.562 | 0.0124 | | |
In order to characterize the long-wavelength portion of the dielectric
function, Fabry-Perot resonators were realized from 1-, 2-, and 3-layer
samples. Representative data for the 3-layer resonator stack is presented in
Fig. 2. A multilayer transfer matrix analysis [13] is used to extract the
dielectric function using the measured SiNx (2.3 $\mu$m) and silicon spacer
(998 $\mu$m) thicknesses. The circular symbols at 1.5 THz and 2.5 THz
indicated in Fig. 3 were computed from a composite analysis of the 3 Fabry-
Perot measurement sets. The horizontal range indicates the data used in each
fit. The best estimates are $\overset{\hat{}}{\varepsilon_{r}}\approx
7.6+i0.08$ over the range 2-3 THz and
$\overset{\hat{}}{\varepsilon_{r}}\approx 7.6+i0.04$ over 0.4-2 THz. The real
component of the static dielectric function derived from the data is in
agreement with prior reported parameters for this stoichiometry [4]. As shown
in Fig. 3, the measurements are internally consistent and represent roughly a
factor-of-three reduction in uncertainty relative to prior infrared SiNx
measurements identified by the authors [7, 6, 5]. The dielectric parameters
reported here are representative of low-stress SiNx membranes encountered in
our fabrication and test efforts.
[width=0.47]fig3_eps_5osc_Final.pdf
Figure 3: (Color online) Real and imaginary parts (solid red lines) of the
dielectric function as extracted from the data shown in Fig. 1. The line
thickness is indicative of the propagated $\sim$ 4% error band. The numerical
Hilbert transform of the modeled $\varepsilon_{r}^{\prime\prime}(\omega)$ is
indicated in the upper panel (dashed blue line) to facilitate comparison with
$\varepsilon_{r}^{\prime}(\omega)$. The filled symbols indicate the parameters
derived from the data presented in Fig. 2.
## References
* [1] D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
* [2] G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Superconductivity 21 (3), 232–235 (2011).
* [3] J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95 (21), 210503 (2005).
* [4] H. Paik and K. D. Osborn, Appl. Phys. Lett. 96 (7), 072505 (2010).
* [5] T. Eriksson, S. Jiang, and C. Granqvist, Appl. Opt. 24, 745–746 (1985).
* [6] E. A. Taft, J. Electrochem. Soc. 118, 1341–1346 (1971).
* [7] E. D. Palik, _Handbook of Optical Constants of Solids_ , Vol. 1 (Elsevier, 1998), pp. 771–774.
* [8] M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21 (4), 1017–1021 (1982).
* [9] T. Makino, J. Electrochem. Soc. 130 (2), 450–455 (1983).
* [10] Addison Engineering, 150 Nortech Parkway, San Jose, CA 95134. (Orientation $<$100$>$, Czochralski, p-type B doped, bulk resistivity $<$ 0.005 $\Omega\cdot$cm)
* [11] F. Gervais, “High-Temperature Infrared Reflectivity Spectroscopy by Scanning Interferometry” in _Electromagnetic Waves in Matter_ , Part I, Vol. 8 (Infrared and Millimeter Waves), K. J. Button, eds. (Academic Press, London, 1983), pp. 284–287.
* [12] C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45 (20), 11749 (1992).
* [13] P. Yeh, _Optical Waves in Layered Media_ (Wiley, New York, 1988), pp. 102–111.
* [14] M. C. Biggs, “Constrained Minimization Using Recursive Quadratic Programming,” in _Towards Global Optimization_ , L. C. W. Dixon and G. P. Szergo, eds. (North-Holland, 1975), pp. 341–349.
* [15] M. J. D. Powell, “Variable Metric Methods for Constrained Optimization,” in _Mathematical Programming: The State of the Art_ , A. Bachem, M. Grotschel and B. Korte, eds. (Springer Verlag, 1983), pp. 288–311.
* [16] L. D. Landau and E. M. Lifshitz, _Electrodyamics of Continuous Media_ , Vol. 8 (Pergamon Press, 1960), pp. 253–262.
* [17] M. Mori and T. Ooura, “Double Exponential Formulas for Fourier Type Integrals with a Divergent Integrand,” in _Applicable Analysis_ , Vol. 2 (World Scientific Series, 1993), pp. 301–308.
* [18] R. Nitsche and T. Fritz, Phys. Rev. B 70 (19), 195432 (2004).
* [19] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, _Numerical Recipes - The Art of Scientific Computing_ (Cambridge University Press, 2007), pp. 799–806.
## Informational Fourth Page
In this section, please provide full versions of citations to assist reviewers
and editors (OL publishes a short form of citations) or any other information
that would aid the peer-review process.
## References
* [1] D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, “Ultra-low-noise MoCu transition edge sensors for space applications,” Appl. Phys. 109, 084507 (2011).
* [2] G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, “Thermal Properties of Silicon Nitride Beams Below One Kelvin,” IEEE Trans. Appl. Superconductivity 21 (3), 232–235 (2011).
* [3] J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, “Decoherence in Josephson Qubits from Dielectric Loss”, Phys. Rev. Lett. 95 (21), 210503 (2005).
* [4] H. Paik and K. D. Osborn, “Reducing Quantum-Regime Dielectric Loss of Silicon Nitride for Superconducting Quantum Circuits,” Appl. Phys. Lett. 96 (7), 072505 (2010).
* [5] T. Eriksson, S. Jiang, and C. Granqvist, “Dielectric function of sputter-deposited silicon dioxide and silicon nitride films in the thermal infrared,” Appl. Opt. 24, 745–746 (1985).
* [6] E. A. Taft, “Characterization of Silicon Nitride Films,” J. Electrochem. Soc. 118, 1341–1346 (1971).
* [7] E. D. Palik, _Handbook of Optical Constants of Solids_ , Vol. 1 (Elsevier, 1998), pp. 771–774.
* [8] M. Sekimoto, H. Yoshihara, and T. Ohkubo, “Silicon Nitride Single-Layer X-Ray Mask,” J. Vac. Sci. Technol. 21 (4), 1017–1021 (1982).
* [9] T. Makino, “Composition and Structure Control by Source Gas Ratio in LPCVD SiNx,” J. Electrochem. Soc. 130 (2), 450–455 (1983).
* [10] Addison Engineering, 150 Nortech Parkway, San Jose, CA 95134. (Orientation $<$100$>$, Czochralski, p-type B doped, bulk resistivity $<$ 0.005 $\Omega\cdot$cm)
* [11] F. Gervais, “High-Temperature Infrared Reflectivity Spectroscopy by Scanning Interferometry” in _Electromagnetic Waves in Matter_ , Part I, Vol. 8 (Infrared and Millimeter Waves), K. J. Button, eds. (Academic Press, London, 1983), pp. 284–287.
* [12] C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: Extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45 (20), 11749 (1992).
* [13] P. Yeh, _Optical Waves in Layered Media_ (Wiley, New York, 1988), pp. 102–111.
* [14] M. C. Biggs, “Constrained Minimization Using Recursive Quadratic Programming,” in _Towards Global Optimization_ , L. C. W. Dixon and G. P. Szergo, eds. (North-Holland, 1975), pp. 341–349.
* [15] M. J. D. Powell, “Variable Metric Methods for Constrained Optimization,” in _Mathematical Programming: The State of the Art_ , A. Bachem, M. Grotschel and B. Korte, eds. (Springer Verlag, 1983), pp. 288–311.
* [16] L. D. Landau and E. M. Lifshitz, _Electrodyamics of Continuous Media_ , Vol. 8 (Pergamon Press, 1960), pp. 253–262.
* [17] M. Mori and T. Ooura, “Double Exponential Formulas for Fourier Type Integrals with a Divergent Integrand,” in _Applicable Analysis_ , Vol. 2 (World Scientific Series, 1993), pp. 301–308.
* [18] R. Nitsche and T. Fritz, “Determination of model-free Kramers-Kronig consistent optical constants of thin absorbing films from just one spectral measurement: Application to organic semiconductors,” Phys. Rev. B 70 (19), 195432 (2004).
* [19] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, _Numerical Recipes - The Art of Scientific Computing_ (Cambridge University Press, 2007), pp. 799–806.
|
arxiv-papers
| 2012-09-13T18:35:44 |
2024-09-04T02:49:35.068618
|
{
"license": "Public Domain",
"authors": "Giuseppe Cataldo, James A. Beall, Hsiao-Mei Cho, Brendan McAndrew,\n Michael D. Niemack, and Edward J. Wollack",
"submitter": "Edward Wollack",
"url": "https://arxiv.org/abs/1209.2987"
}
|
1209.2995
|
# Contraherent cosheaves
Leonid Positselski Faculty of Mathematics and Laboratory of Algebraic
Geometry, National Research University Higher School of Economics, Moscow
117312; and
Sector of Algebra and Number Theory, Institute for Information Transmission
Problems, Moscow 127994, Russia [email protected]
###### Abstract.
Contraherent cosheaves are globalizations of cotorsion (or similar) modules
over commutative rings obtained by gluing together over a scheme. The category
of contraherent cosheaves over a scheme is a Quillen exact category with exact
functors of infinite product. Over a quasi-compact semi-separated scheme or a
Noetherian scheme of finite Krull dimension (in a different version—over any
locally Noetherian scheme), it also has enough projectives. We construct the
derived co-contra correspondence, meaning an equivalence between appropriate
derived categories of quasi-coherent sheaves and contraherent cosheaves, over
a quasi-compact semi-separated scheme and, in a different form, over a
Noetherian scheme with a dualizing complex. The former point of view allows us
to obtain an explicit construction of Neeman’s extraordinary inverse image
functor $f^{!}$ for a morphism of quasi-compact semi-separated schemes
$f\colon X\longrightarrow Y$. The latter approach provides an expanded version
of the covariant Serre–Grothendieck duality theory and leads to Deligne’s
extraordinary inverse image functor $f^{!}$ (which we denote by $f^{+}$) for a
morphism of finite type $f$ between Noetherian schemes. Semi-separated
Noetherian stacks, affine Noetherian formal schemes, and ind-affine ind-
schemes (together with the noncommutative analogues) are briefly discussed in
the appendices.
###### Contents
1. 1 Contraadjusted and Cotorsion Modules
1. 1.1 Contraadjusted and very flat modules
2. 1.2 Affine geometry of contraadjusted and very flat modules
3. 1.3 Cotorsion modules
4. 1.4 Exact categories of contraadjusted and cotorsion modules
5. 1.5 Very flat and cotorsion dimensions
6. 1.6 Coherent rings, finite morphisms, and coadjusted modules
7. 1.7 Very flat morphisms of schemes
2. 2 Contraherent Cosheaves over a Scheme
1. 2.1 Cosheaves of modules over a sheaf of rings
2. 2.2 Exact category of contraherent cosheaves
3. 2.3 Direct and inverse images of contraherent cosheaves
4. 2.4 $\operatorname{\mathfrak{Cohom}}$ from a quasi-coherent sheaf to a contraherent cosheaf
5. 2.5 Contraherent cosheaves of $\operatorname{\mathfrak{Hom}}$ between quasi-coherent sheaves
6. 2.6 Contratensor product of sheaves and cosheaves
3. 3 Locally Contraherent Cosheaves
1. 3.1 Exact category of locally contraherent cosheaves
2. 3.2 Contraherent and locally contraherent cosheaves
3. 3.3 Direct and inverse images of locally contraherent cosheaves
4. 3.4 Coflasque contraherent cosheaves
5. 3.5 Contrahereable cosheaves and the contraherator
6. 3.6 $\operatorname{\mathfrak{Cohom}}$ into a locally derived contrahereable cosheaf
7. 3.7 Contraherent tensor product
8. 3.8 Compatibility of direct and inverse images with the tensor operations
4. 4 Quasi-compact Semi-separated Schemes
1. 4.1 Contraadjusted and cotorsion quasi-coherent sheaves
2. 4.2 Colocally projective contraherent cosheaves
3. 4.3 Colocally flat contraherent cosheaves
4. 4.4 Projective contraherent cosheaves
5. 4.5 Homology of locally contraherent cosheaves
6. 4.6 The “naïve” co-contra correspondence
7. 4.7 Homotopy locally injective complexes
8. 4.8 Derived functors of direct and inverse image
9. 4.9 Finite flat and locally injective dimension
10. 4.10 Morphisms of finite flat dimension
11. 4.11 Finite injective and projective dimension
12. 4.12 Derived tensor operations
5. 5 Noetherian Schemes
1. 5.1 Projective locally cotorsion contraherent cosheaves
2. 5.2 Flat contraherent cosheaves
3. 5.3 Homology of locally cotorsion locally contraherent cosheaves
4. 5.4 Background equivalences of triangulated categories
5. 5.5 Co-contra correspondence over a regular scheme
6. 5.6 Co-contra correspondence over a Gorenstein scheme
7. 5.7 Co-contra correspondence over a scheme with a dualizing complex
8. 5.8 Co-contra correspondence over a non-semi-separated scheme
9. 5.9 Compact generators
10. 5.10 Homotopy projective complexes
11. 5.11 Special inverse image of contraherent cosheaves
12. 5.12 Derived functors of direct and special inverse image
13. 5.13 Adjoint functors and bounded complexes
14. 5.14 Compatibilities for a smooth morphism
15. 5.15 Compatibilities for finite and proper morphisms
16. 5.16 The extraordinary inverse image
6. A Derived Categories of Exact Categories and Resolutions
1. A.1 Derived categories of the second kind
2. A.2 Fully faithful functors
3. A.3 Infinite left resolutions
4. A.4 Homotopy adjusted complexes
5. A.5 Finite left resolutions
6. A.6 Finite homological dimension
7. B Co-Contra Correspondence over a Flat Coring
1. B.1 Contramodules over a flat coring
2. B.2 Base rings of finite weak dimension
3. B.3 Gorenstein base rings
4. B.4 Corings with dualizing complexes
5. B.5 Base ring change
8. C Affine Noetherian Formal Schemes
1. C.1 Torsion modules and contramodules
2. C.2 Contraadjusted and cotorsion contramodules
3. C.3 Very flat contramodules
4. C.4 Affine geometry of $(R,I)$-contramodules
5. C.5 Noncommutative Noetherian rings
9. D Ind-Affine Ind-Schemes
1. D.1 Flat and projective contramodules
2. D.2 Co-contra correspondence
3. D.3 Very flat and contraadjusted contramodules
4. D.4 Cotorsion contramodules
## Introduction
Quasi-coherent sheaves resemble comodules. Both form abelian categories with
exact functors of infinite direct sum (and in fact, even of filtered inductive
limit) and with enough injectives. When one restricts to quasi-coherent
sheaves over Noetherian schemes and comodules over (flat) corings over
Noetherian rings, both abelian categories are locally Noetherian. Neither has
projective objects or exact functors of infinite product, in general.
In fact, quasi-coherent sheaves _are_ comodules. Let $X$ be a quasi-compact
semi-separated scheme and $\\{U_{\alpha}\\}$ be its finite affine open
covering. Denote by $T$ the disconnected union of the schemes $U_{\alpha}$; so
$T$ is also an affine scheme and the natural morphism $T\longrightarrow X$ is
affine. Then quasi-coherent sheaves over $X$ can be described as quasi-
coherent sheaves ${\mathcal{F}}$ over $T$ endowed with an isomorphism
$\phi\colon p_{1}^{*}({\mathcal{F}})\simeq p_{2}^{*}({\mathcal{F}})$ between
the two inverse images under the natural maps $p_{1}$, $p_{2}\colon
T\times_{X}T\birarrow T$. The isomorphism $\phi$ has to satisfy a natural
associativity constraint.
In other words, this means that the ring of functions
${\mathcal{C}}={\mathcal{O}}(T\times_{X}T)$ has a natural structure of a
coring over the ring $A={\mathcal{O}}(T)$. The quasi-coherent sheaves over $X$
are the same thing as (left or right) comodules over this coring. The quasi-
coherent sheaves over a (good enough) stack can be also described in such way
[36].
There are two kinds of module categories over a coalgebra or coring: in
addition to the more familiar comodules, there are also _contramodules_.
Introduced originally by Eilenberg and Moore [16] in 1965 (see also the
notable paper [4]), they were all but forgotten for four decades, until the
author’s preprint and then monograph [50] attracted some new attention to them
towards the end of 2000’s.
Assuming a coring ${\mathcal{C}}$ over a ring $A$ is a projective left
$A$-module, the category of left ${\mathcal{C}}$-contramodules is abelian with
exact functors of infinite products and enough projectives. Generally,
contramodules are “dual-analogous” to comodules in most respects, i. e., they
behave as though they formed two opposite categories—which in fact they don’t
(or otherwise it wouldn’t be interesting).
On the other hand, there is an important homological phenomenon of _comodule-
contramodule correspondence_ , or a _covariant_ equivalence between
appropriately defined (“exotic”) derived categories of left comodules and left
contramodules over the same coring. This equivalence is typically obtained by
deriving certain adjoint functors which act between the abelian categories of
comodules and contramodules and induce a covariant equivalence between their
appropriately picked exact or additive subcategories (e. g., the equivalence
of _Kleisli categories_ , which was empasized in the application to comodules
and contramodules in the paper [7]).
_Contraherent cosheaves_ are geometric module objects over a scheme that are
similar to (and, sometimes, particular cases of) contramodules in the same way
as quasi-coherent sheaves are similar to (or particular cases of) comodules.
Thus the simplest way to define contramodules would be to assume one’s scheme
$X$ to be quasi-compact and semi-separated, pick its finite affine covering
$\\{U_{\alpha}\\}$, and consider contramodules over the related coring
${\mathcal{C}}$ over the ring $A$ as constructed above.
This indeed largely agrees with our approach in this paper, but there are
several problems to be dealt with. First of all, ${\mathcal{C}}$ is not a
projective $A$-module, but only a flat one. The most immediate consequence is
that one cannot hope for an abelian category of ${\mathcal{C}}$-contramodules,
but at best for an exact category. This is where the _cotorsion modules_ [63,
19] (or their generalizations which we call the _contraadjusted_ modules) come
into play. Secondly, it turns out that the exact category of
${\mathcal{C}}$-contramodules, however defined, depends on the choice of an
affine covering $\\{U_{\alpha}\\}$.
In the exposition below, we strive to make our theory as similar (or rather,
dual-analogous) to the classical theory of quasi-coherent sheaves as possible,
while refraining from the choice of a covering to the extent that it remains
practicable. We start with defining cosheaves of modules over a sheaf of rings
on a topological space, and proceed to introducing the exact subcategory of
contraherent cosheaves in the exact category of cosheaves of
${\mathcal{O}}_{X}$-modules on an arbitrary scheme $X$.
Several attempts to develop a theory of cosheaves have been made in the
literature over the years (see, e. g., [9, 59, 11]). The main difficulty
arising in this connection is that the conventional theory of sheaves depends
on the exactness property of filtered inductive limits in the categories of
abelian groups or sets in an essential way. E. g., the most popular approach
is based on the wide use of the construction of stalks, which are defined as
filtered inductive limits. It is important that the functors of stalks are
exact on both the categories of presheaves and sheaves.
The problem is that the costalks of a co(pre)sheaf would be constructed as
filtered projective limits, and filtered projective limits of abelian groups
are not exact. One possible way around this difficulty is to restrict oneself
to (co)constructible cosheaves, for which the projective limits defining the
costalks may be stabilizing and hence exact. This is apparently the approach
taken in the notable recent preprint [13] (see the discussion with further
references in [13, Section 3.5]). The present work is essentially based on the
observation that one does not really need the (co)stalks in the quasi-
coherent/contraherent (co)sheaf theory, as _the functors of (co)sections over
affine open subschemes are already exact on the (co)sheaf categories_.
Let us explain the main definition in some detail. A quasi-coherent sheaf
${\mathcal{F}}$ on a scheme $X$ can be simply defined as a correspondence
assigning to every affine open subscheme $U\subset X$ an
${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ and to every pair of embedded
affine open subschemes $V\subset U\subset X$ an isomorphism of
${\mathcal{O}}_{X}(V)$-modules
${\mathcal{F}}(V)\simeq{\mathcal{O}}_{X}(V)\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{F}}(U).$
The obvious compatibility condition for three embedded affine open subschemes
$W\subset V\subset U\subset X$ needs to be imposed.
Analogously, a contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$ is a
correspondence assigning to every affine open subscheme $U\subset X$ an
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$ and to every pair of
embedded affine open subschemes $V\subset U\subset X$ an isomorphism of
${\mathcal{O}}_{X}(V)$-modules
${\mathfrak{P}}[V]\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U]).$
The difference with the quasi-coherent case is that the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{O}}_{X}(V)$ is always flat, but not
necessarily projective. So to make one’s contraherent cosheaves well-behaved,
one has to impose the additional Ext-vanishing condition
$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}^{1}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U])=0$
for all affine open subschemes $V\subset U\subset X$. Notice that the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{O}}_{X}(V)$ always has projective
dimension not exceeding $1$, so the condition on $\operatorname{Ext}^{1}$ is
sufficient.
This nonprojectivity problem is the reason why one does not have an abelian
category of contraherent cosheaves. Imposing the Ext-vanishing requirement
allows to obtain, at least, an exact one. Given a rule assigning to affine
open subschemes $U\subset X$ the ${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{P}}[U]$ together with the isomorphisms for embedded affine open
subschemes $V\subset U$ as above, and assuming that the
$\operatorname{Ext}^{1}$-vanishing condition holds, one can show that
${\mathfrak{P}}$ satisfies the cosheaf axioms for coverings of affine open
subschemes of $X$ by other affine open subschemes. Then a general result from
[25] says that ${\mathfrak{P}}$ extends uniquely from the base of affine open
subschemes to a cosheaf of ${\mathcal{O}}_{X}$-modules defined, as it should
be, on all the open subsets of $X$.
A module $P$ over a commutative ring $R$ is called _contraadjusted_ if
$\operatorname{Ext}^{1}_{R}(R[s^{-1}],P)\allowbreak=0$ for all $s\in R$. This
is equivalent to the vanishing of $\operatorname{Ext}^{1}_{R}(S,P)$ for all
the $R$-algebras $S$ of functions on the affine open subschemes of
$\operatorname{Spec}R$. More generally, a left module $P$ over a (not
necessarily commutative) ring $R$ is said to be _cotorsion_ if
$\operatorname{Ext}^{1}_{R}(F,P)=0$ (or equivalently,
$\operatorname{Ext}^{>0}_{R}(F,P)=0$) for any flat left $R$-module $F$. It has
been proven that cotorsion modules are “numerous enough” (see [17, 6]); one
can prove the same for contraadjusted modules in the similar way.
Any quotient module of a contraadjusted module is contraadjusted, so the
“contraadjusted dimension” of any module does not exceed $1$; while the
cotorsion dimension of a module may be infinite if the projective dimensions
of flat modules are. This makes the contraadjusted modules useful when working
with schemes that are not necessarily Noetherian of finite Krull dimension.
Modules of the complementary class to the contraadjusted ones (in the same way
as the flats are complementary to the cotorsion modules) we call _very flat_.
All very flat modules have projective dimensions not exceeding $1$. The wide
applicability of contraadjusted and very flat modules (cf. [53, Remark 2.6])
implies the importance of very flat morphisms of schemes, and we initiate the
study of these (though our results in this direction are still far from what
one would hope for).
The above-discussed phenomenon of dependence of the category of contramodules
over the coring ${\mathcal{C}}={\mathcal{O}}(T\times_{X}T)$ over the ring
$A={\mathcal{O}}(T)$ on the affine covering $T\longrightarrow X$ used to
construct it manifests itself in our approach in the unexpected predicament of
the _contraherence property_ of a cosheaf of ${\mathcal{O}}_{X}$-modules
_being not local_. So we have to deal with the _locally contraherent
cosheaves_ , and the necessity to control the extension of this locality
brings the coverings back.
Once an open covering in restriction to which our cosheaf becomes contraherent
is safely fixed, though, many other cosheaf properties that we consider in
this paper become indeed local. And any locally contraherent cosheaf on a
quasi-compact semi-separated scheme has a finite left Čech resolution by
contraherent cosheaves.
One difference between homological theories developed in the settings of exact
and abelian categories is that whenever a functor between abelian categories
isn’t exact, a similar functor between exact categories will tend to have a
shrinked domain. Any functor between abelian categories that has an everywhere
defined left or right derived functor will tend to be everywhere defined
itself, if only because one can always pass to the degree-zero cohomology of
the derived category objects. Not so with exact categories, in which complexes
may have no cohomology objects in general.
Hence the (sometimes annoying) necessity to deal with multitudes of domains of
definitions of various functors in our exposition. On the other hand, a
functor with the domain consisting of adjusted objects is typically exact on
the exact subcategory where it is defined.
Another difference between the theory of comodules and contramodules over
corings as developed in [50] and our present setting is that in _loc. cit._ we
considered corings ${\mathcal{C}}$ over base rings $A$ of finite homological
dimension. On the other hand, the ring $A={\mathcal{O}}(T)$ constructed above
has infinite homological dimension in most cases, while the coring
${\mathcal{C}}={\mathcal{O}}(T\times_{X}T)$ can be said to have “finite
homological dimension relative to $A$”. For this reason, while the comodule-
contramodule correspondence theorem [50, Theorem 5.4] was stated for the
derived categories of the second kind, one of the most general of our co-
contra correspondence results in this paper features an equivalence of the
conventional derived categories.
One application of this equivalence is a new construction of the extraordinary
inverse image functor $f^{!}$ on the derived categories of quasi-coherent
sheaves for any morphism of quasi-compact semi-separated schemes $f\colon
Y\longrightarrow X$. To be more precise, this is a construction of what we
call _Neeman’s extraordinary inverse image functor_ , that is the right
adjoint functor $f^{!}$ to the derived direct image functor
${\mathbb{R}}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
on the derived categories of quasi-coherent sheaves (see discussion below). In
fact, we construct a _right derived_ functor ${\mathbb{R}}f^{!}$, rather than
just a triangulated functor $f^{!}$, as it was usually done before [29, 45].
To a morphism $f$ one assigns the direct and inverse image functors
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ and $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ between the abelian categories of quasi-
coherent sheaves on $X$ and $Y$; the functor $f^{*}$ is left adjoint to the
functor $f_{*}$. To the same morphism, one also assigns the direct image
functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ between the exact categories
of colocally projective contraherent cosheaves and the inverse image functor
$f^{!}\colon X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ between the exact categories
of locally injective locally contraherent cosheaves on $X$ and $Y$. The
functor $f^{!}$ is “partially” right adjoint to the functor $f_{!}$.
Passing to the derived functors, one obtains the adjoint functors
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\allowbreak\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathbb{L}}f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
between the (conventional unbounded) derived categories of quasi-coherent
cosheaves. One also obtains the adjoint functors
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
between the derived categories of contraherent cosheaves on $X$ and $Y$.
The derived co-contra correspondence (for the conventional derived categories)
provides equivalences of triangulated categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
transforming the direct image functor ${\mathbb{R}}f_{*}$ into the direct
image functor ${\mathbb{L}}f_{!}$. So the two inverse image functors
${\mathbb{L}}f^{*}$ and ${\mathbb{R}}f^{!}$ can be viewed as the adjoints on
the two sides to the same triangulated functor of direct image. This finishes
our construction of the triangulated functor
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
right adjoint to ${\mathbb{R}}f_{*}$.
As usually in homological algebra, the tensor product and Hom-type operations
on the quasi-coherent sheaves and contraherent cosheaves play an important
role in our theory. First of all, under appropriate adjustness assumptions one
can assign a contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ to a
quasi-coherent sheaf ${\mathcal{F}}$ and a contraherent cosheaf
${\mathfrak{P}}$ over a scheme $X$. This operation is the analogue of the
tensor product of quasi-coherent sheaves in the contraherent world.
Secondly, to a quasi-coherent sheaf ${\mathcal{F}}$ and a cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ over a scheme $X$ one can assign
a cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathfrak{P}}$. Under our duality-
analogy, this corresponds to taking the sheaf of
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}$ from a quasi-coherent
sheaf to a sheaf of ${\mathcal{O}}_{X}$-modules. When the scheme $X$ is
Noetherian, the sheaf ${\mathcal{F}}$ is coherent, and the cosheaf
${\mathfrak{P}}$ is contraherent, the cosheaf
${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathfrak{P}}$ is contraherent, too.
Under some other assumptions one can apply the (derived or underived)
_contraherator_ functor ${\mathbb{L}}\operatorname{\mathfrak{C}}$ or
$\operatorname{\mathfrak{C}}$ to the cosheaf
${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathfrak{P}}$ to obtain the
(complex of) contraherent cosheaves
${\mathcal{F}}\otimes_{X{\operatorname{\mathrm{-ct}}}}^{\mathbb{L}}{\mathfrak{P}}$
or ${\mathcal{F}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{P}}$.
These are the analogues of the quasi-coherent internal Hom functor
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}$
on the quasi-coherent sheaves, which can be obtained by applying the coherator
functor $\operatorname{\mathcal{Q}}$ [61, Appendix B] to the
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({-},{-})$
sheaf.
The remaining two operations are harder to come by. Modelled after the
comodule-contramodule correspondence functors $\Phi_{\mathcal{C}}$ and
$\Psi_{\mathcal{C}}$ from [50], they play a similarly crucial role in our
present co-contra correspondence theory. Given a quasi-coherent sheaf
${\mathcal{F}}$ and a cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$
on a quasi-separated scheme $X$, one constructs a quasi-coherent sheaf
${\mathcal{F}}\odot_{X}{\mathfrak{P}}$ on $X$. Given two quasi-coherent
sheaves ${\mathcal{F}}$ and ${\mathcal{P}}$ on a quasi-separated scheme, under
certain adjustness assuptions one can constuct a contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})$.
_Derived categories of the second kind_ , whose roots go back to the work of
Husemoller, Moore, and Stasheff on two kinds of differential derived functors
[31] and Hinich’s paper about DG-coalgebras [30], were introduced in their
present form in the author’s monograph [50] and memoir [51]. The most
important representatives of this class of derived category constructions are
known as the _coderived_ and the _contraderived_ categories; the difference
between them consists in the use of the closure with respect to infinite
direct sums in one case and with respect to infinite products in the other.
Here is a typical example of how they occur.
According to Iyengar and Krause [33], the homotopy category of complexes of
projective modules over a Noetherian commutative ring with a dualizing complex
is equivalent to the homotopy category of complexes of injective modules. This
theorem was extended to semi-separated Noetherian schemes with dualizing
complexes by Neeman [47] and Murfet [42] in the following form: the derived
category ${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ of
the exact category of flat quasi-coherent sheaves on such a scheme $X$ is
equivalent to the homotopy category of injective quasi-coherent sheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$. These
results are known as the _covariant Serre–Grothendieck duality_ theory.
One would like to reformulate this equivalence so that it connects certain
derived categories of modules/sheaves, rather than just subcategories of
resolutions. In other words, it would be nice to have some procedure assigning
complexes of projective, flat, and/or injective modules/sheaves to arbitrary
complexes.
In the case of modules, the homotopy category of projectives is identified
with the contraderived category of the abelian category of modules, while the
homotopy category of injectives is equivalent to the coderived category of the
same abelian category. Hence the Iyengar–Krause result is interpreted as an
instance of the “co-contra correspondence”—in this case, an equivalence
between the coderived and contraderived categories of the same abelian
category.
In the case of quasi-coherent sheaves, however, only a half of the above
picture remains true. The homotopy category of injectives
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$ is still
equivalent to the coderived category of quasi-coherent sheaves
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$. But the
attempt to similarly describe the derived category of flats runs into the
problem that the infinite products of quasi-coherent sheaves are not exact, so
the contraderived category construction does not make sense for them.
This is where the contraherent cosheaves come into play. The covariant
Serre–Grothendieck duality for a nonaffine (but semi-separated) scheme with a
dualizing complex is an equivalence of _four_ triangulated categories, rather
than just two. In addition to the derived category of flat quasi-coherent
sheaves ${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ and
the homotopy category of injective quasi-coherent sheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$, there are
also the homotopy category of projective contraherent cosheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$ and the
derived category of locally injective contraherent cosheaves
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$.
Just as the homotopy category of injectives
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$ is equivalent
to the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$, the homotopy
category of projectives
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$ is identified
with the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$ of
contraherent cosheaves. The equivalence between the two “injective” categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ and
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$ does not
depend on the dualizing complex, and neither does the equivalence between the
two “projective” (or “flat”) categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{fl}})$ and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$. The
equivalences connecting the “injective” categories with the “projective” ones
do.
So far, our discussion in this introduction was essentially limited to semi-
separated schemes. Such a restriction of generality is not really that natural
or desirable. Indeed, the affine plane with a double point (which is not semi-
separated) is no less a worthy object of study than the line with a double
point (which is). There is a problem, however, that appears to stand in the
way of a substantial development of the theory of contraherent cosheaves on
the more general kinds of schemes.
The problem is that even when the quasi-coherent sheaves on schemes remain
well-behaved objects, the complexes of such sheaves may be no longer so. More
precisely, the trouble is with the derived functors of direct image of
complexes of quasi-coherent sheaves, which do not always have good properties
(such as locality along the base, etc.) Hence the common wisdom that for the
more complicated schemes $X$ one is supposed to consider complexes of sheaves
of ${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology sheaves, rather
than complexes of quasi-coherent sheaves as such (see, e. g., [56]). As we do
not know what is supposed to be either a second kind or a contraherent
analogue of the construction of the derived category of complexes of sheaves
of ${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology sheaves, we have
to restrict our exposition to, approximately, those situations where the
derived category of the abelian category of quasi-coherent sheaves on $X$ is
still a good category to work in.
There are, basically, two such situations [61, Appendix B]: (1) the quasi-
compact semi-separated schemes and (2) the Noetherian or, sometimes, locally
Noetherian schemes (which, while always quasi-separated, do not have to be
semi-separated). Accordingly, our exposition largely splits in two streams
corresponding to the situations (1) and (2), where different techniques are
applicable. The main difference in the generality level with the quasi-
coherent case is that in the contraherent context one often needs also to
assume one’s schemes to have finite Krull dimension, in order to use Raynaud
and Gruson’s homological dimension results [55].
In particular, we prove the equivalence of the conventional derived categories
of quasi-coherent sheaves and contraherent cosheaves not only for quasi-
compact semi-separated schemes, but also, separately, for all Noetherian
schemes of finite Krull dimension. As to the covariant Serre–Grothendieck
duality theorem, it remains valid in the case of a non-semi-separated
Noetherian scheme with a dualizing complex in the form of an equivalence
between two derived categories of the second kind: the coderived category of
quasi-coherent sheaves
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$
and the contraderived category of contraherent cosheaves
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$.
Before describing another application of our theory, let us have a more
detailed discussion of the extraordinary inverse images of quasi-coherent
sheaves. There are, in fact, _two_ different functors going by the name of
“the functor $f^{!}$” in the literature. One of them, which we name after
Neeman, is simply the functor right adjoint to the derived direct image
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$,
which we were discussing above. Neeman proved its existence for an arbitrary
morphism of quasi-compact semi-separated schemes, using the techniques of
compact generators and Brown representability [45]. Similar arguments apply in
the case of an arbitrary morphism of Noetherian schemes $f\colon
Y\longrightarrow X$.
The other functor, which we call _Deligne’s extraordinary inverse image_ and
denote (to avoid ambiguity) by $f^{+}$, coincides with Neeman’s functor in the
case of a proper morphism $f$. In the case when $f$ is an open embedding, on
the other hand, the functor $f^{+}$ coincides with the conventional
restriction (inverse image) functor $f^{*}$ (which is left adjoint to
${\mathbb{R}}f_{*}$, rather than right adjoint). More generally, in the case
of a smooth morphism $f$ the functor $f^{+}$ only differs from $f^{*}$ by a
dimensional shift and a top form bundle twist. This is the functor that was
constructed in Hartshorne’s book [29] and Deligne’s appendix to it [14] (hence
the name). It is Deligne’s, rather than Neeman’s, extraordinary inverse image
functor that takes a dualizing complex on $X$ to a dualizing complex on $Y$,
i. e.,
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=f^{+}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
It is an important idea apparently due to Gaitsgory [21] that Deligne’s
extraordinary inverse image functor actually acts between the coderived
categories of quasi-coherent sheaves, rather than between their conventional
derived categories. To be sure, Hartshorne and Deligne only construct their
functor for bounded below complexes (for which there is no difference between
the coderived and derived categories). Gaitsgory’s “ind-coherent sheaves” are
closely related to our coderived category of quasi-coherent sheaves.
Concerning Neeman’s right adjoint functor $f^{!}$, it can be shown to exist on
both the derived and (in the case of Noetherian schemes) the coderived
categories. The problem arises when one attempts to define the functor $f^{+}$
by decomposing a morphism $f\colon Y\longrightarrow X$ into proper morphisms
$g$ and open embeddings $h$ and subsequently composing the functors $g^{!}$
for the former with the functors $h^{*}$ for the latter. It just so happens
that the functor
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
obtained in this way depends on the chosen decomposition of the morphism $f$.
This was demonstrated by Neeman in his counterexample [45, Example 6.5].
The arguments of Deligne [14], if extended to the conventional unbounded
derived categories, break down on a rather subtle point: while it is true that
the restriction of an injective quasi-coherent sheaf to an open subscheme of a
Noetherian scheme remains injective, the restriction of a homotopy injective
complex of quasi-coherent sheaves to such a subscheme may no longer be
homotopy injective. On the other hand, Deligne computes the Hom into the
object produced by his extraordinary inverse image functor from an arbitrary
bounded complex of coherent sheaves, which is essentially sufficient to make a
functor between the coderived categories well-defined, as these are compactly
generated by bounded complexes of coherents.
The above discussion of the functor $f^{+}$ is to be compared with the remark
that the conventional derived inverse image functor ${\mathbb{L}}f^{*}$ is not
defined on the coderived categories of quasi-coherent sheaves (but only on
their derived categories), except in the case of a morphism $f$ of finite flat
dimension [53, 21]. On the other hand, the conventional inverse image $f^{*}$
is perfectly well defined on the derived categories of flat quasi-coherent
sheaves (where one does not even need to resolve anything in order to
construct a triangulated functor). We show that the functor
$f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is transformed into the functor
$f^{+}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
by the above-described equivalence of triangulated categories.
Let us finally turn to the connection between Deligne’s extrardinary inverse
image functor and our contraherent cosheaves. Given a morphism of Noetherian
schemes $f\colon Y\longrightarrow X$, just as the direct image functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
has a right adjoint functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$,
so does the direct image functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
has a left adjoint functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$.
This is the contraherent analogue of Neeman’s extraordinary inverse image
functor for quasi-coherent sheaves.
Now assume that $f$ is a morphism of finite type and the scheme $X$ has a
dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$; set
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=f^{+}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
As we mentioned above, the choice of the dualizing complexes induces
equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and similarly for $Y$. Those equivalences of categories transform the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
into a certain functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$.
It is the latter functor that turns out to be isomorphic to Deligne’s
extraordinary inverse image functor (which we denote here by $f^{+}$).
Two proofs of this result, working on slightly different generality levels,
are given in this paper. One applies to “compactifiable” separated morphisms
of finite type between semi-separated Noetherian schemes and presumes
comparison with a Deligne-style construction of the functor $f^{+}$, involving
a decomposition of the morphism $f$ into an open embedding followed by a
proper morphism [14]. The other one is designed for “embeddable” morphisms of
finite type between Noetherian schemes and the comparison with a Hartshorne-
style construction of $f^{+}$ based on a factorization of $f$ into a finite
morphism followed by a smooth one [29].
To end, let us describe some prospects for future research and applications of
contraherent cosheaves. One of such expected applications is related to the
${\boldsymbol{\mathcal{D}}}$–${\boldsymbol{\Omega}}$ duality theory. Here
${\boldsymbol{\mathcal{D}}}$ stands for the sheaf of rings of differential
operators on a smooth scheme and ${\boldsymbol{\Omega}}$ denotes the de Rham
DG-algebra. The derived ${\boldsymbol{\mathcal{D}}}$–${\boldsymbol{\Omega}}$
duality, as formulated in [51, Appendix B] (see also [57] for a further
development), happens on two sides. On the “co” side, the functor
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({\boldsymbol{\Omega}},{-})$
takes right ${\boldsymbol{\mathcal{D}}}$-modules to right DG-modules over
${\boldsymbol{\Omega}}$, and there is the adjoint functor
${-}\otimes_{{\mathcal{O}}_{X}}{\boldsymbol{\mathcal{D}}}$. These functors
induce an equivalence between the derived category of
${\boldsymbol{\mathcal{D}}}$-modules and the coderived category of DG-modules
over ${\boldsymbol{\Omega}}$.
On the “contra” side, over an affine scheme $X$ the functor
${\boldsymbol{\Omega}}(X)\otimes_{{\mathcal{O}}(X)}{-}$ takes left
${\boldsymbol{\mathcal{D}}}$-modules to left DG-modules over
${\boldsymbol{\Omega}}$, and the adjoint functor is
$\operatorname{Hom}_{{\mathcal{O}}(X)}({\boldsymbol{\mathcal{D}}}(X),{-})$.
The induced equivalence is between the derived category of
${\boldsymbol{\mathcal{D}}}(X)$-modules and the contraderived category of DG-
modules over ${\boldsymbol{\Omega}}(X)$. One would like to extend the “contra”
side of the story to nonaffine schemes using the contraherent
cosheafification, as opposed to the quasi-coherent sheafification on the “co”
side. The need for the contraherent cosheaves arises, once again, because the
contraderived category construction does not make sense for quasi-coherent
sheaves. The contraherent cosheaves are designed for being plugged into it.
Another direction in which we would like to extend the theory presented below
is that of Noetherian formal schemes, and more generally, ind-schemes of ind-
finite or even ind-infinite type. The idea is to define an exact category of
contraherent cosheaves of contramodules over a formal scheme that would serve
as the natural “contra”-side counterpart to the abelian category of quasi-
coherent torsion sheaves on the “co” side. (For a taste of the contramodule
theory over complete Noetherian rings, the reader is referred to [54, Appendix
B].)
Moreover, one would like to have a “semi-infinite” version of the homological
formalism of quasi-coherent sheaves and contraherent cosheaves (see Preface to
[50] for the related speculations). A possible setting might be that of an
ind-scheme flatly fibered over an ind-Noetherian ind-scheme with quasi-compact
schemes as the fibers. One would define the semiderived categories of quasi-
coherent sheaves and contraherent cosheaves on such an ind-scheme as mixtures
of the co/contraderived categories along the base ind-scheme and the
conventional derived categories along the fibers.
The idea is to put two our (present) co-contra correspondence theorems “on top
of” one another. That is, join the equivalence of conventional derived
categories of quasi-coherent sheaves and contraherent cosheaves on a quasi-
compact semi-separated scheme together with the equivalence between the
coderived category of quasi-coherent (torsion) sheaves and the contraderived
category of contraherent cosheaves (of contramodules) on an (ind-)Noetherian
(ind-)scheme with a dualizing complex into a single “semimodule-
semicontramodule correspondence” theorem claiming an equivalence between the
two semiderived categories.
What we have in mind here is a theory that might bear the name of a “semi-
infinite homological theory of doubly infinite-dimensional algebraic
varieties”, or _semi-infinite algebraic geometry_. In addition to the above-
described correspondence between the complexes of sheaves and cosheaves, this
theory would also naturally feature a double-sided derived functor of
semitensor (or “mixed tensor-cotensor”) product of complexes of quasi-coherent
torsion sheaves (cf. the discussion of cotensor products of quasi-coherent
sheaves in [53, Section B.2]), together with its contraherent analogue.
Furthermore, one might hope to join the de Rham DG-module and the contramodule
stories together in a single theory by considering contraherent cosheaves of
DG-contramodules over the de Rham–Witt complex, with an eye to applications to
crystalline sheaves and the $p$-adic Hodge theory. As compared so such high
hopes, our real advances in this paper are quite modest.
The very possibility of having a meaningful theory of contraherent cosheaves
on a scheme rests on there being enough contraadjusted or cotorsion modules
over a commutative or associative ring. The contemporary-style set-theoretical
proofs of these results [17, 6, 19, 58, 5] only seem to work in abelian (or
exact) categories with exact functors of filtered inductive limits. Thus the
first problem one encounters when trying to build up the theories of
contraherent cosheaves of contramodules is the necessity of developing
applicable techniques for constructing flat, cotorsion, contraadjusted, and
very flat contramodules to be used in the resolutions.
In the present version of the paper, we partially overcome this obstacle by
constructing enough flat, cotorsion, contraadjusted, and very flat
contramodules over a Noetherian ring in the adic topology, enough flat and
cotorsion contramodules over a pro-Noetherian topological ring of totally
finite Krull dimension, and also enough contraadjusted and very flat
contramodules over the projective limit of a sequence of commutative rings and
surjective morphisms between them with finitely generated kernel ideals. The
constructions are based on the existence of enough objects of the respective
classes in the conventional categories of modules, which is used as a given
fact, and also on the older, more explicit construction [63] of cotorsion
resolutions of modules over commutative Noetherian rings of finite Krull
dimension.
This provides the necessary background for possible definitions of locally
contraadjusted or locally cotorsion contraherent cosheaves of contramodules
over Noetherian formal schemes or ind-Noetherian ind-schemes of totally finite
Krull dimension, and also of locally contraadjusted contraherent cosheaves
over ind-schemes of a more general nature, such as, e. g., the
projectivization of an infinite-dimensional discrete vector space, or the
total space of such projective space’s cotangent bundle. The latter might be a
typical kind of example for which one would like to have the “semi-infinite
algebraic geometry” worked out. And among the ind-Noetherian ind-schemes of
totally finite Krull dimension there are, e. g., the spectra of the (sheaves
of) rings of Witt vectors of algebraic varieties in finite characteristic.
This kind of work, intended to prepare ground for future theory building, is
presently relegated to appendices. There we also construct the derived co-
contra correspondence (an equivalence between the coderived category of
torsion modules and the contraderived category of contramodules) over affine
Noetherian formal schemes and ind-affine ind-Noetherian ind-schemes with
dualizing complexes. Another appendix is devoted to the co-contra
correspondence over noncommutative semi-separated stacks (otherwise known as
flat corings [36]).
Among the many people whose questions, remarks, and suggestions contributed to
the present research, I should mention Sergey Arkhipov, Roman Bezrukavnikov,
Alexander Polishchuk, Sergey Rybakov, Alexander Efimov, Amnon Neeman, Henning
Krause, Mikhail Bondarko, Alexey Gorodentsev, Paul Bressler, Vladimir Hinich,
Greg Stevenson, Michael Finkelberg, and Pierre Deligne. Parts of this paper
were written when I was visiting the University of Bielefeld, and I want to
thank Collaborative Research Center 701 and Prof. Krause for the invitation.
This work was supported in part by RFBR grants.
## 1\. Contraadjusted and Cotorsion Modules
### 1.1. Contraadjusted and very flat modules
Let $R$ be a commutative ring. We will say that an $R$-module $P$ is
_contraadjusted_ if the $R$-module $\operatorname{Ext}_{R}^{1}(R[r^{-1}],P)$
vanishes for every element $r\in R$. An $R$-module $F$ is called _very flat_
if one has $\operatorname{Ext}_{R}^{1}(F,P)=0$ for every contraadjusted
$R$-module $P$.
By the definition, any injective $R$-module is contraadjusted and any
projective $R$-module is very flat. Notice that the projective dimension of
the $R$-module $R[r^{-1}]$ never exceeds $1$, as it has a natural two-term
free resolution
$0\longrightarrow\bigoplus_{n=0}^{\infty}R\longrightarrow\bigoplus_{n=0}^{\infty}R\longrightarrow
R[r^{-1}]\longrightarrow 0$. It follows that any quotient module of a
contraadjusted module is contraadjusted, and one has
$\operatorname{Ext}_{R}^{>0}(F,P)=0$ for any very flat $R$-module $F$ and
contraadjusted $R$-module $P$.
Computing the $\operatorname{Ext}^{1}$ in terms of the above resolution, one
can more explicitly characterize contraadjusted $R$-modules as follows. An
$R$-module $P$ is contraadjusted if and only if for any sequence of elements
$p_{0}$, $p_{1}$, $p_{2}$, $\dotsc\in P$ and $r\in R$ there exists a (not
necessarily unique) sequence of elements $q_{0}$, $q_{1}$, $q_{2}$, $\dotsc\in
P$ such that $q_{i}=p_{i}+rq_{i+1}$ for all $i\ge 0$.
Furthermore, the projective dimension of any very flat module is equal to $1$
or less. Indeed, any $R$-module $M$ has a two-term right resolution by
contraadjusted modules, which can be used to compute
$\operatorname{Ext}^{*}_{R}(F,M)$ for a very flat $R$-module $F$. (The
converse assertion is _not_ true, however; see Example 1.7.7 below.)
It is also clear that the classes of contraadjusted and very flat modules are
closed under extensions. Besides, the class of very flat modules is closed
under the passage to the kernel of a surjective morphism (i. e., the kernel of
a surjective morphism of very flat $R$-modules is very flat). In addition, we
notice that the class of contraadjusted $R$-modules is closed under infinite
products, while the class of very flat $R$-modules is closed under infinite
direct sums.
###### Theorem 1.1.1.
(a) Any $R$-module can be embedded into a contraadjusted $R$-module in such a
way that the quotient module is very flat.
(b) Any $R$-module admits a surjective map onto it from a very flat $R$-module
such that the kernel is contraadjusted.
###### Proof.
Both assertions follow from the results of Eklof and Trlifaj [17, Theorem 10].
It suffices to point out that all the $R$-modules of the form $R[r^{-1}]$ form
a set rather than a proper class. For the reader’s convenience and our future
use, parts of the argument from [17] are reproduced below. ∎
The following definition will be used in the sequel. Let ${\mathsf{A}}$ be an
abelian category with exact functors of inductive limit, and let
${\mathsf{C}}\subset{\mathsf{A}}$ be a class of objects. An object
$X\in{\mathsf{A}}$ is said to be a _transfinitely iterated extension_ of
objects from ${\mathsf{C}}$ if there exist a well-ordered set $\Gamma$ and a
family of subobjects $X_{\gamma}\subset X$, $\gamma\in\Gamma$, such that
$X_{\delta}\subset X_{\gamma}$ whenever $\delta<\gamma$, the union (inductive
limit) $\varinjlim_{\gamma\in\Gamma}X_{\gamma}$ of all $X_{\gamma}$ coincides
with $X$, and the quotient objects
$X_{\gamma}/\varinjlim_{\delta<\gamma}X_{\delta}$ belong to ${\mathsf{C}}$ for
all $\gamma\in\Gamma$ (cf. Section 4.1).
By [17, Lemma 1], any transfinitely iterated extension of the $R$-modules
$R[r^{-1}]$, with arbitrary $r\in R$, is a very flat $R$-module. Proving a
converse assertion will be one of our goals. The following lemma is a
particular case of [17, Theorem 2].
###### Lemma 1.1.2.
Any $R$-module can be embedded into a contraadjusted $R$-module in such a way
that the quotient module is a transfinitely iterated extension of the
$R$-modules $R[r^{-1}]$.
###### Proof.
The proof is a set-theoretic argument based on the fact that the Cartesian
square of any infinite cardinality $\lambda$ is equicardinal to $\lambda$. In
our case, let $\lambda$ be any infinite cardinality no smaller than the
cardinality of the ring $R$. For an $R$-module $L$ of the cardinality not
exceeding $\lambda$ and an $R$-module $M$ of the cardinality $\mu$, the set
$\operatorname{Ext}^{1}_{R}(L,M)$ has the cardinality at most $\mu^{\lambda}$,
as one can see by computing the $\operatorname{Ext}^{1}$ in terms of a
projective resolution of the first argument. In particular; set
$\aleph=2^{\lambda}$; then for any $R$-module $M$ of the cardinality not
exceeding $\aleph$ and any $r\in R$ the cardinality of the set
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],M)$ does not exceed $\aleph$, either.
Let $\bethl$ be the smallest cardinality that is larger than $\aleph$ and let
$\Delta$ be the smallest ordinal of the cardinality $\bethl$. Notice that the
natural map
$\varinjlim_{\delta\in\Delta}\operatorname{Ext}^{*}_{R}(L,Q_{\delta})\longrightarrow\operatorname{Ext}^{*}_{R}(L,\>\varinjlim_{\delta\in\Delta}Q_{\delta})$
is an isomorphism for any $R$-module $L$ of the cardinality not exceeding
$\lambda$ (or even $\aleph$) and any inductive system of $R$-modules
$Q_{\delta}$ indexed by $\Delta$. Indeed, the functor of filtered inductive
limit of abelian groups is exact and the natural map
$\varinjlim_{\delta\in\Delta}\operatorname{Hom}_{R}(L,Q_{\delta})\longrightarrow\operatorname{Hom}_{R}(L,\>\varinjlim_{\delta\in\Delta}Q_{\delta})$
is an isomorphism for any $R$-module $L$ of the cardinality not exceeding
$\aleph$. The latter assertion holds because the image of any map of sets
$L\longrightarrow\Delta$ is contained in a proper initial segment
$\\{\delta^{\prime}\mid\delta^{\prime}<\delta\\}\subset\Delta$ for some
$\delta\in\Delta$.
We proceed by induction on $\Delta$ constructing for every element
$\delta\in\Delta$ an $R$-module $P_{\delta}$ and a well-ordered set
$\Gamma_{\delta}$. For any $\delta^{\prime}<\delta$, we will have an embedding
of $R$-modules $P_{\delta^{\prime}}\longrightarrow P_{\delta}$ (such that the
three embeddings form a commutative diagram for any three elements
$\delta^{\prime\prime}<\delta^{\prime}<\delta$) and an ordered embedding
$\Gamma_{\delta^{\prime}}\subset\Gamma_{\delta}$ (making
$\Gamma_{\delta^{\prime}}$ an initial segment of $\Gamma_{\delta}$).
Furthermore, for every element $\gamma\in\Gamma_{\delta}$, a particular
extension class
$c(\gamma,\delta)\in\operatorname{Ext}^{1}_{R}(R[r(\gamma)^{-1}],P_{\delta})$,
where $r(\gamma)\in R$, will be defined. For every $\delta^{\prime}<\delta$
and $\gamma\in\Gamma_{\delta^{\prime}}$, the class $c(\gamma,\delta)$ will be
equal to the image of the class $c(\gamma,\delta^{\prime})$ with respect to
the natural map
$\operatorname{Ext}^{1}_{R}(R[r(\gamma)^{-1}],P_{\delta^{\prime}})\longrightarrow\operatorname{Ext}^{1}_{R}(R[r(\gamma)^{-1}],P_{\delta})$
induced by the embedding $P_{\delta^{\prime}}\longrightarrow P_{\delta}$.
At the starting point $0\in\Delta$, the module $P_{0}$ is our original
$R$-module $M$ and the set $\Gamma_{0}$ is the disjoint union of all the sets
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],M)$ with $r\in R$, endowed with an
arbitrary well-ordering. The elements $r(\gamma)$ and the classes
$c(\gamma,0)$ for $\gamma\in\Gamma_{0}$ are defined in the obvious way.
Given $\delta=\delta^{\prime}+1\in\Delta$ and assuming that the $R$-module
$P_{\delta^{\prime}}$ and the set $\Gamma_{\delta^{\prime}}$ have been
constructed already, we produce the module $P_{\delta}$ and the set
$\Gamma_{\delta}$ as follows. It will be clear from the construction below
that $\delta^{\prime}$ is always smaller than the well-ordering type of the
set $\Gamma_{\delta^{\prime}}$. So there is a unique element
$\gamma_{\delta^{\prime}}\in\Gamma_{\delta^{\prime}}$ corresponding to the
ordinal $\delta^{\prime}$ (i. e., such that the well-ordering type of the
subset all the elements in $\Gamma_{\delta^{\prime}}$ that are smaller than
$\gamma_{\delta^{\prime}}$ is equivalent to $\delta^{\prime}$).
Define the $R$-module $P_{\delta}$ as the middle term of the extension
corresponding to the class
$c(\gamma_{\delta^{\prime}},\delta^{\prime})\in\operatorname{Ext}^{1}_{R}(R[r(\gamma_{\delta^{\prime}})^{-1}],P_{\delta^{\prime}})$.
There is a natural embedding $P_{\delta^{\prime}}\longrightarrow P_{\delta}$,
as required. Set $\Gamma_{\delta}$ to be the disjoint union of
$\Gamma_{\delta^{\prime}}$ and the sets
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta})$ with $r\in R$, well-ordered
so that $\Gamma_{\delta^{\prime}}$ is an initial segment, while the well-
ordering of the remaining elements is chosen arbitrarily. The elements
$r(\gamma)$ for $\gamma\in\Gamma_{\delta^{\prime}}$ have been defined already
on the previous steps and the classes $c(\gamma,\delta)$ for such $\gamma$ are
defined in the unique way consistent with the previous step, while for the
remaining $\gamma\in\Gamma_{\delta}\setminus\Gamma_{\delta^{\prime}}$ these
elements and classes are defined in the obvious way.
When $\delta$ is a limit ordinal, set
$P_{\delta}=\varinjlim_{\delta^{\prime}<\delta}P_{\delta}^{\prime}$. Let
$\Gamma_{\delta}$ be the disjoint union of
$\bigcup_{\delta^{\prime}<\delta}\Gamma_{\delta^{\prime}}$ and the sets
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta})$ with $r\in R$, well-ordered
so that $\bigcup_{\delta^{\prime}<\delta}\Gamma_{\delta^{\prime}}$ is an
initial segment. The elements $r(\gamma)$ and $c(\gamma,\delta)$ for
$\gamma\in\Gamma_{\delta}$ are defined as above.
Arguing by transfinite induction, one easily concludes that the cardinality of
the $R$-module $P_{\delta}$ never exceeds $\aleph$ for $\delta\in\Delta$, and
neither does the cardinality of the set $\Gamma_{\delta}$. It follows that the
well-ordering type of the set
$\Gamma=\bigcup_{\delta\in\Delta}\Gamma_{\delta}$ is equal to $\Delta$. So for
every $\gamma\in\Gamma$ there exists $\delta\in\Delta$ such that
$\gamma=\gamma_{\delta}$.
Set $P=\varinjlim_{\delta\in\Delta}P_{\delta}$. By construction, there is a
natural embedding of $R$-modules $M\longrightarrow P$ and the cokernel is a
transfinitely iterated extension of the $R$-modules $R[r^{-1}]$. As every
class $c\in\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta})$ corresponds to an
element $\gamma\in\Gamma_{\delta}$, has the corresponding ordinal
$\delta^{\prime}\in\Delta$ such that $\gamma=\gamma_{\delta^{\prime}}$, and
dies in $\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta^{\prime}+1})$, we
conclude that $\operatorname{Ext}^{1}_{R}(R[r^{-1}],P)=0$. ∎
###### Lemma 1.1.3.
Any $R$-module admits a surjective map onto it from a transfinitely iterated
extension of the $R$-modules $R[r^{-1}]$ such that the kernel is
contraadjusted.
###### Proof.
The proof follows the second half of the proof of Theorem 10 in [17].
Specifically, given an $R$-module $M$, pick a surjective map onto it from a
free $R$-module $L$. Denote the kernel by $K$ and embed it into a
contraadjusted $R$-module $P$ so that the quotient module $Q$ is a
transfinitely iterated extension of the $R$-modules $R[r^{-1}]$. Then the
fibered coproduct $F$ of the $R$-modules $L$ and $P$ over $K$ is an extension
of the $R$-modules $Q$ and $L$. It also maps onto $M$ surjectively with the
kernel $P$. ∎
Both assertions of Theorem 1.1.1 are now proven.
###### Corollary 1.1.4.
An $R$-module is very flat if and only if it is a direct summand of a
transfinitely iterated extension of the $R$-modules $R[r^{-1}]$.
###### Proof.
The “if” part has been explained already; let us prove “only if”. Given a very
flat $R$-module $F$, present it as the quotient module of a transfinitely
iterated extension $E$ of the $R$-modules $R[r^{-1}]$ by a contraadjusted
$R$-module $P$. Since $\operatorname{Ext}^{1}_{R}(F,P)=0$, we can conclude
that $F$ is a direct summand of $E$. ∎
In particular, we have proven that any very flat $R$-module is flat.
###### Corollary 1.1.5.
(a) Any very flat $R$-module can be embedded into a contraadjusted very flat
$R$-module in such a way that the quotient module is very flat.
(b) Any contraadjusted $R$-module admits a surjective map onto it from a very
flat contraadjusted $R$-module such that the kernel is contraadjusted.
###### Proof.
Follows from Theorem 1.1.1 and the fact that the classes of contraadjusted and
very flat $R$-modules are closed under extensions. ∎
### 1.2. Affine geometry of contraadjusted and very flat modules
The results of this section form the module-theoretic background of our main
definitions and constructions in Sections 2–3.
###### Lemma 1.2.1.
(a) The class of very flat $R$-modules is closed with respect to the tensor
products over $R$.
(b) For any very flat $R$-module $F$ and contraadjusted $R$-module $P$, the
$R$-module $\operatorname{Hom}_{R}(F,P)$ is contraadjusted.
###### Proof.
One approach is to prove both assertions simultaneously using the adjunction
isomorphism
$\operatorname{Ext}_{R}^{1}(F\otimes_{R}G,\>P)\simeq\operatorname{Ext}_{R}^{1}(G,\operatorname{Hom}_{R}(F,P))$,
which clearly holds for any $R$-module $G$, any very flat $R$-module $F$, and
contraadjusted $R$-module $P$, and raising the generality step by step. Since
$R[r^{-1}]\otimes_{R}R[s^{-1}]\simeq R[(rs)^{-1}]$, it follows that the
$R$-module $\operatorname{Hom}_{R}(R[s^{-1}],P)$ is contraadjusted for any
contraadjusted $R$-module $P$ and $s\in R$. Using the same adjunction
isomorphism, one then concludes that the $R$-module $R[s^{-1}]\otimes_{R}G$ is
very flat for any very flat $R$-module $G$. From this one can deduce in full
generality the assertion (b), and then the assertion (a).
Alternatively, one can use the full strength of Corollary 1.1.4 and check that
the tensor product of two transfinitely iterated extensions of flat modules is
a transfinitely iterated extension of the pairwise tensor products. Then
deduce (b) from (a). ∎
###### Lemma 1.2.2.
Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings. Then
(a) any contraadjusted $S$-module is also a contraadjusted $R$-module in the
$R$-module structure obtained by the restriction of scalars via $f$;
(b) if $F$ is a very flat $R$-module, then the $S$-module $S\otimes_{R}F$
obtained by the extension of scalars via $f$ is also very flat;
(c) if $F$ is a very flat $R$-module and $Q$ is a contraadjusted $S$-module,
then $\operatorname{Hom}_{R}(F,Q)$ is also a contraadjusted $S$-module;
(d) if $F$ is a very flat $R$-module and $G$ is a very flat $S$-module, then
$F\otimes_{R}G$ is also a very flat $S$-module.
###### Proof.
Part (a): one has
$\operatorname{Ext}_{R}^{*}(R[r^{-1}],P)\simeq\operatorname{Ext}_{S}^{*}(S[f(r)^{-1}],P)$
for any $R$-module $P$ and $r\in R$. Part (b) follows from part (a), or
alternatively, from Corollary 1.1.4. To prove part (c), notice that
$\operatorname{Hom}_{R}(F,Q)\simeq\operatorname{Hom}_{S}(S\otimes_{R}F,\>Q)$
and use part (b) together with Lemma 1.2.1(b) (applied to the ring $S$).
Similarly, part (d) follows from part (b) and Lemma 1.2.1(a). ∎
###### Lemma 1.2.3.
Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings such
that the localization $S[s^{-1}]$ is a very flat $R$-module for any element
$s\in S$. Then
(a) the $S$-module $\operatorname{Hom}_{R}(S,P)$ obtained by the coextension
of scalars via $f$ is contraadjusted for any contraadjusted $R$-module $P$;
(b) any very flat $S$-module is also a very flat $R$-module (in the $R$-module
structure obtained by the restriction of scalars via $f$);
(c) the $S$-module $\operatorname{Hom}_{R}(G,P)$ is contraadjusted for any
very flat $S$-module $G$ and contraadjusted $R$-module $P$.
###### Proof.
Part (a): one has
$\operatorname{Ext}_{S}^{1}(S[s^{-1}],\operatorname{Hom}_{R}(S,P))\simeq\operatorname{Ext}_{R}^{1}(S[s^{-1}],P)$
for any $R$-module $P$ such that $\operatorname{Ext}_{R}^{1}(S,P)=0$ and any
$s\in S$. Part (b) follows from part (a), or alternatively, from Corollary
1.1.4. Part (c): one has
$\operatorname{Ext}_{S}^{1}(S[s^{-1}],\operatorname{Hom}_{R}(G,P))\simeq\operatorname{Ext}_{R}^{1}(G\otimes_{S}S[s^{-1}],\>P)$.
By Lemma 1.2.1(a), the $S$-module $G\otimes_{S}S[s^{-1}]$ is very flat; by
part (b), it is also a very flat $R$-module; so the desired vanishing follows.
∎
###### Lemma 1.2.4.
Let $R\longrightarrow S$ be a homomorphism of commutative rings such that the
related morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding. Then $S$ is a very flat $R$-module.
###### Proof.
The open subset $\operatorname{Spec}S\subset\operatorname{Spec}R$, being
quasi-compact, can be covered by a finite number of principal affine open
subsets $\operatorname{Spec}R[r_{\alpha}^{-1}]\subset\operatorname{Spec}R$,
where $\alpha=1$, …, $N$. The Čech sequence
(1) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}R[r_{\alpha}^{-1}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}R[(r_{\alpha}r_{\beta})^{-1}]\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muR[(r_{1}\dotsm
r_{N})^{-1}]\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu0$
is an exact sequence of $S$-modules, since is its localization by every
element $r_{\alpha}$ is exact. It remains to recall that the class of very
flat $R$-modules is closed under the passage to the kernels of surjective
morphisms. ∎
###### Corollary 1.2.5.
The following assertions hold in the assumptions of Lemma 1.2.4.
(a) The $S$-module $S\otimes_{R}F$ is very flat for any very flat $R$-module
$F$.
(b) An $S$-module $G$ is very flat if and only if it is very flat as an
$R$-module.
(c) The $S$-module $\operatorname{Hom}_{R}(S,P)$ is contraadjusted for any
contraadjusted $R$-module $P$.
(d) An $S$-module $Q$ is contraadjusted if and only if it is contraadjusted as
an $R$-module.
###### Proof.
Part (a) is a particular case of Lemma 1.2.2(b). Part (b): if $G$ is a very
flat $S$-module, then it is also very flat as an $R$-module by Lemma 1.2.3(b)
and Lemma 1.2.4. Conversely, if $G$ is very flat as an $R$-module, then
$G\simeq S\otimes_{R}G$ is also a very flat $S$-module by part (a).
Part (c) follows from Lemma 1.2.3(a) and Lemma 1.2.4. Part (d): if $Q$ is a
contraadjusted $S$-module, then it is also contraadjusted as an $R$-module by
Lemma 1.2.2(a). Conversely, for any $S$-module $Q$ there are natural
isomorphisms of $S$-modules
$Q\simeq\operatorname{Hom}_{S}(S,Q)\simeq\operatorname{Hom}_{S}(S\otimes_{R}S,\>Q)\simeq\operatorname{Hom}_{R}(S,Q)$;
and if $Q$ is contraadjusted as an $R$-module, then it is also a
contraadjusted $S$-module by part (c). ∎
###### Lemma 1.2.6.
Let $R\longrightarrow S_{\alpha}$, $\alpha=1$, …, $N$, be a collection of
homomorphisms of commutative rings for which the corresponding collection of
morphisms of affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Then
(a) an $R$-module $F$ is very flat if and only if all the $S_{\alpha}$-modules
$S_{\alpha}\otimes_{R}F$ are very flat;
(b) for any contraadjusted $R$-module $P$, the Čech sequence
(2) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(S_{1}\otimes_{R}\dotsb\otimes_{R}S_{N},\>P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}\operatorname{Hom}_{R}(S_{\alpha}\otimes_{R}S_{\beta},\>P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}\operatorname{Hom}_{R}(S_{\alpha},\>P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
is an exact sequence of $R$-modules.
###### Proof.
Part (a): by Corollary 1.2.5(a-b), all the $R$-modules
$S_{\alpha_{1}}\otimes_{R}\dotsb\otimes_{R}S_{\alpha_{k}}\otimes_{R}F$ are
very flat whenever the $S_{\alpha}$-modules $S_{\alpha}\otimes_{R}F$ are very
flat. For any $R$-module $F$ the Čech sequence
(3) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muF\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}S_{\alpha}\otimes_{R}F\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}S_{\alpha}\otimes_{R}S_{\beta}\otimes_{R}F\\\
\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS_{1}\otimes_{R}\dotsb\otimes_{R}S_{N}\otimes_{R}F\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
is an exact sequence of $R$-modules (since its localization at any prime ideal
of $R$ is). It remains to recall that the class of very flat $R$-modules is
closed with respect to the passage to the kernels of surjections.
Part (b): the exact sequence of $R$-modules
(4) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muR\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}S_{\alpha}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}S_{\alpha}\otimes_{R}S_{\beta}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS_{1}\otimes_{R}\dotsb\otimes_{R}S_{N}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
is composed from short exact sequences of very flat $R$-modules, so the
functor $\operatorname{Hom}_{R}({-},P)$ into a contraadjusted $R$-module $P$
preserves its exactness. ∎
###### Lemma 1.2.7.
Let $f_{\alpha}\colon S\longrightarrow T_{\alpha}$, $\alpha=1$, …, $N$, be a
collection of homomorphisms of commutative rings for which the corresponding
collection of morphisms of affine schemes
$\operatorname{Spec}T_{\alpha}\longrightarrow\operatorname{Spec}S$ is a finite
open covering, and let $R\longrightarrow S$ be a homomorphism of commutative
rings. Then
(a) if all the $R$-modules
$T_{\alpha_{1}}\otimes_{S}\dotsb\otimes_{S}T_{\alpha_{k}}$ are very flat, then
the $R$-module $S$ is very flat;
(b) the $R$-modules $T_{\alpha}[t_{\alpha}^{-1}]$ are very flat for all
$t_{\alpha}\in T_{\alpha}$, $1\le\alpha\le N$, if and only if the $R$-module
$S[s^{-1}]$ is very flat for all $s\in S$.
###### Proof.
Part (a) follows from the Čech exact sequence (cf. (4))
$\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}T_{\alpha}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}T_{\alpha}\otimes_{S}T_{\beta}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT_{1}\otimes_{S}\dotsb\otimes_{S}T_{N}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0.$
To prove the “if” assertion in (b), notice that
$\operatorname{Spec}T_{\alpha}[t_{\alpha}^{-1}]$ as an open subscheme in
$\operatorname{Spec}S$ can be covered by a finite number of principal affine
open subschemes $\operatorname{Spec}S[s^{-1}]$, and the intersections of these
are also principal open affines. The “only if” assertion follows from part (a)
applied to the covering of the affine scheme $\operatorname{Spec}S[s^{-1}]$ by
the affine open subschemes
$\operatorname{Spec}T_{\alpha}[f_{\alpha}(s)^{-1}]$, together with the fact
that the intersection of every subset of these open affines can be covered by
a finite number of principal affine open subschemes of one of the schemes
$\operatorname{Spec}T_{\alpha}$. ∎
###### Lemma 1.2.8.
Let $f_{\alpha}\colon S\longrightarrow T_{\alpha}$, $\alpha=1$, …, $N$, be a
collection of homomorphisms of commutative rings for which the corresponding
collection of morphisms of affine schemes
$\operatorname{Spec}T_{\alpha}\longrightarrow\operatorname{Spec}S$ is a finite
open covering, and let $R\longrightarrow S$ be a homomorphism of commutative
rings. Let $F$ be an $S$-module. Then
(a) if all the $R$-modules
$T_{\alpha_{1}}\otimes_{S}\dotsb\otimes_{S}T_{\alpha_{k}}\otimes_{S}F$ are
very flat, then the $R$-module $F$ is very flat;
(b) the $R$-modules $T_{\alpha}[t_{\alpha}^{-1}]\otimes_{S}F$ are very flat
for all $t_{\alpha}\in T_{\alpha}$, $1\le\alpha\le N$, if and only if the
$R$-module $F[s^{-1}]$ is very flat for all $s\in S$.
###### Proof.
Just as in the previous lemma, part (a) follows from the Čech exact sequence
(3) constructed for the collection of morphisms of rings $S\longrightarrow
T_{\alpha}$ and the $S$-module $F$. The proof of part (b) is similar to that
of Lemma 1.2.7(b). ∎
### 1.3. Cotorsion modules
Let $R$ be an associative ring. A left $R$-module $P$ is said to be
_cotorsion_ [63, 19] if $\operatorname{Ext}^{1}_{R}(F,P)=0$ for any flat left
$R$-module $F$, or equivalently, $Ext^{>0}_{R}(F,P)=0$ for any flat left
$R$-module $F$. Clearly, the class of cotorsion left $R$-modules is closed
under extensions and the passage to the cokernels of embeddings, and also
under infinite products.
The following theorem, previously known essentially as the “flat cover
conjecture”, was proven by Eklof–Trlifaj [17] and Bican–Bashir–Enochs [6] (cf.
our Theorem 1.1.1). The case of a Noetherian commutative ring $R$ of finite
Krull dimension was previously treated by Xu [63] (cf. Lemma 1.3.9 below).
###### Theorem 1.3.1.
(a) Any $R$-module can be embedded into a cotorsion $R$-module in such a way
that the quotient module is flat.
(b) Any $R$-module admits a surjective map onto it from a flat $R$-module such
that the kernel is cotorsion. ∎
The following results concerning cotorsion (and injective) modules are similar
to the results about contraadjusted modules presented in Section 1.2. With a
possible exception of the last lemma, all of these are very well known.
###### Lemma 1.3.2.
Let $R$ be a commutative ring. Then
(a) for any flat $R$-module $F$ and cotorsion $R$-module $P$, the $R$-module
$\operatorname{Hom}_{R}(F,P)$ is cotorsion;
(b) for any $R$-module $M$ and any injective $R$-module $J$, the $R$-module
$\operatorname{Hom}_{R}(M,J)$ is cotorsion;
(c) for any flat $R$-module $M$ and any injective $R$-module $J$, the
$R$-module $\operatorname{Hom}_{R}(F,J)$ is injective.
###### Proof.
One has
$\operatorname{Ext}_{R}^{1}(G,\operatorname{Hom}_{R}(F,P))\simeq\operatorname{Ext}_{R}^{1}(F\otimes_{R}G,\>P)$
for any $R$-modules $F$, $G$, and $P$ such that
$\operatorname{Ext}_{R}^{1}(F,P)=0=\operatorname{Tor}^{R}_{1}(F,G)$. All the
three assertions follow from this simple observation. ∎
Our next lemma is a generalization of Lemma 1.3.2 to the noncommutative case.
###### Lemma 1.3.3.
Let $R$ and $S$ be associative rings. Then
(a) for any $R$-flat $R$-$S$-bimodule $F$ and any cotorsion left $R$-module
$P$, the left $S$-module $\operatorname{Hom}_{R}(F,P)$ is cotorsion;
(b) for any $R$-$S$-bimodule $M$ and injective left $R$-module $J$, the left
$S$-module $\operatorname{Hom}_{R}(M,J)$ is cotorsion;
(c) for any $S$-flat $R$-$S$-bimodule $F$ and any injective left $R$-module
$J$, the left $S$-module $\operatorname{Hom}_{R}(F,J)$ is injective.
###### Proof.
One has
$\operatorname{Ext}_{S}^{1}(G,\operatorname{Hom}_{R}(F,P))\simeq\operatorname{Ext}_{R}^{1}(F\otimes_{S}G,\>P)$
for any $R$-$S$-bimodule $F$, left $S$-module $G$, and left $R$-module $P$
such that $\operatorname{Ext}_{R}^{1}(F,P)=0=\operatorname{Tor}^{S}_{1}(F,G)$.
Besides, the tensor product $F\otimes_{S}G$ is flat over $R$ if $F$ is flat
over $R$ and $G$ is flat over $S$. This proves (a); and (b-c) are even easier.
∎
###### Lemma 1.3.4.
Let $f\colon R\longrightarrow S$ be a homomorphism of associative rings. Then
(a) any cotorsion left $S$-module is also a cotorsion left $R$-module in the
$R$-module structure obtained by the restriction of scalars via $f$;
(b) the left $S$-module $\operatorname{Hom}_{R}(S,J)$ obtained by coextension
of scalars via $f$ is injective for any injective left $S$-module $J$.
###### Proof.
Part (a): one has
$\operatorname{Ext}_{R}^{1}(F,Q)\simeq\operatorname{Ext}_{S}^{1}(S\otimes_{R}F,\>Q)$
for any flat left $R$-module $F$ and any left $S$-module $Q$. Part (b) is left
to reader. ∎
###### Lemma 1.3.5.
Let $f\colon R\longrightarrow S$ be an associative ring homomorphism such that
$S$ is a flat left $R$-module in the induced $R$-module structure. Then
(a) the left $S$-module $\operatorname{Hom}_{R}(S,P)$ obtained by coextension
of scalars via $f$ is cotorsion for any cotorsion left $R$-module $P$;
(b) any injective right $S$-module is also an injective right $R$-module in
the $R$-module structure obtained by the restriction of scalars via $f$.
###### Proof.
Part (a): one has
$\operatorname{Ext}_{S}^{1}(F,\operatorname{Hom}_{R}(S,P))\simeq\operatorname{Ext}_{R}^{1}(F,P)$
for any left $R$-module $P$ such that $\operatorname{Ext}_{R}^{1}(S,P)=0$ and
any left $S$-module $F$. In addition, in the assumptions of Lemma any flat
left $S$-module $F$ is also a flat left $R$-module. ∎
###### Lemma 1.3.6.
Let $R\longrightarrow S_{\alpha}$ be a collection of commutative ring
homomorphisms such that the corresponding collection of morphisms of affine
schemes $\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is
an open covering. Then
(a) a contraadjusted $R$-module $P$ is cotorsion if and only if all the
contraadjusted $S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$ are
cotorsion;
(b) a contraadjusted $R$-module $J$ is injective if and only if all the
contraadjusted $S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},J)$ are
injective.
###### Proof.
Part (a): the assertion “only if” follows from Lemma 1.3.5(a). To prove “if”,
use the Čech exact sequence (2) from Lemma 1.2.6(b). By Lemmas 1.3.4(a) and
1.3.5(a), all the terms of the sequence, except perhaps the rightmost one, are
cotorsion $R$-modules, and since the class of cotorsion $R$-modules is closed
under the cokernels of embeddings, it follows that the rightmost term is
cotorsion as well.
Part (b) is proven in the similar way using parts (b) of Lemmas 1.3.4–1.3.5. ∎
Let $R$ be a Noetherian commutative ring, ${\mathfrak{p}}\subset R$ be a prime
ideal, $R_{\mathfrak{p}}$ denote the localization of $R$ at ${\mathfrak{p}}$,
and $\widehat{R}_{\mathfrak{p}}$ be the completion of the local ring
$R_{\mathfrak{p}}$. We recall the notion of a _contramodule_ over a
topological ring, and in particular over a complete Noetherian local ring,
defined in [54, Section 1 and Appendix B], and denote the abelian category of
contramodules over a topological ring $T$ by
$T{\operatorname{\mathsf{--contra}}}$.
The restriction of scalars with respect to the natural ring homomorphism
$R\longrightarrow\widehat{R}_{\mathfrak{p}}$ provides an exact forgetful
functor
$\widehat{R}_{\mathfrak{p}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$. It follows from [54, Theorem B.1.1(1)] that
this functor is fully faithful. The following proposition is a particular case
of the assertions of [54, Propositions B.10.1 and B.9.1].
###### Proposition 1.3.7.
(a) Any $\widehat{R}_{\mathfrak{p}}$-contramodule is a cotorsion $R$-module.
(b) Any free/projective $\widehat{R}_{\mathfrak{p}}$-contramodule is a flat
cotorsion $R$-module.
###### Proof.
In addition to the cited results from [54], take into account Lemma 1.3.4(a)
and the fact that any flat $R_{\mathfrak{p}}$-module is also a flat
$R$-module. ∎
The following theorem is a restatement of the main result of Enochs’ paper
[18].
###### Theorem 1.3.8.
Let $R$ be a Noetherian commutative ring. Then an $R$-module is flat and
cotorsion if and only if it is isomorphic to an infinite product
$\prod_{\mathfrak{p}}F_{\mathfrak{p}}$ of free contramodules
$F_{\mathfrak{p}}=\widehat{R}_{\mathfrak{p}}[[X_{\mathfrak{p}}]]=\varprojlim_{n}\widehat{R}_{\mathfrak{p}}/{\mathfrak{p}}^{n}[X]$
over the complete local rings $\widehat{R}_{\mathfrak{p}}$. Here
$X_{\mathfrak{p}}$ are some sets, and the direct product is taken over all
prime ideals ${\mathfrak{p}}\subset R$. ∎
The following explicit construction of an injective morphism with flat
cokernel from a flat $R$-module $G$ to a flat cotorsion $R$-module
$\operatorname{\mathrm{FC}}_{R}(G)$ was given in the book [63]. For any prime
ideal ${\mathfrak{p}}\subset R$, consider the localization
$G_{\mathfrak{p}}=R_{\mathfrak{p}}\otimes_{R}G$ of the $R$-module $G$ at
${\mathfrak{p}}$, and take its ${\mathfrak{p}}$-adic completion
$\widehat{G}_{\mathfrak{p}}=\varprojlim_{n}G_{\mathfrak{p}}/{\mathfrak{p}}^{n}G_{\mathfrak{p}}$.
By [54, Lemma 1.3.2 and Corollary B.8.2(b)], the $R$-module
$\widehat{G}_{\mathfrak{p}}$ is a free
$\widehat{R}_{\mathfrak{p}}$-contramodule. Furthermore, there is a natural
$R$-module map $G\longrightarrow\widehat{G}_{\mathfrak{p}}$. Set
$\operatorname{\mathrm{FC}}_{R}(G)=\prod_{\mathfrak{p}}\widehat{G}_{\mathfrak{p}}$.
###### Lemma 1.3.9.
The natural $R$-module morphism
$G\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)$ is injective, and its
cokernel $\operatorname{\mathrm{FC}}_{R}(G)/G$ is a flat $R$-module.
###### Proof.
The following argument can be found in [63, Proposition 4.2.2 and Lemma
3.1.6]. Any $R$-module morphism from $G$ to an
$R_{\mathfrak{p}}/{\mathfrak{p}}^{n}$-module, and hence also to a projective
limit of such modules, factorizes through the morphism
$G\longrightarrow\widehat{G}_{\mathfrak{p}}$. Consequently, in view of Theorem
1.3.8 any morphism from $G$ to a flat cotorsion $R$-module factorizes through
the morphism $G\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)$.
Now one could apply Theorem 1.3.1(a), but it is more instructive to argue
directly as follows. Let $E$ be an injective cogenerator of the abelian
category of $R$-modules; e. g.,
$E=\operatorname{Hom}_{\mathbb{Z}}(R,{\mathbb{Q}}/{\mathbb{Z}})$. The
$R$-module $\operatorname{Hom}_{R}(G,E)$ being injective by Lemma 1.3.2(c),
the $R$-module $\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$ is flat
and cotorsion by part (b) of the same lemma and by Lemma 1.6.1(b) below.
The natural morphism
$G\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$ is
injective, and moreover, for any finitely generated/presented $R$-module $M$
the induced map
$G\otimes_{R}M\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)\otimes_{R}M\simeq\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G\otimes_{R}M,\>E),\>E)$
is injective, too. The map
$G\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$
factorizes through the map
$G\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)$, and it follows that the
map
$G\otimes_{R}M\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)\otimes_{R}M$ is
injective as well. ∎
### 1.4. Exact categories of contraadjusted and cotorsion modules
Let $R$ be a commutative ring. As full subcategories of the abelian category
of $R$-modules closed under extensions, the categories of contraadjusted and
very flat $R$-modules have natural exact category structures. In the exact
category of contraadjusted $R$-modules every morphism has a cokernel, which
is, in addition, an admissible epimorphism.
In the exact category of contraadjusted $R$-modules the functors of infinite
product are everywhere defined and exact; they also agree with the infinite
products in the abelian category of $R$-modules. In the exact category of very
flat $R$-modules, the functors of infinite direct sum are everywhere defined
and exact, and agree with the infinite direct sums in the abelian category of
$R$-modules.
It is clear from Corollary 1.1.5(b) that there are enough projective objects
in the exact category of very flat $R$-modules; these are precisely the very
flat contraadjusted $R$-modules. Similarly, by Corollary 1.1.5(a) in the exact
category of very flat $R$-modules there are enough injective objects; these
are also precisely the very flat contraadjusted modules.
Denote the exact category of contraadjusted $R$-modules by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ and the exact category of
very flat $R$-modules by $R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$.
The tensor product of two very flat $R$-modules is an exact functor of two
arguments $R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$. The $\operatorname{Hom}_{R}$
from a very flat $R$-module into a contraadjusted $R$-module is an exact
functor of two arguments
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}})^{\mathsf{op}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ (where
${\mathsf{C}}^{\mathsf{op}}$ denotes the opposite category to a category
${\mathsf{C}}$).
For any homomorphism of commutative rings $f\colon R\longrightarrow S$, the
restriction of scalars with respect to $f$ is an exact functor
$S{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$. The extension of scalars
$F\longmapsto S\otimes_{R}F$ is an exact functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$.
For any homomorphism of commutative rings $f\colon R\longrightarrow S$
satisfying the condition of Lemma 1.2.3, the restriction of scalars with
respect to $f$ is an exact functor
$S{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$. The coextension of scalars
$P\longmapsto\operatorname{Hom}_{R}(S,P)$ is an exact functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$. In particular, these
assertions hold for any homomorphism of commutative rings $R\longrightarrow S$
satysfying the assumption of Lemma 1.2.4.
###### Lemma 1.4.1.
Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is an open
covering. Then
(a) a pair of homomorphisms of contraadjusted $R$-modules $K\longrightarrow
L\longrightarrow M$ is a short exact sequence if and only if such are the
induced sequences of contraadjusted $S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},K)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},L)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},M)$
for all $\alpha$;
(b) a homomorphism of contraadjusted $R$-modules $P\longrightarrow Q$ is an
admissible epimorphism in $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ if
and only if the induced homomorphisms of contraadjusted $S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
are admissible epimorphisms in
$S_{\alpha}{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ for all $\alpha$.
###### Proof.
Part (a): the “only if” assertion follows from Lemma 1.2.4. For the same
reason, if the sequences
$0\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},K)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},L)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},M)\longrightarrow
0$ are exact, then so are the sequences obtained by applying the functors
$\operatorname{Hom}_{R}(S_{\alpha_{1}}\otimes_{R}\dotsb\otimes_{R}S_{\alpha_{k}},\>{-})$,
$k\ge 1$, to the sequence $K\longrightarrow L\longrightarrow M$. Now it
remains to make use of Lemma 1.2.6(b) in order to deduce exactness of the
original sequence $0\longrightarrow K\longrightarrow L\longrightarrow
M\longrightarrow 0$.
Part (b): it is clear from the very right segment of the exact sequence (2)
that surjectivity of the maps
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
implies surjectivity of the map $P\longrightarrow Q$. It remains to check that
the kernel of the latter morphism is a contraadjusted $R$-module. Denote this
kernel by $K$. Since the morphisms
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
are admissible epimorphisms, so are all the morphisms obtained by applying the
coextension of scalars with respect to the ring homomorphisms
$R\longrightarrow S_{\alpha_{1}}\otimes_{R}\dotsb\otimes_{R}S_{\alpha_{k}}$,
$k\ge 1$, to the morphism $P\longrightarrow Q$.
Now Lemma 1.2.6(b) applied to both sides of the morphism $P\longrightarrow Q$
provides a termwise surjective morphism of finite exact sequences of
$R$-modules. The corresponding exact sequence of kernels has $K$ as its
rightmost nontrivial term, while by Lemma 1.2.2(a) all the other terms are
contraadjusted $R$-modules. It follows that the $R$-module $K$ is also
contraadjusted. ∎
Let $R$ be an associative ring. As a full subcategory of the abelian category
of $R$-modules closed under extensions, the category of cotorsion left
$R$-modules has a natural exact category structure.
The functors of infinite product are everywhere defined and exact in this
exact category, and agree with the infinite products in the abelian category
of $R$-modules. Similarly, the category of flat $R$-modules has a natural
exact category structure with exact functors of infinite direct sum.
It follows from Theorem 1.3.1 that there are enough projective objects in the
exact category of cotorsion $R$-modules; these are precisely the flat
cotorsion $R$-modules. Similarly, there are enough injective objects in the
exact category of flat $R$-modules, and these are also precisely the flat
cotorsion $R$-modules.
Denote the exact category of cotorsion left $R$-modules by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ and the exact category of
flat left $R$-modules by $R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}$. The
abelian category of left $R$-modules will be denoted simply by
$R{\operatorname{\mathsf{--mod}}}$, and the additive category of injective
$R$-modules (with the trivial exact category structure) by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}$.
For any commutative ring $R$, the $\operatorname{Hom}_{R}$ from a flat
$R$-module into a cotorsion $R$-module is an exact functor of two arguments
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}})^{\mathsf{op}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. Analogously, the
$\operatorname{Hom}_{R}$ from an arbitrary $R$-module into an injective
$R$-module is an exact functor
$(R{\operatorname{\mathsf{--mod}}})^{\mathsf{op}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. The functors
$\operatorname{Hom}$ over a noncommutative ring $R$ mentioned in Lemma 1.3.3
have similar exactness properties.
For any associative ring homomorphism $f\colon R\longrightarrow S$, the
restriction of scalars via $f$ is an exact functor
$S{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. For any associative ring
homomorphism $f\colon R\longrightarrow S$ making $S$ a flat left $R$-module,
the coextension of scalars $P\longmapsto\operatorname{Hom}_{R}(S,P)$ is an
exact functor $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.
###### Lemma 1.4.2.
Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is an open
covering. Then
(a) a pair of morphisms of cotorsion $R$-modules $K\longrightarrow
L\longrightarrow M$ is a short exact sequence if and only if such are the
sequences of cotorsion $S_{\alpha}$-modules
$\operatorname{Hom}(S_{\alpha},K)\longrightarrow\operatorname{Hom}(S_{\alpha},L)\longrightarrow\operatorname{Hom}(S_{\alpha},M)$
for all $\alpha$;
(b) a morphism of cotorsion $R$-modules $P\longrightarrow Q$ is an admissible
epimorphism if and only if such are the morphisms of cotorsion
$S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
for all $\alpha$.
###### Proof.
Part (a) follows from Lemma 1.4.1(a); part (b) can be proven in the way
similar to Lemma 1.4.1(b). ∎
### 1.5. Very flat and cotorsion dimensions
Let $R$ be a commutative ring. By analogy with the definition of the flat
dimension of a module, define the _very flat dimension_ of an $R$-module $M$
as the minimal length of its very flat left resolution.
Clearly, the very flat dimension of an $R$-module $M$ is equal to the supremum
of the set of all integers $d$ for which there exists a contraadjusted
$R$-module $P$ such that $\operatorname{Ext}_{R}^{d}(M,P)\neq 0$. The very
flat dimension of a module cannot differ from its projective dimension by more
than $1$.
Similarly, the _cotorsion dimension_ of a left module $M$ over an associative
ring $R$ is conventionally defined as the minimal length of its right
resolution by cotorsion $R$-modules. The cotorsion dimension of a left
$R$-module $M$ is equal to the supremum of the set of all integers $d$ for
which there exists a flat left $R$-module $F$ such that
$\operatorname{Ext}_{R}^{d}(F,M)\neq 0$.
Both the very flat and the cotorsion dimensions of a module do not depend on
the choice of a particular very flat/cotorsion resolution in the same sense as
the familiar projective, flat, and injective dimensions do not (see Corollary
A.5.2 for the general assertion of this kind).
###### Lemma 1.5.1.
Let $R\longrightarrow S$ be a morphism of associative rings. Then any left
$S$-module $Q$ of cotorsion dimension $\le d$ over $S$ has cotorsion dimension
$\le d$ over $R$.
###### Proof.
Follows from Lemma 1.3.4(a). ∎
###### Lemma 1.5.2.
Let $R\longrightarrow S$ be a morphism of associative rings such that $S$ is a
left $R$-module of flat dimension $\le D$. Then
(a) any left $S$-module $G$ of flat dimension $\le d$ over $S$ has flat
dimension $\le d+D$ over $R$;
(b) any right $S$-module $Q$ of injective dimension $\le d$ over $S$ has
injective dimension $\le\nobreak d+D$ over $R$.
###### Proof.
Part (a) follows from the spectral sequence
$\operatorname{Tor}^{S}_{p}(\operatorname{Tor}^{R}_{q}(M,S),G)\Longrightarrow\operatorname{Tor}^{R}_{p+q}(M,G)$,
which holds for any right $R$-module $M$. Part (b) follows from the spectral
sequence
$\operatorname{Ext}_{S^{\mathrm{op}}}^{p}(\operatorname{Tor}^{R}_{q}(N,S),Q)\Longrightarrow\operatorname{Ext}_{R^{\mathrm{op}}}^{p+q}(N,Q)$,
which holds for any right $R$-module $N$ (where $S^{\mathsf{op}}$ and
$R^{\mathsf{op}}$ denote the rings opposite to $S$ and $R$). ∎
###### Lemma 1.5.3.
(a) Let $R\longrightarrow S$ be a morphism of commutative rings such that $S$
is an $R$-module of very flat dimension $\le D$. Then any $S$-module $G$ of
very flat dimension $\le d$ over $S$ has very flat dimension $\le d+1+D$ over
$R$.
(b) Let $R\longrightarrow S$ be a morphism of commutative rings such that
$S[s^{-1}]$ is an $R$-module of very flat dimension $\le D$ for any element
$s\in S$. Then any $S$-module $G$ of very flat dimension $\le d$ over $S$ has
very flat dimension $\le d+D$ over $R$.
###### Proof.
Part (a): the $S$-module $G$ has a left projective resolution of length $\le
d+1$, and any projective $S$-module has very flat dimension $\le D$ over $R$,
which implies the desired assertion (see Corollary A.5.5(a)).
Part (b): by Corollary 1.1.4, the $S$-module $G$ has a left resolution of
length $\le d$ by direct summands of transfinitely iterated extensions of the
$S$-modules $S[s^{-1}]$. Hence it suffices to show that the very flat
dimension of $R$-modules is not raised by the transfinitely iterated
extension.
More generally, we claim that one has $\operatorname{Ext}^{n}_{R}(M,P)=0$
whenever a module $M$ over an associative ring $R$ is a transfinitely iterated
extension of $R$-modules $M_{\alpha}$ and
$\operatorname{Ext}^{n}_{R}(M_{\alpha},P)=0$ for all $\alpha$. The case $n=0$
is easy; the case $n=1$ is the result of [17, Lemma 1]; and the case $n>1$ is
reduced to $n=1$ by replacing the $R$-module $P$ with an $R$-module $Q$
occuring at the rightmost end of a resolution $0\longrightarrow
P\longrightarrow J^{0}\longrightarrow\dotsb\longrightarrow
J^{n-2}\longrightarrow Q\longrightarrow 0$ with injective $R$-modules $J^{i}$.
∎
###### Lemma 1.5.4.
Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes is a finite open covering. Then
(a) the flat dimension of an $R$-module $F$ is equal to the supremum of the
flat dimensions of the $S_{\alpha}$-modules $S_{\alpha}\otimes_{R}F$;
(b) the very flat dimension of an $R$-module $F$ is equal to the supremum of
the very flat dimensions of the $S_{\alpha}$-modules $S_{\alpha}\otimes_{R}F$;
(c) the cotorsion dimension of a contraadjusted $R$-module $P$ is equal to the
supremum of the cotorsion dimensions of the contraadjusted
$S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$;
(d) the injective dimension of a contraadjusted $R$-module $P$ is equal to the
supremum of the injective dimensions of the contraadjusted
$S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$.
###### Proof.
Part (b) follows easily from Lemma 1.2.6(a), and the proof of part (a) is
similar. Parts (c-d) analogously follow from Lemma 1.3.6(a-b). ∎
The following lemma will be needed in Section 4.10.
###### Lemma 1.5.5.
(a) Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings
and $P$ be an $R$-module such that $\operatorname{Ext}^{1}_{R}(S[s^{-1}],P)=0$
for all elements $s\in S$. Then the $S$-module $\operatorname{Hom}_{R}(S,P)$
is contraadjusted.
(b) Let $f\colon R\longrightarrow S$ be a homomorphism of associative rings
and $P$ be a left $R$-module such that $\operatorname{Ext}^{1}_{R}(G,P)=0$ for
all flat left $S$-modules $G$. Then the $S$-module
$\operatorname{Hom}_{R}(S,P)$ is cotorsion.
###### Proof.
See the proofs of Lemmas 1.2.3(a) and 1.3.5(a). ∎
The following theorem is due to Raynaud and Gruson [55, Corollaire II.3.2.7].
###### Theorem 1.5.6.
Let $R$ be a commutative Noetherian ring of Krull dimension $D$. Then the
projective dimension of any flat $R$-module does not exceed $D$. Consequently,
the very flat dimension of any flat $R$-module also does not exceed $D$. ∎
###### Corollary 1.5.7.
Let $R$ be a commutative Noetherian ring of Krull dimension $D$. Then the
cotorsion dimension of any $R$-module does not exceed $D$.
###### Proof.
For any associative ring $R$, the supremum of the projective dimensions of
flat left $R$-modules and the supremum of the cotorsion dimensions of
arbitrary left $R$-modules are equal to each other. Indeed, both numbers are
equal to the supremum of the set of all integers $d$ for which there exist a
flat left $R$-module $F$ and a left $R$-module $P$ such that
$\operatorname{Ext}_{R}^{d}(F,P)\neq 0$. ∎
### 1.6. Coherent rings, finite morphisms, and coadjusted modules
Recall that an associative ring $R$ is called _left coherent_ if all its
finitely generated left ideals are finitely presented. Finitely presented left
modules over a left coherent ring $R$ form an abelian subcategory in
$R{\operatorname{\mathsf{--mod}}}$ closed under kernels, cokernes, and
extensions. The definition of a right coherent associative ring is similar.
###### Lemma 1.6.1.
Let $R$ and $S$ be associative rings. Then
(a) Assuming that the ring $R$ is left Noetherian, for any $R$-injective
$R$-$S$-bimodule $J$ and any flat left $S$-module $F$ the left $R$-module
$J\otimes_{S}F$ is injective.
(b) Assuming that the ring $S$ is right coherent, for any $S$-injective
$R$-$S$-bimodule $I$ and any injective left $R$-module $J$ the left $S$-module
$\operatorname{Hom}_{R}(I,J)$ is flat.
###### Proof.
Part (a) holds due to the natural isomorphism
$\operatorname{Hom}_{R}(M,\>J\otimes_{S}F)\simeq\operatorname{Hom}_{R}(M,J)\otimes_{S}F$
for any finitely presented left module $M$ over an associative ring $R$, any
$R$-$S$-bimodule $J$, and any flat left $S$-module $F$. Part (b) follows from
the natural isomorphism
$N\otimes_{S}\operatorname{Hom}_{R}(I,J)\simeq\operatorname{Hom}_{R}(\operatorname{Hom}_{S^{\mathrm{op}}}(N,I),J))$
for any finitely presented right module $N$ over an associative ring $S$, any
$R$-$S$-bimodule $I$, and any injective left $R$-module $J$ (where
$S^{\mathrm{op}}$ denotes the ring opposite to $S$).
Here we use the facts that injectivity of a left module $I$ over a left
Noetherian ring $R$ is equivalent to exactness of the functor
$\operatorname{Hom}_{R}({-},I)$ on the category of finitely generated left
$R$-modules, while flatness of a left module $F$ over a right coherent ring
$S$ is equivalent to exactness of the functor ${-}\otimes_{S}F$ on the
category of finitely presented right $S$-modules (cf. the proof of the next
Lemma 1.6.2). ∎
###### Lemma 1.6.2.
Let $R$ and $S$ be associative rings such that $S$ is left coherent. Let $F$
be a left $R$-module of finite projective dimension, $P$ be an $S$-flat
$R$-$S$-bimodule such that $\operatorname{Ext}_{R}^{>0}(F,P)=0$, and $M$ be a
finitely presented left $S$-module. Then one has
$\operatorname{Ext}_{R}^{>0}(F,\>P\otimes_{S}M)=0$, the natural map of abelian
groups
$\operatorname{Hom}_{R}(F,P)\otimes_{S}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{S}M)$
is an isomorphism, and the right $S$-module $\operatorname{Hom}_{R}(F,P)$ is
flat.
###### Proof.
Let $L_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow M$ be a
left resolution of $M$ by finitely generated projective $S$-modules. Then
$P\otimes_{S}L_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
P\otimes_{S}M$ is a left resolution of the $R$-module $P\otimes_{S}M$ by
$R$-modules annihilated by $\operatorname{Ext}^{>0}(F,{-})$. Since the
$R$-module $F$ has finite projective dimension, it follows that
$\operatorname{Ext}^{>0}(F,\>P\otimes_{S}M)=0$.
Consequently, the functor
$M\longmapsto\operatorname{Hom}_{R}(F,\>P\otimes_{S}M)$ is exact on the
abelian category of finitely presented left $S$-modules $M$. Obviously, the
functor $M\longmapsto\operatorname{Hom}_{R}(F,P)\otimes_{R}M$ is right exact.
Since the morphism of functors
$\operatorname{Hom}_{R}(F,P)\otimes_{S}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{S}M)$
is an isomorphism for finitely generated projective $S$-modules $M$, we can
conclude that it is an isomorphism for all finitely presented left
$S$-modules.
Now we have proven that the functor
$M\longmapsto\operatorname{Hom}_{R}(F,P)\otimes_{S}M$ is exact on the abelian
category of finitely presented left $S$-modules. Since any left $S$-module is
a filtered inductive limit of finitely presented ones and the inductive limits
commute with tensor products, it follows that the $S$-module
$\operatorname{Hom}_{R}(F,P)$ is flat. ∎
###### Corollary 1.6.3.
Let $R$ be a commutative ring. Then
(a) for any finitely generated $R$-module $M$ and any contraadjusted
$R$-module $P$, the $R$-module $M\otimes_{R}P$ is contraadjusted;
(b) if the ring $R$ is coherent, then for any very flat $R$-module $F$ and any
flat contraadjusted $R$-module $P$, the $R$-module
$\operatorname{Hom}_{R}(F,P)$ is flat and contraadjusted;
(c) in the situation of (b), for any finitely presented $R$-module $M$ the
natural morphism of $R$-modules
$\operatorname{Hom}_{R}(F,P)\otimes_{R}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{R}M)$
is an isomorphism.
###### Proof.
Part (a) immediately follows from the facts that the class of contraadjusted
$R$-modules is closed under finite direct sums and quotients. Part (b) is
provided Lemma 1.6.2 together with Lemma 1.2.1(b), and part (c) is also Lemma
1.6.2. ∎
###### Corollary 1.6.4.
Let $R$ be either a coherent commutative ring such that any flat $R$-module
has finite projective dimension, or a Noetherian commutative ring. Then
(a) for any finitely presented $R$-module $M$ and any flat cotorsion
$R$-module $P$, the $R$-module $M\otimes_{R}P$ is cotorsion;
(b) for any flat $R$-module $F$ and flat cotorsion $R$-module $P$, the
$R$-module $\operatorname{Hom}_{R}(F,P)$ is flat and cotorsion;
(c) in the situation of (a) and (b), the natural morphism of $R$-modules
$\operatorname{Hom}_{R}(F,P)\allowbreak\otimes_{R}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{R}M)$
is an isomorphism.
###### Proof.
In the former set of assumptions about the ring $R$, parts (a) and (c) follow
from Lemma 1.6.2, and part (b) is provided by the same Lemma together with
Lemma 1.3.2(a). In the latter case, the argument is based on Proposition 1.3.7
and Theorem 1.3.8. Part (a): since the functor $M\otimes_{R}{-}$ preserves
infinite products, it suffices to show that the $R$-module
$M\otimes_{R}Q_{\mathfrak{p}}$ has an
$\widehat{R}_{\mathfrak{p}}$-contramodule structure for any
$\widehat{R}_{\mathfrak{p}}$-contramodule $Q_{\mathfrak{p}}$. Indeed, the full
subcategory
$\widehat{R}_{\mathfrak{p}}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed with respect to finite direct sums
and cokernels.
Part (c): the morphism in question is clearly an isomorphism for a finitely
generated projective $R$-module $M$. Hence it suffices to show that the
functor $M\longmapsto\operatorname{Hom}_{R}(F,\>P\otimes_{R}M)$ is right
exact. In fact, it is exact, since the functor $\operatorname{Hom}_{R}(F,{-})$
preserves exactness of short sequences of
$\widehat{R}_{\mathfrak{p}}$-contramodules. Therefore, the functor
$M\longmapsto\operatorname{Hom}_{R}(F,P)\otimes_{R}M$ is also exact, and we
have proven part (b) as well. ∎
###### Corollary 1.6.5.
(a) Let $R\longrightarrow S$ be a homomorphism of commutative rings such that
the related morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding. Assume that the ring $R$ is coherent. Then the $S$-module
$\operatorname{Hom}_{R}(S,P)$ is flat and contraadjusted for any flat
contraadjusted $R$-module $P$.
(b) Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Assume that either the ring $R$ is Noetherian and an $R$-module
$P$ is cotorsion, or the ring $R$ is Noetherian of finite Krull dimension and
an $R$-module $P$ is contraadjusted. Then the $R$-module $P$ is flat if and
only if all the $S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$
are flat.
###### Proof.
Part (a): the $S$-module $\operatorname{Hom}_{R}(S,P)$ is contraadjusted by
Corollary 1.2.5(c). The $R$-module $\operatorname{Hom}_{R}(S,P)$ is flat by
Lemma 1.2.4 and Corollary 1.6.3(b). Since
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding, it follows that $\operatorname{Hom}_{R}(S,P)$ is also flat as an
$S$-module (cf. Corollary 1.2.5(b)).
The “only if” assertion in part (b) is provided by part (a). The proof of the
“if” is postponed to Section 5. The cotorsion case will follow from Corollary
5.1.4, while the finite Krull dimension case will be covered by Corollary
5.2.2(b). ∎
Let $R{\operatorname{\mathsf{--mod}}}_{\mathsf{fp}}$ denote the abelian
category of finitely presented left modules over a left coherent ring $R$. For
a coherent commutative ring $R$, the tensor product of a finitely presented
$R$-module with a flat contraadjusted $R$-module is an exact functor
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fp}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ (where the exact category
structure on $R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ is induced from
$R{\operatorname{\mathsf{--mod}}}$). The $\operatorname{Hom}_{R}$ from a very
flat $R$-module into a flat contraadjusted $R$-module is an exact functor
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}})^{\mathsf{op}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.
Let $R$ be either a coherent commutative ring such that any flat $R$-module
has finite projective dimension, or a Noetherian commutative ring. Then the
tensor product of a finitely presented $R$-module with a flat cotorsion
$R$-module is an exact functor
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fp}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. The $\operatorname{Hom}_{R}$
from a flat $R$-module to a flat cotorsion $R$-module is an exact functor
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}})^{\mathsf{op}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. Here the additive category
of flat cotorsion $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ is endowed with a trivial
exact category structure.
###### Lemma 1.6.6.
Let $R$ be a commutative ring and $I\subset R$ be an ideal. Then
(a) an $R/I$-module $Q$ is a contraadjusted $R/I$-module if and only if it is
a contraadjusted $R$-module;
(b) the $R/I$-module $P/IP$ is contraadjusted for any contraadjusted
$R$-module $P$;
(c) assuming that the ring $R$ is coherent and the ideal $I$ is finitely
generated, for any very flat $R$-module $F$ and flat contraadjusted $R$-module
$P$ the natural morphism of $R/I$-modules
$\operatorname{Hom}_{R}(F,P)/I\operatorname{Hom}_{R}(F,P)\longrightarrow\operatorname{Hom}_{R/I}(F/IF,\>P/IP)$
is an isomorphism.
###### Proof.
Part (a): the characterization of contraadjusted modules given in the
beginning of Section 1.1 shows that the contraadjustness property of a module
depends only on its abelian group structure and the operators by which the
ring acts in it (rather than on the ring indexing such operators).
Alternatively, the “only if” assertion is a particular case of Lemma 1.2.2(a),
and one can deduce the “if” from the observation that any element $\bar{r}\in
R/I$ can be lifted to an element $r\in R$ so that one has an isomorphism of
$R/I$-modules $R/I[\bar{r}^{-1}]\simeq R/I\otimes_{R}R[r^{-1}]$.
Part (b): the $R$-module $P/IP$ is contraadjusted as a quotient module of a
contraadjusted $R$-module. By part (a), $P/IP$ is also a contraadjusted
$R/I$-module. Part (c): by Corollary 1.6.3(c), the natural morphism of
$R$-modules
$\operatorname{Hom}_{R}(F,P)/I\operatorname{Hom}_{R}(F,P)\allowbreak\longrightarrow\operatorname{Hom}_{R}(F,\>P/IP)$
is an isomorphism. ∎
Recall that a morphism of Noetherian rings $R\longrightarrow S$ is called
_finite_ if $S$ is a finitely generated $R$-module in the induced $R$-module
structure.
###### Lemma 1.6.7.
Let $R\longrightarrow S$ be a finite morphism of Noetherian commutative rings.
Then
(a) the $S$-module $S\otimes_{R}P$ is flat and cotorsion for any flat
cotorsion $R$-module $P$;
(b) assuming that the ring $S$ has finite Krull dimension, an $S$-module $Q$
is a cotorsion $S$-module if and only if it is a cotorsion $R$-module;
(c) for any flat $R$-module $F$ and flat cotorsion $R$-module $P$, the natural
morphism of $S$-modules
$S\otimes_{R}\operatorname{Hom}_{R}(F,P)\longrightarrow\operatorname{Hom}_{S}(S\otimes_{R}F,\>S\otimes_{R}P)$
is an isomorphism.
###### Proof.
The proof of parts (a-b) is based on Theorem 1.3.8. Given a prime ideal
${\mathfrak{p}}\subset R$, consider all the prime ideals
${\mathfrak{q}}\subset S$ whose full preimage in $R$ coincides with
${\mathfrak{p}}$. Such ideals form a nonempty finite set, and there are no
inclusions between them [40, Theorems 9.1 and 9.3, and Exercise 9.3]. Let us
denote these ideals by ${\mathfrak{q}}_{1}$, …, ${\mathfrak{q}}_{m}$. By [40,
Theorems 9.4(i), 8.7, and 8.15], we have
$S\otimes_{R}\widehat{R}_{\mathfrak{p}}\simeq\widehat{S}_{{\mathfrak{q}}_{1}}\oplus\dotsb\oplus\widehat{S}_{{\mathfrak{q}}_{m}}$.
Since the functor $S\otimes_{R}{-}$ preserves infinite products, in order to
prove (a) it suffices to show that the $S$-module
$S\otimes_{R}F_{\mathfrak{p}}$ is a finite direct sum of certain free
$\widehat{S}_{{\mathfrak{q}}_{i}}$-contramodules $F_{{\mathfrak{q}}_{i}}$.
This can be done either by noticing that $F_{\mathfrak{p}}$ is a direct
summand of an infinite product of copies of $\widehat{R}_{\mathfrak{p}}$ (see
[54, Section 1.3]), or by showing that the natural map
$S\otimes_{R}\widehat{R}_{\mathfrak{p}}[[X]]\longrightarrow\bigoplus_{i=1}^{m}\widehat{S}_{{\mathfrak{q}}_{i}}[[X]]$
is an isomorphism for any set $X$ (see [54, proof of the first assertion of
Proposition B.9.1]). In addition to the assertion of part (a), we have also
proven that any flat cotorsion $S$-module $Q$ is a direct summand of an
$S$-module $S\otimes_{R}P$ for a certain flat cotorsion $R$-module $P$.
The “only if” assertion in part (b) is a particular case of Lemma 1.3.4(a).
Let us prove the “if”. According to Corollary 1.5.7, the cotorsion dimension
of any $R$-module is finite. By Theorem 1.3.1(a), it follows that any flat
$S$-module admits a finite right resolution by flat cotorsion $S$-modules (cf.
the dual version of Corollary A.5.3). Hence it suffices to prove that
$\operatorname{Ext}_{S}^{>0}(G,Q)=0$ for a flat cotorsion $S$-module $G$. This
allows us to assume that $G=S\otimes_{R}F$, where $F$ is a flat (cotorsion)
$R$-module. It remains to recall the $\operatorname{Ext}$ isomorphism from the
proof of Lemma 1.3.4(a).
Part (c): by Corollary 1.6.4(c), the natural morphism of $R$-modules
$S\otimes_{R}\operatorname{Hom}_{R}(F,P)\longrightarrow\operatorname{Hom}_{R}(F,\>S\otimes_{R}P)$
is an isomorphism. ∎
###### Lemma 1.6.8.
Let $R$ be a commutative ring and $I\subset R$ be a finitely generated
nilpotent ideal. Then
(a) an $R$-module $P$ is contraadjusted if and only if the $R/I$-module $P/IP$
is contraadjusted;
(b) a flat $R$-module $F$ is very flat if and only if the $R/I$-module $F/IF$
is very flat;
(c) assuming that the ring $R$ is Noetherian, a flat $R$-module $P$ is
cotorsion if and only if the $R/I$-module $P/IP$ is cotorsion;
(d) assuming that the ring $R$ is coherent and any flat $R$-module has finite
projective dimension, a flat $R$-module $P$ is cotorsion whenever the
$R/I$-module $P/IP$ is cotorsion.
###### Proof.
Part (a): the “only if” assertion is a particular case of Lemma 1.6.6(b). To
prove the “if”, notice that in our assumptions about $I$ the $R$-module $P$
has a finite decreasing filtration by its submodules $I^{n}P$. Furthermore,
the successive quotients $I^{n}P/I^{n+1}P$ are the targets of the natural
surjective homomorphisms of $R/I$-modules
$I^{n}/I^{n+1}\otimes_{R/I}P/IP\longrightarrow I^{n}P/I^{n+1}P$. Since the
$R/I$-module $I^{n}/I^{n+1}$ is finitely generated, the $R/I$-module
$I^{n}P/I^{n+1}P$ is a quotient module of a finite direct sum of copies of the
$R/I$-module $P/IP$. It remains to use the facts that the class of
contraadjusted modules over a given commutative ring is closed under
extensions and quotients, together with the result of Lemma 1.6.6(a).
Part (b): the “only if” is a particular case of Lemma 1.2.2(b); let us prove
the “if”. Let $P$ be a contraadjusted $R$-module; we have to show that
$\operatorname{Ext}^{1}_{R}(F,P)=0$. According to the above proof of part (a),
the $R$-module $P$ has a finite filtration whose successive quotients are
contraadjusted $R/I$-modules with the $R$-module structures obtained by
restriction of scalars. So it suffices to check that
$\operatorname{Ext}^{1}_{R}(F,Q)=0$ for any contraadjusted $R/I$-module $Q$.
Now, the $R$-module $F$ being flat, one has
$\operatorname{Ext}^{1}_{R}(F,Q)=\operatorname{Ext}^{1}_{R/I}(F/IF,\>Q)$ for
any $R/I$-module $Q$ (see the proof of Lemma 1.3.4(a)).
Parts (c-d): in “only if” assertion in (c) is a particular case of Lemma
1.6.7(a). To prove the “if”, notice the isomorphisms of $R/I$-modules
$I^{n}P/I^{n+1}P\simeq I^{n}/I^{n+1}\otimes_{R}P\simeq
I^{n}/I^{n+1}\otimes_{R/I}P/IP$ (the former of which holds, since the
$R$-module $P$ is flat). The $R/I$-module $P/IP$ being flat and cotorsion, the
$R/I$-modules $I^{n}/I^{n+1}\otimes_{R/I}P/IP$ are cotorsion by Corollary
1.6.4(a), the $R$-modules $I^{n}/I^{n+1}\otimes_{R/I}P/IP$ are cotorsion by
Lemma 1.3.4(a), and the $R$-module $P$ is cotorsion, since the class of
cotorsion $R$-modules is closed under extensions. ∎
Let $R$ be a commutative ring and $I\subset R$ be an ideal. Then the reduction
$P\longmapsto P/IP$ of a flat contraadjusted $R$-module $P$ modulo $I$ is an
exact functor $R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R/I{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R/I{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.
Let $f\colon R\longrightarrow S$ be a finite morphism of Noetherian
commutative rings. Then the extension of scalars $P\longmapsto S\otimes_{R}P$
of a flat cotorsion $R$-module $P$ with respect to $f$ is an additive functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
S{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ (between additive categories
naturally endowed with trivial exact category structures).
Let $R$ be a commutative ring. We will say that an $R$-module $K$ is
_coadjusted_ if the functor of tensor product with $K$ over $R$ preserves the
class of contraadjusted $R$-modules. By Corollary 1.6.3(a), any finitely
generated $R$-module is coadjusted.
An $R$-module $K$ is coadjusted if and only if the $R$-module $K\otimes_{R}P$
is contraadjusted for every flat (or very flat) contraadjusted $R$-module $P$.
Indeed, by Corollary 1.1.5(b), any contraajusted $R$-module is a quotient
module of a very flat contraadjusted $R$-module; so it remains to recall that
any quotient module of a contraadjusted $R$-module is contraadjusted.
Clearly, any quotient module of a coadjusted $R$-module is coadjusted.
Furthermore, the class of coadjusted $R$-modules is closed under extensions.
One can see this either by applying the above criterion of coadjustness in
terms of tensor products with flat contraadjusted $R$-modules, or
straightforwardly from the right exactness property of the functor of tensor
product together with the facts that the class of contraadjusted $R$-modules
is closed under quotients and extensions.
Consequently, there is the induced exact category structure on the full
subcategory of coadjusted $R$-modules in the abelian category
$R{\operatorname{\mathsf{--mod}}}$. We denote this exact category by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{coa}}$. The tensor product of a
coadjusted $R$-module with a flat contraadjusted $R$-module is an exact
functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{coa}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.
Over a Noetherian commutative ring $R$, any injective module $J$ is
coadjusted. Indeed, for any $R$-module $P$, the tensor product $J\otimes_{R}P$
is a quotient module of an infinite direct sum of copies of $J$, which means a
quotient module of an injective module, which is contraadjusted. Hence any
quotient module of an injective module is coadjusted, too, as is any extension
of such modules.
###### Lemma 1.6.9.
(a) Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings
such that the related morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding. Then the $S$-module $S\otimes_{R}K$ obtained by the extension of
scalars via $f$ is coadjusted for any coadjusted $R$-module $K$.
(b) Let $f_{\alpha}\colon R\longrightarrow S_{\alpha}$ be a collection of
homomorphisms of commutative rings for which the corresponding collection of
morphisms of affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Then an $R$-module $K$ is coadjusted if and only if all the
$S_{\alpha}$-modules $S_{\alpha}\otimes_{R}K$ are coadjusted.
###### Proof.
Part (a): any contraadjusted $S$-module $Q$ is also contraadjusted as an
$R$-module, so the tensor product $(S\otimes_{R}K)\otimes_{S}Q\simeq
K\otimes_{R}Q$ is a contraadjusted $R$-module. By Corollary 1.2.5(d), it is
also a contraadjusted $S$-module.
Part (b): the “only if” assertion is provided by part (a); let us prove “if”.
Let $P$ be a contraadjusted $R$-module. Applying the functor $K\otimes_{R}{-}$
to the Čech exact sequence (2) from Lemma 1.2.6(b), we obtain a sequence of
$R$-modules that is exact at its rightmost nontrivial term. So it suffices to
show that the $R$-modules $K\otimes_{R}\operatorname{Hom}_{R}(S_{\alpha},P)$
are contraadjusted.
Now one has
$K\otimes_{R}\operatorname{Hom}_{R}(S_{\alpha},P)\simeq(S_{\alpha}\otimes_{R}K)\otimes_{S_{\alpha}}\operatorname{Hom}_{R}(S_{\alpha},P)$,
the $S_{\alpha}$-module $\operatorname{Hom}_{R}(S_{\alpha},P)$ is
contraadjusted by Corollary 1.2.5(c), and the restriction of scalars from
$S_{\alpha}$ to $R$ preserves contraadjustness by Lemma 1.2.2(a). ∎
### 1.7. Very flat morphisms of schemes
A morphism of schemes $f\colon Y\longrightarrow X$ is called _very flat_ if
for any two affine open subschemes $V\subset Y$ and $U\subset X$ such that
$f(V)\subset U$ the ring of regular functions ${\mathcal{O}}(V)$ is a very
flat module over the ring ${\mathcal{O}}(U)$. By Lemma 1.2.4, any embedding of
an open subscheme is a very flat morphism.
According to Lemmas 1.2.6(a) and 1.2.7(b), the property of a morphism to be
very flat is local in both the source and the target schemes. A morphism of
affine schemes $\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is
very flat if and only if the morphism of commutative rings $R\longrightarrow
S$ satisfies the condition of Lemma 1.2.3. By Lemma 1.2.3(b), the composition
of very flat morphisms of schemes is a very flat morphism.
It does not seem to follow from anything that the base change of a very flat
morphism of schemes should be a very flat morphism. Here is a partial result
in this direction.
###### Lemma 1.7.1.
The base change of a very flat morphism with respect to any locally closed
embedding or universal homeomorphism of schemes is a very flat morphism.
###### Proof.
Essentially, given a very flat morphism $f\colon Y\longrightarrow X$ and a
morphism of schemes $g\colon x\longrightarrow X$, in order to conclude that
the morphism $f^{\prime}\colon y=x\times_{x}Y\longrightarrow x$ is very flat
it suffices to know that the morphism $g^{\prime}\colon y\longrightarrow Y$ is
injective and the topology of $y$ is induced from the topology of $Y$.
Indeed, the very flatness condition being local, one can assume all the four
schemes to be affine. Now any affine open subscheme $v\subset y$ is the full
preimage of a certain open subscheme $V\subset Y$. Covering $V$ with open
affines if necessary and using the locality again, one can assume that $V$ is
affine, too. Finally, if the ${\mathcal{O}}(X)$-module ${\mathcal{O}}(V)$ is
very flat, then by Lemma 1.2.2(b) so is the ${\mathcal{O}}(x)$-module
${\mathcal{O}}(v)={\mathcal{O}}(x\times_{X}V)$. ∎
A quasi-coherent sheaf ${\mathcal{F}}$ over a scheme $X$ is called _very flat_
if the ${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ is very flat for any
affine open subscheme $U\subset X$. According to Lemma 1.2.6(a), very flatness
of a quasi-coherent sheaf over a scheme is a local property. By Lemma
1.2.2(b), the inverse image of a very flat quasi-coherent sheaf under any
morphism of schemes is very flat. By Lemma 1.2.3(b), the direct image of a
very flat quasi-coherent sheaf under a very flat affine morphism of schemes is
very flat.
More generally, given a morphism of schemes $f\colon Y\longrightarrow X$, a
quasi-coherent sheaf ${\mathcal{F}}$ on $Y$ is said to be _very flat over $X$_
if for any affine open subschemes $U\subset X$ and $V\subset Y$ such that
$f(V)\subset U$ the module of sections ${\mathcal{F}}(V)$ is very flat over
the ring ${\mathcal{O}}_{X}(U)$. According to Lemmas 1.2.6(a) and 1.2.8(b),
the property of very flatness of ${\mathcal{F}}$ over $X$ is local in both $X$
and $Y$. By Lemma 1.2.3(b), if the scheme $Y$ is very flat over $X$ and a
quasi-coherent sheaf ${\mathcal{F}}$ is very flat on $Y$, then ${\mathcal{F}}$
is also very flat over $X$.
The following conjecture looks natural. Some evidence in its support is
gathered below in this section.
###### Conjecture 1.7.2.
Any flat morphism of finite type between Noetherian schemes is very flat.
It is well-known that any flat morphism of finite type between Noetherian
schemes (or of finite presentation between arbitrary schemes) is an open map
[28, Théorème 2.4.6]. We will see below that the similar result about very
flat morphisms does not require any finiteness conditions at all.
Given a commutative ring $R$ and an $R$-module $M$, we define its support
$\operatorname{Supp}M\subset\operatorname{Spec}R$ as the set of all prime
ideals ${\mathfrak{p}}\subset R$ for which the tensor product
$k({\mathfrak{p}})\otimes_{R}M$, where $k({\mathfrak{p}})$ denotes the residue
field of the ideal ${\mathfrak{p}}$, does not vanish.
###### Lemma 1.7.3.
Let $R$ be a commutative ring without nilpotent elements and $F$ be a nonzero
very flat $R$-module. Then the support of $F$ contains a nonempty open subset
in $\operatorname{Spec}R$.
###### Proof.
By Corollary 1.1.4, any very flat $R$-module $F$ is a direct summand of a
transfinitely iterated extension $M=\varinjlim_{\alpha}M_{\alpha}$ of certain
$R$-modules of the form $R[s_{\alpha}^{-1}]$, where $s_{\alpha}\in R$. In
particular, $F$ is an $R$-submodule in $M$; consider the minimal index
$\alpha$ for which the intersection $F\cap M_{\alpha}\subset M$ is nonzero.
Denote this intersection by $G$ and set $s=s_{\alpha}$; then $G$ is a nonzero
submodule in $F$ and in $R[s^{-1}]$ simultaneously.
Hence the localization $G[s^{-1}]$ is a nonzero ideal in $R[s^{-1}]$. The
support $\operatorname{Supp}G[s^{-1}]\subset\operatorname{Spec}R[s^{-1}]$ of
the $R[s^{-1}]$-module $G[s^{-1}]$ is equal to the intersection of
$\operatorname{Supp}G\subset\operatorname{Spec}R$ with
$\operatorname{Spec}R[s^{-1}]\subset\operatorname{Spec}R$. By right exactness
of the tensor product functor, the subset $\operatorname{Supp}G[s^{-1}]$
contains the complement $V(s,G)\subset\operatorname{Spec}R[s^{-1}]$ to the
support of the quotient module $R[s^{-1}]/G[s^{-1}]$ in
$\operatorname{Spec}R[s^{-1}]$. The latter is a closed subset in
$\operatorname{Spec}R[s^{-1}]$ corresponding to the ideal $G[s^{-1}]$; if
there are no nilpotents in $R$ then the open subset
$V(s,G)\subset\operatorname{Spec}R[s^{-1}]$ is nonempty.
Let us show that the support of $F$ contains $V(s,G)$. Let ${\mathfrak{p}}\in
V(s,G)\subset\operatorname{Spec}R$ be a prime ideal in $R$ and
$k({\mathfrak{p}})$ be its residue field. Then the map
$k({\mathfrak{p}})\otimes_{R}G=k({\mathfrak{p}})\otimes_{R}G[s^{-1}]\longrightarrow
k({\mathfrak{p}})\otimes_{R}R[s^{-1}]=k({\mathfrak{p}})$ is surjective by the
above argument, the map $k({\mathfrak{p}})\otimes_{R}M_{\alpha}\longrightarrow
k({\mathfrak{p}})\otimes_{R}M$ is injective because the $R$-module
$M/M_{\alpha}$ is flat, and the map
$k({\mathfrak{p}})\otimes_{R}F\longrightarrow k({\mathfrak{p}})\otimes_{R}M$
is injective since $F$ is a direct summand in $M$ (we do not seem to really
use the latter observation). Finally, both the composition $G\longrightarrow
F\longrightarrow M$ and the map $G\longrightarrow R[s^{-1}]$ factorize through
the same $R$-module morphism $G\longrightarrow M_{\alpha}$.
Now it follows from the commutativity of the triangle $G\longrightarrow
M_{\alpha}\longrightarrow R[s^{-1}]$ that the map
$k({\mathfrak{p}})\otimes_{R}G\longrightarrow
k({\mathfrak{p}})\otimes_{R}M_{\alpha}$ is nonzero, it is clear from the
diagram $G\longrightarrow M_{\alpha}\longrightarrow M$ that the composition
$k({\mathfrak{p}})\otimes_{R}G\longrightarrow k({\mathfrak{p}})\otimes_{R}M$
is nonzero, and it follows from commutativity of the diagram $G\longrightarrow
F\longrightarrow M$ that the map $k({\mathfrak{p}})\otimes_{R}G\longrightarrow
k({\mathfrak{p}})\otimes_{R}F$ is nonzero. The assertion of Lemma is proven. ∎
###### Proposition 1.7.4.
Let $F$ be a very flat module over a commutative ring $R$ and
$Z\subset\operatorname{Spec}R$ be a closed subset. Suppose that the
intersection $\operatorname{Supp}F\cap Z\subset\operatorname{Spec}R$ is
nonempty. Then $\operatorname{Supp}F$ contains a nonempty open subset in $Z$.
###### Proof.
Endow $Z$ with the structure of a reduced closed subscheme in
$\operatorname{Spec}R$ and set $S={\mathcal{O}}(Z)$. Then the intersection
$Z\cap\operatorname{Supp}F$ coincides with the support of the $S$-module
$S\otimes_{R}F$ in $\operatorname{Spec}S=Z\subset\operatorname{Spec}R$. By
Lemma 1.2.2(b), the $S$-module $S\otimes_{R}F$ is very flat. Now if this
$S$-module vanishes, then the intersection $Z\cap\operatorname{Supp}F$ is
empty, while otherwise it contains a nonempty open subset in
$\operatorname{Spec}S$ by Lemma 1.7.3. ∎
###### Remark 1.7.5.
It is clear from the above argument that one can replace the condition that
the ring $R$ has no nilpotent elements in Lemma 1.7.3 by the condition of
nonvanishing of the tensor product $S\otimes_{R}F$ of the $R$-module $F$ with
the quotient ring $S=R/J$ of the ring $R$ by its nilradical $J$. In
particular, this condition holds automatically if (the $R$-module $F$ does not
vanish and) the nilradical $J\subset R$ is a nilpotent ideal, i. e., there
exists an integer $N\ge 1$ such that $J^{N}=0$. This includes all Noetherian
commutative rings $R$.
On the other hand, it is not difficult to demonstrate an example of a flat
module over a commutative ring that is annihilated by the reduction modulo the
nilradical. E. g., take $R$ to be the ring of polynomials in $x$, $x^{1/2}$,
$x^{1/4}$, …, $x^{1/2^{N}}$, … over a field $k=S$ with the imposed relation
$x^{r}=0$ for $r>1$, and $F=J$ to be the nilradical (i. e., the kernel of the
augmentation morphism to $k$) in $R$. The $R$-module $F$ is flat as the
inductive limit of free $R$-modules $R\otimes_{k}kx^{1/2^{N}}$ for
$N\to\infty$, and one clearly has $S\otimes_{R}F=R/J\otimes_{R}J=J/J^{2}=0$.
###### Theorem 1.7.6.
The support of any very flat module over a commutative ring $R$ is an open
subset in $\operatorname{Spec}R$.
###### Proof.
Let $F$ be a very flat $R$-module. Denote by $Z$ the closure of the complement
$\operatorname{Spec}R\setminus\operatorname{Supp}F$ in $\operatorname{Spec}R$.
Then, by the definition, $\operatorname{Supp}F$ contains the complement
$\operatorname{Spec}R\setminus Z$, and no open subset in (the induced topology
of) $Z$ is contained in $\operatorname{Supp}F$. By Proposition 1.7.4, it
follows that $\operatorname{Supp}F$ does not intersect $Z$, i. e.,
$\operatorname{Supp}F=\operatorname{Spec}R\setminus Z$ is an open subset in
$\operatorname{Spec}R$. ∎
###### Example 1.7.7.
In particular, we have shown that the ${\mathbb{Z}}$-module ${\mathbb{Q}}$ is
_not_ very flat, even though it is a flat module of projective dimension $1$.
###### Corollary 1.7.8.
Any very flat morphism of schemes is an open map of their underlying
topological spaces.
###### Proof.
Given a very flat morphism $f\colon Y\longrightarrow X$ and an open subset
$W\subset Y$, cover $W$ with open affines $V\subset Y$ for which there exist
open affines $U\subset X$ such that $f(V)\subset U$, and apply Theorem 1.7.6
to the very flat ${\mathcal{O}}(U)$-modules ${\mathcal{O}}(V)$. ∎
In the rest of the section we prove several (rather weak) results about
morphisms from certain classes being always very flat.
###### Theorem 1.7.9.
Any finite étale morphism of Noetherian schemes is very flat.
###### Proof.
The argument is based on the Galois theory of finite étale morphisms of
(Noetherian) schemes [43] (see also [39]). Clearly, one can assume both
schemes to be affine and connected. Let
$\operatorname{Spec}S\to\operatorname{Spec}R$ be our morphism; we have to show
that for any $s\in S$ the $R$-module $S[s^{-1}]$ is very flat.
First let us reduce the question to the case when our morphism is a Galois
covering. Let $\operatorname{Spec}T\longrightarrow\operatorname{Spec}S$ be a
finite étale morphism from a nonempty scheme $\operatorname{Spec}T$ such that
the composition $\operatorname{Spec}T\longrightarrow\operatorname{Spec}R$ is
Galois. Notice that the $S[s^{-1}]$-module $T[s^{-1}]$ is flat and finitely
presented, and consequently projective. If it is known that the $R$-module
$T[s^{-1}]$ is very flat, then it remains to show that the $R$-module
$S[s^{-1}]$ is a direct summand of a (finite) direct sum of copies of
$T[s^{-1}]$.
For this purpose, it suffices to check that the $S[s^{-1}]$-module $T[s^{-1}]$
is a projective generator of the abelian category of $S[s^{-1}]$-modules. In
other words, we have to show that there are no nonzero $S[s^{-1}]$-modules $M$
for which any morphism of $S[s^{-1}]$-modules $T[s^{-1}]\longrightarrow M$
vanishes. Since the functor Hom from a finitely presented module over a
commutative ring commutes with localizations, and finitely generated
projective modules are locally free in the Zariski topology, the desired
property follows from the assumption of the scheme $\operatorname{Spec}S$
being connected.
Now let $G$ be the Galois group of $\operatorname{Spec}S$ over
$\operatorname{Spec}R$. For any subset $\Gamma\subset G$ consider the element
$t_{\Gamma}=\prod_{g\in\Gamma}g(s)\in S$. We will prove the assertion that
$S[t_{\Gamma}^{-1}]$ is a very flat $R$-module by decreasing induction in the
cardinality of $\Gamma$ (for a fixed group $G$, but varying rings $R$ and
$S$). The induction base: if $\Gamma=G$, then the element
$t_{\Gamma}=t_{G}=\prod_{g\in G}g(s)$ belongs to $R\subset S$, and the ring
$S[t_{G}^{-1}]$, being a projective module over $R[t_{G}^{-1}]$, is a very
flat module over $R$.
The induction step: let $H\subset G$ be the stabilizer of the subset
$\Gamma\subset G$ with respect to the action of $G$ in itself by
multiplications on the left, and let $g_{1}$, …, $g_{n}\in G$ be some
representatives of the left cosets $G/H$. The union of the open subschemes
$\operatorname{Spec}S[t_{g_{i}(\Gamma)}^{-1}]$, being a $G$-invariant open
subset in $\operatorname{Spec}S$, is the full preimage of a certain open
subscheme $U\subset\operatorname{Spec}R$. Replacing the scheme
$\operatorname{Spec}R$ by its connected affine open subschemes covering $U$,
we can assume that $\operatorname{Spec}R=U$ and $\operatorname{Spec}S$ is the
union of its open subschemes $\operatorname{Spec}S[t_{g_{i}(\Gamma)}^{-1}]$.
Consider the Čech exact sequence (4) for the covering of the affine scheme
$\operatorname{Spec}S$ by its principal affine open subschemes
$\operatorname{Spec}S[t_{g_{i}(\Gamma)}^{-1}]$. The leftmost nontrivial term
is the ring $S$, the next one is the direct sum of the rings
$S[t_{g_{i}(\Gamma)}^{-1}]$, and the further ones are direct sums of the rings
$S[t_{\Delta}^{-1}]$ for subsets $\Delta\subset G$ which, being unions of two
or more subsets $g_{i}(\Gamma)\subset G$, $i=1$, …, $n$, have cardinality
greater than that of $\Gamma$. It remains to use the induction assumption
together with the facts that the class of very flat $R$-modules is closed with
respect to extensions, the passage to the kernels of surjective morphisms, and
direct summands. ∎
###### Remark 1.7.10.
The second half of the above argument essentially proves the following more
general result. Suppose that a finite group $G$ acts by automorphisms of a
commutative ring $S$ in such a way that $S$ is a finitely generated projective
(or, which is the same, a finitely presented flat) module over its subring of
$G$-invariant elements $S^{G}$. Then the natural morphism
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}S^{G}$ is very flat.
Here, to convince oneself that the above reasoning is applicable, one only
needs to notice that $G$ acts transitively in the fibers of the projection
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}S^{G}$ [3, Theorem 5.10
and Exercise 5.13] and the image of any $G$-invariant open subset in
$\operatorname{Spec}S$ is open in $\operatorname{Spec}S^{G}$ (since any
$G$-invariant ideal in $S$ is contained in the nilradical of the extension in
$S$ of its contraction to $S^{G}$).
###### Proposition 1.7.11.
Any finite flat set-theoretically bijective morphism of Noetherian schemes is
very flat.
###### Proof.
A flat morphism of finite presentation is an open map (see above), so any
morphism satisfying the assumptions of Proposition is a homeomorphism.
Obviously, one can assume both schemes to be affine. Let
$\operatorname{Spec}S\to\operatorname{Spec}R$ be our morphism; it suffices to
show that an open subset is affine in $\operatorname{Spec}R$ if it is affine
in $\operatorname{Spec}S$.
Moreover, we can restrict ourselves to principal affine open subsets in
$\operatorname{Spec}S$. So it suffices to check that
$\operatorname{Spec}S[s^{-1}]=\operatorname{Spec}S[\operatorname{Norm}_{S/R}(s)^{-1}]$
for any $s\in S$ (where $\operatorname{Norm}_{S/R}(s)\in R$ is the determinant
of the $R$-linear operator of multiplication with $s$ in $S$). The latter
question reduces to the case when $R$ is the spectrum of a field, so $S$ is an
Artinian local ring. In this situation, the assertion is obvious (as the norm
of an invertible element is invertible, and that of a nilpotent one is
nilpotent). ∎
The following lemma, claiming that the very flatness property is local with
respect a certain special class of very flat coverings, is to be compared with
Lemma 1.7.1. According to Theorem 1.7.9 and Proposition 1.7.11 (and the proof
of the former), any surjective finite étale morphism or finite flat set-
theoretically bijective morphism of Noetherian schemes $g\colon
x\longrightarrow X$ satisfies its conditions.
###### Lemma 1.7.12.
Let $g\colon x\longrightarrow X$ be a very flat affine morphism of schemes
such that for any (small enough) affine open subscheme $U\subset X$ the ring
${\mathcal{O}}(g^{-1}(U))$ is a projective module over ${\mathcal{O}}(U)$ and
a projective generator of the abelian category of ${\mathcal{O}}(U)$-modules.
Then a morphism of schemes $f\colon Y\longrightarrow X$ is very flat whenever
the morphism $f^{\prime}\colon y=x\times_{X}Y\longrightarrow x$ is very flat.
###### Proof.
Let $V\subset Y$ and $U\subset X$ be affine open subschemes such that
$f(V)\subset U$. Set $u=g^{-1}(U)$ and assume that the
${\mathcal{O}}(u)$-module
${\mathcal{O}}(u)\otimes_{{\mathcal{O}}(U)}{\mathcal{O}}(V)$ is very flat.
Then, the morphism $g$ being very flat, the ring
${\mathcal{O}}(u)\otimes_{{\mathcal{O}}(U)}{\mathcal{O}}(V)$ is very flat as
an ${\mathcal{O}}(U)$-module, too. Finally, since ${\mathcal{O}}(u)$ is a
projective generator of the category of ${\mathcal{O}}(U)$-modules, we can
conclude that the ring ${\mathcal{O}}(V)$ is a very flat
${\mathcal{O}}(U)$-module. ∎
For any scheme $X$, we denote by ${\mathbb{A}}^{n}_{X}$ the $n$-dimensional
relative affine space over $X$; so if $X=\operatorname{Spec}R$ then
${\mathbb{A}}^{n}_{X}=\operatorname{Spec}R[x_{1},\dotsc,x_{n}]={\mathbb{A}}^{n}_{R}$.
###### Theorem 1.7.13.
For any scheme $X$ of finite type over a field $k$ and any $n\ge 1$, the
natural projection $\mathbb{A}^{n}_{X}\longrightarrow X$ is a very flat
morphism.
###### Proof.
Since the very flatness property is local, and the class of very flat
morphisms is preserved by compositions and base changes with respect to closed
embeddings (see Lemma 1.7.1), it suffices to show that the projection
morphisms
$\pi_{n}\colon{\mathbb{A}}_{k}^{n+1}\longrightarrow{\mathbb{A}}_{k}^{n}$ are
very flat for all $n\ge 0$. Furthermore, we will now prove that the field $k$
can be assumed to be algebraically closed.
Indeed, let us check that for any algebraic field extension $L/k$ the morphism
$g\colon
X_{L}=\operatorname{Spec}L\times_{\operatorname{Spec}k}X\longrightarrow X$
satisfies the assumptions of Lemma 1.7.12. All the other conditions being
obvious, we only have to show that the morphism $g$ is very flat. We can
assume the scheme $X$ to be affine. Then any principal affine open subscheme
in $X_{L}$ is defined over some subfield $l\subset L$ finite over $k$. This
reduces the question to the case of a finite field extension $l/k$, when one
can apply Theorem 1.7.9 (in the case of a separable field extension) and
Proposition 1.7.11 (for a purely inseparable one).
Assuming the field $k$ to be (at least) infinite, we proceed by induction in
$n$. The case $n=0$ is obvious. Let $U$ be a principal affine open subscheme
in ${\mathbb{A}}_{k}^{n+1}$; we have to show that ${\mathcal{O}}(U)$ is a very
flat $k[x_{1},\dotsc,x_{n}]$-module.
The complement ${\mathbb{A}}_{k}^{n+1}\setminus U$ is an affine hypersurface,
i. e., the zero locus of a polynomial $f\in k[x_{1},\dotsc,x_{n+1}]$. Let us
decompose the polynomial $f$ into a product of irreducible ones and separate
those factors which do not depend on $x_{n+1}$. So the complement
${\mathbb{A}}_{k}^{n+1}\setminus U$ is presented as the union
$Z\cup\pi_{n}^{-1}(W)$, where $W$ is a hypersurface in ${\mathbb{A}}_{k}^{n}$
and $Z$ is a hypersurface in ${\mathbb{A}}_{k}^{n+1}$ whose projection
$\pi_{n}|_{Z}\colon Z\longrightarrow{\mathbb{A}}_{k}^{n}$, outside of a full
preimage $Y=\pi_{n}|_{Z}^{-1}(X)\subset Z$ of a proper closed subvariety
$X\subset{\mathbb{A}}_{k}^{n}$, is the composition of a finite flat
homeomorphism and a finite étale map. The dimensions of both $X$ and $Y$ do
not exceed $n-1$.
Since the class of very flat $R$-modules is closed with respect to the tensor
products over $R$, it suffices to show that the $k[x_{1},\dotsc,x_{n}]$-module
${\mathcal{O}}({\mathbb{A}}_{k}^{n+1}\setminus Z)$ is very flat; so we can
assume $W$ to be empty. Furthermore, we may presume the dimensions of all the
irreducible components of the varieties $X$ and $Y$ to be equal to $n-1$
exactly.
Consider the $(n+1)$-dimensional vector space $k^{n+1}$ over $k$ with the
coordinate linear functions $x_{1}$, …, $x_{n+1}$. A _line_ (one-dimensional
vector subspace) in $k^{n+1}$ will be called _vertical_ if it contains the
vector $(0,\dotsc,0,1)$ and _horizonal_ if it is generated by a vector whose
last coordinate vanishes. Let us choose a nonvertical line, and draw an affine
line through every point in $Y\subset{\mathbb{A}}_{k}^{n+1}$ in the chosen
direction.
After the passage to the Zariski closure of this whole set of points, we will
obtain (the chosen direction being generic) a certain hypersurface
$H\subset{\mathbb{A}}_{k}^{n+1}$. A linear coordinate change affecting only
$x_{1}$, …, $x_{n}$ will transform the chosen nonvertical line into a
horizontal one. Let it be the line spanned by the vector $(0,\dotsc,0,1,0)\in
k^{n+1}$. In these new coordinates, the hypersurface $H$ is the full preimage
of a hypersurface in $\operatorname{Spec}k[x_{1},\dotsc,x_{n-1},x_{n+1}]$ with
respect to the “horizontal” projection
$\pi_{n}^{\prime}\colon\operatorname{Spec}k[x_{1},\dotsc,x_{n+1}]\longrightarrow\operatorname{Spec}k[x_{1},\dotsc,x_{n-1},x_{n+1}]$
forgetting the coordinate $x_{n}$.
Given a fixed point $q\in{\mathbb{A}}^{n+1}(k)\setminus Y(k)$, the lines in
the directions from $q$ to the points in $Y$ form, at most, an
$(n-1)$-dimensional subvariety in the $n$-dimensional projective space of all
lines in $k^{n+1}$. Hence there is a finite set of nonvertical lines
$p_{i}\subset k^{n+1}$ such that the intersection of the corresponding
hypersurfaces $H_{i}\subset{\mathbb{A}}_{k}^{n+1}$ coincides with $Y$. The
very flatness property being local, it suffices to show that the
$k[x_{1},\dotsc,x_{n}]$-modules
${\mathcal{O}}({\mathbb{A}}_{k+1}^{n}\setminus(H\cup Z))$ are very flat. By
the induction assumption, we know that such are the
$k[x_{1},\dotsc,x_{n}]$-modules ${\mathcal{O}}({\mathbb{A}}_{k+1}^{n}\setminus
H)$.
By Theorem 1.7.9 and Proposition 1.7.11, the ring ${\mathcal{O}}(Z\setminus
H)$ is a very flat ${\mathcal{O}}({\mathbb{A}}_{k}^{n}\setminus X)$-module,
hence also a very flat ${\mathcal{O}}({\mathbb{A}}_{k}^{n})$-module. To
complete the proof, it remains to make use of the following lemma (applied to
the rings $R=k[x_{1},\dotsc,x_{n}]$ and
$S={\mathcal{O}}({\mathbb{A}}_{k}^{n+1}\setminus H)$, and the equation of $Z$
in the role of the element $s$). ∎
###### Lemma 1.7.14.
Let $R\longrightarrow S$ be a homomorphism of commutative rings, $s\in S$ be
an element, and $G$ be an $S$-module without $s$-torsion (i. e., $s$ acts in
$G$ by an injective operator). Suppose that the $R$-modules $G$ and $G/sG$ are
very flat. Then the $R$-module $G[s^{-1}]$ is also very flat.
In particular, if $s$ is a nonzero-divisor in $S$ and the $R$-modules $S$ and
$S/sS$ are very flat, then the $R$-module $S[s^{-1}]$ is also very flat.
###### Proof.
The formula $G[s^{-1}]=\varinjlim_{n\in{\mathbb{N}}}s^{-n}G$ makes $G[s^{-1}]$
a transfinitely iterated extension of one copy of $G$ and a countable set of
copies of $G/sG$. ∎
For any scheme $X$, denote by ${\mathbb{A}}^{\infty}_{X}$ the infinite-
dimensional relative affine space (of countable relative dimension) over $X$.
So if $X=\operatorname{Spec}R$, then
${\mathbb{A}}^{\infty}_{X}=\operatorname{Spec}R[x_{1},x_{2},x_{3},\dotsc]$.
###### Corollary 1.7.15.
For any scheme $X$ of finite type over a field $k$, the natural projection
${\mathbb{A}}^{\infty}_{X}\longrightarrow X$ is a very flat morphism.
###### Proof.
One can assume $X$ to be affine. Then any principal affine open subscheme in
${\mathbb{A}}_{X}^{\infty}$ is the complement to the subscheme of zeroes of an
equation depending on a finite number of variables $x_{i}$ only. Thus the
assertion follows from Theorem 1.7.13. ∎
###### Proposition 1.7.16.
(a) Any finite flat morphism of one-dimensional schemes of finite type over a
field $k$ is very flat.
(b) Any flat (or, which is equivalent, quasi-finite) morphism of finite type
from a reduced one-dimensional scheme to a smooth one-dimensional scheme of
finite type over a field $k$ is very flat.
###### Proof.
Part (a): let $f\colon Y\longrightarrow X$ be a finite flat morphism of affine
one-dimensional schemes over $k$ and $V\subset Y$ be an open subscheme (notice
that any open subscheme in an affine one-dimensional scheme is affine).
Assume first that $V$ contains the general points of all the irreducible
components of $Y$, i. e., $V=Y\setminus Z$, where $Z\subset Y$ is a finite set
of closed points. Consider the open subscheme
$W=Y\setminus(f^{-1}(f(Z))\setminus Z)$. Then $V\cup W=Y$ and $V\cap
W=Y\setminus f^{-1}(f(Z))$, so $V\cup W$ is flat and finite over $X$, while
$V\cap W$ is flat and finite over $U=X\setminus f(Z)$.
Now in the Mayer–Vietoris exact sequence $0\longrightarrow{\mathcal{O}}(V\cup
W)\longrightarrow{\mathcal{O}}(V)\oplus{\mathcal{O}}(W)\longrightarrow{\mathcal{O}}(V\cap
W)\longrightarrow 0$ the left and right terms, being respectively a projective
${\mathcal{O}}(X)$-module and a projective ${\mathcal{O}}(U)$-module, are both
very flat ${\mathcal{O}}(X)$-modules. Hence the middle term is also very flat
over ${\mathcal{O}}(X)$, and so is its direct summand ${\mathcal{O}}(V)$.
In the general case, set $Z=Y\setminus V$ and define an open subscheme
$W\subset Y$ as the complement to the Zariski closure of
$f^{-1}(f(Z))\setminus Z$ in $Y$. Then we have again $V\cap W=Y\setminus
f^{-1}(f(Z))$, which is a flat and finite scheme over $U=X\setminus f(Z)$,
hence ${\mathcal{O}}(V\cap W)$ is a very flat ${\mathcal{O}}(X)$-module. On
the other hand, the union $V\cup W$ is an open subscheme containing all the
general points of irreducible components in $Y$, so the
${\mathcal{O}}(X)$-module ${\mathcal{O}}(V\cup W)$ is very flat according to
the above. It remains to use the very same Mayer–Vietoris sequence once again.
To deduce part (b) from part (a), one can apply Zariski’s main theorem,
embedding a quasi-finite morphism into a finite one. The equivalence of
flatness and quasi-finiteness (a particular case of the general description of
flat morphisms from reduced schemes to smooth one-dimensional ones) mentioned
in the formulation of part (b) plays a key role in this argument. ∎
## 2\. Contraherent Cosheaves over a Scheme
### 2.1. Cosheaves of modules over a sheaf of rings
Let $X$ be a topological space. A _copresheaf of abelian groups_ on $X$ is a
covariant functor from the category of open subsets of $X$ (with the identity
embeddings as morphisms) to the category of abelian groups.
Given a copresheaf of abelian groups ${\mathfrak{P}}$ on $X$, we will denote
the abelian group it assigns to an open subset $U\subset X$ by
${\mathfrak{P}}[U]$ and call it the group of _cosections_ of ${\mathfrak{P}}$
over $U$. For a pair of embedded open subsets $V\subset U\subset X$, the map
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]$ that the copresheaf
${\mathfrak{P}}$ assigns to $V\subset U$ will be called the _corestriction_
map.
A copresheaf of abelian groups ${\mathfrak{P}}$ on $X$ is called a _cosheaf_
if for any open subset $U\subset X$ and its open covering
$U=\bigcup_{\alpha}U_{\alpha}$ the following sequence of abelian groups is
exact
(5) $\textstyle\bigoplus_{\alpha,\beta}{\mathfrak{P}}[U_{\alpha}\cap
U_{\beta}]\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}{\mathfrak{P}}[U_{\alpha}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{P}}[U]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0.$
Let ${\mathcal{O}}$ be a sheaf of associative rings on $X$. A copresheaf of
abelian groups ${\mathfrak{P}}$ on $X$ is said to be a _copresheaf of(left)
${\mathcal{O}}$-modules_ if for each open subset $U\subset X$ the abelian
group ${\mathfrak{P}}[U]$ is endowed with the structure of a (left) module
over the ring ${\mathcal{O}}(U)$, and for each pair of embedded open subsets
$V\subset U\subset X$ the map of corestriction of cosections
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]$ in the copresheaf
${\mathfrak{P}}$ is a homomorphism of ${\mathcal{O}}(U)$-modules. Here the
${\mathcal{O}}(U)$-module structure on ${\mathfrak{P}}[V]$ is obtained from
the ${\mathcal{O}}(V)$-module structure by the restriction of scalars via the
ring homomorphism ${\mathcal{O}}(U)\longrightarrow{\mathcal{O}}(V)$.
A copresheaf of ${\mathcal{O}}$-modules on $X$ is called a _cosheaf of
${\mathcal{O}}$-modules_ if its underlying copresheaf of abelian groups is a
cosheaf of abelian groups.
###### Remark 2.1.1.
One can define copresheaves with values in any category, and cosheaves with
values in any category that has coproducts. In particular, one can speak of
cosheaves of sets, etc. Notice, however, that, unlike for (pre)sheaves, the
underlying copresheaf of sets of a cosheaf of abelian groups is _not_ a
cosheaf of sets in general, as the forgetful functor from the abelian groups
to sets preserves products, but not coproducts. Thus cosheaves of sets (as
developed, e. g., in [11]) and cosheaves of abelian groups or modules are two
quite distinct theories.
Let ${\mathbf{B}}$ be a base of open subsets of $X$. We will consider
covariant functors from ${\mathbf{B}}$ (viewed as a full subcategory of the
category of open subsets in $X$) to the category of abelian groups. We say
that such a functor ${\mathfrak{Q}}$ is _endowed with an
${\mathcal{O}}$-module structure_ if the abelian group ${\mathfrak{Q}}[U]$ is
endowed with an ${\mathcal{O}}(U)$-module structure for each
$U\in{\mathbf{B}}$ and the above compatibility condition holds for the
corestriction maps ${\mathfrak{Q}}[V]\longrightarrow{\mathfrak{Q}}[U]$
assigned by the functor ${\mathfrak{Q}}$ to all $V$, $U\in{\mathbf{B}}$ such
that $V\subset U$.
The following result is essentially contained in [25, Section 0.3.2], as is
its (more familiar) sheaf version, to which we will turn in due order.
###### Theorem 2.1.2.
A covariant functor ${\mathfrak{Q}}$ with an ${\mathcal{O}}$-module structure
on a base ${\mathbf{B}}$ of open subsets of $X$ can be extended to a cosheaf
of ${\mathcal{O}}$-modules ${\mathfrak{P}}$ on $X$ if and only if the
following condition holds. For any open subset $V\in{\mathbf{B}}$, any its
covering $V=\bigcup_{\alpha}V_{\alpha}$ by open subsets
$V_{\alpha}\in{\mathbf{B}}$, and any (or, equivalently, some particular)
covering $V_{\alpha}\cap V_{\beta}=\bigcup_{\gamma}W_{\alpha\beta\gamma}$ of
the intersections $V_{\alpha}\cap V_{\beta}$ by open subsets
$W_{\alpha\beta\gamma}\in{\mathbf{B}}$ the sequence of abelian groups (or
${\mathcal{O}}(V)$-modules)
(6)
$\textstyle\bigoplus_{\alpha,\beta,\gamma}{\mathfrak{Q}}[W_{\alpha\beta\gamma}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}{\mathfrak{Q}}[V_{\alpha}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{Q}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
must be exact. The functor of restriction of cosheaves of
${\mathcal{O}}$-modules to a base ${\mathbf{B}}$ is an equivalence between the
category of cosheaves of ${\mathcal{O}}$-modules on $X$ and the category of
covariant functors on ${\mathbf{B}}$, endowed with ${\mathcal{O}}$-module
structures and satisfying (6).
###### Proof.
The elementary approach taken in the exposition below is to pick an
appropriate stage at which one can dualize and pass to (pre)sheaves, where our
intuitions work better. First we notice that if the functor ${\mathfrak{Q}}$
(with its ${\mathcal{O}}$-module structure) has been extended to a cosheaf of
${\mathcal{O}}$-modules ${\mathfrak{P}}$ on $X$, then for any open subset
$U\subset X$ there is an exact sequence of ${\mathcal{O}}(U)$-modules
(7)
$\textstyle\bigoplus_{W,V^{\prime},V^{\prime\prime}}{\mathfrak{Q}}[W]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{V}{\mathfrak{Q}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{P}}[U]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0,$
where the summation in the middle term runs over all open subsets
$V\in{\mathbf{B}}$, $V\subset U$, while the summation in the leftmost term is
done over all triples of open subsets $W$, $V^{\prime}$,
$V^{\prime\prime}\in{\mathbf{B}}$, $W\subset V^{\prime}$,
$V^{\prime\prime}\subset U$. Conversely, given a functor ${\mathfrak{Q}}$ with
an ${\mathcal{O}}$-module structure one can recover the
${\mathcal{O}}(U)$-module ${\mathfrak{P}}[U]$ as the cokernel of the left
arrow.
Clearly, the modules ${\mathfrak{P}}[U]$ constructed in this way naturally
form a copresheaf of ${\mathcal{O}}$-modules on $X$. Before proving that it is
a cosheaf, one needs to show that for any open covering
$U=\bigcup_{\alpha}V_{\alpha}$ of an open subset $U\subset X$ by open subsets
$V_{\alpha}\in{\mathbf{B}}$ and any open coverings $V_{\alpha}\cap
V_{\beta}=\bigcup_{\gamma}W_{\alpha\beta\gamma}$ of the intersections
$V_{\alpha}\cap V_{\beta}$ by open subsets
$W_{\alpha\beta\gamma}\in{\mathbf{B}}$ the natural map from the cokernel of
the morphism
(8)
$\textstyle\bigoplus_{\alpha,\beta,\gamma}{\mathfrak{Q}}[W_{\alpha\beta\gamma}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}{\mathfrak{Q}}[V_{\alpha}]$
to the (above-defined) ${\mathcal{O}}(U)$-module ${\mathfrak{P}}[U]$ is an
isomorphism. In particular, it will follow that
${\mathfrak{P}}[V]\simeq{\mathfrak{Q}}[V]$ for $V\in{\mathbf{B}}$.
Notice that it suffices to check both assertions for co(pre)sheaves of abelian
groups (though it will not matter in the subsequent argument). Notice also
that a copresheaf of ${\mathcal{O}}$-modules ${\mathfrak{P}}$ is a cosheaf if
and only if the dual presheaf of ${\mathcal{O}}$-modules
$U\longmapsto\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{P}}[U],I)$ is a sheaf
on $X$ for every abelian group $I$ (or specifically for
$I={\mathbb{Q}}/{\mathbb{Z}}$). Similarly, the condition (6) holds for a
covariant functor ${\mathfrak{Q}}$ on a base ${\mathbf{B}}$ if and only if the
dual condition (9) below holds for the contravariant functor
$V\longmapsto\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{Q}}[V],I)$ on
${\mathbf{B}}$. So it remains to prove the following Proposition 2.1.3. ∎
Now we will consider contravariant functors ${\mathcal{G}}$ from
${\mathbf{B}}$ to the category of abelian groups, and say that such a functor
is endowed with an ${\mathcal{O}}$-module structure if the abelian group
${\mathcal{G}}(U)$ is an ${\mathcal{O}}(U)$-module for every
$U\in{\mathbf{B}}$ and the restriction morphisms
${\mathcal{G}}(U)\longrightarrow{\mathcal{G}}(V)$ are morphisms of
${\mathcal{O}}(U)$-modules for all $V$, $U\in{\mathbf{B}}$ such that $V\subset
U$.
###### Proposition 2.1.3.
A contravariant functor ${\mathcal{G}}$ with an ${\mathcal{O}}$-module
structure on a base ${\mathbf{B}}$ of open subsets of $X$ can be extended to a
sheaf of ${\mathcal{O}}$-modules ${\mathcal{F}}$ on $X$ if and only if the
following condition holds. For any open subset $V\in{\mathbf{B}}$, any its
covering $V=\bigcup_{\alpha}V_{\alpha}$ by open subsets
$V_{\alpha}\in{\mathbf{B}}$, and any (or, equivalently, some particular)
covering $V_{\alpha}\cap V_{\beta}=\bigcup_{\gamma}W_{\alpha\beta\gamma}$ of
the intersections $V_{\alpha}\cap V_{\beta}$ by open subsets
$W_{\alpha\beta\gamma}\in{\mathbf{B}}$ the sequence of abelian groups (or
${\mathcal{O}}(V)$-modules)
(9) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathcal{G}}(V)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod_{\alpha}{\mathcal{G}}(V_{\alpha})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod_{\alpha,\beta,\gamma}{\mathcal{G}}(W_{\alpha\beta\gamma})$
must be exact. The functor of restriction of sheaves of
${\mathcal{O}}$-modules to a base ${\mathbf{B}}$ is an equivalence between the
category of sheaves of ${\mathcal{O}}$-modules on $X$ and the category of
contravariant functors on ${\mathbf{B}}$, endowed with ${\mathcal{O}}$-module
structures and satisfying (9).
###### Sketch of proof.
As above, we notice that if the functor ${\mathcal{G}}$ (with its
${\mathcal{O}}$-module structure) has been extended to a sheaf of
${\mathcal{O}}$-modules ${\mathcal{G}}$ on $X$, then for any open subset
$U\subset X$ there is an exact sequence of ${\mathcal{O}}(U)$-modules
$\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathcal{F}}(U)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod_{V}{\mathcal{G}}(V)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod_{W,V^{\prime},V^{\prime\prime}}{\mathcal{G}}(W),$
the summation rules being as in (7). Conversely, given a functor
${\mathcal{G}}$ with an ${\mathcal{O}}$-module structure one can recover the
${\mathcal{O}}(U)$-module ${\mathcal{F}}(U)$ as the kernel of the right arrow.
The rest is a conventional argument with (pre)sheaves and coverings. Recall
that a presheaf ${\mathcal{F}}$ on $X$ is called _separated_ if the map
${\mathcal{F}}(U)\longrightarrow\prod_{\alpha}{\mathcal{F}}(U_{\alpha})$ is
injective for any open covering $U=\bigcup_{\alpha}U_{\alpha}$ of an open
subset $U\subset X$. Similarly, a contravariant functor ${\mathcal{G}}$ on a
base ${\mathbf{B}}$ is said to be separated if its sequences (9) are exact at
the leftmost nontrivial term.
For any open covering $U=\bigcup_{\alpha}V_{\alpha}$ of an open subset
$U\subset X$ by open subsets $V_{\alpha}\in{\mathbf{B}}$ and any open
coverings $V_{\alpha}\cap V_{\beta}=\bigcup_{\gamma}W_{\alpha\beta\gamma}$ of
the intersections $V_{\alpha}\cap V_{\beta}$ by open subsets
$W_{\alpha\beta\gamma}\in{\mathbf{B}}$ there is a natural map from the (above-
defined) ${\mathcal{O}}(U)$-module ${\mathcal{F}}(U)$ to the kernel of the
morphism
(10) $\textstyle\prod_{\alpha}{\mathcal{G}}(V_{\alpha})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod_{\alpha,\beta,\gamma}{\mathcal{G}}(W_{\alpha\beta\gamma}).$
Let us show that this map is an isomorphism provided that ${\mathcal{G}}$
satisfies (9). In particular, it will follow that
${\mathcal{F}}(V)={\mathcal{G}}(V)$ for $V\in{\mathbf{B}}$.
Clearly, when ${\mathcal{G}}$ is separated, the kernel of (10) does not depend
on the choice of the open subsets $W_{\alpha\beta\gamma}$. So we can assume
that the collection $\\{W_{\alpha\beta\gamma}\\}$ for fixed $\alpha$ and
$\beta$ consists of all open subsets $W\in{\mathbf{B}}$ such that $W\subset
V_{\alpha}\cap V_{\beta}$.
Furthermore, one can easily see that the map from ${\mathcal{F}}(U)$ to the
kernel of (10) is injective whenever ${\mathcal{G}}$ is separated. To check
surjectivity, suppose that we are given a collection of sections
$\phi_{\alpha}\in{\mathcal{G}}(V_{\alpha})$ representing an element of the
kernel.
Fix an open subset $V\in{\mathbf{B}}$, $V\subset U$, and consider its covering
by all the open subsets $W\in{\mathbf{B}}$ such that $W\subset V\cap
V_{\alpha}$ for some $\alpha$. Set
$\psi_{W}=\phi_{\alpha}|_{W}\in{\mathcal{G}}(W)$ for every such $W$; by
assumption, if $W\subset V_{\alpha}\cap V_{\beta}$, then
$\phi_{\alpha}|_{W}=\phi_{\beta}|_{W}$, so the element $\psi_{W}$ is well-
defined. Applying (9), we conclude that there exists a unique element
$\phi_{V}\in{\mathcal{G}}(V)$ such that $\phi_{V}|_{W}=\psi_{W}$ for any
$W\subset V\cap V_{\alpha}$. The collection of sections $\phi_{V}$ represents
an element of ${\mathcal{F}}(U)$ that is a preimage of our original element of
the kernel of (10).
Now let us show that ${\mathcal{F}}$ is a sheaf. Let
$U=\bigcup_{\alpha}U_{\alpha}$ be a open covering of an open subset $U\subset
X$. First let us see that ${\mathcal{F}}$ is separated provided that
${\mathcal{G}}$ is. Let $s\in{\mathcal{F}}(U)$ be a section whose restriction
to all the open subsets $U_{\alpha}$ vanishes. The element $s$ is represented
by a collection of sections $\phi_{V}\in{\mathcal{G}}(V)$ defined for all open
subsets $V\subset U$, $V\in{\mathbf{B}}$. The condition $s|_{U_{\alpha}}=0$
means that $\phi_{W}=0$ whenever $W\subset U_{\alpha}$, $W\in{\mathbf{B}}$. To
check that $\phi_{V}=0$ for all $V$, we notice that open subsets $W\subset V$,
$W\in{\mathbf{B}}$ for which there exists $\alpha$ such that $W\subset
U_{\alpha}$ form an open covering of $V$.
Finally, let $s_{\alpha}\in{\mathcal{F}}(U_{\alpha})$ be a collection of
sections such that $s_{\alpha}|_{U_{\alpha}\cap
U_{\beta}}=s_{\beta}|_{U_{\alpha}\cap U_{\beta}}$ for all $\alpha$ and
$\beta$. Every element $s_{\alpha}$ is represented by a collection of sections
$\phi_{V}\in{\mathcal{G}}(V)$ defined for all open subsets $V\subset
U_{\alpha}$, $V\in{\mathbf{B}}$. Clearly, the element $\phi_{V}$ does not
depend on the choice of a particular $\alpha$ for which $V\subset U_{\alpha}$,
so our notation is consistent. All the open subsets $V\subset U$,
$V\in{\mathbf{B}}$ for which there exists some $\alpha$ such that $V\subset
U_{\alpha}$ form an open covering of the open subset $U\subset X$. The
collection of sections $\phi_{V}$ represents an element of the kernel of the
morphism (10) for this covering, hence it corresponds to an element of
${\mathcal{F}}(U)$. ∎
###### Remark 2.1.4.
Let $X$ be a topological space with a topology base ${\mathbf{B}}$ consisting
of quasi-compact open subsets (in the induced topology) for which the
intersection of any two open subsets from ${\mathbf{B}}$ that are contained in
a third open subset from ${\mathbf{B}}$ is quasi-compact as well. E. g., any
scheme $X$ with the base of all affine open subschemes has these properties.
Then it suffices to check both the conditions (6) and (9) for _finite_
coverings $V_{\alpha}$ and $W_{\alpha\beta\gamma}$ only.
Indeed, let us explain the sheaf case. Obviously, injectivity of the left
arrow in (9) for any given covering $V=\bigcup_{\alpha}V_{\alpha}$ follows
from such injectivity for a subcovering $V=\bigcup_{i}V_{i}$,
$\\{V_{i}\\}\subset\\{V_{\alpha}\\}$. Assuming ${\mathcal{G}}$ is separated,
one checks that exactness of the sequence (9) for any given covering follows
from the same exactness for a subcovering.
It follows that for a topological space $X$ with a fixed topology base
${\mathbf{B}}$ satisfying the above condition there is another duality
construction relating sheaves to cosheaves in addition to the one we used in
the proof of Theorem 2.1.2. Given a sheaf of ${\mathcal{O}}$-modules
${\mathcal{F}}$ on $X$, one restricts it to the base ${\mathbf{B}}$, obtaining
a contravariant functor ${\mathcal{G}}$ with an ${\mathcal{O}}$-module
structure, defines the dual covariant functor ${\mathfrak{Q}}$ with an
${\mathcal{O}}$-module structure on ${\mathbf{B}}$ by the rule
${\mathfrak{Q}}[V]=\operatorname{Hom}_{\mathbb{Z}}({\mathcal{G}}(V),I)$, where
$I$ is an injective abelian group, and extends the functor ${\mathfrak{Q}}$ to
a cosheaf of ${\mathcal{O}}$-modules ${\mathfrak{P}}$ on $X$.
It is this second duality functor, rather than the one from the proof of
Theorem 2.1.2, that will play a role in the sequel (Section 3.4 being a rare
exception).
### 2.2. Exact category of contraherent cosheaves
Let $X$ be a scheme and ${\mathcal{O}}={\mathcal{O}}_{X}$ be its structure
sheaf. A cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ is called
_contraherent_ if for any pair of embedded affine open subschemes $V\subset
U\subset X$
1. (i)
the morphism of ${\mathcal{O}}_{X}(V)$-modules
${\mathfrak{P}}[V]\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U])$
induced by the corestriction morphism
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]$ is an isomorphism; and
2. (ii)
one has
$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}^{>0}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U])=0$.
It follows from Lemma 1.2.4 that the ${\mathcal{O}}_{X}(U)$-module
${\mathcal{O}}_{X}(V)$ has projective dimension at most $1$, so it suffices to
require the vanishing of $\operatorname{Ext}^{1}$ in the condition (ii). We
will call (ii) the _contraadjustness condition_ , and (i) the _contraherence
condition_.
###### Theorem 2.2.1.
The restriction of cosheaves of ${\mathcal{O}}_{X}$-modules to the base of all
affine open subschemes of $X$ induces an equivalence between the category of
contraherent cosheaves on $X$ and the category of covariant functors
${\mathfrak{Q}}$ with ${\mathcal{O}}_{X}$-module structures on the category of
affine open subschemes of $X$, satisfying the conditions (i-ii) for any pair
of embedded affine open subschemes $V\subset U\subset X$.
###### Proof.
According to Theorem 2.1.2, a cosheaf of ${\mathcal{O}}_{X}$-modules is
determined by its restriction to the base of affine open subsets of $X$. The
contraadjustness and contraherence conditions depend only on this restriction.
By Lemma 1.2.4, given any affine scheme $U$, a module $P$ over
${\mathcal{O}}(U)$ is contraadjusted if and only if
$\operatorname{Ext}^{1}_{{\mathcal{O}}(U)}({\mathcal{O}}(V),P)=0$ for all
affine open subschemes $V\subset U$. Finally, the key observation is that the
contraadjustness and contraherence conditions (i-ii) for a covariant functor
with an ${\mathcal{O}}_{X}$-module structure on the category of affine open
subschemes of $X$ imply the cosheaf condition (6). This follows from Lemma
1.2.6(b) and Remark 2.1.4. ∎
###### Remark 2.2.2.
Of course, one can similarly define quasi-coherent sheaves ${\mathcal{F}}$ on
$X$ as sheaves of ${\mathcal{O}}_{X}$-modules such that for any pair of
embedded affine open subschemes $V\subset U\subset X$ the morphism of
${\mathcal{O}}_{X}(V)$-modules
${\mathcal{O}}_{X}(V)\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{F}}(U)\longrightarrow{\mathcal{F}}(V)$
induced by the restriction morphism
${\mathcal{F}}(U)\longrightarrow{\mathcal{F}}(V)$ is an isomorphism. Since the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{O}}_{X}(V)$ is always flat, no
version of the condition (ii) is needed in this case. The analogue of Theorem
2.2.1 is well-known for quasi-coherent sheaves (and can be proven in the same
way).
A short sequence of contraherent cosheaves
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ is said to be exact if the sequence of cosection modules
$0\longrightarrow{\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{R}}[U]\longrightarrow
0$ is exact for every affine open subscheme $U\subset X$. Notice that if
$U_{\alpha}$ is an affine open covering of an affine scheme $U$ and
${\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}$ is
a sequence of contraherent cosheaves on $U$, then the sequence of
${\mathcal{O}}(U)$-modules
$0\longrightarrow{\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{R}}[U]\longrightarrow
0$ is exact if and only if all the sequences of
${\mathcal{O}}(U_{\alpha})$-modules
$0\longrightarrow{\mathfrak{P}}[U_{\alpha}]\longrightarrow{\mathfrak{Q}}[U_{\alpha}]\longrightarrow{\mathfrak{R}}[U_{\alpha}]\longrightarrow
0$ are. This follows from Lemma 1.4.1(a).
We denote the exact category of contraherent cosheaves on a scheme $X$ by
$X{\operatorname{\mathsf{--ctrh}}}$. By the definition, the functors of
cosections over affine open subschemes are exact on this exact category. It is
also has exact functors of infinite product, which commute with cosections
over affine open subschemes (and in fact, over any quasi-compact quasi-
separated open subschemes as well). For a more detailed discussion of this
exact category structure, we refer the reader to Section 3.1.
###### Corollary 2.2.3.
The functor assigning the ${\mathcal{O}}(U)$-module ${\mathfrak{P}}[U]$ to a
contraherent cosheaf ${\mathfrak{P}}$ on an affine scheme $U$ is an
equivalence between the exact category $U{\operatorname{\mathsf{--ctrh}}}$ of
contraherent cosheaves on $U$ and the exact category
${\mathcal{O}}(U){\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ of
contraadjusted modules over the commutative ring ${\mathcal{O}}(U)$.
###### Proof.
Clear from the above arguments together with Lemmas 1.2.1(b) and 1.2.4. ∎
###### Lemma 2.2.4.
Let ${\mathfrak{P}}$ be a cosheaf of ${\mathcal{O}}_{U}$-modules and
${\mathfrak{Q}}$ be a contraherent cosheaf on an affine scheme $U$. Then the
group of morphisms of cosheaves of ${\mathcal{O}}_{U}$-modules
${\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$ is isomorphic to the group of
morphisms of ${\mathcal{O}}(U)$-modules
${\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]$.
###### Proof.
Any morphism of cosheaves of ${\mathcal{O}}_{U}$-modules
${\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$ induces a morphism of the
${\mathcal{O}}(U)$-modules of global cosections. Conversely, given a morphism
of ${\mathcal{O}}(U)$-modules
${\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]$ and an affine open
subscheme $V\subset U$, the composition
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]$
is a morphism of ${\mathcal{O}}(U)$-modules from an ${\mathcal{O}}(V)$-module
${\mathfrak{P}}[V]$ to an ${\mathcal{O}}(U)$-module ${\mathfrak{Q}}[U]$. It
induces, therefore, a morphism of ${\mathcal{O}}(V)$-modules
${\mathfrak{P}}[V]\longrightarrow\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{O}}(V),{\mathfrak{Q}}[U])\simeq{\mathfrak{Q}}[V]$.
Now a morphism between the restrictions of two cosheaves of
${\mathcal{O}}_{U}$-modules to the base of affine open subschemes of $U$
extends uniquely to a morphism between the whole cosheaves. ∎
A contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$ is said to be _locally
cotorsion_ if for any affine open subscheme $U\subset X$ the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}(U)$ is cotorsion. By Lemma
1.3.6(a), the property of a contraherent cosheaf on an affine scheme to be
cotorsion is indeed a local, so our terminology is constent.
A contraherent cosheaf ${\mathfrak{J}}$ on a scheme $X$ is called _locally
injective_ if for any affine open subscheme $U\subset X$ the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{J}}(U)$ is injective. By Lemma
1.3.6(b), local injectivity of a contraherent cosheaf is indeed a local
property.
Just as above, one defines the exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ of locally cotorsion and
locally injective contraherent cosheaves on $X$. These are full subcategories
closed under extensions, infinite products, and cokernels of admissible
monomorphisms in $X{\operatorname{\mathsf{--ctrh}}}$, with the induced exact
category structures.
The exact category $U{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ of
locally cotorsion contraherent cosheaves on an affine scheme $U$ is equivalent
to the exact category
${\mathcal{O}}(U){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ of cotorsion
${\mathcal{O}}(U)$-modules. The exact category
$U{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ of locally injective
contraherent cosheaves on $U$ is equivalent to the additive category
${\mathcal{O}}(U){\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}$ of injective
${\mathcal{O}}(U)$-modules endowed with the trivial exact category structure.
###### Remark 2.2.5.
Notice that a morphism of contraherent cosheaves on $X$ is an admissible
monomorphism if and only if it acts injectively on the cosection modules over
all the affine open subschemes on $X$. At the same time, the property of a
morphism of contraherent cosheaves on $X$ to be an admissible monomorphism is
_not_ local in $X$, and _neither_ is the property of a cosheaf of
${\mathcal{O}}_{X}$-modules to be contraherent (see Section 3.2 below). The
property of a morphism of contraherent cosheaves to be an admissible
epimorphism is local, though (see Lemma 1.4.1(b)). All of the above applies to
locally cotorsion and locally injective contraherent cosheaves as well.
Notice also that a morphism of locally injective or locally cotorsion
contraherent cosheaves that is an admissible epimorphism in
$X{\operatorname{\mathsf{--ctrh}}}$ may _not_ be an admissible epimorphism in
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$. On the other hand, if a
morphism of locally injective or locally cotorsion contraherent cosheaves is
an admissible monomorphism in $X{\operatorname{\mathsf{--ctrh}}}$, then it is
also an admissible monomorphism in
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$, as it is clear from the
above.
### 2.3. Direct and inverse images of contraherent cosheaves
Let ${\mathcal{O}}_{X}$ be a sheaf of associative rings on a topological space
$X$ and ${\mathcal{O}}_{Y}$ be such a sheaf on a topological space $Y$.
Furthermore, let $f\colon Y\longrightarrow X$ be a morphism of ringed spaces,
i. e., a continuous map $Y\longrightarrow X$ together with a morphism
${\mathcal{O}}_{X}\longrightarrow f_{*}{\mathcal{O}}_{Y}$ of sheaves of rings
over $X$. Then for any cosheaf of ${\mathcal{O}}_{Y}$-modules ${\mathfrak{Q}}$
the rule $(f_{!}{\mathfrak{Q}})[W]={\mathfrak{Q}}[f^{-1}(W)]$ for all open
subsets $W\subset X$ defines a cosheaf of ${\mathcal{O}}_{X}$-modules
$f_{!}{\mathfrak{Q}}$.
Let ${\mathcal{O}}_{X}$ be a sheaf of associative rings on a topological space
$X$ and $Y\subset X$ be an open subspace. Denote by
${\mathcal{O}}_{Y}={\mathcal{O}}_{X}|_{Y}$ the restriction of the sheaf of
rings ${\mathcal{O}}_{X}$ onto $Y$, and by $j\colon Y\longrightarrow X$ the
corresponding morphism (open embedding) of ringed spaces. Given a cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on $X$, the restriction
${\mathfrak{P}}|_{Y}$ of ${\mathfrak{P}}$ onto $Y$ is a cosheaf of
${\mathcal{O}}_{Y}$-modules defined by the rule
${\mathfrak{P}}|_{Y}(V)={\mathfrak{P}}(V)$ for any open subset $V\subset Y$.
One can easily see that the restriction functor
${\mathfrak{P}}\longmapsto{\mathfrak{P}}|_{Y}$ is right adjoint to the direct
image functor ${\mathfrak{Q}}\longmapsto j_{!}{\mathfrak{Q}}$ between the
categories of cosheaves of ${\mathcal{O}}_{Y}$\- and
${\mathcal{O}}_{X}$-modules, that is the adjunction isomorphism
(11)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}(j_{!}{\mathfrak{Q}},{\mathfrak{P}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{Y}}({\mathfrak{Q}},{\mathfrak{P}}|_{Y})$
holds for any cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ and
cosheaf of ${\mathcal{O}}_{Y}$-modules ${\mathfrak{Q}}$, where
$\operatorname{Hom}^{{\mathcal{O}}_{X}}$ and
$\operatorname{Hom}^{{\mathcal{O}}_{Y}}$ denote the abelian groups of
morphisms in the categories of cosheaves of modules over the sheaves of rings
${\mathcal{O}}_{X}$ and ${\mathcal{O}}_{Y}$. Since one has
$(j_{!}{\mathfrak{Q}})_{Y}\simeq{\mathfrak{Q}}$, it follows, in particular,
that the functor $j_{!}$ is fully faithful.
Let $f\colon Y\longrightarrow X$ be an affine morphism of schemes, and let
${\mathfrak{Q}}$ be a contraherent cosheaf on $Y$. Then $f_{!}{\mathfrak{Q}}$
is a contraherent cosheaf on $X$. Indeed, for any affine open subscheme
$U\subset X$ the ${\mathcal{O}}_{X}(U)$-module
$(f_{!}{\mathfrak{Q}})[U]={\mathfrak{Q}}[(f^{-1}(U)]$ is contraadjusted
according to Lemma 1.2.2(a) applied to the morphism of commutative rings
${\mathcal{O}}_{X}(U)\longrightarrow{\mathcal{O}}_{Y}(f^{-1}(U))$. For any
pair of embedded affine open subschemes $V\subset U\subset X$ we have natural
isomorphisms of ${\mathcal{O}}_{X}(U)$-modules
$(f_{!}{\mathfrak{Q}})[V]={\mathfrak{Q}}[f^{-1}(V)]\simeq\operatorname{Hom}_{{\mathcal{O}}_{Y}(f^{-1}(U))}({\mathcal{O}}_{Y}(f^{-1}(V)),{\mathfrak{Q}}[f^{-1}(U)])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{Q}}[f^{-1}(U)])=\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),(f_{!}{\mathfrak{Q}})[U]),$
since
${\mathcal{O}}_{Y}(f^{-1}(V))\simeq{\mathcal{O}}_{Y}(f^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{X}(V)$.
Recall that a scheme $X$ is called _semi-separated_ [61, Appendix B], if it
admits an affine open covering with affine pairwise intersections of the open
subsets belonging to the covering. Equivalently, a scheme $X$ is semi-
separated if and only if the diagonal morphism $X\longrightarrow
X\times_{\operatorname{Spec}{\mathbb{Z}}}X$ is affine, and if and only if the
intersection of any two affine open subschemes of $X$ is affine. Any morphism
from an affine scheme to a semi-separated scheme is affine, and the fibered
product of any two affine schemes over a semi-separated base scheme is an
affine scheme.
We will say that a morphism of schemes $f\colon Y\longrightarrow X$ is
_coaffine_ if for any affine open subscheme $V\subset Y$ there exists an
affine open subscheme $U\subset X$ such that $f(V)\subset U$, and for any two
such affine open subschemes $f(V)\subset U^{\prime}$, $U^{\prime\prime}\subset
X$ there exists a third affine open subscheme $U\subset X$ such that
$f(V)\subset U\subset U^{\prime}\cap U^{\prime\prime}$. If the scheme $X$ is
semi-separated, then the second condition is trivial. (We will see below in
Section 3.3 that the second condition is not actually necessary for our
constructions.)
Any morphism into an affine scheme is coaffine. Any embedding of an open
subscheme is coaffine. The composition of two coaffine morphisms between semi-
separated schemes is a coaffine morphism.
Let $f\colon Y\longrightarrow X$ be a very flat coaffine morphism of schemes
(see Section 1.7 for the definition and discussion of the former property),
and let ${\mathfrak{P}}$ be a contraherent cosheaf on $X$. Define a
contraherent cosheaf $f^{!}{\mathfrak{P}}$ on $Y$ as follows.
Let $V\subset Y$ be an affine open subscheme. Pick an affine open subscheme
$U\subset X$ such that $f(V)\subset U$, and set
$(f^{!}{\mathfrak{P}})[V]=\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(V),{\mathfrak{P}}[U])$.
Due to the contraherence condition on ${\mathfrak{P}}$, this definition of the
${\mathcal{O}}_{Y}(V)$-module $(f^{!}{\mathfrak{P}})[V]$ does not depend on
the choice of an affine open subscheme $U\subset X$. Since $f$ is a very flat
morphism, the ${\mathcal{O}}_{Y}(V)$-module $(f^{!}{\mathfrak{P}})[V]$ is
contraadjusted by Lemma 1.2.3(a). The contraherence condition obviously holds
for $f^{!}{\mathfrak{P}}$.
Let $f\colon Y\longrightarrow X$ be a flat coaffine morphism of schemes, and
${\mathfrak{P}}$ be a locally cotorsion contraherent cosheaf on $X$. Then the
same rule as above defines a locally cotorsion contraherent cosheaf
$f^{!}{\mathfrak{P}}$ on $Y$. One just uses Lemma 1.3.5(a) in place of Lemma
1.2.3(a). For any coaffine morphism of schemes $f\colon Y\longrightarrow X$
and a locally injective contraherent cosheaf ${\mathfrak{J}}$ on $X$ the very
same rule defines a locally injective contraherent cosheaf
$f^{!}{\mathfrak{J}}$ on $Y$.
For an open embedding of schemes $j\colon Y\longrightarrow X$ and a
contraherent cosheaf ${\mathfrak{P}}$ on $X$ one clearly has
$j^{!}{\mathfrak{P}}\simeq{\mathfrak{P}}|_{Y}$.
If $f\colon Y\longrightarrow X$ is an affine morphism of schemes and
${\mathfrak{Q}}$ is a locally cotorsion contraherent cosheaf on $Y$, then
$f_{!}{\mathfrak{Q}}$ is a locally cotorsion contraherent cosheaf on $X$. If
$f\colon Y\longrightarrow X$ is a flat affine morphism and ${\mathfrak{I}}$ is
a locally injective contraherent cosheaf on $Y$, then $f_{!}{\mathfrak{I}}$ is
a locally injective contraherent cosheaf on $X$.
Let $f\colon Y\longrightarrow X$ be an affine coaffine morphism of schemes.
Then for any contraherent cosheaf ${\mathfrak{Q}}$ on $Y$ and any locally
injective contraherent cosheaf ${\mathfrak{P}}$ on $X$ there is a natural
adjunction isomorphism
$\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{P}})\simeq\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{P}})$,
where $\operatorname{Hom}^{X}$ and $\operatorname{Hom}^{Y}$ denote the abelian
groups of morphisms in the categories of contraherent cosheaves on $X$ and
$Y$.
If, in addition, the morphism $f$ is flat, then such an isomorphism holds for
any contraherent cosheaf ${\mathfrak{Q}}$ on $Y$ and any locally cotorsion
contraherent cosheaf ${\mathfrak{P}}$ on $X$; in particular, $f_{!}$ and
$f^{!}$ form an adjoint pair of functors between the exact categories of
locally cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. Their restrictions also
act as adjoint functors between the exact categories of locally injective
contraherent cosheaves $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ and
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$.
If the morphism $f$ is very flat, then the functor $f^{!}\colon
X{\operatorname{\mathsf{--ctrh}}}\longrightarrow
Y{\operatorname{\mathsf{--ctrh}}}$ is right adjoint to the functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$. In all the mentioned cases, both abelian
groups $\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{P}})$ and
$\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{P}})$ are identified
with the group whose elements are the collections of homomorphisms of
${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{Q}}(V)\longrightarrow{\mathfrak{P}}(U)$, defined for all affine
open subschemes $U\subset X$ and $V\subset Y$ such that $f(V)\subset U$ and
compatible with the corestriction maps.
All the functors between exact categories of contraherent cosheaves
constructed in the above section are exact and preserve infinite products. For
a construction of the direct image functor $f_{!}$ (acting between appropriate
exact subcategories of the exact categories of adjusted objects in the exact
categories of contraherent cosheaves) for a nonaffine morphism of schemes $f$,
see Section 4.5 below.
### 2.4. $\operatorname{\mathfrak{Cohom}}$ from a quasi-coherent sheaf to a
contraherent cosheaf
Let $X$ be a scheme over an affine scheme $\operatorname{Spec}R$. Let
${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and $J$ be an injective
$R$-module. Then the rule
$U\longmapsto\operatorname{Hom}_{R}({\mathcal{M}}(U),J)$ for affine open
subschemes $U\subset X$ defines a contraherent cosheaf over $X$ (cf. Remark
2.1.4). We will denote it by
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)$. Since the
${\mathcal{O}}_{X}(U)$-module $\operatorname{Hom}_{R}({\mathcal{M}}(U),J)$ is
cotorsion by Lemma 1.3.3(b), it is even a locally cotorsion contraherent
cosheaf. When ${\mathcal{F}}$ is a flat quasi-coherent sheaf on $X$ and $J$ is
an injective $R$-module, the contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{F}},J)$ is locally injective.
We recall the definitions of a very flat morphism of schemes and a very flat
quasi-coherent sheaf on a scheme from Section 1.7. If
$X\longrightarrow\operatorname{Spec}R$ is a very flat morphism of schemes and
${\mathcal{F}}$ is a very flat quasi-coherent sheaf on $X$, then for any
contraadjusted $R$-module $P$ the rule
$U\longmapsto\operatorname{Hom}_{R}({\mathcal{F}}(U),P)$ for affine open
subschemes $U\subset X$ defines a contraherent cosheaf on $X$. The
contraadjustness condition on the ${\mathcal{O}}_{X}(U)$-modules
$\operatorname{Hom}_{R}({\mathcal{F}}(U),P)$ holds by Lemma 1.2.3(c). We will
denote the cosheaf so constructed by
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{F}},P)$.
Analogously, if a scheme $X$ is flat over $\operatorname{Spec}R$ and a quasi-
coherent sheaf ${\mathcal{F}}$ on $X$ is flat (or, more generally, the quasi-
coherent sheaf ${\mathcal{F}}$ on $X$ is flat over $\operatorname{Spec}R$, in
the obvious sense), then for any cotorsion $R$-module $P$ the rule
$U\longmapsto\operatorname{Hom}_{R}({\mathcal{F}}(U),P)$ for affine open
subschemes $U\subset X$ defines a contraherent cosheaf on $X$. In fact, the
${\mathcal{O}}_{X}(U)$-modules $\operatorname{Hom}_{R}({\mathcal{F}}(U),P)$
are cotorsion by Lemma 1.3.3(a), hence the contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{F}},P)$ constructed in this way
is locally cotorsion.
Let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on a scheme $X$ and
${\mathfrak{P}}$ be a contraherent cosheaf on $X$. Then the contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is
defined by the rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for all affine open subschemes $U\subset X$. For any two embedded affine open
subschemes $V\subset U\subset X$ one has
$\operatorname{Hom}_{{\mathcal{O}}_{X}(V)}({\mathcal{F}}(V),{\mathfrak{P}}[V])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(V)}({\mathcal{O}}_{X}(V)\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{F}}(U),\>\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U]))\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])),$
so the contraherence condition holds. The contraadjustness condition follows
from Lemma 1.2.1(b).
Similarly, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is locally cotorsion contraherent cosheaf on $X$, then the
contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is defined
by the same rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for all affine open subschemes $U\subset X$. By Lemma 1.3.2(a),
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is a
locally cotorsion contraherent cosheaf on $X$.
Finally, if ${\mathcal{M}}$ is a quasi-coherent sheaf on $X$ and
${\mathfrak{J}}$ is a locally injective contraherent cosheaf, then the
contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ is defined
by the very same rule. One checks the contraherence condition in the same way
as above. By Lemma 1.3.2(b),
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ is a
locally cotorsion contraherent cosheaf on $X$. If ${\mathcal{F}}$ is a flat
quasi-coherent sheaf and ${\mathfrak{J}}$ is a locally injective contraherent
cosheaf on $X$, then the contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})$ is locally
injective.
For any contraadjusted module $P$ over a commutative ring $R$, denote by
$\widecheck{P}$ the corresponding contraherent cosheaf on
$\operatorname{Spec}R$. Let $f\colon X\longrightarrow\operatorname{Spec}R$ be
a morphism of schemes and ${\mathcal{F}}$ be a quasi-coherent sheaf on $X$.
Then whenever ${\mathcal{F}}$ is a very flat quasi-coherent sheaf and $f$ is a
very flat morphism, there is a natural isomorphism of contraherent cosheaves
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{F}},P)\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},f^{!}\widecheck{P})$
on $X$. Indeed, for any affine open subscheme $U\subset X$ one has
$\operatorname{Hom}_{R}({\mathcal{F}}(U),P)\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),\operatorname{Hom}_{R}({\mathcal{O}}_{X}(U),P))\simeq\operatorname{Hom}_{R}({\mathcal{F}}(U),(f^{!}\widecheck{P})[U]).$
The same isomorphism holds whenever ${\mathcal{F}}$ is a flat quasi-coherent
sheaf, $f$ is a flat morphism, and $P$ is a cotorsion $R$-module. Finally, for
any quasi-coherent sheaf ${\mathcal{M}}$ on $X$, any morphism $f\colon
X\longrightarrow\operatorname{Spec}R$, and any injective $R$-module $J$ there
is a natural isomorphism of locally cotorsion contraherent cosheaves
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},f^{!}\widecheck{J})$
on $X$.
### 2.5. Contraherent cosheaves of $\operatorname{\mathfrak{Hom}}$ between
quasi-coherent sheaves
A quasi-coherent sheaf ${\mathcal{P}}$ on a scheme $X$ is said to be
_cotorsion_ [20] if
$\operatorname{Ext}_{X}^{1}({\mathcal{F}},{\mathcal{P}})=0$ for any flat
quasi-coherent sheaf ${\mathcal{F}}$ on $X$. Here $\operatorname{Ext}_{X}$
denotes the $\operatorname{Ext}$ groups in the abelian category of quasi-
coherent sheaves on $X$. A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is
called _contraadjusted_ if one has
$\operatorname{Ext}^{1}_{X}({\mathcal{F}},{\mathcal{P}})=0$ for any very flat
quasi-coherent sheaf ${\mathcal{F}}$ on $X$ (see Section 1.7 for the
definition of the latter).
Clearly the two classes of quasi-coherent sheaves on $X$ so defined are closed
under extensions, so they form full exact subcategories in the abelian
category of quasi-coherent sheaves. Also, these exact subcategories are closed
under the passage to direct summands of objects.
For any affine morphism of schemes $f\colon Y\longrightarrow X$, any flat
quasi-coherent sheaf ${\mathcal{F}}$ on $X$, and any quasi-coherent sheaf
${\mathcal{P}}$ on $Y$ there is a natural isomorphism of the extension groups
$\operatorname{Ext}_{Y}^{1}(f^{*}{\mathcal{F}},{\mathcal{P}})\simeq\operatorname{Ext}_{X}^{1}({\mathcal{F}},f_{*}{\mathcal{P}})$.
Hence the classes of contraadjusted and cotorsion quasi-coherent sheaves on
schemes are preserved by the direct images with respect to affine morphisms.
Let ${\mathcal{F}}$ be a quasi-coherent sheaf on a scheme $X$. Suppose that an
associative ring $R$ acts on $X$ from the right by quasi-coherent sheaf
endomorphisms. Let $M$ be a left $R$-module. Define a contravariant functor
${\mathcal{F}}\otimes_{R}M$ from the category of affine open subschemes
$U\subset X$ to the category of abelian groups by the rule
$({\mathcal{F}}\otimes_{R}M)(U)={\mathcal{F}}(U)\otimes_{R}M$. The natural
${\mathcal{O}}_{X}(U)$-module structures on the groups
$({\mathcal{F}}\otimes_{R}M)(U)$ are compatible with the restriction maps
$({\mathcal{F}}\otimes_{R}M)(U)\longrightarrow({\mathcal{F}}\otimes_{R}M)(V)$
for embedded affine open subschemes $V\subset U\subset X$, and the quasi-
coherence condition
$({\mathcal{F}}\otimes_{R}M)(V)\simeq{\mathcal{O}}_{X}(V)\otimes_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}\otimes_{R}M)(U)$
holds (see Remark 2.2.2). Therefore, the functor ${\mathcal{F}}\otimes_{R}M$
extends uniquely to a quasi-coherent sheaf on $X$, which we will denote also
by ${\mathcal{F}}\otimes_{R}M$.
Let ${\mathcal{P}}$ be a quasi-coherent sheaf on $X$. Then the abelian group
$\operatorname{Hom}_{X}({\mathcal{F}},{\mathcal{P}})$ of morphisms in the
category of quasi-coherent sheaves on $X$ has a natural left $R$-module
structure. One can easily construct a natural isomorphism of abelian groups
$\operatorname{Hom}_{X}({\mathcal{F}}\otimes_{R}M,\>{\mathcal{P}})\simeq\operatorname{Hom}_{R}(M,\operatorname{Hom}_{X}({\mathcal{F}},{\mathcal{P}}))$.
###### Lemma 2.5.1.
Suppose that $\operatorname{Ext}_{X}^{i}({\mathcal{F}},{\mathcal{P}})=0$ for
$0<i\le i_{0}$ and either
(a) $M$ is a flat left $R$-module, or
(b) the right $R$-modules ${\mathcal{F}}(U)$ are flat for all affine open
subschemes $U\subset X$.
Then there is a natural isomorphism of abelian groups
$\operatorname{Ext}_{X}^{i}({\mathcal{F}}\otimes_{R}M,\>{\mathcal{P}})\simeq\operatorname{Ext}_{R}^{i}(M,\operatorname{Hom}_{X}({\mathcal{F}},{\mathcal{P}}))$
for all $0\le i\le i_{0}$.
###### Proof.
Replace $M$ by its left projective $R$-module resolution
$L_{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Then
$\operatorname{Ext}_{X}^{i}({\mathcal{F}}\otimes_{R}L_{j},\>{\mathcal{P}})=0$
for all $0<i\le i_{0}$ and all $j$. Due to the flatness condition (a) or (b),
the complex of quasi-coherent sheaves
${\mathcal{F}}\otimes_{R}L_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
a left resolution of the sheaf ${\mathcal{F}}\otimes_{R}M$. Hence the complex
of abelian groups
$\operatorname{Hom}_{X}({\mathcal{F}}\otimes_{R}L_{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{P}})$
computes
$\operatorname{Ext}_{X}^{i}({\mathcal{F}}\otimes_{R}M,\>{\mathcal{P}})$ for
$0\le i\le i_{0}$. On the other hand, this complex is isomorphic to the
complex
$\operatorname{Hom}_{R}(L_{\text{\smaller\smaller$\scriptstyle\bullet$}},\operatorname{Hom}_{R}({\mathcal{F}},{\mathcal{P}}))$,
which computes
$\operatorname{Ext}_{R}^{i}(M,\operatorname{Hom}_{X}({\mathcal{F}},{\mathcal{P}}))$.
∎
Let ${\mathcal{F}}$ be a quasi-coherent sheaf with a right action of a ring
$R$ on a scheme $X$, and let $f\colon Y\longrightarrow X$ be a morphism of
schemes. Then $f^{*}{\mathcal{F}}$ is a quasi-coherent sheaf on $Y$ with a
right action of $R$, and for any left $R$-module $M$ there is a natural
isomorphism of quasi-coherent sheaves $f^{*}({\mathcal{F}}\otimes_{R}M)\simeq
f^{*}{\mathcal{F}}\otimes_{R}M$. Analogously, if ${\mathcal{G}}$ is a quasi-
coherent sheaf on $Y$ with a right action of $R$ and $f$ is a quasi-compact
quasi-separated morphism, then $f_{*}{\mathcal{G}}$ is a quasi-coherent sheaf
on $X$ with a right action of $R$, and for any left $R$-module $M$ there is a
natural morphism of quasi-coherent sheaves
$f_{*}{\mathcal{G}}\otimes_{R}M\longrightarrow
f_{*}({\mathcal{G}}\otimes_{R}M)$ on $X$. If the morphism $f$ is affine or the
$R$-module $M$ is flat, then this map is an isomorphism of quasi-coherent
sheaves on $X$.
Let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on a semi-separated
scheme $X$, and let ${\mathcal{P}}$ be a contraadjusted quasi-coherent sheaf
on $X$. Define a contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})$ by the rule
$U\longmapsto\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}})$
for any affine open subscheme $U\subset X$, where $j\colon U\longrightarrow X$
denotes the identity open embedding. Given two embedded affine open subschemes
$V\subset U\subset X$ with the identity embeddings $j\colon U\longrightarrow
X$ and $k\colon V\longrightarrow X$, the adjunction provides a natural map of
quasi-coherent sheaves $j_{*}j^{*}{\mathcal{F}}\longrightarrow
k_{*}k^{*}{\mathcal{F}}$. There is also a natural action of the ring
${\mathcal{O}}_{X}(U)$ on the quasi-coherent sheaf $j_{*}j^{*}{\mathcal{F}}$.
Thus our rule defines a covariant functor with an ${\mathcal{O}}_{X}$-module
structure on the category of affine open subschemes in $X$.
Let us check that the contraadjustness and contraherence conditions are
satisfied. For a very flat ${\mathcal{O}}_{X}(U)$-module $G$, we have
$\operatorname{Ext}^{1}_{{\mathcal{O}}_{X}(U)}(G,\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}}))\\\
\simeq\operatorname{Ext}^{1}_{X}((j_{*}j^{*}{\mathcal{F}})\otimes_{{\mathcal{O}}_{X}(U)}G,\>{\mathcal{P}})\simeq\operatorname{Ext}^{1}_{X}(j_{*}(j^{*}{\mathcal{F}}\otimes_{{\mathcal{O}}_{X}(U)}G),\>{\mathcal{P}})=0,$
since $j_{*}(j^{*}{\mathcal{F}}\otimes_{{\mathcal{O}}_{X}(U)}G)$ is a very
flat quasi-coherent sheaf on $X$. For a pair of embedded affine open
subschemes $V\subset U\subset X$, we have
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}}))\simeq\operatorname{Hom}_{X}((j_{*}j^{*}{\mathcal{F}})\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{X}(V),\>{\mathcal{P}})\\\
\simeq\operatorname{Hom}_{X}(j_{*}(j^{*}{\mathcal{F}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{X}(V)),\>{\mathcal{P}})\simeq\operatorname{Hom}_{X}(k_{*}k^{*}{\mathcal{F}},{\mathcal{P}}).$
Similarly one defines a locally cotorsion contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})$ for a flat
quasi-coherent sheaf ${\mathcal{F}}$ and a cotorsion quasi-coherent sheaf
${\mathcal{P}}$ on $X$. When ${\mathcal{F}}$ is a flat quasi-coherent sheaf
and ${\mathcal{J}}$ is an injective quasi-coherent sheaf on $X$, the
contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{J}})$ is locally
injective.
Finally, let ${\mathcal{M}}$ be a quasi-coherent sheaf on a quasi-separated
scheme $X$, and let ${\mathcal{J}}$ be an injective quasi-coherent sheaf on
$X$. Then a locally cotorsion contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}})$ is defined by
the very same rule. The proof of the cotorsion and contraherence conditions is
the same as above.
###### Lemma 2.5.2.
Let $Y\subset X$ be a quasi-compact open subscheme in a semi-separated scheme
such that the identity open embedding $j\colon Y\longrightarrow X$ is an
affine morphism. Then
(a) for any very flat quasi-coherent sheaf ${\mathcal{F}}$ and contraaadjusted
quasi-coherent sheaf ${\mathcal{P}}$ on $X$, there is a natural isomorphism of
${\mathcal{O}}(Y)$-modules
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[Y]\simeq\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}})$;
(b) for any flat quasi-coherent sheaf ${\mathcal{F}}$ and cotorsion quasi-
coherent sheaf ${\mathcal{P}}$ on $X$, there is a natural isomorphism of
${\mathcal{O}}(Y)$-modules
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[Y]\simeq\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}})$;
(c) for any quasi-coherent sheaf ${\mathcal{M}}$ and injective quasi-coherent
sheaf ${\mathcal{J}}$ on $X$, there is a natural isomorphism of
${\mathcal{O}}(Y)$-modules
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}})[Y]\simeq\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{M}},{\mathcal{J}})$.
Now let $Y\subset X$ be any quasi-compact open subscheme in a quasi-separated
scheme; let $j\colon Y\longrightarrow X$ denote the identity open embedding.
Then
(d) for any flasque quasi-coherent sheaf ${\mathcal{M}}$ and injective quasi-
coherent sheaf ${\mathcal{J}}$ on $X$, there is a natural isomorphism of
${\mathcal{O}}(Y)$-modules
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}})[Y]\simeq\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{M}},{\mathcal{J}})$.
###### Proof.
Let $Y=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering of a
quasi-separated scheme and ${\mathcal{G}}$ be a quasi-coherent sheaf on $Y$.
Denote by $k_{\alpha_{1},\dotsc,\alpha_{i}}$ the open embeddings
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{i}}\longrightarrow Y$. Then there is
a finite Čech exact sequence
(12) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathcal{G}}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}k_{\alpha}{}_{*}k_{\alpha}^{*}{\mathcal{G}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}k_{\alpha,\beta}{}_{*}k_{\alpha,\beta}^{*}{\mathcal{G}}\\\
\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk_{1,\dotsc,N}{}_{*}k_{1,\dotsc,N}^{*}{\mathcal{G}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
of quasi-coherent sheaves on $Y$ (to check the exactness, it suffices to
consider the restrictions of this sequence to the open subschemes
$U_{\alpha}$, over each of which it is contractible). Set
${\mathcal{G}}=j^{*}{\mathcal{F}}$ or $j^{*}{\mathcal{M}}$.
When the embedding morphism $j\colon Y\longrightarrow X$ is affine, the
functor $j_{*}$ preserves exactness of sequences of quasi-coherent sheaves.
When the sheaf ${\mathcal{M}}$ is flasque, so are the sheaves constituting the
sequence (12), which therefore remains exact after taking the direct images
with respect to any morphism. In both cases we obtain a finite exact sequence
of quasi-coherent sheaves on $X$
$\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{*}j^{*}{\mathcal{F}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}h_{\alpha}{}_{*}h_{\alpha}^{*}{\mathcal{F}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}h_{\alpha,\beta}{}_{*}h_{\alpha,\beta}^{*}{\mathcal{F}}\\\
\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muh_{1,\dotsc,N}{}_{*}h_{1,\dotsc,N}^{*}{\mathcal{F}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
or similarly for ${\mathcal{M}}$, where $h_{\alpha_{1},\dots,\alpha_{i}}$
denote the open embedings $U_{\alpha_{1}}\cap\dotsb\cap
U_{\alpha_{i}}\longrightarrow X$.
It is a sequence of very flat quasi-coherent sheaves in the case (a) and a
sequence of flat quasi-coherent sheaves in the case (b). The functor
$\operatorname{Hom}_{X}({-},{\mathcal{P}})$ transforms it into an exact
sequence of ${\mathcal{O}}(Y)$-modules ending in
$\textstyle\bigoplus_{\alpha<\beta}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[U_{\alpha}\cap
U_{\beta}]\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[U_{\alpha}]\\\
\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
and it remains to compare it with the construction (8) of the
${\mathcal{O}}(Y)$-module
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[Y]$ in terms
of the modules
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[U_{\alpha}]$
and
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})[U_{\alpha}\cap
U_{\beta}]$. The proofs of parts (c) and (d) are finished in the similar way.
∎
For any affine morphism $f\colon Y\longrightarrow X$ and any quasi-coherent
sheaves ${\mathcal{M}}$ on $X$ and ${\mathcal{N}}$ on $Y$ there is a natural
isomorphism
(13)
$f_{*}(f^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{N}})\simeq{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}f_{*}{\mathcal{N}}$
of quasi-coherent sheaves on $X$ (“the projection formula”). In particular,
for any quasi-coherent sheaves ${\mathcal{M}}$ and ${\mathcal{K}}$ on $X$
there is a natural isomorphism
(14)
$f_{*}f^{*}({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{K}})\simeq{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}f_{*}f^{*}{\mathcal{K}}$
of quasi-coherent sheaves on $X$. Assuming that the quasi-coherent sheaf
${\mathcal{M}}$ on $X$ is flat, the same isomorphisms hold for any quasi-
compact quasi-separated morphism of schemes $f\colon Y\longrightarrow X$.
For any embedding $j\colon U\longrightarrow X$ of an affine open subscheme
into a semi-separated scheme $X$ and any quasi-coherent sheaves
${\mathcal{K}}$ and ${\mathcal{M}}$ on $X$ there is a natural isomorphism
(15) $j_{*}j^{*}({\mathcal{K}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}})\simeq
j_{*}j^{*}{\mathcal{K}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{M}}(U)$
of quasi-coherent sheaves on $X$. Assuming that the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{M}}(U)$ is flat, the same isomorphism
holds a quasi-separated scheme $X$.
Recall that the _quasi-coherent internal Hom_ sheaf
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{M}},{\mathcal{P}})$
for quasi-coherent sheaves ${\mathcal{M}}$ and ${\mathcal{P}}$ on a scheme $X$
is defined as the quasi-coherent sheaf for which there is a natural
isomorphism of abelian groups
$\operatorname{Hom}_{X}({\mathcal{K}},\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{M}},{\mathcal{P}}))\simeq\operatorname{Hom}_{X}({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{K}},\>{\mathcal{P}})$
for any quasi-coherent sheaf ${\mathcal{K}}$ on $X$. The sheaf
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{M}},{\mathcal{P}})$
can be constructed by applying the coherator functor [61, Sections B.12–B.14]
to the sheaf of ${\mathcal{O}}_{X}$-modules
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({\mathcal{M}},{\mathcal{P}})$.
###### Lemma 2.5.3.
Let $X$ be a scheme. Then
(a) for any very flat quasi-coherent sheaf ${\mathcal{F}}$ and contraadjusted
quasi-coherent sheaf ${\mathcal{P}}$ on $X$, the quasi-coherent sheaf
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{F}},{\mathcal{P}})$
on $X$ is contraadjusted;
(b) for any flat quasi-coherent sheaf ${\mathcal{F}}$ and cotorsion quasi-
coherent sheaf ${\mathcal{P}}$ on $X$, the quasi-coherent sheaf
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{F}},{\mathcal{P}})$
on $X$ is cotorsion;
(c) for any quasi-coherent sheaf ${\mathcal{M}}$ and any injective quasi-
coherent sheaf ${\mathcal{J}}$ on $X$, the quasi-coherent sheaf
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{M}},{\mathcal{J}})$
on $X$ is cotorsion;
(d) for any flat quasi-coherent sheaf ${\mathcal{F}}$ and any injective quasi-
coherent sheaf ${\mathcal{J}}$ on $X$, the quasi-coherent sheaf
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{F}},{\mathcal{J}})$
is injective.
###### Proof.
We will prove part (a); the proofs of the other parts are similar. Let
${\mathcal{G}}$ be a very flat quasi-coherent sheaf on $X$. We will show that
the functor
$\operatorname{Hom}_{X}({-},\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{F}},{\mathcal{P}}))$
transforms any short exact sequence of quasi-coherent sheaves
$0\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{L}}\longrightarrow{\mathcal{G}}\longrightarrow
0$ into a short exact sequence of abelian groups. Indeed, the sequence of
quasi-coherent sheaves
$0\longrightarrow{\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{K}}\longrightarrow{\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{L}}\longrightarrow{\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}}\longrightarrow
0$ is exact, because ${\mathcal{F}}$ is flat (or because ${\mathcal{G}}$ is
flat). Since ${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}}$ is very
flat by Lemma 1.2.1(a) and ${\mathcal{P}}$ is contraadjusted, the functor
$\operatorname{Hom}_{X}({-},{\mathcal{P}})$ transforms the latter sequence of
sheaves into a short exact sequence of abelian groups. ∎
It follows from the isomorphism (14) that for any very flat quasi-coherent
sheaves ${\mathcal{F}}$ and ${\mathcal{G}}$ on a semi-separated scheme $X$ and
any contraadjusted quasi-coherent sheaf ${\mathcal{P}}$ on $X$ there is a
natural isomorphism of contraherent cosheaves
(16)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}},\>{\mathcal{P}})\simeq\operatorname{\mathfrak{Hom}}_{X}({\mathcal{G}},\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{F}},{\mathcal{P}})).$
Similarly, for any flat quasi-coherent sheaves ${\mathcal{F}}$ and
${\mathcal{G}}$ and a cotorsion quasi-coherent sheaf ${\mathcal{P}}$ on $X$
there is a natural isomorphism (16) of locally cotorsion contraherent
cosheaves. Finally, for any flat quasi-coherent sheaf ${\mathcal{F}}$, quasi-
coherent sheaf ${\mathcal{M}}$, and injective quasi-coherent sheaf
${\mathcal{J}}$ on $X$ there are natural isomorphisms of locally cotorsion
contraherent cosheaves
(17)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}},\>{\mathcal{J}})\simeq\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{F}},{\mathcal{J}}))\simeq\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{M}},{\mathcal{J}})).$
The left isomorphism holds over any quasi-separated scheme $X$.
It follows from the isomorphism (15) that for any very flat quasi-coherent
sheaves ${\mathcal{F}}$ and ${\mathcal{G}}$ and a contraadjusted quasi-
coherent sheaf ${\mathcal{P}}$ on a semi-separated scheme $X$ there is a
natural isomorphism of contraherent cosheaves
(18)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{G}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}},\>{\mathcal{P}})\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{G}},{\mathcal{P}})).$
Similarly, for any flat quasi-coherent sheaves ${\mathcal{F}}$ and
${\mathcal{G}}$ and a cotorsion quasi-coherent sheaf ${\mathcal{P}}$ on $X$
there is a natural isomorphism (16) of locally cotorsion contraherent
cosheaves. Finally, for any flat quasi-coherent sheaf ${\mathcal{F}}$, quasi-
coherent sheaf ${\mathcal{K}}$, and injective quasi-coherent sheaf
${\mathcal{J}}$ on $X$ there are natural isomorphisms of locally cotorsion
contraherent cosheaves
(19)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{K}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}},\>{\mathcal{J}})\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{K}},{\mathcal{J}}))\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{K}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{J}})).$
The left isomorphism holds over any quasi-separated scheme $X$.
###### Remark 2.5.4.
One can slightly generalize the constructions and results of this section by
weakening the definitions of contraadjusted and cotorsion quasi-coherent
sheaves. Namely, a quasi-coherent sheaf ${\mathcal{P}}$ on $X$ may be called
weakly cotorsion if the functor $\operatorname{Hom}_{X}({-},{\mathcal{P}})$
transforms short exact sequences of flat quasi-coherent sheaves on $X$ into
short exact sequences of abelian groups. The weakly contraadjusted quasi-
coherent sheaves are defined similarly (with the flat quasi-coherent sheaves
replaced by very flat ones). Appropriate versions of Lemmas 2.5.1 and 2.5.3
can be proven in this setting, and the contraherent cosheaves
$\operatorname{\mathfrak{Hom}}$ can be defined.
On a quasi-compact semi-separated scheme $X$ (or more generally, on a scheme
where there are enough flat or very flat quasi-coherent sheaves), there is no
difference between the weak and ordinary cotorsion/contraadjusted quasi-
coherent sheaves (see Section 4.1 below; cf. [50, Sections 5.1.4 and 5.3]).
One reason why we chose to use the stronger versions of these conditions here
rather that the weaker ones is that it is not immediately clear whether the
classes of weakly cotorsion/contraadjusted quasi-coherent sheaves are closed
under extensions, or how the exact categories of such sheaves should be
defined.
### 2.6. Contratensor product of sheaves and cosheaves
Let $X$ be a quasi-separated scheme and ${\mathbf{B}}$ be an (initially fixed)
base of open subsets of $X$ consisting of some affine open subschemes. Let
${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and ${\mathfrak{P}}$ be a
cosheaf of ${\mathcal{O}}_{X}$-modules.
The _contratensor product_ ${\mathcal{M}}\odot_{X}{\mathfrak{P}}$ (computed on
the base ${\mathbf{B}}$) is a quasi-coherent sheaf on $X$ defined as the
(nonfiltered) inductive limit of the following diagram of quasi-coherent
sheaves on $X$ indexed by affine open subschemes $U\in{\mathbf{B}}$ (cf. [25,
Section 0.3.2] and Section 2.1 above).
To any affine open subscheme $U\in{\mathbf{B}}$ with the identity open
embedding $j\colon U\longrightarrow X$ we assign the quasi-coherent sheaf
$j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$ on
$X$. For any pair of embedded affine open subschemes $V\subset U$, $V$,
$U\in{\mathbf{B}}$ with the embedding maps $j\colon U\longrightarrow X$ and
$k\colon V\longrightarrow X$ there is the morphism of quasi-coherent sheaves
$k_{*}k^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(V)}{\mathfrak{P}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$
defined in terms of the natural isomorphism $k_{*}k^{*}{\mathcal{M}}\simeq
j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{X}(V)$ of
quasi-coherent sheaves on $X$ and the ${\mathcal{O}}_{X}(U)$-module morphism
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]$.
Let ${\mathcal{M}}$ and ${\mathcal{J}}$ be quasi-coherent sheaves on a quasi-
separated scheme $X$ for which the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}})$ is defined
(i. e., one of the sufficient conditions given in Section 2.5 for the
construction of $\operatorname{\mathfrak{Hom}}$ to make sense is satisfied).
Then for any cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ there is
a natural isomorphism of abelian groups
(20)
$\operatorname{Hom}_{X}({\mathcal{M}}\odot_{X}{\mathfrak{P}},\>{\mathcal{J}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}})).$
In other words, the functor ${\mathcal{M}}\odot_{X}{-}$ is left adjoint to the
functor $\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{-})$ “wherever the
latter is defined”.
Indeed, both groups of homomorphisms consist of all the compatible collections
of morphisms of quasi-coherent sheaves
$j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathcal{J}}$
on $X$, or equivalently, all the compatible collections of morphisms of
${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{P}}[U]\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{M}},{\mathcal{J}})$
defined for all the identity embeddings $j\colon U\longrightarrow X$ of affine
open subschemes $U\in{\mathbf{B}}$. The compatibility is with respect to the
identity embeddings of affine open subschemes $h\colon V\longrightarrow U$,
$V$, $U\in{\mathbf{B}}$, into one another.
In particular, the adjunction isomorphism (20) holds for any quasi-coherent
sheaf ${\mathcal{M}}$, cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}$, and injective quasi-coherent sheaf ${\mathcal{J}}$. Since
there are enough injective quasi-coherent sheaves, it follows that the quasi-
coherent sheaf of contratensor product ${\mathcal{M}}\odot_{X}{\mathfrak{P}}$
does not depend on the base of open affines ${\mathbf{B}}$ that was used to
construct it.
More generally, let ${\mathbf{D}}$ be a partially ordered set endowed with an
order-preserving map into the set of all affine open subschemes of $X$, which
we will denote by $a\longmapsto U(a)$, i. e., one has $U(b)\subset U(a)$
whenever $b\le a\in{\mathbf{D}}$. Suppose that
$X=\bigcup_{a\in{\mathbf{D}}}U(a)$ and for any $a$, $b\in{\mathbf{D}}$ the
intersection $U(a)\cap U(b)\subset X$ is equal to the union $\bigcup_{c\le
a,b}U(c)$. Then the inductive limit of the diagram
$j_{a}{}_{*}j_{a}^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U_{a})}{\mathfrak{P}}[U_{a}]$
indexed by $a\in{\mathbf{D}}$, where $j_{a}$ denotes the open embedding
$U_{a}\longrightarrow X$, is naturally isomorphic to the contratensor product
${\mathcal{M}}\odot_{X}{\mathfrak{P}}$.
Indeed, given a cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ and a
contraherent cosheaf ${\mathfrak{Q}}$ on $X$, an arbitrary collection of
morphisms of ${\mathcal{O}}_{X}(U_{a})$-modules
${\mathfrak{P}}[U_{a}]\longrightarrow{\mathfrak{Q}}[U_{a}]$ compatible with
the corestriction maps for $b\le a$ uniquely determines a morphism of
cosheaves of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$ (see Lemma 2.2.4). In
particular, this applies to the case of a contraherent cosheaf
${\mathfrak{Q}}=\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}})$.
The isomorphism
$j_{*}j^{*}({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{K}})\simeq{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}j_{*}j^{*}{\mathcal{K}}$
for an embedding of affine open subscheme $j\colon U\longrightarrow X$ and
quasi-coherent sheaves ${\mathcal{M}}$ and ${\mathcal{K}}$ on $X$ (see (14))
allows to construct a natural isomorphism of quasi-coherent sheaves
(21)
${\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}({\mathcal{K}}\odot_{X}{\mathfrak{P}})\simeq({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{K}})\odot_{X}{\mathfrak{P}}$
for any quasi-coherent sheaves ${\mathcal{M}}$ and ${\mathcal{K}}$ and any
cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on a semi-separated
scheme $X$. The same isomorphism holds over a quasi-separated scheme $X$,
assuming that the quasi-coherent sheaf ${\mathcal{M}}$ is flat.
## 3\. Locally Contraherent Cosheaves
### 3.1. Exact category of locally contraherent cosheaves
A cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on a scheme $X$ is
called _locally contraherent_ if every point $x\in X$ has an open neighborhood
$x\in W\subset X$ such that the cosheaf of ${\mathcal{O}}_{W}$-modules
${\mathfrak{P}}|_{W}$ is contraherent.
Given an open covering ${\mathbf{W}}=\\{W\\}$ a scheme $X$, a cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ is called _${\mathbf{W}}$
-locally contraherent_ if for any open subscheme $W\subset X$ belonging to
${\mathbf{W}}$ the cosheaf of ${\mathcal{O}}_{W}$-modules
${\mathfrak{P}}|_{W}$ is contraherent on $W$. Obviously, a cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ is locally contraherent if and
only if there exists an open covering ${\mathbf{W}}$ of the scheme $X$ such
that ${\mathfrak{P}}$ is ${\mathbf{W}}$-locally contraherent.
Let us call an open subscheme of a scheme $X$ _subordinate_ to an open
covering ${\mathbf{W}}$ if it is contained in one of the open subsets of $X$
belonging to ${\mathbf{W}}$. Notice that, by the definition of a contraherent
cosheaf, the property of a cosheaf of ${\mathcal{O}}_{X}$-modules to be
${\mathbf{W}}$-locally contraherent only depends on the collection of all
affine open subschemes $U\subset X$ subordinate to ${\mathbf{W}}$.
###### Theorem 3.1.1.
Let ${\mathbf{W}}$ be an open covering of a scheme $X$. Then the restriction
of cosheaves of ${\mathcal{O}}_{X}$-modules to the base of open subsets of $X$
consisting of all the affine open subschemes subordinate to ${\mathbf{W}}$
induces an equivalence between the category of ${\mathbf{W}}$-locally
contraherent cosheaves on $X$ and the category of covariant functors with
${\mathcal{O}}_{X}$-module structures on the category of affine open
subschemes of $X$ subordinate to ${\mathbf{W}}$, satisfying the
contraadjustness and contraherence conditions (i-ii) of Section 2.2 for all
affine open subschemes $V\subset U\subset X$ subordinate to ${\mathbf{W}}$.
###### Proof.
The same as in Theorem 2.2.1, except that the base of affine open subschemes
of $X$ subordinate to ${\mathbf{W}}$ is considered throughout. ∎
Let $X$ be a scheme and ${\mathbf{W}}$ be its open covering. By Theorem 2.1.2,
the category of cosheaves of ${\mathcal{O}}_{X}$-modules is a full subcategory
of the category of covariant functors with ${\mathcal{O}}_{X}$-module
structures on the category of affine open subschemes of $X$ subordinate to
${\mathbf{W}}$. The category of such functors with ${\mathcal{O}}_{X}$-module
structures is clearly abelian, has exact functors of infinite direct sum and
infinite product, and the functors of cosections over a particular affine open
subscheme subordinate to ${\mathbf{W}}$ are exact on it and preserve infinite
direct sums and products.
The full subcategory of cosheaves of ${\mathcal{O}}_{X}$-modules in this
abelian category is closed under extensions, cokernels, and infinite direct
sums. For the quasi-compactness reasons explained in Remark 2.1.4, it is also
closed under infinite products.
Therefore, the category of cosheaves of ${\mathcal{O}}_{X}$-modules acquires
the induced exact category structure with exact functors of infinite direct
sum and product, and exact functors of cosections on affine open subschemes
subordinate to ${\mathbf{W}}$. Let us denote the category of cosheaves of
${\mathcal{O}}_{X}$-modules endowed with this exact category structure (which,
of course, depends on the choice of a covering ${\mathbf{W}}$) by
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$. Along the
way we have proven that infinite products exist in the additive category of
cosheaves of ${\mathcal{O}}_{X}$-modules on a scheme $X$, and the functors of
cosections over quasi-compact quasi-separated open subschemes of $X$ preserve
them.
The full subcategory of ${\mathbf{W}}$-locally contraherent cosheaves is
closed under extensions, cokernels of admissible monomorphisms, and infinite
products in the exact category
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$. Thus the
category of ${\mathbf{W}}$-locally contrantraherent cosheaves has the induced
exact category structure with exact functors of infinite product, and exact
functors of cosections over affine open subschemes subordinate to
${\mathbf{W}}$. We denote this exact category of ${\mathbf{W}}$-locally
contraherent cosheaves on a scheme $X$ by
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
More explicitly, a short sequence of ${\mathbf{W}}$-locally contraherent
cosheaves
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ is exact in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ if the
sequence of cosection modules
$0\longrightarrow{\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{R}}[U]\longrightarrow
0$ is exact for every affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. Passing to the inductive limit with respect to refinements of
the coverings ${\mathbf{W}}$, we obtain the exact category structure on the
category of locally contraherent cosheaves $X{\operatorname{\mathsf{--lcth}}}$
on the scheme $X$.
A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ is said
to be _locally cotorsion_ if for any affine open subscheme $U\subset X$
subordinate to ${\mathbf{W}}$ the ${\mathcal{O}}_{X}(U)$-module
${\mathfrak{P}}[U]$ is cotorsion. By Lemma 1.3.6(a), this definition can be
equivalently rephrased by saying that a locally contraherent cosheaf
${\mathfrak{P}}$ on $X$ is locally cotorsion if and only if for any affine
open subscheme $U\subset X$ such that the cosheaf ${\mathfrak{P}}|_{U}$ is
contraherent on the scheme $U$ the ${\mathcal{O}}(U)$-module
${\mathfrak{P}}[U]$ is cotorsion.
A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{J}}$ on $X$ is
called _locally injective_ if for any affine open subscheme $U\subset X$
subordinate to ${\mathbf{W}}$ the ${\mathcal{O}}_{X}(U)$-module
${\mathfrak{J}}[U]$ is injective. By Lemma 1.3.6(b), a locally contraherent
cosheaf ${\mathfrak{J}}$ on $X$ is locally injective if and only if for any
affine open subscheme $U\subset X$ such that the cosheaf ${\mathfrak{J}}|_{U}$
is contraherent on the scheme $U$ the ${\mathcal{O}}(U)$-module
${\mathfrak{J}}[U]$ is injective.
One defines the exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ of locally
cotorsion and locally injective ${\mathbf{W}}$-locally contraherent cosheaves
on $X$ in the same way as above. These are full subcategories closed under
extensions, infinite products, and cokernels of admissible monomorphisms in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, with the induced exact
category structures. Passing to the inductive limit with respect to
refinements, we obtain the exact categories
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ of locally cotorsion and
locally injective locally contraherent cosheaves on $X$.
### 3.2. Contraherent and locally contraherent cosheaves
By Lemma 1.4.1(a), a short sequence of ${\mathbf{W}}$-locally contraherent
cosheaves on $X$ is exact in $X{\operatorname{\mathsf{--lcth}}}$ (i. e., after
some refinement of the covering) if and only if it is exact in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. By Lemma 1.4.1(b), a
morphism of ${\mathbf{W}}$-locally contraherent cosheaves is an admissible
epimorphism in $X{\operatorname{\mathsf{--lcth}}}$ if and only if it is an
admissible epimorphism in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
Analogously, by Lemma 1.4.2(a), a short sequence of locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves on $X$ is exact in
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ if and only if it is exact
in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. By Lemma
1.4.2(b), a morphism of locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaves on $X$ is an admissible epimorphism in
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ if and only if it is an
admissible epimorphism in
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\mathbf{W}}$. The similar
assertions hold for locally injective locally contraherent cosheaves, and they
are provable in the same way.
On the other hand, a morphism in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$,
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$, or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is an
admissible monomorphism if and only if it acts injectively on the modules of
cosections over all the affine open subschemes $U\subset X$ subordinate to
${\mathbf{W}}$. The following counterexample shows that this condition _does_
change when the covering ${\mathbf{W}}$ is refined.
In other words, the full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$ is closed under the passage to the kernels
of admissible epimorphisms, but not to the cokernels of admissible
monomorphisms in $X{\operatorname{\mathsf{--lcth}}}$. Once we show that, it
will also follow that there _do_ exist locally contraherent cosheaves that are
not contraherent. The locally cotorsion and locally injective contraherent
cosheaves have all the same problems.
###### Example 3.2.1.
Let $R$ be a commutative ring and $f$, $g\in R$ be two elements generating the
unit ideal. Let $M$ be an $R$-module containing no $f$-divisible or
$g$-divisible elements, i. e.,
$\operatorname{Hom}_{R}(R[f^{-1}],M)=0=\operatorname{Hom}_{R}(R[g^{-1}],M)$.
Let $M\longrightarrow P$ be an embedding of $M$ into a contraadjusted
$R$-module $P$, and let $Q$ be the cokernel of this embedding. Then $Q$ is
also a contraadjusted $R$-module. One can take $R$ to be a Dedekind domain, so
that it has homological dimension $1$; then whenever $P$ is a cotorsion or
injective $R$-module, $Q$ has the same property.
Consider the morphism of contraherent cosheaves
$\widecheck{P}\longrightarrow\widecheck{Q}$ on $\operatorname{Spec}R$ related
to the surjective morphism of contraadjusted (cotorsion, or injective)
$R$-modules $P\longrightarrow Q$. In restriction to the covering of
$\operatorname{Spec}R$ by the two principal affine open subsets
$\operatorname{Spec}R[f^{-1}]$ and $\operatorname{Spec}R[g^{-1}]$, we obtain
two morphisms of contraherent cosheaves related to the two morphisms of
contraadjusted modules
$\operatorname{Hom}_{R}(R[f^{-1}],P)\longrightarrow\operatorname{Hom}_{R}(R[f^{-1}],Q)$
and
$\operatorname{Hom}_{R}(R[g^{-1}],P)\longrightarrow\operatorname{Hom}_{R}(R[g^{-1}],Q)$
over the rings $\operatorname{Spec}R[f^{-1}]$ and
$\operatorname{Spec}R[g^{-1}]$.
Due to the condition imposed on $M$, the latter two morphisms of
contraadjusted modules are injective. On the other hand, the morphism of
contraadjusted $R$-modules $P\longrightarrow Q$ is not. It follows that the
cokernel ${\mathfrak{R}}$ of the morphism of contraherent cosheaves
$\widecheck{P}\longrightarrow\widecheck{Q}$ taken in the category of all
cosheaves of ${\mathcal{O}}_{\operatorname{Spec}R}$-modules (or equivalently,
in the category of copresheaves of
${\mathcal{O}}_{\operatorname{Spec}R}$-modules) is contraherent in restriction
to $\operatorname{Spec}R[f^{-1}]$ and $\operatorname{Spec}R[g^{-1}]$, but not
over $\operatorname{Spec}R$. In fact, one has
${\mathfrak{R}}[\operatorname{Spec}R]=0$ (since the morphism $P\longrightarrow
Q$ is surjective).
Let us point out that for any cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}$ on a scheme $X$ such that the ${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{P}}[U]$ are contraadjusted for all affine open subschemes
$U\subset X$ subordinate to a particular open covering ${\mathbf{W}}$, the
${\mathcal{O}}_{X}(U)$-modules ${\mathfrak{P}}[U]$ are contraadjusted for
_all_ affine open subschemes $U\subset X$. This is so simply because the class
of contraadjusted modules is closed under finite direct sums, restrictions of
scalars, and cokernels. So the contraadjustness condition (ii) of Section 2.2
is, in fact, local; it is the contraherence condition (i) that isn’t.
In the rest of the section we will explain how to distinguish the contraherent
cosheaves among all the locally contraherent ones. Let $X$ be a semi-separated
scheme, ${\mathbf{W}}$ be its open covering, and $\\{U_{\alpha}\\}$ be an
affine open covering subordinate to ${\mathbf{W}}$ (i. e., consisting of
affine open subschemes subordinate to ${\mathbf{W}}$).
Let ${\mathfrak{P}}$ be a ${\mathbf{W}}$-locally contraherent cosheaf on $X$.
Consider the homological Čech complex of abelian groups (or
${\mathcal{O}}(X)$-modules)
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
of the form
(22) $\textstyle\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta<\gamma}{\mathfrak{P}}[U_{\alpha}\cap
U_{\beta}\cap U_{\gamma}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}{\mathfrak{P}}[U_{\alpha}\cap U_{\beta}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}{\mathfrak{P}}[U_{\alpha}].$
Here (as in the sequel) our notation presumes the indices $\alpha$ to be
linearly ordered. More generally, the complex (22) can be considered for any
open covering $U_{\alpha}$ of a topological space $X$ and any cosheaf of
abelian groups ${\mathfrak{P}}$ on $X$. Let
$\Delta(X,{\mathfrak{P}})={\mathfrak{P}}[X]$ denote the functor of global
cosections of (locally contraherent) cosheaves on $X$; then, by the
definition, we have $\Delta(X,{\mathfrak{P}})\simeq
H_{0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$.
###### Lemma 3.2.2.
Let $U$ be an affine scheme with an open covering ${\mathbf{W}}$ and a finite
affine open covering $\\{U_{\alpha}\\}$ subordinate to ${\mathbf{W}}$. Then a
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $U$ is
contraherent if and only if
$H_{>0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})=0$.
###### Proof.
The “only if” part is provided by Lemma 1.2.6(b). Let us prove “if”. If the
Čech complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
has no higher homology, then it is a finite left resolution of the
${\mathcal{O}}(U)$-module ${\mathfrak{P}}[U]$ by contraadjusted
${\mathcal{O}}(U)$-modules. As we have explained above, the
${\mathcal{O}}(U)$-module ${\mathfrak{P}}[U]$ is contraadjusted, too.
For any affine open subscheme $V\subset U$, consider the Čech complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},\>{\mathfrak{P}}|_{V})$ related to the restrictions of our
cosheaf ${\mathfrak{P}}$ and our covering $U_{\alpha}$ to the open subscheme
$V$. The complex $C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},\>{\mathfrak{P}}|_{V})$ can be obtained from the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
by applying the functor
$\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{O}}(V),{-})$. We have
$H_{0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})\simeq{\mathfrak{P}}[U]\quad\text{and}\quad
H_{0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},\>{\mathfrak{P}}|_{V})\simeq{\mathfrak{P}}[V].$
Since the functor
$\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{O}}(V),{-})$ preserves
exactness of short sequences of contraadjusted ${\mathcal{O}}(U)$-modules, we
conclude that
${\mathfrak{P}}[V]\simeq\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{O}}(V),{\mathfrak{P}}[U])$.
Both the contraadjustness and contraherence conditions have been now verified.
∎
###### Corollary 3.2.3.
If a ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on an affine
scheme $U$ is an extension of two contraherent cosheaves ${\mathfrak{P}}$ and
${\mathfrak{R}}$ in the exact category
$U{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (or
${\mathcal{O}}_{U}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$), then
${\mathfrak{Q}}$ is also a contraherent cosheaf on $U$.
###### Proof.
Pick a finite affine open covering $\\{U_{\alpha}\\}$ of the affine scheme $U$
subordinate to the covering ${\mathbf{W}}$. Then the complex of abelian groups
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{Q}})$
is an extension of the complexes of abelian groups
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
and
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{R}})$.
Hence whenever the latter two complexes have no higher homology, neither does
the former one. ∎
###### Corollary 3.2.4.
For any scheme $X$ and any its open covering ${\mathbf{W}}$, the full exact
subcategory of ${\mathbf{W}}$-locally contraherent cosheaves on $X$ is closed
under extensions in the exact category of locally contraherent cosheaves on
$X$. In particular, the full exact subcategory of contraherent cosheaves on
$X$ is closed under extensions in the exact category of locally contraherent
(or ${\mathbf{W}}$-locally contraherent) cosheaves on $X$.
###### Proof.
Follows easily from Corollary 3.2.3. ∎
### 3.3. Direct and inverse images of locally contraherent cosheaves
Let ${\mathbf{W}}$ be an open covering of a scheme $X$ and ${\mathbf{T}}$ be
an open covering of a scheme $Y$. A morphism of schemes $f\colon
Y\longrightarrow X$ is called _$({\mathbf{W}},{\mathbf{T}})$ -affine_ if for
any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ the open
subscheme $f^{-1}(U)\subset Y$ is affine and subordinate to ${\mathbf{T}}$.
Any $({\mathbf{W}},{\mathbf{T}})$-affine morphism is affine.
Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
morphism of schemes and ${\mathfrak{Q}}$ be a ${\mathbf{T}}$-locally
contraherent cosheaf on $Y$. Then the cosheaf of ${\mathcal{O}}_{X}$-modules
$f_{!}{\mathfrak{Q}}$ on $X$ is ${\mathbf{W}}$-locally contraherent. The proof
of this assertion is similar to that of its global version in Section 2.3. We
have constructed an exact functor of direct image $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ between the exact categories
of ${\mathbf{T}}$-locally contraherent cosheaves on $Y$ and
${\mathbf{W}}$-locally contraherent cosheaves on $X$.
A morphism of schemes $f\colon Y\longrightarrow X$ is called
_$({\mathbf{W}},{\mathbf{T}})$ -coaffine_ if for any affine open subscheme
$V\subset Y$ subordinate to ${\mathbf{T}}$ there exists an affine open
subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ such that $f(V)\subset
U$. Notice that for any fixed open covering ${\mathbf{W}}$ of a semi-separated
scheme $X$ and any morphism of schemes $f\colon Y\longrightarrow X$ the
covering ${\mathbf{T}}$ of the scheme $Y$ consisting of all the full preimages
$f^{-1}(U)$ of affine open subschemes $U\subset X$ has the property that the
morphism $f\colon Y\longrightarrow X$ is
$({\mathbf{W}},{\mathbf{T}})$-coaffine.
If the morphism $f$ is affine, it is also $({\mathbf{W}},{\mathbf{T}})$-affine
with respect to the covering ${\mathbf{T}}$ constructed in this way. A
morphism $f\colon Y\longrightarrow X$ is simultaneously
$({\mathbf{W}},{\mathbf{T}})$-affine and
$({\mathbf{W}},{\mathbf{T}})$-coaffine if and only if it is affine and the set
of all affine open subschemes $V\subset Y$ subordinate to ${\mathbf{T}}$
consists precisely of all affine open subschemes $V$ for which there exists an
affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ such that
$f(V)\subset U$.
Let $f\colon Y\longrightarrow X$ be a very flat
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism, and let ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Define a
${\mathbf{T}}$-locally contraherent cosheaf $f^{!}{\mathfrak{P}}$ on $Y$ in
the following way. Let $V\subset Y$ be an affine open subscheme subordinate to
${\mathbf{T}}$. For any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$ and such that $f(V)\subset U$, we set
$(f^{!}{\mathfrak{P}})[V]_{U}=\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(V),{\mathfrak{P}}[U])$.
The ${\mathcal{O}}_{Y}(V)$-module $(f^{!}{\mathfrak{P}})[V]_{U}$ is
contraadjusted by Lemma 1.2.3(a). The contraherence isomorphism
$(f^{!}{\mathfrak{P}})[V^{\prime}]_{U}\simeq\operatorname{Hom}_{{\mathcal{O}}_{Y}(V)}({\mathcal{O}}_{Y}(V^{\prime}),(f^{!}{\mathfrak{P}})[V]_{U})$
clearly holds for any affine open subscheme $V^{\prime}\subset V$. The
(${\mathbf{W}}$-local) contraherence condition on ${\mathfrak{P}}$ implies a
natural isomorphism of ${\mathcal{O}}_{Y}(V)$-modules
$(f^{!}{\mathfrak{P}})[V]_{U^{\prime}}\simeq(f^{!}{\mathfrak{P}})[V]_{U^{\prime\prime}}$
for any embedded affine open subschemes $U^{\prime}\subset U^{\prime\prime}$
in $X$ subordinate to ${\mathbf{W}}$ and containing $f(V)$ (cf. Section 2.3).
It remains to construct such an isomorphism for any two (not necessarily
embedded) affine open subschemes $U^{\prime}$, $U^{\prime\prime}\subset X$.
The case of a semi-separated scheme $X$ is clear. In the general case, let
$U^{\prime}\cap U^{\prime\prime}=\bigcup_{\alpha}U_{\alpha}$ be an affine open
covering of the intersection. Since $V$ is quasi-compact, the image $f(V)$ is
covered by a finite subset of the affine open schemes $U_{\alpha}$. Let
$V_{\alpha}$ denote the preimages of $U_{\alpha}$ with respect to the morphism
$V\longrightarrow U^{\prime}\cap U^{\prime\prime}$; then
$V=\bigcup_{\alpha}V_{\alpha}$ is an affine open covering of the affine scheme
$V$.
The restrictions $f_{U^{\prime}}\colon V\longrightarrow U^{\prime}$ and
$f_{U^{\prime\prime}}\colon V\longrightarrow U^{\prime\prime}$ of the morphism
$f$ are very flat morphisms of affine schemes, while the restrictions
${\mathfrak{P}}|_{U^{\prime}}$ and ${\mathfrak{P}}|_{U^{\prime\prime}}$ of the
cosheaf ${\mathfrak{P}}$ are contraherent cosheaves on $U^{\prime}$ and
$U^{\prime\prime}$. Consider the contraherent cosheaves
$f_{U^{\prime}}^{!}{\mathfrak{P}}|_{U^{\prime}}$ and
$f_{U^{\prime\prime}}^{!}{\mathfrak{P}}_{U^{\prime\prime}}$ on $V$ (as defined
in Section 2.3). Their cosection modules
$(f_{U^{\prime}}^{!}{\mathfrak{P}}|_{U^{\prime}})[V_{\alpha}]$ and
$(f_{U^{\prime\prime}}^{!}{\mathfrak{P}}|_{U^{\prime\prime}})[V_{\alpha}]$ are
naturally isomorphic for all $\alpha$, since $f(V_{\alpha})\subset
U_{\alpha}\subset U^{\prime}\cap U^{\prime\prime}$. Similarly, there are
natural isomorphisms
$(f_{U^{\prime}}^{!}{\mathfrak{P}}|_{U^{\prime}})[V_{\alpha}\cap
V_{\beta}]\simeq(f_{U^{\prime\prime}}^{!}{\mathfrak{P}}|_{U^{\prime\prime}})[V_{\alpha}\cap
V_{\beta}]$ forming commutative diagrams with the corestrictions from
$V_{\alpha}\cap V_{\beta}$ to $V_{\alpha}$ and $V_{\beta}$, since
$f(V_{\alpha}\cap V_{\beta})\subset U_{\alpha}\cap U_{\beta}$ and the
intersections $U_{\alpha}\cap U_{\beta}$ are affine schemes.
Now the cosheaf axiom (5) for contraherent cosheaves
$f_{U^{\prime}}^{!}{\mathfrak{P}}|_{U^{\prime}}$ and
$f_{U^{\prime\prime}}^{!}{\mathfrak{P}}|_{U^{\prime\prime}}$ and the covering
$V=\bigcup_{\alpha}V_{\alpha}$ provides the desired isomorphism between the
${\mathcal{O}}_{Y}(V)$-modules
$(f^{!}{\mathfrak{P}})[V]_{U^{\prime}}=(f_{U^{\prime}}^{!}{\mathfrak{P}}|_{U^{\prime}})[V]$
and
$(f^{!}{\mathfrak{P}})[V]_{U^{\prime\prime}}=(f_{U^{\prime\prime}}^{!}{\mathfrak{P}}|_{U^{\prime\prime}})[V]$.
One can easily see that such isomorphisms form a commutative diagram for any
three affine open subschemes $U^{\prime}$, $U^{\prime\prime}$,
$U^{\prime\prime\prime}\subset X$ containing $f(V)$.
The ${\mathbf{T}}$-locally contraherent cosheaf $f^{!}{\mathfrak{P}}$ on $Y$
is constructed. We have obtained an exact functor of inverse image
$f^{!}\colon X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muY{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$.
Let $f\colon Y\longrightarrow X$ be a flat
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism of schemes, and let
${\mathfrak{P}}$ be a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$. Then the same procedure as above defines a locally cotorsion
${\mathbf{T}}$-locally contraherent cosheaf $f^{!}{\mathfrak{P}}$ on $Y$. So
we obtain an exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$. Finally, for
any $({\mathbf{W}},{\mathbf{T}})$-coaffine morphism of schemes $f\colon
Y\longrightarrow X$ and any locally injective ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{J}}$ on $X$ the same rule defines a locally
injective ${\mathbf{T}}$-locally contraherent cosheaf $f^{!}{\mathfrak{J}}$ on
$Y$. We obtain an exact functor of inverse image $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$.
Passing to the inductive limits of exact categories with respects to the
refinements of coverings and taking into account the above remark about
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphisms, we obtain an exact functor
of inverse image $f^{!}\colon X{\operatorname{\mathsf{--lcth}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}$ for any very flat morphism of schemes
$f\colon Y\longrightarrow X$.
For an open embedding $j\colon Y\longrightarrow X$, the direct image $j^{!}$
coincides with the restriction functor
${\mathfrak{P}}\longmapsto{\mathfrak{P}}|_{Y}$ on the locally contraherent
cosheaves ${\mathfrak{P}}$ on $X$. For a flat morphism $f$, we have an exact
functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$, and for an arbitrary
morphism of schemes $f\colon Y\longrightarrow Y$ there is an exact functor of
inverse image of locally injective locally contraherent cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$.
If $f\colon Y\longrightarrow X$ is a $({\mathbf{W}},{\mathbf{T}})$-affine
morphism and ${\mathfrak{Q}}$ is a locally cotorsion ${\mathbf{T}}$-locally
contraherent cosheaf on $Y$, then $f_{!}{\mathfrak{Q}}$ is a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$. So the direct image
functor $f_{!}$ restricts to an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. If $f$ is a
flat $({\mathbf{W}},{\mathbf{T}})$-affine morphism and ${\mathfrak{I}}$ is a
locally injective ${\mathbf{T}}$-locally contraherent cosheaf on $Y$, then
$f_{!}{\mathfrak{I}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$. Hence in this case the direct image also
restricts to an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$.
Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism. Then for any
${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on $Y$ and
locally injective ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{J}}$
on $X$ there is an adjunction isomorphism
(23)
$\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{J}})\simeq\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{J}}),$
where $\operatorname{Hom}^{X}$ and $\operatorname{Hom}^{Y}$ denote the abelian
groups of morphisms in the categories of locally contraherent cosheaves on $X$
and $Y$.
If the morphism $f$ is, in addition, flat, then the isomorphism
(24)
$\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{P}})\simeq\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{P}})$
holds for any ${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ and
locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$
on $X$. In particular, $f_{!}$ and $f^{!}$ form an adjoint pair of functors
between the exact categories
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. When the
morphism $f$ is very flat, the functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$ is right adjoint to the
functor $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
Most generally, there is an adjunction isomorphism
(25)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}(f_{!}{\mathfrak{Q}},{\mathfrak{J}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{Y}}({\mathfrak{Q}},f^{!}{\mathfrak{J}})$
for any morphism of schemes $f$, a cosheaf of ${\mathcal{O}}_{Y}$-modules
${\mathfrak{Q}}$, and a locally injective locally contraherent cosheaf
${\mathfrak{J}}$ on $X$. Similarly, there is an isomorphism
(26)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}(f_{!}{\mathfrak{Q}},{\mathfrak{P}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{Y}}({\mathfrak{Q}},f^{!}{\mathfrak{P}})$
for any flat morphism $f$, a cosheaf of ${\mathcal{O}}_{Y}$-modules
${\mathfrak{Q}}$, and a locally cotorsion locally contraherent cosheaf
${\mathfrak{P}}$ on $X$, and also for a very flat morphism $f$, a cosheaf of
${\mathcal{O}}_{Y}$-modules ${\mathfrak{Q}}$, and a locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$.
In all the mentioned cases, both abelian groups
$\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{P}})$ or
$\operatorname{Hom}^{{\mathcal{O}}_{X}}(f_{!}{\mathfrak{Q}},{\mathfrak{P}})$
(etc.) and $\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{P}})$ or
$\operatorname{Hom}^{{\mathcal{O}}_{Y}}({\mathfrak{Q}},f^{!}{\mathfrak{P}})$
(etc.) are identified with the group of all the compatible collections of
homomorphisms of ${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{Q}}[V]\longrightarrow{\mathfrak{P}}[U]$ defined for all affine
open subschemes $U\subset X$ and $V\subset Y$ subordinate to, respectively,
${\mathbf{W}}$ and ${\mathbf{T}}$, and such that $f(V)\subset U$. In other
words, the functor $f^{!}$ is right adjoint to the functor $f_{!}$ “wherever
the former functor is defined”.
All the functors between exact categories of locally contraherent cosheaves
constructed above are exact and preserve infinite products. The functor of
direct image of cosheaves of ${\mathcal{O}}$-modules $f_{!}$ preserves
infinite products whenever a morphism of schemes $f\colon Y\longrightarrow X$
is quasi-compact and quasi-separated.
Let $f\colon Y\longrightarrow X$ be a morphism of schemes and $j\colon
U\longrightarrow X$ be an open embedding. Set $V=U\times_{X}Y$, and denote by
$j^{\prime}\colon V\longrightarrow Y$ and $f^{\prime}\colon V\longrightarrow
U$ the natural morphisms. Then for any cosheaf of ${\mathcal{O}}_{Y}$-modules
${\mathfrak{Q}}$, there is a natural isomorphism of cosheaves of
${\mathcal{O}}_{U}$-modules $(f_{!}{\mathfrak{Q}})|_{U}\simeq
f^{\prime}_{!}({\mathfrak{Q}}|_{V})$.
In particular, suppose $f$ is a $({\mathbf{W}},{\mathbf{T}})$-affine morphism.
Define the open coverings ${\mathbf{W}}|_{U}$ and ${\mathbf{T}}|_{V}$ as the
collections of all intersections of the open subsets $W\in{\mathbf{W}}$ and
$T\in{\mathbf{T}}$ with $U$ and $V$, respectively. Then $f^{\prime}$ is a
$({\mathbf{W}}|_{U},{\mathbf{T}}|_{V}$)-affine morphism. For any
${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on $Y$ there is a
natural isomorphism of ${\mathbf{W}}|_{U}$-locally contraherent cosheaves
$j^{!}f_{!}{\mathfrak{Q}}\simeq f^{\prime}_{!}j^{\prime}{}^{!}{\mathfrak{Q}}$
on $U$.
More generally, let $f\colon Y\longrightarrow X$ and $g\colon x\longrightarrow
X$ be morphisms of schemes. Set $y=x\times_{X}Y$; let $f^{\prime}\colon
y\longrightarrow x$ and $g^{\prime}\colon y\longrightarrow Y$ be the natural
morphisms. Let ${\mathbf{W}}$, ${\mathbf{T}}$, and ${\mathbf{w}}$ be open
coverings of, respectively, $X$, $Y$, and $x$ such that the morphism $f$ is
$({\mathbf{W}},{\mathbf{T}})$-affine, while the morphism $g$ is
$({\mathbf{W}},{\mathbf{w}})$-coaffine.
Define two coverings ${\mathbf{t}}^{\prime}$ and ${\mathbf{t}}^{\prime\prime}$
of the scheme $y$ by the rules that ${\mathbf{t}}^{\prime}$ consists of all
the full preimages of affine open subschemes in $x$ subordinate to
${\mathbf{w}}$, while ${\mathbf{t}}^{\prime\prime}$ is the collection of all
the full preimages of affine open subschemes in $Y$ subordinate to
${\mathbf{T}}$. One can easily see that the covering ${\mathbf{t}}^{\prime}$
is subordinate to ${\mathbf{t}}^{\prime\prime}$. Let ${\mathbf{t}}$ be any
covering of $y$ such that ${\mathbf{t}}^{\prime}$ is subordinate to
${\mathbf{t}}$ and ${\mathbf{t}}$ is subordinate to
${\mathbf{t}}^{\prime\prime}$. Then the former condition guarantees that the
morphism $f^{\prime}$ is $({\mathbf{w}},{\mathbf{t}})$-affine, while under the
latter condition the morphism $g^{\prime}$ is
$({\mathbf{T}},{\mathbf{t}})$-coaffine.
Assume that the morphisms $g$ and $g^{\prime}$ are very flat. Then for any
${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $Y$ there is a
natural isomorphism $g^{!}f_{!}{\mathfrak{P}}\simeq
f^{\prime}_{!}g^{\prime}{}^{!}{\mathfrak{P}}$ of ${\mathbf{w}}$-locally
contraherent cosheaves on $x$.
Alternatively, assume that the morphism $g$ is flat. Then for any locally
cotorsion ${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $Y$
there is a natural isomorphism $g^{!}f_{!}{\mathfrak{P}}\simeq
f^{\prime}_{!}g^{\prime}{}^{!}{\mathfrak{P}}$ of locally cotorsion
${\mathbf{w}}$-locally contraherent cosheaves on $x$.
As a third alternative, assume that the morphism $f$ is flat (while $g$ may be
arbitrary). Then for any locally injective ${\mathbf{T}}$-locally contraherent
cosheaf ${\mathfrak{J}}$ on $Y$ there is a natural isomorphism
$g^{!}f_{!}{\mathfrak{J}}\simeq f^{\prime}_{!}g^{\prime}{}^{!}{\mathfrak{J}}$
of locally injective ${\mathbf{w}}$-locally contraherent cosheaves on $x$.
All these isomorphisms of locally contraherent cosheaves are constructed using
the natural isomorphism of $r$-modules
$\operatorname{Hom}_{R}(r,P)\simeq\operatorname{Hom}_{S}(S\otimes_{R}r,\>P)$
for any commutative ring homomorphisms $R\longrightarrow S$ and
$R\longrightarrow r$, and any $S$-module $P$.
In other words, the direct images of ${\mathbf{T}}$-locally contraherent
cosheaves under $({\mathbf{W}},{\mathbf{T}})$-affine morphisms commute with
the inverse images in those base change situations when all the functors
involved are defined.
The following particular case will be important for us. Let ${\mathbf{W}}$ be
an open covering of a scheme $X$ and $j\colon Y\longrightarrow X$ be an affine
open embedding subordinate to ${\mathbf{W}}$ (i. e., $Y$ is contained in one
of the open subsets of $X$ belonging to ${\mathbf{W}}$). Then one can endow
the scheme $Y$ with the open covering ${\mathbf{T}}=\\{Y\\}$ consisting of the
only open subset $Y\subset Y$. This makes the embedding $j$ both
$({\mathbf{W}},{\mathbf{T}})$-affine and
$({\mathbf{W}},{\mathbf{T}})$-coaffine. Also, the morphism $j$, being an open
embedding, is very flat.
Therefore, the inverse and direct images $j^{!}$ and $j_{!}$ form a pair of
adjoint exact functors between the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ of ${\mathbf{W}}$-locally
contraherent cosheaves on $X$ and the exact category
$Y{\operatorname{\mathsf{--ctrh}}}$ of contraherent cosheaves on $Y$.
Moreover, the image of the functor $j_{!}$ is contained in the full exact
subcategory of (globally) contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. Both functors preserve the
subcategories of locally cotorsion and locally injective cosheaves.
Now let ${\mathbf{W}}$ be an open covering of a quasi-compact semi-separated
scheme $X$ and let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine
covering of $X$ subordinate to ${\mathbf{W}}$. Denote by
$j_{\alpha_{1},\dotsc,\alpha_{k}}$ the open embeddings
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{k}}\longrightarrow X$. Then for any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ the
cosheaf Čech sequence (cf. (12))
(27) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{1,\dotsc,N}{}_{!}j_{1,\dotsc,N}^{!}{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}j_{\alpha,\beta}{}_{!}j_{\alpha,\beta}^{!}{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}j_{\alpha}{}_{!}j_{\alpha}^{!}{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
is exact in the exact category of ${\mathbf{W}}$-locally contraherent
cosheaves on $X$. Indeed, the corresponding sequence of cosections over every
affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ is exact by
Lemma 1.2.6(b). We have constructed a finite left resolution of a
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ by contraherent
cosheaves. When ${\mathfrak{P}}$ is a locally cotorsion or locally injective
${\mathbf{W}}$-locally contraherent cosheaf, the sequence (27) is exact in the
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$, respectively.
### 3.4. Coflasque contraherent cosheaves
Let $X$ be a topological space and ${\mathfrak{F}}$ be a cosheaf of abelian
groups on $X$. A cosheaf ${\mathfrak{F}}$ is called _coflasque_ if for any
open subsets $V\subset U\subset X$ the corestriction map
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective. Obviously,
the class of coflasque cosheaves of abelian groups is preserved by the
restrictions to open subsets and the direct images with respect to continuous
maps.
###### Lemma 3.4.1.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be an open covering. Then
(a) a cosheaf ${\mathfrak{F}}$ on $X$ is coflasque if and only if its
restriction ${\mathfrak{F}}|_{U_{\alpha}}$ to each open subset $U_{\alpha}$ is
coflasque;
(b) if the cosheaf ${\mathfrak{F}}$ is coflasque, then the Čech complex (22)
has no higher homology groups,
$H_{>0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}})=0$.
###### Proof.
One can either check these assertions directly or deduce them from the similar
results for flasque sheaves of abelian groups using the construction of the
sheaf $U\longmapsto\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}[U],I)$ from
the proof of Theorem 2.1.2. Here $I$ is an injective abelian group; clearly,
the sheaf so obtained is flasque for all $I$ if and only if the original
cosheaf ${\mathfrak{F}}$ was coflasque. The sheaf (dual) versions of
assertions (a-b) are well-known [22, Section II.3.1 and Théorème II.5.2.3(a)].
∎
###### Corollary 3.4.2.
Let $X$ be a scheme with an open covering ${\mathbf{W}}$ and ${\mathfrak{F}}$
be a ${\mathbf{W}}$-locally contraherent cosheaf on $X$. Suppose that the
cosheaf ${\mathfrak{F}}$ is coflasque. Then ${\mathfrak{F}}$ is a (globally)
contraherent cosheaf on $X$.
###### Proof.
Follows from Lemmas 3.2.2 and 3.4.1(b). ∎
Assume that the topological space $X$ has a base of the topology consisting of
quasi-compact open subsets. Then one has
${\mathfrak{F}}[Y]\simeq\varinjlim_{U\subset Y}{\mathfrak{F}}[U]$ for any
cosheaf of abelian groups ${\mathfrak{F}}$ on $X$ and any open subset
$Y\subset X$, where the filtered inductive limit is taken over all the quasi-
compact open subsets $U\subset Y$. It follows easily that a cosheaf
${\mathfrak{F}}$ is coflasque if and only if the corestriction map
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective for any pair
of embedded quasi-compact open subsets $V\subset U\subset X$.
Moreover, if $X$ is a scheme then it follows from Lemma 3.4.1(a) that a
cosheaf of abelian groups ${\mathfrak{F}}$ on $X$ is coflasque if and only if
the corestriction map ${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is
injective for any affine open subscheme $U\subset X$ and quasi-compact open
subscheme $V\subset U$. It follows that an infinite product of a family of
coflasque contraherent cosheaves on $X$ is coflasque. The following
counterexample shows, however, that coflasqueness of contraherent cosheaves on
schemes cannot be checked on the pairs of embedded affine open subschemes.
###### Example 3.4.3.
Let $X$ be a Noetherian scheme. It is well-known that any injective quasi-
coherent sheaf ${\mathcal{J}}$ on $X$ is a flasque sheaf of abelian groups.
Let ${\mathcal{F}}$ be a quasi-coherent sheaf on $X$ and
${\mathcal{F}}\longrightarrow{\mathcal{J}}$ be an injective morphism. Then one
has $({\mathcal{J}}/{\mathcal{F}})(U)\simeq{\mathcal{J}}(U)/{\mathcal{F}}(U)$
for any affine open subscheme $U\subset X$, so the map
$({\mathcal{J}}/{\mathcal{F}})(U)\longrightarrow({\mathcal{J}}/{\mathcal{F}})(V)$
is surjective for any pair of embedded affine open subschemes $V\subset U$. On
the other hand, if the quotient sheaf ${\mathcal{J}}/{\mathcal{F}}$ were
flasque, one would have $H^{i+1}(X,{\mathcal{F}})\simeq
H^{i}(X,{\mathcal{J}}/{\mathcal{F}})=0$ for $i\ge 1$, which is clearly not the
case in general.
Now let $X$ be a scheme over an affine scheme $\operatorname{Spec}R$. Let
${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and $J$ be an injective
$R$-module. Then for any quasi-compact quasi-separated open subscheme
$Y\subset X$ one has
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)[Y]\simeq\operatorname{Hom}_{R}({\mathcal{M}}(U),J)$.
For a Noetherian scheme $X$, it follows that the cosheaf
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)$ is coflasque if and
only if the sheaf ${\mathcal{M}}$ is flasque (cf. Lemma 3.4.6(c) below).
###### Corollary 3.4.4.
Let $X$ be a scheme with an open covering ${\mathbf{W}}$ and
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ be a short exact sequence in
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$ (e. g., a
short exact sequence of ${\mathbf{W}}$-locally contraherent cosheaves on $X$).
Then
(a) the cosheaf ${\mathfrak{Q}}$ is coflasque whenever both the cosheaves
${\mathfrak{P}}$ and ${\mathfrak{R}}$ are;
(b) the cosheaf ${\mathfrak{P}}$ is cosflasque whenever both the cosheaves
${\mathfrak{Q}}$ and ${\mathfrak{R}}$ are;
(c) if the cosheaf ${\mathfrak{R}}$ is coflasque, then the short sequence
$0\longrightarrow{\mathfrak{P}}[Y]\longrightarrow{\mathfrak{Q}}[Y]\longrightarrow{\mathfrak{R}}[Y]\longrightarrow
0$ is exact for any open subscheme $Y\subset X$.
###### Proof.
The assertion actually holds for any short exact sequence in the exact
category of cosheaves of abelian groups on a scheme $X$ with the exact
category structure related to the base of affine open subschemes subordinate
to a covering ${\mathbf{W}}$.
Part (c): let us first consider the case of a semi-separated open subscheme
$Y$. Pick an affine open covering $Y=\bigcup_{\alpha}U_{\alpha}$ subordinate
to ${\mathbf{W}}$. The intersection of any nonempty finite subset of
$U_{\alpha}$ being also an affine open subscheme in $X$, the short sequence of
complexes $0\longrightarrow
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}}|_{Y})\longrightarrow
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{Q}}|_{Y})\longrightarrow
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{R}}|_{Y})\longrightarrow
0$ is exact. Recall that ${\mathfrak{F}}[Y]\simeq
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}|_{Y})$
for any cosheaf of abelian groups ${\mathfrak{F}}$ on $X$. By Lemma 3.4.1, one
has
$H_{>0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{R}}|_{Y})=0$,
and it follows that the short sequence
$0\longrightarrow{\mathfrak{P}}[Y]\longrightarrow{\mathfrak{Q}}[Y]\longrightarrow{\mathfrak{R}}[Y]\longrightarrow
0$ is exact.
Now for any open subscheme $Y\subset X$, pick an open covering
$Y=\bigcup_{\alpha}U_{\alpha}$ of $Y$ by semi-separated open subschemes
$U_{\alpha}$. The intersection of any nonempty finite subset of $U_{\alpha}$
being semi-separated, the short sequence of Čech complexes for the covering
$U_{\alpha}$ and the short exact sequence of cosheaves
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ is exact according to the above, and the same argument concludes the proof.
Parts (a-b): let $U\subset V\subset X$ be any pair of embedded open
subschemes. Assuming that the cosheaf ${\mathfrak{R}}$ is coflasque, according
to part (c) we have short exact sequences of the modules of cosections
$0\longrightarrow{\mathfrak{P}}[V]\longrightarrow{\mathfrak{Q}}[V]\longrightarrow{\mathfrak{R}}[V]\longrightarrow
0$ and
$0\longrightarrow{\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{R}}[U]\longrightarrow
0$. If the morphism from the former sequence to the latter is injective on the
rightmost terms, then it is injective on the middle terms if and only if it is
injective on the leftmost terms. ∎
###### Remark 3.4.5.
Let $U$ be an affine scheme. Then a cosheaf of abelian groups ${\mathfrak{F}}$
on $U$ is coflasque if and only if the following three conditions hold:
1. (i)
for any affine open subscheme $V\subset U$, the corestriction map
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective;
2. (ii)
for any two affine open subschemes $V^{\prime\prime}$,
$V^{\prime\prime}\subset U$, the image of the map
${\mathfrak{F}}[V^{\prime}\cap
V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[U]$ is equal to the
intersection of the images of the maps
${\mathfrak{F}}[V^{\prime}]\longrightarrow{\mathfrak{F}}[U]$ and
${\mathfrak{F}}[V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[U]$;
3. (iii)
for any finite collection of affine open subschemes $V_{\alpha}\subset U$, the
images of the maps
${\mathfrak{F}}[V_{\alpha}]\longrightarrow{\mathfrak{F}}[U]$ generate a
distributive sublattice in the lattice of all subgroups of the abelian group
${\mathfrak{F}}[U]$.
Indeed, the condition (i) being assumed, in view of the exact sequence
${\mathfrak{F}}[V^{\prime}\cap
V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[V^{\prime}]\oplus
gF[V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[V^{\prime}\cup
V^{\prime\prime}]\longrightarrow 0$ the condition (ii) is equivalent to
injectivity of the corestriction map ${\mathfrak{F}}[V^{\prime}\cup
V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[U]$.
Furthermore, set $W=\bigcup_{\alpha}V_{\alpha}$. The condition (iii) for any
proper subcollection of $V_{\alpha}$ and the conditions (i-ii) being assumed,
the condition (iii) for the collection $\\{V_{\alpha}\\}$ becomes equivalent
to vanishing of the higher homology of the Čech complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}}|_{W})$
together with injectivity of the map
${\mathfrak{F}}[W]\longrightarrow{\mathfrak{F}}[U]$. Indeed, the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}}|_{W})\longrightarrow{\mathfrak{F}}[U]$
can be identified with (the abelian group version of) the “cobar complex”
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}({\mathfrak{F}}[U];\>{\mathfrak{F}}[V_{\alpha}])$
from [49, Proposition 7.2(c*) of Chapter 1] (see also [50, Lemma
11.4.3.1(c*)]).
###### Lemma 3.4.6.
(a) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathcal{G}}$
be a flat quasi-coherent sheaf on a locally Noetherian scheme $X$. Then the
quasi-coherent sheaf ${\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}}$
on $X$ is flasque.
(b) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathfrak{J}}$
be a locally injective contraherent cosheaf on a scheme $X$. Then the
contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ on $X$ is
coflasque.
(c) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathcal{J}}$
be an injective quasi-coherent sheaf on an affine Noetherian scheme $U$. Then
the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{U}({\mathcal{M}},{\mathcal{J}})$ on $U$ is
coflasque.
(d) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathfrak{G}}$
be a flat cosheaf of ${\mathcal{O}}_{U}$-modules on an affine scheme $U$. Then
the quasi-coherent sheaf ${\mathcal{M}}\odot_{U}{\mathfrak{G}}$ on $U$ is
flasque.
###### Proof.
The (co)flasqueness of (co)sheaves being a local property, it suffices to
consider the case of an affine scheme $X=U$ in all assertions (a-d). Then (c)
becomes a restatement of (b) and (d) a restatement of (a). It also follows
that one can extend the assertion (c) to locally injective locally
contraherent cosheaves ${\mathfrak{J}}$ (cf. Section 3.6).
To prove part (a) in the affine case, one notices the isomorphism
$({\mathcal{M}}\otimes_{{\mathcal{O}}_{U}}{\mathcal{G}})(V)\simeq{\mathcal{M}}(V)\otimes_{{\mathcal{O}}(U)}{\mathcal{G}}(U)$
holding for any quasi-compact open subscheme $V$ in an affine scheme $U$,
quasi-coherent sheaf ${\mathcal{M}}$, and a flat quasi-coherent sheaf
${\mathcal{G}}$ on $U$. More generally, one has
$({\mathcal{M}}\otimes_{R}G)(Y)\simeq{\mathcal{M}}(Y)\otimes_{R}G$ for any
quasi-coherent sheaf ${\mathcal{M}}$ with a right $R$-module structure on a
quasi-compact quasi-separated scheme $Y$ and a flat left $R$-module $G$. Part
(b) follows from the similar isomorphism
$\operatorname{\mathfrak{Cohom}}_{U}({\mathcal{M}},{\mathfrak{J}})[V]\simeq\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{M}}(V),{\mathfrak{J}}[U])$
holding for any quasi-coherent sheaf ${\mathcal{M}}$ and a (locally) injective
contraherent cosheaf ${\mathfrak{J}}$ on $U$. More generally,
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)[Y]\simeq\operatorname{Hom}_{R}({\mathcal{M}}(Y),J)$
for any quasi-compact quasi-separated scheme $Y$ over $\operatorname{Spec}R$,
quasi-coherent sheaf ${\mathcal{M}}$ on $Y$, and an injective $R$-module $J$.
∎
###### Lemma 3.4.7.
(a) Let $X$ be a Noetherian topological space of finite Krull dimension $\le
d+1$, and let
$0\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{F}}^{0}\longrightarrow\dotsb\longrightarrow{\mathcal{F}}^{d}\longrightarrow{\mathcal{F}}\longrightarrow
0$ be an exact sequence of sheaves of abelian groups on $X$. Suppose that the
sheaves ${\mathcal{F}}^{i}$ are flasque. Then the sheaf ${\mathcal{F}}$ is
flasque.
(b) Let $X$ be a scheme with an open covering ${\mathbf{W}}$ and
$0\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{F}}_{d}\longrightarrow\dotsb\longrightarrow{\mathfrak{F}}_{0}\longrightarrow{\mathfrak{E}}\longrightarrow
0$ be an exact sequence in
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$. Suppose that
the cosheaves ${\mathfrak{F}}_{i}$ are coflasque and the underlying
topological space of the scheme $X$ is Noetherian of finite Krull dimension
$\le d+1$. Then the cosheaf ${\mathfrak{F}}$ is coflasque.
###### Proof.
Part (a) is a version of Grothendieck’s vanishing theorem [24, Théorème
3.6.5]; it can be either proven directly along the lines of Grothendieck’s
proof, or, making a slightly stronger assumption that the dimension of $X$
does not exceed $d$, deduced from the assersion of Grothendieck’s theorem.
Indeed, let ${\mathcal{G}}$ denote the image of the morphism of sheaves
${\mathcal{F}}^{d-1}\longrightarrow{\mathcal{F}}^{d}$; by Grothendieck’s
theorem, one has $H^{1}(V,{\mathcal{G}}|_{V})\simeq
H^{d+1}(V,{\mathcal{E}}|_{V})=0$ for any open subset $V\subset X$. Hence the
map ${\mathcal{F}}^{d}(V)\longrightarrow{\mathcal{F}}(V)$ is surjective, and
it follows that the map ${\mathcal{F}}(X)\longrightarrow{\mathcal{F}}(V)$ is
surjective, too.
Part (b): given an injective abelian group $I$, the sequence
$0\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{E}}[U],I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{0}[U],I)\longrightarrow\dotsb\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{d}[U],I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}[U],I)\longrightarrow
0$ is exact for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. Hence the construction of the sheaf
$V\longmapsto\operatorname{Hom}_{\mathbb{Z}}({-}[V],I)$, where $V\subset X$
are arbitrary open subschemes, transforms our sequence of cosheaves into an
exact sequence of sheaves of ${\mathcal{O}}_{X}$-modules
$0\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{E}},I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{0},I)\longrightarrow\dotsb\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{d},I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}},I)\longrightarrow
0$. Now the sheaves $\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}^{i},I)$
are flasque, and by part (a) it follows that the sheaf
$\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}},I)$ is. Therefore, the cosheaf
${\mathfrak{F}}$ is coflasque. (Cf. Section A.5.) ∎
We have shown, in particular, that coflasque contraherent cosheaves on a
scheme $X$ form a full subcategory of $X{\operatorname{\mathsf{--ctrh}}}$
closed under extensions, kernels of admissible epimorphisms, and infinite
products. Hence this full subcategory acquires an induced exact category
structure, which we will denote by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$.
Similarly, coflasque locally cotorsion contraherent cosheaves on $X$ form a
full subcategory of $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ closed
under extensions, kernels of admissible epimorphisms, and infinite products.
We denote the induced exact category structure on this full subcategory by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$.
###### Corollary 3.4.8.
Let $f\colon Y\longrightarrow X$ be a quasi-compact quasi-separated morphism
of schemes. Then
(a) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes coflasque contraherent cosheaves on $Y$ to coflasque
contraherent cosheaves on $X$, and induces an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$ between these exact
categories;
(b) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes coflasque locally cotorsion contraherent cosheaves on $Y$ to
coflasque locally cotorsion contraherent cosheaves on $X$, and induces an
exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$ between these
exact categories.
###### Proof.
Since the contraadjustness/contraherence conditions, local cotorsion, and
exactness of short sequences in $X{\operatorname{\mathsf{--ctrh}}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ only depend on the
restrictions to affine open subschemes, while the coflasqueness is preserved
by restrictions to any open subschemes, we can assume that the scheme $X$ is
affine. Then the scheme $Y$ is quasi-compact and quasi-separated.
Let $Y=\bigcup_{\alpha}V_{\alpha}$ be a finite affine open covering and
${\mathfrak{F}}$ be a coflasque contraherent cosheaf on $Y$. Then the Čech
complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}})$
is a finite left resolution of the ${\mathcal{O}}(X)$-module
${\mathfrak{F}}[Y]$. Let us first consider the case when $Y$ is semi-
separated, so the intersection of any nonempty subset of $V_{\alpha}$ is
affine.
By Lemma 1.2.2(a), our resolution consists of contraadjusted
${\mathcal{O}}(X)$-modules. When ${\mathfrak{F}}$ is a locally cotorsion
contraherent cosheaf, by Lemma 1.3.4(a) this resolution even consists of
cotorsion ${\mathcal{O}}(X)$-modules. It follows that the
${\mathcal{O}}(X)$-module $(f_{!}{\mathfrak{F}})[X]={\mathfrak{F}}[Y]$ is
contraadjusted in the former case and cotorsion in the latter one.
Now let $U\subset X$ be an affine open subscheme. Then one has
${\mathfrak{F}}[f^{-1}(U)\cap
V]\simeq\operatorname{Hom}_{{\mathcal{O}}_{Y}(V)}({\mathcal{O}}_{Y}(f^{-1}(U)\cap
V),\>{\mathfrak{F}}[V])\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{\mathfrak{F}}[V])$
for any affine open subscheme $V\subset Y$; so the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{f^{-1}(U)\cap
V_{\alpha}\\},\>{\mathfrak{F}}|_{f^{-}1(U)})$ can be obtained by applying the
functor $\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{-})$ to
the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}})$.
It follows that
$(f_{!}{\mathfrak{F}})[U]\simeq{\mathfrak{F}}[f^{-1}(U)]\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{\mathfrak{F}}[Y])$
and the contraherence condition holds for $f_{!}{\mathfrak{F}}$.
Finally, let us turn to the general case. According to the above, for any
semi-separated quasi-compact open subscheme $V\subset Y$ the
${\mathcal{O}}(X)$-module ${\mathfrak{F}}[V]$ is contraadjusted (and even
cotorsion if ${\mathfrak{F}}$ is locally cotorsion), and for any affine open
subscheme $U\subset X$ one has ${\mathfrak{F}}[f^{-1}(U)\cap
V]\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{\mathfrak{F}}[V])$.
Given that the intersection of any nonempty subset of $V_{\alpha}$ is
separated (being quasi-affine) and quasi-compact, the same argument as above
goes through. ∎
###### Corollary 3.4.9.
Let $f\colon Y\longrightarrow X$ be a quasi-compact quasi-separated morphism
of schemes and ${\mathbf{T}}$ be an open covering of $Y$. Then
(a) for any complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of flasque
quasi-coherent sheaves on $Y$ that is acyclic as a complex over
$Y{\operatorname{\mathsf{--qcoh}}}$, the complex
$f_{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of flasque
quasi-coherent sheaves on $X$ is acyclic as a complex over
$X{\operatorname{\mathsf{--qcoh}}}$;
(b) for any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of coflasque
contraherent cosheaves on $Y$ that is acyclic as a complex over
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$, the complex
$f_{!}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
coflasque contraherent cosheaves on $X$ is acyclic as a complex over
$X{\operatorname{\mathsf{--ctrh}}}$;
(c) for any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of coflasque
locally cotorsion contraherent cosheaves on $Y$ that is acyclic as a complex
over $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$, the
complex $f_{!}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of coflasque locally cotorsion contraherent cosheaves on $X$ is acyclic as a
complex over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.
###### Proof.
Let us prove part (c), parts (a-b) being analogous. In view of the results of
Section 3.2, it suffices to consider the case of an affine scheme $X$ and a
quasi-compact quasi-separated scheme $Y$. We have to show that the complex of
cotorsion ${\mathcal{O}}(X)$-modules
$\Delta(Y,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ is
acyclic over ${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.
Considering the case of a semi-separated scheme $Y$ first and the general case
second, one can assume that $Y$ has a finite open covering
$Y=\bigcup_{\alpha=1}^{N}U_{\alpha}$ by quasi-compact quasi-separated schemes
$U_{\alpha}$ subordinate to ${\mathbf{T}}$ such that for any intersection $V$
of a nonempty subset of $U_{\alpha}$ the complex
$\Delta(V,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ is
acyclic over ${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.
Consider the Čech bicomplex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of the complex of cosheaves
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on the scheme
$Y$ with the open covering $U_{\alpha}$. There is a natural morphism of
bicomplexes
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\Delta(X,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$;
and for every degree $i$ the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{i})\longrightarrow\Delta(X,{\mathfrak{F}}^{i})$
is a finite (and uniformly bounded) acyclic complex of cotorsion
${\mathcal{O}}(X)$-modules. It follows that the induced morphism of total
complexes is a quasi-isomorphism (in fact, a morphism with an absolutely
acyclic cone) of complexes over
${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. On the other
hand, for every $k$ the complex
$C_{k}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is by assumption acyclic over over
${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. It follows
that the total complex of
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
and hence also the complex
$\Delta(Y,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$, is
acyclic over ${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.
∎
### 3.5. Contrahereable cosheaves and the contraherator
A cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on a scheme $X$ is
said to be _derived contrahereable_ if
1. (i∘)
for any affine open subscheme $U\subset X$ and its finite affine open covering
$U=\bigcup_{\alpha=1}^{N}U_{\alpha}$ the homological Čech sequence (cf. (22))
(28) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{P}}[U_{1}\cap\dotsb\cap U_{N}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}{\mathfrak{P}}[U_{\alpha}\cap U_{\beta}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}{\mathfrak{P}}[U_{\alpha}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{P}}[U]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
is exact; and
2. (ii)
for any affine open subscheme $U\subset X$, the ${\mathcal{O}}_{X}(U)$-module
${\mathfrak{P}}[U]$ is contraadjusted.
We will call (i∘) the _exactness condition_ and (ii) the _contraadjustness
condition_.
Notice that the present contraadjustness condition (ii) is an equivalent
restatement of the contraadjustness condition (ii) of Section 2.2, while the
exactness condition (i∘) is weaker than the contraherence condition (i) of
Section 2.2 (provided that the condition (ii) is assumed). The condition (i∘)
is also weaker than the coflasqueness condition on a cosheaf of
${\mathcal{O}}_{X}$-modules discussed in Section 3.4.
By Remark 2.1.4, the exactness condition (i∘) can be thought of as a
strengthening of the cosheaf property (6) of a covariant functor with an
${\mathcal{O}}_{X}$-module structure on the category of affine open subschemes
of $X$. Any such functor satisfying (i∘) can be extended to a cosheaf of
${\mathcal{O}}_{X}$-modules in a unique way.
Let ${\mathbf{W}}$ be an open covering of a scheme $X$. A cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on $X$ is called _${\mathbf{W}}$
-locally derived contrahereable_ if its restrictions ${\mathfrak{P}}|_{W}$ to
all the open subschemes $W\in{\mathbf{W}}$ are derived contrahereable on $W$.
In other words, this means that the conditions (i∘) and (ii) must hold for all
the affine open subschemes $U\subset X$ subordinate to ${\mathbf{W}}$. A
cosheaf of ${\mathcal{O}}_{X}$-modules is called locally derived
contrahereable if it is ${\mathbf{W}}$-locally derived contrahereable for some
open covering ${\mathbf{W}}$. Any contraherent cosheaf is derived
contrahereable, and any ${\mathbf{W}}$-locally contraherent cosheaf is
${\mathbf{W}}$-locally derived contrahereable. Conversely, according to Lemma
3.2.2, if a ${\mathbf{W}}$-locally contraherent cosheaf is derived
contrahereable, then it is contraherent.
A ${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{P}}$ on
$X$ is called _locally cotorsion_ (respectively, _locally injective_) if the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$ is cotorsion (resp.,
injective) for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. A locally derived contrahereable cosheaf ${\mathfrak{P}}$ is
locally cotorsion (resp., locally injective) if and only if the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$ is cotorsion (resp.,
injective) for every affine open subscheme $U\subset X$ such that the cosheaf
${\mathfrak{P}}|_{U}$ is derived contrahereable.
In the exact category of cosheaves of ${\mathcal{O}}_{X}$-modules
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$ defined in
Section 3.1, the ${\mathbf{W}}$-locally derived contrahereable cosheaves form
an exact subcategory closed under extensions, infinite products, and cokernels
of admissible monomorphisms. Hence there is the induced exact category
structure on the category of ${\mathbf{W}}$-locally derived contrahereable
cosheaves on $X$.
Let $U$ be an affine scheme and ${\mathfrak{Q}}$ be a derived contrahereable
cosheaf on $U$. The _contraherator_
$\operatorname{\mathfrak{C}}{\mathfrak{Q}}$ of the cosheaf ${\mathfrak{Q}}$ is
defined in this simplest case as the contraherent cosheaf on $U$ corresponding
to the contraadjusted ${\mathcal{O}}(U)$-module ${\mathfrak{Q}}[U]$, that is
$\operatorname{\mathfrak{C}}{\mathfrak{Q}}=\widecheck{{\mathfrak{Q}}[U]}$.
There is a natural morphism of derived contrahereable cosheaves
${\mathfrak{Q}}\longrightarrow\operatorname{\mathfrak{C}}{\mathfrak{Q}}$ on
$U$ (see Lemma 2.2.4). For any affine open subscheme $V\subset U$ there is a
natural morphism of contraherent cosheaves
$\operatorname{\mathfrak{C}}({\mathfrak{Q}}|_{V})\longrightarrow(\operatorname{\mathfrak{C}}{\mathfrak{Q}})|_{V}$
on $V$. Our next goal is to extend this construction to an appropriate class
of cosheaves of ${\mathcal{O}}_{X}$-modules on quasi-compact semi-separated
schemes $X$ (cf. [61, Appendix B]).
Let $X$ be such a scheme with an open covering ${\mathbf{W}}$ and a finite
affine open covering $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ subordinate to
${\mathbf{W}}$. The _contraherator complex_
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
of a ${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{P}}$ on
$X$ is a finite Čech complex of contraherent cosheaves on $X$ of the form
(29) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{1,\dotsc,N}{}_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{U_{1}\cap\dotsb\cap
U_{N}})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb\\\ \textstyle\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}j_{\alpha,\beta}{}_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{U\alpha\cap
U_{\beta}})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}j_{\alpha}{}_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{U_{\alpha}}),$
where $j_{\alpha_{1},\dotsc,\alpha_{k}}$ is the open embedding
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{k}}\longrightarrow X$ and the
notation $\operatorname{\mathfrak{C}}({\mathfrak{Q}})$ was explained above.
The differentials in this complex are constructed using the adjunction of the
direct and inverse images of contraherent cosheaves and the above morphisms
$\operatorname{\mathfrak{C}}({\mathfrak{Q}}|_{V})\longrightarrow(\operatorname{\mathfrak{C}}{\mathfrak{Q}})|_{V}$.
###### Lemma 3.5.1.
Let ${\mathfrak{P}}$ be a ${\mathbf{W}}$-locally derived contrahereable
cosheaf on a quasi-compact semi-separated scheme $X$. Then the object of the
bounded derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}})$ of the exact
category of contraherent cosheaves on $X$ represented by the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
does not depend on the choice of a finite affine open covering
$\\{U_{\alpha}\\}$ of $X$ subordinate to the covering ${\mathbf{W}}$.
###### Proof.
Let us adjoin another affine open subscheme $V\subset X$, subordinate to
${\mathbf{W}}$, to the covering $\\{U_{\alpha}\\}$. Then the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
embeds into the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V,U_{\alpha}\\},{\mathfrak{P}})$
by a termwise split morphism of complexes with the cokernel isomorphic to the
complex
$k_{!}\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},{\mathfrak{P}}|_{V})\longrightarrow
k_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{V})$.
The complex of contraherent cosheaves
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},{\mathfrak{P}}|_{V})\longrightarrow\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{V})$
on $V$ corresponds to the acyclic complex of contraadjusted
${\mathcal{O}}(V)$-modules (28) for the covering of an affine open subscheme
$V\subset X$ by the affine open subschemes $V\cap U_{\alpha}$. Hence the
cokernel of the admissible monomorphism of complexes
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})\longrightarrow\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V,U_{\alpha}\\},{\mathfrak{P}})$
is an acyclic complex of contraherent cosheaves on $X$.
Now, given two affine open coverings
$X=\bigcup_{\alpha}U_{\alpha}=\bigcup_{\beta}V_{\beta}$ subordinate to
${\mathbf{W}}$, one compares both complexes
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
and
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\beta}\\},{\mathfrak{P}})$
with the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha},V_{\beta}\\},{\mathfrak{P}})$
corresponding to the union of the two coverings $\\{U_{\alpha},V_{\beta}\\}$.
∎
A ${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{P}}$ on a
quasi-compact semi-separated scheme $X$ is called _${\mathbf{W}}$ -locally
contrahereable_ if the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
for some particular (or equivalently, for any) finite affine open covering
$X=\bigcup_{\alpha}U_{\alpha}$ is quasi-isomorphic in
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ to
a ${\mathbf{W}}$-locally contraherent cosheaf on $X$ (viewed as a complex
concentrated in homological degree $0$). The ${\mathbf{W}}$-locally
contraherent cosheaf that appears here is called the (_${\mathbf{W}}$ -local_)
_contraherator_ of ${\mathfrak{P}}$ and denoted by
$\operatorname{\mathfrak{C}}{\mathfrak{P}}$. A derived contrahereable cosheaf
${\mathfrak{P}}$ on $X$ is called _contrahereable_ if it is locally
contrahereable with respect to the covering $\\{X\\}$.
Any derived contrahereable cosheaf ${\mathfrak{Q}}$ on an affine scheme $U$ is
contrahereable, because the contraherator complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U\\},{\mathfrak{Q}})$
is concentrated in homological degree $0$. The contraherator cosheaf
$\operatorname{\mathfrak{C}}{\mathfrak{Q}}$ constructed in this way coincides
with the one defined above specifically in the affine scheme case; so our
notation and terminology is consistent. Any ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{P}}$ on a quasi-compact semi-separated scheme
$X$ is ${\mathbf{W}}$-locally contrahereable; the corresponding contraherator
complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
is the contraherent Čech resolution (27) of a ${\mathbf{W}}$-locally
contraherent cosheaf
${\mathfrak{P}}=\operatorname{\mathfrak{C}}{\mathfrak{P}}$.
The contraherator complex construction
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{-})$
is an exact functor from the exact category of ${\mathbf{W}}$-locally
contrahereable cosheaves to the exact category of finite complexes of
contraherent cosheaves on $X$, or to the bounded derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}})$. The full
subcategory of ${\mathbf{W}}$-locally contrahereable cosheaves in the exact
category of ${\mathbf{W}}$-locally derived contrahereable cosheaves is closed
under extensions and infinite products. Hence it acquires the induced exact
category structure. The contraherator $\operatorname{\mathfrak{C}}$ is an
exact functor from the exact category of ${\mathbf{W}}$-locally contrahereable
cosheaves to that of ${\mathbf{W}}$-locally contraherent ones.
All the above applies to locally cotorsion and locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaves as well. These form
full exact subcategories closed under extensions, infinite products, and
cokernels of admissible monomorphisms in the exact category of
${\mathbf{W}}$-locally derived contrahereable cosheaves. The contraherator
complex construction
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{-})$
takes locally cotorsion (resp., locally injective) ${\mathbf{W}}$-locally
derived contrahereable cosheaves to finite complexes of locally cotorsion
(resp., locally injective) contraherent cosheaves on $X$.
A locally cotorsion (resp., locally injective) ${\mathbf{W}}$-locally derived
contrarehreable cosheaf is called ${\mathbf{W}}$-locally contrahereable if it
is ${\mathbf{W}}$-locally contrahereable as a ${\mathbf{W}}$-locally derived
contrahereable cosheaf. The contraherator functor
$\operatorname{\mathfrak{C}}$ takes locally cotorsion (resp., locally
injective) ${\mathbf{W}}$-locally contrahereable cosheaves to locally
cotorsion (resp., locally injective) ${\mathbf{W}}$-locally contraherent
cosheaves.
Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
morphism of schemes. Then the direct image functor $f_{!}$ takes
${\mathbf{T}}$-locally derived contrahereable cosheaves on $Y$ to
${\mathbf{W}}$-locally derived contrahereable cosheaves on $X$. For a
$({\mathbf{W}},{\mathbf{T}})$-affine morphism $f$ of quasi-compact semi-
separated schemes, the functor $f_{!}$ also commutes with the contraherator
complex construction, as there is a natural isomorphism of complexes of
contraherent cosheaves
$f_{!}\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{f^{-1}(U_{\alpha})\\},{\mathfrak{P}})\simeq\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},f_{!}{\mathfrak{P}})$
for any finite affine open covering $U_{\alpha}$ of $X$ subordinate to
${\mathbf{W}}$. It follows that the functor $f_{!}$ takes
${\mathbf{T}}$-locally contrahereable cosheaves to ${\mathbf{W}}$-locally
contrahereable cosheaves and commutes with the functor
$\operatorname{\mathfrak{C}}$.
For any ${\mathbf{W}}$-locally contrahereable cosheaf ${\mathfrak{P}}$ and
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on a quasi-
compact semi-separated scheme $X$, there is a natural isomorphism of the
groups of morphisms
(30)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},{\mathfrak{Q}})\simeq\operatorname{Hom}^{X}(\operatorname{\mathfrak{C}}{\mathfrak{P}},{\mathfrak{Q}}).$
In other words, the functor $\operatorname{\mathfrak{C}}$ is left adjoint to
the identity embedding functor of the category of ${\mathbf{W}}$-locally
contrahereable cosheaves into the category of ${\mathbf{W}}$-locally
contraherent ones. Indeed, applying the contraherator complex construction
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{-})$
to a morphism ${\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$ and passing to the
zero homology, we obtain the corresponding morphism
$\operatorname{\mathfrak{C}}{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$.
Conversely, any cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ is the
cokernel of the rightmost arrow of the complex
(31) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{1,\dotsc,N}{}_{!}({\mathfrak{P}}|_{U_{1}\cap\dotsb\cap U_{N}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}j_{\alpha,\beta}{}_{!}({\mathfrak{P}}|_{U\alpha\cap
U_{\beta}})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}j_{\alpha}{}_{!}({\mathfrak{P}}|_{U_{\alpha}})$
in the additive category of cosheaves of ${\mathcal{O}}_{X}$-modules. A
cosheaf ${\mathfrak{P}}$ satisfying the exactness condition (i∘) for affine
open subschemes $U\subset X$ subordinate to ${\mathbf{W}}$ is also quasi-
isomorphic to the whole complex (31) in the exact category
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$. Passing to
the zero homology of the natural morphism between the complexes (31) and (29),
we produce the desired adjunction morphism
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{C}}{\mathfrak{P}}$.
For any ${\mathbf{W}}$-locally contrahereable cosheaf ${\mathfrak{P}}$ and any
quasi-coherent sheaf ${\mathcal{M}}$ on $X$ the natural morphism of cosheaves
of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{C}}{\mathfrak{P}}$
induces an isomorphism of the contratensor products
(32)
${\mathcal{M}}\odot_{X}{\mathfrak{P}}\simeq{\mathcal{M}}\odot_{X}\operatorname{\mathfrak{C}}({\mathfrak{P}}).$
Indeed, for any injective quasi-coherent sheaf ${\mathcal{J}}$ on $X$ one has
$\operatorname{Hom}_{X}({\mathcal{M}}\odot_{X}{\mathfrak{P}},\>{\mathcal{J}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}}))\\\
\simeq\operatorname{Hom}^{X}(\operatorname{\mathfrak{C}}{\mathfrak{P}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},{\mathcal{J}}))\simeq\operatorname{Hom}_{X}({\mathcal{M}}\odot_{X}\operatorname{\mathfrak{C}}{\mathfrak{P}},\>{\mathcal{J}})$
in view of the isomorphism (20).
### 3.6. $\operatorname{\mathfrak{Cohom}}$ into a locally derived
contrahereable cosheaf
We start with discussing the $\operatorname{\mathfrak{Cohom}}$ from a quasi-
coherent sheaf to a locally contraherent cosheaf.
Let ${\mathbf{W}}$ be an open covering of a scheme $X$. Let ${\mathcal{F}}$ be
a very flat quasi-coherent sheaf on $X$, and let ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. The ${\mathbf{W}}$-locally
contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ on the
scheme $X$ is defined by the rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$. The
contraadjustness and ${\mathbf{W}}$-local contraherence conditions can be
verified in the same way as it was done in Section 2.4.
Similarly, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$, then the locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is
defined by the same rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$.
Finally, if ${\mathcal{M}}$ is a quasi-coherent sheaf and ${\mathfrak{J}}$ is
a locally injective ${\mathbf{W}}$-locally contraherent cosheaf on $X$, then
the locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ is defined
by the same rule as above. If ${\mathcal{F}}$ is a flat quasi-coherent sheaf
and ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$, then the ${\mathbf{W}}$-locally contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})$ is
locally injective.
For any two very flat quasi-coherent sheaves ${\mathcal{F}}$ and
${\mathcal{G}}$ on a scheme $X$ and any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ there is a natural isomorphism of
${\mathbf{W}}$-locally contraherent cosheaves
(33)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}},\>{\mathfrak{P}})\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{G}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})).$
Similarly, for any two flat quasi-coherent sheaves ${\mathcal{F}}$ and
${\mathcal{G}}$ and a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ there is a natural isomorphism (33) of locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves. Finally, for any flat
quasi-coherent sheaf ${\mathcal{F}}$, quasi-coherent sheaf ${\mathcal{M}}$,
and locally injective ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{J}}$ on $X$ there are natural isomorphisms of locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves
(34)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}},\>{\mathfrak{J}})\\\
\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}}))\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})).$
More generally, let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on
$X$, and let ${\mathfrak{P}}$ be a ${\mathbf{W}}$-locally derived
contrahereable cosheaf on $X$. The ${\mathbf{W}}$-locally derived
contrahereable cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ on the
scheme $X$ is defined by the rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$. For
any pair of embedded affine open subschemes $V\subset U$ the restriction and
corestriction morphisms ${\mathcal{F}}(U)\longrightarrow{\mathcal{F}}(V)$ and
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]$ induce a morphism of
${\mathcal{O}}_{X}(U)$-modules
$\operatorname{Hom}_{{\mathcal{O}}_{X}(V)}({\mathcal{F}}(V),{\mathfrak{P}}[V])\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U]),$
so our rule defines a covariant functor with ${\mathcal{O}}_{X}$-module
structure on the category of affine open subschemes $U\subset X$ subordinate
to ${\mathbf{W}}$. The contraherence condition clearly holds; and to check the
exactness condition (i∘) of Section 3.5 for this covariant functor, it
suffices to apply the functor
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{-})$ to the exact
sequence of contraadjusted ${\mathcal{O}}_{X}(U)$-modules (28) for the cosheaf
${\mathfrak{P}}$.
Similarly, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally derived
contrahereable cosheaf on $X$, then the locally cotorsion
${\mathbf{W}}$-locally derived contrahereable cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is defined
by the same rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}\allowbreak({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$.
Finally, if ${\mathcal{M}}$ is a quasi-coherent sheaf and ${\mathfrak{J}}$ is
a locally injective ${\mathbf{W}}$-locally derived contrahereable cosheaf on
$X$, then the locally cotorsion ${\mathbf{W}}$-locally derived contrahereable
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ is
defined by the very same rule. If ${\mathcal{F}}$ is a flat quasi-coherent
sheaf and ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
derived contrahereable cosheaf on $X$, then the ${\mathbf{W}}$-locally derived
cotrahereable cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})$ is locally
injective.
The associativity isomorphisms (33–34) hold for the
$\operatorname{\mathfrak{Cohom}}$ into ${\mathbf{W}}$-locally derived
contrahereable cosheaves under the assumptions similar to the ones made above
in the locally contraherent case.
### 3.7. Contraherent tensor product
Let ${\mathfrak{P}}$ be a cosheaf of ${\mathcal{O}}_{X}$-modules on a scheme
$X$ and ${\mathcal{M}}$ be a quasi-coherent cosheaf on $X$. Define a covariant
functor with an ${\mathcal{O}}_{X}$-module structure on the category of affine
open subschemes of $X$ by the rule
$U\longmapsto{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$.
To a pair of embedded affine open subschemes $V\subset U\subset X$ this
functor assigns the ${\mathcal{O}}_{X}(U)$-module homomorphism
${\mathcal{M}}(V)\otimes_{{\mathcal{O}}_{X}(V)}{\mathfrak{P}}[V]\simeq{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U].$
Obviously, this functor satisfies the condition (6) of Theorem 2.1.2 (since
the restriction of the cosheaf ${\mathfrak{P}}$ to affine open subschemes of
$X$ does). Hence the functor
$U\longmapsto{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$
extends uniquely to a cosheaf of ${\mathcal{O}}_{X}$-modules on $X$, which we
will denote by ${\mathcal{M}}\otimes_{X}{\mathfrak{P}}$.
For any quasi-coherent sheaf ${\mathcal{M}}$, cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$, and locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{J}}$ on a
scheme $X$ there is a natural isomorphism of abelian groups
(35)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathcal{M}}\otimes_{X}{\mathfrak{P}},\>{\mathfrak{J}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})).$
The analogous adjunction isomorphism holds in the other cases mentioned in
Section 3.6 when the functor $\operatorname{\mathfrak{Cohom}}$ from a quasi-
coherent sheaf to a locally derived contrahereable cosheaf is defined. In
other words, the functors
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{-})$ and
${\mathcal{M}}\otimes_{X}{-}$ between subcategories of the category of
cosheaves of ${\mathcal{O}}_{X}$-modules are adjoint “wherever the former
functor is defined”.
For a locally free sheaf of finite rank ${\mathcal{E}}$ and a
${\mathbf{W}}$-locally contraherent (resp., ${\mathbf{W}}$-locally derived
contrahereable) cosheaf ${\mathfrak{P}}$ on a scheme $X$, the cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathcal{E}}\otimes_{X}{\mathfrak{P}}$ is
${\mathbf{W}}$-locally contraherent (resp., ${\mathbf{W}}$-locally derived
contrahereable). There is a natural isomorphism of ${\mathbf{W}}$-locally
contraherent (resp., ${\mathbf{W}}$-locally derived contrahereable) cosheaves
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({\mathcal{E}},{\mathcal{O}}_{X})\otimes_{X}{\mathfrak{P}}\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{E}},{\mathfrak{P}})$
on $X$.
The isomorphism
$j_{*}j^{*}({\mathcal{K}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}})\simeq
j_{*}j^{*}{\mathcal{K}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{M}}(U)$ (15)
for quasi-coherent sheaves ${\mathcal{K}}$ and ${\mathcal{M}}$ and the
embedding of an affine open subscheme $j\colon U\longrightarrow X$ allows to
construct a natural isomorphism of quasi-coherent sheaves
(36)
$({\mathcal{K}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}})\odot_{X}{\mathfrak{P}}\simeq{\mathcal{K}}\odot_{X}({\mathcal{M}}\otimes_{X}{\mathfrak{P}})$
for any quasi-coherent sheaves ${\mathcal{K}}$ and ${\mathcal{M}}$ and a
cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on a semi-separated
scheme $X$.
Let ${\mathbf{W}}$ be an open covering of a scheme $X$. We will call a cosheaf
of ${\mathcal{O}}_{X}$-modules ${\mathfrak{F}}$ _${\mathbf{W}}$ -flat_ if the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{F}}[U]$ is flat for every affine
open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$. A cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{F}}$ is said to be _flat_ if it is
$\\{X\\}$-flat. Clearly, the direct image of a ${\mathbf{T}}$-flat cosheaf of
${\mathcal{O}}_{Y}$-modules with respect to a flat
$({\mathbf{W}},{\mathbf{T}})$-affine morphism of schemes $f\colon
Y\longrightarrow X$ is ${\mathbf{W}}$-flat.
One can easily see that whenever a cosheaf ${\mathfrak{F}}$ is
${\mathbf{W}}$-flat and satisfies the “exactness condition” (i∘) of Section
3.5 for finite affine open coverings of affine open subschemes $U\subset X$
subordinate to ${\mathbf{W}}$, the cosheaf
${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$ also satisfies the condition (i∘) for
such open affines $U\subset X$. Similarly, whenever a quasi-coherent sheaf
${\mathcal{F}}$ is flat and a cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}$ satisfies the condition (i∘), so does the cosheaf
${\mathcal{F}}\otimes_{X}{\mathfrak{P}}$.
The full subcategory of ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally
contraherent cosheaves is closed under extensions and kernels of admissible
epimorphisms in the exact category of ${\mathbf{W}}$-locally contraherent
cosheaves $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ on $X$. Hence it
acquires the induced exact category structure, which we will denote by
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}$. The category
$X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{\mathsf{fl}}$ will be denoted by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. Similarly, there is the
exact category structure on the category of ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally derived contrahereable cosheaves on $X$ induced from
the exact category of ${\mathbf{W}}$-locally derived contrahereable cosheaves.
A quasi-coherent sheaf ${\mathcal{K}}$ on a scheme $X$ is called _coadjusted_
if the ${\mathcal{O}}_{X}(U)$-module ${\mathcal{K}}(U)$ is coadjusted (see
Section 1.6) for every affine open subscheme $U\subset X$. By Lemma 1.6.9, the
coadjustness of a quasi-coherent sheaf is a local property. By the definition,
if a cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on $X$ satisfies
the contraherence condition (ii) of Section 2.2 or 3.5 and a quasi-coherent
sheaf ${\mathcal{K}}$ on $X$ is coadjusted, then the cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathcal{K}}\otimes_{X}{\mathfrak{P}}$ also
satisfies the condition (ii).
The full subcategory of coadjusted quasi-coherent sheaves is closed under
extensions and the passage to quotient objects in the abelian category of
quasi-coherent sheaves $X{\operatorname{\mathsf{--qcoh}}}$ on $X$. Hence it
acquires the induced exact category structure, which we will denote by
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{coa}}$.
Let ${\mathcal{K}}$ be a coadjusted quasi-coherent sheaf and ${\mathfrak{F}}$
be a ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent (or more
generally, ${\mathbf{W}}$-locally derived contrahereable) cosheaf on a scheme
$X$. Then the tensor product ${\mathcal{K}}\otimes_{X}{\mathfrak{F}}$
satisfies both conditions (i∘) and (ii) for affine open subschemes $U\subset
X$ subordinate to ${\mathbf{W}}$, i. e., it is ${\mathbf{W}}$-locally derived
contrahereable. (Of course, the cosheaf
${\mathcal{K}}\otimes_{X}{\mathfrak{F}}$ is _not_ in general locally
contraherent, even if the cosheaf ${\mathfrak{F}}$ was ${\mathbf{W}}$-locally
contraherent.)
Assuming the scheme $X$ is quasi-compact and semi-separated, the contraherator
complex construction now allows to assign to this cosheaf of
${\mathcal{O}}_{X}$-modules a complex of contraherent cosheaves
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},\>{\mathcal{K}}\otimes_{X}{\mathfrak{F}})$
on $X$. To a short exact sequence of coadjusted quasi-coherent sheaves or
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent (or
${\mathbf{W}}$-locally derived contrahereable) cosheaves on $X$, the functor
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},\>{-}\otimes_{X}{-})$
assigns a short exact sequence of complexes of contraherent cosheaves.
By Lemma 3.5.1, the corresponding object of the bounded derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}})$ does not depend
on the choice of a finite affine open covering $\\{U_{\alpha}\\}$. We will
denote it by
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}^{\mathbb{L}}{\mathfrak{F}}$
and call the _derived contraherent tensor product_ of a coadjusted quasi-
coherent sheaf ${\mathcal{K}}$ and a ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on a quasi-
compact semi-separated scheme $X$.
When the derived category object
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}^{\mathbb{L}}{\mathfrak{F}}$,
viewed as an object of the derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
via the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, turns out to be isomorphic to
an object of the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, we say that the (underived)
_contraherent tensor product_ of ${\mathcal{K}}$ and ${\mathfrak{F}}$ is
defined, and denote the corresponding object by
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}\mskip
1.5mu\in X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. In other words, for
a coadjusted quasi-coherent sheaf ${\mathcal{K}}$ and a ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on a quasi-
compact semi-separated scheme $X$ one sets
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}=\operatorname{\mathfrak{C}}({\mathcal{K}}\otimes_{X}{\mathfrak{F}})$
whenever the right-hand side is defined (where $\operatorname{\mathfrak{C}}$
denotes the ${\mathbf{W}}$-local contraherator).
Now assume that the scheme $X$ is locally Noetherian. Then, by Corollary
1.6.5(a), a contraherent cosheaf ${\mathfrak{F}}$ on an affine open subscheme
$U\subset X$ is flat if and only if the contraadjusted
${\mathcal{O}}(U)$-module ${\mathfrak{F}}[U]$ is flat. Besides, the full
subcategory of ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent
cosheaves in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is closed under
infinite products. In addition, any coherent sheaf on $X$ is coadjusted, as is
any injective quasi-coherent sheaf and any quasi-coherent quotient sheaf of an
injective one.
For any injective quasi-coherent sheaf ${\mathcal{J}}$ and any
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally derived contrahereable cosheaf
${\mathfrak{F}}$ on $X$, the tensor product
${\mathcal{J}}\otimes_{X}{\mathfrak{F}}$ is a locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaf on $X$. For any flat
quasi-coherent sheaf ${\mathcal{F}}$ and any locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{J}}$ on $X$,
the tensor product ${\mathcal{F}}\otimes_{X}{\mathfrak{J}}$ is a locally
injective ${\mathbf{W}}$-locally derived contrahereable cosheaf. These
assertions hold since the tensor product of a flat module and an injective
module over a Noetherian ring is injective.
Let ${\mathcal{M}}$ be a coherent sheaf and ${\mathfrak{F}}$ be a
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then
the cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$ is ${\mathbf{W}}$-locally
contraherent. Indeed, by Corollary 1.6.3(a), the ${\mathcal{O}}_{X}(U)$-module
${\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}(U)$ is
contraadjusted for any affine open subscheme $U\subset X$. For a pair of
embedded affine open subschemes $V\subset U\subset X$ subordinate to the
covering ${\mathbf{W}}$, one has
${\mathcal{M}}(V)\otimes_{{\mathcal{O}}_{X}(V)}{\mathfrak{F}}[V]\simeq{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[V]\\\
\simeq{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{F}}[U])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),\>{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U])$
according to Corollary 1.6.3(c). If the scheme $X$ is semi-separated and
Noetherian, the contraherent tensor product
${\mathcal{M}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$ is
defined and isomorphic to the tensor product
${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$.
Similarly, it follows from Corollary 1.6.4 that for any coherent sheaf
${\mathcal{M}}$ and ${\mathbf{W}}$-flat locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ the tensor
product ${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$ is a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$. (See Sections 5.1–5.2 and
Lemma 5.7.2 for further discussion.)
### 3.8. Compatibility of direct and inverse images with the tensor
operations
Let ${\mathbf{W}}$ be an open covering of a scheme $X$ and ${\mathbf{T}}$ be
an open covering of a scheme $Y$. Let $f\colon Y\longrightarrow X$ be a
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism.
Let ${\mathcal{F}}$ be a flat quasi-coherent sheaf and ${\mathfrak{J}}$ be a
locally injective ${\mathbf{W}}$-locally contraherent cosheaf on the scheme
$X$. Then there is a natural isomorphism of locally injective
${\mathbf{T}}$-locally contraherent cosheaves
(37)
$f^{!}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})\simeq\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{F}},f^{!}{\mathfrak{J}})$
on the scheme $Y$.
Assume additionally that $f$ is a flat morphism. Let ${\mathcal{M}}$ be a
quasi-coherent sheaf and ${\mathfrak{J}}$ be a locally injective
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{T}}$-locally contraherent cosheaves
(38)
$f^{!}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})\simeq\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{M}},f^{!}{\mathfrak{J}})$
on $Y$. Analogously, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$, then there is a natural isomorphism of locally cotorsion
${\mathbf{T}}$-locally contraherent cosheaves
(39)
$f^{!}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})\simeq\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{F}},f^{!}{\mathfrak{P}})$
on the scheme $Y$.
Assume that, moreover, $f$ is a very flat morphism. Let ${\mathcal{F}}$ be a
very flat quasi-coherent sheaf and ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is the natural
isomorphism (39) of ${\mathbf{T}}$-locally contraherent cosheaves on $Y$.
Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism. Let ${\mathcal{N}}$ be a
quasi-coherent cosheaf on $Y$ and ${\mathfrak{J}}$ be a locally injective
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves
(40)
$\operatorname{\mathfrak{Cohom}}_{X}(f_{*}{\mathcal{N}},{\mathfrak{J}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}({\mathcal{N}},f^{!}{\mathfrak{J}})$
on the scheme $X$. This is one version of the projection formula for the
$\operatorname{\mathfrak{Cohom}}$ from a quasi-coherent sheaf to a
contraherent cosheaf.
Assume additionally that $f$ is a flat morphism. Let ${\mathcal{G}}$ be a flat
quasi-coherent sheaf on $Y$ and ${\mathfrak{P}}$ be a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves
(41)
$\operatorname{\mathfrak{Cohom}}_{X}(f_{*}{\mathcal{G}},{\mathfrak{P}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}({\mathcal{G}},f^{!}{\mathfrak{P}})$
on the scheme $X$.
Assume that, moreover, $f$ is a very flat morphism. Let ${\mathcal{G}}$ be a
very flat quasi-coherent sheaf on $Y$ and ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is the natural
isomorphism (41) of ${\mathbf{W}}$-locally contraherent cosheaves on $X$.
Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
morphism. Let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on $X$ and
${\mathfrak{Q}}$ be a ${\mathbf{T}}$-locally contraherent cosheaf on $Y$. Then
there is a natural isomorphism of ${\mathbf{W}}$-locally contraherent
cosheaves
(42)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},f_{!}{\mathfrak{Q}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{F}},{\mathfrak{Q}})$
on the scheme $X$. The similar isomorphism of locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves on $X$ holds for any flat quasi-
coherent sheaf ${\mathcal{F}}$ on $X$ and locally cotorsion
${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on $Y$. This is
another version of the projection formula for
$\operatorname{\mathfrak{Cohom}}$.
Assume additionally that $f$ is a flat morphism. Let ${\mathcal{M}}$ be a
quasi-coherent sheaf on $X$ and ${\mathfrak{I}}$ be a locally injective
${\mathbf{T}}$-locally contraherent cosheaf on $Y$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves
(43)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},f_{!}{\mathfrak{I}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{M}},{\mathfrak{I}})$
on the scheme $X$.
Let $f\colon Y\longrightarrow X$ be either an affine morphism of semi-
separated schemes, or a morphism of quasi-compact semi-separated schemes. Let
${\mathcal{F}}$ be a very flat quasi-coherent sheaf on $X$ and ${\mathcal{Q}}$
be a contraadjusted quasi-coherent sheaf on $Y$. Then there is a natural
isomorphism of contraherent cosheaves
(44)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},f_{*}{\mathcal{Q}})\simeq
f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{F}},{\mathcal{Q}})$
on $X$. Here the quasi-coherent sheaf $f_{*}{\mathcal{Q}}$ on $X$ is
contraadjusted according to Section 2.5 above (if $f$ is affine) or Corollary
4.1.13 below (if $X$ and $Y$ are quasi-compact). The right-hand side is, by
construction, a contraherent cosheaf if the morphism $f$ is affine, and a
cosheaf of ${\mathcal{O}}_{X}$-modules otherwise (see Section 2.3). Both sides
are, in fact, contraherent in the general case, because the isomorphism holds
and the left-hand side is. This is a version of the projection formula for
$\operatorname{\mathfrak{Hom}}$.
Indeed, let $j\colon U\longrightarrow X$ be an embedding of an affine open
subscheme; set $V=U\times_{X}Y$. Let $j^{\prime}\colon V\longrightarrow Y$ and
$f^{\prime}\colon V\longrightarrow U$ be the natural morphisms. Then one has
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},f_{*}{\mathcal{Q}})[U]\simeq\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},f_{*}{\mathcal{Q}})\simeq\operatorname{Hom}_{Y}(f^{*}\\!\mskip
1.5muj_{*}j^{*}{\mathcal{F}},{\mathcal{Q}})\\\
\simeq\operatorname{Hom}_{Y}(j^{\prime}_{*}f^{\prime}{}^{*}\\!\mskip
1.5muj^{*}{\mathcal{F}},{\mathcal{Q}})\simeq\operatorname{Hom}_{Y}(j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{F}},{\mathcal{Q}})\simeq\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{F}},{\mathcal{Q}})[V].$
Here we are using the fact that the direct images of quasi-coherent sheaves
with respect to affine morphisms of schemes commute with the inverse images in
the base change situations. Notice that, when the morphism $f$ is not affine,
neither is the scheme $V$; however, the scheme $V$ is quasi-compact and the
open embedding morphism $j^{\prime}\colon V\longrightarrow Y$ is affine, so
Lemma 2.5.2(a) applies. The similar isomorphism of locally cotorsion
contraherent cosheaves on $X$ holds for any flat quasi-coherent sheaf
${\mathcal{F}}$ on $X$ and any cotorsion quasi-coherent sheaf ${\mathcal{Q}}$
on $Y$.
Now let $f$ be a flat quasi-compact morphism of semi-separated schemes. Let
${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and ${\mathcal{I}}$ be an
injective quasi-coherent sheaf on $Y$. Then there is a natural isomorphism of
locally cotorsion contraherent cosheaves
(45)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},f_{*}{\mathcal{I}})\simeq
f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{M}},{\mathcal{I}})$
on the scheme $Y$. The proof is similar to the above. For any flat quasi-
compact morphism of quasi-separated schemes $f\colon Y\longrightarrow X$, a
quasi-coherent sheaf ${\mathcal{M}}$ on $X$, and an injective quasi-coherent
sheaf ${\mathcal{I}}$ on $Y$, there is a natural morphism of cosheaves of
${\mathcal{O}}_{X}$-modules from the right-hand side to the left-hand side of
(45); this morphism is an isomorphism whenever the morphism $f$ is also
affine.
Finally, let $f\colon Y\longrightarrow X$ be a quasi-compact open embedding of
quasi-separated schemes. Then the isomorphism (45) holds for any flasque
quasi-coherent sheaf ${\mathcal{M}}$ on $X$ and injective quasi-coherent sheaf
${\mathcal{I}}$ on $Y$. In fact, the direct images of quasi-coherent sheaves
with respect to quasi-compact quasi-separated morphisms commute with the
inverse images with respect to flat morphisms of schemes in the base change
situations, while the restrictions to open subschemes also preserve the
flasqueness; so one can apply Lemma 2.5.2(d).
Let $f\colon Y\longrightarrow X$ be an affine morphism of semi-separated
schemes. Let ${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and
${\mathfrak{Q}}$ be a cosheaf of ${\mathcal{O}}_{Y}$-modules. Then there is a
natural isomorphism of quasi-coherent sheaves
(46) ${\mathcal{M}}\odot_{X}f_{!}{\mathfrak{Q}}\simeq
f_{*}(f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}})$
on the scheme $X$. This is a version of the projection formula for the
contratensor product of quasi-coherent sheaves and cosheaves of
${\mathcal{O}}_{X}$-modules.
Indeed, in the notation above, for any affine open subscheme $U\subset X$ we
have
$j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}(f_{!}{\mathfrak{Q}})[U]\simeq
j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[V]\\\
\simeq(j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(V))\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\simeq
j_{*}(j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(V))\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\\\
\simeq j_{*}f^{\prime}_{*}f^{\prime}{}^{*}\\!\mskip
1.5muj^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\simeq
f_{*}j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\\\
\simeq f_{*}(j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]).$
As it was explained Section 2.6, the contratensor product
$f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}}$ can be computed as the inductive
limit over the diagram ${\mathbf{D}}$ formed by the affine open subschemes
$V\subset Y$ of the form $V=U\times_{X}Y$, where $U$ are affine open
subschemes in $X$. It remains to use the fact that the direct image of quasi-
coherent sheaves with respect to an affine morphism is an exact functor. The
same isomorphism (46) holds for a flat affine morphism $f$ of quasi-separated
schemes.
Furthermore, there is a natural morphism from the left-hand side to the right-
hand side of (46) for any quasi-compact morphism of quasi-separated schemes
$f\colon Y\longrightarrow X$. It is constructed as the composition
$\textstyle{\mathcal{M}}\odot_{X}f_{!}{\mathfrak{Q}}\mskip 1.5mu=\mskip
1.5mu\varinjlim_{U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[f^{-1}(U)]\mskip
1.5mu\simeq\mskip
1.5mu\varinjlim_{U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}(\varinjlim_{V\subset
f^{-1}(U)}{\mathfrak{Q}}[V])\\\ \textstyle\simeq\mskip
1.5mu\varinjlim_{U}\varinjlim_{f(V)\subset
U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[V]\mskip
1.5mu\simeq\mskip 1.5mu\varinjlim_{f(V)\subset
U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\\\
\textstyle\varinjlim_{f(V)\subset
U}j_{*}(j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(V))\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\mskip
1.5mu\simeq\mskip 1.5mu\varinjlim_{f(V)\subset
U}f_{*}j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muf_{*}\varinjlim_{f(V)\subset U}j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\mskip
1.5mu\simeq\mskip 1.5muf_{*}(f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}}).$
Here the inductive limit is taken firstly over affine open subschemes
$U\subset X$, then over affine open subschemes $V\subset Y$ such that
$f(V)\subset U$, and eventually over pairs of affine open subschemes $U\subset
X$ and $V\subset Y$ such that $f(V)\subset U$. The open embeddings
$U\longrightarrow X$ and $V\longrightarrow Y$ are denoted by $j$ and
$j^{\prime}$, while the morphism $V\longrightarrow U$ is denoted by
$f^{\prime}$. The final isomorphism holds, since the contratensor product
$f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}}$ can be computed over the diagram
${\mathbf{D}}$ formed by all the pairs of affine open subschemes $(U,V)$ such
that $f(V)\subset U$.
In particular, the isomorphism
(47) ${\mathcal{M}}\odot_{X}h_{!}{\mathfrak{F}}\simeq
h_{*}(h^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{F}})$
holds for any open embedding $h\colon Y\longrightarrow X$ of an affine scheme
$Y$ into a quasi-separated scheme $X$, any quasi-coherent sheaf
${\mathcal{M}}$ on $X$, and any flat cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathfrak{F}}$ on $Y$. Indeed, in this case one has
$\textstyle\varinjlim_{h(V)\subset
U}h_{*}j^{\prime}_{*}j^{\prime}{}^{*}h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{F}}[V]\simeq\varinjlim_{V}h_{*}j^{\prime}_{*}j^{\prime}{}^{*}h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{F}}[V]\\\
\simeq
h_{*}h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(Y)}{\mathfrak{F}}[Y]\simeq
h_{*}(h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(Y)}{\mathfrak{F}}[Y]),$
where the $\varinjlim_{V}$ is taken over all affine open subschemes $V\subset
Y$, which is clearly equivalent to considering $V=Y$ only.
Let $f\colon Y\longrightarrow X$ be an affine morphism of schemes. Then for
any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ and any cosheaf of
${\mathcal{O}}_{Y}$-modules ${\mathfrak{Q}}$ there is a natural isomorphism of
cosheaves of ${\mathcal{O}}_{X}$-modules
(48)
$f_{!}(f^{*}{\mathcal{M}}\otimes_{Y}{\mathfrak{Q}})\simeq{\mathcal{M}}\otimes_{X}f_{!}{\mathfrak{Q}}.$
This is a version of the projection formula for the tensor product of quasi-
coherent sheaves and cosheaves of ${\mathcal{O}}_{X}$-modules.
## 4\. Quasi-compact Semi-separated Schemes
### 4.1. Contraadjusted and cotorsion quasi-coherent sheaves
Recall that the definition of a very flat quasi-coherent sheaf was given in
Section 1.7 and the definition of a contraadjusted quasi-coherent sheaf in
Section 2.5 (cf. Remark 2.5.4).
In particular, a quasi-coherent sheaf ${\mathcal{P}}$ over an affine scheme
$U$ is very flat (respectively, contraadjusted) if and only if the
${\mathcal{O}}(U)$-module ${\mathcal{P}}(U)$ is very flat (respectively,
contraadjusted). The class of very flat quasi-coherent sheaves is preserved by
inverse images with respect to arbitrary morphisms of schemes and direct
images with respect to very flat affine morphisms (which includes affine open
embeddings). The class of contraadjusted quasi-coherent sheaves is preserved
by direct images with respect to affine morphisms of schemes.
The class of very flat quasi-coherent sheaves on any scheme $X$ is closed
under the passage to the kernel of a surjective morphism. Both the full
subcategories of very flat and contraadjusted quasi-coherent sheaves are
closed under extensions in the abelian category of quasi-coherent sheaves.
Hence they acquire the induced exact category structures, which we denote by
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$, respectively.
Let us introduce one bit of categorical terminology. Given an exact category
${\mathsf{E}}$ and a class of objects ${\mathsf{C}}\subset{\mathsf{E}}$, we
say that an object $X\in{\mathsf{E}}$ is a _finitely iterated extension_ of
objects from ${\mathsf{C}}$ if there exists a nonnegative integer $N$ and a
sequence of admissible monomorphisms $0=X_{0}\longrightarrow
X_{1}\longrightarrow\dotsb\longrightarrow X_{N-1}\longrightarrow X_{N}=X$ in
${\mathsf{E}}$ such that the cokernels of all the morphisms
$X_{i-1}\longrightarrow X_{i}$ belong to ${\mathsf{C}}$ (cf. Section 1.1).
Let $X$ be a quasi-compact semi-separated scheme.
###### Lemma 4.1.1.
Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{M}}\longrightarrow
0$, where ${\mathcal{F}}$ is a very flat quasi-coherent sheaf and
${\mathcal{P}}$ is a finitely iterated extension of the direct images of
contraadjusted quasi-coherent sheaves from affine open subschemes in $X$.
###### Proof.
The proof is based on the construction from [53, Section A.1] and Theorem
1.1.1(b). We argue by a kind of induction in the number of affine open
subschemes covering $X$. Assume that for some open subscheme $h\colon
V\longrightarrow X$ there is a short exact sequence
$0\longrightarrow{\mathcal{Q}}\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{M}}\longrightarrow
0$ of quasi-coherent sheaves on $X$ such that the restriction
$h^{*}{\mathcal{K}}$ of the sheaf ${\mathcal{K}}$ to the open subscheme $V$ is
very flat, while the sheaf ${\mathcal{Q}}$ is a finitely iterated extension of
the direct images of contraadjusted quasi-coherent sheaves from affine open
subschemes in $X$. Let $j\colon U\longrightarrow X$ be an affine open
subscheme; we will construct a short exact sequence
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{M}}\longrightarrow
0$ having the same properties with respect to the open subscheme $U\cup
V\subset X$.
Pick an short exact sequence
$0\longrightarrow{\mathcal{R}}\longrightarrow{\mathcal{G}}\longrightarrow
j^{*}{\mathcal{K}}\longrightarrow 0$ of quasi-coherent sheaves on the affine
scheme $U$ such that the sheaf ${\mathcal{G}}$ is very flat and the sheaf
${\mathcal{R}}$ is contraadjusted. Consider its direct image $0\longrightarrow
j_{*}{\mathcal{R}}\longrightarrow j_{*}{\mathcal{G}}\longrightarrow
j_{*}j^{*}{\mathcal{K}}\longrightarrow 0$ with respect to the affine open
embedding $j$, and take its pull-back with respect to the adjunction morphism
${\mathcal{K}}\longrightarrow j_{*}j^{*}{\mathcal{K}}$. Let ${\mathcal{F}}$
denote the middle term of the resulting short exact sequence of quasi-coherent
sheaves on $X$.
By Lemma 1.2.6(a), it suffices to show that the restrictions of
${\mathcal{F}}$ to $U$ and $V$ are very flat in order to conclude that the
restriction to $U\cup V$ is. We have $j^{*}{\mathcal{F}}\simeq{\mathcal{G}}$,
which is very flat by the construction. On the other hand, the sheaf
$j^{*}{\mathcal{K}}$ is very flat over $V\cap U$, hence so is the sheaf
${\mathcal{R}}$, as the kernel of a surjective map
${\mathcal{G}}\longrightarrow j^{*}{\mathcal{K}}$. The embedding $U\cap
V\longrightarrow V$ is a very flat affine morphism, so the sheaf
$j_{*}{\mathcal{R}}$ is very flat over $V$. Now it is clear from the short
exact sequence $0\longrightarrow
j_{*}{\mathcal{R}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{K}}\longrightarrow
0$ that the sheaf ${\mathcal{F}}$ is very flat over $V$.
Finally, the kernel ${\mathcal{P}}$ of the composition of surjective morphisms
${\mathcal{F}}\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{M}}$ is an
extension of the sheaves ${\mathcal{Q}}$ and $j_{*}{\mathcal{R}}$, the latter
of which is the direct image of a contraadjustested quasi-coherent sheaf from
an affine open subscheme of $X$, and the former is a finitely iterated
extension of such. ∎
###### Corollary 4.1.2.
(a) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is contraadjusted if and
only if the functor $\operatorname{Hom}_{X}({-},{\mathcal{P}})$ takes short
exact sequences of very flat quasi-coherent sheaves on $X$ to short exact
sequences of abelian groups.
(b) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is contraadjusted if and
only if $\operatorname{Ext}_{X}^{>0}({\mathcal{F}},{\mathcal{P}})=0$ for any
very flat quasi-coherent sheaf ${\mathcal{F}}$ on $X$.
(c) The class of contraadjusted quasi-coherent sheaves on $X$ is closed with
respect to the passage to the cokernels of injective morphisms.
###### Proof.
While the condition in part (a) is _a priori_ weaker and the condition in part
(b) is _a priori_ stronger than our definition of a contraherent cosheaf
${\mathcal{P}}$ by the condition
$\operatorname{Ext}_{X}^{1}({\mathcal{F}},{\mathcal{P}})=0$ for any very flat
${\mathcal{F}}$, all the three conditions are easily seen to be equivalent
provided that every quasi-coherent sheaf on $X$ is the quotient sheaf of a
very flat one. That much we know from Lemma 4.1.1. The condition in (b)
clearly has the property (c). ∎
###### Lemma 4.1.3.
Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{M}}\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow
0$, where ${\mathcal{F}}$ is a very flat quasi-coherent sheaf and
${\mathcal{P}}$ is a finitely iterated extension of the direct images of
contraadjusted quasi-coherent sheaves from affine open subschemes in $X$.
###### Proof.
Any quasi-coherent sheaf on a quasi-compact quasi-separated scheme can be
embedded into a finite direct sum of direct images of injective quasi-coherent
sheaves from affine open subschemes constituting a finite covering. So an
embedding ${\mathcal{M}}\longrightarrow{\mathcal{J}}$ of a sheaf
${\mathcal{M}}$ into a sheaf ${\mathcal{J}}$ with the desired (an even
stronger) properties exists, and it remains to make sure that the quotient
sheaf has the desired properties.
One does this using Lemma 4.1.1 and (the dual version of) the procedure used
in the second half of the proof of Theorem 10 in [17] (see the proof of Lemma
1.1.3). Present the quotient sheaf ${\mathcal{J}}/{\mathcal{M}}$ as the
quotient sheaf of a very flat sheaf ${\mathcal{F}}$ by a subsheaf
${\mathcal{Q}}$ representable as a finitely iterated extension of the desired
kind. Set ${\mathcal{P}}$ to be the fibered product of ${\mathcal{J}}$ and
${\mathcal{F}}$ over ${\mathcal{J}}/{\mathcal{M}}$; then ${\mathcal{P}}$ is an
extension of ${\mathcal{J}}$ and ${\mathcal{Q}}$, and there is a natural
injective morphism ${\mathcal{M}}\longrightarrow{\mathcal{P}}$ with the
cokernel ${\mathcal{F}}$. ∎
###### Corollary 4.1.4.
(a) Any quasi-coherent sheaf on $X$ admits a surjective map onto it from a
very flat quasi-coherent sheaf such that the kernel is contraadjusted.
(b) Any quasi-coherent sheaf on $X$ can be embedded into a contraadjusted
quasi-coherent sheaf in such a way that the cokernel is very flat.
(c) A quasi-coherent sheaf on $X$ is contraadjusted if and only if it is a
direct summand of a finitely iterated extension of the direct images of
contraadjusted quasi-coherent sheaves from affine open subschemes of $X$.
###### Proof.
Parts (a) and (b) follow from Lemmas 4.1.1 and 4.1.3, respectively. The proof
of part (c) uses (the dual version of) the argument from the proof of
Corollary 1.1.4. Given a contraadjusted quasi-coherent sheaf ${\mathcal{P}}$,
use Lemma 4.1.3 to embed it into a finitely iterated extension ${\mathcal{Q}}$
of the desired kind in such a way that the cokernel ${\mathcal{F}}$ is a very
flat quasi-coherent sheaf. Since
$\operatorname{Ext}^{1}_{X}({\mathcal{F}},{\mathcal{P}})=0$ by the definition,
we can conclude that ${\mathcal{P}}$ is a direct summand of ${\mathcal{Q}}$. ∎
###### Lemma 4.1.5.
A quasi-coherent sheaf on $X$ is very flat and contraadjusted if and only if
it is a direct summand of a finite direct sum of the direct images of very
flat contraadjusted quasi-coherent sheaves from affine open subschemes of $X$.
###### Proof.
The “if” assertion is clear. To prove “only if”, notice that the very flat
contraadjusted quasi-coherent sheaves are the injective objects of the exact
category of very flat quasi-coherent sheaves (cf. Section 1.4). So it remains
to show that there are enough injectives of the kind described in the
formulation of Lemma in the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$.
Indeed, let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on $X$ and
$X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Denote by
$j_{\alpha}$ the identity open embeddings $U_{\alpha}\longrightarrow X$. For
each $\alpha$, pick an injective morphism
$j_{\alpha}^{*}{\mathcal{F}}\longrightarrow{\mathcal{G}}_{\alpha}$ from a very
flat quasi-coherent sheaf $j_{\alpha}^{*}{\mathcal{F}}$ to a very flat
contraadjusted quasi-coherent sheaf ${\mathcal{G}}_{\alpha}$ on $U_{\alpha}$
such that the cokernel
${\mathcal{G}}_{\alpha}/j_{\alpha}^{*}{\mathcal{F}}_{\alpha}$ is a very flat.
Then $\bigoplus_{\alpha}j_{\alpha}{}_{*}{\mathcal{G}}_{\alpha}$ is a very flat
contraadjusted quasi-coherent sheaf on $X$ and the cokernel of the natural
morphism
${\mathcal{F}}\longrightarrow\bigoplus_{\alpha}j_{\alpha}{}_{*}{\mathcal{G}}_{\alpha}$
is very flat (since its restriction to each $U_{\alpha}$ is). ∎
###### Lemma 4.1.6.
A quasi-coherent sheaf on $X$ is flat and contraadjusted if and only if it is
a direct summand of a finitely iterated extension of the direct images of flat
contraadjusted quasi-coherent sheaves from affine open subschemes of $X$.
###### Proof.
Given a flat quasi-coherent sheaf ${\mathcal{E}}$, we apply the constructions
of Lemmas 4.1.1 and 4.1.3 in order to obtain a short exact sequence of quasi-
coherent sheaves
$0\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow
0$ with a very flat quasi-coherent sheaf ${\mathcal{F}}$. One can verify step
by step that the whole construction is performed entirely inside the exact
category of flat quasi-coherent sheaves on $X$, so the quasi-coherent sheaf
${\mathcal{P}}$ it produces is a finitely iterated extension of the direct
images of flat contraadjusted quasi-coherent sheaves from affine open
subschemes of $X$. Now if the sheaf ${\mathcal{E}}$ was also contraadjusted,
then the short exact sequence splits by Corollary 4.1.2(b), providing the
desired result. ∎
The following corollary provides equivalent definitions of contraadjusted and
very flat quasi-coherent sheaves on a quasi-compact semi-separated scheme
resembling the corresponding definitions for modules over a ring in Section
1.1.
###### Corollary 4.1.7.
(a) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is contraadjusted if and
only if
$\operatorname{Ext}_{X}^{>0}(j_{*}j^{*}{\mathcal{O}}_{X},{\mathcal{P}})=0$ for
any affine open embedding of schemes $j\colon Y\longrightarrow X$.
(b) A quasi-coherent sheaf ${\mathcal{F}}$ on $X$ is very flat if and only if
$\operatorname{Ext}_{X}^{1}({\mathcal{F}},{\mathcal{P}})=0$ for any
contraadjusted quasi-coherent sheaf ${\mathcal{P}}$ on $X$.
###### Proof.
Part (a): the “only if” assertion follows from Corollary 4.1.2(b). To prove
“if”, notice that any very flat sheaf ${\mathcal{F}}$ on $X$ has a finite
right Čech resolution by finite direct sums of sheaves of the form
$j_{*}j^{*}{\mathcal{F}}$, where $j\colon U\longrightarrow X$ are embeddings
of affine open subschemes. Hence the condition
$\operatorname{Ext}_{X}^{>0}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}})=0$ for all
such $j$ implies $\operatorname{Ext}_{X}^{>0}({\mathcal{F}},{\mathcal{P}})=0$.
Furthermore, a very flat quasi-coherent sheaf $j^{*}{\mathcal{F}}$ on $U$ is a
direct summand of a transfinitely iterated extension of the direct images of
the structure sheaves of principal affine open subschemes $V\subset U$ (by
Corollary 1.1.4). Since the direct images with respect to affine morphisms
preserve transfinitely iterated extensions, it remains to use the quasi-
coherent sheaf version of the result that
$\operatorname{Ext}^{1}$-orthogonality is preserved by transfinitely iterated
extensions in the first argument [17, Lemma 1].
Part (b): “only if” holds by the definition of contraadjusted sheaves, and
“if” can be deduced from Corollary 4.1.4(a) by an argument similar to the
proof of Corollary 1.1.4 (and dual to that of Corollary 4.1.4(c)). ∎
Now we proceed to formulate the analogues of the above assertions for
cotorsion quasi-coherent sheaves. The definition of these was given in Section
2.5. The class of cotorsion quasi-coherent sheaves is closed under extensions
in the abelian category of quasi-coherent sheaves on a scheme and under the
direct images with respect to affine morphisms of schemes. We denote the
induced exact category structure on the category of of cotorsion quasi-
coherent sheaves on a scheme $X$ by
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$.
As above, in the sequel $X$ denotes a quasi-compact semi-separated scheme.
###### Lemma 4.1.8.
Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{M}}\longrightarrow
0$, where ${\mathcal{F}}$ is a flat quasi-coherent sheaf and ${\mathcal{P}}$
is a finitely iterated extension of the direct images of cotorsion quasi-
coherent sheaves from affine open subschemes in $X$.
###### Proof.
Similar to that of Lemma 4.1.1, except that Theorem 1.3.1(b) is being used in
place of Theorem 1.1.1(b). ∎
###### Corollary 4.1.9.
(a) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is cotorsion if and only if
the functor $\operatorname{Hom}_{X}({-},{\mathcal{P}})$ takes short exact
sequences of flat quasi-coherent sheaves on $X$ to short exact sequences of
abelian groups.
(b) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is cotorsion if and only if
$\operatorname{Ext}_{X}^{>0}({\mathcal{F}},{\mathcal{P}})=0$ for any flat
quasi-coherent sheaf ${\mathcal{F}}$ on $X$.
(c) The class of cotorsion quasi-coherent sheaves on $X$ is closed with
respect to the passage to the cokernels of injective morphisms.
###### Proof.
Similar to that of Corollary 4.1.2. ∎
###### Lemma 4.1.10.
Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{M}}\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow
0$, where ${\mathcal{F}}$ is a flat quasi-coherent sheaf and ${\mathcal{P}}$
is a finitely iterated extension of the direct images of cotorsion quasi-
coherent sheaves from affine open subschemes in $X$.
###### Proof.
Similar to that of Lemma 4.1.3. ∎
###### Corollary 4.1.11.
(a) Any quasi-coherent sheaf on $X$ admits a surjective map onto it from a
flat quasi-coherent sheaf such that the kernel is cotorsion.
(b) Any quasi-coherent sheaf on $X$ can be embedded into a cotorsion quasi-
coherent sheaf in such a way that the cokernel is flat.
(c) A quasi-coherent sheaf on $X$ is cotorsion if and only if it is a direct
summand of a finitely iterated extension of the direct images of cotorsion
quasi-coherent sheaves from affine open subschemes of $X$.
###### Proof.
Similar to that of Corollary 4.1.4. ∎
###### Lemma 4.1.12.
A quasi-coherent sheaf on $X$ is flat and cotorsion if and only if it is a
direct summand of a finite direct sum of the direct images of flat cotorsion
quasi-coherent sheaves from affine open subschemes of $X$.
###### Proof.
Similar to that of Lemma 4.1.5. ∎
The following result shows that contraadjusted (and in particular, cotorsion)
quasi-coherent sheaves are adjusted to direct images with respect to nonaffine
morphisms of quasi-compact semi-separated schemes (cf. Corollary 4.5.3 below).
###### Corollary 4.1.13.
Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then
(a) the functor $f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ takes the full exact subcategory
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
Y{\operatorname{\mathsf{--qcoh}}}$ into the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$, and induces an exact functor between these
exact categories;
(b) the functor $f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ takes the full exact subcategory
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
Y{\operatorname{\mathsf{--qcoh}}}$ into the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
X{\operatorname{\mathsf{--qcoh}}}$, and induces an exact functor between these
exact categories.
###### Proof.
For any affine morphism $g\colon V\longrightarrow Y$ into a quasi-compact
semi-separated scheme $Y$, the inverse image functor $g^{*}$ takes quasi-
coherent sheaves that can be represented as finitely iterated extensions of
the direct images of quasi-coherent sheaves from affine open subschemes in $Y$
to quasi-coherent sheaves of the similar type on $V$. This follows easily from
the fact that direct images of quasi-coherent sheaves with respect to affine
morphisms of schemes commute with inverse images in the base change
situations. In particular, it follows from Corollary 4.1.4(c) that the functor
$g^{*}$ takes contraadjusted quasi-coherent sheaves on $Y$ to quasi-coherent
sheaves that are direct summands of finitely iterated extensions of the direct
images of quasi-coherent sheaves from affine open subschemes $W\subset V$.
The quasi-coherent sheaves on $V$ that can be represented as such iterated
extensions form a full exact subcategory in the abelian category of quasi-
coherent sheaves. The functor of global sections $\Gamma(V,{-})$ is exact on
this exact category. Indeed, there is a natural isomorphism of the Ext groups
$\operatorname{Ext}_{V}^{*}({\mathcal{F}},h_{*}{\mathcal{G}})\simeq\operatorname{Ext}_{W}^{*}(h^{*}{\mathcal{F}},{\mathcal{G}})$
for any quasi-coherent sheaves ${\mathcal{F}}$ on $V$ and ${\mathcal{G}}$ on
$W$, and a flat affine morphism $h\colon W\longrightarrow V$. Applying this
isomorphism in the case when $h$ is the embedding of an affine open subscheme
and ${\mathcal{F}}={\mathcal{O}}_{V}$, one concludes that
$\operatorname{Ext}_{V}^{>0}({\mathcal{O}}_{V},{\mathcal{G}})=0$ for all
quasi-coherent sheaves ${\mathcal{G}}$ from the exact category in question.
Specializing to the case of the open subschemes $V=U\times_{X}Y\subset Y$,
where $U$ are affine open subschemes in $X$, we deduce the assertion that the
functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is exact. It remains to recall that the
direct images of contraadjusted quasi-coherent sheaves with respect to affine
morphisms of schemes are contraadjusted in order to show that $f_{*}$ takes
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$. Since the direct images of
cotorsion quasi-coherent sheaves with respect to affine morphisms of schemes
are cotorsion, it similarly follows that $f_{*}$ takes
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$. ∎
### 4.2. Colocally projective contraherent cosheaves
Let $X$ be a scheme and ${\mathbf{W}}$ be its open covering. A
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ is called
_colocally projective_ if for any short exact sequence
$0\longrightarrow{\mathfrak{I}}\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{K}}\longrightarrow
0$ of locally injective ${\mathbf{W}}$-locally contraherent cosheaves on $X$
the short sequence of abelian groups
$0\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}},{\mathfrak{I}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}},{\mathfrak{J}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}},{\mathfrak{K}})\longrightarrow
0$ is exact.
Obviously, the class of colocally projective ${\mathbf{W}}$-locally
contraherent cosheaves on $X$ is closed under direct summands. It follows from
the adjunction isomorphism (23) of Section 3.3 that the functor of direct
image of ${\mathbf{T}}$-locally contraherent cosheaves $f_{!}$ with respect to
any $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism of schemes $f\colon
Y\longrightarrow X$ takes colocally projective ${\mathbf{T}}$-locally
contraherent cosheaves on $Y$ to colocally projective ${\mathbf{W}}$-locally
contraherent cosheaves on $X$. It is also clear that _any_ contraherent
cosheaf on an affine scheme $U$ with the covering $\\{U\\}$ is colocally
projective.
###### Lemma 4.2.1.
On any scheme $X$ with an open covering ${\mathbf{W}}$, any coflasque
contraherent cosheaf is colocally projective.
###### Proof.
We will prove a somewhat stronger assertion: any short exact sequence
$0\longrightarrow{\mathfrak{I}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ with ${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$ and ${\mathfrak{I}}\in
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ splits. It will
follow easily that the functor $\operatorname{Hom}^{X}({\mathfrak{F}},{-})$
takes any short exact sequence in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ to a short
exact sequence of abelian groups.
We proceed by applying Zorn’s lemma to the partially ordered set of sections
$\phi_{Y}\colon{\mathfrak{F}}|_{Y}\longrightarrow{\mathfrak{Q}}|_{Y}$ of the
morphism of cosheaves ${\mathfrak{Q}}\longrightarrow{\mathfrak{F}}$ defined
over open subsets $Y\subset X$. Since it suffices to define a morphism of
cosheaves on the modules of cosections over affine open subschemes, which are
quasi-compact, a compatible system of sections $\phi_{Y_{i}}$ defined over a
linearly ordered family of open subsets $Y_{i}\subset X$ extends uniquely to a
section over the union $\bigcup_{i}Y_{i}$.
Now let $\phi_{Y}$ be a section over $Y$ and $U\subset X$ be an affine open
subscheme subordinate to ${\mathbf{W}}$. Set $V=Y\cap U$; by assumption, the
${\mathcal{O}}(U)$-module homomorphism
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective and the
${\mathcal{O}}(U)$-module ${\mathfrak{I}}[U]$ is injective. The short exact
sequence of ${\mathcal{O}}(U)$-modules
$0\longrightarrow{\mathfrak{I}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{F}}[U]\longrightarrow
0$ splits, and the difference between two such splittings
${\mathfrak{F}}[U]\birarrow{\mathfrak{Q}}[U]$ is an arbitrary
${\mathcal{O}}(U)$-module morphism
${\mathfrak{F}}[U]\longrightarrow{\mathfrak{I}}[U]$. The composition
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{Q}}[U]$ of the morphism
$\phi_{Y}[V]\colon{\mathfrak{F}}[V]\longrightarrow{\mathfrak{Q}}[V]$ and the
corestriction morphism ${\mathfrak{Q}}[V]\longrightarrow{\mathfrak{Q}}[U]$ can
therefore be extended to an ${\mathcal{O}}(U)$-linear section
${\mathfrak{F}}[U]\longrightarrow{\mathfrak{Q}}[U]$ of the surjection
${\mathfrak{Q}}[U]\longrightarrow{\mathfrak{F}}[U]$.
We have constructed a morphism of contraherent cosheaves
$\phi_{U}\colon{\mathfrak{F}}|_{U}\longrightarrow{\mathfrak{Q}}|_{U}$ whose
restriction to $V$ coincides with the restriction of the morphism
$\phi_{Y}\colon{\mathfrak{F}}|_{Y}\longrightarrow{\mathfrak{Q}}|_{Y}$. Set
$Z=Y\cup U$; the pair of morphisms of cosheaves $\phi_{Y}$ and $\phi_{U}$
extends uniquely to a morphism of cosheaves
$\phi_{Z}\colon{\mathfrak{F}}|_{Z}\longrightarrow{\mathfrak{Q}}|_{Z}$. Since
the morphisms $\phi_{Y}$ and $\phi_{U}$ were some sections of of the
surjection ${\mathfrak{Q}}\longrightarrow{\mathfrak{F}}$ over $Y$ and $U$, the
morphism $\phi_{Z}$ is a section of this surjection over $Z$. ∎
Generally speaking, according to the above definition the colocal projectivity
property of a locally contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$
may depend not only on the cosheaf ${\mathfrak{P}}$ itself, but also on the
covering ${\mathbf{W}}$. No such dependence occurs on quasi-compact semi-
separated schemes. Indeed, we will see below in this section that on such a
scheme any colocally projective ${\mathbf{W}}$-locally contraherent cosheaf is
(globally) contraherent. Moreover, the class of colocally projective
${\mathbf{W}}$-locally contraherent cosheaves coincides with the class of
colocally projective contraherent cosheaves and does not depend on the
covering ${\mathbf{W}}$.
Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
open covering.
###### Lemma 4.2.2.
Let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then
(a) any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$
can be included in an exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves from the affine open
subschemes $U_{\alpha}\subset X$;
(b) any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of locally cotorsion contraherent cosheaves
from the affine open subschemes $U_{\alpha}\subset X$.
###### Proof.
The argument is a dual version of the proofs of Lemmas 4.1.1 and 4.1.8. Let us
prove part (a); the proof of part (b) is completely analogous.
Arguing by induction in $1\le\beta\le N$, we consider the open subscheme
$V=\bigcup_{\alpha<\beta}U_{\alpha}$ with the induced covering
${\mathbf{W}}|_{V}=\\{V\cap W\mid W\in{\mathbf{W}}\\}$ and the identity
embedding $h\colon V\longrightarrow X$. Assume that we have constructed an
exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{Q}}\longrightarrow
0$ of ${\mathbf{W}}$-locally contraherent cosheaves on $X$ such that the
restriction $h^{!}{\mathfrak{K}}$ of the ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{K}}$ to the open subscheme $V\subset X$ is locally
injective, while the cosheaf ${\mathfrak{Q}}$ on $X$ is a finitely iterated
extension of the direct images of contraherent cosheaves from the affine open
subschemes $U_{\alpha}\subset X$, $\alpha<\beta$. When $\beta=1$, it suffices
to take ${\mathfrak{K}}={\mathfrak{M}}$ and ${\mathfrak{Q}}=0$ for the
induction base. Set $U=U_{\beta}$ and denote by $j\colon U\longrightarrow X$
the identity open embedding.
Let $0\longrightarrow
j^{!}{\mathfrak{K}}\longrightarrow{\mathfrak{I}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ be an exact triple of contraherent cosheaves on the affine scheme $U$ such
that the contraherent cosheaf ${\mathfrak{I}}$ is (locally) injective.
Consider its direct image $0\longrightarrow
j_{!}j^{!}{\mathfrak{K}}\longrightarrow j_{!}{\mathfrak{I}}\longrightarrow
j_{!}{\mathfrak{R}}\longrightarrow 0$ with respect to the affine open
embedding $j$, and take its push-forward with respect to the adjunction
morphism $j_{!}j^{!}{\mathfrak{K}}\longrightarrow{\mathfrak{K}}$. Let us show
that in the resulting exact triple
$0\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{J}}\longrightarrow
j_{!}{\mathfrak{R}}\longrightarrow 0$ the ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{J}}$ is locally injective in the restriction to $U\cup V$.
By Lemma 1.3.6(b), it suffices to show that the restrictions of
${\mathfrak{J}}$ to $U$ and $V$ are locally injective.
Indeed, in the restriction to $U$ we have $j^{!}j_{!}j^{!}{\mathfrak{K}}\simeq
j^{!}{\mathfrak{K}}$, hence $j^{!}{\mathfrak{J}}\simeq
j^{!}j_{!}{\mathfrak{I}}\simeq{\mathfrak{I}}$ is a (locally) injective
contraherent cosheaf. On the other hand, if $j^{\prime}\colon U\cap
V\longrightarrow V$ and $h^{\prime}\colon U\cap V\longrightarrow U$ denote the
embeddings of $U\cap V$, then $h^{!}j_{!}{\mathfrak{R}}\simeq
j^{\prime}_{!}h^{\prime}{}^{!}{\mathfrak{R}}$ (as explained in the end of
Section 3.3). Notice that the contraherent cosheaf
$h^{\prime}{}^{!}j^{!}{\mathfrak{K}}\simeq
j^{\prime}{}^{!}h^{!}{\mathfrak{K}}$ is locally injective, hence the
contraherent cosheaf $h^{\prime}{}^{!}{\mathfrak{R}}$ is locally injective as
the cokernel of the admissible monomorphism of locally injective contraherent
cosheaves $h^{\prime}{}^{!}j^{!}{\mathfrak{K}}\longrightarrow
h^{\prime}{}^{!}{\mathfrak{I}}$. Since the local injectivity of
${\mathbf{T}}$-locally contraherent cosheaves is preserved by the direct
images with respect to flat $({\mathbf{W}},{\mathbf{T}})$-affine morphisms,
the contraherent cosheaf $j^{\prime}_{!}h^{\prime!}{\mathfrak{R}}$ is locally
injective, too. Now in the exact triple $0\longrightarrow
h^{!}{\mathfrak{K}}\longrightarrow h^{!}{\mathfrak{J}}\longrightarrow
h^{!}j_{!}{\mathfrak{R}}\longrightarrow 0$ of ${\mathbf{W}}|_{V}$-locally
contraherent cosheaves on $V$ the middle term is locally injective, because so
are the other two terms.
Finally, the composition of admissible monomorphisms of ${\mathbf{W}}$-locally
contraherent cosheaves
${\mathfrak{M}}\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{J}}$ on
$X$ is an admissible monomorphism with the cokernel isomorphic to an extension
of the contraherent cosheaves $j_{!}{\mathfrak{R}}$ and ${\mathfrak{Q}}$,
hence also a finitely iterated extension of the direct images of contraherent
cosheaves from the affine open subschemes $U_{\alpha}\subset X$,
$\alpha\le\beta$. The induction step is finished, and the whole lemma is
proven. ∎
We denote by $\operatorname{Ext}^{X,*}({-},{-})$ the $\operatorname{Ext}$
groups in the exact category of ${\mathbf{W}}$-locally contraherent cosheaves
on $X$. Notice that these do not depend on the covering ${\mathbf{W}}$ and
coincide with the $\operatorname{Ext}$ groups in the whole category of locally
contraherent cosheaves $X{\operatorname{\mathsf{--lcth}}}$. Indeed, the full
exact subcategory $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is closed
under extensions and the passage to kernels of admissible epimorphisms in
$X{\operatorname{\mathsf{--lcth}}}$ (see Section 3.2), and for any object
${\mathfrak{P}}\in X{\operatorname{\mathsf{--lcth}}}$ there exists an
admissible epimorphism onto ${\mathfrak{P}}$ from an object of
$X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see the resolution (27) in
Section 3.3).
For the same reasons (up to duality), the $\operatorname{Ext}$ groups computed
in the exact subcategories of locally cotorsion and locally injective
${\mathbf{W}}$-locally contraherent cosheaves
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ agree with
those in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (and also in
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$). Indeed, the full exact
subcategories $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
and $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ are closed
under extensions and the passage to cokernels of admissible monomorphisms in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Section 3.1), and we
have just constructed in Lemma 4.2.2 an admissible monomorphism from any
${\mathbf{W}}$-locally contraherent cosheaf to a locally injective one. We
refer to Sections A.2–A.3 for further details.
###### Corollary 4.2.3.
(a) A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ is
colocally projective if and only if
$\operatorname{Ext}^{X,1}({\mathfrak{P}},{\mathfrak{J}})=0$ and if and only if
$\operatorname{Ext}^{X,>0}({\mathfrak{P}},{\mathfrak{J}})=0$ for all locally
injective ${\mathbf{W}}$-locally contraherent cosheaves ${\mathfrak{J}}$ on
$X$.
(b) The class of colocally projective ${\mathbf{W}}$-locally contraherent
cosheaves on $X$ is closed under extensions and the passage to kernels of
admissible epimorphisms in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
###### Proof.
Part (a) follows from the existence of an admissible monomorphism from any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ into a locally injective
${\mathbf{W}}$-locally contraherent cosheaf (a weak form of Lemma 4.2.2(a)).
Part (b) follows from part (a). ∎
###### Lemma 4.2.4.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then
(a) any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$
can be included in an exact triple
$0\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves from the affine open
subschemes $U_{\alpha}\subset X$;
(b) any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of locally cotorsion contraherent cosheaves
from the affine open subschemes $U_{\alpha}\subset X$.
###### Proof.
There is an admissible epimorphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}j_{\alpha}^{!}{\mathfrak{M}}\longrightarrow{\mathfrak{M}}$
(see (27) for the notation and explanation) onto any ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{M}}$ from a finite direct sum of the direct
images of contraherent cosheaves from the affine open subschemes $U_{\alpha}$.
When ${\mathfrak{M}}$ is a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf, this is an admissible epimorphism in the category of
locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves, and
$j_{\alpha}^{!}{\mathfrak{M}}$ are locally cotorsion contraherent cosheaves on
$U_{\alpha}$.
Given that, the desired exact triples in Lemma can be obtained from those of
Lemma 4.2.2 by the construction from the second half of the proof of Theorem
10 in [17] (see the proof of Lemma 1.1.3; cf. the proofs of Lemmas 4.1.3 and
4.1.10). ∎
###### Corollary 4.2.5.
(a) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible monomorphism from ${\mathfrak{M}}$ into a
locally injective ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{J}}$
on $X$ such that the cokernel ${\mathfrak{P}}$ is a colocally projective
${\mathbf{W}}$-locally contraherent cosheaf.
(b) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible epimorphism onto ${\mathfrak{M}}$ from a
colocally projective ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{P}}$ on $X$ such that the kernel ${\mathfrak{J}}$ is a locally
injective ${\mathbf{W}}$-locally contraherent cosheaf.
(c) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ is colocally projective if and only if it is (a contraherent
cosheaf and) a direct summand of a finitely iterated extension of the direct
images of contraherent cosheaves from the affine open subschemes
$U_{\alpha}\subset X$.
###### Proof.
The “if” assertion in part (c) follows from Corollary 4.2.3(b) together with
our preliminary remars in the beginning of this section. This having been
shown, part (a) follows from Lemma 4.2.2(a) and part (b) from Lemma 4.2.4(a).
The “only if” assertion in (c) follows from Corollary 4.2.3(a) and Lemma
4.2.4(a) by the argument from the proof of Corollary 1.1.4 (cf. Corollaries
4.1.4(c) and 4.1.11(c)). Notice that the functors of direct image with respect
to the open embeddings $U_{\alpha}\longrightarrow X$ take contraherent
cosheaves to contraherent cosheaves, and the full subcategory of contraherent
cosheaves $X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}$ is closed under extensions. ∎
By a colocally projective locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf we will mean a ${\mathbf{W}}$-locally contraherent
cosheaf that is simultaneously colocally projective and locally cotorsion.
###### Corollary 4.2.6.
(a) For any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ there exists an admissible monomorphism from
${\mathfrak{M}}$ into a locally injective ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{J}}$ on $X$ such that the cokernel ${\mathfrak{P}}$ is a
colocally projective locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf.
(b) For any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ there exists an admissible epimorphism onto
${\mathfrak{M}}$ from a colocally projective locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ such that
the kernel ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf.
(c) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ is colocally projective if and only if it is (a
contraherent cosheaf and) a direct summand of a finitely iterated extension of
the direct images of locally cotorsion contraherent cosheaves from the affine
open subschemes $U_{\alpha}\subset X$.
###### Proof.
Same as Corollary 4.2.5, except that parts (b) of Lemmas 4.2.2 and 4.2.4 need
to be used. Parts (a-b) can be also easily deduced from Corollary 4.2.5(a-b).
∎
###### Corollary 4.2.7.
The full subcategory of colocally projective ${\mathbf{W}}$-locally
contraherent cosheaves in the exact category of all locally contraherent
cosheaves on $X$ does not depend on the choice of the open covering
${\mathbf{W}}$.
###### Proof.
Given two open coverings ${\mathbf{W}}^{\prime}$ and
${\mathbf{W}}^{\prime\prime}$ of the scheme $X$, pick a finite affine open
covering $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ subordinate to both
${\mathbf{W}}^{\prime}$ and ${\mathbf{W}}^{\prime\prime}$, and apply Corollary
4.2.5(c). ∎
As a full subcategory closed under extensions and kernels of admissible
epimorphisms in $X{\operatorname{\mathsf{--ctrh}}}$, the category of colocally
projective contraherent cosheaves on $X$ acquires the induced exact category
structure. We denote this exact category by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. The (similarly
constructed) exact category of colocally projective locally cotorsion
contraherent cosheaves on $X$ is denoted by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$.
The full subcategory of contraherent sheaves that are simultaneously colocally
projective and locally injective will be denoted by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$. Clearly,
any extension of two objects from this subcategory is trivial in
$X{\operatorname{\mathsf{--ctrh}}}$, so the category of colocally projective
locally injective contraherent cosheaves is naturally viewed as an additive
category endowed with the trivial exact category structure.
It follows from Corollary 4.2.5(a-b) that the additive category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ is
simultaneously the category of projective objects in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ and the
category of injective objects in
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, and that it has enough of
both such projectives and injectives.
###### Corollary 4.2.8.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Then a
contraherent cosheaf on $X$ is colocally projective and locally injective if
and only if it is isomorphic to a direct summand of a finite direct sum of the
direct images of (locally) injective contraherent cosheaves from the open
embeddings $U_{\alpha}\longrightarrow X$.
###### Proof.
For any locally injective ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{J}}$ on $X$, the map
$\bigoplus_{\alpha}j_{\alpha}{}_{!}j_{\alpha}^{!}{\mathfrak{J}}\longrightarrow{\mathfrak{J}}$
is an admissible epimorphism in the category of locally injective
${\mathbf{W}}$-locally contraherent cosheaves. Now if ${\mathfrak{J}}$ is also
colocally projective, then the extension splits. ∎
###### Corollary 4.2.9.
The three full subcategories of colocally projective cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$,
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$, and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$ are closed
with respect to infinite products in the category
$X{\operatorname{\mathsf{--ctrh}}}$.
###### Proof.
The assertions are easily deduced from the descriptions of the full
subcategories of colocally projective cosheaves given in Corollaries 4.2.5(c),
4.2.6(c), and 4.2.8 together with the fact that the functor of direct image of
contraherent cosheaves with respect to an affine morphism of schemes preserves
infinite products. ∎
###### Corollary 4.2.10.
Let $f\colon Y\longrightarrow X$ be an affine morphism of quasi-compact semi-
separated schemes. Then
(a) the functor of inverse image of locally injective locally contraherent
cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ takes the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$ into
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$;
(b) assuming that the morphism $f$ is also flat, the functor of inverse image
of locally cotorsion locally contraherent cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ takes the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$ into
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$;
(c) assuming that the morphism $f$ is also very flat, the functor of inverse
image of locally contraherent cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}$ takes the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ into
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.
###### Proof.
Parts (a-c) follow from Corollaries 4.2.5(c), 4.2.6(c), and 4.2.8,
respectively, together with the base change results from the second half of
Section 3.3. ∎
### 4.3. Colocally flat contraherent cosheaves
Let $X$ be a scheme and ${\mathbf{W}}$ be its open covering. A
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ is called
_colocally flat_ if for any short exact sequence
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves on $X$
the short sequence of abelian groups
$0\longrightarrow\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{P}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{Q}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{R}})\longrightarrow
0$ is exact.
Let us issue a _warning_ that our terminology is misleading: the colocal
flatness is, by the definition, a stronger condition that the colocal
projectivity. It follows from the adjunction isomorphism (24) that the functor
of direct image of ${\mathbf{T}}$-locally contraherent cosheaves $f_{!}$ with
respect a flat $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism of schemes $f\colon
Y\longrightarrow X$ takes colocally flat ${\mathbf{T}}$-locally contraherent
cosheaves on $Y$ to colocally flat ${\mathbf{W}}$-locally contraherent
cosheaves on $X$. Clearly, a contraherent cosheaf ${\mathfrak{F}}$ on an
affine scheme $U$ with the covering $\\{U\\}$ is colocally flat whenever the
contraadjusted ${\mathcal{O}}(U)$-module ${\mathfrak{F}}[U]$ is flat; the
converse assertion can be deduced from Theorem 1.3.1(b) (cf. Corollary
4.3.4(c) below).
Let $X$ be a quasi-compact semi-separated scheme. It follows from the results
of Section 4.2 that any colocally flat ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ is contraherent. We will see below in this section that the
class of colocally flat ${\mathbf{W}}$-locally contraherent cosheaves on $X$
coincides with the class of colocally flat contraherent cosheaves and does not
depend on the covering ${\mathbf{W}}$.
###### Lemma 4.3.1.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$, where ${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{F}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves on $U_{\alpha}$
corresponding to flat contraadjusted ${\mathcal{O}}(U_{\alpha})$-modules.
###### Proof.
Similar to the proof of Lemma 4.2.2, except that Theorem 1.3.1(a) needs to be
used to resolve a contraherent cosheaf on an affine open subscheme $U\subset
X$ (cf. the proof of Lemma 4.1.1). Besides, one has to use Lemma 1.3.6(a) and
the fact that the class of locally cotorsion contraherent cosheaves is
preserved by direct images with respect to affine morphisms. ∎
###### Corollary 4.3.2.
(a) A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ is
colocally flat if and only if
$\operatorname{Ext}^{X,1}({\mathfrak{F}},{\mathfrak{P}})=0$ and if and only if
$\operatorname{Ext}^{X,>0}({\mathfrak{F}},{\mathfrak{P}})=0$ for all locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves ${\mathfrak{P}}$ on
$X$.
(b) The class of colocally flat ${\mathbf{W}}$-locally contraherent cosheaves
on $X$ is closed under extensions and the passage to kernels of admissible
epimorphisms in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
###### Proof.
Similar to the proof of Corollary 4.2.3. ∎
###### Lemma 4.3.3.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where ${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{F}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves on $U_{\alpha}$
corresponding to flat contraadjusted ${\mathcal{O}}(U_{\alpha})$-modules.
###### Proof.
The proof is similar to that of Lemma 4.2.4 and based on Lemma 4.3.1. The key
is to show that there is an admissible epimorphism in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ onto any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ from a finitely
iterated extension (in fact, even a finite direct sum) of the direct images of
contraherent cosheaves on $U_{\alpha}$ corresponding to flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.
Here it suffices to pick admissible epimorphisms from such contraherent
cosheaves ${\mathfrak{F}}_{\alpha}$ on $U_{\alpha}$ onto the restrictions
$j_{\alpha}^{!}{\mathfrak{M}}$ of ${\mathfrak{M}}$ to $U_{\alpha}$ and
consider the corresponding morphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{M}}$
of ${\mathbf{W}}$-locally contraherent cosheaves on $X$. To check that this is
an admissible epimorphism, one can, e. g., notice that it is so in the
restriction to each $U_{\alpha}$ and recall that being an admissible
epimorphism of ${\mathbf{W}}$-locally contraherent cosheaves is a local
property (see Section 3.2). ∎
###### Corollary 4.3.4.
(a) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible monomorphism from ${\mathfrak{M}}$ into a
locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$
on $X$ such that the cokernel ${\mathfrak{F}}$ is a colocally flat
${\mathbf{W}}$-locally contraherent cosheaf.
(b) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible epimorphism onto ${\mathfrak{M}}$ from a
colocally flat ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on
$X$ such that the kernel ${\mathfrak{P}}$ is a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf.
(c) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ is colocally flat if and only if it is a direct summand of a
finitely iterated extension of the direct images of contraherent cosheaves on
$U_{\alpha}$ corresponding to flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.
###### Proof.
Similar to the proof of Corollary 4.2.5 and based on Lemmas 4.3.1, 4.3.3 and
Corollary 4.3.2. ∎
###### Corollary 4.3.5.
The full subcategory of colocally flat ${\mathbf{W}}$-locally contraherent
cosheaves in the exact category $X{\operatorname{\mathsf{--lcth}}}$ does not
depend on the open covering ${\mathbf{W}}$.
###### Proof.
Similar to the proof of Corollary 4.2.7 and based on Corollary 4.3.4(c). ∎
As a full subcategory closed under extensions and kernels of admissible
epimorphisms in $X{\operatorname{\mathsf{--ctrh}}}$, the category of colocally
flat contraherent cosheaves on $X$ acquires the induced exact category
structure, which we denote by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$.
The following corollary is to be compared with Corollaries 5.2.2(a) and
5.2.9(a).
###### Corollary 4.3.6.
Any colocally flat contraherent cosheaf over a semi-separated Noetherian
scheme is flat.
###### Proof.
Follows from Corollary 4.3.4(c) together with the remarks about flat
contraherent cosheaves over affine Noetherian schemes and the direct images of
flat cosheaves of ${\mathcal{O}}$-modules in Section 3.7. ∎
###### Corollary 4.3.7.
Over a semi-separated Noetherian scheme $X$, the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ is closed with respect to
infinite products in $X{\operatorname{\mathsf{--ctrh}}}$.
###### Proof.
In addition to what has been said in the proof of Corollary 4.2.9, it is also
important here that infinite products of flat modules over a coherent ring are
flat. ∎
###### Corollary 4.3.8.
Let $X$ be a semi-separated Noetherian scheme and $j\colon Y\longrightarrow X$
be an affine open embedding. Then the inverse image functor $j^{!}$ takes
colocally flat contraherent cosheaves to colocally flat contraherent
cosheaves.
###### Proof.
Similar to the proof of Corollary 4.2.10 and based on Corollary 4.3.4(c); the
only difference is that one also has to use Corollary 1.6.5(a). ∎
### 4.4. Projective contraherent cosheaves
Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
affine open covering.
###### Lemma 4.4.1.
(a) The exact category of ${\mathbf{W}}$-locally contraherent cosheaves on $X$
has enough projective objects.
(b) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ is projective if and only if it is a direct summand of a direct
sum over $\alpha$ of the direct images of contraherent cosheaves on
$U_{\alpha}$ corresponding to very flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.
###### Proof.
The assertion “if” in part (b) follows from the adjunction of the direct and
inverse image functors for the embeddings $U_{\alpha}\longrightarrow X$
together with the fact that the very flat contraadjusted modules are the
projective objects of the exact categories of contraadjusted modules over
${\mathcal{O}}(U_{\alpha})$ (see Section 1.4).
It remains to show that there exists an admissible epimorphism onto any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$ from a
direct sum of the direct images of contraherent cosheaves on $U_{\alpha}$
corresponding to very flat contraadjusted modules. The construction is similar
to the one used in the proof of Lemma 4.3.3 and based on Theorem 1.1.1(b). One
picks admissible epimorphisms from contraherent cosheaves
${\mathfrak{F}}_{\alpha}$ of the desired kind on the affine schemes
$U_{\alpha}$ onto the restrictions $j_{\alpha}^{!}{\mathfrak{M}}$ of the
cosheaf ${\mathfrak{M}}$ and considers the corresponding morphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{M}}$.
∎
###### Corollary 4.4.2.
(a) There are enough projective objects in the exact category
$X{\operatorname{\mathsf{--lcth}}}$ of locally contraherent cosheaves on $X$,
and all these projective objects belong to the full subcategory of
contraherent cosheaves $X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}$.
(b) The full subcategories of projective objects in the three exact categories
$X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$ coincide. ∎
###### Lemma 4.4.3.
(a) The exact category of locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaves on $X$ has enough projective objects.
(b) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ is projective if and only if it is a direct
summand of a direct sum over $\alpha$ of the direct images of locally
cotorsion contraherent cosheaves on $U_{\alpha}$ corresponding to flat
cotorsion ${\mathcal{O}}(U_{\alpha})$-modules.
###### Proof.
Similar to the proof of Lemma 4.4.1 and based on Theorem 1.3.1(b). ∎
###### Corollary 4.4.4.
(a) There are enough projective objects in the exact category
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ of locally cotorsion
locally contraherent cosheaves on $X$, and all these projective objects belong
to the full subcategory of locally cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$.
(b) The full subcategories of projective objects in the three exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ coincide. ∎
We denote the additive category of projective (objects in the category of)
contraherent cosheaves on $X$ by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$, and the additive category
of projective (objects in the category of) locally cotorsion contraherent
cosheaves $X$ by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
Let us issue a _warning_ that both the terminology and notation are misleading
here: a projective locally cotorsion contraherent cosheaf on $X$ does _not_
have to be a projective contraherent cosheaf. Indeed, a flat cotorsion module
over a commutative ring would not be in general very flat.
On the other hand we notice that, by the definition, both additive categories
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ are
contained in the exact category of colocally flat contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ (and consequently also in
the exact category of colocally projective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$). Moreover, by the
definition one clearly has
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}=X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\cap
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$. Finally, we notice that
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ is the category of
_projective_ objects in $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$,
while $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is the
category of _injective_ objects in
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ (and there are enough of
both).
A version of part (a) of the following corollary that is valid in a different
generality will be obtained in Section 5.2.
###### Corollary 4.4.5.
Let $X$ be a semi-separated Noetherian scheme. Then
(a) any cosheaf from $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ is
flat;
(b) any cosheaf from
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is flat.
###### Proof.
Follows from Corollary 4.3.6. ∎
More general versions of the next corollary and of part (b) of the previous
one will be obtained in Section 5.1. In both cases, we will see that the semi-
separatedness and quasi-compactness assumptions can be dropped.
###### Corollary 4.4.6.
Over a semi-separated Noetherian scheme $X$, the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ of
projective locally cotorsion contraherent cosheaves is closed under infinite
products in $X{\operatorname{\mathsf{--ctrh}}}$.
###### Proof.
Follows from Corollary 4.3.7. ∎
A more general version of parts (a-b) of the following Corollary will be
proven in the next Section 4.5, while more general versions of part (b-c) will
be also obtained in Section 5.1. In both cases, we will see that the
affineness assumption on the morphism $f$ is unnecessary.
###### Corollary 4.4.7.
Let $f\colon Y\longrightarrow X$ be an affine morphism of quasi-compact semi-
separated schemes. Then
(a) if the morphism $f$ is very flat, then the direct image functor $f_{!}$
takes projective contraherent cosheaves to projective contraherent cosheaves;
(b) if the morphism $f$ is flat, then the direct image functor $f_{!}$ takes
projective locally cotorsion contraherent cosheaves to projective locally
cotorsion contraherent cosheaves;
(c) if the scheme $X$ is Noetherian and the morphism $f$ is an open embedding,
then the inverse image functor $f^{!}$ takes projective locally cotorsion
contraherent cosheaves to projective locally cotorsion contraherent cosheaves.
###### Proof.
Part (a) holds, since in its assumptions the functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ is “parially left adjoint” to the exact
functor $f^{!}\colon X{\operatorname{\mathsf{--lcth}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}$. The proof of part (b) is similar
(alternatively, it can be deduced from the facts that the functor $f_{!}$
takes $Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$). Part (c) follows from
Corollary 4.3.8. ∎
### 4.5. Homology of locally contraherent cosheaves
The functor $\Delta(X,{-})$ of global cosections of locally contraherent
cosheaves on a scheme $X$, which assigns to a cosheaf ${\mathfrak{E}}$ the
abelian group (or even the ${\mathcal{O}}(X)$-module) ${\mathfrak{E}}[X]$, is
right exact as a functor on the exact category of locally contraherent
cosheaves $X{\operatorname{\mathsf{--lcth}}}$ on $X$. In other words, if
$0\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$ is a short exact sequence of locally contraherent cosheaves on $X$, then
the sequence of abelian groups
$\Delta(X,{\mathfrak{K}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\Delta(X,{\mathfrak{L}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\Delta(X,{\mathfrak{M}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$
is exact. Indeed, the procedure recovering the groups of cosections of
cosheaves ${\mathcal{F}}$ on $X$ from their groups of cosections over affine
open subschemes $U\subset X$ subordinate to a particular covering
${\mathbf{W}}$ and the corestriction maps between such groups uses the
operations of the infinite direct sum and the cokernel of a morphism (or in
other words, the nonfiltered inductive limit) only (see (5), (7), or (22)).
Recall that for any $({\mathbf{W}},{\mathbf{T}})$-affine morphism of schemes
$f\colon Y\longrightarrow X$ the functor of direct image $f_{!}$ takes
${\mathbf{T}}$-locally contraherent cosheaves on $Y$ to ${\mathbf{W}}$-locally
contraherent cosheaves on $X$. By the definition, there is a natural
isomorphism of ${\mathcal{O}}(X)$-modules
${\mathfrak{E}}[Y]\simeq(f_{!}{\mathfrak{E}})[X]$ for any cosheaf of
${\mathcal{O}}_{Y}$-modules ${\mathfrak{E}}$.
Now let $X$ be a quasi-compact semi-separated scheme. Then the left derived
functor of the functor of global cosections of locally contraherent cosheaves
on $X$ can be defined in the conventional way using left projective
resolutions in the exact category $X{\operatorname{\mathsf{--lcth}}}$ (see
Lemma 4.4.1 and Corollary 4.4.2). Notice that the derived functors of
$\Delta(X,{-})$ (and in fact, any left derived functors) computed in the exact
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ for a particular
open covering ${\mathbf{W}}$ and in the whole category
$X{\operatorname{\mathsf{--lcth}}}$ agree. We denote this derived functor by
${\mathbb{L}}_{*}\Delta(X,{-})$. The groups
${\mathbb{L}}_{i}\Delta(X,{\mathfrak{E}})$ are called the _homology groups_ of
a locally contraherent cosheaf ${\mathfrak{E}}$ on the scheme $X$.
Let us point out that the functor $\Delta(U,{-})$ of global cosections of
contraherent cosheaves on an affine scheme $U$ is exact, so the groups
${\mathbb{L}}_{>0}\Delta(U,{\mathfrak{E}})$ vanish when $U$ is affine and
${\mathfrak{E}}$ is contraherent.
By Corollary 4.4.7(a), for any very flat $({\mathbf{W}},{\mathbf{T}})$-affine
morphism of quasi-compact semi-separated schemes $f\colon Y\longrightarrow X$
the exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ takes projective contraherent
cosheaves on $Y$ to projective contraherent cosheaves on $X$. It also makes a
commutative diagram with the restrictions of the functors $\Delta(X,{-})$ and
$\Delta(Y,{-})$ to the categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$. Hence one has
${\mathbb{L}}_{*}\Delta(Y,{\mathfrak{E}})\simeq{\mathbb{L}}_{*}\Delta(X,f_{!}{\mathfrak{E}})$
for any ${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{E}}$ on $Y$.
In particular, the latter assertion applies to the embeddings of affine open
subschemes $j\colon U\longrightarrow X$, so
${\mathbb{L}}_{>0}\Delta(X,j_{!}{\mathfrak{E}})=0$ for all contraherent
cosheaves ${\mathfrak{E}}$ on $U$. Since the derived functor
${\mathbb{L}}_{*}\Delta$ takes short exact sequences of locally contraherent
cosheaves to long exact sequences of abelian groups, it follows from Corollary
4.2.5(c) that ${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{P}})=0$ for any colocally
projective contraherent cosheaf ${\mathfrak{P}}$ on $X$.
Therefore, the derived functor ${\mathbb{L}}_{*}\Delta$ can be computed using
colocally projective left resolutions. Now we also see that the derived
functors ${\mathbb{L}}_{*}\Delta$ defined in the theories of arbitrary (i. e.,
locally contraadjusted) contraherent cosheaves and of locally cotorsion
contraherent cosheaves agree.
Let ${\mathfrak{E}}$ be a ${\mathbf{W}}$-locally contraherent cosheaf on $X$,
and let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering of $X$
subordinate to ${\mathbf{W}}$. Then the contraherent Čech resolution (27) for
${\mathfrak{E}}$ is a colocally projective left resolution of a locally
contraherent cosheaf ${\mathfrak{E}}$, and one can use it to compute the
derived functor ${\mathbb{L}}_{*}(X,{\mathfrak{E}})$. In other words, the
homology of a ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{E}}$ on
a quasi-compact semi-separated scheme $X$ are computed by the homological Čech
complex $C_{*}(\\{U_{\alpha}\\},{\mathfrak{E}})$ (see (22)) related to any
finite affine open covering $X=\bigcup_{\alpha}U_{\alpha}$ subordinate to
${\mathbf{W}}$.
The following result is to be compared with the cohomological criterion of
affineness of schemes [26, Théorème 5.2.1].
###### Corollary 4.5.1.
A locally contraherent cosheaf ${\mathfrak{E}}$ on an affine scheme $U$ is
contraherent if and only if its higher homology
${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{E}})$ vanish.
###### Proof.
See Lemma 3.2.2. ∎
###### Corollary 4.5.2.
Let $R$ be a commutative ring, $I\subset R$ be a nilpotent ideal, and $S=R/I$
be the quotient ring. Let
$i\colon\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ denote the
corresponding homeomorphic closed embedding of affine schemes. Then a locally
injective locally contraherent cosheaf ${\mathfrak{J}}$ on
$U=\operatorname{Spec}R$ is contraherent if and only if its inverse image
$i^{!}{\mathfrak{J}}$ on $V=\operatorname{Spec}S$ is contraherent.
###### Proof.
Since any morphism into an affine scheme is coaffine, the “only if” assertion
is obvious. To prove the “if”, we apply again Lemma 3.2.2. Assuming that
${\mathfrak{J}}$ is ${\mathbf{W}}$-locally contraherent on $U$ and
$U=\bigcup_{\alpha}U_{\alpha}$ is a finite affine open covering of $U$
subordinate to ${\mathbf{W}}$, the homological Čech complex
$C_{*}(\\{U_{\alpha}\\},{\mathfrak{J}})$ is a finite complex of injective
$R$-modules. Set $V_{\alpha}=i^{-1}(U_{\alpha})\subset V$; then the complex
$C_{*}(\\{V_{\alpha}\\},{\mathfrak{J}})$ is the maximal subcomplex of
$R$-modules in $C_{*}(\\{U_{\alpha}\\},{\mathfrak{J}})$ annihilated by the
action of $I$.
Since the maximal submodule ${}_{I}M\subset M$ annihilated by $I$ is nonzero
for any nonzero $R$-module $M$, one easily proves by induction that a finite
complex $K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of injective
$R$-modules is acyclic at all its terms except perhaps the rightmost one
whenever so is the complex of injective $R/I$-modules
${}_{I}K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Notice that this
argument does not seem to apply to the maximal reduced closed subscheme
$\operatorname{Spec}R/J$ of an arbitrary affine scheme $\operatorname{Spec}R$
in general, as the maximal submodule ${}_{J}K\subset K$ annihilated by the
nilradical $J\subset R$ may well be zero even for a nonzero injective
$R$-module $K$ (set $K=\operatorname{Hom}_{k}(F,k)$ in the example from Remark
1.7.5). ∎
The following result is to be compared with Corollaries 3.4.8 and 4.1.13 (for
another comparison, see Corollaries 4.2.10 and 4.3.8).
###### Corollary 4.5.3.
Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then
(a) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes colocally projective contraherent cosheaves on $Y$ to colocally
projective contraherent cosheaves on $X$, and induces an exact functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ between these exact
categories;
(b) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes colocally projective locally cotorsion contraherent cosheaves on
$Y$ to colocally projective locally cotorsion contraherent cosheaves on $X$,
and induces an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ between these
exact categories;
(c) if the morphism $f$ is flat, then the functor of direct image of cosheaves
of ${\mathcal{O}}$-modules $f_{!}$ takes colocally flat contraherent cosheaves
on $Y$ to colocally flat contraherent cosheaves on $X$, and induces and exact
functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ between these exact
categories;
(d) if the morphism $f$ is flat, then the functor of direct image of cosheaves
of ${\mathcal{O}}$-modules $f_{!}$ takes colocally projective locally
injective contraherent cosheaves on $Y$ to colocally projective locally
injective contraherent cosheaves on $X$.
###### Proof.
Part (a): by Corollary 4.2.10(a), the inverse image of a colocally projective
contraherent cosheaf on $Y$ with respect to an affine open embedding $j\colon
V\longrightarrow Y$ is colocally projective. As we have seen above, the global
cosections of colocally projective contraherent cosheaves is an exact functor.
It follows that the functor $f_{!}$ takes short exact sequences in
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ to short exact sequences in
the exact category of cosheaves of ${\mathcal{O}}_{X}$-modules (with the exact
category structure
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\\{X\\}}$ related to the
covering $\\{X\\}$ of the scheme $X$; see Section 3.1).
Since $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is a full exact
subcategory closed under extensions in $X{\operatorname{\mathsf{--ctrh}}}$,
and the latter exact category is such an exact subcategory in
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\\{X\\}}$, in view of
Corollary 4.2.5(c) it remains to recall that the direct images of colocally
projective contraherent cosheaves with respect to affine morphisms of schemes
are colocally projective (see the remarks in the beginning of Section 4.2).
Part (b) is similar; the proof of part (c) is also similar and based on the
remarks about direct images in the beginning of Section 4.3 together with
Corollary 4.3.8; and to prove part (d) one only needs to recall that the
direct images of locally injective contraherent cosheaves with respect to flat
affine morphisms of schemes are locally injective and use Corollary 4.2.8. ∎
Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. By the result of Section 3.3 (see (25)), the adjunction isomorphism
(23) holds, in particular, for any colocally projective contraherent cosheaf
${\mathfrak{Q}}$ on $Y$ and any locally injective locally contraherent cosheaf
${\mathfrak{J}}$ on $X$.
If the morphism $f$ is flat then, according to (26), the adjunction
isomorphism
(49)
$\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{M}})\simeq\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{M}})$
holds for any colocally projective contraherent cosheaf ${\mathfrak{Q}}$ on
$Y$ and any locally cotorsion locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$. When the morphism $f$ is also affine, the restrictions of $f_{!}$ and
$f^{!}$ form an adjoint pair of functors between the exact categories
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$. In
addition, these functors take the additive categories
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ into one
another.
If the morphism $f$ is very flat, the same adjunction isomorphism (49) holds
for any colocally projective contraherent cosheaf ${\mathfrak{Q}}$ on $Y$ and
any locally contraherent cosheaf ${\mathfrak{E}}$ on $X$. When the morphism
$f$ is also affine, the restrictions of $f_{!}$ and $f^{!}$ form an adjoint
pair of functors between the exact categories
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.
###### Corollary 4.5.4.
Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then
(a) if the morphism $f$ is very flat, then the direct image functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ takes projective
contraherent cosheaves to projective contraherent cosheaves;
(b) if the morphism $f$ is flat, then the direct image functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ takes
projective locally cotorsion contraherent cosheaves to projective locally
cotorsion contraherent cosheaves.
###### Proof.
Follows from the above partial adjunctions (49) between the exact functors
$f_{!}$ and $f^{!}$. Part (b) can be also deduced from Corollary 4.5.3(b-c). ∎
### 4.6. The “naïve” co-contra correspondence
Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
open covering. We refer to Section A.1 for the definitions of the derived
categories mentioned below.
Recall the definition of the left homological dimension
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E$ of an object $E$ of an exact
category ${\mathsf{E}}$ with respect to a full exact subcategory
${\mathsf{F}}\subset{\mathsf{E}}$, given (under a specific set of assumptions
on ${\mathsf{F}}$ and ${\mathsf{E}}$) in Section A.5. The _right homological
dimension with respect to an exact subcategory ${\mathsf{F}}$_ (or the right
${\mathsf{F}}$-homological dimension)
$\operatorname{rd}_{{\mathsf{F}}/{\mathsf{E}}}E$ is defined in the dual way
(and under the dual set of assumptions).
###### Lemma 4.6.1.
(a) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the right homological dimension of any quasi-coherent sheaf on $X$ with
respect to the exact subcategory of contraadjusted quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ (is well-defined and) does not exceed $N$.
(b) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the left homological dimension of any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of colocally projective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ does not exceed $N-1$.
Consequently, the same bound holds for the left homological dimension of any
object of $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ with respect to the
exact subcategory $X{\operatorname{\mathsf{--ctrh}}}$.
###### Proof.
Part (a): first of all, the assumptions about the pair of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ making the right homological dimension
well-defined hold by Corollaries 4.1.2(c) and 4.1.4(b).
Furthermore, the right homological dimension of any module over a commutative
ring $R$ with respect to the exact category of contraadjusted $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\subset
R{\operatorname{\mathsf{--mod}}}$ does not exceed $1$. It follows easily that
the homological dimension of any quasi-coherent sheaf of the form
$j_{*}{\mathcal{G}}$, where $j\colon U\longrightarrow X$ is an affine open
subscheme, with respect to the exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ does not exceed $1$, either. Now any quasi-
coherent sheaf ${\mathcal{F}}$ on $X$ has a Čech resolution (12) of length
$N-1$ by finite direct sums of quasi-coherent sheaves of the above form. It
remains to use the dual version of Corollary A.5.5(a).
Part (b): the conditions on the exact categories
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ making the left homological
dimension well-defined hold by Corollaries 4.2.3(b) and 4.2.5(b). The pair of
exact categories $X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ satisfies the same assumptions
for the reasons explained in Section 3.2. It remains to recall the resolution
(27). ∎
###### Lemma 4.6.2.
Let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then
(a) the left homological dimension of any locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of colocally projective locally cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ does not exceed
$N-1$. Consequently, the same bound holds for the left homological dimension
of any object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ with respect
to the exact subcategory $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$;
(b) the left homological dimension of any locally injective
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of colocally projective locally injective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ does not exceed
$N-1$. Consequently, the same bound holds for the left homological dimension
of any object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ with respect
to the exact subcategory $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$.
###### Proof.
Similar to the proof of Lemma 4.6.1(b). ∎
###### Corollary 4.6.3.
(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is an equivalence of
triangulated categories.
(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}$ is an equivalence of triangulated
categories.
The reason why most unbounded derived categories aren’t mentioned in part (b)
is because one needs a uniform restriction on the extension of locality of
locally contraherent cosheaves in order to work simultaneously with infinite
collections of these. In particular, infinite products exist in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, but not necessarily in
$X{\operatorname{\mathsf{--lcth}}}$, so the contraderived category of the
latter exact category is not well-defined.
###### Proof of Corollary 4.6.3.
Part (a) follows from Proposition A.5.6 together with Lemma 4.6.1(b). Part (b)
in the case $\star={\mathsf{b}}$ is obtained from part (a) by passing to the
inductive limit over refinements of coverings, while in the case $\star=-$ it
is provided by Proposition A.3.1(a). ∎
The following two corollaries are similar to the previous one. The only
difference in the proofs is that Lemma 4.6.2 is being used in place of Lemma
4.6.1(b).
###### Corollary 4.6.4.
(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is an
equivalence of triangulated categories.
(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ is an equivalence of
triangulated categories. ∎
###### Corollary 4.6.5.
(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is an
equivalence of triangulated categories.
(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ is an equivalence of
triangulated categories. ∎
The next theorem is the main result of this section.
###### Theorem 4.6.6.
For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$ there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$.
These equivalences of derived categories form commutative diagrams with the
natural functors
${\mathsf{D}}^{\mathsf{b}}\longrightarrow{\mathsf{D}}^{\pm}\longrightarrow{\mathsf{D}}$,
${\mathsf{D}}^{\mathsf{b}}\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}\longrightarrow{\mathsf{D}}^{\mathsf{abs}}$,
${\mathsf{D}}^{{\mathsf{abs}}\pm}\longrightarrow{\mathsf{D}}^{\pm}$,
${\mathsf{D}}^{\mathsf{abs}}\longrightarrow{\mathsf{D}}$ between different
versions of derived categories of the same exact category.
Notice that Theorem 4.6.6 does not say anything about the coderived and
contraderived categories ${\mathsf{D}}^{\mathsf{co}}$ and
${\mathsf{D}}^{\mathsf{ctr}}$ of quasi-coherent sheaves and contraherent
cosheaves (neither does Corollary 4.6.3 mention the coderived categories). The
reason is that infinite products are not exact in the abelian category of
quasi-coherent sheaves and infinite direct sums may not exist in the exact
category of contraherent cosheaves. So only the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ and the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$ are well-
defined. Comparing these two requires a different approach; the entire Section
5 will be devoted to that.
###### Proof of Theorem 4.6.6.
By Proposition A.5.6 and its dual version, together with Lemma 4.6.1, the
functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
induced by the corresponding embedings of exact categories are all
equivalences of triangulated categories. Hence it suffices to construct a
natural equivalence of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ in order to prove all
assertions of Theorem.
According to Sections 2.5 and 2.6, there are natural functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})\colon
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}$
and
${\mathcal{O}}_{X}\odot_{X}{-}\colon X{\operatorname{\mathsf{--lcth}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--qcoh}}}$
related by the adjunction isomorphism (20), which holds for those objects for
which the former functor is defined. So it remains to prove the following
lemma. ∎
###### Lemma 4.6.7.
On a quasi-compact semi-separated scheme $X$, the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ takes
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, the functor
${\mathcal{O}}_{X}\odot_{X}{-}$ takes
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$, and the restrictions of
these functors to these subcategories are mutually inverse equivalences of
exact categories.
###### Proof.
Obviously, on an affine scheme $U$ the functor
$\operatorname{\mathfrak{Hom}}_{U}({\mathcal{O}}_{U},{-})$ takes a
contraadjusted quasi-coherent sheaf ${\mathcal{Q}}$ with the contraadjusted
${\mathcal{O}}(U)$-module of global sections ${\mathcal{Q}}(U)$ to the
contraherent cosheaf ${\mathfrak{Q}}$ with the contraadjusted
${\mathcal{O}}(U)$-module of global cosections
${\mathfrak{Q}}[U]={\mathcal{Q}}(U)$. Furthermore, if $j\colon
U\longrightarrow X$ is the embedding of an affine open subscheme, then by the
formula (44) of Section 3.8 there is a natural isomorphism
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},j_{*}{\mathcal{Q}})\simeq
j_{!}{\mathfrak{Q}}$ of contraherent cosheaves on $X$.
Analogously, the functor ${\mathcal{O}}_{U}\odot_{U}{-}$ takes a contraherent
cosheaf ${\mathfrak{Q}}$ with the contraadjusted ${\mathcal{O}}(U)$-module of
global cosections ${\mathfrak{Q}}[U]$ to the contraadjusted quasi-coherent
sheaf ${\mathcal{Q}}$ with the ${\mathcal{O}}(U)$-module of global sections
${\mathcal{Q}}(U)$ on $U$. If an embedding of affine open subscheme $j\colon
U\longrightarrow X$ is given, then by the formula (46) there is a natural
isomorphism ${\mathcal{O}}_{X}\odot_{X}j_{!}{\mathfrak{Q}}\simeq
j_{*}{\mathcal{Q}}$ of quasi-coherent sheaves on $X$.
By Corollary 4.1.4(c), any sheaf from
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ is a direct summand of a
finitely iterated extension of the direct images of contraadjusted quasi-
coherent sheaves from affine open subschemes of $X$. It is clear from the
definition of the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ that it preserves
exactness of short sequences of contraadjusted quasi-coherent cosheaves; hence
it preserves, in particular, such iterated extensions.
By Corollary 4.2.5(c), any cosheaf from
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is a direct summand of a
finitely iterated extension of the direct images of contraherent cosheaves
from affine open subschemes of $X$. Let us show that the functor
${\mathcal{O}}_{X}\odot_{X}{-}$ preserves exactness of short sequences of
colocally projective contraherent cosheaves on $X$, and therefore, in
particular, preserves such extensions. Indeed, the adjunction isomorphism
$\operatorname{Hom}_{X}({\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}},\>{\mathcal{F}})\simeq\operatorname{Hom}^{X}({\mathfrak{P}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{F}}))$
holds for any contraherent cosheaf ${\mathfrak{P}}$ and contraadjusted quasi-
coherent sheaf ${\mathcal{F}}$. Besides, the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}})$ is
locally injective for any injective quasi-coherent sheaf ${\mathcal{J}}$ on
$X$. By Corollary 4.2.3(a), it follows that the functor
${\mathfrak{P}}\longmapsto\operatorname{Hom}_{X}({\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}},\>{\mathcal{J}})$
preserves exactness of short sequences of colocally projective contraherent
cosheaves on $X$, and consequently so does the functor
${\mathfrak{P}}\longmapsto{\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}}$.
Now one can easily deduce that the adjunction morphisms
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},\>{\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}})\quad\text{and}\quad{\mathcal{O}}_{X}\odot_{X}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{F}})\longrightarrow{\mathcal{F}}$
are isomorphisms for any colocally projective contraherent cosheaf
${\mathfrak{P}}$ and contraadjusted quasi-coherent sheaf ${\mathcal{F}}$, as a
morphism of finitely filtered objects inducing an isomorphism of the
associated graded objects is also itself an isomorphism. The proof of Lemma,
and hence also of Theorem 4.6.6, is finished. ∎
Given an exact category ${\mathsf{E}}$, let
$\mathsf{Hot}^{\star}({\mathsf{E}})$ denote the homotopy category
$\mathsf{Hot}({\mathsf{E}})$ if $\star={\mathsf{abs}}$, ${\mathsf{co}}$,
${\mathsf{ctr}}$, or $\empt$; the category $\mathsf{Hot}^{+}({\mathsf{E}})$,
if $\star={\mathsf{abs}}+$ or $+$; the category
$\mathsf{Hot}^{-}({\mathsf{E}})$, if $\star={\mathsf{abs}}-$ or $-$; and the
category $\mathsf{Hot}^{\mathsf{b}}({\mathsf{E}})$ if $\star={\mathsf{b}}$.
The following two corollaries provide, essentially, several restricted
versions of Theorem 4.6.6.
###### Corollary 4.6.8.
(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$.
(b) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, there is a natural
equivalence of triangulated categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$.
###### Proof.
By Proposition A.5.6 together with Lemma 4.6.2, the functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
induced by the corresponding embeddings of exact categories are equivalences
of triangulated categories. Hence it suffices to show that the equivalence of
exact categories from Lemma 4.6.7 identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$. The former
of these assertions follows from Corollaries 4.1.11(c) and 4.2.6(c), while the
latter one is obtained from Corollary 4.2.8 together with the fact that any
injective quasi-coherent sheaf on $X$ is a direct summand of a finite direct
sum of the direct images of injective quasi-coherent sheaves from open
embeddings $U_{\alpha}\longrightarrow X$ forming a covering. ∎
###### Lemma 4.6.9.
Let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering.
Then
(a) the right homological dimension of any very flat quasi-coherent sheaf on
$X$ with respect to the exact subcategory of contraadjusted very flat quasi-
coherent sheaves $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ does not exceed $N$;
(b) the right homological dimension of any flat quasi-coherent sheaf on $X$
with respect to the exact subcategory of contraadjusted flat quasi-coherent
sheaves $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ does not exceed $N$.
###### Proof.
The right homological dimension is well-defined due to Corollary 4.1.4(b), so
it remains to apply Lemma 4.6.1(a) and the dual version of Corollary A.5.3. ∎
###### Corollary 4.6.10.
(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$, there is a natural
equivalence of triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$.
(b) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})$.
(c) There is a natural equivalence of triangulated categories
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.
###### Proof.
Part (a): assuming $\star\neq{\mathsf{co}}$, by Lemma 4.6.9(a) together with
the dual version of Proposition A.5.6 the triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
is an equivalence of categories. In particular, we have proven that
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})={\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$,
so the previous assertion holds for $\star={\mathsf{co}}$ as well. Hence it
suffices to show that the equivalence of exact categories from Lemma 4.6.7
identifies $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$. This follows from Lemmas
4.1.5 and 4.4.1(b). The proof of part (b) is similar: in view of Lemma
4.6.9(b), the triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is an equivalence of categories, and it remains to show that the equivalence
of exact categories from Lemma 4.6.7 identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$. Here one applies Lemma
4.1.6 and Corollary 4.3.4(c).
To prove part (c), notice that the triangulated functor
$\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is an equivalence of categories by Corollary 4.1.11(b) and the dual version of
Proposition A.3.1(a). So it remains to show that the equivalence of exact
categories $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. This
follows from Lemmas 4.1.12 and 4.4.3(b). Alternatively, one can prove that the
functor
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})$
is an equivalence of categories by applying directly the dual version of
Proposition A.3.1(a) together with Corollaries 4.3.4(a) and 4.3.2(b). ∎
### 4.7. Homotopy locally injective complexes
Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
open covering. The goal of this section is to construct a full subcategory in
the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
that would be equivalent to the unbounded derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$. The
significance of this construction is best illustrated using the duality-
analogy between the contraherent cosheaves and the quasi-coherent sheaves.
As usually, the notation
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ refers to the
unbounded derived category of the exact category of flat quasi-coherent
sheaves on $X$. The full triangulated subcategory
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
of _homotopy flat complexes_ of flat quasi-coherent sheaves on $X$ is defined
as the minimal triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ containing the
objects of $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and closed under
infinite direct sums (cf. Section A.4). The following result is essentially
due to Spaltenstein [60].
###### Theorem 4.7.1.
(a) The composition of natural triangulated functors
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}\allowbreak\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.
(b) A complex of flat quasi-coherent sheaves
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ belongs
to
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
if and only if its tensor product
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with any acyclic complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ is also
an acyclic complex of quasi-coherent sheaves.
###### Proof.
Part (a) is a particular case of Proposition A.4.3. To prove part (b), let us
first show that the tensor product of any complex of quasi-coherent sheaves
and a complex of sheaves acyclic with respect to the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ is acyclic. Indeed, any
complex in an abelian category is a locally stabilizing inductive limit of
finite complexes; so it suffices to notice that the tensor product of any
quasi-coherent sheaf with a complex acyclic with respect to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ is acyclic. Hence the class
of all complexes of flat quasi-coherent sheaves satisfying the condition in
part (b) can be viewed as a strictly full triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.
Now the “only if” assertion easily follows from the facts that the tensor
products of quasi-coherent sheaves preserve infinite direct sums and the
tensor product with a flat quasi-coherent sheaf is an exact functor. In view
of (the proof of) part (a), it suffices to show that any complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ satisfying the tensor
product condition of part (b) and acyclic with respect to
$X{\operatorname{\mathsf{--qcoh}}}$ is also acyclic with respect to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ in order to prove “if”.
Notice that the tensor product of a bounded above complex of flat quasi-
coherent sheaves and an acyclic complex of quasi-coherent sheaves is an
acyclic complex. Since any quasi-coherent sheaf ${\mathcal{M}}$ over $X$ has a
flat left resolution, it follows that the complex of quasi-coherent sheaves
${\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic. One easily concludes that the complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic with
respect to $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$. ∎
The full triangulated subcategory
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$
of _homotopy locally injective complexes_ of localy injective
${\mathbf{W}}$-locally contraherent cosheaves on $X$ is defined as the minimal
full triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
containing the objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ and closed
under infinite products. Given a complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and a complex
of ${\mathbf{W}}$-locally contraherent cosheaves
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ such
that the ${\mathbf{W}}$-locally contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}^{i},{\mathfrak{P}}^{j})$ is
defined for all $i$, $j\in{\mathbb{Z}}$ (see Sections 2.4 and 3.6), we define
the complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
as the total complex of the bicomplex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}^{i},{\mathfrak{P}}^{j})$
constructed by taking infinite products of ${\mathbf{W}}$-locally contraherent
cosheaves along the diagonals.
###### Theorem 4.7.2.
(a) The composition of natural triangulated functors
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}\allowbreak\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories.
(b) A complex of locally injective ${\mathbf{W}}$-locally contraherent
cosheaves ${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on
$X$ belongs to
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
if and only if the complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ into it
from any acyclic complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is an acyclic
complex in the exact category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$
(or, at one’s choice, in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$.
###### Proof.
Part (a): the argument goes along the lines of the proof of Theorem 4.7.1(a),
but Proposition A.4.3 is not directly applicable, the category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ being not
abelian; so there are some complications. First of all, we will need another
definition. The full triangulated subcategory
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
of homotopy locally injective complexes of colocally projective locally
injective contraherent cosheaves on $X$ is defined as the minimal full
triangulated subcategory containing the objects of
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ and closed
under infinite products.
It was shown in Section 4.6 that the natural functor
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories. Analogously one shows (using, e.
g., Corollary A.5.3) that the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
is an equivalence of categories, as are the similar functors
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
and
$\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$.
Therefore, the equivalence
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
identifies the subcategories generated by bounded or bounded below complexes.
Thus the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$
is also an equivalence of triangulated categories, and it remains to show that
the functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$
is an equivalence of categories.
We will show that any complex over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ admits a quasi-isomorphism
with respect to the exact category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ into a complex belonging to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}$.
In particular, by the dual version of [51, Lemma 1.6] applied to the homotopy
category $\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$ with
the full triangulated subcategory
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
and the thick subcategory of complexes acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ it will follow that the
category ${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$ is
equivalent to the localization of
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
by the thick subcategory of complexes acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. By the dual version of
Corollary A.4.2, the latter subcategory is semiorthogonal to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}$
in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$.
In view of the same construction of a quasi-isomorphism with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, these two subcategories
form a semiorthogonal decomposition of
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$,
which implies the desired assertion.
###### Lemma 4.7.3.
There exists a positive integer $d$ such that for any complex
${\mathfrak{P}}^{0}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}^{d+1}$
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ there exists a complex
${\mathfrak{Q}}^{0}\longrightarrow\dotsb\longrightarrow{\mathfrak{Q}}^{d+1}$
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ together with a
morphism of complexes
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
such that ${\mathfrak{Q}}^{0}=0$, while the morphisms of contraherent
cosheaves ${\mathfrak{P}}^{d}\longrightarrow{\mathfrak{Q}}^{d}$ and
${\mathfrak{P}}^{d+1}\longrightarrow{\mathfrak{Q}}^{d+1}$ are isomorphisms.
###### Proof.
In view of Lemma 4.6.7, it suffices to prove the assertion of Lemma for a
complex
${\mathcal{P}}^{0}\longrightarrow\dotsb\longrightarrow{\mathcal{P}}^{d+1}$
over the category $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$. Set
${\mathcal{Q}}^{0}=0$. Consider the quasi-coherent sheaf
${\mathcal{R}}^{1}=\operatorname{coker}({\mathcal{P}}^{0}\to{\mathcal{P}}^{1})$
and embed it into a contraadjusted quasi-coherent sheaf ${\mathcal{Q}}^{1}$.
Denote by ${\mathcal{R}}^{2}$ the fibered coproduct of the quasi-coherent
sheaves ${\mathcal{Q}}^{1}$ and ${\mathcal{P}}^{2}$ over ${\mathcal{R}}^{1}$,
embed it into a contraadjusted quasi-coherent sheaf ${\mathcal{Q}}^{2}$, and
proceed by applying the dual version of the construction of Lemma A.3.2(a) up
to producing the quasi-coherent sheaves ${}{\mathcal{Q}}^{d-2}$ and
${\mathcal{R}}^{d-1}$.
The sequence
$0\longrightarrow{\mathcal{R}}^{1}\longrightarrow{\mathcal{Q}}^{1}\oplus{\mathcal{P}}^{2}\longrightarrow{\mathcal{Q}}^{2}\oplus{\mathcal{P}}^{3}\longrightarrow\dotsb\longrightarrow{\mathcal{Q}}^{d-2}\oplus{\mathcal{P}}^{d-1}\longrightarrow{\mathcal{R}}^{d-1}\longrightarrow
0$ is a right resolution of the quasi-coherent sheaf ${\mathcal{R}}^{1}$, all
of whose terms, except perhaps the rightmost one, are contraadjusted quasi-
coherent sheaves. By Lemma 4.6.1(a) and the dual version of Corollary A.5.2,
for $d$ large enough the quasi-coherent sheaf ${\mathcal{R}}^{d-1}$ will be
contraadjusted. It remains to set ${\mathcal{Q}}^{d-1}={\mathcal{R}}^{d-1}$,
${\mathcal{Q}}^{d}={\mathcal{P}}^{d}$, and
${\mathcal{Q}}^{d+1}={\mathcal{P}}^{d+1}$. ∎
Now let ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
complex over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. For each
fragment of $d+2$ consequtive terms
${\mathfrak{P}}^{i-d-1}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}^{i}$
in ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ we
construct the corresponding complex
${}^{(i)}{\mathfrak{Q}}^{i-d-1}\longrightarrow\dotsb\longrightarrow{}^{(i)}{\mathfrak{Q}}^{i}$
as in Lemma 4.7.3. Pick an admissible monomorphism
${}^{(i)}{\mathfrak{Q}}^{i-d}\longrightarrow{}^{(i)}{\mathfrak{I}}^{i-d}$ from
a colocally projective contraherent cosheaf ${}^{(i)}{\mathfrak{Q}}^{i-d}$
into a colocally projective locally injective contraherent cosheaf
${}^{(i)}{\mathfrak{I}}^{i-d}$ on $X$ (see Corollaries 4.2.5(b) and 4.2.3(a)).
Proceeding with the dual version of the construction of Lemma A.3.2(a) (see
also the above proof of Lemma 4.7.3), we obtain a termwise admissible
monomorphism with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ from the complex
${}^{(i)}{\mathfrak{Q}}^{i-d}\longrightarrow\dotsb\longrightarrow{}^{(i)}{\mathfrak{Q}}^{i}$
into a complex
${}^{(i)}{\mathfrak{I}}^{i-d}\longrightarrow\dotsb\longrightarrow{}^{(i)}{\mathfrak{I}}^{i}$
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ such
that the cone of this morphism is quasi-isomorphic to an object of
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ placed in the cohomological
degree $i$.
Set ${}^{(i)}{\mathfrak{I}}^{j}=0$ for $j$ outside of the segment $[i-d,\>i]$.
We obtain a finite complex
${}^{(i)}{\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ endowed with
a morphism of complexes
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{(i)}{\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the following property. For any affine open subscheme $U\subset X$, the
induced morphism of cohomology modules
$H^{i}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U])\longrightarrow
H^{i}({\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U])$ is
injective.
Denote by
${}^{0}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the
direct product of all the complexes
${}^{(i)}{\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The
morphism of complexes
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{0}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over the exact category $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is
a termwise admissible monomorphism. Consider the corresponding complex of
cokernels and apply the same procedure to it, constructing a termwise acyclic
complex of complexes
$0\longrightarrow{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{0}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{1}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\dotsb$
over the exact category $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ in
which all the complexes
${}^{i}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
infinite products of finite complexes over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.
By Lemma A.3.4 applied to the projective system of quotient complexes of silly
filtration with respect to the left index of the bicomplex
${}^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
the total complex of
${}^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
constructed by taking infinite products along the diagonals belongs to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}$.
It remains to show that the cone
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the
morphism from ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
to the total complex of
${}^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic with respect to $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.
Indeed, by the dual version of Lemma A.4.4, the complex of cosections
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U]$ is an
acyclic complex of ${\mathcal{O}}_{X}(U)$-modules for any affine open
subscheme $U\subset X$. Since the ${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{E}}^{i}[U]$ are contraadjusted and quotient modules of
contraadjusted modules are contraadjusted, the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U]$ is also
acyclic with respect to the exact category
${\mathcal{O}}(U){\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.
Since the functor
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{-})$
preserves exactness of short sequences of contraadjusted
${\mathcal{O}}_{X}(U)$-modules for any pair of embedded affine open subschemes
$V\subset U\subset X$, one easily concludes that the rules
$U\longmapsto\operatorname{coker}({\mathfrak{E}}^{i-1}[U]\to{\mathfrak{E}}^{i}[U])$
define contraherent cosheaves on $X$. Hence the complex of contrahent
cosheaves ${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic over $X{\operatorname{\mathsf{--ctrh}}}$. Since the contraherent
cosheaves ${\mathfrak{E}}^{i}$ belong to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, this complex is also
acyclic over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ by Lemma
4.6.1(b) and Corollary A.5.2.
Part (a) is proven; let us prove part (b). The argument is similar to the
proof of Theorem 4.7.1(b). First we show that
$\operatorname{\mathfrak{Cohom}}_{X}$ from any complex of quasi-coherent
sheaves into a complex of locally contraherent cosheaves acyclic with respect
to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is acyclic
with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Indeed, any
complex of quasi-coherent sheaves is a locally stabilizing inductive limit of
a sequence of finite complexes. So it remains to recall that
$\operatorname{\mathfrak{Cohom}}_{X}$ from a quasi-coherent sheaf into a
complex acyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is a complex
acyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$, and use Lemma
A.3.4 again.
Hence the class of all complexes over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ satisfying the
$\operatorname{\mathfrak{Cohom}}$ condition in part (b) can be viewed as a
strictly full triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$.
Now the “only if” assertion follows from the preservation of infinite products
in the second argument by the functor $\operatorname{\mathfrak{Cohom}}_{X}$
and its exactness as a functor on the category
$X{\operatorname{\mathsf{--qcoh}}}$ for any fixed locally injective locally
contraherent cosheaf in the second argument. In view of (the proof of) part
(a), in order to prove “if” it suffices to show that any complex
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ satisfying the
$\operatorname{\mathfrak{Cohom}}$ condition in (b) and acyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is also acyclic with respect
to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$.
Notice that the $\operatorname{\mathfrak{Cohom}}_{X}$ from a bounded above
complex of very flat quasi-coherent sheaves into an acyclic complex of
${\mathbf{W}}$-locally contraherent cosheaves is an acyclic complex of
${\mathbf{W}}$-locally contraherent cosheaves. Since any quasi-coherent sheaf
${\mathcal{M}}$ on $X$ has a very flat left resolution (see Lemma 4.1.1), it
follows that the complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic with respect to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
Now let $U\subset X$ be an affine open subscheme subordinate to
${\mathbf{W}}$, let $N$ be an ${\mathcal{O}}(U)$-module, viewed also as a
quasi-coherent sheaf on $U$, and ${\mathcal{M}}$ be any quasi-coherent
extension (e. g., the direct image) of $N$ to $X$. Then acyclicity of the
complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
with respect to the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ implies, in particular,
exactness of the complex of ${\mathcal{O}}_{X}(U)$-modules
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}(N,{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U])$.
Since this holds for all ${\mathcal{O}}_{X}(U)$-modules $N$, it follows that
all the ${\mathcal{O}}_{X}(U)$-modules of cocycles in the acyclic complex of
${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U]$ are
injective. ∎
The following lemma will be needed in Section 4.8.
###### Lemma 4.7.4.
Let ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
complex over the exact category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex
over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ belonging
to
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$.
Then the natural morphism of graded abelian groups
$H^{*}\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}_{{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is an isomorphism (in other words, the complex
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
computes the groups of morphisms in the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$).
###### Proof.
Since any complex over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ admits
a quasi-isomorphism into it from a complex over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and any complex over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ acyclic over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is also acyclic over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, it suffices to show that
the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ acyclic with
respect to $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and any complex
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$.
For a complex ${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
obtained from objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ by iterating
the operations of cone and infinite direct sum the latter assertion is obvious
(see Corollary 4.2.3(a)), so it remains to consider the case of a complex
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ acyclic with
respect to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$. In
this case we will show that the complex
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.
Let $i$ be an integer. Applying Lemma 4.7.3 to the fragment
${\mathfrak{P}}^{i-d-1}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}^{i}$
of the complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, we obtain a
morphism of complexes from
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to a finite
complex ${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ such that the morphisms
${\mathfrak{P}}^{i-1}\longrightarrow{\mathfrak{Q}}^{i-1}$ and
${\mathfrak{P}}^{i}\longrightarrow{\mathfrak{Q}}^{i}$ are isomorphisms. The
cocone of this morphism splits naturally into a direct sum of two complexes
concentrated in cohomological degrees $\le i$ and $\ge i$, respectively. We
are interested in the former complex. Its subcomplex of silly truncation
${\mathfrak{R}}(j,i)^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
finite complex over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$
concentrated in the cohomological degrees between $j$ and $i$ and endowed with
a morphism of complexes
${\mathfrak{R}}(j,i)^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
which is a termwise isomorphism in the degrees between $j$ and $i-d$.
The complex ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
a termwise stabilizing inductive limit of the sequence of complexes
${\mathfrak{R}}(j,i)^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as the
degree $j$ decreases, while the degree $i$ increases (fast enough). It remains
to recall that the functor $\operatorname{Hom}^{X}$ from a colocally
projective contraherent cosheaf takes acyclic complexes over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ to acyclic complexes of
abelian groups, and, e. g., use Lemma A.3.4 once again. ∎
### 4.8. Derived functors of direct and inverse image
For the rest of Section 4, the upper index $\star$ in the notation for derived
and homotopy categories stands for one of the symbols ${\mathsf{b}}$, $+$,
$-$, $\empt$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{co}}$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$.
Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then for any symbol $\star\neq{\mathsf{ctr}}$ the right derived
functor of direct image
(50)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
is constructed in the following way. By Lemma 4.6.1(a) together with the dual
version of Proposition A.5.6, the natural functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories (as is the similar functor for
sheaves over $X$). By Corollary 4.1.13(a), the restriction of the functor of
direct image $f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ provides an exact functor
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$. Now the derived functor
${\mathbb{R}}f_{*}$ is defined by by restricting the functor of direct image
$f_{*}\colon\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})$
to the full subcategory of complexes of contraadjusted quasi-coherent sheaves
on $Y$ (with the appropriate boundedness conditions).
For any symbol $\star\neq{\mathsf{co}}$, the left derived functor of direct
image
(51)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
is constructed in the following way. By Lemma 4.6.1(b) (for the covering
$\\{Y\\}$ of the scheme $Y$) together with Proposition A.5.6, the natural
functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})$
is an equivalence of triangulated categories (as is the similar functor for
cosheaves over $X$). By Corollary 4.5.3(a), there is an exact functor of
direct image $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. The derived functor
${\mathbb{L}}f_{!}$ is defined as the induced functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$.
Similarly one defines the left derived functor of direct image
(52)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}).$
###### Theorem 4.8.1.
For any symbol $\star\neq{\mathsf{co}}$, ${\mathsf{ctr}}$, the equivalences of
categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
from Theorem 4.6.6 transform the right derived functor ${\mathbb{R}}f_{*}$
(50) into the left derived functor ${\mathbb{L}}f_{!}$ (51).
###### Proof.
It suffices to show that the equivalences of exact categories
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ from Lemma 4.6.7 transform
the functor $f_{*}$ into the functor $f_{!}$. The isomorphism (44) of Section
3.8 proves as much. ∎
Let $f\colon Y\longrightarrow X$ be a morphism of schemes. Let ${\mathbf{W}}$
and ${\mathbf{T}}$ be open coverings of the schemes $X$ and $Y$, respectively,
for which the morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-coaffine.
According to Section 3.3, there is an exact functor of inverse image
$f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$; for a flat
morphism $f$, there is also an exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$, and for a very
flat morphism $f$, an exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$.
For a quasi-compact semi-separated scheme $X$, it follows from Corollary
4.1.11(a) and Proposition A.3.1(a) that the natural functor
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories. Similarly, it follows from Corollary 4.2.5(a)
and the dual version of Proposition A.3.1(a) that the natural functor
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories. This allows to define, for any
morphism $f\colon Y\longrightarrow X$ into a quasi-compact semi-separated
scheme $X$ and coverings ${\mathbf{W}}$, ${\mathbf{T}}$ as above, the derived
functors of inverse image
(53)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--qcoh}}})$
and
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$
by applying the functors $f^{*}$ and $f^{!}$ to (appropriately bounded)
complexes of flat sheaves and locally injective cosheaves. When both schemes
are quasi-compact and semi-separated, one can take into account the
equivalences of categories from Corollary 4.6.3(a) in order to produce the
right derived functor
(54)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--ctrh}}}),$
which clearly does not depend on the choice of the coverings ${\mathbf{W}}$
and ${\mathbf{T}}$.
According to Section 4.7, the natural functors
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of categories for any quasi-compact semi-separated scheme $X$
with an open covering ${\mathbf{W}}$. This allows to define, for any morphism
$f\colon Y\longrightarrow X$ into a quasi-compact semi-separated scheme $X$
and coverings ${\mathbf{W}}$, ${\mathbf{T}}$ as above, the derived functors of
inverse image
(55)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
and
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$
by applying the functors
$f^{*}\colon\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}})$
and
$f^{!}\colon\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\allowbreak\longrightarrow\mathsf{Hot}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}})$
to homotopy flat complexes of flat quasi-coherent sheaves and homotopy locally
injective complexes of locally injective ${\mathbf{W}}$-locally contraherent
cosheaves, respectively. Of course, this construction is well-known for quasi-
coherent sheaves [60, 45]; we discuss here the sheaf and cosheaf situations
together in order to emphasize the duality-analogy between them.
When both schemes are quasi-compact and semi-separated, one can use the
equivalences of categories from Corollary 4.6.3(a) in order to obtain the
right derived functor
(56)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
which does not depend on the choice of the coverings ${\mathbf{W}}$ and
${\mathbf{T}}$. Notice also that the restriction of the functor $f^{*}$ takes
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
into
$\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
and the restriction of the functor $f^{!}$ takes
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$
into
$\mathsf{Hot}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}})^{\mathsf{lin}}$.
It is easy to see that for any morphism of quasi-compact semi-separated
schemes $f\colon Y\longrightarrow X$ the functor ${\mathbb{L}}f^{*}$ (53) is
left adjoint to the functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})$
(50) and the functor ${\mathbb{R}}f^{!}$ (54) is right adjoint to the functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--ctrh}}})\allowbreak\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}})$
(51). Essentially, one uses the partial adjunctions on the level of exact
categories together with the fact that the derived functor constructions are
indeed those of the “left” and “right” derived functors, as stated (cf. [50,
Lemma 8.3]).
Similarly, one concludes from the construction in the proof of Theorem 4.7.2
that the functor ${\mathbb{L}}f^{*}$ (55) is left adjoint to the functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
(50). And in order to show that the functor ${\mathbb{R}}f^{!}$ (56) is right
adjoint to the functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
(51), one can use Lemma 4.7.4. So we have obtained a new proof of the
following classical result [29, 45].
###### Corollary 4.8.2.
For any morphism of quasi-compact semi-separated schemes $f\colon
Y\longrightarrow X$, the derived direct image functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
has a right adjoint functor
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$.
###### Proof.
We have (more or less) explicitly constructed the functor $f^{!}$ as the right
derived functor
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
(56) of the exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ between exact subcategories
of the exact categories of locally contraherent cosheaves on $X$ and $Y$. The
above construction of the functor $f^{!}$ for bounded below complexes (54) is
particularly explicit. In either case, the construction is based on the
identification of the functor ${\mathbb{R}}f_{*}$ of derived direct image of
quasi-coherent sheaves with the functor ${\mathbb{L}}f_{!}$ of derived direct
image of contraherent cosheaves, which is provided by Theorems 4.6.6 and
4.8.1. ∎
### 4.9. Finite flat and locally injective dimension
A quasi-coherent sheaf ${\mathcal{F}}$ on a scheme $X$ is said to have _flat
dimension not exceeding $d$_ if the flat dimension of the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ does not exceed $d$ for any
affine open subscheme $U\subset X$. If a quasi-coherent sheaf ${\mathcal{F}}$
on $X$ admits a left resolution by flat quasi-coherent sheaves (e. g., $X$ is
quasi-compact and semi-separated), then the flat dimension of ${\mathcal{F}}$
is equal to the minimal length of such resolution.
The property of a quasi-coherent sheaf to have flat dimension not exceeding
$d$ is local, since so is the property of a quasi-coherent sheaf to be flat.
Quasi-coherent sheaves of finite flat dimension form a full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions and kernels of
surjective morphisms; the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ of quasi-coherent sheaves of
flat dimension not exceeding $d$ is closed under the same operations, and also
under infinite direct sums.
Let us say that a quasi-coherent sheaf ${\mathcal{F}}$ on a scheme $X$ has
_very flat dimension not exceeding $d$_ if the very flat dimension of the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ does not exceed $d$ for any
affine open subscheme $U\subset X$ (see Section 1.5 for the definition). Over
a quasi-compact semi-separated scheme $X$, a quasi-coherent sheaf has very
flat dimension $\le d$ if and only if it admits a very flat left resolution of
length $\le\nobreak d$.
Since the property of a quasi-coherent sheaf to be very flat is local, so is
its property to have flat dimension not exceeding $d$ (cf. Lemma 1.5.4).
Quasi-coherent sheaves of very flat dimension $\le\nobreak d$ form a full
subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions, kernels of
surjective morphisms, and infinite direct sums. We denote the inductive limit
of the exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ as
$d\to\infty$ by $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}}$.
A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$
is said to have _locally injective dimension not exceeding $d$_ if the
injective dimension of the ${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$
does not exceed $d$ for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. Over a quasi-compact semi-separated scheme $X$, a
${\mathbf{W}}$-locally contraherent cosheaf has locally injective dimension
$\le d$ if and only if it admits a locally injective right resolution of
length $\le d$ in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
The property of a ${\mathbf{W}}$-locally contraherent cosheaf to have locally
injective dimension not exceeding $d$ is local and refinements of the covering
${\mathbf{W}}$ do not change it. ${\mathbf{W}}$-locally contraherent cosheaves
of finite locally injective dimension form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$ closed under
extensions and cokernels of admissible monomorphisms; the full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$ of quasi-
coherent sheaves of locally injective dimension not exceeding $d$ is closed
under the same operations, and also under infinite products. We set
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{\mathsf{flid}}$
and
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{{\operatorname{\mathsf{flid--}}}d}$.
For the rest of the section, let $X$ be a quasi-compact semi-separated scheme
with an open covering ${\mathbf{W}}$.
###### Corollary 4.9.1.
(a) For any symbol $\star\neq{\mathsf{ctr}}$ and any (finite) integer $d\ge
0$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ is an
equivalence of triangulated categories.
(b) For any symbol $\star\neq{\mathsf{ctr}}$ and any (finite) integer $d\ge
0$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ is an
equivalence of triangulated categories.
(c) For any symbol $\star\neq{\mathsf{co}}$ and any (finite) integer $d\ge 0$,
the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
is an equivalence of triangulated categories.
###### Proof.
Parts (a-b) follow from Proposition A.5.6, while part (c) follows from the
dual version of the same. ∎
###### Corollary 4.9.2.
(a) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ is an equivalence of
triangulated categories.
(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}}$ is an equivalence of
triangulated categories.
(c) For any symbol $\star={\mathsf{b}}$ or $+$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$ is an
equivalence of triangulated categories.
###### Proof.
The assertions concerning the case $\star={\mathsf{b}}$ follow from the
respective assertions of Corollary 4.9.1 by passage to the inductive limit as
$d\to\infty$. The assertions concerning the case $\star=-$ in parts (a-b)
follow from Proposition A.3.1(a), while the assertion about $\star=+$ in part
(c) follows from the dual version of it. ∎
###### Lemma 4.9.3.
If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the left homological dimension of any
object of the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
with respect to the full exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ does
not exceed $N-1$.
###### Proof.
In view of Corollary A.5.3, it suffices to show that any object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
admits an admissible epimorphism with respect to the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
from an object of
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$. We
will do more and show that the exact sequence (27) is a left resolution of an
object ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
by objects of
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$.
Indeed, the functor of inverse image with respect to a very flat
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism $f\colon Y\longrightarrow X$
takes
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
into
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{{\operatorname{\mathsf{flid--}}}d}$,
while the functor of direct image with respect to a flat
$({\mathbf{W}},{\mathbf{T}})$-affine morphism $f$ takes
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{{\operatorname{\mathsf{flid--}}}d}$
into
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$.
The sequence (27) is exact over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$,
since it is exact over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
is closed under admissible monomorphisms in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. ∎
###### Lemma 4.9.4.
(a) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the homological dimension of the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ does
not exceed $N+d$.
(b) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the homological dimension of the exact
category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
does not exceed $N-1+d$.
###### Proof.
Part (a): in fact, one proves the stronger assertion that
$\operatorname{Ext}^{>N+d}_{X}({\mathcal{F}},{\mathcal{M}})=0$ for any quasi-
coherent sheaf ${\mathcal{M}}$ and any quasi-coherent sheaf of very flat
dimension $\le d$ over $X$ (also, the $\operatorname{Ext}$ groups in the exact
category
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ agree
with those in the abelian category $X{\operatorname{\mathsf{--qcoh}}}$). Since
any object of
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ has a
finite left resolution of length $\le d$ by objects
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$, it suffices to consider
the case of ${\mathcal{F}}\in
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$.
The latter can be dealt with using the Čech resolution (12) of the sheaf
${\mathcal{M}}$ and the adjunction of exact functors $j^{*}$ and $j_{*}$ for
the embedding of an affine open subscheme $j\colon U\longrightarrow X$,
inducing the similar adjunction on the level of $\operatorname{Ext}$ groups
(cf. the proof of Lemma 5.4.1(b) below). Alternatively, the desired assertion
can be deduced from Lemma 4.6.1(a). The proof of part (b) is similar and can
be based either on the Čech resolution (27), or on Lemma 4.6.1(b). ∎
###### Corollary 4.9.5.
(a) The natural triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
and
${\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
are equivalences of triangulated categories. In particular, such functors
between the derived categories of the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ are equivalences of
categories.
(b) The natural triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
and
${\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
are equivalences of triangulated categories. In particular, such functors
between the derived categories of the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ are
equivalences of categories.
###### Proof.
Follows from the respective parts of Lemma 4.9.4 together with the result of
[50, Remark 2.1]. ∎
As a matter of notational convenience, set the triangulated category
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$ to be
the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
as $d\to\infty$ for any symbol $\star\neq{\mathsf{ctr}}$. Furthermore, set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$ to
be the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
as $d\to\infty$. For any morphism $f\colon Y\longrightarrow X$ into a quasi-
compact semi-separated scheme $X$ one constructs the left derived functor
(57)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
as the functor on the derived categories induced by the exact functor
$f^{*}\colon X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$. The left derived functor
(58)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$
is constructed in the similar way.
Analogously, set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
to be the inductive limit of (the equivalences of categories)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
as $d\to\infty$ for any symbol $\star\neq{\mathsf{co}}$. For any morphism
$f\colon Y\longrightarrow X$ into a quasi-compact semi-separated scheme $X$
and any open coverings ${\mathbf{W}}$ and ${\mathbf{T}}$ of the schemes $Y$
and $X$ for which the morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-coaffine,
the right derived functor
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{flid}})$
is constructed as the functor on the derived categories induced by the exact
functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$.
As usually, we set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})={\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{\mathsf{flid}})$.
Now Lemma 4.9.3 together with Proposition A.5.6 provide a natural equivalence
of triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$.
For a morphism $f\colon Y\longrightarrow X$ of quasi-compact semi-separated
schemes, such equivalences allow to define the derived functor
(59)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}),$
which clearly does not depend on the choice of the coverings ${\mathbf{W}}$
and ${\mathbf{T}}$.
### 4.10. Morphisms of finite flat dimension
A morphism of schemes $f\colon Y\longrightarrow X$ is said to have _flat
dimension not exceeding $D$_ if for any affine open subschemes $U\subset X$
and $V\subset Y$ such that $f(V)\subset U$ the ${\mathcal{O}}_{X}(U)$-module
${\mathcal{O}}_{Y}(V)$ has flat dimension not exceeding $D$. The morphism $f$
has _very flat dimension not exceeding $D$_ if the similar bound holds for the
very flat dimension of the ${\mathcal{O}}_{X}(U)$-modules
${\mathcal{O}}_{Y}(V)$.
For any morphism $f\colon Y\longrightarrow X$ of finite flat dimension $\le D$
into a quasi-compact semi-separated scheme $X$ and any symbol
$\star\neq{\mathsf{ctr}}$, the left derived functor
(60)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
is constructed in the following way. Let us call a quasi-coherent sheaf
${\mathcal{F}}$ on $X$ _adjusted to $f$_ if for any affine open subschemes
$U\subset X$ and $V\subset Y$ such that $f(V)\subset U$ one has
$\operatorname{Tor}^{{\mathcal{O}}_{X}(U)}_{>0}({\mathcal{O}}_{Y}(V),{\mathcal{F}}(U))=0$.
Quasi-coherent sheaves ${\mathcal{F}}$ on $X$ adjusted to $f$ form a full
subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions, kernels of
surjective morphisms and infinite direct sums, and such that any quasi-
coherent sheaf on $X$ has a finite left resolution of length $\le D$ by
sheaves from
$X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}}$. By
Proposition A.5.6, it follows that the natural functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.
The right exact functor $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ restricts to an exact functor $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$. In view of the above equivalence of
categories, the induced functor on the derived categories
$f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
provides the desired derived functor ${\mathbb{L}}f^{*}$. For any morphism of
finite flat dimension $f\colon Y\longrightarrow X$ between quasi-compact semi-
separated schemes $Y$ and $X$, the functor ${\mathbb{L}}f^{*}$ is left adjoint
to the functor ${\mathbb{R}}f_{*}$ (50) from Section 4.8 (cf. [53, Section
1.9]).
For any morphism $f\colon Y\longrightarrow X$ of finite very flat dimension
$\le D$ into a quasi-compact semi-separated scheme $X$, any open coverings
${\mathbf{W}}$ and ${\mathbf{T}}$ of the schemes $X$ and $Y$ for which the
morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-coaffine, and any symbol
$\star\neq{\mathsf{co}}$, the right derived functor
(61)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$
is constructed in the following way. Let us call a ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{P}}$ on $X$ _adjusted to $f$_ if for any
affine open subschemes $U\subset X$ and $V\subset Y$ such that $U$ is
subordinate to ${\mathbf{W}}$ and $f(V)\subset U$ one has
$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}^{>0}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U])=0$.
One can easily see that the adjustness condition does not change when
restricted to open subschemes $V$ subordinate to ${\mathbf{T}}$, nor it is
changed by a refinement of the covering ${\mathbf{W}}$.
Locally contraherent cosheaves on $X$ adjusted to $f$ form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}^{f{\operatorname{\mathsf{--adj}}}}\subset
X{\operatorname{\mathsf{--lcth}}}$ closed under extensions and cokernels of
admissible monomorphisms; the category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}=X{\operatorname{\mathsf{--lcth}}}^{f{\operatorname{\mathsf{--adj}}}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is also closed under infinite
products in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and such that any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ has a finite right
resolution of length $\le D$ by objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}$.
By the dual version of Proposition A.5.6, the natural functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories. The construction of the exact
functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$ from Section
3.3 extends without any changes to the case of cosheaves from
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}$,
defining an exact functor
$f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muY{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}.$
Instead of Lemma 1.2.3(a) used in Sections 2.3 and 3.3, one can use Lemma
1.5.5(a) in order to check that the contraadjustness condition is preserved
here.
In view of the above equivalence of triangulated categories, the induced
functor
$f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$
provides the desired functor ${\mathbb{R}}f^{!}$ (61). When both schemes $X$
and $Y$ are quasi-compact and semi-separated, one can use the equivalences of
categories from Corollary 4.6.3(a) in order to obtain the right derived
functor
(62)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}),$
which is right adjoint to the functor ${\mathbb{L}}f_{!}$ (51) from Section
4.8.
For a morphism $f\colon Y\longrightarrow X$ of flat dimension $\le D$ into a
quasi-compact semi-separated scheme $X$, any open coverings ${\mathbf{W}}$ and
${\mathbf{T}}$ of the schemes $X$ and $Y$ for which the morphism $f$ is
$({\mathbf{W}},{\mathbf{T}})$-coaffine, and any symbol
$\star\neq{\mathsf{co}}$, one can similarly construct the right derived
functor
(63)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathbf{W}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathbf{T}}^{\mathsf{lct}}).$
More precisely, a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ is said to be _adjusted to $f$_ if for any
affine open subschemes $U\subset X$ and $V\subset Y$ such that $U$ is
subordinate to ${\mathbf{W}}$ and $f(V)\subset U$, and for any flat
${\mathcal{O}}_{Y}(V)$-module $G$, one has
$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}(G,{\mathfrak{P}}[U])=0$. This
condition does not change when restricted to open subschemes $V\subset Y$
subordinate to ${\mathbf{T}}$, nor is it changed by any refinement of the
covering ${\mathbf{W}}$ of the scheme $X$.
As above, locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves on
$X$ adjusted to $f$ form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ closed under extensions,
cokernels of admissible monomorphisms, and infinite products. It follows from
Lemma 1.5.2(a) that any locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ has a finite right resolution of length $\le D$ by objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}}$.
Hence the natural functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is an equivalence of triangulated categories.
Now Lemma 1.5.5(b) allows to construct an exact functor
$f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muY{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}},$
and the induced functor
$f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}})$
provides the desired derived functor (63). When both schemes $X$ and $Y$ are
quasi-compact and semi-separated, one can use Corollary 4.6.4(a) in order to
obtain the right derived functor
(64)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}),$
which is right adjoint to the functor ${\mathbb{L}}f_{!}$ (52).
Let $X$ be a quasi-compact semi-separated scheme with an open covering
${\mathbf{W}}$.
###### Lemma 4.10.1.
(a) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the right homological dimension of any quasi-coherent sheaf of flat
dimension $\le d$ on $X$ with respect to the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ does not
exceed $N$.
(b) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the right homological dimension of any quasi-coherent sheaf of very flat
dimension $\le d$ on $X$ with respect to the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ does
not exceed $N$.
(c) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the left homological dimension of any
${\mathbf{W}}$-locally contraherent cosheaf of locally injective dimension
$\le d$ on $X$ with respect to the full exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
does not exceed $N-1$.
###### Proof.
Part (a): in view of Lemma 4.6.1(a) and the dual version of Corollary A.5.3,
it suffices to show that there exists an injective morphism from any given
quasi-coherent sheaf belonging to
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ into a
quasi-coherent sheaf belonging to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ with the
cokernel belonging to
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$. This
follows from Corollary 4.1.4(b) or 4.1.11(b). The proof of part (b) is
similar.
The proof of part (c) is analogous up to duality, and based on Lemma 4.6.1(b)
and Corollary 4.2.5(b) (alternatively, the argument from the proof of Lemma
4.9.3 is sufficient in this case). ∎
###### Lemma 4.10.2.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Then
(a) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$;
(b) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$;
(c) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$;
(d) a contraherent cosheaf on $X$ belongs to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of contraherent cosheaves from
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$.
###### Proof.
One has to repeat the arguments in Sections 4.1 and 4.2 working with,
respectively, quasi-coherent sheaves of (very) flat dimension $\le d$ only or
locally contraherent cosheaves of locally injective dimension $\le d$ only
throughout (cf. Lemma 4.1.6). ∎
Let $f\colon Y\longrightarrow X$ be a morphism of finite flat dimension $\le
D$ between quasi-compact semi-separated schemes $X$ and $Y$.
###### Corollary 4.10.3.
(a) The exact functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ takes objects of
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ to
objects of $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}(d+D)}$.
(b) If the morphism $f$ has very flat dimension not exceeding $D$, then the
exact functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ takes objects of
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ to
objects of $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}(d+D)}$.
(c) The exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ takes objects of
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ to
objects of $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}(d+D)}$.
###### Proof.
Part (a) follows from Lemma 4.10.2(a) together with the fact that the direct
image with respect to an affine morphism of flat dimension $\le D$ takes
quasi-coherent sheaves of flat dimension $\le d$ to quasi-coherent sheaves of
flat dimension $\le d+D$. The latter is provided by Lemma 1.5.2(a). The proof
of part (b) is similar and based on Lemmas 4.10.2(c) and 1.5.3(b).
Finally, part (c) follows from Lemma 4.10.2(d) together with the fact that the
direct image with respect to a $({\mathbf{W}},{\mathbf{T}})$-affine morphism
of flat dimension $\le D$ takes ${\mathbf{T}}$-locally contraherent cosheaves
of locally injective dimension $\le d$ to ${\mathbf{W}}$-locally contraherent
cosheaves of locally injective dimension $\le d+D$. The latter is provided by
Lemma 1.5.2(b). ∎
According to Lemma 4.10.1(a) and the dual version of Proposition A.5.6, for
any symbol $\star\neq{\mathsf{ctr}}$ the natural functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
is an equivalence of triangulated categories (as is the similar functor for
sheaves over $X$). So one can construct the right derived functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}(d+D)})$
as the functor on the derived categories induced by the exact functor
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}(d+D)}$ from
Corollary 4.10.3(a). Passing to the inductive limits as $d\to\infty$, we
obtain the right derived functor
(65)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}),$
which is right adjoint to the functor ${\mathbb{L}}f^{*}$ (57). For a morphism
$f$ of finite very flat dimension, the right derived functor
(66)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$
right adjoint to the functor ${\mathbb{L}}f^{*}$ (58) is constructed in the
similar way.
Analogously, according to Lemma 4.10.1(c) and Proposition A.5.6, for any
symbol $\star\neq{\mathsf{co}}$ the natural functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d})$
is an equivalence of triangulated categories (as is the similar functor for
cosheaves over $X$). Thus one can construct the left derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}(d+D)})$
as the functor on the derived categories induced by the exact functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}(d+D)}$
from Corollary 4.10.3(c). Passing to the inductive limits as $d\to\infty$, we
obtain the left derived functor
(67)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}),$
which is left adjoint to the functor ${\mathbb{R}}f^{!}$ (59).
### 4.11. Finite injective and projective dimension
For any scheme $X$, we denote the full subcategory of objects of injective
dimension $\le d$ in the abelian category $X{\operatorname{\mathsf{--qcoh}}}$
by $X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$. For
a quasi-compact semi-separated scheme $X$, the full subcategory of objects of
projective dimension $\le d$ in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is denoted by
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$
and the full subcategory of objects of projective dimension $\le d$ in the
exact category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
by
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$.
Set
$X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},\,{{\operatorname{\mathsf{fpd--}}}d}}$
and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\\{X\\},\,{{\operatorname{\mathsf{fpd--}}}d}}$.
Furthermore, the _colocally flat dimension_ of a ${\mathbf{W}}$-locally
contraherent cosheaf on a quasi-compact semi-separated scheme $X$ is defined
as its left homological dimension with respect to the exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Section A.5). The
colocally flat dimension is well-defined by Corollaries 4.3.2(b) and 4.3.4(b).
A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ has
colocally flat dimension $\le d$ if and only if
$\operatorname{Ext}^{>d}({\mathfrak{F}},{\mathfrak{P}})=0$ for any locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$.
The full subcategory of objects of colocally flat dimension $\le d$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is denoted by
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}}$.
We set
$X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{clfd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},\,{{\operatorname{\mathsf{clfd--}}}d}}$.
Since the subcategories of projective objects in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ do not depend
on the covering ${\mathbf{W}}$, and the full subcategories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ are closed under kernels of
admissible epimorphisms, the projective dimension of an object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ does not
change when the open covering ${\mathbf{W}}$ is replaced by its refinement.
Similarly, the colocally flat dimension of a ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ does not depend on the covering ${\mathbf{W}}$.
One can easily see that the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}$ is closed under extensions and cokernels of
admissible monomorphisms, while the full subcategories
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$,
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, and
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ are closed under extensions
and kernels of admissible epimorphisms.
###### Corollary 4.11.1.
(a) For any scheme $X$, the natural triangulated functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$
are equivalences of categories.
(b) For any quasi-compact semi-separated scheme $X$, the natural triangulated
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$
are equivalences of categories.
(c) For any quasi-compact semi-separated scheme $X$ and any symbol
$\star\neq{\mathsf{co}}$, the natural triangulated functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}})$
are equivalences of categories.
(d) For any quasi-compact semi-separated scheme $X$, the natural triangulated
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$
are equivalences of categories.
###### Proof.
Parts (b-d) follow from Proposition A.5.6, while part (a) follows from the
dual version of it. ∎
###### Corollary 4.11.2.
(a) For any scheme $X$, the natural functor
$\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories. For any symbol
$\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{co}}$,
or ${\mathsf{abs}}$, the natural triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
is fully faithful.
(b) For any quasi-compact semi-separated scheme $X$, the natural functors
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$
are equivalences of triangulated categories. For any scheme $X$ and any symbol
$\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{ctr}}$,
or ${\mathsf{abs}}$, the natural triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
is fully faithful.
(c) For any quasi-compact semi-separated scheme $X$, the natural functors
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
are equivalences of triangulated categories. For any scheme $X$ and any symbol
$\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{ctr}}$,
or ${\mathsf{abs}}$, the natural triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
is fully faithful.
###### Proof.
In each part (a-c), the first assertion follows from there being enough
injective/projective objects in the respective abelian/exact categories,
together with Proposition A.3.1(a) (or the dual version of it). In the second
assertions of parts (b-c), the notation
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ stands for
the full additive subcategories of projective objects in the exact categories
$X{\operatorname{\mathsf{--ctrh}}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ on an arbitrary scheme $X$.
Irrespectively of there being enough such projectives or injectives, these
kind of assertions hold in any exact category (or in any exact category with
infinite direct sums or products, as appropriate) by Lemma A.1.3. ∎
Let $X$ be a quasi-compact semi-separated scheme.
###### Lemma 4.11.3.
The equivalence of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ from Lemma 4.6.7 identifies
the exact subcategories
(a) $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$,
(b) $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$,
(c) $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{clfd--}}}d}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$,
(d) $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.
###### Proof.
Part (a): since the functor ${\mathcal{O}}_{X}\odot_{X}{-}$ takes short exact
sequences in $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ to short exact
sequences in $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ and commutes
with the direct images of contraherent cosheaves and quasi-coherent sheaves
from the affine open subschemes of $X$, it follows from Lemma 4.10.2(d) that
this functor takes $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ into
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$. To
prove the converse, consider a contraadjusted quasi-coherent sheaf
${\mathcal{P}}$ of injective dimension $d$ on $X$, and let
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{J}}^{0}\longrightarrow\dotsb\longrightarrow{\mathcal{J}}^{d}\longrightarrow
0$ be its right injective resolution of length $d$ in
$X{\operatorname{\mathsf{--qcoh}}}$.
This resolution is an exact sequence over the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$, so it is transformed to an
exact sequence over the exact category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ by the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$. The contraherent
cosheaves
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}}^{i})$ being
locally injective according to the proof of Corollary 4.6.8(b), it follows
that the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{P}})$ admits a
right resolution of length $d$ by locally injective contraherent cosheaves, i.
e., has locally injective dimension $\le\nobreak d$.
Part (d): by the proof of Corollary 4.6.8(a), the equivalence of exact
categories $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ with
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{lct}}_{\mathsf{clp}}$.
Furthermore, it follows from Lemma 4.10.2(b) that the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ takes
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ into
$X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}$,
since a cotorsion module of flat dimension $\le d$ over a commutative ring $R$
corresponds to a (locally) cotorsion contraherent cosheaf of projective
dimension $\le d$ over $\operatorname{Spec}R$. Conversely, a projective
resolution of length $d$ of an object of
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$ is
transformed by the functor ${\mathcal{O}}_{X}\odot_{X}{-}$ into a flat
resolution of length $d$ of the corresponding cotorsion quasi-coherent sheaf
(see the proof of Corolary 4.6.10(c)).
The proof of part (b) is similar and based on Lemma 4.10.2(c) and the proof of
Corollary 4.6.10(a), while the proof of part (c) is based on Lemma 4.10.2(a)
and the proof of Corollary 4.6.10(b). In both cases it is important that every
projective (or, respectively, colocally flat) contraherent cosheaf is
colocally projective. ∎
###### Corollary 4.11.4.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Then
(a) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$;
(b) a contraherent cosheaf on $X$ belongs to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of contraherent cosheaves from
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
U_{\alpha}{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}$;
(c) a contraherent cosheaf on $X$ belongs to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{clfd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of contraherent cosheaves from
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
U_{\alpha}{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{clfd--}}}d}$;
(d) a contraherent cosheaf on $X$ belongs to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}$
if and only if it is a direct summand of a finitely iterated extension of the
direct images of contraherent cosheaves from
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}$.
###### Proof.
Follows from Lemmas 4.10.2 and 4.11.3. ∎
Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes.
###### Lemma 4.11.5.
(a) Whenever the flat dimension of the morphism $f$ does not exceed $D$, the
functor of direct image $f_{*}$ takes injective quasi-coherent sheaves on $Y$
to quasi-coherent sheaves of injective dimension $\le D$ on $X$.
(b) Whenever the very flat dimension of the morphism $f$ does not exceed $D$,
the functor of direct image $f_{!}$ takes projective contraherent cosheaves on
$Y$ to contraherent cosheaves of projective dimension $\le D$ on $X$.
(c) Whenever the flat dimension of the morphism $f$ does not exceed $D$, the
functor of direct image $f_{!}$ takes colocally flat contraherent cosheaves on
$Y$ to contraherent cosheaves of colocally flat dimension $\le D$ on $X$.
(d) Whenever the flat dimension of the morphism $f$ does not exceed $D$, the
functor of direct image $f_{!}$ takes projective locally cotorsion
contraherent cosheaves on $Y$ to locally cotorsion contraherent cosheaves of
projective dimension $\le D$ on $X$.
###### Proof.
In part (a) it actually suffices to assume that the scheme $Y$ is quasi-
compact and quasi-separated (while the scheme $X$ has to be quasi-compact and
semi-separated). Let us prove part (c), parts (a), (b), (d) being analogous.
The functor $f_{!}$ takes $Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$
to $X{\operatorname{\mathsf{--ctrh}}}$ by Corollary 4.5.3(a), so it remains to
check that $\operatorname{Ext}^{X,>D}(f_{!}{\mathfrak{F}},{\mathfrak{P}})=0$
for any cosheaves ${\mathfrak{F}}\in
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ and ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.
According to the adjunction of derived functors ${\mathbb{L}}f_{!}$ (51) and
${\mathbb{R}}f^{!}$ (62), one has
$\operatorname{Ext}^{X,*}(f_{!}{\mathfrak{F}},{\mathfrak{P}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{b}}(Y{\operatorname{\mathsf{--lcth}}})}({\mathfrak{F}},\>{\mathbb{R}}f^{!}({\mathfrak{P}})[*])$.
Hence it suffices to show that the object
${\mathbb{R}}f^{!}({\mathfrak{P}})\in{\mathsf{D}}^{\mathsf{b}}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
can be represented by a finite complex over
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ concentrated in the
cohomological degrees $\le D$. The latter is true because any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ admits a finite right
resolution of length $\le D$ by $f$-adjusted locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves. ∎
Set the triangulated category
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fid}})$ to be
the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$
as $d\to\infty$ for any $\star\neq{\mathsf{co}}$, ${\mathsf{ctr}}$. For any
morphism of finite flat dimension $f\colon Y\longrightarrow X$ between quasi-
compact semi-separated schemes, one constructs the right derived functor
(68)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fid}})$
as the functor on the homotopy/derived categories induced by the additive
functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D}$ from
Lemma 4.11.5(a).
Analogously, set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{fpd}})$ to be
the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d})$
as $d\to\infty$ for any $\star\neq{\mathsf{co}}$, ${\mathsf{ctr}}$. Besides,
set
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clfd}})$
to be the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{clfd--}}}d})$
as $d\to\infty$, and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{fpd}})$
to be the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d})$
as $d\to\infty$. For any morphism of finite flat dimension $f\colon
Y\longrightarrow X$ between quasi-compact semi-separated schemes, one
constructs the left derived functor
(69)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{fpd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{fpd}})$
as the functor on the homotopy/derived categories induced by the additive
functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}D}$
from Lemma 4.11.5(d). The left derived functor
(70)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clfd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clfd}})$
is constructed in the similar way. Finally, for a morphism $f$ of finite very
flat dimension, one can similarly construct the left derived functor
(71)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{fpd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{fpd}}).$
The following corollary provides three restricted versions of Theorem 4.8.1.
###### Corollary 4.11.6.
(a) Assume that the morphism $f$ has finite flat dimension. Then for any
symbol $\star\neq{\mathsf{co}}$ the equivalences of categories
$\mathsf{Hot}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
and
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
from Corollary 4.6.8(b) transform the right derived functor
${\mathbb{R}}f_{*}$ (68) into the left derived functor ${\mathbb{L}}f_{!}$
(67).
(b) Assume that the morphism $f$ has finite very flat dimension. Then for any
symbol $\star\neq{\mathsf{ctr}}$ the equivalences of categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq\mathsf{Hot}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$
from Corollary 4.6.10(a) transform the right derived functor
${\mathbb{R}}f_{*}$ (66) into the left derived functor ${\mathbb{L}}f_{!}$
(71).
(c) Assume that the morphism $f$ has finite flat dimension. Then for any
symbol $\star\neq{\mathsf{co}}$, ${\mathsf{ctr}}$ the equivalences of
categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})$
from Corollary 4.6.10(b) transform the right derived functor
${\mathbb{R}}f_{*}$ (65) into the left derived functor ${\mathbb{L}}f_{!}$
(70).
###### Proof.
Part (a): Clearly, one can assume $\star\neq{\mathsf{ctr}}$. The composition
of equivalences of triangulated categories
${\mathsf{D}}^{\star}(X\allowbreak{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}D})$
can be constructed directly in terms of the equivalence of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{flid--}}}D}$ from
Lemma 4.11.3(a). Now this equivalence of exact categories together with the
equivalence of additive categories
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}\simeq
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$ used in the
proof of Corollary 4.6.8(b) transform the functor $f_{*}$ into the functor
$f_{!}$, which implies the desired assertion. Parts (b) and (c) are similarly
proved using Lemma 4.11.3(b-c). ∎
### 4.12. Derived tensor operations
Let $X$ be a quasi-compact semi-separated scheme. The functor of tensor
product
(72)
$\otimes_{{\mathcal{O}}_{X}}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\times{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is well-defined for any symbol $\star={\mathsf{co}}$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, since the tensor product of a complex
coacyclic (respectively, absolutely acyclic) over the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and any complex over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ is coacyclic (resp.,
absolutely acyclic) over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$.
The functor of tensor product
(73)
$\otimes_{{\mathcal{O}}_{X}}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\times{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
is well-defined for the similar reasons (though in this case the situation is
actually simpler; see below).
Furthermore, the functor of tensor product
(74)
$\otimes_{{\mathcal{O}}_{X}}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\times{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
is well-defined for $\star={\mathsf{co}}$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, since the tensor product of a complex
coacyclic (resp., absolutely acyclic) over the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and any complex over
$X{\operatorname{\mathsf{--qcoh}}}$ is coacyclic (resp., absolutely acyclic)
over $X{\operatorname{\mathsf{--qcoh}}}$, as is the tensor product of any
complex over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and a complex
coacyclic (resp., absolutely acyclic) over
$X{\operatorname{\mathsf{--qcoh}}}$. In view of Lemma A.1.2, the functors
(72–74) are also well-defined for $\star=+$; and it is a standard fact that
they are well-defined for $\star={\mathsf{b}}$ or $-$. Thus
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
become tensor triangulated categories for any $\star\neq\empt$,
${\mathsf{ctr}}$, and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$ is a triangulated
module category over
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.
Recall the definition of a homotopy flat complex of flat quasi-coherent
sheaves from Section 4.7. The left derived functor of tensor product
(75)
$\otimes_{{\mathcal{O}}_{X}}^{\mathbb{L}}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\times{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
is constructed for $\star=\empt$ by applying the functor
$\otimes_{{\mathcal{O}}_{X}}$ of tensor product of complexes of quasi-coherent
sheaves to homotopy flat complexes of flat quasi-coherent sheaves in one of
the arguments and arbitrary complexes of quasi-coherent sheaves in the other
one. This derived functor is well-defined by Theorem 4.7.1. In the case of
$\star=-$, the derived functor (75) can be constructed by applying the functor
$\otimes_{{\mathcal{O}}_{X}}$ to bounded above complexes of flat quasi-
coherent sheaves in one of the arguments. So
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$ is a tensor
triangulated category for any symbol $\star=-$ or $\empt$.
The full triangulated subcategory of _homotopy very flat complexes_ of very
flat quasi-coherent sheaves
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})^{\mathsf{vfl}}$
in the unbounded derived category of the exact category of very flat quasi-
coherent sheaves
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$ on $X$ is
defined as the minimal triangulated subcategory containing the objects of
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ and closed under infinite
direct sums (cf. Section 4.7). By Corollary 4.1.4(a) and Proposition A.4.3,
the composition of natural triangulated functors
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})^{\mathsf{vfl}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.
By Corollary 4.9.5(a), any acyclic complex over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ is absolutely acyclic over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$. Hence the composition of
functors
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})^{\mathsf{vfl}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
provides a natural functor
(76)
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}),$
which is clearly a tensor triangulated functor between these tensor
triangulated categories. In fact, this functor is also fully faithful, and
left adjoint to the composition of Verdier localization functors
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
(see Corollary A.4.8). Composing the functor (76) with the tensor action
functor (74), we obtain the left derived functor
(77)
$\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\times{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
making ${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ a
triangulated module category over the triangulated tensor category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ (cf. [21, Section 1.4]).
Given a symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, we
set $\star^{\prime}$ to be the “dual”symbol ${\mathsf{b}}$, $-$, $+$, $\empt$,
${\mathsf{abs}}-$, ${\mathsf{abs}}+$, ${\mathsf{ctr}}$, ${\mathsf{co}}$, or
${\mathsf{abs}}$, respectively. Furthermore, let ${\mathbf{W}}$ be an open
covering of the scheme $X$. Given two complexes
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ and
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ such that the
${\mathbf{W}}$-locally contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}}^{i},{\mathfrak{J}}^{j})$ is
well-defined by the constructions of Section 3.6 for every pair
$(i,j)\in{\mathbb{Z}}^{2}$, we set
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
to be the total complex of the bicomplex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}}^{i},{\mathfrak{J}}^{j})$
over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ constructed by taking
infinite products of ${\mathbf{W}}$-locally contraherent cosheaves along the
diagonals of the bicomplex.
The functor of cohomomorphisms
(78)
$\operatorname{\mathfrak{Cohom}}_{X}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{op}}\times{\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is well-defined for any symbol $\star={\mathsf{co}}$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, since the
$\operatorname{\mathfrak{Cohom}}$ from a complex coacyclic (respectively,
absolutely acyclic) over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$
into any complex over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is a
contraacyclic (resp., absolutely acyclic) complex over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$, as is the
$\operatorname{\mathfrak{Cohom}}$ from any complex over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ into a complex contraacyclic
(resp., absolutely acyclic) over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. The functor
of cohomomorphisms
(79)
$\operatorname{\mathfrak{Cohom}}_{X}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})^{\mathsf{op}}\times{\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is well-defined for the similar reasons. In view of Lemma A.1.2, the functors
(78–79) are also well-defined for $\star=+$; and one can straightforwardly
check that they are well-defined for $\star={\mathsf{b}}$ or $-$. Thus the
category opposite to
${\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
becomes a triangulated module category over the tensor triangulated category
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ and
the category opposite to
${\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is a triangulated module category over the tensor triangulated category
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$ for
any symbol $\star\neq\empt$, ${\mathsf{ctr}}$.
Recall the definition of a homotopy locally injective complex of locally
injective ${\mathbf{W}}$-locally contraherent cosheaves from Section 4.7. The
right derived functor of cohomomorphisms
(80)
${\mathbb{R}}\operatorname{\mathfrak{Cohom}}_{X}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{op}}\times{\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is constructed for $\star=\empt$ by applying the functor
$\operatorname{\mathfrak{Cohom}}_{X}$ to homotopy very flat complexes of very
flat quasi-coherent sheaves in the first argument and arbitrary complexes of
${\mathbf{W}}$-locally contraherent cosheaves in the second argument, or
alternatively, to arbitrary complexes of quasi-coherent sheaves in the first
argument and homotopy locally injective complexes of locally injective
${\mathbf{W}}$-locally contraherent cosheaves in the second argument. This
derived functor is well-defined by (the proof of) Theorem 4.7.2 (cf. [50,
Lemma 2.7]). In the case of $\star=-$, the derived functor (80) can be
constructed by applying the functor $\operatorname{\mathfrak{Cohom}}_{X}$ to
bounded bounded above complexes of very flat quasi-coherent sheaves in the
first argument or bounded below complexes of locally injective
${\mathbf{W}}$-locally contraherent cosheaves in the second argument. So
${\mathsf{D}}^{\star^{\prime}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})^{\mathsf{op}}$
is a triangulated module category over the tensor triangulated category
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$ for any symbol
$\star=-$ or $\empt$.
Composing the functor (76) with the tensor action functor (78), one obtains
the right derived functor
(81)
${\mathbb{R}}^{\prime}\operatorname{\mathfrak{Cohom}}_{X}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{op}}\times{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
making
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})^{\mathsf{op}}$
a triangulated module category over the triangulated tensor category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$. Similarly, the composition
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})^{\mathsf{vfl}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
provides a natural functor
(82)
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}),$
which is a tensor triangulated functor between these tensor triangulated
categories. It is also fully faithful, and left adjoint to the composition of
Verdier localization functors
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.
Composing the functor (82) with the tensor action functor (79), one obtains
the left derived functor
(83)
${\mathbb{R}}^{\prime}\operatorname{\mathfrak{Cohom}}_{X}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{op}}\times{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
making
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})^{\mathsf{op}}$
a triangulated module category over the triangulated tensor category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.
## 5\. Noetherian Schemes
### 5.1. Projective locally cotorsion contraherent cosheaves
Let $X$ be a (not necessarily semi-separated) locally Noetherian scheme and
${\mathbf{W}}$ be its open covering. The following theorem is to be compared
to the classification of injective quasi-coherent sheaves on $X$ [29,
Proposition II.7.17].
###### Theorem 5.1.1.
(a) There are enough projective objects in the exact categories of locally
cotorsion locally contraherent cosheaves
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$, and all these projective
objects belong to the full subcategory of locally cotorsion contraherent
cosheaves $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. The full
subcategories of projective objects in the three exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ coincide.
(b) For any scheme point $x\in X$, denote by $\widehat{\mathcal{O}}_{x,X}$ the
completion of the local ring ${\mathcal{O}}_{x,X}$ and by
$\iota_{x}\colon\operatorname{Spec}{\mathcal{O}}_{x,X}\longrightarrow X$ the
natural morphism. Then a locally cotorsion contraherent cosheaf
${\mathfrak{F}}$ on $X$ is projective if and only if it is isomorphic to an
infinite product $\prod_{x\in X}\iota_{x}{}_{!}\widecheck{F}_{x}$ of the
direct images $\iota_{x}{}_{!}\widecheck{F}_{x}$ of the contraherent cosheaves
$\widecheck{F}_{x}$ on $\operatorname{Spec}{\mathcal{O}}_{x,X}$ corresponding
to some free contramodules $F_{x}$ over the complete Noetherian local rings
$\widehat{\mathcal{O}}_{x,X}$ (viewed as ${\mathcal{O}}_{x,X}$-modules via the
restriction of scalars).
###### Proof.
For a quasi-compact semi-separated scheme $X$, the assertions of part (a) were
proven in Section 4.4. In the general case, we will prove parts (a) and (b)
simultaneously. The argument is based on Theorem 1.3.8.
First of all, let us show that the cosheaf of ${\mathcal{O}}_{X}$-modules
$\iota_{x}{}_{!}\widecheck{P}_{x}$ is a locally cotorsion contraherent cosheaf
on $X$ for any contramodule $P_{x}$ over $\widehat{\mathcal{O}}_{x,X}$. It
suffices to check that the restriction of $\iota_{x}{}_{!}\widecheck{P}_{x}$
to any affine open subscheme $U\subset X$ is a (locally) cotorsion
contraherent cosheaf. If the point $x$ belongs to $U$, then the morphism
$\iota_{x}$ factorizes through $U$; denoting the morphism
$\operatorname{Spec}{\mathcal{O}}_{x,X}\longrightarrow U$ by $\kappa_{x}$, one
has
$(\iota_{x}{}_{!}\widecheck{P}_{x})|_{U}\simeq\kappa_{x}{}_{!}\widecheck{P}_{x}$.
The morphism $\kappa_{x}$ being affine and flat, and the contraherent cosheaf
$\widecheck{P}_{x}$ over $\operatorname{Spec}{\mathcal{O}}_{x,X}$ being
(locally) cotorsion by Proposition 1.3.7(a),
$\kappa_{x}{}_{!}\widecheck{P}_{x}$ is a (locally) cotorsion contraherent
cosheaf over $U$.
If the point $x$ does not belong to $U$, we will show that the
${\mathcal{O}}_{X}(U)$-module
$(\iota_{x}{}_{!}\widecheck{P}_{x})[U]\simeq\widecheck{P}_{x}[\iota_{x}^{-1}(U)]$
vanishes. So it will follow that the restriction of
$\iota_{x}{}_{!}\widecheck{P}_{x}$ to the open complement of the closure of
the point $x$ in $X$ is a zero cosheaf. Indeed, it suffices to check that the
module of cosections $\widecheck{P}_{x}[V]$ vanishes for any principal affine
open subscheme $V\subset\operatorname{Spec}{\mathcal{O}}_{x,X}$ that does not
contain the closed point. In other words, it has to be shown that
$\operatorname{Hom}_{{\mathcal{O}}_{x,X}}({\mathcal{O}}_{x,X}[s^{-1}],P_{x})=0$
for any element $s$ from the maximal ideal of the local ring
${\mathcal{O}}_{x,X}$. This holds for any
$\widehat{\mathcal{O}}_{x,X}$-contramodule $P_{x}$; see [54, Theorem
B.1.1(2c)].
It follows that cosheaves of the form
$\prod_{x}\iota_{x}{}_{!}\widecheck{P}_{x}$ on $X$, where $P_{x}$ are some
$\widehat{\mathcal{O}}_{x,X}$-contramodules, belong to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. Let us show that cosheaves
${\mathfrak{F}}=\prod_{x}\iota_{x}{}_{!}\widecheck{F}_{x}$, where $F_{x}$
denote some free $\widehat{\mathcal{O}}_{x,X}$-contramodules, are projective
objects in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Pick an
affine open covering $U_{\alpha}$ of the scheme $X$ subordinate to the
covering ${\mathbf{W}}$, and choose a well-ordering of the set of indices
$\\{\alpha\\}$.
Given an index $\alpha$, denote by ${\mathfrak{F}}_{\alpha}$ the product of
the cosheaves $\iota_{z}{}_{!}\widecheck{F}_{z}$ on $X$ taken over all points
$z\in S_{\alpha}=U_{\alpha}\setminus\bigcup_{\beta<\alpha}U_{\beta}$. Clearly,
one has ${\mathfrak{F}}\simeq\prod_{\alpha}{\mathfrak{F}}_{\alpha}$.
Furthermore, since $X$ is locally Noetherian, for any affine (or even quasi-
compact) open subscheme $U\subset X$ one has $U\cap S_{\alpha}=\empt$, and
consequently ${\mathfrak{F}}_{\alpha}[U]=0$, for all but a finite number of
indices $\alpha$. We conclude that the cosheaf ${\mathfrak{F}}$ is also the
direct sum of the cosheaves ${\mathfrak{F}}_{\alpha}$ (taken in the category
of cosheaves of ${\mathcal{O}}_{X}$-modules). Hence it suffices to check that
each ${\mathfrak{F}}_{\alpha}$ is a projective object in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$.
Denoting by $j_{\alpha}$ the open embedding $U_{\alpha}\longrightarrow X$ and
by $\kappa_{z}$ the natural morphisms
$\operatorname{Spec}{\mathcal{O}}_{z,X}\longrightarrow U_{\alpha}$, we notice
that ${\mathfrak{F}}_{\alpha}=j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}$, where
${\mathfrak{G}}_{\alpha}=\prod_{z\in
S_{\alpha}}\kappa_{z}{}_{!}\widecheck{F}_{z}$. By Theorem 1.3.8,
${\mathfrak{G}}_{\alpha}$ is a projective object in
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$; and by the
adjunction (24) it follows that ${\mathfrak{F}}_{\alpha}$ is a projective
object in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$.
Now let us construct for any locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{Q}}$ on $X$ an admissible epimorphism onto
${\mathfrak{Q}}$ from an object ${\mathfrak{F}}=\prod_{x\in
X}\iota_{x}{}_{!}\widecheck{F}_{x}$ in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Choose an
affine open covering $U_{\alpha}$ as above, and proceed by transfinite
induction in $\alpha$, constructing contraherent cosheaves
${\mathfrak{F}}_{\alpha}=\prod_{z\in
S_{\alpha}}\iota_{z}{}_{!}\widecheck{F}_{z}$ and morphisms of locally
contraherent cosheaves ${\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{Q}}$.
Suppose that such cosheaves and morphisms have been constructed for all
$\alpha<\beta$; then, as it was explained above, there is the induced morphism
of locally contraherent cosheaves
$\prod_{\alpha<\beta}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{Q}}$.
Assume that the related morphism of the modules of cosections
$\prod_{\alpha<\beta}{\mathfrak{F}}_{\alpha}[U]=\bigoplus_{\alpha<\beta}{\mathfrak{F}}_{\alpha}[U]\longrightarrow{\mathfrak{Q}}[U]$
is an admissible epimorphism of cotorsion ${\mathcal{O}}_{X}(U)$-modules for
any affine open subscheme $U\subset\bigcup_{\alpha<\beta}U_{\alpha}\subset X$
subordinate to ${\mathbf{W}}$. We are going to construct a contraherent
cosheaf ${\mathfrak{F}}_{\beta}=\prod_{z\in
S_{\beta}}\iota_{z}{}_{!}\widecheck{F}_{z}$ and a morphism of locally
contraherent cosheaves ${\mathfrak{F}}_{\beta}\longrightarrow{\mathfrak{Q}}$
such that the induced morphism
$\prod_{\alpha\le\beta}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{Q}}$
satisfies the above condition for any affine open subscheme
$U\subset\bigcup_{\alpha\le\beta}U_{\alpha}\subset X$ subordinate to
${\mathbf{W}}$.
Pick an admissible epimorphism in
$U_{\beta}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ onto the (locally)
cotorsion contraherent cosheaf $j_{\beta}^{!}{\mathfrak{Q}}$ from a projective
(locally) cotorsion contraherent cosheaf ${\mathfrak{G}}$ on $U_{\beta}$. By
Theorem 1.3.8, the cosheaf ${\mathfrak{G}}$ decomposes into a direct product
${\mathfrak{G}}\simeq\prod_{x\in U_{\beta}}\kappa_{x}{}_{!}\widecheck{G}_{x}$,
where $G_{x}$ are some free $\widehat{\mathcal{O}}_{x,X}$-contramodules. Let
us rewrite this product as ${\mathfrak{G}}=\prod_{z\in
S_{\beta}}\kappa_{z}{}_{!}\widecheck{G}_{z}\oplus\prod_{y\in
U_{\beta}\setminus S_{\beta}}\kappa_{y}{}_{!}\widecheck{G}_{y}$; denote the
former direct summand by ${\mathfrak{G}}_{\beta}$ and the latter one by
${\mathfrak{E}}$. Set
${\mathfrak{F}}=\prod_{\alpha<\beta}{\mathfrak{F}}_{\alpha}$ and
${\mathfrak{F}}_{\beta}=j_{\beta}{}_{!}{\mathfrak{G}}_{\beta}$.
The property of a morphism of locally cotorsion ${\mathbf{T}}$-locally
contraherent cosheaves to be an admissible epimorphism being local, we only
need to show that the natural morphism
${\mathfrak{F}}\oplus{\mathfrak{F}}_{\beta}\longrightarrow{\mathfrak{Q}}$
becomes an admissible epimorphism in
$U_{\beta}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ when restricted to
$U_{\beta}$. In other words, this means that the morphism
$j_{\beta}^{!}{\mathfrak{F}}\oplus{\mathfrak{G}}_{\beta}\longrightarrow
j_{\beta}^{!}{\mathfrak{Q}}$ should be an admissible epimorphism in
$U_{\beta}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. It suffices to
check that the admissible epimorphism ${\mathfrak{G}}\longrightarrow
j_{\beta}^{!}{\mathfrak{Q}}$ factorizes through the morphism in question, or
that the morphism ${\mathfrak{E}}\longrightarrow j_{\beta}^{!}{\mathfrak{Q}}$
factorizes through the morphism $j_{\beta}^{!}{\mathfrak{F}}\longrightarrow
j_{\beta}^{!}{\mathfrak{Q}}$.
Denote by $j$ the open embedding $U_{\beta}\setminus S_{\beta}\longrightarrow
U_{\beta}$; then one has ${\mathfrak{E}}\simeq j_{!}{\mathfrak{L}}$, where
${\mathfrak{L}}$ is a projective object in $(U_{\beta}\setminus
S_{\beta}){\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ (as we have proven
above). Hence the morphism ${\mathfrak{L}}\longrightarrow
j^{!}j_{\beta}^{!}{\mathfrak{Q}}$ factorizes through an admissible epimorphism
of locally cotorsion contraherent cosheaves
$j^{!}j_{\beta}^{!}{\mathfrak{F}}\longrightarrow
j^{!}j_{\beta}^{!}{\mathfrak{Q}}$, as desired.
We have proven part (a); and to finish the proof of part (b) it remains to
show that the class of contraherent cosheaves of the form
$\prod_{x}\iota_{x}{}_{!}\widecheck{F}_{x}$ on $X$ is closed under the passage
to direct summands. The argument is based on the following lemma.
###### Lemma 5.1.2.
(a) Suppose that the set of all scheme points of $X$ is presented as a union
of two nonintersecting subsets $X=S\sqcup T$ such that for any points $z\in S$
and $y\in T$ the closure of $z$ in $X$ does not contain $y$. Then for any
cosheaves of ${\mathcal{O}}$-modules ${\mathfrak{P}}_{y}$ over
$\operatorname{Spec}{\mathcal{O}}_{y,X}$ and any contramodules $P_{z}$ over
$\widehat{\mathcal{O}}_{z,X}$ one has
$\operatorname{Hom}^{{\mathcal{O}}_{X}}(\prod_{y\in
T}\iota_{y}{}_{!}{\mathfrak{P}}_{y},\>\prod_{z\in
S}\iota_{z}{}_{!}\widecheck{P}_{z})=0$.
(b) For any scheme point $x\in X$, the functor assigning to an
$\widehat{\mathcal{O}}_{x,X}$-contramodule $P_{x}$ the locally cotorsion
contraherent cosheaf $\iota_{x}{}_{!}\widecheck{P}_{x}$ on $X$ is fully
faithful.
###### Proof.
Part (a): by the definition of the infinite product, it suffices to show that
$\operatorname{Hom}^{{\mathcal{O}}_{X}}(\prod_{y\in
T}\iota_{y}{}_{!}{\mathfrak{P}}_{y},\>\iota_{z}{}_{!}\widecheck{P}_{z})=0$ for
any $z\in S$. Let $Z$ be the closure of $z$ in $X$ and $Y=X\setminus Z$ be its
complement; then one has $T\subset Y$ and $Y$ is an open subscheme in $X$. Let
$j$ denote the open embedding $Y\longrightarrow X$. Given $y\in T$, denote the
natural morphism $\operatorname{Spec}{\mathcal{O}}_{y,X}\longrightarrow Y$ by
$\kappa_{y}$, so $\iota_{y}=j\circ\kappa_{y}$.
Now we have $\prod_{y\in T}\iota_{y}{}_{!}{\mathfrak{P}}_{y}\simeq
j_{!}\prod_{y\in T}\kappa_{y}{}_{!}{\mathfrak{P}}_{y}$ and, according to the
adjunction (26),
$\textstyle\operatorname{Hom}^{{\mathcal{O}}_{X}}(j_{!}\prod_{y\in
T}\kappa_{y}{}_{!}{\mathfrak{P}}_{y},\>\iota_{z}{}_{!}\widecheck{P}_{z})\simeq\operatorname{Hom}^{{\mathcal{O}}_{Y}}(\prod_{y\in
T}\kappa_{y}{}_{!}{\mathfrak{P}}_{y},\>j^{!}\iota_{z}{}_{!}\widecheck{P}_{z}).$
It was shown above that $j^{!}\iota_{z}{}_{!}\widecheck{P}_{z}=0$, so we are
done.
Part (b): it was explained in Section 1.3 that the functor assigning to an
$\widehat{\mathcal{O}}_{x,X}$-contramodule $P_{x}$ the locally cotorsion
contraherent cosheaf $\widecheck{P}_{x}$ on
$\operatorname{Spec}{\mathcal{O}}_{x,X}$ is fully faithful. The morphism
$\iota_{x}$ being flat and coaffine, the adjunction (26) applies and it
suffices to show that the adjunction morphism
$\widecheck{P}_{x}\longrightarrow\iota_{x}^{!}\iota_{x}{}_{!}\widecheck{P}_{x}$
is an isomorphism in
$\operatorname{Spec}{\mathcal{O}}_{x,X}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.
One can replace the scheme $X$ by any affine open subscheme $U\subset X$
containing $x$, and it remains to use the isomorphism
${\mathcal{O}}_{x,X}\otimes_{{\mathcal{O}}(U)}{\mathcal{O}}_{x,X}\simeq{\mathcal{O}}_{x,X}$.
∎
Now we can finish the proof of Theorem. Let $S$ be a subset of scheme points
of $X$ closed under specialization. It follows from Lemma 5.1.2(a) that any
contraherent cosheaf ${\mathfrak{F}}$ on $X$ isomorphic to $\prod_{x\in
X}\iota_{x}{}_{!}\widecheck{F}_{x}$ is endowed with a natural projection onto
its direct summand ${\mathfrak{F}}(S)=\prod_{z\in
S}\iota_{z}{}_{!}\widecheck{F}_{z}$, and such projections commute with any
morphisms between contraherent cosheaves ${\mathfrak{F}}$ of this form. Given
two subsets $S^{\prime}\subset S\subset X$ closed under specialization, there
is a natural projection
${\mathfrak{F}}(S)\longrightarrow{\mathfrak{F}}(S^{\prime})$, and the diagram
formed by all such projections is commutative.
Any idempotent endomorphism $e$ of the contraherent cosheaf ${\mathfrak{F}}$
acts on this diagram. When the complement $S\setminus S^{\prime}$ consists of
a single scheme point $x\in X$, the kernel of the projection
${\mathfrak{F}}(S)\longrightarrow{\mathfrak{F}}(S^{\prime})$ is isomorphic to
the direct image $\iota_{X}{}_{!}\widecheck{F}_{x}$. The endomorphism $e$
induces an idempotent endomorphism of this kernel; by Lemma 5.1.2(b), the
latter endomorphism comes from an idempotent endomorphism $e_{x}$ of the free
$\widehat{\mathcal{O}}_{x,X}$-contramodule $F_{x}$. By [54, Lemma 1.3.2], the
image $e_{x}F_{x}$ is also a free $\widehat{\mathcal{O}}_{x,X}$-contramodule.
Using appropriate transfinite induction procedures, it is not difficult to
show that the infinite product $\prod_{x\in
X}\iota_{x}{}_{!}\widecheck{(e_{x}F_{x})}$ is isomorphic to the image of the
idempotent endomorphism $e$ of the contraherent cosheaf ${\mathfrak{F}}$. ∎
As in Section 4.4, we denote the category of projective locally cotorsion
contraherent cosheaves on $X$ by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. The
following corollary says that being a projective locally cotorsion
contraherent cosheaf on a locally Noetherian scheme is a local property.
Besides, all such cosheaves are coflasque (see Section 3.4). The similar
properties of injective quasi-coherent sheaves are usually deduced from [29,
Theorem II.7.18].
###### Corollary 5.1.3.
(a) Let $Y\subset X$ be an open subscheme. Then for any cosheaf
${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ the cosheaf
${\mathfrak{F}}|_{Y}$ belongs to
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
(b) In the situation of (a), the corestriction map
${\mathfrak{F}}[Y]\longrightarrow{\mathfrak{F}}[X]$ is injective. If the
scheme $X$ is affine, then
$0\longrightarrow{\mathfrak{F}}[Y]\longrightarrow{\mathfrak{F}}[X]\longrightarrow{\mathfrak{F}}[X]/{\mathfrak{F}}[Y]\longrightarrow
0$ is a split short exact sequence of flat cotorsion
${\mathcal{O}}(X)$-modules.
(c) Let $X=\bigcup_{\alpha}Y_{\alpha}$ be an open covering. Then a locally
contraherent cosheaf on $X$ belongs to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ if and only
if its restrictions to $Y_{\alpha}$ belong to
$Y_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ for
all $\alpha$.
###### Proof.
Part (a): let $j$ denote the open embedding $Y\longrightarrow X$. Then by
Theorem 5.1.1 one has $j^{!}{\mathfrak{F}}\simeq j^{!}(\prod_{x\in
X}\iota_{x}{}_{!}\widecheck{F}_{x})\simeq\prod_{x\in
X}j^{!}\iota_{x}{}_{!}\widecheck{F}_{x}$. Furthermore, it was explained in the
proof of the same Theorem that $j^{!}\iota_{z}{}_{!}\widecheck{F}_{z}=0$ for
any $z\in X\setminus Y$. Denoting by $\kappa_{y}$ the natural morphism
$\operatorname{Spec}{\mathcal{O}}_{y,X}\longrightarrow Y$ for a point $y\in
Y$, one clearly has
$j^{!}\iota_{y}{}_{!}\widecheck{F}_{y}\simeq\kappa_{y}{}_{!}\widecheck{F}_{y}$.
Hence the isomorphism $j^{!}{\mathfrak{F}}\simeq\prod_{y\in
Y}\kappa_{y}\widecheck{F}_{y}$, proving that $j^{!}{\mathfrak{F}}\in
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
Part (b): by the definition, one has
$(\iota_{x}{}_{!}\widecheck{F}_{x})[X]\simeq F_{x}$. Since the cosections over
a quasi-compact quasi-separated scheme commute with infinite products of
contraherent cosheaves, we conclude from the above computation that
${\mathfrak{F}}[U]\simeq\prod_{x\in U}F_{x}$ for any quasi-compact open
subscheme $U\subset X$. The cosheaf axiom (5) now implies the isomorphism
${\mathfrak{F}}[X]\simeq\varinjlim_{U\subset X}{\mathfrak{F}}[U]$, the
filtered inductive limit being taken over all quasi-compact open subschemes
$U\subset X$, and similarly for ${\mathfrak{F}}[Y]$. So the corestriction map
${\mathfrak{F}}[Y]\longrightarrow{\mathfrak{F}}[X]$ is the embedding of a
direct summand.
Part (c): the “only if” assertion follows from part (a); let us prove the
“if”. A transfinite induction in (any well-ordering of) the set of indices
$\alpha$ reduces the question to the following assertion. Suppose $X$ is
presented as the union of two open subschemes $W\cup Y$. Furthermore, the
restriction of a locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ onto $W$
is identified with the direct product $\prod_{w\in
W}\kappa_{w}{}_{!}\widecheck{F}_{w}$, where $F_{w}$ are some free
$\widehat{\mathcal{O}}_{w,X}$-contramodules, while $\kappa_{w}$ are the
natural morphisms $\operatorname{Spec}{\mathcal{O}}_{w,X}\longrightarrow W$.
Assume also that the restriction of ${\mathfrak{F}}$ onto $Y$ is a projective
locally cotorsion contraherent cosheaf. Then there exist some free
$\widehat{\mathcal{O}}_{x,X}$-contramodules $F_{z}$ defined for all $z\in
X\setminus W$ and an isomorphism of locally contraherent cosheaves
${\mathfrak{F}}\simeq\prod_{x\in X}\iota_{x}{}_{!}\widecheck{F}_{x}$ whose
restriction to $W$ coincides with the given isomorphism
${\mathfrak{F}}|_{W}\simeq\prod_{w\in W}\kappa_{w}{}_{!}\widecheck{F}_{w}$.
Indeed, by Theorem 5.1.1 we have ${\mathfrak{F}}|_{Y}\simeq\prod_{y\in
Y}\iota^{\prime}_{y}{}_{!}\widecheck{F}^{\prime}_{y}$, where
$\iota^{\prime}_{y}$ denote the natural morphisms
$\operatorname{Spec}{\mathcal{O}}_{y,X}\longrightarrow Y$ and $F^{\prime}_{y}$
are some free contramodules over $\widehat{\mathcal{O}}_{y,X}$. Restricting to
the intersection $V=W\cap Y$, we obtain an isomorphism of contraherent
cosheaves $\prod_{v\in
V}\kappa^{\prime}_{v}{}_{!}\widecheck{F}_{v}\simeq\prod_{v\in
V}\kappa^{\prime}_{v}{}_{!}\widecheck{F}^{\prime}_{v}$ on $V$, where
$\kappa^{\prime}_{v}$ denotes the natural morphisms
$\operatorname{Spec}{\mathcal{O}}_{v,X}\longrightarrow V$. It is clear from
the arguments in the second half of the proof of Theorem 5.1.1(b) that such an
isomorphism of infinite products induces an isomorphism of
$\widehat{\mathcal{O}}_{v,X}$-contramodules $F_{v}\simeq F^{\prime}_{v}$. Let
us identify $F^{\prime}_{v}$ with $F_{v}$ using this isomorphism, and set
$F_{y}=F^{\prime}_{y}$ for $y\in X\setminus W\subset Y$. Then our isomorphism
of infinite products can be viewed as an automorphism $\phi$ of the
contraherent cosheaf $\prod_{v\in
V}\kappa^{\prime}_{v}{}_{!}\widecheck{F}_{v}$ on $V$. We would like to show
that the cosheaf ${\mathfrak{F}}$ is isomorphic to $\prod_{x\in
X}\iota_{x}{}_{!}\widecheck{F}_{x}$.
Since a cosheaf of ${\mathcal{O}}_{X}$-modules is determined by its
restrictions to $W$ and $Y$ together with the induced isomorphism between the
restrictions of these to the intersection $V=W\cap Y$, it suffices to check
that the automorphism $\phi$ can be lifted to an automorphism of the
contraherent cosheaf $\prod_{v\in V}\iota^{\prime}_{v}\widecheck{F}_{v}$ on
$Y$. In fact, the rings of endomorphisms of the two contraherent cosheaves are
isomorphic. Indeed, the functor of direct image of cosheaves of
${\mathcal{O}}$-modules $j_{!}$ with respect to the open embedding $j\colon
V\longrightarrow Y$ is fully faithful by (11); the morphism $j$ being quasi-
compact and quasi-separated, this functor also preserves infinite products. ∎
###### Corollary 5.1.4.
The classes of projective locally cotorsion contraherent cosheaves and
${\mathbf{W}}$-flat locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaves on $X$ coincide. In particular, any ${\mathbf{W}}$-flat locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaf on $X$ is flat,
contraherent, and colocally flat.
###### Proof.
The inclusion in one direction is provided by Corollaries 4.4.5(b) and
5.1.3(a). To prove the converse, pick an affine open covering $U_{\alpha}$ of
the scheme $X$ subordinate to ${\mathbf{W}}$. Then for any ${\mathbf{W}}$-flat
locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$
on $X$ the ${\mathcal{O}}_{X}(U_{\alpha})$-modules
${\mathfrak{F}}[U_{\alpha}]$ are, by the definition, flat and cotorsion. Hence
the locally cotorsion contraherent cosheaves ${\mathfrak{F}}|_{U_{\alpha}}$
are projective, and it remains to apply Corollary 5.1.3(c). ∎
###### Corollary 5.1.5.
The full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is closed
with respect to infinite products in
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--ctrh}}}$.
###### Proof.
Easily deduced either from Corollary 5.1.4, or directly from Theorem 5.1.1(b)
(cf. [54, Lemma 1.3.7]). ∎
###### Corollary 5.1.6.
Let $f\colon Y\longrightarrow X$ be a quasi-compact morphism of locally
Noetherian schemes. Then
(a) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes projective locally cotorsion contraherent cosheaves on $Y$ to
locally cotorsion contraherent cosheaves on $X$;
(b) if the morphism $f$ is flat, then the functor of direct image of cosheaves
of ${\mathcal{O}}$-modules $f_{!}$ takes projective locally cotorsion
contraherent cosheaves on $Y$ to projective locally cotorsion contraherent
cosheaves on $X$.
###### Proof.
Part (a): let us show that the functor $f_{!}$ takes locally cotorsion
contraherent cosheaves of the form ${\mathfrak{P}}=\prod_{y\in
Y}\kappa_{y}{}_{!}\widecheck{P}_{y}$ on $Y$, where $\kappa_{y}$ are the
natural morphisms $\operatorname{Spec}{\mathcal{O}}_{y,Y}\longrightarrow Y$
and $P_{y}$ are some $\widehat{\mathcal{O}}_{y,Y}$-contramodules, to locally
cotorsion contraherent cosheaves of the same form on $X$. More precisely, one
has $f_{!}{\mathfrak{P}}\simeq\prod_{x\in X}\iota_{x}{}_{!}\widecheck{P}_{x}$,
where $\iota_{x}$ are the natural morphisms
$\operatorname{Spec}{\mathcal{O}}_{x,X}\longrightarrow X$ and
$P_{x}=\prod_{f(y)=x}P_{y}$, the $\widehat{\mathcal{O}}_{x,X}$-contramodule
structures on $P_{y}$ being obtained by the (contra)restriction of scalars
with respect to the homomorphisms of complete Noetherian rings
$\widehat{\mathcal{O}}_{x,X}\longrightarrow\widehat{\mathcal{O}}_{y,Y}$ (see
[54, Section 1.8]).
Indeed, the morphism $f$ being quasi-compact and quasi-separated, the functor
$f_{!}$ preserves infinite products of cosheaves of ${\mathcal{O}}$-modules.
So it suffices to consider the case of a locally cotorsion contraherent
cosheaf $\kappa_{y}{}_{!}\widecheck{P}_{y}$ on $Y$. Now the composition of
morphisms of schemes $\operatorname{Spec}{\mathcal{O}}_{y,Y}\longrightarrow
Y\longrightarrow X$ is equal to the composition
$\operatorname{Spec}{\mathcal{O}}_{y,Y}\longrightarrow\operatorname{Spec}{\mathcal{O}}_{x,X}\longrightarrow
X$, and it remains to use the compatibility of the direct images of cosheaves
of ${\mathcal{O}}$-modules with the compositions of morphisms of schemes.
Alternatively, part (a) is a particular case of Corollary 3.4.8(b).
To deduce part (b), one can apply the assertion that an
$\widehat{\mathcal{O}}_{x,X}$-contramodule is projective if and only if it is
a flat ${\mathcal{O}}_{x,X}$-module [54, Corollary B.8.2(c)]. Alternatively,
use the adjunction (26) together with exactness of the inverse image of
locally cotorsion locally contraherent cosheaves with respect to a flat
morphism of schemes. ∎
### 5.2. Flat contraherent cosheaves
In this section we complete our treatment of flat contraherent cosheaves on
locally Noetherian schemes, which was started in Sections 1.6 and 3.7 and
continued in Sections 4.3 and 5.1.
A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on a scheme $X$
is said to have _locally cotorsion dimension not exceeding $d$_ if the
cotorsion dimension of the ${\mathcal{O}}_{X}(U)$-module ${\mathfrak{M}}[U]$
does not exceed $d$ for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$ (cf. Sections 1.5 and 4.9). Clearly, the locally cotorsion
dimension of a ${\mathbf{W}}$-locally contraherent cosheaf does not change
when the covering ${\mathbf{W}}$ is replaced by its refinement (see Lemma
1.5.4). If a ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ has
a right resolution by locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaves in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, then its
locally cotorsion dimension is equal to the minimal length of such resolution.
###### Lemma 5.2.1.
Let $X$ be a semi-separated Noetherian scheme of Krull dimension $D$ with an
open covering ${\mathbf{W}}$. Then
(a) the right homological dimension of any ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ with respect to the exact subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ does not exceed $D$;
(b) the right homological dimension of any ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of projective locally cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}$ does not exceed
$D$.
###### Proof.
The locally cotorsion dimension of any locally contraherent cosheaf on a
locally Noetherian scheme of Krull dimension $D$ does not exceed $D$ by
Corollary 1.5.7. The right homological dimension in part (a) is well-defined
due to Corollary 4.2.5(a) or 4.3.4(a) and the results of Section 3.1, hence it
is obviously equal to the locally cotorsion dimension. The right homological
dimension in part (b) is well-defined by Corollaries 4.3.4(a), 4.3.5–4.3.6,
and 5.1.4, so (b) follows from (a) in view of the dual version of Corollary
A.5.3. ∎
###### Corollary 5.2.2.
(a) On a semi-separated Noetherian scheme $X$ of finite Krull dimension, the
classes of ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaves
and colocally flat contraherent cosheaves coincide.
(b) On a locally Noetherian scheme $X$ of finite Krull dimension, any
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf is flat and
contraherent, so the categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ coincide.
(c) On a locally Noetherian scheme $X$ of finite Krull dimension, any flat
contraherent cosheaf ${\mathfrak{F}}$ is coflasque, so one has
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$. If $X$ is affine and
$Y\subset X$ is an open subscheme, then
$0\longrightarrow{\mathfrak{F}}[Y]\longrightarrow{\mathfrak{F}}[X]\longrightarrow{\mathfrak{F}}[X]/{\mathfrak{F}}[Y]\longrightarrow
0$ is a short exact sequence of flat contraadjusted
${\mathcal{O}}(X)$-modules.
###### Proof.
Part (a): any colocally flat contraherent cosheaf on $X$ is flat by Corollary
4.3.6 (see also Corollary 4.3.5). On the other hand, by Lemma 5.2.1(b), any
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{F}}$ on $X$ has a finite right resolution by cosheaves from
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, which are
colocally flat by definition. By Corollary 4.3.2(b), it follows that
${\mathfrak{F}}$ is colocally flat.
Part (b): given an affine open subscheme $U\subset X$, denote by
${\mathbf{W}}|_{U}$ the collection of all open subsets $U\cap W$ with
$W\in{\mathbf{W}}$. Then the restriction ${\mathfrak{F}}|_{U}$ of any
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{F}}$ on $X$ onto $U$ is, by definition, ${\mathbf{W}}|_{U}$-flat
and ${\mathbf{W}}|_{U}$-locally contraherent. Applying part (a), we conclude
that the cosheaf ${\mathfrak{F}}|_{U}$ is contraherent and flat. This being
true for any affine open subscheme $U\subset X$ means precisely that
${\mathfrak{F}}$ is contraherent and flat on $X$.
Part (c): coflasqueness being a local property by Lemma 3.4.1(a), it suffices
to consider the case of an affine scheme $X$. Then we have seen above that
${\mathfrak{F}}$ has a finite right resolution by cosheaves from
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, which have
the properties listed in part (c) by Corollary 5.1.3(b). By Corollary
3.4.4(b), our resolution is an exact sequence in
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$ and the cosheaf
${\mathfrak{F}}$ is coflasque.
Furthermore, by Corollary 3.4.4(c), the related sequence of cosections over
any open subscheme $Y\subset X$ is exact. Passing to the corestriction maps
related to the pair of embedded open subschemes $Y\subset X$, we obtain an
injective morphism of exact sequences of ${\mathcal{O}}(X)$-modules; so the
related sequence of cokernels is also exact. All of its terms except perhaps
the leftmost one being flat, it follows that the lefmost term
${\mathfrak{F}}[X]/{\mathfrak{F}}[Y]$ is a flat ${\mathcal{O}}(X)$-module,
too. The ${\mathcal{O}}(X)$-module ${\mathfrak{F}}[Y]$ is contraadjusted by
Corollary 3.4.8(a) applied to the embedding morphism $Y\longrightarrow X$. ∎
###### Lemma 5.2.3.
Let $f\colon Y\longrightarrow X$ be a flat quasi-compact morphism from a semi-
separated locally Noetherian scheme $Y$ of finite Krull dimension to a locally
Noetherian scheme $X$. Then the functor of direct image of cosheaves of
${\mathcal{O}}$-modules $f_{!}$ takes flat contraherent cosheaves on $Y$ to
flat contraherent cosheaves on $X$, and induces an exact functor
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$.
###### Proof.
Clearly, it suffices to consider the case of a Noetherian affine scheme $X$.
Then the scheme $Y$ is Noetherian and semi-separated. By Corollary 5.2.2(a),
any flat contraherent cosheaf on $Y$ is colocally flat. By Corollary 4.5.3(c),
the functor $f_{!}$ restricts to an exact functor
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$. By Corollary 4.3.6, any
colocally flat contraherent cosheaf on $X$ is flat. (A proof working in a
greater generality will be given below in Corollary 5.2.10(b).) ∎
Let $X$ be a Noetherian scheme of finite Krull dimension $D$ with an open
covering ${\mathbf{W}}$ and a finite affine open covering $U_{\alpha}$
subordinate to ${\mathbf{W}}$.
###### Corollary 5.2.4.
(a) There are enough projective objects in the exact categories of locally
contraherent cosheaves $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}$ on $X$, and all these projective objects
belong to the full subcategory of contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}$. The full subcategories of projective
objects in the three exact categories
$X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$ coincide.
(b) A contraherent cosheaf on $X$ is projective if and only if it is
isomorphic to a finite direct sum of the direct images of projective
contraherent cosheaves from $U_{\alpha}$. In particular, any projective
contraherent cosheaf on $X$ is flat (and consequently, coflasque).
###### Proof.
The argument is similar to the proofs of Lemmas 4.3.3, 4.4.1, and 4.4.3; the
only difference is that, the scheme $X$ being not necessarily semi-separated,
one has to also use Lemma 5.2.3. Let $j_{\alpha}\colon
U_{\alpha}\longrightarrow X$ denote the open embedding morphisms. According to
Corollary 4.4.5(a) and Lemma 5.2.3, the cosheaf of ${\mathcal{O}}_{X}$-modules
$j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}$ is flat and contraherent for any
projective contraherent cosheaf ${\mathfrak{F}}_{\alpha}$ on $U_{\alpha}$. The
adjunction (11) or (26) shows that it is also a projective object in
$X{\operatorname{\mathsf{--lcth}}}$.
It remains to show that there are enough projective objects of the form
$\bigoplus_{\alpha}j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. Let ${\mathfrak{Q}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. For every $\alpha$, pick
an admissible epimorphism onto the contraherent cosheaf
$j_{\alpha}^{!}{\mathfrak{Q}}$ from a projective contraherent cosheaf
${\mathfrak{F}}_{\alpha}$ on $U_{\alpha}$. Then the same adjunction provides a
natural morphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{Q}}$.
This morphism is an admissible epimorphism of locally contraherent cosheaves,
because it is so in restriction to each open subscheme $U_{\alpha}\subset X$.
∎
As in Section 4.4, we denote the category of projective contraherent cosheaves
on $X$ by $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$. Clearly, the
objects of $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ are the
projective objects of the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ (and there are enough of
them). We will see below in this section that the objects of
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ are the
_injective_ objects of $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ (and
there are enough of them).
###### Lemma 5.2.5.
(a) Any coflasque contraherent cosheaf ${\mathfrak{E}}$ on $X$ can be included
in an exact triple
$0\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ in $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$, where
${\mathfrak{P}}$ is a coflasque locally cotorsion contraherent cosheaf on $X$
and ${\mathfrak{F}}$ is a finitely iterated extension of the direct images of
flat contraherent cosheaves from $U_{\alpha}$.
(b) The right homological dimension of any coflasque contraherent cosheaf on
$X$ with respect to the exact subcategory of coflasque locally cotorsion
contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}\subset
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{cfq}}$ does not exceed $D$.
###### Proof.
Part (a): the proof is similar to that of Lemma 4.3.1; the only difference is
that, the scheme $X$ being not necessarily semi-separated, one has to also use
Lemma 3.4.8. Notice that flat contraherent cosheaves on $U_{\alpha}$ are
coflasque by Lemma 5.2.2(c) and finitely iterated extensions of coflasque
contraherent cosheaves in $X{\operatorname{\mathsf{--ctrh}}}$ are coflasque by
Lemma 3.4.4(a).
One proceeds by induction in a linear ordering of the indices $\alpha$,
considering the open subscheme $V=\bigcup_{\alpha<\beta}U_{\alpha}$. Assume
that we have constructed an exact triple
${\mathfrak{E}}\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{L}}$ of
coflasque contraherent cosheaves on $X$ such that the contraherent cosheaf
${\mathfrak{K}}|_{V}$ is locally cotorsion, while the cosheaf ${\mathfrak{L}}$
on $X$ is a finitely iterated extension of the direct images of flat
contraherent cosheaves from the affine open subschemes $U_{\alpha}$ with
$\alpha<\beta$. Let $j\colon U=U_{\beta}\longrightarrow X$ be the identity
open embedding.
Pick an exact triple
$j^{!}{\mathfrak{K}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{G}}$
of contraherent cosheaves on $U$ such that ${\mathfrak{G}}$ is a flat
contraherent cosheaf (see Theorem 1.3.1(a)); then the cosheaf ${\mathfrak{G}}$
is coflasque by Lemma 5.2.2(c) and the cosheaf ${\mathfrak{Q}}$ is coflasque
by Lemma 3.4.4(a). By Lemma 3.4.8(a), the related sequence of direct images
$j_{!}j^{!}{\mathfrak{K}}\longrightarrow j_{!}{\mathfrak{Q}}\longrightarrow
j_{!}{\mathfrak{G}}$ is an exact triple of coflasque contraherent cosheaves on
$X$; by part (b) of the same lemma, the cosheaf $j_{!}{\mathfrak{Q}}$ belongs
to $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$.
Let ${\mathfrak{K}}\longrightarrow{\mathfrak{R}}\longrightarrow
j_{!}{\mathfrak{G}}$ denote the push-forward of the exact triple
$j_{!}j^{!}{\mathfrak{K}}\longrightarrow j_{!}{\mathfrak{Q}}\longrightarrow
j_{!}{\mathfrak{G}}$ with respect to the natural morphism
$j_{!}j^{!}{\mathfrak{K}}\longrightarrow{\mathfrak{K}}$. We will show that the
coflasque contraherent cosheaf ${\mathfrak{R}}$ on $X$ is locally cotorsion in
restriction to $U\cup V$.
Indeed, in the restriction to $U$ one has
$j^{!}{\mathfrak{R}}\simeq{\mathfrak{Q}}$. On the other hand, denoting by
$j^{\prime}$ the embedding $U\cap V\longrightarrow V$, one has
$(j_{!}{\mathfrak{G}})|_{V}\simeq j^{\prime}_{!}({\mathfrak{G}}|_{U\cap V})$.
The contraherent cosheaf ${\mathfrak{K}}|_{U\cap V}$ being locally cotorsion,
so is the cokernel ${\mathfrak{G}}|_{U\cap V}$ of the admissible monomorphism
of locally cotorsion contraherent cosheaves ${\mathfrak{K}}|_{U\cap
V}\longrightarrow{\mathfrak{Q}}|_{U\cap V}$.
By Lemma 3.4.8(b), $j^{\prime}_{!}({\mathfrak{G}}|_{U\cap V})$ is a coflasque
locally cotorsion contaherent cosheaf on $V$. Now in the exact triple
${\mathfrak{K}}|_{V}\longrightarrow{\mathfrak{R}}|_{V}\longrightarrow(j_{!}{\mathfrak{G}})_{V}$
the middle term is locally cotorsion, since the two other terms are.
Finally, the composition
${\mathfrak{E}}\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{R}}$ of
admissible monomorphisms in $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$
is again an admissible monomorphism with the cokernel isomorphic to an
extension of the flat contraherent cosheaves $j_{!}{\mathfrak{G}}$ and
${\mathfrak{L}}$.
Part (b): the right homological dimension is well-defined by part (a) and does
not exceed $D$ for the reasons explained in the proof of Lemma 5.2.1. ∎
###### Corollary 5.2.6.
(a) Any flat contraherent cosheaf ${\mathfrak{G}}$ on $X$ can be included in
an exact triple
$0\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$, where
${\mathfrak{P}}$ is a projective locally cotorsion contraherent cosheaf on $X$
and ${\mathfrak{F}}$ is a finitely iterated extension of the direct images of
flat contraherent cosheaves from $U_{\alpha}$.
(b) The right homological dimension of any flat contraherent cosheaf on $X$
with respect to the exact subcategory of projective locally cotorsion
contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ does not exceed $D$; the
homological dimension of the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ does not exceed $D$; and the
left homological dimension of any flat contraherent cosheaf on $X$ with
respect to the exact category of projective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ does not exceed $D$.
###### Proof.
Part (a) follows from Lemma 5.2.5(a) together with Corollaries 5.2.2(c) and
5.1.4. Part (b): the right homological dimension is well-defined by part (a)
and does not exceed $D$ by Lemma 5.2.5(b) and the dual version of Corollary
A.5.3. By the dual version of Proposition A.3.1(a) or A.5.6, it follows that
the natural functor
$\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
is fully faithful (the exact category structure on
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ being
trivial). Applying the first assertion of part (b) again, we conclude that the
homological dimension of the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ does not exceed $D$, and
consequently that the left homological dimension of any object of
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ with respect to its
subcategory of projective objects
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ does not exceed $D$. ∎
###### Lemma 5.2.7.
Any flat contraherent cosheaf ${\mathfrak{G}}$ on $X$ can be included in an
exact triple
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{G}}\longrightarrow
0$ in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$, where
${\mathfrak{P}}$ is a projective locally cotorsion contraherent cosheaf on $X$
and ${\mathfrak{F}}$ is a finitely iterated extension of the direct images of
flat contraherent cosheaves from $U_{\alpha}$.
###### Proof.
According to Corollary 5.2.4, there is an admissible epimorphism
${\mathfrak{E}}\longrightarrow{\mathfrak{G}}$ in the exact category
$X{\operatorname{\mathsf{--ctrh}}}$ onto any given contraherent cosheaf
${\mathfrak{G}}$ from a finite direct sum ${\mathfrak{E}}$ of the direct
images of flat (and even projective) contraherent cosheaves from $U_{\alpha}$.
Furthermore, any admissible epimorphism in $X{\operatorname{\mathsf{--ctrh}}}$
between objects from $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is also
an admissible epimorphism in
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. The rest of the argument is
similar to the proofs of Lemmas 4.3.3 and 4.2.4, and based on Corollary
5.2.6(a) (applied to the kernel of the morphism
${\mathfrak{E}}\longrightarrow{\mathfrak{G}}$). ∎
The following corollary is to be compared with Corollary 4.11.2 above; see
also Corollaries 5.3.3, 5.4.4, and Theorem 5.4.10 below.
###### Corollary 5.2.8.
(a) For any Noetherian scheme $X$ of finite Krull dimension, the natural
functors $\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$,
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
are equivalences of triangulated categories, as are the natural functors
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$,
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
.
(b) For any locally Noetherian scheme $X$ with an open covering
${\mathbf{W}}$, the natural functors
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
are equivalences of triangulated categories.
(c) For any Noetherian scheme $X$ of finite Krull dimension with an open
covering ${\mathbf{W}}$, the natural functors
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$
are equivalences of triangulated categories.
(d) For any Noetherian scheme $X$ of finite Krull dimension with an open
covering ${\mathbf{W}}$, the natural functors
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
and
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$
are equivalences of triangulated categories.
###### Proof.
In view of Corollary 5.2.6(b), all assertions of part (a) follow from
Proposition A.5.6 (together with its dual version) and [50, Remark 2.1]. Part
(b) is provided by Proposition A.3.1(a) together with Theorem 5.1.1(a), while
part (c) follows from the same Proposition together with Corollary 5.2.4.
Finally, part (d) is obtained by comparing parts (a-c). ∎
It follows from Corollary 5.2.8 that the $\operatorname{Ext}$ groups computed
in the exact categories $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ agree with
each other and with the $\operatorname{Ext}$ groups computed in the exact
categories $X{\operatorname{\mathsf{--lcth}}}$ and
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$. Besides, these also agree
with the $\operatorname{Ext}$ groups computed in the exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ (the latter
being endowed with the trivial exact category structure). As in Section 4.2,
we denote these $\operatorname{Ext}$ groups by
$\operatorname{Ext}^{X,*}({-},{-})$.
###### Corollary 5.2.9.
(a) Let $X$ be a Noetherian scheme of finite Krull dimension with an open
covering ${\mathbf{W}}$. Then one has
$\operatorname{Ext}^{X,>0}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any flat
contraherent cosheaf ${\mathfrak{F}}$ and locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on $X$.
Consequently, any flat contraherent cosheaf on $X$ is colocally flat.
(b) Let $X$ be a Noetherian scheme of finite Krull dimension with a finite
affine open covering $X=\bigcup_{\alpha}U_{\alpha}$. Then a contraherent
cosheaf on $X$ is flat if and only if it is a direct summand of a finitely
iterated extension of the direct images of flat contraherent cosheaves from
$U_{\alpha}$.
###### Proof.
According to Corollary 5.2.6(b), any flat contraherent cosheaf on $X$ has a
finite right resolution by projective locally cotorsion contraherent
cosheaves. Since the $\operatorname{Ext}$ groups in the exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ agree, the
assertion (a) follows. Now “only if” assertion in part (b) is easily deduced
from Lemma 5.2.7 together with part (a), while the “if” is provided by Lemma
5.2.3. ∎
The following corollary is to be compared with Corollaries 3.4.8, 4.4.7,
4.5.3, 4.5.4, and 5.1.6.
###### Corollary 5.2.10.
Let $f\colon Y\longrightarrow X$ be a quasi-compact morphism of locally
Noetherian schemes such that the scheme $Y$ has finite Krull dimension. Then
(a) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes flat contraherent cosheaves on $Y$ to contraherent cosheaves on
$X$, and induces an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ between these exact categories;
(b) if the morphism $f$ is flat, then the functor of direct image of cosheaves
of ${\mathcal{O}}$-modules $f_{!}$ takes flat contraherent cosheaves on $Y$ to
flat contraherent cosheaves on $X$, and induces an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ between these exact
categories;
(c) if the scheme $Y$ is Noetherian and the morphism $f$ is very flat, then
the functor of direct image of cosheaves of ${\mathcal{O}}$-modules $f_{!}$
takes projective contraherent cosheaves on $Y$ to projective contraherent
cosheaves on $X$.
###### Proof.
Part (a) is a particular case of Corollary 3.4.8(a). In part (b), one can
assume that $X$ is an affine scheme, so the scheme $Y$ is Noetherian of finite
Krull dimension. It suffices to show that the ${\mathcal{O}}(X)$-module
$(f_{!}{\mathfrak{F}})[X]={\mathfrak{F}}[Y]$ is flat for any flat contraherent
cosheaf ${\mathfrak{F}}$ on $X$. For this purpose, consider a finite right
resolution of the cosheaf ${\mathfrak{F}}$ by objects from
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ in
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$, and apply the functor
$\Delta(Y,{-})$ to it. It follows from Corollaries 5.2.2(c) and 3.4.4(c) that
the sequence will remain exact. By Corollary 5.1.6(b), we obtain a finite
right resolution of the module ${\mathfrak{F}}[Y]$ by flat
${\mathcal{O}}(X)$-modules, implying that the ${\mathcal{O}}(X)$-module
${\mathfrak{F}}[Y]$ is also flat. Part (c) follows from part (a) or (b)
together with the adjunction (26). ∎
### 5.3. Homology of locally cotorsion locally contraherent cosheaves
Let $X$ be a locally Noetherian scheme. Then the left derived functor of the
functor of global cosections $\Delta(X,{-})$ of locally cotorsion locally
contraherent cosheaves on $X$ is defined using left projective resolutions in
the exact category $X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ (see
Theorem 5.1.1(a)).
Notice that the derived functors ${\mathbb{L}}_{*}\Delta(X,{-})$ computed in
the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ for a
particular open covering ${\mathbf{W}}$ and in the whole category
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ agree. The groups
${\mathbb{L}}_{i}\Delta(X,{\mathfrak{E}})$ are called the _homology groups_ of
a locally cotorsion locally contraherent cosheaf ${\mathfrak{E}}$ on the
scheme $X$.
Let us show that ${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{F}})=0$ for any
coflasque locally cotorsion contraherent cosheaf ${\mathfrak{F}}$. By
Corollary 5.1.3(b), any projective locally cotorsion contraherent cosheaf on
$X$ is coflasque. In view of Corollary 3.4.4(b), any resolution of an object
of $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$ by
objects of $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$
in the category $X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ is exact
with respect to the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$. By part (c)
of the same Corollary, the functor $\Delta(X,{-})$ preserves exactness of such
sequences. Hence one can compute the derived functor
${\mathbb{L}}_{*}\Delta(X,{-})$ using coflasque locally cotorsion contraherent
resolutions.
Similarly, let $X$ be a Noetherian scheme of finite Krull dimension. Then the
left derived functor of the functor of global cosections $\Delta(X,{-})$ of
locally contraherent cosheaves on $X$ is defined using left projective
resolutions in the exact category $X{\operatorname{\mathsf{--lcth}}}$ (see
Corollary 5.2.4). The derived functors ${\mathbb{L}}_{*}\Delta(X,{-})$
computed in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ for any prarticular open
covering ${\mathbf{W}}$ and in the whole category
$X{\operatorname{\mathsf{--lcth}}}$ agree. Furthermore, one can compute the
derived functor ${\mathbb{L}}_{*}\Delta(X,{-})$ using coflasque left
resolutions. The groups ${\mathbb{L}}_{i}\Delta(X,{\mathfrak{E}})$ are called
the _homology groups_ of a locally contraherent cosheaf ${\mathfrak{E}}$ on
the scheme $X$.
It is clear from the above that the two definitions agree when they are both
applicable; they also agree with the definitions given in Section 4.5 (cf.
Lemma 4.2.1).
Using injective resolutions, one can similarly define the cohomology of quasi-
coherent sheaves ${\mathbb{R}}^{*}\Gamma(X,{-})$ on a locally Noetherian
scheme $X$. Injective quasi-coherent sheaves on $X$ being flasque, this
definition agrees with the conventional sheaf-theoretical one. The full
subcategory of flasque quasi-coherent sheaves in
$X{\operatorname{\mathsf{--qcoh}}}$ is closed under extensions, cokernels of
injective morphisms and infinite direct sums; we denote the induced exact
category structure on it by $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}}$.
The following lemma is to be compared with Lemmas 4.6.1 and 4.6.2.
###### Lemma 5.3.1.
(a) Let $X$ be a locally Noetherian scheme of Krull dimension $D$. Then the
right homological dimension of any quasi-coherent sheaf on $X$ with respect to
the exact subcategory of flasque quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ does not exceed $D$.
(b) Let $X$ be a locally Noetherian scheme of Krull dimension $D$. Then the
left homological dimension of any locally cotorsion locally contraherent
cosheaf on $X$ with respect to the exact subcategory of coflasque locally
cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ does not exceed $D$.
Consequently, the same bound holds for the left homological dimension of any
object of $X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ with respect to
the exact subcategory $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.
(c) Let $X$ be a Noetherian scheme of Krull dimension $D$. Then the left
homological dimension of any locally contraherent cosheaf on $X$ with respect
to the exact subcategory of coflasque contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}\subset
X{\operatorname{\mathsf{--lcth}}}$ does not exceed $D$. Consequently, the same
bound holds for the left homological dimension of any object of
$X{\operatorname{\mathsf{--lcth}}}$ with respect to the exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}$.
###### Proof.
Follows from Lemma 3.4.7 and Corollary 3.4.2. ∎
###### Corollary 5.3.2.
(a) Let $X$ be a locally Noetherian scheme of finite Krull dimension. Then for
any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$ the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is an equivalence of triangulated
categories.
(b) Let $X$ be a locally Noetherian scheme of finite Krull dimension. Then for
any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$ the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is an
equivalence of triangulated categories.
For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
is an equivalence of categories.
(c) Let $X$ be a Noetherian scheme of finite Krull dimension. Then for any
symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$ the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is an equivalence of
triangulated categories.
For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})$
is an equivalence of categories.
###### Proof.
Follows from Lemma 5.3.1 together with Proposition A.5.6. ∎
###### Corollary 5.3.3.
(a) Let $X$ be a locally Noetherian scheme of finite Krull dimension. Then for
any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is an
equivalence of categories.
For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
is an equivalence of categories.
(b) Let $X$ be a Noetherian scheme of finite Krull dimension. Then for any
symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is an equivalence of
triangulated categories.
For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})$
is an equivalence of categories.
###### Proof.
Follows from Lemma 5.3.1(b-c) (see Corollaries 4.6.3–4.6.5 for comparison). ∎
The results below in this section purport to replace the above homological
dimension estimates based on the Krull dimension with the ones based on the
numbers of covering open affines.
###### Lemma 5.3.4.
Let $X$ be a Noetherian scheme and $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be its
finite affine open covering. For each subset
$1\le\alpha_{1}<\dotsb<\alpha_{k}\le N$ of indices $\\{\alpha\\}$, let
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{k}}=\bigcup_{\beta=1}^{n}V_{\beta}$,
where $n=n_{\alpha_{1},\dotsc,\alpha_{k}}$, be a finite affine open covering
of the intersection. Let $M$ denote the supremum of the expressions
$k-1+n_{\alpha_{1},\dotsc,\alpha_{k}}$ taken over all the nonempty subsets of
indices $\alpha_{1}$, …, $\alpha_{k}$. Then one has
(a) ${\mathbb{R}}^{\ge M}\Gamma(X,{\mathcal{E}})=0$ for any quasi-coherent
sheaf ${\mathcal{E}}$ on $X$;
(b) ${\mathbb{L}}_{\ge M}\Delta(X,{\mathfrak{E}})=0$ for any locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{E}}$ on $X$, provided
that the affine open covering $\\{U_{\alpha}\\}$ is subordinate to
${\mathbf{W}}$. Assuming additionally that the Krull dimension of $X$ is
finite, the same bound holds for any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{E}}$ on $X$.
###### Proof.
We will prove part (b). The first assertion: let
${\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}$
be a left projective resolution of an object ${\mathfrak{E}}\in
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Consider the
Čech resolution (27) for each cosheaf ${\mathfrak{F}}_{i}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. By
Corollaries 5.1.3(a) and 5.1.6(b), this is a sequence of projective locally
cotorsion contraherent cosheaves. Its restriction to each open subset
$U_{\alpha}\subset X$ being naturally contractible, this finite sequence is
exact in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$, and consequently
also exact (i. e., even contractible) in
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
Denote the bicomplex of cosheaves we have obtained (without the rightmost term
that is being resolved) by
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and the corresponding bicomplex of the groups of global cosections by
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Now the total complex of the bicomplex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is quasi-isomorphic (in fact, in this case even homotopy equivalent) to the
complex
$\Delta(X,{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
computing the homology groups ${\mathbb{L}}_{*}\Delta(X,{\mathfrak{E}})$.
On the other hand, for each $1\le k\le N$, the complex
$C_{k}(\\{U_{\alpha}\\},{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
computes the direct sum of the homology of the cosheaves
$j_{\alpha_{1},\dotsc,\alpha_{k}}^{!}{\mathfrak{E}}$ on
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{N}}$ over all
$1\le\alpha_{1}<\dotsb<\alpha_{k}\le N$. The schemes
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{N}}$ being quasi-compact and
separated, and the cosheaves
$j_{\alpha_{1},\dotsc,\alpha_{k}}^{!}{\mathfrak{E}}$ being contraherent, the
latter homology can be also computed by the Čech complexes
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\beta}\\},j_{\alpha_{1},\dotsc,\alpha_{k}}^{!}{\mathfrak{E}})$
(see Section 4.5) and consequently vanish in the homological degrees $\ge
n_{1}+\dotsb+n_{k}$.
To prove the second assertion, one uses a flat (or more generally, coflasque)
left resolution of a contraherent cosheaf ${\mathfrak{E}}$ and argues as above
using Corollary 5.2.10(b) (or Corollary 3.4.8(a), respectively). ∎
Let us say that a locally cotorsion locally contraherent cosheaf
${\mathfrak{P}}$ on a locally Noetherian scheme $X$ is _acyclic_ if
${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{P}})=0$. Acyclic locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves form a full subcategory in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ closed under
extensions and kernels of admissible epimorphisms; when $X$ is Noetherian (i.
e., quasi-compact), this subcategory is also closed under infinite products.
Hence it acquires the induced exact category structure, which we denote by
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}^{\mathsf{lct}}$.
Similarly, we say that a locally contraherent cosheaf ${\mathfrak{P}}$ on a
Noetherian scheme $X$ of finite Krull dimension is _acyclic_ if
${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{P}})=0$. Acyclic ${\mathbf{W}}$-locally
contraherent cosheaves form a full subcategory in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ closed under extensions,
kernels of admissible epimorphisms, and infinite products. The induced exact
category structure on this subcategory is denote by
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}$. Clearly,
one has
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}^{\mathsf{lct}}$
and $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}\subset
X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}$ (under the
respective assumptions).
Finally, a quasi-coherent sheaf ${\mathcal{P}}$ on a locally Noetherian scheme
$X$ is said to be _acyclic_ if ${\mathbb{R}}^{>0}\Gamma(X,{\mathcal{P}})=0$.
Acyclic quasi-coherent sheaves form a full subcategory in
$X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions and cokernels of
injective morphisms; when $X$ is Noetherian, this subcategory is also closed
under infinite direct sums. The induced exact category structure on it is
denoted by $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ac}}$.
###### Corollary 5.3.5.
Let $X$ be a Noetherian scheme with an open covering ${\mathbf{W}}$ and $M$ be
the minimal possible value of the nonnegative integer defined in Lemma 5.3.4
(depending on an affine covering $X=\bigcup_{\alpha}X_{\alpha}$ subordinate to
${\mathbf{W}}$ and affine coverings of its intersections
$X_{\alpha_{1}}\cap\dotsb\cap X_{\alpha_{k}}$). Then
(a) the right homological dimension of any quasi-coherent sheaf on $X$ with
respect to the exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ac}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ does not exceed $M-1$;
(b) the left homological dimension of any locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ does not exceed
$M-1$;
(c) assuming $X$ has finite Krull dimension, the left homological dimension of
any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf on $X$ with
respect to the exact subcategory
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ does not exceed $M-1$.
###### Proof.
Part (b): since there are enough projectives in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and these
belong to
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}^{\mathsf{lct}}$,
the left homological dimension is well-defined. Now if
$0\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{P}}_{M-2}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}_{0}\longrightarrow{\mathfrak{E}}\longrightarrow
0$ is an exact sequence in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ with
${\mathfrak{P}}_{i}\in
X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}^{\mathsf{lct}}$,
then it is clear from Lemma 5.3.4(b) that ${\mathfrak{Q}}\in
X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{\mathsf{ac}}}^{\mathsf{lct}}$.
The proofs of parts (a) and (c) are similar. ∎
Let $f\colon Y\longrightarrow X$ be a quasi-compact morphism of locally
Noetherian schemes. By Corollary 5.2.8(b), the natural functor
$\mathsf{Hot}^{-}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
is an equivalence of triangulated categories. The derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
is constructed by applying the functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ from Corollary 5.1.6(a)
termwise to bounded above complexes of projective locally cotorsion
contraherent cosheaves. By Corollary 3.4.8(b), one can compute the derived
functor ${\mathbb{L}}f_{!}$ using left resolutions of objects of
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ by coflasque locally
cotorsion contraherent cosheaves.
Similarly, if the scheme $Y$ is Noetherian of finite Krull dimension, by
Corollary 5.2.8(a), the natural functor
$\mathsf{Hot}^{-}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--lcth}}})$
is an equivalence of triangulated categories. The derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--lcth}}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$
is constructed by applying the functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ from Corollary 5.2.10(a) termwise to
bounded above complexes of projective contraherent cosheaves. By Corollary
3.4.8(a), one can compute the derived functor ${\mathbb{L}}f_{!}$ using left
resolutions by coflasque contraherent cosheaves.
Let ${\mathbf{W}}$ and ${\mathbf{T}}$ be open coverings of the schemes $X$ and
$Y$. We will call a locally cotorsion ${\mathbf{T}}$-locally contraherent
cosheaf ${\mathfrak{Q}}$ on $Y$ _acyclic with respect to $f$ over
${\mathbf{W}}$_ (or _$f/{\mathbf{W}}$ -acyclic_) if the object
${\mathbb{L}}f_{!}({\mathfrak{Q}})\in{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
belongs to the full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}\subset{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$.
In other words, the complex ${\mathbb{L}}f_{!}({\mathfrak{Q}})$ should have
left homological dimension not exceeding $0$ with respect to the exact
subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ (in the sense of Section
A.5).
Similarly, if the scheme $Y$ is Noetherian of finite Krull dimension, we will
call a ${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on $Y$
_acyclic with respect to $f$ over ${\mathbf{W}}$_ (or _$f/{\mathbf{W}}$
-acyclic_ if the object
${\mathbb{L}}f_{!}({\mathfrak{Q}})\in{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$
belongs to the full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}\subset{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$.
In other words, the complex ${\mathbb{L}}f_{!}({\mathfrak{Q}})$ must have left
homological dimension at most $0$ with respect to the exact subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$. According to Corollary A.5.3, an object of
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ is
$f/{\mathbf{W}}$-acyclic if and only if it is $f/{\mathbf{W}}$-acyclic as an
object of $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$. Any coflasque
(locally cotorsion) contrarent cosheaf on $Y$ is $f/{\mathbf{W}}$-acyclic.
It is easy to see that the full subcategory of $f/{\mathbf{W}}$-acyclic
locally cotorsion ${\mathbf{T}}$-locally contraherent cosheaves in
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is closed
under extensions, kernels of admissible epimorphisms, and infinite products.
The full subcategory of $f/{\mathbf{W}}$-acyclic ${\mathbf{T}}$-locally
contraherent cosheaves in $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$
(defined under the appropriate assumptions above) has the same properties. We
denote these subcategories with their induced exact category structures by
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$
and
$Y{\operatorname{\mathsf{--lcth}}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$,
respectively.
Finally, the natural functor
$\mathsf{Hot}^{+}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories by Corollary 4.11.2(a); and the
derived functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$
is constructed by applying the functor $f_{*}$ termwise to bounded below
complexes of injective quasi-coherent sheaves.
A quasi-coherent sheaf ${\mathcal{Q}}$ on $Y$ is called _acyclic with respect
to $f$_ (or _$f$ -acyclic_) if the object
${\mathbb{R}}f_{*}({\mathcal{Q}})\in{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$
belongs to the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}\subset{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$.
The full subcategory of $f$-acyclic quasi-coherent sheaves in
$Y{\operatorname{\mathsf{--qcoh}}}$ is closed under extensions, cokernels of
injective morphisms, and infinite direct sums. We denote this exact
subcategory by
$Y{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--ac}}}}$.
###### Lemma 5.3.6.
(a) Let ${\mathfrak{Q}}$ be an $f/{\mathbf{W}}$-acyclic locally cotorsion
${\mathbf{T}}$-locally contraherent cosheaf on $Y$. Then the cosheaf of
${\mathcal{O}}_{X}$-modules $f_{!}{\mathfrak{Q}}$ is locally cotorsion
${\mathbf{W}}$-locally contraherent, and the object represented by it in the
derived category
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$ is
naturally isomorphic to ${\mathbb{L}}f_{!}{\mathfrak{Q}}$.
(b) Assuming that the scheme $Y$ is Noetherian of finite Krull dimension, let
${\mathfrak{Q}}$ be an $f/{\mathbf{W}}$-acyclic ${\mathbf{T}}$-locally
contraherent cosheaf on $Y$. Then the cosheaf of ${\mathcal{O}}_{X}$-modules
$f_{!}{\mathfrak{Q}}$ is ${\mathbf{W}}$-locally contraherent, and the object
represented by it in the derived category
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$ is naturally isomorphic
to ${\mathbb{L}}f_{!}{\mathfrak{Q}}$.
###### Proof.
We will prove part (a), the proof of part (b) being similar. A complex
$\dotsb\longrightarrow{\mathfrak{P}}_{2}\longrightarrow{\mathfrak{P}}_{1}\longrightarrow{\mathfrak{P}}_{0}$
over $X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\mathbf{W}}$ being
isomorphic to an object ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\mathbf{W}}$ in
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$ means
that for each affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$
the complex of cotorsion ${\mathcal{O}}(U)$-modules
$\dotsb\longrightarrow{\mathfrak{P}}_{2}[U]\longrightarrow{\mathfrak{P}}_{1}[U]\longrightarrow{\mathfrak{P}}_{0}[U]$
is acyclic except at the rightmost term, its ${\mathcal{O}}(U)$-modules of
cocycles are cotorsion, and the cokernel of the morphism
${\mathfrak{P}}_{1}\longrightarrow{\mathfrak{P}}_{0}$ taken in the category of
cosheaves of ${\mathcal{O}}_{X}$-modules, that is the cosheaf
$U\longmapsto\operatorname{coker}({\mathfrak{P}}_{1}[U]\to{\mathfrak{P}}_{0}[U])$,
is identified with ${\mathfrak{P}}$.
Now let
$\dotsb\longrightarrow{\mathfrak{F}}_{2}\longrightarrow{\mathfrak{F}}_{1}\longrightarrow{\mathfrak{F}}_{0}$
be a left projective resolution of the object ${\mathfrak{Q}}$ in the exact
category $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$; then
the cosheaves $f_{!}{\mathfrak{F}}_{i}$ on $X$ belong to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. It remains to
notice that the functor $f_{!}$ preserves cokernels taken in the categories of
cosheaves of ${\mathcal{O}}_{Y}$\- and ${\mathcal{O}}_{X}$-modules. ∎
###### Lemma 5.3.7.
Let $f\colon Y\longrightarrow X$ be a morphism of Noetherian schemes with open
coverings ${\mathbf{T}}$ and ${\mathbf{W}}$. Assume that either
(a) ${\mathbf{W}}$ is a finite affine open covering of $X$, or
(b) one of the schemes $X$ or $Y$ has finite Krull dimension.
Then any locally cotorsion ${\mathbf{T}}$-locally contraherent cosheaf on $Y$
has finite left homological dimension with respect to the exact subcategory
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}\subset
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$.
###### Proof.
Since $Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\subset
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$,
the left homological dimension with respect to the exact subcategory
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}\subset
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ is well-defined
for any quasi-compact morphism of locally Noetherian schemes $f\colon
Y\longrightarrow X$. In view of Corollaries 5.1.6(a) and A.5.2, the left
homological dimension of a cosheaf ${\mathfrak{E}}\in
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ with respect to
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$
does not exceed (in fact, is equal to) the left homological dimension of the
complex
${\mathbb{L}}f_{!}({\mathfrak{E}})\in{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$.
To prove part (a), denote by $M(Z,{\mathbf{t}})$ the minimal value of the
nonnegative integer $M$ defined in Lemma 5.3.4 and Corollary 5.3.5 for a given
Noetherian scheme $Z$ with an open covering ${\mathbf{t}}$. Let $M$ be the
maximal value of $M(f^{-1}(W),{\mathbf{T}}|_{f^{-1}(W)})$ taken over all
affine open subschemes $W\in{\mathbf{W}}$. We will show that the left
homological dimension of any object of
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ with respect
to
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$
does not exceed $M-1$.
Set $Z=f^{-1}(W)$ and ${\mathbf{t}}={\mathbf{T}}|_{Z}$. It suffices to check
that for any Noetherian affine scheme $W$, a Noetherian scheme $Z$ with an
open covering ${\mathbf{t}}$, a morphism of schemes $g\colon Z\longrightarrow
W$, and a cosheaf ${\mathfrak{E}}\in
Z{\operatorname{\mathsf{--lcth}}}_{\mathbf{t}}^{\mathsf{lct}}$, the complex of
cotorsion ${\mathcal{O}}(W)$-modules $({\mathbb{L}}g_{!}{\mathfrak{E}})[W]$ is
isomorphic to a complex of cotorsion ${\mathcal{O}}(W)$-modules concentrated
in the homological degrees $\le M-1$ in the derived category
${\mathcal{O}}(W){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. The argument
below follows the proof of Lemma 5.3.4.
Let
${\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}$
be a left projective resolution of the locally cotorsion locally contraherent
cosheaf ${\mathfrak{E}}$ and $U_{\alpha}$ be a finite affine covering of the
scheme $Z$ subordinate to ${\mathbf{t}}$. Then the total complex of the
bicomplex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is homotopy equivalent to the complex
$(g_{!}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})[W]$
computing the object
$({\mathbb{L}}g_{!}{\mathfrak{E}})[W]\in{\mathsf{D}}^{-}({\mathcal{O}}(W){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$.
On the other hand, for each $1\le k\le N$ the complex
$C_{k}(\\{U_{\alpha}\\},{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is the direct sum of the complexes
$(g_{\alpha_{1},\dotsc,\alpha_{k}!}j^{!}_{\alpha_{1},\dotsc,\alpha_{k}}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})[W]$,
where $g_{\alpha_{1},\dotsc,\alpha_{k}}$ denotes the composition $g\circ
j_{\alpha_{1},\dotsc,\alpha_{k}}\colon U_{\alpha_{1}}\cap\dotsb\cap
U_{\alpha_{k}}\longrightarrow W$.
Let $V_{\beta}$ be an affine open covering of the intersection
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{k}}$. Then the complex
$(g_{\alpha_{1},\dotsc,\alpha_{k}!}j^{!}_{\alpha_{1},\dotsc,\alpha_{k}}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})[W]$
is homotopy equivalent to the total complex of the Čech bicomplex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\beta}\\},j_{\alpha_{1},\dotsc,\alpha_{k}}^{!}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
For each $1\le l\le n_{\alpha_{1},\dotsc,\alpha_{k}}$, the complex
$C_{l}(\\{V_{\beta}\\},j^{!}_{\alpha_{1},\dotsc,\alpha_{k}}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is the direct sum of the complexes
$(h_{\beta_{1},\dotsc,\beta_{l}!}e^{!}_{\beta_{1},\dotsc,\beta_{l}}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})[W]$,
where $e_{\beta_{1},\dotsc,\beta_{l}}$ are the embeddings
$V_{\beta_{1}}\cap\dotsb\cap V_{\beta_{l}}\longrightarrow Z$ and
$h_{\beta_{1},\dotsc,\beta_{l}}$ are the compositions
$g_{\alpha_{1},\dotsc,\alpha_{k}}\circ e_{\beta_{1},\dotsc,\beta_{l}}\colon
V_{\beta_{1}}\cap\dotsb\cap V_{\beta_{l}}\longrightarrow W$.
Finally, the schemes $V_{\beta_{1}}\cap\dotsb\cap V_{\beta_{l}}$ being affine
and
$e_{\beta_{1},\dotsc,\beta_{l}}^{!}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
being a projective resolution of a locally cotorsion contraherent cosheaf
$e_{\beta_{1},\dotsc,\beta_{l}}^{!}{\mathfrak{E}}$ on
$V_{\beta_{1}}\cap\dotsb\cap V_{\beta_{l}}$, the complex of cotorsion
${\mathcal{O}}(W)$-modules
$(h_{\beta_{1},\dotsc,\beta_{l}!}e^{!}_{\beta_{1},\dotsc,\beta_{l}}{\mathfrak{F}}_{\text{\smaller\smaller$\scriptstyle\bullet$}})[W]$
is isomorphic to the cotorsion ${\mathcal{O}}(W)$-module
${\mathfrak{E}}[V_{\beta_{1}}\cap\dotsb\cap
V_{\beta_{l}}]\simeq(h_{\beta_{1},\dotsc,\beta_{l}!}e^{!}_{\beta_{1},\dotsc,\beta_{l}}{\mathfrak{E}})[W]$
in
${\mathsf{D}}^{-}({\mathcal{O}}(W){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$.
Part (a) is proven. Similarly one can show that the left homological dimension
of any object of $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$ with respect
to
$Y{\operatorname{\mathsf{--lcth}}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$
does not exceed $M-1$ (assuming that the Krull dimension of $Y$ is finite).
Now if the scheme $X$ has finite Krull dimension $D$, pick a finite affine
open covering $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ subordinate to
${\mathbf{W}}$. Then, by Lemma 5.3.1(b), the left homological dimension of any
object of
$X{\operatorname{\mathsf{--lcth}}}_{\\{U_{\alpha}\\}}^{\mathsf{lct}}$ with
respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\\{U_{\alpha}\\}}^{\mathsf{lct}}$ does not
exceed $D$. It follows that the left homological dimension of any complex from
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$ with
respect to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
does not exceed its left homological dimension with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\\{U_{\alpha}\\}}^{\mathsf{lct}}$ plus
$D$. Hence the left homological dimension of any locally cotorsion locally
contraherent cosheaf from
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ with respect
to
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$
does not exceed its left homological dimension with respect to
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,f/\\{U_{\alpha}\\}{{\operatorname{\mathsf{--ac}}}}}$
plus $D$. Here the former summand is finite by part (a).
If the scheme $Y$ has finite Krull dimension $D$, then the left homological
dimension in question does not exceed $D$ by Lemma 5.3.1(a), since
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}\subset
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}$.
∎
###### Lemma 5.3.8.
Let $f\colon Y\longrightarrow X$ be a morphism of Noetherian schemes. Then any
quasi-coherent sheaf on $Y$ has finite right homological dimension with
respect to the exact subcategory
$Y{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--ac}}}}\subset
Y{\operatorname{\mathsf{--qcoh}}}$.
###### Proof.
Similar to (and simpler than) Lemma 5.3.7. ∎
###### Corollary 5.3.9.
Let $f\colon Y\longrightarrow X$ be a morphism of Noetherian schemes. Then
(a) for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--ac}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
induced by the embedding of exact categories
$Y{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--ac}}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ is an equivalence of categories;
(b) for any symbol $\star=-$ or ${\mathsf{ctr}}$, the triangulated functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ is an
equivalence of categories;
(c) assuming that one of the conditions of Lemma 5.3.7 holds, for any symbol
$\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$ the triangulated functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{T}},\,{f/{\mathbf{W}}{\operatorname{\mathsf{--ac}}}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$ is an
equivalence of categories.
###### Proof.
Part (a) follows from Lemma 5.3.8 together with the dual version of
Proposition A.5.6. Part (b) is provided by Proposition A.3.1. Part (c) follows
from Lemma 5.3.7 together with Proposition A.5.6. ∎
### 5.4. Background equivalences of triangulated categories
The results of this section complement those of Sections 4.6 and 4.9–4.11.
Let $X$ be a semi-separated Noetherian scheme. The _cotorsion dimension_ of a
quasi-coherent sheaf on $X$ is defined as its right homological dimension with
respect to the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
X{\operatorname{\mathsf{--qcoh}}}$, i. e., the minimal length of a right
resolution by cotorsion quasi-coherent sheaves. For the definition of the
_very flat dimension_ of a quasi-coherent sheaf on $X$, we refer to Section
4.9.
###### Lemma 5.4.1.
Let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering, and
let $D$ denote the Krull dimension of the scheme $X$. Then
(a) the very flat dimension of any flat quasi-coherent sheaf on $X$ does not
exceed $D$;
(b) the homological dimension of the exact category of flat quasi-coherent
sheaves on $X$ does not exceed $N-1+D$;
(c) the cotorsion dimension of any quasi-coherent sheaf on $X$ does not exceed
$N-1+D$;
(d) the right homological dimension of any flat quasi-coherent sheaf on $X$
with respect to the exact subcategory of flat cotorsion quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ does not exceed $N-1+D$;
###### Proof.
Part (a) follows from Theorem 1.5.6. Part (b): one proves that
$\operatorname{Ext}_{X}^{>N-1+D}({\mathcal{F}},\allowbreak{\mathcal{M}})=0$
for any flat quasi-coherent sheaf ${\mathcal{F}}$ and any quasi-coherent
${\mathcal{M}}$ on $X$. One can use the Čech resolution (12) of the sheaf
${\mathcal{M}}$ and the natural isomorphisms
$\operatorname{Ext}_{X}^{*}({\mathcal{F}},j_{*}{\mathcal{G}})\simeq\operatorname{Ext}_{U}^{*}(j^{*}{\mathcal{F}},{\mathcal{G}})$
for the embeddings of affine open subschemes $j\colon U\longrightarrow X$ and
any quasi-coherent sheaves ${\mathcal{F}}$, ${\mathcal{G}}$ in order to reduce
the question to the case of an affine scheme $U$. Then it remains to apply the
same result from [55] discussed in Section 1.5. Taking into account
Corollaries 4.1.9(c) and 4.1.11(b) (guaranteeing that the cotorsion dimension
is well-defined), the Ext vanishing that we have proven implies part (c) as
well. The right homological dimension in part (d) is well-defined due to
Corollary 4.1.11(b), so (d) follows from (c) in view of the dual version of
Corollary A.5.3 (cf. Lemma 4.6.9). ∎
###### Corollary 5.4.2.
Let $X$ be a semi-separated Noetherian scheme of finite Krull dimension and
$d\ge 0$ be any (finite) integer. Then the natural triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
and
${\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\allowbreak\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
are equivalences of triangulated categories. In particular, such functors
between the derived categories of the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ are equivalences of
categories.
###### Proof.
Follows from Lemma 5.4.1(b) together with the result of [50, Remark 2.1]. ∎
###### Corollary 5.4.3.
Let $X$ be a semi-separated Noetherian scheme of finite Krull dimension. Then
for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ is an equivalence of
triangulated categories.
###### Proof.
Follows from Lemma 5.4.1(a) together with Proposition A.5.6. ∎
###### Corollary 5.4.4.
Let $X$ be a Noetherian scheme of finite Krull dimension. Then
(a) for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$ is an equivalence of
triangulated categories;
(b) for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is an equivalence of
triangulated categories;
(c) for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}$ is an equivalence of triangulated
categories.
###### Proof.
Part (a) is provided by Lemma 5.2.5(b) together with the dual version of
Proposition A.5.6. Parts (b-c) follow from part (a) and Corollary 5.3.2(b-c).
Alternatively, in the case of a semi-separated Noetherian scheme $X$ of finite
Krull dimension, the assertions (b-c) can be obtained directly from Lemma
5.2.1(a). ∎
The following corollary is another restricted version of Theorem 4.6.6; it is
to be compared with Corollary 4.6.10.
###### Corollary 5.4.5.
Let $X$ be a semi-separated Noetherian scheme of finite Krull dimension. Then
for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$ there is a natural
equivalence of triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.
###### Proof.
Assuming $\star\neq{\mathsf{co}}$, by Lemma 5.4.1(d) together with the dual
version of Proposition A.5.6 the triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is an equivalence of categories. In view of Corollary 5.4.2, the same
assertion holds for $\star={\mathsf{co}}$. Hence it remains to recall that the
equivalence of categories from Lemma 4.6.7 identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ (see the
proof of Corollary 4.6.10(c)). ∎
###### Corollary 5.4.6.
Let $f\colon Y\longrightarrow X$ be a morphism of finite flat dimension
between semi-separated Noetherian schemes. Then for any symbol
$\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$,
${\mathsf{co}}$, or ${\mathsf{abs}}$ the equivalences of triangulated
categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{\star}(Y\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
from Corollary 5.4.5 transform the right derived functor ${\mathbb{R}}f_{*}$
(65) into the left derived functor ${\mathbb{L}}f_{!}$ (69).
###### Proof.
Can be either deduced from Corollary 4.11.6(c), or proven directly in the
similar way using Lemma 4.11.3(d). ∎
Let $X$ be a locally Noetherian scheme with an open covering ${\mathbf{W}}$.
As in Section 4.11, we denote by
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$ the
full subcategory of objects of injective dimension $\le d$ in
$X{\operatorname{\mathsf{--qcoh}}}$ and by
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$
the full subcategory of objects of projective dimension $\le d$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. For a
Noetherian scheme $X$ of finite Krull dimension, let
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$
denote the full subcategory of objects of projective dimension $\le d$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. We set
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},{{\operatorname{\mathsf{fpd--}}}d}}^{\mathsf{lct}}$
and
$X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},{{\operatorname{\mathsf{fpd--}}}d}}$.
Clearly, the projective dimension of an object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ does not change when the open
covering ${\mathbf{W}}$ is replaced by its refinement.
The full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}$ is closed under extensions, cokernels of
admissible monomoprhisms, and infinite direct sums. The full subcategory
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is closed under
extensions, kernels of admissible epimorphisms, and infinite products (see
Corollary 5.1.5).
###### Corollary 5.4.7.
(a) Let $X$ be a locally Noetherian scheme. Then the natural triangulated
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$
are equivalences of categories.
(b) Let $X$ be a locally Noetherian scheme. Then the natural triangulated
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$
are equivalences of categories.
(c) Let $X$ be a Noetherian scheme of finite Krull dimension. Then the natural
triangulated functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$
are equivalences of categories.
###### Proof.
Part (a) follows from Corollary 4.11.1(a) and [50, Remark 2.1], while parts
(b-c) follow from Proposition A.5.6 and the same Remark. ∎
A cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{G}}$ on a scheme $X$ is
said to have _${\mathbf{W}}$ -flat dimension not exceeding $d$_ if the flat
dimension of the ${\mathcal{O}}_{X}(U)$-module ${\mathfrak{G}}[U]$ does not
exceed $d$ for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. The flat dimension of a cosheaf of ${\mathcal{O}}_{X}$-modules
is defined as its $\\{X\\}$-flat dimension. ${\mathbf{W}}$-locally
contraherent cosheaves of ${\mathbf{W}}$-flat dimension not exceeding $d$ on a
locally Noetherian scheme $X$ form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ closed under extensions,
kernels of admissible epimorphisms, and infinite products. We set
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{ffd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{{\operatorname{\mathsf{ffd--}}}d}$.
The flat dimension of a contraherent cosheaf ${\mathfrak{G}}$ on an affine
Noetherian scheme $U$ is equal to the flat dimension of the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{G}}[U]$ (see Section 3.7). Over a
semi-separated Noetherian scheme $X$, a ${\mathbf{W}}$-locally contraherent
cosheaf has ${\mathbf{W}}$-flat dimension $\le d$ if and only if it admits a
left resolution of length $\le d$ by ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaves (see Corollary 4.4.5(a)).
Hence it follows from Corollary 5.2.2(b) (applied to affine open subschemes
$U\subset X$) that the ${\mathbf{W}}$-flat dimension of a
${\mathbf{W}}$-locally contraherent cosheaf on a locally Noetherian scheme $X$
of finite Krull dimension does not change when the covering ${\mathbf{W}}$ is
replaced by its refinement. According to part (a) of the same Corollary, on a
semi-separated Noetherian scheme of finite Krull dimension the
${\mathbf{W}}$-flat dimension of a ${\mathbf{W}}$-locally contraherent cosheaf
is equal to its colocally flat dimension; so
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}}$.
By Corollary 5.1.4, the ${\mathbf{W}}$-flat dimension of a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on a locally Noetherian scheme $X$
coincides with its projective dimension in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ (and also does
not depend on ${\mathbf{W}}$). So one has
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}=X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$.
###### Lemma 5.4.8.
Let $X$ be a Noetherian scheme of finite Krull dimension $D$. Then a
${\mathbf{W}}$-locally contraherent cosheaf on $X$ has finite projective
dimension in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ if and only if it has finite
${\mathbf{W}}$-flat dimension. More precisely, the inclusions of full
subcategories
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}(d+D)}}$
hold in the category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
###### Proof.
The inclusion
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}$
holds due to Corollary 5.2.4. Conversely, by the same Corollary any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$ has a left
resolution by flat contraherent cosheaves, so the ${\mathbf{W}}$-flat
dimension of ${\mathfrak{M}}$ is equal to its left homological dimension with
respect to $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Corollary 5.2.2(b)). It
remains to apply the last assertion of Corollary 5.2.6(b). ∎
###### Corollary 5.4.9.
For any Noetherian scheme $X$ of finite Krull dimension and any (finite)
integer $d\ge 0$, the natural triangulated functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})$
are equivalences of triangulated categories.
###### Proof.
It is clear from Lemma 5.4.8 that the homological dimension of the exact
category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}$
is finite, so it remains to apply [50, Remark 2.1] (to obtain the equivalences
between various derived categories of this exact category) and Proposition
A.5.6 (to identify the absolute derived categories with the homotopy
categories of projective objects). Alternatively, one can use [51, Theorem 3.6
and Remark 3.6]. ∎
The following theorem is the main result of this section.
###### Theorem 5.4.10.
(a) For any locally Noetherian scheme $X$, the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.
(b) For any locally Noetherian scheme $X$, the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}^{\mathsf{lct}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is an equivalence of triangulated categories.
(c) For any semi-separated Noetherian scheme $X$, the natural functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories.
(d) For any Noetherian scheme $X$ of finite Krull dimension, the natural
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of triangulated categories.
###### Proof.
Part (a) is a standard result (see, e. g., [53, Lemma 1.7(b)]) which is a
particular case of [51, Theorem 3.7 and Remark 3.7] and can be also obtained
from the dual version of Proposition A.3.1(b). The key observation is that
there are enough injectives in $X{\operatorname{\mathsf{--qcoh}}}$ and the
full subcategory $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ they form
is closed under infinite direct sums. Similarly, part (b) can be obtained
either from Proposition A.3.1(b), or from the dual version of [51, Theorem 3.7
and Remark 3.7] (see also [51, Section 3.8]). In any case the argument is
based on Theorem 5.1.1(a) and Corollary 5.1.5. Part (c) holds by Proposition
A.3.1(b) together with Lemma 4.3.3, 4.4.1(a), or 4.4.3(a).
Finally, in part (d) the functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
are equivalences of categories by Corollary 5.4.9, and the functor
${\mathsf{D}}^{\mathsf{ctr}}(X\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of categories by Proposition A.3.1(b) together with
Corollary 5.2.4. A direct proof of the equivalence
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is also possible; it proceeds along the following lines.
One has to use the more advanced features of the results of [51, Sections
3.7–3.8] involving the full generality of the conditions (${*}$)–(${*}{*}$).
Alternatively, one can apply the more general Corollary A.6.2. Specifically,
let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering; then it
follows from Corollary 5.2.4(b) that an infinite product of projective
contraherent cosheaves on $X$ is a direct summand of a direct sum over
$\alpha$ of the direct images of contraherent cosheaves on $U_{\alpha}$
correspoding to infinite products of very flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.
Infinite products of such modules may not be very flat, but they are certainly
flat and contraadjusted. By the last assertion of Corollary 5.2.6(b), one can
conclude that the projective dimensions of infinite products of projective
objects in $X{\operatorname{\mathsf{--ctrh}}}$ do not exceed the Krull
dimension $D$ of the scheme $X$. So the contraherent cosheaf analogue of the
condition (${*}{*}$) holds for $X{\operatorname{\mathsf{--ctrh}}}$, or in
other words, the assumption of Corollary A.6.2 is satisfied by the pair of
exact categories $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--ctrh}}}$. ∎
The following corollary is to be compared with Corollaries 5.2.8(b) and
5.3.3(b).
###### Corollary 5.4.11.
For any locally Noetherian scheme $X$ with an open covering ${\mathbf{W}}$,
the triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is an
equivalence of triangulated categories.
###### Proof.
Follows from Theorem 5.4.10(b) applied to the coverings $\\{X\\}$ and
${\mathbf{W}}$ of the scheme $X$. Alternatively, one can apply directly
Proposition A.3.1(b) together with Theorem 5.1.1(a). ∎
### 5.5. Co-contra correspondence over a regular scheme
Let $X$ be a regular semi-separated Noetherian scheme of finite Krull
dimension.
###### Theorem 5.5.1.
(a) The triangulated functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is an equivalence of triangulated
categories.
(b) The triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ is an equivalence of triangulated
categories.
(c) There is a natural equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
provided by the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ and
${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}$.
###### Proof.
Part (a) actually holds for any symbol $\star\neq{\mathsf{ctr}}$ in the upper
indices of the derived category signs, and is a particular case of Corollary
4.9.1(a). Indeed, one has
$X{\operatorname{\mathsf{--qcoh}}}=X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$
provided that $d$ is greater or equal to the Krull dimension of $X$.
Similarly, part (b) actually holds for any symbol $\star\neq{\mathsf{co}}$ in
the upper indices, and is a particular case of Corollary 4.9.1(c). Indeed, one
has
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}=X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
provided that $d$ is greater or equal to the Krull dimension of $X$.
To prove part (c), notice that all the triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of categories by Corollary 4.9.5 (since one also has
$X{\operatorname{\mathsf{--qcoh}}}=X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$
provided that $d$ is greater or equal to the Krull dimension of $X$). So it
remains to apply Theorem 4.6.6. ∎
### 5.6. Co-contra correspondence over a Gorenstein scheme
Let $X$ be a Gorenstein semi-separated Noetherian scheme of finite Krull
dimension. We will use the following formulation of the Gorenstein condition:
for any affine open subscheme $U\subset X$, the classes of
${\mathcal{O}}_{X}(U)$-modules of finite flat dimension, of finite projective
dimension, and of finite injective dimension coincide.
Notice that neither of these dimensions can exceed the Krull dimension $D$ of
the scheme $X$. Accordingly, the class of ${\mathcal{O}}_{X}(U)$-modules
defined by the above finite homological dimension conditions is closed under
both infinite direct sums and infinite products. It is also closed under
extensions and the passages to the cokernels of embeddings and the kernels of
surjections.
Moreover, since the injectivity of a quasi-coherent sheaf on a Noetherian
scheme is a local property, the full subcategories of quasi-coherent sheaves
of finite flat dimension and of finite injective dimension coincide in
$X{\operatorname{\mathsf{--qcoh}}}$. Similarly, the full subcategories of
locally contraherent cosheaves of finite flat dimension and of finite locally
injective dimension coincide in $X{\operatorname{\mathsf{--lcth}}}$. Neither
of these dimensions can exceed $D$.
###### Theorem 5.6.1.
(a) The triangulated functors
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embeddings of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ are equivalences of triangulated
categories.
(b) The triangulated functors
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
induced by the embeddings of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ are equivalences of triangulated
categories.
(c) There is a natural equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})$
provided by the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ and
${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}$.
###### Proof.
Parts (a-b): by Corollary 4.9.1(a,c), the functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X\allowbreak{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
are equivalences of categories for any symbol $\star\neq{\mathsf{ctr}}$ and
the functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
are equivalences of categories for any symbol $\star\neq{\mathsf{co}}$.
To prove that the functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories, notice that one has
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D}=X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$
and the functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D})$
is an equivalence of categories by Corollary 5.4.7(a), while the composition
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories by Theorem 5.4.10(a).
Similarly, to prove that the functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equialence of categories, notice that one has
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}D}=X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$
and the functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}D})$
is an equivalence of categories by Corolary 5.4.9, while the composition
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}D})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of categories by Theorem 5.4.10(d).
To prove part (c), notice that the functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
are equivalences of categories by Corollary 5.4.2, while the functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
are equivalences of categories by Corollary 4.9.5(b).
Furthermore, consider the intersections
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$. As was
explained in Section 4.10, the functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
is an equivalence of triangulated categories for any
$\star\neq{\mathsf{ctr}}$, while the functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
is an equivalence of triangulated categories for any $\star\neq{\mathsf{co}}$.
Finally, it is clear from Lemma 4.10.2(a,d) (see also Lemma 4.11.3) that the
equivalence of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ of Lemma 4.6.7 identifies
their full exact subcategories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}$. So the induced equivalence
of the derived categories ${\mathsf{D}}^{\mathsf{abs}}$ or ${\mathsf{D}}$
provides the desired equivalence of triangulated categories in part (c). ∎
### 5.7. Co-contra correspondence over a scheme with a dualizing complex
Let $X$ be a semi-separated Noetherian scheme with a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ [29], which
we will view as a finite complex of injective quasi-coherent sheaves on $X$.
The following result complements the covariant Serre–Grothendieck duality
theory as developed in the papers and the thesis [33, 47, 42, 53].
###### Theorem 5.7.1.
There are natural equivalences between the four triangulated categories
${\mathsf{D}}^{{\mathsf{abs}}={\mathsf{co}}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$,
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$,
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$, and
${\mathsf{D}}^{{\mathsf{abs}}={\mathsf{ctr}}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$.
(Here the notation ${\mathsf{abs}}={\mathsf{co}}$ and
${\mathsf{abs}}={\mathsf{ctr}}$ presumes the assertions that the corresponding
derived categories of the second kind coincide for the exact category in
question.) Among these, the equivalences
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
do not require a dualizing complex and do not depend on it; all the remaining
equivalences do and do.
###### Proof.
For any quasi-compact semi-separated scheme $X$ with an open covering
${\mathbf{W}}$, one has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})={\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
by Corollary 4.9.5(b). For any Noetherian scheme $X$ of finite Krull
dimension, one has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})={\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
by Corollary 5.4.2.
For any semi-separated Noetherian scheme $X$, one has
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq\mathsf{Hot}(X\allowbreak{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$
by Theorem 5.4.10(a) and
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
by Corollary 4.6.8(b). Hence the desired equivalence
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
and
${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}).$
For any semi-separated Noetherian scheme $X$ of finite Krull dimension, one
has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
by Corollary 5.4.5,
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
by Theorem 5.4.10(b), and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
by Corollary 5.4.4(b). Alternatively, one can refer to the equivalence
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
holding by Corollary 5.4.3,
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$
by Corollary 4.6.10(a), and
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
by Theorem 5.4.10(d). Either way, one gets the same desired equivalence
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
and
${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}).$
Now we are going to construct a commutative diagram of equivalences of
triangulated categories
$\dgARROWLENGTH=7em\begin{diagram}$
for any symbol $\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, or
${\mathsf{abs}}$.
The exterior vertical functors are constructed by applying the additive
functors ${\mathcal{O}}_{X}\odot_{X}{-}$ and
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ to the given
complexes termwise. The interior (derived) vertical functors have been defined
in Corollaries 4.6.8(b) and 5.4.5. All the functors invoking the dualizing
complex ${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
constructed by applying the respective exact functors of two arguments to
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the
given unbounded complex termwise and totalizing the bicomplexes so obtained.
First of all, one notices that the functors in the interior upper triangle are
right adjoint to the ones in the exterior. This follows from the adjunction
(20) together with the adjunction of the tensor product of quasi-coherent
sheaves and the quasi-coherent internal Hom.
The upper horizontal functors
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{-}$
and
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
are mutually inverse for the reasons explained in [42, Theorem 8.4 and
Proposition 8.9] and [53, Theorem 2.5]. The argument in [53] is based on the
observations that the morphism of finite complexes of flat quasi-coherent
sheaves
${\mathcal{F}}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}})$
is a quasi-isomorphism for any sheaf ${\mathcal{F}}\in
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and the morphism of finite
complexes of injective quasi-coherent sheaves
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathcal{J}}$
is a quasi-isomorphism for any sheaf ${\mathcal{J}}\in
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$.
Let us additionally point out that, according to Lemma 2.5.3(c) and [42, Lemma
8.7], the complex
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a complex of flat cotorsion quasi-coherent sheaves for any complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. So the functor
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
actually lands in
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ (as does the functor
${\mathcal{O}}_{X}\odot_{X}{-}$ on the diagram, according to the proof of
Corollary 5.4.5). The interior upper triangle is commutative due to the
natural isomorphism (17). The exterior upper triangle is commutative due to
the natural isomorphism (21).
In order to discuss the equivalence of categories in the lower horizontal
line, we will need the following lemma. It is based on the definitions of the
$\operatorname{\mathfrak{Cohom}}$ functor in Section 3.6 and the contraherent
tensor product functor $\otimes_{X{\operatorname{\mathrm{-ct}}}}$ in Section
3.7.
###### Lemma 5.7.2.
Let ${\mathcal{J}}$ be an injective quasi-coherent sheaf on a semi-separated
Noetherian scheme $X$ with an open covering ${\mathbf{W}}$. Then there are two
well-defined exact functors
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{-})\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$
and
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}\colon
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$
between the exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ of locally injective
${\mathbf{W}}$-locally contraherent cosheaves and colocally flat contraherent
cosheaves on $X$. The functor
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$ is left adjoint to
the functor $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{-})$. Besides,
the functor $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{-})$ takes
values in the additive subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$, while the functor
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$ takes values in the
additive subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$. For any quasi-
coherent sheaf ${\mathcal{M}}$ and any colocally flat contraherent cosheaf
${\mathfrak{F}}$ on $X$ there is a natural isomorphism
(84)
${\mathcal{M}}\odot_{X}({\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})\simeq({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}})\odot_{X}{\mathfrak{F}}$
of quasi-coherent sheaves on $X$.
###### Proof.
Let us show that the locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{\mathfrak{K}})$ is
projective for any locally injective ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{K}}$ on $X$. Indeed, ${\mathcal{J}}$ is a direct summand
of a finite direct sum of the direct images of injective quasi-coherent
sheaves ${\mathcal{I}}$ from the embeddings of affine open subschemes $j\colon
U\longrightarrow X$ subordinate to ${\mathbf{W}}$. So it suffices to consider
the case of ${\mathcal{J}}=j_{*}{\mathcal{I}}$.
According to (40), there is a natural isomorphism of locally cotorsion
(${\mathbf{W}}$-locally) contraherent cosheaves
$\operatorname{\mathfrak{Cohom}}_{X}(j_{*}{\mathcal{I}},{\mathfrak{K}})\simeq
j_{!}\operatorname{\mathfrak{Cohom}}_{U}({\mathcal{I}},j^{!}{\mathfrak{K}})$
on $X$. The ${\mathcal{O}}(U)$-modules ${\mathcal{I}}(U)$ and
${\mathfrak{K}}[U]$ are injective, so
$\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{I}}(U),{\mathfrak{K}}[U])$ is
a flat cotorsion ${\mathcal{O}}(U)$-module. In other words, the locally
cotorsion contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{U}({\mathcal{I}},j^{!}{\mathfrak{K}})$ is
projective on $U$, and therefore its direct image with respect to $j$ is
projective on $X$ (see Lemma 4.4.3(b) or Corollary 4.4.7(b)).
Now let ${\mathfrak{F}}$ be a colocally flat contraherent cosheaf on $X$.
Then, in particular, ${\mathfrak{F}}$ is a flat contraherent cosheaf
(Corollary 4.3.6), so the tensor product
${\mathcal{J}}\otimes_{X}{\mathfrak{F}}$ is a locally injective derived
contrahereable cosheaf on $X$.
Moreover, by Corollary 4.3.4(c), ${\mathfrak{F}}$ is a direct summand of a
finitely iterated extension of the direct images of flat contraherent
cosheaves from affine open subschemes of $X$. It was explained in Section 3.5
that derived contrahereable cosheaves on affine schemes are contrahereable and
the direct images of cosheaves with respect to affine morphisms preserve
contrahereability. Besides, the full subcategory of contrahereable cosheaves
on $X$ is closed under extensions in the exact category of derived
contrahereable cosheaves, and the functor ${\mathcal{J}}\otimes_{X}{-}$ takes
short exact sequences of flat contraherent cosheaves to short exact sequences
of derived contrahereable cosheaves on $X$ (see Section 3.7).
So it follows from the isomorphism (48) that
${\mathcal{J}}\otimes_{X}{\mathfrak{F}}$ is a locally injective contrahereable
cosheaf. Its contraherator
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}=\operatorname{\mathfrak{C}}({\mathcal{J}}\otimes_{X}{\mathfrak{F}})$
is consequently a locally injective contraherent cosheaf on $X$. Furthermore,
according to Section 3.5 the (global) contraherator construction is an exact
functor commuting with the direct images with respect to affine morphisms.
Hence the contraherent cosheaf
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$ is a
direct summand of a finitely iterated extension of the direct images of
(locally) injective contraherent cosheaves from affine open subschemes of $X$,
i. e., ${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$
is a colocally projective locally injective contraherent cosheaf.
We have constructed the desired exact functors. A combination of the
adjunction isomorphisms (35) and (30) makes them adjoint to each other.
Finally, for any ${\mathcal{M}}\in X{\operatorname{\mathsf{--qcoh}}}$ and
${\mathfrak{F}}\in X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ one has
${\mathcal{M}}\odot_{X}({\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})={\mathcal{M}}\odot_{X}\operatorname{\mathfrak{C}}({\mathcal{J}}\otimes_{X}{\mathfrak{F}})\simeq{\mathcal{M}}\odot_{X}({\mathcal{J}}\otimes_{X}{\mathfrak{F}})\simeq({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}})\odot_{X}{\mathfrak{F}}$
according to the isomorphisms (32) and (36). ∎
Now we can return to the proof of Theorem 5.7.1. The functors in the interior
lower triangle are left adjoint to the ones in the exterior, as it follows
from the adjunction (20) and Lemma 5.7.2. Let us show that the lower
horizontal functors are mutually inverse.
According to the proof of Corollary 4.6.8(b), the functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
induced by the embedding
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is an
equivalence of triangulated categories. Therefore, it suffices to show that
for any cosheaf ${\mathfrak{J}}\in
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ the morphism
of complexes of contraherent cosheaves
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathfrak{J}}$
is a homotopy equivalence (or just a quasi-isomorphism in
$X{\operatorname{\mathsf{--ctrh}}}$), and for any cosheaf ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ the morphism
of complexes of contraherent cosheaves
${\mathfrak{P}}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{P}})$
is a homotopy equivalence (or just a quasi-isomorphism in
$X{\operatorname{\mathsf{--ctrh}}}$).
According to Corollaries 4.2.8 and 4.4.3, any object of
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is a direct
summand of a finite direct sum of direct images of objects in the similar
categories on affine open subschemes of $X$. According to (43) and (48)
together with the results of Section 3.5, both functors
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$
commute with such direct images. So the question reduces to the case of an
affine scheme $U$, for which the distinction between quasi-coherent sheaves
and contraherent cosheaves mostly loses its significance, as both are
identified with (appropriate classes of) ${\mathcal{O}}(U)$-modules. For this
reason, the desired quasi-isomorphisms follow from the similar quasi-
isomorphisms for quasi-coherent sheaves obtained in [53, proof of Theorem 2.5]
(as quoted above).
Alternatively, one can argue in the way similar to the proof in [53].
Essentially, this means using an “inverse image localization” procedure in
place of the “direct image localizaton” above. The argument proceeds as
follows.
Let
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a quasi-isomorphism between a finite complex
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of coherent sheaves over $X$ and the complex of injective quasi-coherent
sheaves ${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then the tensor product
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}$
is a finite complex of contraherent cosheaves for any flat contraherent
cosheaf ${\mathfrak{F}}$ on $X$. Furthermore, the morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a quasi-isomorphism of finite complexes over the exact category of
coadjusted quasi-coherent cosheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{coa}}$ on $X$, hence the induced
morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}$
is a quasi-isomorphism of finite complexes over the exact category of derived
contrahereable cosheaves on $X$. It follows that the morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}\simeq{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$
is a quasi-isomorphism of finite complexes of contraherent cosheaves on $X$
for any flat contraherent cosheaf ${\mathfrak{F}}$.
Let ${\mathfrak{J}}$ be a cosheaf from
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$. In order to
show that the morphism of finite complexes
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\longrightarrow{\mathfrak{J}}$
is a quasi-isomorphism over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$, it suffices
to check that the composition
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\longrightarrow{\mathfrak{J}}$
is a quasi-isomorphism over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
The latter assertion can be checked locally, i. e., it simply means that for
any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ the
morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),{\mathfrak{J}}[U])\longrightarrow{\mathfrak{J}}[U]$
is a quasi-isomorphism of complexes of ${\mathcal{O}}_{X}(U)$-modules. This
can be deduced from the condition that the morphism
${\mathcal{O}}_{X}(U)\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U))$
is a quasi-isomorphism, as explained in the proof in [53].
Let ${\mathfrak{F}}$ be a flat contraherent cosheaf on $X$. Pick a bounded
above complex of very flat quasi-coherent sheaves
${}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X$ together with a quasi-isomorphism
${}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then the bounded below complex of contraherent cosheaves
$\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}})$
is well-defined. The morphisms of bounded below complexes
$\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}})\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})$
and
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})$
are quasi-isomorphisms over $X{\operatorname{\mathsf{--ctrh}}}$. Thus in order
to show that the morphism
${\mathfrak{F}}\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})$
is a quasi-isomorphism in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$,
it suffices to check that the morphism
${\mathfrak{F}}\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}})$
is a quasi-isomorphism of bounded below complexes over
$X{\operatorname{\mathsf{--ctrh}}}$.
The latter is again a local assertion, meaning simply that the morphism
${\mathfrak{F}}[U]\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),\>{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U])$
is a quasi-isomorphism of complexes of ${\mathcal{O}}_{X}(U)$-modules for any
affine open subscheme $U\subset X$. One proves it by replacing
${}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)$
by a quasi-isomorphic bounded above complex
${}^{\prime\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)$
of finitely generated projective ${\mathcal{O}}_{X}(U)$-modules, and reducing
again to the condition that the morphism
${\mathcal{O}}_{X}(U)\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({}^{\prime\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U))$
is a quasi-isomorphism (cf. [53]).
According to the proof of Corollary 4.6.8(b), the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ on the diagram
actually lands in
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
(as does the functor
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$,
according to Lemma 5.7.2). The exterior upper triangle is commutative due to
the natural isomorphism (19). The interior upper triangle is commutative due
to the natural isomorphism (84).
The assertion that the two diagonal functors on the diagram are mutually
inverse follows from the above. It can be also proven directly in the manner
of the former of the above the proofs of the assertion that the two lower
horizontal functors are mutually inverse. One needs to use the natural
isomorphisms (45) and (46) for commutation with the direct images. ∎
### 5.8. Co-contra correspondence over a non-semi-separated scheme
The goal of this section is to obtain partial generalizations of Theorems
4.6.6 and 5.7.1 to the case of a non-semi-separated Noetherian scheme.
###### Theorem 5.8.1.
Let $X$ be a Noetherian scheme of finite Krull dimension. Then for any symbol
$\star={\mathsf{b}}$, $+$, $-$, or $\empt$ there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$.
###### Proof.
According to Corollaries 5.3.3 and 5.4.4, one has
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\allowbreak\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
for any open covering ${\mathbf{W}}$ of the scheme $X$. We will construct an
equivalence of triangulated categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
and then show that it takes the full subcategories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
into the full subcategories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
and back for all symbols $\star={\mathsf{b}}$, $+$, or $-$.
By Lemma 3.4.7(a), the sheaf ${\mathcal{O}}_{X}$ has a finite right resolution
by flasque quasi-coherent sheaves. We fix such a resolution
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the time
being. Given a complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$, we pick a complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ quasi-isomorphic to
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ (see Theorem 5.4.10(a), cf. Theorem 5.10.2
below) and assign to
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the total
complex of the bicomplex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. Given a complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$, we pick a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ quasi-
isomorphic to ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ (see Theorem
5.4.10(b), cf. Theorem 5.10.3(a) below) and assign to
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the total
complex of the bicomplex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}$.
Let us first show that the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is acyclic whenever a
complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ is. For any scheme point
$x\in X$, let ${\mathfrak{m}}_{x,X}$ denote the maximal ideal of the local
ring ${\mathcal{O}}_{x,X}$. By [29, Proposition II.7.17], any injective quasi-
coherent sheaf ${\mathcal{I}}$ on $X$ can be presented as an infinite direct
sum ${\mathcal{I}}=\bigoplus_{x\in X}\iota_{x}{}_{*}\widetilde{I}_{x}$, where
$\iota_{x}\colon\operatorname{Spec}{\mathcal{O}}_{x,X}\longrightarrow X$ are
the natural morphisms and $\widetilde{I}_{x}$ are the quasi-coherent sheaves
on $\operatorname{Spec}{\mathcal{O}}_{x,X}$ corresponding to infinite direct
sums of copies of the injective envelopes of the ${\mathcal{O}}_{x,X}$-modules
${\mathcal{O}}_{x,X}/{\mathfrak{m}}_{x,X}$. Let
$X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering. Set
$S_{\beta}\subset X$ to be the set-theoretic complement to
$\bigcup_{\alpha<\beta}U_{\alpha}$ in $U_{\beta}$, and consider the direct sum
decomposition ${\mathcal{I}}=\bigoplus_{\alpha=1}^{N}{\mathcal{I}}_{\alpha}$
with ${\mathcal{I}}_{\alpha}=\bigoplus_{z\in
S_{\alpha}}\iota_{z}{}_{*}\widetilde{I}_{z}$.
The associated decreasing filtration
${\mathcal{I}}_{\ge\alpha}=\bigoplus_{\beta\ge\alpha}{\mathcal{I}}_{\beta}$ is
preserved by all morphisms of injective quasi-coherent sheaves ${\mathcal{I}}$
on $X$ (cf. Theorem 5.1.1 and Lemma 5.1.2). We obtain a termwise split
filtration
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\ge\alpha}$ on
the complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the associated quotient complexes
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
isomorphic to the direct images
$j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of complexes of injective quasi-coherent sheaves
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ from
the open embeddings $j_{\alpha}\colon U_{\alpha}\longrightarrow X$. Moreover,
for $\alpha=1$ the complex of quasi-coherent sheaves
${\mathcal{K}}_{1}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j_{1}^{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic, since the complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is; and the
complex
$j_{1}{}_{*}{\mathcal{K}}_{1}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic by Corollary 3.4.9(a) or Lemma 3.4.7(a). It follows by induction
that all the complexes
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ are acyclic over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}$.
Now one has
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
by (45). The complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is quasi-
isomorphic to
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
since ${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a complex over $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$,
while the complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$,
since the complex
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic over $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ (see
Corollary 1.5.7 or Lemma 5.4.1(c)). So the complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$;
by Lemma 3.4.6(c), it is also a complex of coflasque contraherent cosheaves.
By Corollary 3.4.9(c), or alternatively by Lemma 3.4.7(b) together with
Corollary 3.4.8(b), it follows that the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.
Therefore, so is the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Similarly one proves that the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic over $X{\operatorname{\mathsf{--qcoh}}}$ whenever a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is acyclic
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. One has to use
Theorem 5.1.1 and Lemma 5.1.2 (see the proof of Theorem 5.9.1(c) below), the
isomorphism (47), and Lemma 3.4.6(d).
We have shown that the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}{-}$
are well defined by the above rules
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
It is a standard fact that the adjunction (20) makes such two triangulated
functors adjoint to each other (cf. [50, Lemma 8.3]). Let us check that the
adjunction morphism
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is an isomorphism in ${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ for any
complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. For the reasons explained
above, one can assume
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
for some complex
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}$. Then
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
Let ${\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a complex over
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$
endowed with a quasi-isomorphism
${\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. Then
$j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, and the
morphism
$j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. So one has
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j_{\alpha}{}_{*}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Both ${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
$j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
being complexes of flasque quasi-coherent sheaves on $U_{\alpha}$, it remains
to show that the natural morphism
$j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a quasi-isomorphism over $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}$. Now
the morphisms
${\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
${\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are quasi-isomorphisms, and the desired assertion follows.
Similarly one shows that the adjunction morphism
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is an isomorphism in
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ for any
complex ${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. This
finishes the construction of the equivalence of categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$.
To show that it does not depend on the choice of a flasque resolution
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the sheaf
${\mathcal{O}}_{X}$, consider an acyclic finite complex
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of flasque
quasi-coherent sheaves on $X$. Then for any injective quasi-coherent sheaf
${\mathcal{J}}$ on $X$ the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is acyclic by
construction. To show that the complex
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}$
is acyclic over $X{\operatorname{\mathsf{--qcoh}}}$ for any cosheaf
${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, one reduces
the question to the case of an affine scheme $X$ using Theorem 5.1.1(b) and
Lemma 3.4.6(d).
Finally, it remains to show that the equivalence of categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
that we have constructed takes bounded above (resp., below) complexes to
bounded above (resp., below) complexes and vice versa (up to quasi-
isomorphism). If a complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ is bounded below, it has bounded below
injective resolution
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is also bounded below.
Now assume that a complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ has bounded above
cohomology.
Arguing as above, consider its decreasing filtration
${\mathcal{J}}_{\ge\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the associatived quotient complexes
${\mathcal{J}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Using Lemma 3.4.7(a), one shows that the cohomology sheaves of the complexes
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ are also bounded
above. By Corollary 1.5.7, the right homological dimension of
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with
respect to $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
U_{\alpha}^{\operatorname{\mathsf{--qcoh}}}$ is finite, and it follows that
the complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{X},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is quasi-isomorphic to a bounded above complex over
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. The complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is quasi-
isomorphic to
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Finally, the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\allowbreak\simeq
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is quasi-isomorphic to a bounded above complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ by Lemma 3.4.6(c) and the
other results of Section 3.4.
Similarly one can show that for any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ quasi-
isomorphic to a bounded below complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}$ has bounded below cohomology sheaves.
∎
Now let $X$ be a Noetherian scheme with a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ [29,
Chapter 5]. As above, we will consider
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as a finite
complex of injective quasi-coherent sheaves on $X$. The following partial
version of the covariant Serre–Grothendieck duality holds without the semi-
separatedness assumption on $X$.
###### Theorem 5.8.2.
The choice of a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ induces a
natural equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$.
###### Proof.
According to Corollary 5.4.4(b), one has
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
for any open covering ${\mathbf{W}}$ of the scheme $X$. By Theorem
5.4.10(a-b), one has
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
and
$\mathsf{Hot}(X\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$.
We will show that the functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{-}$
induce an equivalence of the homotopy categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
for any symbol $\star={\mathsf{b}}$, $+$, $-$, or $\empt$.
Let ${\mathcal{I}}$ be an injective quasi-coherent sheaf on $X$ and $j\colon
U\longrightarrow X$ be the embedding of an affine open subscheme. Then the
results of Section 3.8 provide a natural isomorphism of contraherent cosheaves
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},j_{*}{\mathcal{J}})\simeq
j_{!}\operatorname{\mathfrak{Hom}}_{U}(j^{*}{\mathcal{I}},{\mathcal{J}})$ on
$X$ for any injective quasi-coherent sheaf ${\mathcal{J}}$ on $U$ and a
natural isomorphism of quasi-coherent sheaves
${\mathcal{I}}\odot_{X}j_{!}{\mathfrak{G}}\simeq
j_{*}(j^{*}{\mathcal{I}}\odot_{X}{\mathfrak{G}})$ on $X$ for any flat cosheaf
of ${\mathcal{O}}_{U}$-modules ${\mathfrak{G}}$ on $U$.
Notice that the functor $j_{*}$ takes injective quasi-coherent sheaves to
injective quasi-coherent sheaves and the functor $j_{!}$ takes projective
locally cotorsion contraherent cosheaves to projective locally cotorsion
contraherent cosheaves (Corollary 5.1.6(b)). Furthermore, let
$X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. It is clear
from the classification theorems (see Theorem 5.1.1(b)) that any injective
quasi-coherent sheaf or projective locally cotorsion contraherent cosheaf on
$X$ is a finite direct sum of the direct images of similar (co)sheaves from
$U_{\alpha}$.
It follows that the functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{-})$ and
${\mathcal{I}}\odot_{X}{-}$ take injective quasi-coherent sheaves to
projective locally cotorsion contraherent cosheaves on $X$ and back. By (20),
these are two adjoint functors between the additive categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
Substituting
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in place of
${\mathcal{I}}$ and totalizing the finite complexes of complexes of
(co)sheaves, we obtain two adjoint functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{-}$
between the homotopy categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$ and
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.
In order to show that these are mutually inverse equivalences, it suffices to
check that the adjunction morphisms
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\longrightarrow{\mathcal{J}}$
and
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{P}})$
are quasi-isomorphisms/homotopy equivalences of finite complexes for any
${\mathcal{J}}\in X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. Presenting
${\mathcal{J}}$ and ${\mathfrak{P}}$ as finite direct sums of the direct
images of similar (co)sheaves from affine open subschemes of $X$ and taking
again into account the isomorphisms (45), (47) reduces the question to the
case of an affine scheme, where the assertion is already known.
Alternatively, one can work directly in the greater generality of arbitrary
(not necessarily locally cotorsion) and flat contraherent cosheaves. According
to Theorem 5.4.10(d), one has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$.
Let us show that the functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{-}$
induce an equivalence of triangulated categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
for any symbol $\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, or
${\mathsf{abs}}$.
Given an injective quasi-coherent sheaf ${\mathcal{I}}$ on $X$, let us first
check that the functor
${\mathfrak{F}}\longmapsto{\mathcal{I}}\odot_{X}{\mathfrak{F}}$ takes short
exact sequences of flat contraherent cosheaves to short exact sequences of
quasi-coherent sheaves on $X$. By the adjunction isomorphism (20), for any
injective quasi-coherent sheaf ${\mathcal{J}}$ on $X$ one has
$\operatorname{Hom}_{X}({\mathcal{I}}\odot_{X}{\mathfrak{F}},\>{\mathcal{J}})\simeq\operatorname{Hom}^{X}({\mathfrak{F}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{\mathcal{J}}))$.
The contraherent cosheaf
${\mathfrak{Q}}=\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{\mathcal{J}})$
being locally cotorsion, the functor
${\mathfrak{F}}\longmapsto\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{Q}})$
is exact on $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ by Corollary
5.2.9(a).
Furthermore, by part (b) of the same Corollary any flat contraherent cosheaf
${\mathfrak{F}}$ on $X$ is a direct summand of a finitely iterated extension
of the direct images $j_{!}{\mathfrak{G}}$ of flat contraherent cosheaves
${\mathfrak{G}}$ on affine open subschemes $U\subset X$. Using the isomorphism
(47), we conclude that the quasi-coherent sheaf
${\mathcal{I}}\odot_{X}{\mathfrak{F}}$ is injective. It follows, in
particular, that the complex of quasi-coherent sheaves
${\mathcal{I}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contractible for any acyclic complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the exact
category $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. Therefore, the
same applies to the complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$.
Finally, to prove that the map
${\mathfrak{F}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}})$
is a quasi-isomorphism for any flat contraherent cosheaf ${\mathfrak{F}}$, it
suffices again to consider the case ${\mathfrak{F}}=j_{!}{\mathfrak{G}}$, when
the assertion follows from the isomorphisms (45), (47). Hence the morphism
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
has a cone absolutely acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ for any compex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. ∎
### 5.9. Compact generators
Let ${\mathsf{D}}$ be a triangulated category where arbitrary infinite direct
sums exist. We recall that object $C\in{\mathsf{D}}$ is called _compact_ if
the functor $\operatorname{Hom}_{\mathsf{D}}(C,{-})$ takes infinite direct
sums in ${\mathsf{D}}$ to infinite direct sums of abelian groups [45]. A set
of compact objects ${\mathsf{C}}\subset{\mathsf{D}}$ is said to _generate_
${\mathsf{D}}$ if any object $X\in{\mathsf{D}}$ such that
$\operatorname{Hom}_{\mathsf{D}}(C,X[*])=0$ for all $C\in{\mathsf{C}}$
vanishes in ${\mathsf{D}}$.
Equivalently, this means that any full triangulated subcategory of
${\mathsf{D}}$ containing ${\mathsf{C}}$ and closed under infinite direct sums
coincides with ${\mathsf{D}}$. If ${\mathsf{C}}$ is a set of compact
generators for ${\mathsf{D}}$, then an object of ${\mathsf{D}}$ is compact if
and only if it belongs to the minimal thick subcategory of ${\mathsf{D}}$
containing ${\mathsf{C}}$.
Let $X{\operatorname{\mathsf{--coh}}}$ denote the abelian category of coherent
sheaves on a Noetherian scheme $X$.
###### Theorem 5.9.1.
(a) For any scheme $X$, the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ admits
arbitrary infinite direct sums, while the contraderived categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
admit infinite products.
(b) For any Noetherian scheme $X$, the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ is compactly
generated. The triangulated functor
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--coh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embedding of abelian categories
$X{\operatorname{\mathsf{--coh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is fully faithful, and its image is the
full subcategory of compact objects in
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$.
(c) For any Noetherian scheme $X$ of finite Krull dimension, the contraderived
categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
admit arbitrary infinite direct sums and are compactly generated.
###### Proof.
Part (a) holds, because the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$ admits arbitrary infinite direct sums and
the full subcategory of coacyclic complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})$ is closed under infinite
direct sums (see [46, Proposition 1.2.1 and Lemma 3.2.10]). Analogously, the
exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ admit arbitrary infinite
products and the full subcategories of contraacyclic complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and $\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ are closed
under infinite products.
In the assumption of part (b), the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$ is a locally Noetherian Grothendieck
category, so the assertions hold by Theorem 5.4.10(a) and [37, Proposition
2.3] (see also Lemma A.1.2). A more generally applicable assertion/argument
can be found in [53, Proposition 1.5(d)] and/or [51, Section 3.11].
Part (c): notice first of all that all the categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalent to each other by Corollaries 5.3.3 and 5.4.4(b). Furthermore,
if the scheme $X$ admits a dualizing complex, the assertion of part (c)
follows from Theorem 5.8.2 and part (b). The following more complicated
argument allows to prove the desired assertion in the stated generality.
By Theorem 5.4.10(b), the triangulated category in question is equivalent to
the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.
Let us first consider the case when the scheme $X$ is semi-separated. Then
Corollary 5.4.5 identifies our triangulated category with
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.
It follows immediately that this triangulated category admits arbitrary
infinite direct sums.
In the case of an affine Noetherian scheme $U$ of finite Krull dimension,
another application of Proposition A.5.6 allows to identify
${\mathsf{D}}^{\mathsf{co}}(U{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
with the homotopy category of complexes of projective
${\mathcal{O}}(U)$-modules, which is compactly generated by [34, Theorem 2.4].
More generally, the category
${\mathsf{D}}(U{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ is equivalent
to the homotopy category of projective ${\mathcal{O}}(U)$-modules for any
affine scheme $U$ by [47, Section 8] and is compactly generated for any affine
Noetherian scheme $U$ by [47, Proposition 7.14] (see also [48]). Finally, for
any semi-separated Noetherian scheme $X$ the triangulated category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ is compactly
generated by [42, Theorem 4.10].
Now let us turn to the general case. First we have to show that the category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
admits arbitrary infinite direct sums. Let
$X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering, and let
$S_{\beta}\subset X$ denote the set-theoretic complement to
$\bigcup_{\alpha<\beta}U_{\alpha}$ in $U_{\beta}$. Let $j_{\alpha}\colon
U_{\alpha}\longrightarrow X$ denote the open embedding morphisms; then the
direct image functor
$j_{\alpha}{}_{!}\colon\mathsf{Hot}(U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
is left adjoint to the inverse image functor
$j_{\alpha}^{!}\colon\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow\mathsf{Hot}(U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
(see Corollaries 5.1.3(a) and 5.1.6(b), and the adjunction (26)). Hence the
functor $j_{\alpha}{}_{!}$ preserves infinite direct sums.
As explained in the proof of Theorem 5.1.1, any projective locally cotorsion
contraherent cosheaf ${\mathfrak{F}}$ on $X$ decomposes into a direct sum
${\mathfrak{F}}=\bigoplus_{\alpha=1}^{N}{\mathfrak{F}}_{\alpha}$, where each
direct summand ${\mathfrak{F}}_{\alpha}$ is an infinite product over the
points $z\in S_{\alpha}$ of the direct images of contraherent cosheaves on
$\operatorname{Spec}{\mathcal{O}}_{z,X}$ corresponding to free contramodules
over $\widehat{\mathcal{O}}_{z,X}$. According to Lemma 5.1.2, the associated
increasing filtration
${\mathfrak{F}}_{\le\alpha}=\bigoplus_{\beta\le\alpha}{\mathfrak{F}}_{\beta}$
on ${\mathfrak{F}}$ is preserved by all morphisms of cosheaves
${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
Given a family
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
complexes over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, we now see
that every complex
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
endowed with a finite termwise split filtration
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\le\alpha}$
such that the family of associated quotient complexes
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
can be obtained by applying the direct image functor $j_{\alpha}{}_{!}$ to a
family of complexes over
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. It
follows that the object
$\bigoplus_{i}{}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
exists in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$,
and it remains to apply the following lemma (which is slightly stronger than
[46, Proposition 1.2.1]).
###### Lemma 5.9.2.
Let $A_{i}\longrightarrow B_{i}\longrightarrow C_{i}\longrightarrow A_{i}[1]$
be a family of distinguished triangles in a triangulated category
${\mathsf{D}}$. Suppose that the infinite direct sums $\bigoplus_{i}A_{i}$ and
$\bigoplus_{i}B_{i}$ exist in ${\mathsf{D}}$. Then a cone $C$ of the natural
morphism $\bigoplus_{i}A_{i}\longrightarrow\bigoplus_{i}B_{i}$ is the infinite
direct sum of the family of objects $C_{i}$ in ${\mathsf{D}}$.
###### Proof.
Set $A=\bigoplus_{i}A_{i}$ and $B=\bigoplus_{i}B_{i}$. By one of the
triangulated category axioms, there exist morphisms of distinguished triangles
$(A_{i}\to B_{i}\to C_{i}\to A_{i}[1])\longrightarrow(A\to B\to C\to A[1])$
whose components $A_{i}\longrightarrow A$ and $B_{i}\longrightarrow B$ are the
natural embeddings. For any object $E\in{\mathsf{D}}$, apply the functor
$\operatorname{Hom}_{\mathsf{D}}({-},E)$ to this family of morphisms of
triangles and pass to the infinite product (of abelian groups) over $i$. The
resulting morphism from the long exact sequence
$\dotsb\longrightarrow\operatorname{Hom}_{\mathsf{D}}(A[1],E)\longrightarrow\operatorname{Hom}_{\mathsf{D}}(C,E)\longrightarrow\operatorname{Hom}_{\mathsf{D}}(B,E)\longrightarrow\operatorname{Hom}_{\mathsf{D}}(A,E)\longrightarrow\dotsb$
to the long exact sequence
$\dotsb\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(A_{i}[1],E)\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(C_{i},E)\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(B_{i},E)\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(A_{i},E)\longrightarrow\dotsb$
is an isomorphism at the two thirds of all the terms, and consequently an
isomorphism at the remaining terms, too. ∎
Denote temporarily the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
by ${\mathsf{D}}(X)$. To show that the category ${\mathsf{D}}(X)$ is compactly
generated, we will use the result of [56, Theorem 5.15]. Let $Y\subset X$ be
an open subscheme such that the category ${\mathsf{D}}(Y)$ is compactly
generated (e. g., we already know this to hold when $Y$ is semi-separated).
Let $j\colon Y\longrightarrow X$ denote the open embedding morphism.
The composition $j^{!}j_{!}$ of the direct image and inverse image functors
$j_{!}\colon{\mathsf{D}}(Y)\longrightarrow{\mathsf{D}}(X)$ and
$j^{!}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(Y)$ is isomorphic to
the identity endofunctor of ${\mathsf{D}}(Y)$, so the functor $j_{!}$ is fully
faithful and the functor $j^{!}$ is a Verdier localization functor. Applying
again Lemma 5.1.2, we conclude that the kernel of $j^{!}$ is the homotopy
category of projective locally cotorsion contraherent cosheaves on $X$ with
vanishing restrictions to $Y$. Denote this homotopy category by
${\mathsf{D}}(Z,X)$, where $Z=X\setminus Y$, and its identity embedding
functor by $i_{!}\colon{\mathsf{D}}(Z,X)\longrightarrow{\mathsf{D}}(X)$.
The functor $j_{!}$ is known to preserve infinite products, and the
triangulated category ${\mathsf{D}}(Y)$ is assumed to be compactly generated;
so it follows that there exists a triangulated functor
$j^{*}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(Y)$ left adjoint to
$j_{!}$ (see [46, Remark 6.4.5 and Theorem 8.6.1] and [37, Proposition
3.3(2)]). The existence of the functor $j_{!}$ left adjoint to $j^{!}$ implies
existence of a functor
$i^{*}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(Z,X)$ left adjoint to
$i_{!}$; and the existence of the functor $j^{*}$ left adjoint to $j_{!}$
implies existence of a functor
$i_{+}\colon{\mathsf{D}}(Z,X)\longrightarrow{\mathsf{D}}(X)$ left adjoint to
$i^{*}$.
The functors $j^{*}$ and $i_{+}$ have double right adjoints (i. e., the right
adjoints and the right adjoints to the right adjoints), hence they not only
preserve infinite direct sums, but also take compact objects to compact
objects. Furthermore, for any open subscheme $W\subset X$ with the embedding
morphism $h\colon W\longrightarrow X$ one has the base change isomorphism
$h^{!}j_{!}\simeq j^{\prime}_{!}h^{\prime}{}^{!}$, where $j^{\prime}$ and
$h^{\prime}$ denote the open embeddings $W\cap Y\longrightarrow W$ and $W\cap
Y\longrightarrow Y$. If the triangulated category ${\mathsf{D}}(W\cap Y)$ is
compactly generated, one can pass to the left adjoint functors, obtaining an
isomorphism of triangulated functors $j^{*}h_{!}\simeq h^{\prime}_{!}\mskip
1.5muj^{\prime}{}^{*}$.
Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine (or, more generally,
semi-separated) open covering, $Z_{\alpha}=X\setminus U_{\alpha}$ be the
corresponding closed complements, and
$i_{\alpha}{}_{+}\colon{\mathsf{D}}(Z_{\alpha},X)\longrightarrow{\mathsf{D}}(X)$
be the related fully faithful triangulated functors. It follows from the above
that the images of the functors $i_{\alpha}{}_{+}$ form a collection of
Bousfield subcategories in ${\mathsf{D}}(X)$ pairwise intersecting properly in
the sense of [56, Lemma 5.7(2)]. Furthermore, the category ${\mathsf{D}}(X)$
being generated by the images of the functors $j_{\alpha}{}_{!}$, the
intersection of the kernels of the functors
$j_{\alpha}^{*}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(U_{\alpha})$
is zero. These coincide with the images of the functors $i_{\alpha}{}_{+}$.
Thus the triangulated subcategories
$i_{\alpha}{}_{+}{\mathsf{D}}(Z_{\alpha},X)\subset{\mathsf{D}}(X)$ form a
_cocovering_ (in the sense of [56]).
It remains to check that intersections of the images of
$i_{\beta}{}_{+}{\mathsf{D}}(Z_{\beta},X)$ under the localization morphism
${\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(U_{\alpha})$ are compactly
generated in ${\mathsf{D}}(U_{\alpha})$. Let $Y$ be a semi-separated
Noetherian scheme of finite Krull dimension and $V\subset Y$ be an open
subscheme with the closed complement $Z=Y\setminus V$. We will show that the
image of the fully faithful triangulated functor
$i_{+}\colon{\mathsf{D}}(Z,Y)\longrightarrow{\mathsf{D}}(Y)$ is compactly
generated in ${\mathsf{D}}(Y)$; this is clearly sufficient.
The result of Corollary 5.4.5 identifies
${\mathsf{D}}(Y)=\mathsf{Hot}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
with ${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ and
${\mathsf{D}}(V)=\mathsf{Hot}(V{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
with ${\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.
According to Corollary 5.4.6, this identification transforms the functor
$j_{!}\colon{\mathsf{D}}(V)\longrightarrow{\mathsf{D}}(Y)$ into the derived
functor
${\mathbb{R}}j_{*}\colon{\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
constructed in (65). The latter functor is right adjoint to the functor
$j^{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$,
which is therefore identified with the functor
$j^{*}\colon{\mathsf{D}}(Y)\longrightarrow{\mathsf{D}}(V)$.
Finally, we refer to [42, Proposition 4.5 and Theorem 4.10] for the assertion
that the kernel of the functor
$j^{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is compactly generated in
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$. ∎
###### Theorem 5.9.3.
(a) For any scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ admits infinite direct sums,
while the derived categories
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and ${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ admit
infinite products.
(b) For any quasi-compact semi-separated scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ is compactly generated. The
full triangulated subcategory of perfect complexes in
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\subset{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\subset{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is the full subcategory of compact objects in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.
(c) For any Noetherian scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ is compactly generated. The
full triangulated subcategory of perfect complexes in
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--coh}}})\subset{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is the full subcategory of compact objects in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.
(d) For any quasi-compact semi-separated scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ admits infinite
direct sums and is compactly generated.
(e) For any Noetherian scheme $X$ of finite Krull dimension, the derived
categories
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and ${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ admit
infinite direct sums and are compactly generated.
###### Proof.
The proof of part (a) is similar to that of Theorem 5.9.1(a): the assersions
hold, since the class of acyclic complexes over
$X{\operatorname{\mathsf{--qcoh}}}$ is closed under infinite direct sums, and
the classes of acyclic complexes over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ are closed under infinite
products.
Parts (b) and (c) are particular cases of [56, Theorem 6.8], according to
which the derived category ${\mathsf{D}}(X)$ of complexes of sheaves of
${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology sheaves is
compactly generated for any quasi-compact quasi-separated scheme $X$. Here one
needs to know that the natural functor
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X)$
is an equivalence of categories when $X$ is either quasi-compact and semi-
separated, or else Noetherian (cf. [61, Appendix B]). In the semi-separated
case, this was proven in [8, Sections 5–6]. The proof in the Noetherian case
is similar.
Alternatively, one can prove parts (b) and (c) directly in the way analogous
to the argument in [56]. In either approach, one needs to know that the
functor ${\mathbb{R}}j_{*}$ of derived direct image of complexes over
$Y{\operatorname{\mathsf{--qcoh}}}$ with respect to an open embedding $j\colon
Y\longrightarrow X$ of schemes from the class under consideration is well-
behaved. E. g., it needs to be local in the base, or form a commutative square
with the derived functor of direct image of complexes of
${\mathcal{O}}_{Y}$-modules, etc. (cf. [41, Theorems 31 and 42]).
In the semi-separated case, one can establish such properties using
contraadjusted resolutions and (the proof of) Corollary 4.1.13(a) (see the
construction of the functor ${\mathbb{R}}f_{*}$ in Section 4.8 above). In the
Noetherian case, one needs to use flasque resolutions and Corollary 3.4.9(a)
(see the construction of the functor ${\mathbb{R}}f_{*}$ in Section 5.12
below).
Part (d) follows from part (b) together with Theorem 4.6.6. Part (e) follows
from part (c) together with Theorem 5.8.1 and Corollary 5.4.4(b). ∎
### 5.10. Homotopy projective complexes
Let $X$ be a scheme. A complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of quasi-
coherent sheaves on $X$ is called _homotopy injective_ if the complex of
abelian groups
$\operatorname{Hom}_{X}({\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$. The
full subcategory of homotopy injective complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})$ is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{inj}}$ and the full
subcategory of complexes of injective quasi-coherent sheaves that are also
homotopy injective is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})^{\mathsf{inj}}\subset\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$.
Similarly, a complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves is called _homotopy
projective_ if the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the exact
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. The
full subcategory of homotopy projective complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}$.
We will see below in this section that the property of a complex of locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves on a Noetherian scheme
$X$ of finite Krull dimension to be homotopy projective does not change when
the covering ${\mathbf{W}}$ is replaced by its refinement.
Finally, a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
${\mathbf{W}}$-locally contraherent cosheaves is called _homotopy projective_
if the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the exact
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. The full
subcategory of homotopy projective complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}$.
Let us issue a _warning_ that our terminology is misleading: a homotopy
projective complex of locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaves need not be homotopy projective as a complex of
${\mathbf{W}}$-locally contraherent cosheaves. It will be shown below that the
property of a complex of ${\mathbf{W}}$-locally contraherent cosheaves on a
Noetherian scheme $X$ of finite Krull dimension to be homotopy projective does
not change when the covering ${\mathbf{W}}$ is replaced by its refinement.
###### Lemma 5.10.1.
(a) Let $X$ be a locally Noetherian scheme of finite Krull dimension with an
open covering ${\mathbf{W}}$. Then a complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ belongs to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}$
if and only if the complex
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ acyclic with
respect to $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$.
(b) Let $X$ be a Noetherian scheme of finite Krull dimension with an open
covering ${\mathbf{W}}$. Then a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ belongs to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}$
if and only if the complex
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$.
###### Proof.
We will prove part (a), part (b) being similar. The “only if” assertion holds
by the definition. To check the “if”, consider a complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. By (the proof
of) Theorem 5.4.10(b), there exists a complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ together
with a morphism of complexes of locally contraherent cosheaves
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with a cone contraacyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Moreover, the
complex $\operatorname{Hom}^{X}$ from any complex of projective locally
cotorsion contraherent cosheaves to a contraacyclic complex over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is acyclic.
Hence the morphism
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism. Finally, if the complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic
over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$, then so
is the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and by Lemma
5.3.1(b) it follows that the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is also
acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$. ∎
According to Lemma 5.10.1, the property of a complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$
(respectively, over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$) to
belong to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}$
(resp.,
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}$)
does not depend on the covering ${\mathbf{W}}$ (in the assumptions of the
respective part of the lemma). We will denote the full subcategory in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
(resp., $\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$)
consisting of the homotopy projective complexes by
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})_{\mathsf{prj}}$
(resp.,
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}$).
It is a standard fact that bounded above complexes of projectives are homotopy
projective,
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})_{\mathsf{prj}}$
and
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}$.
The next result is essentially well-known.
###### Theorem 5.10.2.
Let $X$ be a locally Noetherian scheme. Then the natural functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})^{\mathsf{inj}}\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{inj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
are equivalences of triangulated categories.
###### Proof.
It is clear that both functors are fully faithful. The functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{inj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories by [1, Theorem 5.4]; this is applicable to any
Grothendieck abelian category in place of $X{\operatorname{\mathsf{--qcoh}}}$
(for an even more general statement, see [38, Theorem 6]).
To prove that any homotopy injective complex of quasi-coherent sheaves on a
locally Noetherian scheme is homotopy equivalent to a homotopy injective
complex of injective quasi-coherent sheaves, one can use (the proof of)
Theorem 5.4.10(a). From any complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ there exists a closed morphism into a
complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ with a coacyclic cone
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. If the complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ was homotopy
injective, the morphism
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[1]$
is homotopic to zero, hence the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a direct
summand of ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$. Any morphism
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
being also homotopic to zero, it follows that the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
contractible. ∎
###### Theorem 5.10.3.
Let $X$ be a Noetherian scheme of finite Krull dimension with an open covering
${\mathbf{W}}$. Then
(a) the natural functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})_{\mathsf{prj}}\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\mathbf{W}})$
are equivalences of triangulated categories;
(b) the natural functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of triangulated categories.
###### Proof.
We will prove part (b), part (a) being similar. Since both functors are
clearly fully faithful, it suffices to show that the composition
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of categories. This is equivalent to saying that the
localization functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
has a left adjoint whose image is essentially contained in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$.
The functor in question factorizes into the composition of two localization
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$.
The functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
has a left adjoint functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\hookrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
provided by Theorem 5.4.10(d); so it remains to show that the functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
has a left adjoint.
Since the latter functor preserves infinite products, the assertion follows
from Theorem 5.9.1(c) and [37, Proposition 3.3(2)]. Here one also needs to
know that the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
“exists” (i. e., morphisms between any given two objects form a set rather
than a class). This is established by noticing that the classes of quasi-
isomorphisms are locally small in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and $\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ (see [62,
Section 10.3.6 and Proposition 10.4.4]).
Similarly one can prove Theorem 5.10.2 for a Noetherian scheme $X$ using
Theorems 5.4.10(a) and 5.9.1(b); the only difference is that this time one
needs a right adjoint functor, so [37, Proposition 3.3(1)] has to be applied.
(Cf. [51, Section 5.5]). ∎
A complex ${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
${\mathbf{W}}$-locally contraherent cosheaves on a scheme $X$ is called
_homotopy flat_ if the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex of locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaves
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ acyclic over
the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. On a locally
Noetherian scheme $X$ of finite Krull dimension, the latter condition is
equivalent to the acyclicity over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Corollary 1.5.7). Notice
that the property of a complex of ${\mathbf{W}}$-locally contraherent
cosheaves to be homotopy flat may possibly change when the covering
${\mathbf{W}}$ is replaced by its refinement.
###### Lemma 5.10.4.
Let $X$ be a Noetherian scheme of finite Krull dimension with an open covering
${\mathbf{W}}$. Then a complex of flat contraherent cosheaves
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ is
homotopy flat if and only if the complex
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ acyclic with
respect to $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$.
###### Proof.
Similar to that of Lemma 5.10.1. The only difference is that one has to use
Corollary 5.2.9(a) in order to show that the complex $\operatorname{Hom}^{X}$
from any complex of flat contraherent cosheaves on $X$ to a contraacyclic
complex over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
is acyclic. ∎
According to Lemma 5.10.4, the property of a complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ to belong to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})^{\mathsf{fl}}$
does not depend on the covering ${\mathbf{W}}$ (on a Noetherian scheme $X$ of
finite Krull dimension). We denote the full subcategory in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$ consisting of
the homotopy flat complexes by
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$.
One can easily check that bounded above complexes of flat contraherent
cosheaves are homotopy flat,
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$.
###### Theorem 5.10.5.
Let $X$ be a Noetherian scheme of finite Krull dimension. Then the quotient
category of the homotopy category of homotopy flat complexes of flat
contraherent cosheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
on $X$ by its thick subcategory of acyclic complexes over the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is equivalent to the derived
category ${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$.
###### Proof.
By Corollary 5.2.6(b) and [50, Remark 2.1], any acyclic complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is absolutely acyclic; and
one can see from Corollary 5.2.9(a) that any absolutely acyclic complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is homotopy flat. According
to (the proof of) Theorem 5.10.3(b), there is a quasi-isomorphism into any
complex over $X{\operatorname{\mathsf{--ctrh}}}$ from a complex belonging to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$.
In view of [51, Lemma 1.6], it remains to show that any homotopy flat complex
of flat contraherent cosheaves that is acyclic over
$X{\operatorname{\mathsf{--ctrh}}}$ is also acyclic over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$.
According again to Corollary 5.2.6(b) and the dual version of the proof of
Proposition A.5.6, any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ admits a morphism into a
complex ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ with a cone
absolutely acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. If the complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ was homotopy
flat, it follows that the complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopy
flat, too. This means that
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a homotopy
projective complex of locally cotorsion contraherent cosheaves on $X$. If the
complex ${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ was
also acyclic over $X{\operatorname{\mathsf{--ctrh}}}$, so is the complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. It follows
that ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$, and therefore
contractible. We have proven that the complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is absolutely
acyclic over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. ∎
### 5.11. Special inverse image of contraherent cosheaves
Recall that an affine morphism of schemes $f\colon Y\longrightarrow X$ is
called _finite_ if for any affine open subscheme $U\subset X$ the ring
${\mathcal{O}}_{Y}(f^{-1}(U))$ is a finitely generated module over the ring
${\mathcal{O}}_{X}(U)$. One can easily see that this condition on a morphism
$f$ is local in $X$.
Let $f\colon Y\longrightarrow X$ be a finite morphism of locally Noetherian
schemes. Given a quasi-coherent sheaf ${\mathcal{M}}$ on $X$, one defines the
quasi-coherent sheaf $f^{!}{\mathcal{M}}$ on $Y$ by the rule
$(f^{!}{\mathcal{M}})(V)={\mathcal{O}}_{Y}(V)\otimes_{{\mathcal{O}}_{Y}(f^{-1}(U))}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{M}}(U))$
for any affine open subschemes $V\subset Y$ and $U\subset X$ such that
$f(V)\subset U$ [29, Section III.6]. The construction is well-defined, since
for any pair of embedded affine open subschemes $U^{\prime}\subset U\subset X$
one has
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U^{\prime})}({\mathcal{O}}_{Y}(f^{-1}(U^{\prime}),\>{\mathcal{M}}(U^{\prime}))\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U^{\prime})}({\mathcal{O}}_{X}(U^{\prime})\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{O}}_{X}(U^{\prime})\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{M}}(U))\\\
\simeq{\mathcal{O}}_{X}(U^{\prime})\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{M}}(U)).$
Indeed, one has $\operatorname{Hom}_{R}(L,\>F\otimes_{R}M)\simeq
F\otimes_{R}\operatorname{Hom}_{R}(L,M)$ for any module $M$, finitely
presented module $L$, and flat module $F$ over a commutative ring $R$. (See
Section 3.3 for a treatment of the non-semi-separatedness issue.)
The functor $f^{!}\colon X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ is right adjoint to the exact functor
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$. Indeed, it suffices to define a morphism
of quasi-coherent sheaves on $Y$ on the modules of sections over the affine
open subschemes $f^{-1}(U)\subset Y$. So given quasi-coherent sheaves
${\mathcal{M}}$ on $X$ and ${\mathcal{N}}$ on $Y$, both groups of morphisms
$\operatorname{Hom}_{X}(f_{*}{\mathcal{N}},{\mathcal{M}})$ and
$\operatorname{Hom}_{Y}({\mathcal{N}},f^{!}{\mathcal{M}})$ are identified with
the group of all compatible collections of morphisms of
${\mathcal{O}}_{X}(U)$-modules
${\mathcal{N}}(f^{-1}(U))\longrightarrow{\mathcal{M}}(U)$, or equivalently,
compatible collections of morphisms of ${\mathcal{O}}_{Y}(f^{-1}(U))$-modules
${\mathcal{N}}(f^{-1}(U))\allowbreak\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{M}}(U))$.
Let $i\colon Z\longrightarrow X$ be a closed embedding of locally Noetherian
schemes. Let ${\mathbf{W}}$ be an open covering of $X$ and ${\mathbf{T}}$ be
an open covering of $Z$ such that $i$ is a
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism. Given a ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$, one
defines a ${\mathbf{T}}$-flat ${\mathbf{T}}$-locally contraherent cosheaf
$i^{*}{\mathfrak{F}}$ on $Z$ by the rule
$(i^{*}{\mathfrak{F}})[i^{-1}(U)]={\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U]$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$.
Clearly, affine open subschemes of the form $i^{-1}(U)$ constitute a base of
the topology of $Z$. The construction is well-defined, since for any pair of
embedded affine open subschemes $U^{\prime}\subset U\subset X$ subordinate to
${\mathbf{W}}$ one has
${\mathcal{O}}_{Z}(i^{-1}(U^{\prime}))\otimes_{{\mathcal{O}}_{X}(U^{\prime})}{\mathfrak{F}}[U^{\prime}]\\\
\simeq({\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{X}(U^{\prime}))\otimes_{{\mathcal{O}}_{X}(U^{\prime})}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(U^{\prime}),{\mathfrak{F}}[U])\\\
\simeq{\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(U^{\prime}),{\mathfrak{F}}[U])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{Z}(i^{-1}(U))}({\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{X}(U^{\prime}),\>{\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{Z}(i^{-1}(U))}({\mathcal{O}}_{Z}(i^{-1}(U^{\prime})),\>{\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U]),$
where the third isomorphism holds by Lemma 1.6.6(c). The
${\mathcal{O}}_{Z}(i^{-1}(U))$-module
${\mathcal{O}}_{Z}(i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U]$
is contraadjusted by Lemma 1.6.6(b).
Assuming that the morphism $i$ is $({\mathbf{W}},{\mathbf{T}})$-affine and
$({\mathbf{W}},{\mathbf{T}})$-coaffine, the exact functor $i^{*}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}\longrightarrow
Z{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{fl}}$ is “partially
left adjoint” to the exact functor $i_{!}\colon
Z{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. In other words, for any
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{F}}$ on $X$ and any ${\mathbf{T}}$-locally contraherent cosheaf
${\mathfrak{Q}}$ on $Z$ there is a natural adjunction isomorphism
(85)
$\operatorname{Hom}^{X}({\mathfrak{F}},i_{!}{\mathfrak{Q}})\simeq\operatorname{Hom}^{Z}(i^{*}{\mathfrak{F}},{\mathfrak{Q}}).$
Indeed, it suffices to define a morphism of ${\mathbf{T}}$-locally
contraherent cosheaves on $Z$ on the modules of cosections over the open
subschemes $i^{-1}(U)\subset Z$ for all affine open subschemes $U\subset X$
subordinate to ${\mathbf{W}}$. So both groups of morphisms in question are
identified with the group of all compatible collections of morphisms of
${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{F}}[U]\longrightarrow{\mathfrak{Q}}[i^{-1}(U)]$, or equivalently,
compatible collections of morphisms of ${\mathcal{O}}_{Z}((i^{-1}(U))$-modules
${\mathcal{O}}_{Z}((i^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U]\longrightarrow{\mathfrak{Q}}[i^{-1}(U)]$.
Notice that for any open covering ${\mathbf{T}}$ of a closed subscheme
$Z\subset X$ there exists an open covering ${\mathbf{W}}$ of the scheme $X$
for which the embedding morphism $i$ is $({\mathbf{W}},{\mathbf{T}})$-affine
and $({\mathbf{W}},{\mathbf{T}})$-coaffine. For a locally Noetherian scheme
$X$ of finite Krull dimension and its closed subscheme $Z$, one has
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}=X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$
and
$Z{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{fl}}=Z{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$
for any open coverings ${\mathbf{W}}$ and ${\mathbf{T}}$ (see Corollary
5.2.2(b)). In this case, the adjunction isomorphism (85) holds for any flat
contraherent cosheaf ${\mathfrak{F}}$ on $X$ and locally contraherent cosheaf
${\mathfrak{Q}}$ on $Z$. Most generally, the isomorphism
$\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{F}},i_{!}{\mathfrak{Q}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{Z}}(i^{*}{\mathfrak{F}},{\mathfrak{Q}})$
holds for any ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{F}}$ on a locally Noetherian scheme $X$ and any cosheaf of
${\mathcal{O}}_{Z}$-modules ${\mathfrak{Q}}$ on a closed subscheme $Z\subset
X$.
Let $f\colon Y\longrightarrow X$ be a finite morphism of locally Noetherian
schemes. Given a projective locally cotorsion locally contraherent cosheaf
${\mathfrak{P}}$ on $X$, one defines a projective locally contraherent cosheaf
$f^{*}{\mathfrak{P}}$ on $Y$ by the rule
$(f^{*}{\mathfrak{P}})[V]=\operatorname{Hom}_{{\mathcal{O}}_{Y}(f^{-1})(U))}({\mathcal{O}}_{Y}(V),\>{\mathcal{O}}_{Y}(f^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U])$
for any affine open subschemes $V\subset Y$ and $U\subset X$ such that
$f(V)\subset U$. The construction is well-defined, since for any affine open
subschemes there is a natural isomorphism of
${\mathcal{O}}_{Y}(f^{-1}(U^{\prime}))$-modules
${\mathcal{O}}_{Y}(f^{-1}(U^{\prime}))\otimes_{{\mathcal{O}}_{X}(U^{\prime})}{\mathfrak{P}}[U^{\prime}]\simeq\operatorname{Hom}_{{\mathcal{O}}_{Y}(f^{-1}(U))}({\mathcal{O}}_{Y}(f^{-1}(U^{\prime})),\>{\mathcal{O}}_{Y}(f^{-1}(U))\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U])$
obtained in the way similar to the above computation for the closed embedding
case, except that Lemma 1.6.7(c) is being applied. The
${\mathcal{O}}_{Y}(f^{-1}(U))$-module
${\mathcal{O}}_{Y}(f^{-1}(U))\allowbreak\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$
is flat and cotorsion by Lemma 1.6.7(a); hence the
${\mathcal{O}}_{Y}(V)$-module $(f^{*}{\mathfrak{P}})[V]$ is flat and cotorsion
by Corollary 1.6.5(a) and Lemma 1.3.5(a). A contraherent cosheaf that is
locally flat and cotorsion on a locally Noetherian scheme $Y$ belongs to
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ by Corollary
5.1.4.
Assuming that the morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-affine, the
functor $f^{*}\colon
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\longrightarrow
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is “partially
left adjoint” to the exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. In other words, for any
${\mathbf{W}}$-flat locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ and any contraherent cosheaf ${\mathfrak{Q}}$
on $Y$ there is a natural adjunction isomorphism
(86)
$\operatorname{Hom}^{X}({\mathfrak{P}},f_{!}{\mathfrak{Q}})\simeq\operatorname{Hom}^{Y}(f^{*}{\mathfrak{P}},{\mathfrak{Q}}).$
Indeed, it suffices to define a morphism of ${\mathbf{T}}$-locally
contraherent cosheaves on $Y$ on the modules of cosections over the open
subschemes $f^{-1}(U)\subset Y$ for all affine open subschemes $U\subset X$
subordinate to ${\mathbf{W}}$, and the construction proceeds exactly in the
same way as in the above case of a closed embedding $i$. Generally, the
isomorphism
$\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},f_{!}{\mathfrak{Q}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{Y}}(f^{*}{\mathfrak{P}},{\mathfrak{Q}})$
holds for any projective locally cotorsion contraherent cosheaf
${\mathfrak{P}}$ on $X$ and any cosheaf of ${\mathcal{O}}_{Y}$-modules
${\mathfrak{Q}}$ on $Y$.
The functor $f^{!}\colon X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ for a finite morphism of locally Noetherian
schemes $f\colon Y\longrightarrow X$ preserves infinite direct sums of quasi-
coherent cosheaves. The functor $i^{*}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}\longrightarrow
Z{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{fl}}$ for a
$({\mathbf{W}},{\mathbf{T}})$-affine closed embedding of locally Noetherian
schemes $i\colon Z\longrightarrow X$ preserves infinite products of flat
locally contraherent cosheaves. The functor $f^{*}\colon
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\longrightarrow
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ for a finite
morphism of locally Noetherian schemes $f\colon Y\longrightarrow X$ preserves
infinite products of projective locally cotorsion contraherent cosheaves.
### 5.12. Derived functors of direct and special inverse image
For the rest of Section 5, the upper index $\star$ in the notation for derived
and homotopy categories stands for one of the symbols ${\mathsf{b}}$, $+$,
$-$, $\empt$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{co}}$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$.
Let $X$ be a locally Noetherian scheme with an open covering ${\mathbf{W}}$.
The following corollary is to be compared with Corollary 5.3.2.
###### Corollary 5.12.1.
(a) The triangulated functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories. The category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ is equivalent to the
quotient category of the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}})$ by the thick
subcategory of complexes over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}}$ that are acyclic over
$X{\operatorname{\mathsf{--qcoh}}}$.
(b) The triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is an equivalence of categories. The category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is equivalent to the quotient category of the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}})$
by the thick subcategory of complexes over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$ that are
acyclic over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$.
###### Proof.
The first assertions in (a) and (b) hold, because there are enough injective
quasi-coherent sheaves, which are flasque, and enough projective locally
cotorsion contraherent cosheaves, which are coflasque. Since the class of
flasque quasi-coherent sheaves is also closed under infinite direct sums,
while the class of coflasque contraherent cosheaves is closed under infinite
products (see Section 3.4), the desired assertions are provided by Proposition
A.3.1(b) and its dual version. Now we know that there is a quasi-isomorphism
from any complex over $X{\operatorname{\mathsf{--qcoh}}}$ into a complex over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}}$ and onto any complex over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ from a complex
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$, so the
second assertions in (a) and (b) follow by [51, Lemma 1.6]. ∎
Let $f\colon Y\longrightarrow X$ be a quasi-compact morphism of locally
Noetherian schemes. As it was mentioned in Section 5.3, the functor
(87)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$
can be constructed using injective or flasque resolutions. More generally, the
right derived functor
(88)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
can be constructed for $\star={\mathsf{co}}$ using injective (see Theorem
5.4.10(a)) or flasque (see Corollary 5.12.1(a)) resolutions, and for
$\star=\empt$ using homotopy injective (see Theorem 5.10.2) or flasque (cf.
Corollary 3.4.9(a)) resolutions.
When the scheme $Y$ has finite Krull dimension, the functor
${\mathbb{R}}f_{*}$ can be constructed for any symbol
$\star\neq{\mathsf{ctr}}$ using flasque resolutions (see Corollary 5.3.2(a)).
Finally, when both schemes $X$ and $Y$ are Noetherian, the functor (88) can be
constructed for any symbol $\star\neq{\mathsf{ctr}}$ using $f$-acyclic
resolutions (see Corollary 5.3.9(a)).
The functor
(89)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
was constructed in Section 5.3 using projective or coflasque resolutions. More
generally, the left derived functor
(90)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
can be constructed for $\star={\mathsf{ctr}}$ using projective (see Theorem
5.4.10(b)) or coflasque (see Corollary 5.12.1(b)) resolutions; and for
$\star=\empt$ using homotopy projective (when $X$ is Noetherian of finite
Krull dimension, see Theorem 5.10.3(a)) or coflasque (in the general case, cf.
Corollary 3.4.9(c)) resolutions.
When the scheme $Y$ has finite Krull dimension, the functor (90) can be
constructed for any symbol $\star\neq{\mathsf{co}}$ using coflasque
resolutions (see Corollary 5.3.2(b)). Finally, when both schemes $X$ and $Y$
are Noetherian and one of the conditions of Lemma 5.3.7 is satisfied, one can
construct the left derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
for any symbol $\star\neq{\mathsf{co}}$ using $f/{\mathbf{W}}$-acyclic
resolutions (see Corollary 5.3.9(c)).
Now assume that the scheme $Y$ is Noetherian of finite Krull dimension. The
functor
(91)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--lcth}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--lcth}}})$
was constructed in Section 5.3 using projective or coflasque resolutions. More
generally, the left derived functor
(92)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
can be constructed for any symbol $\star\neq{\mathsf{co}}$ using coflasque
resolutions (see Corollary 5.3.2(c); cf. Corollary 5.3.3).
For any quasi-compact morphism $f$ of locally Noetherian schemes and any
symbol $\star=\empt$ or ${\mathsf{co}}$, the functor ${\mathbb{R}}f_{*}$ (88)
preserves infinite direct sums, as it is clear from its construction in terms
of complexes of flasque quasi-coherent sheaves (cf. [45, Lemma 1.4]). By
Theorems 5.9.1(b), 5.9.3(c), and [37, Proposition 3.3(1)], it follows that
whenever the scheme $Y$ is Noetherian there exists a triangulated functor
(93) $f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
right adjoint to ${\mathbb{R}}f_{*}$.
For any symbol $\star=\empt$ or ${\mathsf{ctr}}$, the functor
${\mathbb{L}}f_{!}$ (90) preserves infinite products, as it is clear from its
construction in terms of complexes of coflasque locally cotorsion contraherent
cosheaves. By Theorems 5.9.1(c), 5.9.3(e), and [37, Proposition 3.3(2)], it
follows that whenever the scheme $Y$ is Noetherian of finite Krull dimension
there exists a triangulated functor
(94)
$f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
left adjoint to ${\mathbb{L}}f_{!}$. Assuming again that the scheme $Y$ is
Noetherian of finite Krull dimension, the functor ${\mathbb{L}}f_{!}$ (92)
preserves infinite products, and it follows that there exists a triangulated
functor
(95) $f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})$
left adjoint to ${\mathbb{L}}f_{!}$.
Now let $f\colon Y\longrightarrow X$ be a finite morphism of locally
Noetherian schemes. Notice that any finite morphism is affine, so the functor
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is exact; hence the induced functor
(96) $f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
defined for any symbol $\star\neq{\mathsf{ctr}}$. The right derived functor
(97)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
of the special inverse image functor $f^{!}\colon
X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ from Section 5.11 is constructed for
$\star=+$ or ${\mathsf{co}}$ in terms of injective resolutions (see Theorem
5.4.10(a)), and for $\star=\empt$ in terms of homotopy injective resolutions
(see Theorem 5.10.2). The right derived functor ${\mathbb{R}}f^{!}$ (97) is
right adjoint to the induced functor $f_{*}$ (96).
Similarly, the functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is well-defined and exact,
as is the functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$; hence the induced functors
(98)
$f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
and
(99) $f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
defined for any symbol $\star\neq{\mathsf{co}}$.
The left derived functor
(100)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
of the special inverse image functor $f^{*}\colon
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\longrightarrow
Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ from Section
5.11 is constructed for $\star=-$ or ${\mathsf{ctr}}$ in terms of projective
(locally cotorsion) resolutions (see Theorem 5.4.10(b)). When the scheme $X$
is Noetherian of finite Krull dimension, the functor (100) is constructed for
$\star=\empt$ in terms of homotopy projective resolutions (see Theorem
5.10.3(a)). The left derived functor ${\mathbb{L}}f^{*}$ (100) is left adjoint
to the induced functor $f_{!}$ (98).
Finally, let $i\colon Z\longrightarrow X$ be a closed embedding of Noetherian
schemes of finite Krull dimension. Then the left derived functor
(101)
${\mathbb{L}}i^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Z{\operatorname{\mathsf{--ctrh}}})$
of the special inverse image functor $i^{*}\colon
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\longrightarrow
Z{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ from Section 5.11 is
constructed for $\star=-$ or ${\mathsf{ctr}}$ in terms of flat resolutions
(see Theorem 5.4.10(d)), and for $\star=\empt$ in terms of homotopy projective
(see Theorem 5.10.3(b)) or flat and homotopy flat (see Theorem 5.10.5)
resolutions. The left derived functor ${\mathbb{L}}i^{*}$ (101) is left
adjoint to the induced functor $i_{!}$ (99). Clearly, the constructions of the
derived functors (100) and (101) agree wherever both a defined.
The following theorem generalizes Corollaries 4.11.6 and 5.4.6 to morphisms
$f$ of not necessarily finite flat dimension (between Noetherian schemes).
###### Theorem 5.12.2.
(a) Let $f\colon Y\longrightarrow X$ be a morphism of semi-separated
Noetherian schemes. Then the equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
and
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
from Theorem 5.7.1 transform the triangulated functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(93) into the triangulated functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
(59).
(b) Let $f\colon Y\longrightarrow X$ be a morphism of semi-separated
Noetherian schemes of finite Krull dimension. Then the equivalences of
triangulated categories
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.7.1 transform the triangulated functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
(57) into the triangulated functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
(95).
###### Proof.
Part (a): let us show that our the equivalences of categories transform the
functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(50, 88) into a functor left adjoint to the functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
(59). Let ${\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
complex over $Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
${\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$. Pick
an open covering ${\mathbf{T}}$ of the scheme $Y$ such that $f$ is a
$(\\{X\\},{\mathbf{T}})$-coaffine morphism. We have to construct a natural
isomorphism of abelian groups
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})}({\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}_{\mathbf{T}})}(\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{O}}_{Y},{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),f^{!}{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Let
$f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a morphism from the complex
$f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ (see Corollary 4.1.13(a))
to a complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ with a cone coacyclic
with respect to $X{\operatorname{\mathsf{--qcoh}}}$; so
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Then both
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and ${\mathcal{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
complexes over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$, hence one
has
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})}(\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}})}(\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\\\
\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})}({\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\\\
\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})}(f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})}(\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}).$
Here the first isomorphism holds because
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}})$
(see the proof of Corollary 4.6.8(b)) and the second isomorphism is provided
by the equivalence of categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}\simeq
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. The third isomorphism
follows from the proof of Lemma A.1.3(a) and the fourth one comes from the
equivalence of categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ (see Lemma 4.6.7).
Furthermore, by (44) and (25) one has
$\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})}(\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},f_{*}{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})}(f_{!}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{O}}_{Y},{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\\\
\simeq\operatorname{Hom}_{\mathsf{Hot}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}})}(\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{O}}_{Y},{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),f^{!}{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}_{\mathbf{T}})}(\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{O}}_{Y},{\mathcal{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),f^{!}{\mathfrak{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),$
where the last isomorphism follows from the proof of Lemma A.1.3(b), as the
objects of $Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$
are projective in the exact category
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$.
Part (b): we will show that the equivalences of categories
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\allowbreak\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
transform the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
(57) into the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
(94). Let ${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
complex over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and
${\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex
over $Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. We
have to construct a natural isomorphism of abelian groups
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})}(f^{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{Y}\odot_{Y}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Let
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
morphism into the complex
$f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$ (see
Corollary 4.5.3(b)) from a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ with a cone
contraacyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$; so
${\mathcal{O}}_{X}\otimes_{X}^{\mathbb{L}}f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={\mathcal{O}}_{X}\otimes_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then both ${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{O}}_{X}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are complexes over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$, hence one has
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\\\
\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})}(\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\\\
\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}})}(\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}),f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}).$
Here the first isomorphism holds because
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ (see the proof of Corollary
5.4.5) and the second isomorphism is provided by the equivalence of categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\simeq
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ (see the
proof of Corollary 4.6.10(c)). The third isomorphism follows from the proof of
Lemma A.1.3(b) and the fourth one comes from the equivalence of categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}\simeq
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ (see the proof of Corollary
4.6.8(a)).
Furthermore, one has
$\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{X}\odot_{X}f_{!}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>f_{*}({\mathcal{O}}_{Y}\odot_{Y}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}))\\\
\simeq\operatorname{Hom}_{\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})}(f^{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{Y}\odot_{Y}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})}(f^{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{O}}_{Y}\odot_{Y}{\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}).$
Here the first isomorphism holds according to the proof of Theorem 4.8.1
(since ${\mathfrak{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
complex of colocally projective contraherent cosheaves on $Y$) and the last
one follows from the proof of Lemma A.1.3(a) (as the objects of
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ are injective in the exact
category $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$). ∎
### 5.13. Adjoint functors and bounded complexes
The following theorem is a version of Theorem 4.8.1 for non-semi-separated
Noetherian schemes. One would like to have it for an arbitrary morphism of
Noetherian schemes of finite Krull dimension, but we are only able to present
a proof in the case of a flat morphism.
###### Theorem 5.13.1.
Let $f\colon Y\longrightarrow X$ be a flat morphism of Noetherian schemes of
finite Krull dimension. Then the equivalences of triangulated categories
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.8.1 transform the right derived functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
(88) into the left derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
(92).
###### Proof.
Clearly, for any morphism $f$ of Noetherian schemes of finite Krull dimension
and any symbol $\star\neq{\mathsf{co}}$ the equivalence of categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})$
induced by the embedding of exact categories
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--ctrh}}}$ and the similar equivalence for $X$
transform the left derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
(90) into the functor (92). Hence it remains to check that the equivalences of
triangulated categories
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
and
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
constructed in the proof of Theorem 5.8.1 transform the functor (88) into the
functor (90).
Let
${\mathcal{O}}_{X}\longrightarrow{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a finite resolution of the sheaf ${\mathcal{O}}_{X}$ by flasque quasi-
coherent sheaves on $X$. Then the morphism ${\mathcal{O}}_{Y}\longrightarrow
f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
quasi-isomorphism in $Y{\operatorname{\mathsf{--qcoh}}}$. Pick a finite
resolution
$f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of the complex
$f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ by
flasque quasi-coherent sheaves on $Y$. Then the composition
${\mathcal{O}}_{Y}\longrightarrow{\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is also a quasi-isomorphism.
According to Section 3.8, for any complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ there is a natural morphism
$f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{E}}_{X},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},f_{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes of cosheaves of ${\mathcal{O}}_{X}$-modules. Composing this
morphism with the morphism
$f_{!}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induced by the quasi-isomorphism
$f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
we obtain a natural morphism of complexes of cosheaves of
${\mathcal{O}}_{X}$-modules
$f_{!}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},f_{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Finally, pick a quasi-isomorphism
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes over the exact category
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ acting from a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$ to the
complex
$\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Applying the functor $f_{!}$ and composing again, we obtain a morphism
${\mathbb{L}}f_{!}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=f_{!}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},f_{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. We have
constructed a natural transformation
${\mathbb{L}}f_{!}\,\mskip
1.5mu{\mathbb{R}}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathbb{R}}f_{*}({-}))$
of functors
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$.
One can easily check that such natural transformations are compatible with the
compositions of flat morphisms $f$.
Now one can cover the scheme $X$ with affine open subschemes $U_{\alpha}$ and
the scheme $Y$ with affine open subschemes $V_{\beta}$ so that for any $\beta$
there exists $\alpha$ for which $f(V_{\beta})\subset V_{\alpha}$. The
triangulated category ${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$ is
generated by the derived direct images of objects from
${\mathsf{D}}(V_{\alpha}{\operatorname{\mathsf{--qcoh}}})$. So it suffices to
show that our natural transformation is an isomorphism whenever either both
schemes $X$ and $Y$ are semi-separated, or the morphism $f$ is an open
embedding. The former case is covered by Theorem 4.8.1, while in the latter
situation one can use the isomorphism (45) together with Lemma 3.4.6(c).
Alternatively, according to Section 3.8 for any complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ there is a
natural morphism
${\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}f_{!}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f_{*}(f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes of quasi-coherent sheaves on $X$. Composing it with the morphism
$f_{*}(f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
f_{*}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induced by the quasi-isomorphism
$f^{*}{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
we obtain a natural morphism of complexes of quasi-coherent sheaves
${\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}f_{!}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f_{*}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
on $X$. Finally, pick a quasi-isomorphism
${\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of complexes over $Y{\operatorname{\mathsf{--qcoh}}}$ acting from the complex
${\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
to a complex ${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fq}}$. Applying the functor
$f_{*}$ and composing again, we obtain a morphism
${\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}f_{!}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f_{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={\mathbb{R}}f_{*}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes over $X{\operatorname{\mathsf{--qcoh}}}$. We have constructed a
natural transformation
${\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}{\mathbb{L}}f_{!}({-})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathbb{R}}f_{*}({\mathcal{E}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}^{\mathbb{L}}{-})$
of functors
${\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.
To finish the proof, one continues to argue as above, using the isomorphism
(47) and Lemma 3.4.6(d). ∎
###### Remark 5.13.2.
Let $f\colon Y\longrightarrow X$ be a morphism of Noetherian schemes of finite
Krull dimension. If the conclusion of Theorem 5.13.1 holds for $f$, it follows
by adjunction that the equivalences of categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.8.1 transform the functor
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
(93) into a functor right adjoint to
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
(92) and the functor
$f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
(95) into a functor left adjoint to
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
(88).
Of course, these are supposed to be the conventional derived functors of
inverse image of quasi-coherent sheaves and contraherent cosheaves. One can
notice, however, that the conventional constructions of such functors involve
some difficulties when one is working outside of the situations covered by
Theorems 4.8.1 and 5.13.1. The case of semi-separated schemes $X$ and $Y$ is
covered by our exposition in Section 4 (see (55–56)), and in the case of a
flat morphism $f$ the underived inverse image would do (if one is working with
locally cotorsion contraherent cosheaves).
In the general case, it is not clear if there exist enough flat quasi-coherent
sheaves or locally injective contraherent cosheaves to make the derived
functor constructions work. One can construct the derived functor
${\mathbb{L}}f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
using complexes of sheaves of ${\mathcal{O}}$-modules with quasi-coherent
cohomology sheaves. We do _not_ know how to define a derived functor
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
or
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
for an arbitrary morphism $f$ of Noetherian schemes of finite Krull dimension.
###### Lemma 5.13.3.
(a) For any morphism $f\colon Y\longrightarrow X$ from a Noetherian scheme $Y$
to a locally Noetherian scheme $X$ such that either the scheme $X$ is
Noetherian or the scheme $Y$ has finite Krull dimension, the triangulated
functor
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
(93) takes ${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$ into
${\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})$ and induces a
triangulated functor
$f^{!}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})$
right adjoint to the right derived functor ${\mathbb{R}}f_{*}$ (87).
(b) For any morphism of Noetherian schemes of finite Krull dimension $f\colon
Y\longrightarrow X$, the triangulated functor
$f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
(94) takes
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ into
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ and
induces a triangulated functor
$f^{*}\colon{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
left adjoint to the left derived functor ${\mathbb{L}}f_{!}$ (89).
(c) For any morphism of Noetherian schemes of finite Krull dimension $f\colon
Y\longrightarrow X$, the triangulated functor
$f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
(95) takes ${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})$ into
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}})$ and induces a
triangulated functor
$f^{*}\colon{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}})$
left adjoint to the left derived functor ${\mathbb{L}}f_{!}$ (91).
###### Proof.
In each part (a-c), the second assertion follows immediately from the first
one (since the derived functors of direct image of bounded and unbounded
complexes agree). In view of Corollary 5.4.4(b), part (c) is also equivalent
to part (b).
To prove part (a), notice that a complex of quasi-coherent sheaves
${\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over $Y$ has
its cohomology sheaves concentrated in the cohomological degrees $\ge-N$ if
and only if one has
$\operatorname{Hom}_{{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$
for any complex ${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $Y{\operatorname{\mathsf{--qcoh}}}$ concentrated in the cohomological
degrees $<-N$. This is true for any abelian category in place of
$Y{\operatorname{\mathsf{--qcoh}}}$. Now given a complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$, one has
$\operatorname{Hom}_{{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},f^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})}({\mathbb{R}}f_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$
whenever ${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
concentrated in the cohomological degrees $\ge-N+M$, where $M$ is a certain
fixed constant. Indeed, the functor ${\mathbb{R}}f_{*}$ raises the
cohomological degrees by at most the Krull dimension of the scheme $Y$ (if it
has finite Krull dimension), or by at most the constant from Lemmas
5.3.7–5.3.8 (if both schemes are Noetherian).
In order to deduce parts (b-c), we will use Theorem 5.13.1. The equivalence of
triangulated categories
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.8.1 identifies
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--qcoh}}})$ with
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}})$. Given a complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ from
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})$, we would like to show
that the complex
${\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$Y{\operatorname{\mathsf{--qcoh}}}$ corresponding to the complex
${\mathfrak{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=f^{*}{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $Y{\operatorname{\mathsf{--ctrh}}}$ belongs to
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--qcoh}}})$. This is equivalent to
saying that the complex
$j^{\prime}{}^{*}{\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $V{\operatorname{\mathsf{--qcoh}}}$ belongs to
${\mathsf{D}}^{-}(V{\operatorname{\mathsf{--qcoh}}})$ for the embedding of any
small enough affine open subscheme $j^{\prime}\colon V\longrightarrow Y$.
By Theorem 5.13.1 applied to the morphism $j^{\prime}$, the complex
$j^{\prime}{}^{*}{\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
corresponds to the complex
$j^{\prime}{}^{*}{\mathfrak{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $V{\operatorname{\mathsf{--ctrh}}}$. We can assume that the composition
$f\circ j^{\prime}\colon V\longrightarrow X$ factorizes through the embedding
of an affine open subscheme $j\colon U\longrightarrow X$. Let $f^{\prime}$
denote the related morphism $V\longrightarrow U$. Then one has
$j^{\prime}{}^{*}{\mathfrak{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
f^{*}\\!\mskip
1.5muj^{*}{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
${\mathsf{D}}(V{\operatorname{\mathsf{--ctrh}}})$. Let
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ denote the
complex over $X{\operatorname{\mathsf{--qcoh}}}$ corresponding to
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$; by Theorem
5.8.1, one has
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})$.
Applying again Theorem 5.13.1 to the morphism $j$ and Theorem 4.8.1 to the
morphism $f^{\prime}$, we conclude that the complex $f^{*}\\!\mskip
1.5muj^{*}{\mathfrak{M}}$ over
${\mathsf{D}}(V{\operatorname{\mathsf{--ctrh}}})$ corresponds to the complex
${\mathbb{L}}f^{*}j^{*}{\mathcal{M}}$ over
${\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}})$. The latter is clearly
bounded above, and the desired assertion is proven. ∎
###### Corollary 5.13.4.
(a) For any morphism $f\colon Y\longrightarrow X$ from a Noetherian scheme $Y$
to a locally Noetherian scheme $X$ such that either the scheme $X$ is
Noetherian or the scheme $Y$ has finite Krull dimension, the triangulated
functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(93) takes ${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$ into
${\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})$, and the induced
triangulated functor
$f^{!}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})$
coincides with the one obtained in Lemma 5.13.3.
(b) For any morphism of Noetherian schemes of finite Krull dimension $f\colon
Y\longrightarrow X$, the triangulated functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
(94) takes
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ into
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$, and the
induced triangulated functor
$f^{*}\colon\allowbreak{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
coincides with the one obtained in Lemma 5.13.3.
(c) For any morphism of Noetherian schemes of finite Krull dimension $f\colon
Y\longrightarrow X$, the triangulated functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
(95) takes ${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})$ into
${\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}})$, and the induced
triangulated functor
$f^{*}\colon{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
coincides with the one obtained in Lemma 5.13.3.
###### Proof.
Notice first of all that there are natural fully faithful functors
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$,
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$,
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
(and similarly for $Y$) by Lemma A.1.2. Furthermore, one can prove the first
assertion of part (a) in the way similar to the proof of Lemma 5.13.3(a).
A complex of quasi-coherent sheaves
${\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over $Y$ is
isomorphic in ${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
to a complex whose terms are concentrated in the cohomological degrees $\ge-N$
if and only if one has
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$
for any complex ${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $Y{\operatorname{\mathsf{--qcoh}}}$ whose terms are concentrated in the
cohomological degrees $<N$. This is true for any abelian category with exact
functors of infinite direct sum in place of
$Y{\operatorname{\mathsf{--qcoh}}}$, since complexes with the terms
concentrated in the degrees $\le 0$ and $\ge 0$ form a t-structure on the
coderived category [50, Remark 4.1]. For a more explicit argument, see [37,
Lemma 2.2]. The rest of the proof of the first assertion is similar to that of
Lemma 5.13.3(a); and both assertions can be proven in the way dual-analogous
to the following proof of part (c).
It is clear from the constructions of the functors ${\mathbb{L}}f_{!}$ (92) in
terms of coflasque resolutions that the functors
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
form a commutative diagram with the Verdier localization functors
${\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$.
According to (95) and (the proof of) Theorem 5.10.3(b), all the functors in
this commutative square have left adjoints, which therefore also form a
commutative square.
The functor
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
left adjoint to the localization functor can be constructed as the functor
assigning to a complex over $X{\operatorname{\mathsf{--ctrh}}}$ its homotopy
projective resolution, viewed as an object of
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$ (and
similarly for $Y$). Since the homotopy projective resolution
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of a bounded
above complex ${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--ctrh}}}$ can be chosen to be also bounded
above, and the cone of the morphism
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contraacyclic, part (c) follows from Lemma 5.13.3(c). ∎
###### Proposition 5.13.5.
Let $f\colon Y\longrightarrow X$ be a proper morphism (of finite type and) of
finite flat dimension between semi-separated Noetherian schemes. Assume that
the object $f^{!}{\mathcal{O}}_{X}$ is compact in
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$ (i. e., it is a perfect
complex on $Y$). Then the functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(93) is naturally isomorphic to the functor
$f^{!}{\mathcal{O}}_{X}\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}$,
where
$f^{!}{\mathcal{O}}_{X}\in{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$,
while
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
is the functor constructed in (60) and
$\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}$ is the tensor action
functor (77).
###### Proof.
Instead of proving the desired assertion from scratch, we will deduce it step
by step from the related results of [45, Section 5], using the results above
in this section to brigde the gap between the derived and coderived
categories.
###### Lemma 5.13.6.
Let $f\colon Y\longrightarrow X$ be a morphism of finite flat dimension
between semi-separated Noetherian schemes. Then for any objects
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
and
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
there is a natural isomorphism
${\mathbb{R}}f_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq{\mathbb{R}}f_{*}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
in ${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$, where
${\mathbb{R}}f_{*}$ denotes the derived direct image functors (50).
###### Proof.
For any objects
${\mathcal{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
one easily constructs a natural isomorphism
${\mathbb{L}}f^{*}({\mathcal{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq{\mathbb{L}}f^{*}{\mathcal{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in ${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$ (where
${\mathbb{L}}f^{*}$ denotes the derived functors (55, 60)). Substituting
${\mathcal{K}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={\mathbb{R}}f_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$,
one can consider the composition
${\mathbb{L}}f^{*}({\mathbb{R}}f_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq{\mathbb{L}}f^{*}\mskip
1.5mu{\mathbb{R}}f_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of morphisms in
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$. By
adjunction, we obtain the natural transformation
${\mathbb{R}}f_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathbb{R}}f_{*}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of functors
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\times{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$.
Since all the functor involved preserve infinite direct sums, it suffices to
check that our morphism is an isomorphism for compact generators of the
categories ${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$ and
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$, that is one
can assume ${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to
be a perfect complex on $X$ and
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to be a finite
complex of coherent sheaves on $Y$ (see Theorems 5.9.3(b) and 5.9.1(b)). In
this case all the complexes involved are bounded below and, in view of Lemma
A.1.2, the question reduces to the similar assertion for the conventional
derived categories, which is known due to [45, Proposition 5.3]. ∎
In particular, for any object
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
we have a natural isomorphism
${\mathbb{R}}f_{*}f^{!}{\mathcal{O}}_{X}\otimes_{{\mathcal{O}}_{X}}^{{\mathbb{L}}^{\prime}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq{\mathbb{R}}f_{*}(f^{!}{\mathcal{O}}_{X}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Composing it with the morphism induced by the adjunction morphism
${\mathbb{R}}f_{*}f^{!}{\mathcal{O}}_{X}\longrightarrow{\mathcal{O}}_{X}$, we
obtain a natural morphism
${\mathbb{R}}f_{*}(f^{!}{\mathcal{O}}_{X}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in ${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$. We have
constructed a natural transformation
$f^{!}{\mathcal{O}}_{X}\otimes_{{\mathcal{O}}_{Y}}^{{\mathbb{L}}^{\prime}}{\mathbb{L}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of functors
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$.
Since the morphism $f$ is proper, the functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
takes compact objects to compact objects [27, Théorème 3.2.1]. By [45, Theorem
5.1], it follows that the functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
preserves infinite direct sums. So does the functor in the left-hand side of
our morphism; therefore, it suffices to check that this morphism is an
isomorphism when
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a finite
complex of coherent sheaves on $X$. Since we assume $f^{!}{\mathcal{O}}_{X}$
to be a perfect complex, in this case all the complexes involved are bounded
below, and in view of Corollary 5.13.4(a), the question again reduces to the
similar assertion for the conventional derived categories, which is provided
by [45, Example 5.2 and Theorem 5.4]. ∎
###### Remark 5.13.7.
The condition that $f^{!}{\mathcal{O}}_{X}$ is a perfect complex, which was
not needed in [45, Section 5], cannot be dropped in the above Proposition, as
one can see already in the case when $X$ is the spectrum of a field and
$Y\longrightarrow X$ is a finite morphism. The problem arises because of the
difference between the left and right adjoint functors to the Verdier
localization functor
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$.
To construct a specific counterexample, let $k$ be a field and $R$ be the
quotient ring $R=k[x,y]/(x^{2},xy,y^{2})$, so that the images of the elements
$1$, $x$, and $y$ form a basis in $R$ over $k$. Set $X=\operatorname{Spec}k$
and $Y=\operatorname{Spec}R$. Then the injective coherent sheaf
$\widetilde{R^{*}}$ on $Y$ corresponding to the $R$-module
$R^{*}=\operatorname{Hom}_{k}(R,k)$ represents the object
$f^{!}{\mathcal{O}}_{X}\in{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$. We
have to show that the $R$-module $R^{*}$ is not isomorphic in
${\mathsf{D}}^{\mathsf{co}}(R{\operatorname{\mathsf{--mod}}})$ to its left
projective resolution.
Indeed, otherwise it would follow that the complex
$\operatorname{Hom}_{R}(R^{*},J^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex of injective $R$-modules
$J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Since there is a short
exact sequence of $R$-modules $0\longrightarrow k^{\oplus 3}\longrightarrow
R^{\oplus 2}\longrightarrow R^{*}\longrightarrow 0$, the complex
$\operatorname{Hom}_{R}(k,J^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
would then be also acyclic, making the complex of $R$-modules
$J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ contractible. This would
mean that the coderived category of $R$-modules coincides with their derived
category, which cannot be true, as their subcategories of compact objects are
clearly different.
On the other hand, one can get rid of the semi-separability assumptions in
Proposition 5.13.5 by constructing the functor
${\mathbb{L}}f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
in terms of complexes of sheaves of ${\mathcal{O}}$-modules with quasi-
coherent cohomology sheaves (as it is done in [45]; cf. the proof of Theorem
5.9.3(b-c)) and obtaining the functor
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
from it using the techniques of [21].
### 5.14. Compatibilities for a smooth morphism
Let $f\colon Y\longrightarrow X$ be a smooth morphism of Noetherian schemes.
Let ${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
dualizing complex for $X$; then
$f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for $Y$ [29, Theorem V.8.3]. The complex
$f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ being
not necessarily a complex of injectives, let us pick a finite complex over
$Y{\operatorname{\mathsf{--qcoh}}}$ quasi-isomorphic to
$f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
denote it temporarily by
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
###### Corollary 5.14.1.
Assume the schemes $X$ and $Y$ to be semi-separated. Then
(a) the equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
from Theorem 5.7.1 related to the choice of the dualizing complexes
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ and
$Y$ transform the inverse image functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
(57) into the (underived, as the morphism $f$ is flat) inverse image functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(60);
(b) the equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.7.1 related to the choice of the dualizing complexes
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ and
$Y$ transform the inverse image functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
(59) into the inverse image functor
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
(62).
###### Remark 5.14.2.
It would follow from Conjecture 1.7.2 that the inverse image functor (62) in
the formulation of part (b) of Corollary 5.14.1 is in fact an underived
inverse image functor $f^{!}$, just as in part (a). In any event, the morphism
$f$ being at least flat, there is an underived inverse image functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
(64) (cf. the proof below).
###### Proof of Corollary 5.14.1.
Part (a) (cf. [53, Section 2.14]): notice that for any finite complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of quasi-
coherent sheaves on $Y$ the functor
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
constructed by tensoring complexes over
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathcal{O}}_{Y}$ is well-defined, and replacing
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ by a quasi-
isomorphic complex leads to an isomorphic functor. So it remains to use the
isomorphism
$f^{*}({\mathcal{D}}_{X}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}})\simeq
f^{*}{\mathcal{D}}_{X}\otimes_{{\mathcal{O}}_{X}}f^{*}{\mathcal{F}}$ holding
for any flat (or even arbitrary) quasi-coherent sheaf ${\mathcal{F}}$ on $X$.
Part (b): let ${\mathbf{W}}$ and ${\mathbf{T}}$ be open coverings of the
schemes $X$ and $Y$ such that the morphism $f$ is
$({\mathbf{W}},{\mathbf{T}})$-coaffine. For any finite complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of quasi-
coherent sheaves on $Y$, the functor
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}})$
constructed by taking $\operatorname{\mathfrak{Cohom}}_{Y}$ from
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to complexes
over $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$ is well-
defined, and replacing
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ by a quasi-
isomorphic complex leads to an isomorphic functor. So it remains to use the
isomorphism (38) together with the fact that the equivalence of triangulated
categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and the similar equivalence
for $Y$ transform the functor (63) into the functor (61) for any morphism $f$
of finite very flat dimension between quasi-compact semi-separated schemes. ∎
The following theorem is to be compared with Theorems 4.8.1 and 5.13.1, and
Corollaries 4.11.6 and 5.4.6.
###### Theorem 5.14.3.
The equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.7.1 or 5.8.2 related to the choice of the dualizing complexes
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $Y$ and
$X$ transform the right derived functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
(88) into the left derived functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
(92).
###### Proof.
It suffices to show that the equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\allowbreak\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
from the proof of Theorem 5.8.2 transform the functor (88) into the functor
(90). The latter equivalences were constructed on the level of injective and
projective resolutions as the equivalences of homotopy categories
$\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq\mathsf{Hot}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
induced by the functors
$\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{-}$,
and similarly for $X$. In particular, it was shown that these functors take
complexes over $Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ to complexes
over $Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ and
back.
Furthermore, we notice that such complexes are adjusted to the derived
functors ${\mathbb{R}}f_{*}$ and ${\mathbb{L}}f_{!}$ acting between the
co/contraderived categories. The morphism $f$ being flat, the direct image
functors $f_{*}$ and $f_{!}$ take
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ into
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ into
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ (see
Corollary 5.1.6(b)).
According to Section 3.8, for any complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ there is a natural morphism
$f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},f_{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes of cosheaves of ${\mathcal{O}}_{X}$-modules. Composing this
morphism with the morphism
$f_{!}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induced by the quasi-isomorphism
$f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
we obtain a natural morphism
(102)
$f_{!}\operatorname{\mathfrak{Hom}}_{Y}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},f_{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. Similarly,
according to Section 3.8 for any complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ there is a
natural morphism
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}f_{!}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f_{*}(f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes over $X{\operatorname{\mathsf{--qcoh}}}$. Composing it with the
morphism
$f_{*}(f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow
f_{*}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induced by the quasi-isomorphism
$f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
we obtain a natural morphism
(103)
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}f_{!}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muf_{*}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{Y}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of complexes over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. The
natural morphisms (102–103) are compatible with the adjunction (20) and with
the compositions of the morphisms of schemes $f$.
It suffices to show that the morphism (102) is a homotopy equivalence (or just
a quasi-isomorphism over $X{\operatorname{\mathsf{--ctrh}}}$) for any one-term
complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={\mathcal{J}}$
over $Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$, or equivalently, the
morphism (103) is a homotopy equivalence (or just a quasi-isomorphism over
$X{\operatorname{\mathsf{--qcoh}}}$) for any one-term complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={\mathfrak{P}}$
over $Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
According to (45), the morphism (102) is actually an isomorphism whenever both
schemes $X$ and $Y$ are semi-separated, or the morphism $f$ is affine, or it
is an open embedding. According to (46–47), the morphism (103) is an
isomorphism whenever the morphism $f$ is affine, or it is an open embedding of
an affine scheme.
We have already proven the desired assertion in the case of semi-separated
Noetherian schemes $X$ and $Y$. To handle the general case, cover the scheme
$X$ with semi-separated open subschemes $U_{\alpha}$ and the scheme $Y$ with
semi-separated open subschemes $V_{\beta}$ so that for any $\beta$ there
exists $\alpha$ for which $f(V_{\beta})\subset U_{\alpha}$. Decompose an
object ${\mathcal{J}}\in Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$
into a direct sum of the direct images of injective quasi-coherent sheaves
from $V_{\beta}$. Since we know (102) to be an isomorphism for the morphisms
$V_{\beta}\longrightarrow Y$, $U_{\alpha}\longrightarrow X$, and
$V_{\beta}\longrightarrow U_{\alpha}$, it follows that this map is an
isomorphism of complexes over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ for the
morphism $f$. ∎
### 5.15. Compatibilities for finite and proper morphisms
Let $X$ be a Noetherian scheme with a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, which we
will view as a finite complex over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. Let $f\colon
Y\longrightarrow X$ be a finite morphism of schemes. Then
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=f^{!}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
where $f^{!}$ denotes the special inverse image functor from Section 5.11, is
a dualizing complex for $Y$ [29, Proposition V.2.4].
###### Theorem 5.15.1.
The equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
from Theorem 5.7.1 or 5.8.2 related to the choice of the dualizing complexes
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ and
$Y$ transform the right derived functor
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(97) into the left derived functor
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
(100).
###### Proof.
We will show that the equivalences of homotopy categories
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
and
$\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq\mathsf{Hot}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
from the proof of Theorem 5.8.2 transform the functor $f^{!}$ into the functor
$f^{*}$. Notice that complexes over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ are adjusted
to the derived functors ${\mathbb{R}}f^{!}$ and ${\mathbb{L}}f^{*}$ acting
between the co/contraderived categories. The special inverse image functors
$f^{!}$ and $f^{*}$ take $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ to
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ to
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
First of all, we will need the following base change lemma.
###### Lemma 5.15.2.
Let $g\colon x\longrightarrow X$ be a morphism of locally Noetherian schemes
and $f\colon Y\longrightarrow X$ be a finite morphism. Set $y=x\times_{X}Y$,
and denote the natural morphisms by $f^{\prime}\colon y\longrightarrow x$ and
$g^{\prime}\colon y\longrightarrow Y$. Then
(a) for any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ there is a natural
morphism of quasi-coherent sheaves $g^{\prime}{}^{*}\\!\mskip
1.5muf^{!}{\mathcal{M}}\longrightarrow f^{\prime}{}^{!}g^{*}{\mathcal{M}}$ on
$y$; this map is an isomorphism for any ${\mathcal{M}}$ whenever the morphism
$g$ is flat;
(b) whenever the morphism $g$ is an open embedding, for any projective locally
cotorsion contraherent cosheaf ${\mathfrak{P}}$ on $X$ there is a natural
isomorphism of projective locally cotorsion contraherent cosheaves
$g^{\prime}{}^{!}f^{*}{\mathfrak{P}}\simeq f^{\prime}{}^{*}g^{!}\mskip
1.5mu{\mathfrak{P}}$ on $y$;
(c) whenever the morphism $g$ is quasi-compact, for any quasi-coherent sheaf m
on $x$ there is a natural isomorphism of quasi-coherent sheaves
$g^{\prime}_{*}f^{\prime}{}^{!}\mskip 1.5mu{\text{m}}\simeq
f^{!}g_{*}{\text{m}}$ on $Y$;
(d) whenever the morphism $g$ is flat and quasi-compact, for any projective
locally cotorsion contraherent cosheaf ${\mathfrak{p}}$ on $x$ there is a
natural isomorphism of projective locally cotorsion contraherent cosheaves
$g^{\prime}_{!}f^{\prime}{}^{*}{\mathfrak{p}}\simeq f^{*}g_{!}\mskip
1.5mu{\mathfrak{p}}$ on $Y$.
###### Proof.
Notice that the functors being composed in parts (b) and (d) preserve the
class of projective locally cotorsion contraherent cosheaves by Corollaries
5.1.3(a) and 5.1.6(b). Furthermore, parts (a) and (b) were essentially proven
in Section 5.11. The assertion of part (c) is local in $X$, so one can assume
$X$ to be affine. Then if $x$ is also affine, the assertion is obvious; and
the general case of a Noetherian scheme $x$ is handled by computing the global
sections of $m$ in terms of a finite affine covering $u_{\alpha}$ of $x$ and
finite affine coverings of the intersections $u_{\alpha}\cap u_{\beta}$. The
proof of part (d) is similar to that of part (c). ∎
For an affine open subscheme $U\subset X$ with the open embedding morphism
$j\colon U\longrightarrow X$, denote by $V$ the full preimage
$U\times_{X}Y\subset Y$ and by $j^{\prime}\colon V\longrightarrow Y$ its open
embedding morphism. Let ${\mathcal{I}}$ and ${\mathcal{J}}$ be injective
quasi-coherent sheaves on $X$. Then the composition
$\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{I}},{\mathcal{J}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{Y}(f^{!}j_{*}j^{*}{\mathcal{I}},f^{!}{\mathcal{J}})\mskip
1.5mu\simeq\mskip
1.5mu\operatorname{Hom}_{Y}(j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{!}{\mathcal{I}},f^{!}{\mathcal{J}})$
of the map induced by the functor $f^{!}$ with the isomorphism provided by
Lemma 5.15.2(a,c) induces a morphism of ${\mathcal{O}}(V)$-modules
$(f^{*}\\!\mskip
1.5mu\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{\mathcal{J}}))[V]\mskip
1.5mu=\mskip
1.5mu{\mathcal{O}}(V)\otimes_{{\mathcal{O}}(U)}\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{I}},{\mathcal{J}})\\\
\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{Y}(j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{!}{\mathcal{I}},f^{!}{\mathcal{J}})\mskip 1.5mu=\mskip
1.5mu\operatorname{\mathfrak{Hom}}_{Y}(f^{!}{\mathcal{I}},f^{!}{\mathcal{J}})[V],$
defining a morphism of projective locally cotorsion contraherent cosheaves
$f^{*}\\!\mskip
1.5mu\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{\mathcal{J}})\allowbreak\longrightarrow\operatorname{\mathfrak{Hom}}_{Y}(f^{!}{\mathcal{I}},f^{!}{\mathcal{J}})$
on $Y$.
To show that this morphism is an isomorphism, decompose the sheaf
${\mathcal{J}}$ into a finite direct sum of the direct images of injective
quasi-coherent sheaves ${\mathcal{K}}$ from embeddings $k\colon
W\longrightarrow X$ of affine open subschemes $W\subset X$. Then the
isomorphisms of Lemma 5.15.2(c-d) together with the isomorphism (45) reduce
the question to the case of an affine scheme $X$. In the latter situation we
have
${\mathcal{O}}(Y)\otimes_{{\mathcal{O}}(X)}\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{I}}(X),{\mathcal{J}}(X))\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}(\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}(Y),{\mathcal{I}}(X)),\mskip
1.5mu{\mathcal{J}}(X))\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}(Y)}(\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}(Y),{\mathcal{I}}(X)),\>\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}(Y),{\mathcal{J}}(X))),$
since ${\mathcal{O}}(Y)$ is a finitely presented ${\mathcal{O}}(X)$-module and
${\mathcal{I}}(X)$, ${\mathcal{J}}(X)$ are injective
${\mathcal{O}}(X)$-modules.
Alternatively, let ${\mathcal{M}}$ be a quasi-coherent sheaf and
${\mathfrak{P}}$ be a projective locally cotorsion contraherent cosheaf on
$X$. We keep the notation above for the open embeddings $j\colon
U\longrightarrow X$ and $j^{\prime}\colon V\longrightarrow Y$. Then the
composition
$j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{!}{\mathcal{M}}\otimes_{{\mathcal{O}}(V)}(f^{*}{\mathfrak{P}})[V]\mskip
1.5mu\simeq\mskip 1.5muj^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{!}{\mathcal{M}}\otimes_{{\mathcal{O}}(U)}{\mathfrak{P}}[U]\\\
\simeq\mskip
1.5muf^{!}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(U)}{\mathfrak{P}}[U]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muf^{!}(j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(U)}{\mathfrak{P}}[U])$
provides a natural morphism $j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{!}{\mathcal{M}}\otimes_{{\mathcal{O}}(V)}(f^{*}{\mathfrak{P}})[V]\longrightarrow
f^{!}(j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(U)}{\mathfrak{P}}[U])$ of
quasi-coherent sheaves on $Y$. Passing to the inductive limit over $U$ and
noticing that the contratensor product
$f^{!}{\mathcal{M}}\odot_{Y}f^{*}{\mathfrak{P}}$ can be computed on the
diagram of affine open subschemes $V=U\times_{X}Y\subset Y$ (see Section 2.6),
we obtain a morphism of quasi-coherent sheaves
$f^{!}{\mathcal{M}}\odot_{Y}f^{*}{\mathfrak{P}}\longrightarrow
f^{!}({\mathcal{M}}\odot_{X}{\mathfrak{P}})$ on $Y$. When the quasi-coherent
sheaf ${\mathcal{M}}={\mathcal{I}}$ is injective, this is a morphism of
injective quasi-coherent sheaves.
To show that this morphism is an isomorphism, decompose the cosheaf
${\mathfrak{P}}$ into a finite direct sum of the direct images of projective
locally cotorsion contraherent cosheaves ${\mathfrak{Q}}$ from embeddings
$k\colon W\longrightarrow X$ of affine open subschemes $W\subset X$. Then the
isomorphisms of Lemma 5.15.2(c-d) together with the isomorphism (47) reduce
the question to the case of an affine scheme $X$. In the latter situation we
have
$\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}(Y),{\mathcal{M}}(X))\otimes_{{\mathcal{O}}(Y)}({\mathcal{O}}(Y)\otimes_{{\mathcal{O}}(X)}{\mathfrak{P}}[X])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}(Y),{\mathcal{M}}(X))\otimes_{{\mathcal{O}}(X)}{\mathfrak{P}}[X]\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}(Y),\>{\mathcal{M}}(X)\otimes_{{\mathcal{O}}(X)}{\mathfrak{P}}[X]),$
since the ${\mathcal{O}}(X)$-module ${\mathcal{O}}(Y)$ is finitely presented
and the ${\mathcal{O}}(X)$-module ${\mathfrak{P}}[X]$ is flat. ∎
Now let $X$ be a semi-separated Noetherian scheme with a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Let
$f\colon Y\longrightarrow X$ be a proper morphism (of finite type). Notice
that, the morphism $f$ being separated, the scheme $Y$ is consequently semi-
separated (and Noetherian), too. Set
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to be a
finite complex over $Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ quasi-
isomorphic to
$f^{!}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, where
$f^{!}$ denotes the triangulated functor (93) right adjoint to the derived
direct image functor ${\mathbb{R}}f_{*}$ (cf. Corollary 5.13.4(a) and [29,
Remark before Proposition V.8.5]).
The following theorem is supposed to be applied together with Theorem 5.12.2
and/or Corollary 5.14.1.
###### Theorem 5.15.3.
The equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
from Theorem 5.7.1 transform the triangulated functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
(57) into the triangulated functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(93).
###### Proof.
For any complex of flat quasi-coherent sheaves
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ there is
a natural isomorphism
$f_{*}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq
f_{*}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of complexes over $X{\operatorname{\mathsf{--qcoh}}}$ (see (13)). Hence the
adjunction morphism
$f_{*}{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={\mathbb{R}}f_{*}f^{!}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induces a natural morphism
${\mathbb{R}}f_{*}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})=f_{*}({\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in ${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ (see (74)).
We have constructed a natural transformation
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f^{!}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$.
Since finite complexes of coherent sheaves
${\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $Y$ form a
set of compact generators of
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$ (see Theorem
5.9.1(b)), in order to prove that our morphism is an isomorphism it suffices
to show that the induced morphism of abelian groups
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})}({\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})}({\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>f^{!}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})}({\mathbb{R}}f_{*}{\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is an isomorphism for any
${\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Both
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
being complexes of injective quasi-coherent sheaves, the Hom into them in the
coderived categories coincides with the one taken in the homotopy categories
of complexes of quasi-coherent sheaves.
The complexes
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{N}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ being finite,
the latter Hom only depends on a finite fragment of the complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. This reduces
the question to the case of a single coherent sheaf ${\mathcal{N}}$ on $Y$ and
a single flat quasi-coherent sheaf ${\mathcal{F}}$ on $X$; one has to show
that the natural morphism of complexes of abelian groups
$\operatorname{Hom}_{Y}({\mathcal{N}},\>{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{F}})\longrightarrow\operatorname{Hom}_{X}({\mathbb{R}}f_{*}{\mathcal{N}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}})$
is a quasi-isomorphism. Notice that in the case
${\mathcal{F}}={\mathcal{O}}_{X}$ this is the definition of
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
f^{!}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
In the case when there are enough vector bundles (locally free sheaves of
finite rank) on $X$, one can pick a left resolution
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the flat
quasi-coherent sheaf ${\mathcal{F}}$ by infinite direct sums of vector bundles
and argue as above, reducing the question further to the case when
${\mathcal{F}}$ is an infinite direct sum of vector bundles, when the
assertion follows by compactness. In the general case, using the Čech
resolution (12) of a flat quasi-coherent sheaf ${\mathcal{F}}$, one can assume
${\mathcal{F}}$ to be the direct image of a flat quasi-coherent sheaf
${\mathcal{G}}$ from an affine open subscheme $U\subset X$.
By the same projection formula (13) applied to the open embeddings $j\colon
U\longrightarrow X$ and $j^{\prime}\colon V=U\times_{X}Y\longrightarrow Y$
(which are affine morphisms), one has
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}j_{*}{\mathcal{G}}\simeq
j_{*}(j^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{U}}{\mathcal{G}})$
and
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{Y}}f^{*}\\!\mskip
1.5muj_{*}{\mathcal{G}}\simeq
j^{\prime}_{*}(j^{\prime*}{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{V}}f^{\prime*}{\mathcal{G}})$.
Here we are also using the base change isomorphism $f^{*}\\!\mskip
1.5muj_{*}{\mathcal{G}}\simeq j^{\prime}_{*}f^{\prime*}{\mathcal{G}}$, and
denote by $f^{\prime}$ the morphism $V\longrightarrow U$. Using the adjunction
of the direct image functors $j_{*}$ and $j^{*}$ together with the isomorphism
$j^{*}\mskip
1.5mu{\mathbb{R}}f_{*}{\mathcal{N}}\simeq{\mathbb{R}}f^{\prime}_{*}\,j^{\prime}{}^{*}{\mathcal{N}}$
in ${\mathsf{D}}^{\mathsf{b}}(U{\operatorname{\mathsf{--coh}}})$, one can
replace the morphism $f\colon Y\longrightarrow X$ by the morphism
$f^{\prime}\colon V\longrightarrow U$ into an affine scheme $U$ in the desired
assertion, where there are obviously enough vector bundles.
In the above argument one needs to know that the natural morphism
$j^{\prime*}{\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j^{\prime*}\\!\mskip
1.5muf^{!}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
f^{\prime}{}^{!}j^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is an isomorphism in ${\mathsf{D}}^{+}(V{\operatorname{\mathsf{--qcoh}}})$.
This is a result of Deligne [14]; see Theorem 5.16.1 below. ∎
### 5.16. The extraordinary inverse image
Let $f\colon Y\longrightarrow X$ be a separated morphism of finite type
between Noetherian schemes. (What we call) Deligne’s extaordinary inverse
image functor
$f^{+}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
is a triangulated functor defined by the following rules:
1. (i)
whenever $f$ is an open embedding, $f^{+}=f^{*}$ is the conventional inverse
image functor (induced by the exact functor $f^{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ and left adjoint to the derived direct
image functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
(50, 88);
2. (ii)
whenever $f$ is a proper morphism, $f^{+}=f^{!}$ is (what we call) Neeman’s
extraordinary inverse image functor (93), i. e., the functor right adjoint to
the derived direct image functor ${\mathbb{R}}f_{*}$ (cf. Corollary 4.8.2);
3. (iii)
given two composable separated morphisms of finite type $f\colon
Y\longrightarrow X$ and $g\colon Z\longrightarrow Y$ between Noetherian
schemes, one has a natural isomorphism of triangulated functors
$(fg)^{+}\simeq g^{+}f^{+}$.
Deligne constructs his extraordinary inverse image functor (which we denote
by)
$f^{+}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})$
in [14] (cf. [29, Theorem III.8.7]). Notice that by Lemma 5.13.3(a) and
Corollary 5.13.4(a) the two versions of Neeman’s extraordinary inverse image
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
and
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
have isomorphic restrictions to
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$,
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$, both acting
from ${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$ to
${\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})$ and being right adjoint
to
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}})$
(87); so there is no ambiguity here.
According to a counterexample of Neeman’s [45, Example 6.5], there _cannot_
exist a triangulated functor
$f^{+}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
defined for all, say, locally closed embeddings of affine schemes of finite
type over a fixed field and satisfying (i) for open embeddings, (ii) for
closed embeddings, and (iii) for compositions. Our aim is to show that there
is _no_ similar inconsistency in the rules (i-iii) to prevent existence of a
functor $f^{+}$ acting between the coderived categories of (unbounded
complexes of) quasi-coherent sheaves. Given that, and assuming
compactifiability of separated morphisms of finite type between Noetherian
schemes (a version of Nagata’s theorem, see [14]), one would be able to
proceed with the construction of the functor
$f^{+}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(cf. [21, Sections 5 and 6]).
###### Theorem 5.16.1.
Let $g\colon Y\longrightarrow X$ and $g^{\prime}\colon V\longrightarrow U$ be
proper morphisms (of finite type) between Noetherian schemes, forming a
commutative diagram with open embeddings $h\colon U\longrightarrow X$ and
$h^{\prime}\colon V\longrightarrow Y$. Then there is a natural isomorphism of
triangulated functors $h^{\prime}{}^{*}g^{!}\simeq
g^{\prime}{}^{!}h^{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})$.
###### Proof.
We follow the approach in [14] (with occasional technical points borrowed from
[21]). The argument is based on the results of [27, Chapitre 3].
Set $V^{\prime}=U\times_{X}Y$. Then the natural morphism $V\longrightarrow
V^{\prime}$ is both an open embedding and a proper morphism, that is
$V^{\prime}$ is a disconnected union of $V$ and $V^{\prime}\setminus V$. Hence
one can assume that $V=V^{\prime}$, i. e., the square is Cartesian.
The construction of the derived functor ${\mathbb{R}}f_{*}$ (88) being local
in the base (since the flasqueness/injectivity properties of quasi-coherent
sheaves are local), there is a natural isomorphism of triangulated functors
$h^{*}{\mathbb{R}}g_{*}\simeq{\mathbb{R}}g^{\prime}_{*}\,h^{\prime*}$. Given
an object
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$,
we now have a natural morphism
${\mathbb{R}}g^{\prime}_{*}\,h^{\prime}{}^{*}g^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
h^{*}{\mathbb{R}}g_{*}\mskip
1.5mug^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
${\mathsf{D}}^{\mathsf{co}}(U{\operatorname{\mathsf{--qcoh}}})$, inducing a
natural morphism
$h^{\prime}{}^{*}g^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
g^{\prime}{}^{!}h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in ${\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})$. We have
constructed a morphism of functors $h^{\prime}{}^{*}g^{!}\longrightarrow
g^{\prime}{}^{!}h^{*}$.
In order to prove that this morphism is an isomorphism in
${\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})$, we will show
that the induced morphism
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>h^{\prime}{}^{*}g^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak
g^{\prime}{}^{!}h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is an isomorphism of abelian groups for any finite complex of coherent sheaves
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $V$. This
would be sufficient in view of the fact that finite complexes of coherents
form a set of compact generators of
${\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})$ (see Theorem
5.9.1(b)).
For any category ${\mathsf{C}}$, we denote by
$\operatorname{\mathsf{Pro}}{\mathsf{C}}$ the category of pro-objects in
${\mathsf{C}}$ (for our purposes, it suffices to consider projective systems
indexed by the nonnegative integers). To any open embedding of Noetherian
schemes $j\colon W\longrightarrow Z$ one assigns an exact functor (between
abelian categories) $j_{!}\colon
W{\operatorname{\mathsf{--coh}}}\longrightarrow\operatorname{\mathsf{Pro}}(Z{\operatorname{\mathsf{--coh}}})$
defined by the following rule.
Let ${\mathcal{I}}\subset{\mathcal{O}}_{Z}$ denote the sheaf of ideals
corresponding to some closed subscheme structure on the complement $Z\setminus
W$. Given a coherent sheaf ${\mathcal{F}}$ on $W$, pick coherent sheaf
${\mathcal{F}}^{\prime}$ on $Z$ with
${\mathcal{F}}^{\prime}|_{W}\simeq{\mathcal{F}}$. The functor $j_{!}$ takes
the sheaf $W$ to the projective system formed by the coherent sheaves
$j_{*}({\mathcal{I}}^{n}{\mathcal{F}}^{\prime})$, $n\ge 0$ on $Z$ (and their
natural embeddings). One can check that this pro-object does not depend (up to
a natural isomorphism) on the choice of a coherent extension
${\mathcal{F}}^{\prime}$. In fact, one has
$\operatorname{Hom}_{W}({\mathcal{F}},j^{*}{\mathcal{G}})\simeq\operatorname{Hom}_{Z}(j_{!}{\mathcal{F}},{\mathcal{G}})$
for any ${\mathcal{G}}\in Z{\operatorname{\mathsf{--qcoh}}}$ [14, Proposition
4].
Passing to the bounded derived categories, one obtains a triangulated functor
$j_{!}\colon{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(\operatorname{\mathsf{Pro}}Z{\operatorname{\mathsf{--coh}}})$.
Let
$\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})\subset\operatorname{\mathsf{Pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})$
denote the full subcategory formed by projective systems of complexes with
uniformly bounded cohomology sheaves. The system of cohomology functors
$\operatorname{\mathsf{pro}}H^{i}\colon\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})\longrightarrow\operatorname{\mathsf{Pro}}(Z{\operatorname{\mathsf{--coh}}})$
is conservative (a result applicable to any abelian category [14, Proposition
3]). Furthermore, there is a natural functor
${\mathsf{D}}^{\mathsf{b}}(\operatorname{\mathsf{Pro}}Z{\operatorname{\mathsf{--coh}}})\longrightarrow\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})$.
Composing it with the above functor $j_{!}$ and passing to pro-objects in
$W{\operatorname{\mathsf{--coh}}}$ one constructs the functor
$j_{!}\colon\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})\longrightarrow\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})$.
On the other hand, the functor
$h^{*}\colon{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})$
induces a natural functor
$\operatorname{\mathsf{pro}}h^{*}\colon\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})\longrightarrow\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})$.
###### Lemma 5.16.2.
(a) For any objects
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})$
and
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{\mathsf{co}}(Z{\operatorname{\mathsf{--qcoh}}})$
there is a natural isomorphism of abelian groups
$\operatorname{Hom}_{\operatorname{\mathsf{Pro}}{\mathsf{D}}^{\mathsf{co}}(W{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\operatorname{\mathsf{Pro}}{\mathsf{D}}^{\mathsf{co}}(Z{\operatorname{\mathsf{--qcoh}}})}(h_{!}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
(b) For any objects
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})$
and
${\mathcal{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})$
there is a natural isomorphism of abelian groups
$\operatorname{Hom}_{\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(W{\operatorname{\mathsf{--coh}}})}({\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},h^{*}{\mathcal{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})}(h_{!}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{G}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
###### Proof.
Notice that the assertion of part (a) is _not_ true for the conventional
unbounded derived categories. To prove it for the coderived categories, one
assumes ${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to be a
complex over $Z{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$, so that
$\operatorname{Hom}$ into
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in the
coderived category is isomorphic to the similar $\operatorname{Hom}$ in the
homotopy category of complexes. It is important here that
$h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
complex over $W{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ in these
assumptions. The desired adjunction on the level of (pro)derived categories
then follows from the above adjunction on the level of abelian categories. To
deduce part (b), one can use the fact that the natural functor
${\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Z{\operatorname{\mathsf{--qcoh}}})$
is fully faithful (and similarly for $W$). ∎
Given a proper morphism $f\colon T\longrightarrow Z$ (of finite type) between
Noetherian schemes, there is the direct image functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{b}}(T{\operatorname{\mathsf{--coh}}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})$
[27, Théorème 3.2.1]. Passing to pro-objects, one obtains the induced functor
$\operatorname{\mathsf{pro}}{\mathbb{R}}f_{*}\colon\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(T{\operatorname{\mathsf{--coh}}})\longrightarrow\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(Z{\operatorname{\mathsf{--coh}}})$.
###### Lemma 5.16.3.
Let $V\longrightarrow U$, $Y\longrightarrow X$ be a Cartesian square formed by
proper morphisms of Noetherian schemes $g\colon Y\longrightarrow X$ and
$g^{\prime}\colon V\longrightarrow U$, and open embeddings $h\colon
U\longrightarrow X$ and $h^{\prime}\colon V\longrightarrow Y$ (as above). Then
there is a natural isomorphism of functors
$\operatorname{\mathsf{pro}}{\mathbb{R}}g_{*}\circ h^{\prime}_{!}\simeq
h_{!}\circ\operatorname{\mathsf{pro}}{\mathbb{R}}g^{\prime}_{*}\colon\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(V{\operatorname{\mathsf{--coh}}})\longrightarrow\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--coh}}})$.
###### Proof.
Given an object
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{pro}}{\mathsf{D}}^{b}(V{\operatorname{\mathsf{--coh}}})$,
one has a natural morphism
$\operatorname{\mathsf{pro}}{\mathbb{R}}g^{\prime}_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\allowbreak\longrightarrow\operatorname{\mathsf{pro}}{\mathbb{R}}g^{\prime}_{*}\operatorname{\mathsf{pro}}h^{\prime}{}^{*}\,h^{\prime}_{!}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq\operatorname{\mathsf{pro}}h^{*}\operatorname{\mathsf{pro}}{\mathbb{R}}g_{*}\mskip
1.5muh^{\prime}_{!}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in
$\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(U{\operatorname{\mathsf{--coh}}})$,
inducing a natural morphism
$h_{!}\operatorname{\mathsf{pro}}{\mathbb{R}}g^{\prime}_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathsf{pro}}{\mathbb{R}}g_{*}\mskip
1.5muh^{\prime}_{!}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in
$\operatorname{\mathsf{pro}}{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--coh}}})$
by Lemma 5.16.2(b). We have constructed a morphism of functors
$h_{!}\circ\operatorname{\mathsf{pro}}{\mathbb{R}}g^{\prime}_{*}\longrightarrow\operatorname{\mathsf{pro}}{\mathbb{R}}g_{*}\circ
h^{\prime}_{!}$. In order to check that this morphism is an isomorphism, it
suffices to consider the case of
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}^{\mathsf{b}}(V{\operatorname{\mathsf{--coh}}})$
and show that the morphism in question becomes an isomorphism after applying
the cohomology functors $\operatorname{\mathsf{pro}}H^{i}$ taking values in
$\operatorname{\mathsf{Pro}}(X{\operatorname{\mathsf{--coh}}})$.
Furthermore, one can restrict oneself to the case of a single coherent sheaf
${\mathcal{L}}\in V{\operatorname{\mathsf{--coh}}}$. So we have to show that
the morphisms $h_{!}\mskip
1.5mu{\mathbb{R}}^{i}g^{\prime}_{*}({\mathcal{L}})\longrightarrow\operatorname{\mathsf{Pro}}{\mathbb{R}}^{i}g_{*}(h^{\prime}_{!}{\mathcal{L}})$
are isomorphisms in
$\operatorname{\mathsf{Pro}}(X{\operatorname{\mathsf{--coh}}})$ for any
coherent sheaf ${\mathcal{L}}$ on $V$ and all $i\ge 0$. Finally, one can
assume $X$ to be an affine scheme (as all the constructions of the functors
involved are local in the base, and the property of a morphism in
$\operatorname{\mathsf{Pro}}(X{\operatorname{\mathsf{--coh}}})$ to be an
isomorphism is local in $X$).
Let ${\mathcal{L}}^{\prime}$ be a quasi-coherent sheaf on $Y$ extending
${\mathcal{L}}$. Set $R={\mathcal{O}}(X)$ and
$I={\mathcal{O}}_{X}({\mathcal{I}})$, where ${\mathcal{I}}$ is a sheaf of
ideals in ${\mathcal{O}}_{X}$ corresponding to a closed subscheme structure on
$X\setminus U$. The question reduces to showing that the natural morphism
between the pro-objects represented by the projective systems
$I^{n}H^{i}(Y,{\mathcal{L}}^{\prime})$ and
$H^{i}(Y,I^{n}{\mathcal{L}}^{\prime})$ is an isomorphism in
$\operatorname{\mathsf{Pro}}(R{\operatorname{\mathsf{--mod}}})$. According to
[27, Corollaire 3.3.2], the $(\bigoplus_{n=0}^{\infty}I^{n})$-module
$\bigoplus_{n=0}^{\infty}H^{i}(Y,I^{n}{\mathcal{L}}^{\prime})$ is finitely
generated; the desired assertion is deduced straightforwardly from this
result. ∎
Now we can finish the proof of the theorem. Given a finite complex
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$V{\operatorname{\mathsf{--coh}}}$ and a complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$, one has
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>h^{\prime}{}^{*}g^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\operatorname{\mathsf{Pro}}{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})}(h_{!}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>g^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\operatorname{\mathsf{Pro}}{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})}(\operatorname{\mathsf{pro}}{\mathbb{R}}g_{*}\mskip
1.5muh_{!}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(V{\operatorname{\mathsf{--qcoh}}})}\allowbreak({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>g^{\prime}{}^{!}h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}(U{\operatorname{\mathsf{--qcoh}}})}({\mathbb{R}}g^{\prime}_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>h^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\operatorname{\mathsf{Pro}}{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})}(h_{!}\mskip
1.5mu{\mathbb{R}}g^{\prime}_{*}{\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
by Lemma 5.16.2(a), so it remains to apply Lemma 5.16.3. ∎
The following theorem, which is the main result of this section, essentially
follows from the several previous results. Let $f\colon Y\longrightarrow X$ be
a morphism of finite type between Noetherian schemes. Let
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
dualizing complex on $X$; set
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to be a
finite complex over $Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ quasi-
isomorphic to
$f^{+}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ [29,
Remark before Proposition V.8.5].
###### Theorem 5.16.4.
The equivalences of categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
from Theorem 5.7.1 or 5.8.2 related to the choice of the dualizing complexes
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ and
$Y$ transform Deligne’s extraordinary inverse image functor
$f^{+}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
into the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
(95) left adjoint to the derived functor ${\mathbb{L}}f_{!}$ (92), at least,
in either of the following two situations:
1. (a)
$f$ is a separated morphism of semi-separated Noetherian schemes that can be
factorized into an open embedding followed by a proper morphism;
2. (b)
$f$ is morphism of Noetherian schemes that can be factorized into a finite
morphism followed by a smooth morphism.
###### Proof.
According to Theorem 5.15.1, desired assertion is true for finite morphisms of
Noetherian schemes with dualizing complexes $f\colon Y\longrightarrow X$.
Comparing Theorem 5.12.2(b) with Theorem 5.15.3, one comes to the same
conclusion in the case of a proper morphism $f\colon Y\longrightarrow X$ (of
finite type) between semi-separated Noetherian schemes with dualizing
complexes.
On the other hand, let us consider the case of a smooth morphism $f$. Then it
is essentially known ([29, Corollary VII.4.3], [45, Theorem 5.4], Proposition
5.13.5 above) that the functor
$f^{+}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
only differs from the conventional inverse image functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
by a shift and a twist. Namely, for any complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ one has
$f^{!}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq\omega_{Y/X}[d]\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
where $d$ is the relative dimension and $\omega_{Y/X}$ is the line bundle of
relative top forms on $Y$. In particular, in our present notation one would
have
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq\omega_{Y/X}[n]\otimes_{{\mathcal{O}}_{Y}}f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
(up to a quasi-isomorphism of finite complexes over
$Y{\operatorname{\mathsf{--qcoh}}}$).
By Theorem 5.14.3, the equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
related to the choice of the dualizing complexes
$f^{*}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $Y$ and
$X$ transform the functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
(88) into the functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
(92). Passing to the left adjoint functors, we conclude that the same
equivalences transform the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
(induced by the exact functor $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}\allowbreak\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$, cf. (60)) into the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
(95).
It remains to take the twist and the shift into account in order to deduce the
desired assertion for the smooth morphism $f$. As a particular case, the above
argument also covers the situation when $f$ is an open embedding. ∎
## Appendix A Derived Categories of Exact Categories and Resolutions
In this appendix we recall and review some general results about the derived
categories of the first and the second kind of abstract exact categories and
their full subcategories, in presence of finite or infinite resolutions. There
is nothing essentially new here. Two or three most difficult arguments are
omitted or only briefly sketched with references to the author’s previous
works containing elaborated proofs of similar results in different (but more
concrete) settings given in place of the details.
Note that the present one is still not the full generality for many results
considered here. For most assertions concerning derived categories of the
second kind, the full generality is that of exact DG-categories [51, Section
3.2 and Remark 3.5], which we feel is a bit too abstract to base our
exposition on.
### A.1. Derived categories of the second kind
Let ${\mathsf{E}}$ be an exact category. The homotopy categories of (finite,
bounded above, bounded below, and unbounded) complexes over ${\mathsf{E}}$
will be denoted by $\mathsf{Hot}^{\mathsf{b}}({\mathsf{E}})$,
$\mathsf{Hot}^{-}({\mathsf{E}})$, $\mathsf{Hot}^{+}({\mathsf{E}})$, and
$\mathsf{Hot}({\mathsf{E}})$, respectively. For the definitions of the
conventional derived categories (of the first kind)
${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})$, ${\mathsf{D}}^{-}({\mathsf{E}})$,
${\mathsf{D}}^{+}({\mathsf{E}})$, and ${\mathsf{D}}({\mathsf{E}})$ we refer to
[44, 35] and [52, Section A.7]. Here are the definitions of the derived
categories of the second kind [50, 51, 53].
An (unbounded) complex over ${\mathsf{E}}$ is said to be _absolutely acyclic_
if it belongs to the minimal thick subcategory of $\mathsf{Hot}({\mathsf{E}})$
containing the all the total complexes of short exact sequences of complexes
over ${\mathsf{E}}$. Here a short exact sequence
$0\longrightarrow{}^{\prime}\mskip
1.5mu\\!K^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
K^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime\prime}\\!\mskip
1.5muK^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ of
complexes over ${\mathsf{E}}$ is viewed as a bicomplex with three rows and
totalized as such. The _absolute derived category_
${\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$ of the exact category
${\mathsf{E}}$ is defined as the quotient category of the homotopy category
$\mathsf{Hot}({\mathsf{E}})$ by the thick subcategory of absolutely acyclic
complexes.
Similarly, a bounded above (respectively, below) complex over ${\mathsf{E}}$
is called absolutely acyclic if it belongs to the minimal thick subcategory of
$\mathsf{Hot}^{-}({\mathsf{E}})$ (resp., $\mathsf{Hot}^{+}({\mathsf{E}})$)
containing all the total complexes of short exact sequences of bounded above
(resp., below) complexes over ${\mathsf{E}}$. We will see below that a bounded
above (resp., below) complex over ${\mathsf{E}}$ is absolutely acyclic if and
only if it is absolutely acyclic as an unbounded complex, so there is no
ambiguity in our terminology. The bounded above (resp., below) absolute
derived category of ${\mathsf{E}}$ is defined as the quotient category of
$\mathsf{Hot}^{-}({\mathsf{E}})$ (resp., $\mathsf{Hot}^{+}({\mathsf{E}})$) by
the thick subcategory of absolutely acyclic complexes and denoted by
${\mathsf{D}}^{{\mathsf{abs}}-}({\mathsf{E}})$ (resp.,
${\mathsf{D}}^{{\mathsf{abs}}+}({\mathsf{E}})$).
We do not define the “absolute derived category of finite complexes over
${\mathsf{E}}$”, as it would not be any different from the conventional
bounded derived category ${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})$. Indeed,
any (bounded or unbounded) absolutely acyclic complex is acyclic; and any
finite acyclic complex over an exact category is absolutely acyclic, since it
is composed of short exact sequences. Moreover, any acyclic complex over an
exact category of finite homological dimension is absolutely acyclic [50,
Remark 2.1].
For comparison with the results below, we recall that for any exact category
${\mathsf{E}}$ the natural functors
${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\pm}({\mathsf{E}})\longrightarrow{\mathsf{D}}({\mathsf{E}})$
are all fully faithful [52, Corollary A.11].
###### Lemma A.1.1.
For any exact category ${\mathsf{E}}$, the functors
${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}-}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$
and
${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}+}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$
induced by the natural embeddings of the categories of bounded complexes into
those of unbounded ones are all fully faithful.
###### Proof.
We will show that any morphism in $\mathsf{Hot}({\mathsf{E}})$ (in an
appropriate direction) between a complex bounded in a particular way and a
complex absolutely acyclic with respect to the class of complexes unbounded in
that particular way factorizes through a complex absolutely acyclic with
respect to the class of correspondingly bounded complexes. For this purpose,
it suffices to demonstrate that any absolutely acyclic complex can be
presented as a termwise stabiling filtered inductive (or projective) limit of
complexes absolutely acyclic with respect to the class of complexes bounded
from a particular side.
Indeed, any short exact sequence of complexes over ${\mathsf{E}}$ is the
inductive limit of short exact sequences of their subcomplexes of silly
filtration, which are bounded below. One easily concludes that any absolutely
acyclic complex is a termwise stabilizing inductive limit of complexes
absolutely acyclic with respect to the class of complexes bounded below, and
any absolutely acyclic complex bounded above is a termwise stabilizing
inductive limit of finite acyclic complexes. This proves that the functors
${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}-}({\mathsf{E}})$
and
${\mathsf{D}}^{{\mathsf{abs}}+}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$
are fully faithful.
On the other hand, any absolutely acyclic complex bounded below, being, by
implication, an acyclic complex bounded below, is the inductive limit of its
subcomplexes of canonical filtration, which are finite acyclic complexes. This
shows that the functor
${\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}+}({\mathsf{E}})$
is fully faithful, too. Finally, to prove that the functor
${\mathsf{D}}^{{\mathsf{abs}}-}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$
is fully faithful, one presents any absolutely acyclic complex as a termwise
stabilizing projective limit of complexes absolutely acyclic with respect to
the class of complexes bounded above. ∎
Assume that infinite direct sums exist and are exact functors in the exact
category ${\mathsf{E}}$. Then a complex over ${\mathsf{E}}$ is called
_coacyclic_ if it belongs to the minimal triangulated subcategory of
$\mathsf{Hot}({\mathsf{E}})$ containing the total complexes of short exact
sequences of complexes over ${\mathsf{E}}$ and closed under infinite direct
sums. The _coderived category_ ${\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})$ of
the exact category ${\mathsf{E}}$ is defined as the quotient category of the
homotopy category $\mathsf{Hot}({\mathsf{E}})$ by the thick subcategory of
coacyclic complexes.
Similarly, if the functors of infinite product are everywhere defined and
exact in the exact category ${\mathsf{E}}$, one calls a complex over
${\mathsf{E}}$ _contraacyclic_ if it belongs to the minimal triangulated
subcategory of $\mathsf{Hot}({\mathsf{E}})$ containing the total complexes of
short exact sequences of complexes over ${\mathsf{E}}$ and closed under
infinite products. The _contraderived category_
${\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$ of the exact category
${\mathsf{E}}$ is the quotient category of $\mathsf{Hot}({\mathsf{E}})$ by the
thick subcategory of contraacyclic complexes [50, Sections 2.1 and 4.1].
###### Lemma A.1.2.
(a) For any exact category ${\mathsf{E}}$ with exact functors of infinite
direct sum, the full subcategory of bounded below complexes in
${\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})$ is equivalent to
${\mathsf{D}}^{+}({\mathsf{E}})$.
(b) For any exact category ${\mathsf{E}}$ with exact functors of infinite
product, the full subcategory of bounded above complexes in
${\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$ is equivalent to
${\mathsf{D}}^{-}({\mathsf{E}})$.
###### Proof.
By [50, Lemmas 2.1 and 4.1], any bounded below acyclic complex over
${\mathsf{E}}$ is coacyclic and any bounded above acyclic complex over
${\mathsf{E}}$ is contraacyclic. Hence there are natural triangulated functors
${\mathsf{D}}^{+}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})$
and
${\mathsf{D}}^{-}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$
(in the respective assumptions of parts (a) and (b)). It also follows that the
subcomplexes and quotient complexes of canonical filtration of any
co/contraacyclic complex remain co/contraacyclic. Hence any morphism in
$\mathsf{Hot}({\mathsf{E}})$ from a bounded above complex to a
co/contraacyclic complex factorizes through a bounded above co/contraacyclic
complex, and any morphism from a co/contraacyclic complex to a bounded below
complex factorizes through a bounded below co/contraacyclic complex.
Therefore, our triangulated functors are fully faithful [50, Remark 4.1]. ∎
Denote the full additive subcategory of injective objects in ${\mathsf{E}}$ by
${\mathsf{E}}^{\mathsf{inj}}\subset{\mathsf{E}}$ and the full additive
subcategory of projective objects by
${\mathsf{E}}^{\mathsf{prj}}\subset{\mathsf{E}}$.
###### Lemma A.1.3.
(a) The triangulated functors
$\mathsf{Hot}^{\mathsf{b}}({\mathsf{E}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})$,
$\mathsf{Hot}^{\pm}({\mathsf{E}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}({\mathsf{E}})$,
$\mathsf{Hot}({\mathsf{E}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$,
and
$\mathsf{Hot}({\mathsf{E}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})$
are fully faithful.
(b) The triangulated functors
$\mathsf{Hot}^{\mathsf{b}}({\mathsf{E}}^{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}({\mathsf{E}})$,
$\mathsf{Hot}^{\pm}({\mathsf{E}}^{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}({\mathsf{E}})$,
$\mathsf{Hot}({\mathsf{E}}^{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}({\mathsf{E}})$,
and
$\mathsf{Hot}({\mathsf{E}}^{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$
are fully faithful.
###### Proof.
This is essentially a version of [51, Theorem 3.5] and a particular case of
[51, Remark 3.5]. For any total complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of a short exact sequence
of complexes over ${\mathsf{E}}$ and any complex
$J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathsf{E}}^{\mathsf{inj}}$ the complex of abelian groups
$\operatorname{Hom}_{\mathsf{E}}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},J^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic. Therefore, the same also holds for a complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ that can be obtained from
such total complexes using the operations of cone and infinite direct sum
(irrespectively even of such operations being everywhere defined or exact in
${\mathsf{E}}$). Similarly, for any total complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of a short exact sequence
of complexes over ${\mathsf{E}}$ and any complex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathsf{E}}^{\mathsf{prj}}$ the complex of abelian group is acyclic (hence
the same also holds for any complexes that can be obtained from such total
complexes using the operations of cone and infinite product). This
semiorthogonality implies the assertions of Lemma. ∎
### A.2. Fully faithful functors
Let ${\mathsf{E}}$ be an exact category and ${\mathsf{F}}\subset{\mathsf{E}}$
be a full subcategory closed under extensions and the passage to the kernels
of admissible epimorphisms. We endow ${\mathsf{F}}$ with the induced exact
category structure. Suppose that for any admissible epimorphism
$E\longrightarrow F$ in ${\mathsf{E}}$ from an object $E\in{\mathsf{E}}$ to an
object $F\in{\mathsf{F}}$ there exist an object $G\in{\mathsf{F}}$, an
admissible epimorphism $G\longrightarrow F$ in ${\mathsf{F}}$, and a morphism
$G\longrightarrow E$ in ${\mathsf{E}}$ such that the triangle
$G\longrightarrow E\longrightarrow F$ is commutative (cf. [35, Section 12]).
###### Proposition A.2.1.
For any symbol $\star={\mathsf{b}}$, $-$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\star}({\mathsf{E}})$
induced by the exact embedding functor
${\mathsf{F}}\longrightarrow{\mathsf{E}}$ is fully faithful. When
$\star={\mathsf{ctr}}$, it is presumed here that the functors of infinite
product are everywhere defined and exact in the exact category ${\mathsf{E}}$
and preserve the full subcategory ${\mathsf{F}}\subset{\mathsf{E}}$.
###### Proof.
We use the notation $\mathsf{Hot}^{\star}({\mathsf{E}})$ for the category
$\mathsf{Hot}({\mathsf{E}})$ if $\star=\empt$, ${\mathsf{co}}$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the category
$\mathsf{Hot}^{-}({\mathsf{E}})$ if $\star=-$ or ${\mathsf{abs}}-$, the
category $\mathsf{Hot}^{+}({\mathsf{E}})$ if $\star=+$ or ${\mathsf{abs}}+$,
and the category $\mathsf{Hot}^{\mathsf{b}}({\mathsf{E}})$ if
$\star={\mathsf{b}}$. Let us call an object of
$\mathsf{Hot}^{\star}({\mathsf{E}})$ _$\star$ -acyclic_ if it is annihilated
by the localization functor
$\mathsf{Hot}^{\star}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\star}({\mathsf{E}})$.
In view of Lemma A.1.1, it suffices to consider the cases $\star=-$,
${\mathsf{abs}}$, and ${\mathsf{ctr}}$. In either case, we will show that any
morphism in $\mathsf{Hot}^{\star}({\mathsf{E}})$ from a complex
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathsf{Hot}^{\star}({\mathsf{F}})$
into a $\star$-acyclic complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathsf{Hot}^{\star}({\mathsf{E}})$
factorizes through a complex
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathsf{Hot}^{\star}({\mathsf{F}})$
that is $\star$-acyclic as a complex over ${\mathsf{F}}$.
We start with the case $\star=-$. Assume for simplicity that both complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are concentrated in the
nonpositive cohomological degrees;
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex over
${\mathsf{F}}$ and $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is an
exact complex over ${\mathsf{E}}$. Notice that the morphism
$A^{-1}\longrightarrow A^{0}$ must be an admissible epimorphism in this case.
Set $G^{0}=F^{0}$, and let $B^{-1}\in{\mathsf{E}}$ denote the fibered product
of the objects $F^{0}$ and $A^{-1}$ over $A^{0}$. Then there exists a unique
morphism $A^{-2}\longrightarrow B^{-1}$ having a zero composition with the
morphism $B^{-1}\longrightarrow F^{0}$ and forming a commutative diagram with
the morphisms $A^{-2}\longrightarrow A^{-1}$ and $B^{-1}\longrightarrow
A^{-1}$.
One easily checks that the complex $\dotsb\longrightarrow
A^{-3}\longrightarrow A^{-2}\longrightarrow B^{-1}\longrightarrow
F^{0}\longrightarrow 0$ is exact. Furthemore, there exists a unique morphism
$F^{-1}\longrightarrow B^{-1}$ whose compositions with the morphisms
$B^{-1}\longrightarrow F^{0}$ and $B^{-1}\longrightarrow A^{-1}$ are the
differential $F^{-1}\longrightarrow F^{0}$ and the component
$F^{-1}\longrightarrow A^{-1}$ of the morphism of complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. We have factorized the
latter morphism of complexes through the above exact complex, whose degree-
zero term belongs to ${\mathsf{F}}$.
From this point on we proceed by induction in the homological degree. Suppose
that our morphism of complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has been factorized through
an exact complex $\dotsb\longrightarrow A^{-n-2}\longrightarrow
A^{-n-1}\longrightarrow B^{-n}\longrightarrow
G^{-n+1}\longrightarrow\dotsb\longrightarrow G^{0}\longrightarrow 0$, which
coincides with the complex $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in the degrees $-n-1$ and below, and whose terms belong to ${\mathsf{F}}$ in
the degrees $-n+1$ and above. Since the full subcategory
${\mathsf{F}}\subset{\mathsf{E}}$ is assumed to be closed under extensions,
the image $Z^{-n}$ of the morphism $B^{-n}\longrightarrow G^{-n+1}$ belongs to
${\mathsf{F}}$. Let $G^{-n}\longrightarrow Z^{-n}$ be an admissible
epimorphism in ${\mathsf{F}}$ factorizable through the admissible epimorphism
$B^{-n}\longrightarrow Z^{-n}$ in ${\mathsf{E}}$.
Replacing, if necessary, the object $G^{-n}$ by the direct sum $G^{-n}\oplus
F^{-n}$, we can make the morphism $F^{-n}\longrightarrow B^{-n}$ factorizable
through the morphism $G^{-n}\longrightarrow B^{-n}$. Let $H^{-n-1}$ and
$Y^{-n-1}$ denote the kernels of the morphisms $G^{-n}\longrightarrow Z^{-n}$
and $B^{-n}\longrightarrow Z^{-n}$, respectively; then there is a natural
morphism $H^{-n-1}\longrightarrow Y^{-n-1}$. Denote by $B^{-n-1}$ the fibered
product of the latter morphism with the morphism $A^{-n-1}\longrightarrow
Y^{-n-1}$. There is a unique morphism $A^{-n-2}\longrightarrow B^{-n-1}$
having a zero composition with the morphism $B^{-n-1}\longrightarrow H^{-n-1}$
and forming a commutative diagram with the morphisms $A^{-n-2}\longrightarrow
A^{-n-1}$ and $B^{-n-1}\longrightarrow A^{-n-1}$.
We have constructed an exact complex $\dotsb\longrightarrow
A^{-n-3}\longrightarrow A^{-n-2}\longrightarrow B^{-n-1}\longrightarrow
G^{-n}\longrightarrow G^{-n+1}\longrightarrow\dotsb\longrightarrow
G^{0}\longrightarrow 0$, coinciding with our previous complex in the degrees
$-n+1$ and above and with the complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in the degrees $-n-2$ and
below, and having terms belonging to ${\mathsf{F}}$ in the degrees $-n$ and
above. The morphism from the complex
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ into our previous
intermediate complex factorizes through the new one (since the composition
$F^{-n-1}\longrightarrow F^{-n}\longrightarrow G^{-n}$ factorizes uniquely
through the morphism $H^{-n-1}\longrightarrow G^{-n}$, and then there exists a
unique morphism $F^{-n-1}\longrightarrow B^{-n-1}$ whose compositions with the
morphisms $B^{-n-1}\longrightarrow H^{-n-1}$ and $B^{-n-1}\longrightarrow
A^{-n-1}$ are equal to the required ones). Continuing with this procedure ad
infinitum provides the desired exact complex
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{F}}$.
The proof in the case $\star={\mathsf{abs}}$ is similar to that of [53,
Proposition 1.5], and the case $\star={\mathsf{ctr}}$ is proven along the
lines of [53, Remark 1.5] and [54, Theorem 4.2.1] (cf. the proofs of
Proposition A.6.1 and Theorem B.5.3 below). Not to reiterate here the whole
argument from [53, 54], let us restrict ourselves to a brief sketch.
One has to show that any morphism from a complex over ${\mathsf{F}}$ to a
complex absolutely acyclic (contraacyclic) over ${\mathsf{E}}$ factorizes
through a complex absolutely acyclic (contraacyclic) over ${\mathsf{F}}$ in
the homotopy category $\mathsf{Hot}({\mathsf{E}})$. This is checked by
induction in the transformation rules using which one constructs arbitrary
absolutely acyclic (contraacyclic) complexes over ${\mathsf{E}}$ from the
total complexes of short exact sequences. Finally, the case of a morphism from
a complex over ${\mathsf{F}}$ to the total compex of a short exact sequence
over ${\mathsf{E}}$ is treated using the two lemmas below.
###### Lemma A.2.2.
Let $U^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
V^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
W^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a short exact sequence of
complexes over an exact category ${\mathsf{E}}$ and
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be its total complex. Then
a morphism $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of complexes over
${\mathsf{E}}$ is homotopic to zero whenever its component
$N^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
W^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1]$, which is a morphism of
graded objects in ${\mathsf{E}}$, can be lifted to a morphism of graded
objects $N^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
V^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1]$. ∎
###### Lemma A.2.3.
Let $U^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
V^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
W^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a short exact sequence of
complexes over an exact category ${\mathsf{E}}$ and
$M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be its total complex. Let
$Q$ be a graded object in ${\mathsf{E}}$ and
$\widetilde{Q}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be the
(contractible) complex over ${\mathsf{E}}$ freely generated by $Q$. Then a
morphism of complexes
$\tilde{q}\colon\widetilde{Q}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has its component
$\widetilde{Q}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
W^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1]$ liftable to a morphism
of graded objects
$\widetilde{Q}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
V^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1]$ whenever its restriction
$q\colon Q\longrightarrow M^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to
the graded subobject
$Q\subset\widetilde{Q}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has the
same property. ∎
In order to apply the lemmas, one needs to notice that, in our assumptions on
${\mathsf{E}}$ and ${\mathsf{F}}$, for any admissible epimorphism
$V\longrightarrow W$ in ${\mathsf{E}}$, any object $F\in{\mathsf{F}}$, and any
morphism $F\longrightarrow W$ in ${\mathsf{E}}$ there exist an object
$Q\in{\mathsf{F}}$, an admissible epimorphism $Q\longrightarrow F$ in
${\mathsf{F}}$, and a morphism $Q\longrightarrow V$ in ${\mathsf{E}}$ such
that the square $Q\longrightarrow F$, $V\longrightarrow W$ is commutative.
Otherwise, the argument is no different from the one in [53, 54]. ∎
### A.3. Infinite left resolutions
Let ${\mathsf{E}}$ be an exact category and ${\mathsf{F}}\subset{\mathsf{E}}$
be a full subcategory closed under extensions and the passage to the kernels
of admissible epimorphisms. Suppose further that every object of
${\mathsf{E}}$ is the image of an admissible epimorphism from an object
belonging to ${\mathsf{F}}$. We endow ${\mathsf{F}}$ with the induced
structure of an exact category.
###### Proposition A.3.1.
(a) The triangulated functor
${\mathsf{D}}^{-}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{-}({\mathsf{E}})$
induced by the exact embedding functor
${\mathsf{F}}\longrightarrow{\mathsf{E}}$ is an equivalence of triangulated
categories.
(b) If the infinite products are everywhere defined and exact in the exact
category ${\mathsf{E}}$, and preserve the full subcategory
${\mathsf{F}}\subset{\mathsf{E}}$, then the triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$
induced by the embedding ${\mathsf{F}}\longrightarrow{\mathsf{E}}$ is an
equivalence of categories.
###### Proof.
The proof of part (a) is based on part (b) of the following lemma.
###### Lemma A.3.2.
(a) For any finite complex $E^{-d}\longrightarrow\dotsb\longrightarrow E^{0}$
over ${\mathsf{E}}$ there exists a finite complex
$F^{-d}\longrightarrow\dotsb\longrightarrow F^{0}$ over ${\mathsf{F}}$
together with a morphism of complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{E}}$ such
that the morphisms $F^{i}\longrightarrow E^{i}$ are admissible epimorphisms in
${\mathsf{E}}$ and the cocone (or equivalently, the termwise kernel) of the
morphism $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is quasi-isomorphic to an
object of ${\mathsf{E}}$ placed in the cohomological degree $-d$.
(b) For any bounded above complex $\dotsb\longrightarrow E^{-2}\longrightarrow
E^{-1}\longrightarrow E^{0}$ over ${\mathsf{E}}$ there exists a bounded above
complex $\dotsb\longrightarrow F^{-2}\longrightarrow F^{-1}\longrightarrow
F^{0}$ over ${\mathsf{F}}$ together with a quasi-isomorphism of complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{E}}$ such
that the morphisms $F^{i}\longrightarrow E^{i}$ are admissible epimorphisms in
${\mathsf{E}}$.
###### Proof.
Pick an admissible epimorphism $F^{0}\longrightarrow E^{0}$ with
$F^{0}\in{\mathsf{F}}$ and consider the fibered product $G^{-1}$ of the
objects $E^{-1}$ and $F^{0}$ over $E^{0}$ in ${\mathsf{E}}$. Then there exists
a unique morphism $E^{-2}\longrightarrow G^{-1}$ having a zero composition
with the morphism $G^{-1}\longrightarrow F^{0}$ and forming a commutative
diagram with the morphisms $E^{-2}\longrightarrow E^{-1}$ and
$G^{-1}\longrightarrow E^{-1}$. Continuing the construction, pick an
admissible epimorphism $F^{-1}\longrightarrow G^{-1}$ with
$F^{-1}\in{\mathsf{F}}$, consider the fibered product $G^{-2}$ of $E^{-2}$ and
$F^{-1}$ over $G^{-1}$, etc. In the case (a), proceed in this way until the
object $F^{-d}$ is constructed; in the case (b), proceed indefinitely. The
desired assertions follow from the observation that natural morphism between
the complexes $G^{-d}\longrightarrow
F^{-d+1}\longrightarrow\dotsb\longrightarrow F^{0}$ and $E^{-d}\longrightarrow
E^{-d+1}\longrightarrow\dotsb\longrightarrow E^{0}$ is a quasi-isomorphism for
any $d\ge 1$. ∎
In view of [51, Lemma 1.6], in order to finish the proof of part (a) of
Proposition it remains to show that any bounded above complex over
${\mathsf{F}}$ that is acyclic over ${\mathsf{E}}$ is also acyclic over
${\mathsf{F}}$. This follows immediately from the condition that
${\mathsf{F}}$ is closed with respect to the passage to the kernels of
admissible epimorphisms in ${\mathsf{E}}$.
In the situation of part (b), the functor in question is fully faithful by
Proposition A.2.1. A construction of a morphism with contraacyclic cone onto a
given complex over ${\mathsf{E}}$ from an appropriately chosen complex over
${\mathsf{F}}$ is presented below.
###### Lemma A.3.3.
(a) For any complex $E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathsf{E}}$, there exists a complex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{F}}$
together with a morphism of complexes
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ such that the morphism
$P^{i}\longrightarrow E^{i}$ is an admissible epimorphism for each
$i\in{\mathbb{Z}}$.
(b) For any complex $E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathsf{E}}$, there exists a bicomplex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over ${\mathsf{F}}$ together with a morphism of bicomplexes
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{E}}$ such
that the complexes $P_{j}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
vanish for all $j<0$, while for each $i\in{\mathbb{Z}}$ the complex
$\dotsb\longrightarrow P_{2}^{i}\longrightarrow P_{1}^{i}\longrightarrow
P_{0}^{i}\longrightarrow E^{i}\longrightarrow 0$ is acyclic with respect to
the exact category ${\mathsf{E}}$.
###### Proof.
To prove part (a), pick admissible epimorphisms $F^{i}\longrightarrow E^{i}$
onto all the objects $E^{i}\in{\mathsf{E}}$ from some objects
$F^{i}\in{\mathsf{F}}$. Then the contractible complex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the terms
$P^{i}=F^{i}\oplus F^{i-1}$ (that is the complex freely generated by the
graded object $F^{*}$ over ${\mathsf{F}}$) comes together with a natural
morphism of complexes
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the desired property.
Part (b) is easily deduced from (a) by passing to the termwise kernel of the
morphism of complexes
$P_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and iterating the
construction. ∎
###### Lemma A.3.4.
Let ${\mathsf{A}}$ be an additive category with countable direct products. Let
$\dotsb\longrightarrow
P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(2)\longrightarrow
P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(1)\longrightarrow
P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(0)$
be a projective system of bicomplexes over ${\mathsf{A}}$. Suppose that for
every pair of integers $i$, $j\in{\mathbb{Z}}$ the projective system $\dotsb
P^{ij}(2)\longrightarrow P^{ij}(1)\longrightarrow P^{ij}(0)$ stabilizes, and
let $P^{ij}(\infty)$ denote the corresponding limit. Then the total complex of
the bicomplex
$P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(\infty)$
constructed by taking infinite products along the diagonals is homotopy
equivalent to a complex obtained from the total complexes of the bicomplexes
$P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(n)$
(constructed in the same way) by iterated application of the operations of
shift, cone, and countable product.
###### Proof.
Denote by $T(n)$ and $T(\infty)$ the total complexes of, respectively, the
bicomplexes
$P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(n)$
and
$P^{{\text{\smaller\smaller$\scriptstyle\bullet$}}{\text{\smaller\smaller$\scriptstyle\bullet$}}}(\infty)$.
Then the short sequence of telescope construction
$\textstyle 0\longrightarrow
T(\infty)\longrightarrow\prod_{n}T(n)\longrightarrow\prod_{n}T(n)\longrightarrow
0$
is a termwise split short exact sequence of complexes over ${\mathsf{A}}$.
Indeed, at every term of the complexes the sequences decomposes into a
countable product of sequences corresponding to the projective systems
$P^{ij}(*)$ with fixed indices $i$, $j$ and their limits $P^{ij}(\infty)$. It
remains to notice that the telescope sequence of a stabilizing projective
system is split exact, and a product of split exact sequences is split exact.
∎
Returning to part (b) of Proposition, given a complex
$E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{E}}$, one
applies Lemma A.3.3(b) to obtain a bicomplex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over ${\mathsf{F}}$ mapping onto
$E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Let us show that the cone
of the morphism onto $E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ from
the total complex $T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
constructed by taking infinite products along the diagonals of the bicomplex
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a contraacyclic complex over ${\mathsf{E}}$. For this purpose, augment the
bicomplex
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the complex $E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
represent the resulting bicomplex as the termwise stabilizing projective limit
of its quotient bicomplexes of canonical filtration with respect to the lower
indices. The latter bicomplexes being finite exact sequences of complexes over
${\mathsf{E}}$, the assertion follows from Lemma A.3.4. ∎
### A.4. Homotopy adjusted complexes
The following simple construction (cf. [60]) will be useful for us when
working with the conventional unbounded derived categories in Section 4 (see,
specifically, Section 4.7).
Let ${\mathsf{E}}$ be an exact category. If the functors of infinite direct
sum exist and are exact in ${\mathsf{E}}$, we denote by
${\mathsf{D}}({\mathsf{E}})^{\mathsf{lh}}\subset{\mathsf{D}}({\mathsf{E}})$
the minimal full triangulated subcategory in ${\mathsf{D}}({\mathsf{E}})$
containing the objects of ${\mathsf{E}}$ and closed under infinite direct
sums. Similarly, if the functors of infinite product exist and are exact in
${\mathsf{E}}$, we denote by ${\mathsf{D}}({\mathsf{E}})^{\mathsf{rh}}$ the
minimal full triangulated subcategory of ${\mathsf{D}}({\mathsf{E}})$
containing the objects of ${\mathsf{E}}$ and closed under infinite products.
It is not difficult to show (see the proof of Proposition A.4.3) that, in the
assumptions of the respective definitions,
${\mathsf{D}}^{-}({\mathsf{E}})\subset{\mathsf{D}}({\mathsf{E}})^{\mathsf{lh}}$
and
${\mathsf{D}}^{+}({\mathsf{E}})\subset{\mathsf{D}}({\mathsf{E}})^{\mathsf{rh}}$.
We return temporarily to the assumptions of Section A.2 concerning a pair of
exact categories ${\mathsf{F}}\subset{\mathsf{E}}$.
###### Lemma A.4.1.
Let $B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a bounded above
complex over ${\mathsf{F}}$ and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic complex over
${\mathsf{E}}$. Then any morphism of complexes
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{E}}$
factorizes through a bounded above acyclic complex
$K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{F}}$.
###### Proof.
The canonical truncation of exact complexes over the exact category
${\mathsf{E}}$ allows to assume the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to be bounded above. In
this case, the assertion was established in the proof of Proposition A.2.1.
(For a different argument leading to a slightly weaker conclusion, see [53,
proof of Lemma 2.9].) ∎
###### Corollary A.4.2.
Let $B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a bounded above
complex over ${\mathsf{F}}$ and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex over
${\mathsf{F}}$ that is acyclic as a complex over ${\mathsf{E}}$. Then the
group
$\operatorname{Hom}_{{\mathsf{D}}({\mathsf{F}})}(B^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of morphisms in the derived category ${\mathsf{D}}({\mathsf{F}})$ of the exact
category ${\mathsf{F}}$ vanishes.
###### Proof.
Indeed, any morphism from $B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
to $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
${\mathsf{D}}({\mathsf{F}})$ can be represented as a fraction
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longleftarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, where
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a morphism of complexes over ${\mathsf{F}}$ and
${}^{\prime}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a quasi-isomorphism of
such complexes. Then the complex
${}^{\prime}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is also acyclic
over ${\mathsf{E}}$, and it remains to apply Lemma A.4.1 to the morphism
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
∎
From now on the assumptions of Section A.3 about a full subcategory
${\mathsf{F}}$ in an exact category ${\mathsf{E}}$ are enforced.
###### Proposition A.4.3.
Suppose that the exact category ${\mathsf{E}}$ is actually abelian, and that
infinite direct sums are everywhere defined and exact in the category
${\mathsf{E}}$ and preserve the full exact subcategory
${\mathsf{F}}\subset{\mathsf{E}}$. Then the composition of natural
triangulated functors
${\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}\longrightarrow{\mathsf{D}}({\mathsf{F}})\longrightarrow{\mathsf{D}}({\mathsf{E}})$
is an equivalence of triangulated categories.
###### Proof.
We will show that any complex over ${\mathsf{E}}$ is the target of a quasi-
isomorphism with the source belonging to
${\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}$. By [51, Lemma 1.6], it will
follow, in particular, that ${\mathsf{D}}({\mathsf{E}})$ is isomorphic to the
localization of ${\mathsf{D}}({\mathsf{F}})$ by the thick subcategory of
complexes over ${\mathsf{F}}$ acyclic over ${\mathsf{E}}$. By Corollary A.4.2,
the latter subcategory is semiorthogonal to
${\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}$, so the same constuction of a
quasi-isomorphism with respect to ${\mathsf{E}}$ will also imply that these
two subcategories form a semiorthogonal decomposition of
${\mathsf{D}}({\mathsf{F}})$. This would clearly suffice to prove the desired
assertion.
Let $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex over
${\mathsf{E}}$. Consider all of its subcomplexes of canonical truncation, pick
a termwise surjective quasi-isomorphism onto each of them from a bounded above
complex over ${\mathsf{F}}$, and replace the latter with its finite subcomplex
of silly filtration with, say, only two nonzero terms. Take the direct sum
$B_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of all the obtained
complexes over ${\mathsf{F}}$ and consider the natural morphism of complexes
$B_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. This is a termwise
surjective morphism of complexes which also acts surjectively on all the
objects of coboundaries, cocycles, and cohomology. Next we apply the same
construction to the kernel of this morphism of complexes, etc.
We have constructed an exact complex of complexes $\dotsb\longrightarrow
B_{2}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B_{1}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ which
remains exact after replacing all the complexes
$B_{i}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with their cohomology
objects (taken in the abelian category ${\mathsf{E}}$). All the complexes
$B_{i}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belong to
${\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}$ by the construction. It remains to
show that the totalization of the bicomplex
$B_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
obtained by taking infinite direct sums along the diagonals also belongs to
${\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}$ and maps quasi-isomorphically onto
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
The totalization of the bicomplex
$B_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a direct limit of the totalizations of its subbicomplexes of silly
filtration in the lower indices
$B_{n}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\dotsb\longrightarrow
B_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. By the dual version of
Lemma A.3.4, the former assertion follows. To prove the latter one, it
suffices to apply the following result due to Eilenberg and Moore [15] to the
bicomplex obtained by augmenting
$B_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. ∎
###### Lemma A.4.4.
Let ${\mathsf{A}}$ be an abelian category with exact functors of countable
direct sum, and let
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a bicomplex over ${\mathsf{A}}$ such that the complexes
$D_{j}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ vanish for all $j<0$,
while the complexes $D_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{i}$ are
acyclic for all $i\in{\mathbb{Z}}$, as are the complexes
$H^{i}(D_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Then the total complex of the bicomplex
$D_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
obtained by taking infinite direct sums along the diagonals is acyclic.
###### Proof.
Denote by $S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(\infty)$ the
totalization of the bicomplex
$D_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and by $S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n)$ the totalizations
of its subbicomplexes of silly filtration
$D_{n}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\dotsb\longrightarrow
D_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Consider the telescope
short exact sequence
$\textstyle
0\longrightarrow\bigoplus_{n}S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n)\longrightarrow\bigoplus
S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n)\longrightarrow
S^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$
and pass to the long exact sequence of cohomology associated with this short
exact sequence of complexes. The morphisms
$\bigoplus_{n}H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n))\longrightarrow\bigoplus_{n}H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n))$
in this long exact sequence are the differentials in the two-term complexes
computing the derived functor of inductive limit
$\varinjlim^{*}_{n}H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n))$.
It is clear from the conditions on the bicomplex
$D_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
that the morphisms of cohomology
$H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n-1))\longrightarrow
H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n))$ induced by the
embeddings of complexes
$S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n-1)\longrightarrow
S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n)$ vanish. Hence the
morphisms
$\bigoplus_{n}H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n))\longrightarrow\bigoplus_{n}H^{i}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}}(n))$
are isomorphisms and
$H^{*}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$. ∎
###### Remark A.4.5.
In particular, it follows from Proposition A.4.3 that
${\mathsf{D}}({\mathsf{E}})={\mathsf{D}}({\mathsf{E}})^{\mathsf{lh}}$ for any
abelian category ${\mathsf{E}}$ with exact functors of infinite direct sum. On
the other hand, the following example is instructive. Let
${\mathsf{E}}=R{\operatorname{\mathsf{--mod}}}$ be the abelian category of
left modules over an associative ring $R$ and
${\mathsf{F}}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{prj}}$ be the full
additive subcategory of projective $R$-modules (with the induced trivial exact
category structure). Then we have
${\mathsf{D}}({\mathsf{F}})=\mathsf{Hot}({\mathsf{F}})\neq{\mathsf{D}}({\mathsf{E}})$,
while
${\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}\subneq{\mathsf{D}}({\mathsf{F}})$ is
the full subcategory of homotopy projective complexes in
$\mathsf{Hot}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{prj}})$ [60]. Hence
one can see (by considering
${\mathsf{E}}={\mathsf{F}}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{prj}}$)
that the assertion of Proposition A.4.3 is not generally true when the exact
category ${\mathsf{E}}$ is not abelian.
###### Corollary A.4.6.
In the assumptions of Proposition A.4.3, the fully faithful functor
${\mathsf{D}}({\mathsf{E}})\simeq{\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}\longrightarrow{\mathsf{D}}({\mathsf{F}})$
is left adjoint to the triangulated functor
${\mathsf{D}}({\mathsf{F}})\longrightarrow{\mathsf{D}}({\mathsf{E}})$ induced
by the embedding of exact categories
${\mathsf{F}}\longrightarrow{\mathsf{E}}$.
###### Proof.
Clear from the proof of Proposition A.4.3. ∎
Keeping the assumptions of Proposition A.4.3, assume additionally that the
exact category ${\mathsf{F}}$ has finite homological dimension. Then the
natural functor
${\mathsf{D}}^{\mathsf{co}}({\mathsf{F}})\longrightarrow{\mathsf{D}}({\mathsf{F}})$
is an equivalence of triangulated categories [50, Remark 2.1]. Consider the
composition of triangulated functors
${\mathsf{D}}({\mathsf{E}})\simeq{\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}\longrightarrow{\mathsf{D}}({\mathsf{F}})\simeq{\mathsf{D}}^{\mathsf{co}}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})$.
The following result is a generalization of [53, Lemma 2.9].
###### Corollary A.4.7.
The functor
${\mathsf{D}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})$
so constructed is left adjoint to the Verdier localization functor
${\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})\longrightarrow{\mathsf{D}}({\mathsf{E}})$.
###### Proof.
One has to show that
$\operatorname{Hom}_{{\mathsf{D}}^{\mathsf{co}}({\mathsf{E}})}(B^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$
for any complex
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}({\mathsf{F}})^{\mathsf{lh}}$
and any acyclic complex $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over ${\mathsf{E}}$. This vanishing easily follows from Lemma A.4.1. ∎
Finally, let ${\mathsf{G}}\subset{\mathsf{F}}$ be two full subcategories in an
abelian category ${\mathsf{E}}$, each satisfying the assumptions of Section
A.3 and Proposition A.4.3. Assume that the exact category ${\mathsf{G}}$ has
finite homological dimension. Consider the composition of triangulated
functors
${\mathsf{D}}({\mathsf{E}})\simeq{\mathsf{D}}({\mathsf{G}})^{\mathsf{lh}}\longrightarrow{\mathsf{D}}({\mathsf{G}})\simeq{\mathsf{D}}^{\mathsf{co}}({\mathsf{G}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathsf{F}})$.
The next corollary is a straightforward generalization of the previous one.
###### Corollary A.4.8.
The functor
${\mathsf{D}}({\mathsf{E}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathsf{F}})$
constructed above is left adjoint to the composition of Verdier localization
functors
${\mathsf{D}}^{\mathsf{co}}({\mathsf{F}})\longrightarrow{\mathsf{D}}({\mathsf{F}})\longrightarrow{\mathsf{D}}({\mathsf{E}})$.
∎
### A.5. Finite left resolutions
We keep the assumptions of Section A.3. Assume additionally that the additive
category ${\mathsf{E}}$ is, in the terminology of [44, 52], “semi-saturated”
(i. e., it contains the kernels of its split epimorphisms, or equivalently,
the cokernels of its split monomorphisms). Then the additive category
${\mathsf{F}}$ has the same property. The following results elaborate upon the
ideas of [53, Remark 2.1].
We will say that an object of the derived category
${\mathsf{D}}^{-}({\mathsf{E}})$ has _left ${\mathsf{F}}$-homological
dimension not exceeding $m$_ if its isomorphism class can be represented by a
bounded above complex $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathsf{F}}$ such that $F^{i}=0$ for $i<-m$. By the definition, the full
subcategory of objects of finite left ${\mathsf{F}}$-homological dimension in
${\mathsf{D}}^{-}({\mathsf{E}})$ is the image of the fully faithful
triangulated functor
${\mathsf{D}}^{\mathsf{b}}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{-}({\mathsf{F}})\simeq{\mathsf{D}}^{-}({\mathsf{E}})$.
###### Lemma A.5.1.
If a bounded above complex $G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over the exact subcategory ${\mathsf{F}}$, viewed as an object of the derived
category ${\mathsf{D}}^{-}({\mathsf{E}})$, has left ${\mathsf{F}}$-homological
dimension not exceeding $m$, then the differential $G^{-m-1}\longrightarrow
G^{-m}$ has a cokernel ${}^{\prime}G^{-m}$ in the additive category
${\mathsf{F}}$, and the complex $\dotsb\longrightarrow G^{-m-1}\longrightarrow
G^{-m}\longrightarrow{}^{\prime}G^{-m}\longrightarrow 0$ over ${\mathsf{F}}$
is acyclic. Consequently, the complex
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{F}}$ is
quasi-isomorphic to the finite complex
$0\longrightarrow{}^{\prime}G^{-m}\longrightarrow G^{-m+1}\longrightarrow
G^{-m+2}\longrightarrow\dotsb\longrightarrow 0$.
###### Proof.
In view of the equivalence of categories
${\mathsf{D}}^{-}({\mathsf{F}})\simeq{\mathsf{D}}^{-}({\mathsf{E}})$, the
assertion really depends on the exact subcategory ${\mathsf{F}}$ only. By the
definition of the derived category, two complexes representing isomorphic
objects in it are connected by a pair of quasi-isomorphisms. Thus it suffices
to consider two cases when there is a quasi-isomorphism acting either in the
direction $G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, or
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ (where
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a bounded above complex
over ${\mathsf{F}}$ such that $F^{i}=0$ for $i<-m$).
In the former case, acyclicity of the cone of the morphism
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ implies the existence of
cokernels of its differentials and the acyclicity of canonical truncations,
which provides the desired conclusion.
In the latter case, from acyclicity of the cone of the morphism
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ one can similarly see that
the morphism $G^{-m-2}\longrightarrow G^{-m-1}\oplus F^{-m}$ with the
vanishing component $G^{-m-2}\longrightarrow F^{-m}$ has a cokernel, and it
follows that the morphism $G^{-m-2}\longrightarrow G^{-m-1}$ also does.
Denoting the cokernel of the latter morphism by ${}^{\prime}G^{-m-1}$, one
easily concludes that the complex $\dotsb\longrightarrow
G^{-m-2}\longrightarrow
G^{-m-1}\longrightarrow{}^{\prime}G^{-m-1}\longrightarrow 0$ is acyclic, and
it remains to show that the morphism ${}^{\prime}G^{-m-1}\longrightarrow
G^{-m}$ is an admissible monomorphism in the exact category ${\mathsf{F}}$.
Indeed, the morphism ${}^{\prime}G^{-m-1}\oplus F^{-m}\longrightarrow
G^{-m}\oplus F^{-m+1}$ is; and hence so is its composition with the embedding
of a direct summand
${}^{\prime}G^{-m-1}\longrightarrow{}^{\prime}G^{-m-1}\oplus F^{-m}$. ∎
###### Corollary A.5.2.
Let $G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a finite complex
over the exact category ${\mathsf{E}}$ such that $G^{i}=0$ for $i<-m$ and
$G^{i}\in{\mathsf{F}}$ for $i>-m$. Assume that the object represented by
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
${\mathsf{D}}^{-}({\mathsf{E}})$ has left ${\mathsf{F}}$-homological dimension
not exceeding $m$. Then the object $G^{-m}$ also belongs to ${\mathsf{F}}$.
###### Proof.
Replace the object $G^{-m}$ with its left resolution by objects from
${\mathsf{F}}$ and apply Lemma A.5.1. ∎
We say that an object $E\in{\mathsf{E}}$ has left ${\mathsf{F}}$-homological
dimension not exceeding $m$ if the corresponding object of the derived
category ${\mathsf{D}}^{-}({\mathsf{E}})$ does. In other words, $E$ must have
a left resolution by objects of ${\mathsf{F}}$ of the length not exceeding
$m$. Let us denote the left ${\mathsf{F}}$-homological dimension of an object
$E$ by $\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E$.
###### Corollary A.5.3.
Let ${\mathsf{E}}^{\prime}\subset{\mathsf{E}}$ be a (strictly) full semi-
saturated additive subcategory with an induced exact category structure. Set
${\mathsf{F}}^{\prime}={\mathsf{E}}^{\prime}\cap{\mathsf{F}}$, and assume that
every object of ${\mathsf{E}}^{\prime}$ is the image of an admissible
epimorphism in the exact category ${\mathsf{E}}^{\prime}$ acting from an
object belonging to ${\mathsf{F}}^{\prime}$. Then for any object
$E\in{\mathsf{E}}^{\prime}$ one has
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E=\operatorname{ld}_{{\mathsf{F}}^{\prime}/{\mathsf{E}}^{\prime}}E$.
###### Proof.
Follows from Corollary A.5.2. ∎
###### Lemma A.5.4.
Let $0\longrightarrow E^{\prime}\longrightarrow E\longrightarrow
E^{\prime\prime}\longrightarrow 0$ be an exact triple in ${\mathsf{E}}$. Then
(a) if $\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{\prime}\le m$ and
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{\prime\prime}\le m$, then
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E\le m$;
(b) if $\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E\le m$ and
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{\prime\prime}\le m+1$, then
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{\prime}\le m$;
(c) if $\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E\le m$ and
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{\prime}\le m-1$, then
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{\prime\prime}\le m$.
###### Proof.
Let us prove part (a); the proofs of parts (b) and (c) are similar. The
morphism $E^{\prime\prime}[-1]\longrightarrow E^{\prime}$ in
${\mathsf{D}}^{-}({\mathsf{E}})\simeq{\mathsf{D}}^{-}({\mathsf{F}})$ can be
represented by a morphism of complexes ${}^{\prime\prime}\\!\mskip
1.5muF^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}\\!\mskip
1.5muF^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
$\mathsf{Hot}^{-}({\mathsf{F}})$. By Lemma A.5.1, both complexes can be
replaced by their canonical truncations at the degree $-m$. Obviously, there
is the induced morphism between the complexes truncated in this way, so we can
simply assume that ${}^{\prime\prime}\\!\mskip
1.5muF^{i}=0={}^{\prime}\\!\mskip 1.5muF^{i}$ for $i<-m$. Moreover, the
complex ${}^{\prime\prime}\\!\mskip
1.5muF^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ could be truncated even
one step further, i. e., the morphism ${}^{\prime\prime}\\!\mskip
1.5muF^{-m}\longrightarrow{}^{\prime\prime}\\!\mskip 1.5muF^{-m+1}$ is an
admissible monomorphism in the exact category ${\mathsf{F}}$. From this one
easily concludes that for the cone
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the morphism of
complexes ${}^{\prime\prime}\\!\mskip
1.5muF^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}\\!\mskip
1.5muF^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ one has $G^{i}=0$ for
$i<-m-1$ and the morphism $G^{-m-1}\longrightarrow G^{-m}$ is an admissible
monomorphism in ${\mathsf{F}}$. ∎
###### Corollary A.5.5.
(a) Let $0\longrightarrow E_{n}\longrightarrow\dotsb\longrightarrow
E_{0}\longrightarrow E\longrightarrow 0$ be an exact sequence in
${\mathsf{E}}$. Then the left ${\mathsf{F}}$-homological dimension
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E$ does not exceed the supremum
of the expressions $\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E_{i}+i$ over
$0\le i\le n$ (where we set
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}0=-1$).
(b) Let $0\longrightarrow E\longrightarrow
E^{0}\longrightarrow\dotsb\longrightarrow E^{n}\longrightarrow 0$ be an exact
sequence in ${\mathsf{E}}$. Then the left ${\mathsf{F}}$-homological dimension
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E$ does not exceed the supremum
of the expressions $\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E^{i}-i$ over
$0\le i\le n$.
###### Proof.
Part (a) follows by induction from Lemma A.5.4(c), and part (b) similarly
follows from Lemma A.5.4(b). ∎
###### Proposition A.5.6.
Suppose that the left ${\mathsf{F}}$-homological dimension of all objects
$E\in{\mathsf{E}}$ does not exceed a fixed constant $d$. Then the triangulated
functor
${\mathsf{D}}^{\star}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\star}({\mathsf{E}})$
induced by the exact embedding functor
${\mathsf{F}}\longrightarrow{\mathsf{E}}$ is an equivalence of triangulated
categories for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$,
${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{co}}$, ${\mathsf{ctr}}$, or
${\mathsf{abs}}$.
When $\star={\mathsf{co}}$ (respectively, $\star={\mathsf{ctr}}$), it is
presumed here that the functors of infinite direct sum (resp., infinite
product) are everywhere defined and exact in the category ${\mathsf{E}}$ and
preserve the full subcategory ${\mathsf{F}}\subset{\mathsf{E}}$.
###### Proof.
The cases $\star=-$ or ${\mathsf{ctr}}$ were considered in Proposition A.3.1
(and hold in its weaker assumptions). They can be also treated together with
the other cases, as it is explained below.
In the cases $\star={\mathsf{b}}$, $+$, $-$, or $\empt$ one can argue as
follows. Using the construction of Lemma A.3.3 and taking into account
Corollary A.5.2, one produces for any $\star$-bounded complex
$E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{E}}$ its
finite left resolution
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of length $d$ (in the lower indices) by $\star$-bounded (in the upper indices)
complexes over ${\mathsf{F}}$. The total complex of
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
maps by a quasi-isomophism (in fact, a morphism with an absolutely acyclic
cone) over the exact category ${\mathsf{E}}$ onto the complex
$E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
By [51, Lemma 1.6], it remains to show that any complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over ${\mathsf{F}}$ that is
acyclic as a complex over ${\mathsf{E}}$ is also acyclic as a complex over
${\mathsf{F}}$. For this purpose, we apply the same construction of the
resolution
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
to the complex $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The complex
$P_{0}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic over
${\mathsf{F}}$ and maps by a termwise admissible epimorphism onto the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$; it follows that the
induced morphisms of the objects of cocycles are admissible epimorphisms, too.
The passage to the cocycle objects of acyclic complexes also commutes with the
passage to the kernels of termwise admissible epimorphisms.
We conclude that the cocycle objects of the acyclic complexes
$P_{i}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ form resolutions of the
cocycle objects of the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Since the left
${\mathsf{F}}$-homological dimension of objects of ${\mathsf{E}}$ does not
exceed $d$ and for $i<d$ the cocycle objects of
$P_{i}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belong to
${\mathsf{F}}$, so do the cocycle objects of the complex
$P_{d}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Now the total complex
of
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic over ${\mathsf{F}}$ and maps onto
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with a cone acyclic over
${\mathsf{F}}$.
The rather involved argument in the cases $\star={\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$ is
similar to that in [53, Theorem 1.4] and goes back to the proof of [50,
Theorem 7.2.2]. We do not reiterate the details here. ∎
### A.6. Finite homological dimension
First we return to the assumptions of Section A.2. The following result is a
partial generalization of Lemma A.1.3.
###### Proposition A.6.1.
Assume that infinite products are everywhere defined and exact in the exact
category ${\mathsf{E}}$. Suppose also that the exact category ${\mathsf{F}}$
has finite homological dimension. Then the triangulated functor
${\mathsf{D}}^{\mathsf{abs}}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$
induced by the exact embedding functor
${\mathsf{F}}\longrightarrow{\mathsf{E}}$ is fully faithful.
###### Proof.
A combination of the assertions of Proposition A.2.1 and [50, Remark 2.1]
implies the assertion of Proposition in the case when ${\mathsf{F}}$ is
preserved by the infinite products in ${\mathsf{E}}$. The proof in the general
case consists in a combination of the arguments proving the two mentioned
results.
We will show that any morphism in $\mathsf{Hot}({\mathsf{E}})$ from a complex
over ${\mathsf{F}}$ to a complex contraacyclic over ${\mathsf{E}}$ factorizes
through a complex absolutely acyclic over ${\mathsf{F}}$. For this purpose, it
suffices to check that the class of complexes over ${\mathsf{E}}$ having this
property contains the total complexes of exact triples of complexes over
${\mathsf{E}}$ and is closed with respect to cones and infinite products.
The former two assersions are essentially proven in [53, Section 1.5] (see
also Lemmas A.2.2–A.2.3 above). To prove the latter one, suppose that we are
given a morphism of complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\prod_{\alpha}A_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in $\mathsf{Hot}({\mathsf{E}})$, where
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex over
${\mathsf{F}}$. Suppose further that each component morphism
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ factorizes through
a complex $G_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ that is
absolutely acyclic over ${\mathsf{F}}$. Clearly, the morphism
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\prod_{\alpha}A_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
factorizes through the complex
$\prod_{\alpha}G_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
It follows from the proof in [50, Remark 2.1] (for a more generally applicable
argument, see also [53, Section 1.6]) that there exists an integer $n$ such
that every complex absolutely acyclic over ${\mathsf{F}}$ can be obtained from
totalizations of exact triples of complexes over ${\mathsf{F}}$ by applying
the operation of the passage to a cone at most $n$ times. Therefore, the
product
$\prod_{\alpha}G_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
an absolutely acyclic complex over ${\mathsf{E}}$. Hence it follows from what
we already know that the morphism
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\prod_{\alpha}G_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
factorizes through a complex absolutely acyclic over ${\mathsf{F}}$. ∎
Now the assumptions of Section A.5 are enforced. The assumption of the
following corollary is essentially a generalization of the conditions
(${*}$)–(${*}{*}$) from [51, Sections 3.7–3.8].
###### Corollary A.6.2.
In the situation of Proposition A.6.1, assume additionally that countable
products of objects from ${\mathsf{F}}$ taken in the category ${\mathsf{E}}$
have finite left ${\mathsf{F}}$-homological dimensions. Then the functor
${\mathsf{D}}^{\mathsf{abs}}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$
is an equivalence of triangulated categories.
###### Proof.
Clearly, it suffices to find for any complex over ${\mathsf{E}}$ a morphism
into it from a complex over ${\mathsf{F}}$ with a cone contraacyclic over
${\mathsf{E}}$. The following two-step construction procedure goes back to
[51].
Given a complex $E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathsf{E}}$, we proceed as in the above proof of Proposition A.3.1(b),
applying Lemma A.3.3(b) in order to obtain a bicomplex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over ${\mathsf{F}}$. The total complex
$T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the bicomplex
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
constructed by taking infinite products along the diagonals maps onto
$E^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with a cone contraacyclic
with respect to ${\mathsf{E}}$.
By assumption, the left ${\mathsf{F}}$-homological dimensions of the terms of
the complex $T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are finite, and
in fact bounded by a fixed constant $d$. Applying Lemma A.3.3(b) again
together with Corollary A.5.2, we produce a finite complex of complexes
$Q_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of length $d$ (in the lower indices) over ${\mathsf{F}}$ mapping termwise
quasi-isomorphically onto $T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Now the total complex of
$Q_{\text{\smaller\smaller$\scriptstyle\bullet$}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
maps onto $T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with a cone
absolutely acyclic over ${\mathsf{E}}$. ∎
The following generalization of the last result is straightforward. Let
${\mathsf{G}}\subset{\mathsf{F}}$ be two full subcategories of an exact
category ${\mathsf{E}}$, each satisfying the assumptions of Section A.3.
Suppose that the pair of subcategories ${\mathsf{F}}\subset{\mathsf{E}}$
satisfies the assumptions of Proposition A.6.1, while countable products of
objects from ${\mathsf{G}}$ taken in the category ${\mathsf{E}}$ have finite
left ${\mathsf{F}}$-homological dimension. Then the functor
${\mathsf{D}}^{\mathsf{abs}}({\mathsf{F}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathsf{E}})$
is an equivalence of triangulated categories.
## Appendix B Co-Contra Correspondence over a Flat Coring
The aim of this appendix is to extend the assertion of Theorem 5.7.1 from
semi-separated Noetherian schemes to semi-separated Noetherian stacks. These
are the stacks that can be represented by groupoids with affine schemes of
vertices and arrows (see [2, Section 2.1]). We avoid explicit use of the stack
language by working with what Kontsevich and Rosenberg call “finite covers”
[36] instead. This naturally includes the noncommutative geometry situation.
The corresponding algebraic language is that of flat corings over
noncommutative rings; thus this appendix provides a bridge between the results
of Section 5 and those of [50, Chapter 5].
### B.1. Contramodules over a flat coring
Let $A$ be an associative ring with unit. We refer to the memoir and
monographs [16, 10, 50] for the definitions of a (coassociative) _coring_
(with counit) ${\mathcal{C}}$ over $A$, a _left comodule_ ${\mathcal{M}}$ over
${\mathcal{C}}$, and a _right comodule_ ${\mathcal{N}}$ over ${\mathcal{C}}$.
The definition of a _left contramodule_ ${\mathfrak{P}}$ over ${\mathcal{C}}$
can be found in [16, Section III.5] or [50, Section 0.2.4 or 3.1.1]. We denote
the abelian groups of morphisms in the additive category of left
${\mathcal{C}}$-comodules by
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},{\mathcal{M}})$ and the
similar groups related to the additive category of left
${\mathcal{C}}$-contramodules by
$\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},{\mathfrak{Q}})$.
The category ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ of left
${\mathcal{C}}$-comodules is abelian and the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact if and only if ${\mathcal{C}}$ is a
flat right $A$-module [50, Section 1.1.2]. The category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ of left
${\mathcal{C}}$-contramodules is abelian and the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact if and only if ${\mathcal{C}}$ is a
projective left $A$-module [50, Section 3.1.2]. The following counterexample
shows that the category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ may
be not abelian even though ${\mathcal{C}}$ is a flat left and right
$A$-module.
###### Example B.1.1.
Let us consider corings ${\mathcal{C}}$ of the following form. The coring
${\mathcal{C}}$ decomposes into a direct sum of $A$-$A$-bimodules
${\mathcal{C}}={\mathcal{C}}_{11}\oplus{\mathcal{C}}_{12}\oplus{\mathcal{C}}_{22}$;
the counit map ${\mathcal{C}}\longrightarrow A$ annihilates
${\mathcal{C}}_{12}$ and the comultiplication map takes ${\mathcal{C}}_{ik}$
into the direct sum
$\bigoplus_{j}{\mathcal{C}}_{ij}\otimes_{A}{\mathcal{C}}_{jk}$. Assume further
that the restrictions of the counit map to ${\mathcal{C}}_{11}$ and
${\mathcal{C}}_{22}$ are both isomorphisms ${\mathcal{C}}_{ii}\simeq A$.
Notice that the data of an $A$-$A$-bimodule ${\mathcal{C}}_{12}$ determines
the coring ${\mathcal{C}}$ in this case. A left ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ is the same thing as a pair of left $A$-modules
${\mathfrak{P}}_{1}$ and ${\mathfrak{P}}_{2}$ endowed with an $A$-module
morphism
$\operatorname{Hom}_{A}({\mathcal{C}}_{12},{\mathfrak{P}}_{1})\longrightarrow{\mathfrak{P}}_{2}$.
The kernel of a morphism of left ${\mathcal{C}}$-contramodules
$(f_{1},f_{2})\colon({\mathfrak{P}}_{1},{\mathfrak{P}}_{2})\longrightarrow({\mathfrak{Q}}_{1},{\mathfrak{Q}}_{2})$
(taken in the additive category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$) is the
${\mathcal{C}}$-contramodule $(\ker f_{1},\>\ker f_{2})$. The cokernel of the
morphism $(f_{1},f_{2})$ can be computed as the ${\mathcal{C}}$-contramodule
$(\operatorname{coker}f_{1},\>{\mathfrak{L}})$, where ${\mathfrak{L}}$ is the
cokernel of the morphism from ${\mathfrak{P}}_{2}$ to the fibered coproduct
$\operatorname{Hom}_{A}({\mathcal{C}}_{12},\>\operatorname{coker}f_{1})\sqcup_{\operatorname{Hom}_{A}({\mathcal{C}}_{12},\>{\mathfrak{Q}}_{1})}{\mathfrak{Q}}_{2}$
(where $\ker f$ and $\operatorname{coker}f$ denote the kernel and cokernel of
a morphism of $A$-modules $f$).
Now setting $A$ be the ring of integers ${\mathbb{Z}}$ and
${\mathcal{C}}_{12}$ to be the (bi)module of rational numbers ${\mathbb{Q}}$
over ${\mathbb{Z}}$, one easily checks that the category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ is not abelian. It suffices
to consider ${\mathfrak{P}}_{1}=\bigoplus_{n\in\mathbb{N}}{\mathbb{Z}}$,
${\mathfrak{P}}_{2}=0$, and
${\mathfrak{Q}}_{1}={\mathfrak{Q}}_{2}={\mathbb{Q}}$, the structure morphism
$\operatorname{Hom}_{\mathbb{Z}}({\mathbb{Q}},{\mathfrak{Q}}_{1})\longrightarrow{\mathfrak{Q}}_{2}$
being the identity isomorphism. The morphism
$f_{1}\colon{\mathfrak{P}}_{1}\longrightarrow{\mathfrak{Q}}_{1}$ is chosen to
be surjective, while of course $f_{2}=0$. Then the kernel of $f$ is the
${\mathcal{C}}$-contramodule $(\ker f_{1},\>0)$ and the cokernel of the kernel
of $f$ is $({\mathbb{Q}},0)$, while the cokernel of $f$ is $(0,0)$ and kernel
of the cokernel of $f$ is $({\mathbb{Q}},{\mathbb{Q}})$.
Similarly one can construct a nonflat coring ${\mathcal{C}}$ for which the
category ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ is not abelian. For
a coring
${\mathcal{C}}={\mathcal{C}}_{11}\oplus{\mathcal{C}}_{12}\oplus{\mathcal{C}}_{22}$
with ${\mathcal{C}}_{ii}\simeq A$ as above, a left ${\mathcal{C}}$-comodule is
the same thing as a pair of left $A$-modules ${\mathcal{M}}_{1}$ and
${\mathcal{M}}_{2}$ endowed with an $A$-module morphism
${\mathcal{M}}_{1}\longrightarrow{\mathcal{C}}_{12}\otimes_{A}{\mathcal{M}}_{2}$.
The cokernel and the kernel of an arbitrary morphism of left
${\mathcal{C}}$-comodules can be computed in the way dual-analogous to the
above computation for ${\mathcal{C}}$-contramodules.
Setting $A={\mathbb{Z}}$ and ${\mathcal{C}}_{12}={\mathbb{Z}}/n$ with any
$n\ge 2$, it suffices to consider the morphism of left
${\mathcal{C}}$-comodules
$(g_{1},g_{2})\colon({\mathcal{L}}_{1},{\mathcal{L}}_{2})\longrightarrow({\mathcal{M}}_{1},{\mathcal{M}}_{2})$
with ${\mathcal{L}}_{1}={\mathcal{L}}_{2}={\mathbb{Z}}/n$, the structure
morphism
${\mathcal{L}}_{1}\longrightarrow{\mathcal{L}}_{2}/n{\mathcal{L}}_{2}$ being
the identity map, ${\mathcal{M}}_{1}=0$,
${\mathcal{M}}_{2}={\mathbb{Z}}/n^{2}$, and an injective map
$g_{2}\colon{\mathcal{L}}_{2}\longrightarrow{\mathcal{M}}_{2}$. Then the
cokernel of $g$ is the ${\mathcal{C}}$-comodule $(0,{\mathbb{Z}}/n)$ and the
kernel of the cokernel of $g$ is $(0,{\mathbb{Z}}/n)$, while the kernel of $g$
is $(0,0)$ and the cokernel of the kernel of $g$ is
$({\mathbb{Z}}/n,{\mathbb{Z}}/n)$.
It is clear from the above example that the category of all left contramodules
over a flat coring ${\mathcal{C}}$ does not have good homological properties
in general—at least, unless one restricts the class of exact sequences under
consideration to those whose exactness is preserves by the functor
$\operatorname{Hom}_{A}({\mathcal{C}},{-})$. Our preference is to restrict the
class of contramodules instead (or rather, at the same time). So we will be
interested in the category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
of left ${\mathcal{C}}$-contramodules whose underlying left $A$-modules are
cotorsion modules (see Section 1.3).
Assuming that ${\mathcal{C}}$ is a flat left $A$-module, this category has a
natural exact category structure where a short sequence of contramodules is
exact if and only if its underlying short sequence of $A$-modules is exact in
the abelian category $A{\operatorname{\mathsf{--mod}}}$. We denote the
$\operatorname{Ext}$ groups computed in this exact category by
$\operatorname{Ext}^{{\mathcal{C}},*}({\mathfrak{P}},{\mathfrak{Q}})$.
Assuming that ${\mathcal{C}}$ is a flat right $A$-module, so the category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ is abelian, we denote the
$\operatorname{Ext}$ groups computed in this category by
$\operatorname{Ext}_{\mathcal{C}}^{*}({\mathcal{L}},{\mathcal{M}})$.
Given a left $A$-module $U$, the left ${\mathcal{C}}$-comodule
${\mathcal{C}}\otimes_{A}U$ is said to be _coinduced_ from $U$. For any left
${\mathcal{C}}$-comodule ${\mathcal{L}}$, there is a natural isomorphism
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},\>{\mathcal{C}}\otimes_{A}\nobreak
U)\simeq\operatorname{Hom}_{A}({\mathcal{L}},U)$. Given a left $A$-module $V$,
the left ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{A}({\mathcal{C}},V)$ is said to be _induced_ from $V$.
For any left ${\mathcal{C}}$-contramodule ${\mathfrak{Q}}$, there is a natural
isomorphism
$\operatorname{Hom}^{\mathcal{C}}(\operatorname{Hom}_{A}({\mathcal{C}},V),{\mathfrak{Q}})\simeq\operatorname{Hom}_{A}(V,{\mathfrak{Q}})$
[50, Sections 1.1.2 and 3.1.2]. By Lemma 1.3.3(a), the induced
${\mathcal{C}}$-contramodule $\operatorname{Hom}_{A}({\mathcal{C}},V)$ is
$A$-cotorsion whenever the coring ${\mathcal{C}}$ is a flat left $A$-module
and the $A$-module $V$ is cotorsion.
Notice that, assuming ${\mathcal{C}}$ to be a flat right $A$-module, the
direct summands of ${\mathcal{C}}$-comodules coinduced from injective left
$A$-modules are the injective objects of the abelian category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$, and there are enough of
them. Similarly, assuming ${\mathcal{C}}$ to be a flat left $A$-module, the
direct summands of ${\mathcal{C}}$-contramodules induced from flat cotorsion
left $A$-modules are the projective objects of the exact category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$,
and there are enough of them (as one can see using Theorem 1.3.1(b)).
Recall that the _contratensor product_
${\mathcal{N}}\odot_{\mathcal{C}}{\mathfrak{P}}$ of a right
${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is an abelian group constructed
as the cokernel of the natural pair of maps
${\mathcal{N}}\otimes_{A}\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\birarrow{\mathcal{N}}\otimes_{A}{\mathfrak{P}}$,
one of which is induced by the contraaction map
$\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$,
while the other one is the composition of the maps induced by the coaction map
${\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}$ and the
evaluation map
${\mathcal{C}}\otimes_{A}\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$.
For any right ${\mathcal{C}}$-comodule ${\mathcal{N}}$ and any left $A$-module
$V$, there is a natural isomorphism
${\mathcal{N}}\odot_{\mathcal{C}}\operatorname{Hom}_{A}({\mathcal{C}},V)\simeq{\mathcal{N}}\otimes_{A}V$
[50, Sections 0.2.6 and 5.1.1–2].
Given two corings ${\mathcal{C}}$ and ${\mathcal{E}}$ over associative rings
$A$ and $B$, and a ${\mathcal{C}}$-${\mathcal{E}}$-bicomodule ${\mathcal{K}}$,
the rules
${\mathfrak{P}}\longmapsto{\mathcal{K}}\odot_{\mathcal{E}}{\mathfrak{P}}$ and
${\mathcal{M}}\longmapsto\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{M}})$
define a pair of adjoint functors between the categories of left
${\mathcal{E}}$-contramodules and left ${\mathcal{C}}$-comodules. In the
particular case of ${\mathcal{C}}={\mathcal{D}}={\mathcal{K}}$, the
corresponding functors are denoted by
$\Phi_{\mathcal{C}}\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and
$\Psi_{\mathcal{C}}\colon{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$,
so
$\Phi_{\mathcal{C}}({\mathfrak{P}})={\mathcal{C}}\odot_{\mathcal{C}}{\mathfrak{P}}$
and
$\Psi_{\mathcal{C}}({\mathcal{M}})=\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}},{\mathcal{M}})$.
The functors $\Phi_{\mathcal{C}}$ and $\Psi_{\mathcal{C}}$ transform the
induced left ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{A}({\mathcal{C}},U)$ into the coinduced left
${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}U$ and back, inducing an
equivalence between the full subcategories of comodules and contramodules of
this form in ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ [7, 50].
### B.2. Base rings of finite weak dimension
Let ${\mathcal{C}}$ be a coring over an associative ring $A$. In this section
we assume that ${\mathcal{C}}$ is a flat left and right $A$-module and,
additionally, that $A$ is a ring of finite weak dimension (i. e., the functor
$\operatorname{Tor}^{A}({-},{-})$ has finite homological dimension). Notice
that the injective dimension of any cotorsion $A$-module is finite in this
case.
The content of this section is very close to that of [50, Chapter 5, and,
partly, Section 9.1], the main difference being that the left $A$-module
projectivity assumption on the coring ${\mathcal{C}}$ used in [50] is weakened
here to the flatness assumption. That is why the duality-analogy between
comodules and contramodules is more obscure here than in _loc. cit_. Also, the
form of the presentation below may be more in line with the main body of this
paper than with [50].
###### Lemma B.2.1.
(a) Any ${\mathcal{C}}$-comodule can be presented as the quotient comodule of
an $A$-flat ${\mathcal{C}}$-comodule by a finitely iterated extension of
${\mathcal{C}}$-comodules coinduced from cotorsion $A$-modules.
(b) Any $A$-cotorsion ${\mathcal{C}}$-contramodule has an admissible
monomorphism into an $A$-injective ${\mathcal{C}}$-contramodule such that the
cokernel is a finitely iterated extension of ${\mathcal{C}}$-contramodules
induced from cotorsion $A$-modules.
###### Proof.
Part (a) is proven by the argument of [50, Lemma 1.1.3] used together with the
result of Theorem 1.3.1(b). Part (b) is similar to [50, Lemma 3.1.3(b)]. ∎
A left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ is said to be _cotorsion_ if
the functor $\operatorname{Hom}_{\mathcal{C}}({-},{\mathcal{M}})$ takes short
exact sequences of $A$-flat ${\mathcal{C}}$-comodules to short exact sequences
of abelian groups. In particular, any ${\mathcal{C}}$-comodule coinduced from
a cotorsion $A$-module is cotorsion.
An $A$-cotorsion left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is said to
be _projective relative to $A$_ (_${\mathcal{C}}/A$ -projective_) if the
functor $\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},{-})$ takes short
exact sequences of $A$-injective ${\mathcal{C}}$-contramodules to short exact
sequences of abelian groups. In particular, any ${\mathcal{C}}$-contramodule
induced from a cotorsion $A$-module is ${\mathcal{C}}/A$-projective.
###### Corollary B.2.2.
(a) A left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ is cotorsion if and only
if
$\operatorname{Ext}_{\mathcal{C}}^{>0}({\mathcal{L}},{\mathcal{M}})\allowbreak=0$
for any $A$-flat left ${\mathcal{C}}$-comodule ${\mathcal{L}}$. In particular,
the functor $\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},{-})$ takes short
exact sequences of cotorsion left ${\mathcal{C}}$-comodules to short exact
sequences of abelian groups. The class of cotorsion ${\mathcal{C}}$-comodules
is closed under extensions and the passage to cokernels of injective
morphisms.
(b) A left $A$-cotorsion ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is
${\mathcal{C}}/A$-projective if and only if
$\operatorname{Ext}^{{\mathcal{C}},>0}({\mathfrak{P}},{\mathfrak{Q}})=0$ for
any $A$-injective left ${\mathcal{C}}$-contramodule ${\mathfrak{Q}}$. In
particular, the functor $\operatorname{Hom}^{\mathcal{C}}({-},{\mathfrak{Q}})$
takes short exact sequences of ${\mathcal{C}}/A$-projective $A$-cotorsion
${\mathcal{C}}$-contramodules to short exact sequences of abelian groups. The
class of ${\mathcal{C}}/A$-projective $A$-cotorsion
${\mathcal{C}}$-contramodules is closed under extensions and the passage to
kernels of admissible epimorphisms in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$.
###### Proof.
Follows from there being enough $A$-flat ${\mathcal{C}}$-comodules in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and $A$-injective
${\mathcal{C}}$-contramodules in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$,
i. e., weak forms of the assertions of Lemma B.2.1 (cf. [50, Lemma 5.3.1]). ∎
It follows, in particular, that the full subcategories of cotorsion
${\mathcal{C}}$-comodules and ${\mathcal{C}}/A$-projective $A$-cotorsion
${\mathcal{C}}$-contramodules in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
can be endowed with the induced exact category structures. We denote these
exact categories by
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}^{\mathsf{cot}}$ and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}_{{\mathcal{C}}/A\operatorname{\mathsf{--pr}}}$,
respectively.
###### Lemma B.2.3.
(a) Any ${\mathcal{C}}$-comodule admits an injective morphism into a finitely
iterated extension of ${\mathcal{C}}$-comodules coinduced from cotorsion
$A$-modules such that the quotient ${\mathcal{C}}$-comodule is $A$-flat.
(b) For any $A$-cotorsion ${\mathcal{C}}$-contramodule there exists an
admissible epimorphism onto it from a finitely iterated extension of
${\mathcal{C}}$-contramodules induced from cotorsion $A$-modules such that the
kernel is an $A$-injective ${\mathcal{C}}$-contramodule.
###### Proof.
The assertions follow from Lemma B.2.1 together with the existence of enough
comodules coinduced from cotorsion modules in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and enough contramodules
induced from cotorsion modules in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
by virtue of the argument from the second half of the proof of [17, Theorem
10] (cf. Lemmas 1.1.3, 4.1.3, 4.1.10, 4.2.4, 4.3.3, 5.2.7, etc.) An
alternative argument (providing somewhat weaker assertions) can be found in
[50, Lemma 9.1.2]. ∎
###### Corollary B.2.4.
(a) A ${\mathcal{C}}$-comodule is cotorsion if and only if it is a direct
summand of a finitely iterated extension of ${\mathcal{C}}$-comodules
coinduced from cotorsion $A$-modules.
(b) An $A$-cotorsion ${\mathcal{C}}$-contramodule is
${\mathcal{C}}/A$-projective if and only if it is a direct summand of a
finitely iterated extension of ${\mathcal{C}}$-contramodules induced from
cotorsion $A$-modules.
###### Proof.
Follows from Lemma B.2.3 and Corollary B.2.2. ∎
###### Corollary B.2.5.
(a) A ${\mathcal{C}}$-comodule is simultaneously cotorsion and $A$-flat if and
only if it is a direct summand of a ${\mathcal{C}}$-comodule coinduced from a
flat cotorsion $A$-module.
(b) An $A$-cotorsion ${\mathcal{C}}$-contramodule is simultaneously
${\mathcal{C}}/A$-projective and $A$-injective if and only if it is a direct
summand of a ${\mathcal{C}}$-contramodule induced from an injective
$A$-module.
###### Proof.
Both the “if” assertions are obvious. To prove the “only if” in part (a),
consider an $A$-flat cotorsion ${\mathcal{C}}$-comodule ${\mathcal{M}}$. Using
Theorem 1.3.1(a), pick an injective morphism ${\mathcal{M}}\longrightarrow P$
from ${\mathcal{M}}$ into a cotorsion $A$-module $P$ such that the cokernel
$P/{\mathcal{M}}$ is a flat $A$-module. Clearly, $P$ is a flat cotorsion
$A$-module. The cokernel of the composition of ${\mathcal{C}}$-comodule
morphisms
${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}P$,
being an extension of two $A$-flat ${\mathcal{C}}$-comodules, is also
$A$-flat. According to Corollary B.2.2(a), it follows that the
${\mathcal{C}}$-comodule ${\mathcal{M}}$ is a direct summand of
${\mathcal{C}}\otimes_{A}P$.
To prove the “only if” in part (b), consider a ${\mathcal{C}}/A$-projective
$A$-injective ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$. The natural
morphism of ${\mathcal{C}}$-contramodules
$\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
is surjective with an $A$-injective kernel. It remains to apply Corollary
B.2.2(b) in order to conclude that the extension splits. ∎
###### Theorem B.2.6.
(a) For any cotorsion left ${\mathcal{C}}$-comodule ${\mathcal{M}}$, the left
${\mathcal{C}}$-contramodule $\Psi_{\mathcal{C}}({\mathcal{M}})$ is a
${\mathcal{C}}/A$-projective $A$-cotorsion ${\mathcal{C}}$-contramodule.
(b) For any ${\mathcal{C}}/A$-projective $A$-cotorsion
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, the left
${\mathcal{C}}$-comodule $\Phi_{\mathcal{C}}({\mathfrak{P}})$ is cotorsion.
(c) The functors $\Psi_{\mathcal{C}}$ and $\Phi_{\mathcal{C}}$ restrict to
mutually inverse equivalences between the exact subcategories
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}^{\mathsf{cot}}\subset{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}_{{\mathcal{C}}/A\operatorname{\mathsf{--pr}}}\subset{\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$.
###### Proof.
The functor $\Psi_{\mathcal{C}}$ takes cotorsion left
${\mathcal{C}}$-comodules ${\mathcal{M}}$ to $A$-cotorsion left
${\mathcal{C}}$-contramodules, since the functor
$\operatorname{Hom}_{A}(F,\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}},{\mathcal{M}}))\simeq\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}}\otimes_{A}F,\>{\mathcal{M}})$
is exact on the exact category of flat left $A$-modules $F$. Furthermore, the
functor
$\Psi_{\mathcal{C}}\colon{\mathcal{M}}\longmapsto\operatorname{Hom}_{\mathcal{C}}({\mathcal{C}},{\mathcal{M}})$
takes short exact sequences of cotorsion ${\mathcal{C}}$-comodules to short
exact sequences of $A$-cotorsion ${\mathcal{C}}$-contramodules, since
${\mathcal{C}}$ is a flat left $A$-module.
Similarly, the functor $\Phi_{\mathcal{C}}$ takes short exact sequences of
${\mathcal{C}}/A$-projective $A$-cotorsion ${\mathcal{C}}$-contramodules to
short exact sequences of ${\mathcal{C}}$-comodules, since
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{C}}\odot_{\mathcal{C}}{\mathfrak{P}},\>\allowbreak{\mathbb{Q}}/{\mathbb{Z}})\simeq\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},\operatorname{Hom}_{\mathbb{Z}}({\mathcal{C}},{\mathbb{Q}}/{\mathbb{Z}}))$
and the left ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{C}},{\mathbb{Q}}/{\mathbb{Z}})$ is
$A$-injective (${\mathcal{C}}$ being a flat right $A$-module).
In view of these observations, all the assertions follow from Corollary B.2.4.
Alternatively, one could proceed along the lines of the proof of [50, Theorem
5.3], using the facts that any cotorsion ${\mathcal{C}}$-comodule has finite
injective dimension in ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and
any ${\mathcal{C}}/A$-projective $A$-cotorsion ${\mathcal{C}}$-contramodule
has finite projective dimension in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
(see the proof of the next Theorem B.2.7). ∎
###### Theorem B.2.7.
(a) Assume additionally that countable direct sums of injective left
$A$-modules have finite injective dimensions. Then the triangulated functor
${\mathsf{D}}^{\mathsf{abs}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}}^{\mathsf{cot}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
induced by the embedding of exact categories
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}^{\mathsf{cot}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
is an equivalence of triangulated categories.
(b) The triangulated functor
${\mathsf{D}}^{\mathsf{abs}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}_{{\mathcal{C}}/A\operatorname{\mathsf{--pr}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
induced by the embdding of exact categories
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}_{{\mathcal{C}}/A\operatorname{\mathsf{--pr}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
is an equivalence of triangulated categories.
###### Proof.
To prove part (a), we notice that the injective dimension of any cotorsion
${\mathcal{C}}$-comodule ${\mathcal{M}}$ (as an object of
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$) does not exceed the weak
homological dimension of the ring $A$. Indeed, one can compute the functor
$\operatorname{Ext}_{\mathcal{C}}({-},{\mathcal{M}})$ using $A$-flat left
resolutions of the first argument (which exist by Lemma B.2.1(a)). Given the
description of injective ${\mathcal{C}}$-comodules in Section B.1, it follows
from the assumptions of part (a) that countable direct sums of cotorsion
${\mathcal{C}}$-comodules have finite injective dimensions, too. It remains to
apply the dual version of Corollary A.6.2.
Similarly, to prove part (b) one first notices that the projective dimension
of any ${\mathcal{C}}/A$-projective $A$-cotorsion ${\mathcal{C}}$-contramodule
does not exceed the supremum of the injective dimensions of cotorsion
$A$-modules, i. e., the weak homological dimension of the ring $A$.
Furthermore, it is clear from the description of projective $A$-cotorsion
${\mathcal{C}}$-contramodules in Section B.1 that infinite products of
projective objects have finite projective dimensions in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$.
Hence Corollary A.6.2 (or the subsequent remark) applies. Alternatively, one
could argue in the way similar to the proof of [50, Theorem 5.4] (cf. Theorem
B.3.1 below). ∎
###### Corollary B.2.8.
The coderived category of left ${\mathcal{C}}$-comodules
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
and the contraderived category of $A$-cotorsion left
${\mathcal{C}}$-contramodules
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
are naturally equivalent. The equivalence is provided by the derived functors
of co-contra correspondence ${\mathbb{R}}\Psi_{\mathcal{C}}$ and
${\mathbb{L}}\Phi_{\mathcal{C}}$ constructed in terms of cotorsion right
resolutions of comodules and relatively projective left resolutions of
contramodules.
###### Proof.
Follows from Theorems B.2.6 and B.2.7. ∎
### B.3. Gorenstein base rings
Let ${\mathcal{C}}$ be a coring over an associative ring $A$. We suppose the
ring $A$ to be left Gorenstein, in the sense that the classes of left
$A$-modules of finite injective dimension and of finite flat dimension
coincide. In this case, it is clear that both kinds of dimensions are
uniformly bounded by a constant for those modules for which they are finite.
More generally, it will be sufficient to require that the classes of cotorsion
left $A$-modules of finite injective dimension and of finite flat dimension
coincide, countable direct sums of injective left $A$-modules have finite
injective dimensions, and flat cotorsion $A$-modules have uniformly bounded
injective dimensions. Then it follows that the flat and injective dimensions
of cotorsion $A$-modules of finite flat/injective dimension are also uniformly
bounded.
###### Theorem B.3.1.
(a) Assume that the coring ${\mathcal{C}}$ is a flat right $A$-module. Then
the coderived category
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
of the abelian category of left ${\mathcal{C}}$-comodules is equivalent to the
quotient category of the homotopy category of the additive category of left
${\mathcal{C}}$-comodules coinduced from cotorsion $A$-modules of finite
flat/injective dimension by its minimal thick subcategory containing the total
complexes of short exact sequences of complexes of ${\mathcal{C}}$-comodules
that at every term of the complexes are short exact sequences of
${\mathcal{C}}$-comodules coinduced from short exact sequences of cotorsion
$A$-modules of finite flat/injective dimension.
(b) Assume that the coring ${\mathcal{C}}$ is a flat left $A$-module. Then the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
of the exact category of $A$-cotorsion left ${\mathcal{C}}$-contramodules is
equivalent to the quotient category of the homotopy category of the additive
category of left ${\mathcal{C}}$-contramodules induced from cotorsion
$A$-modules of finite flat/injective dimension by its minimal thick
subcategory cointaining the total complexes of short exact sequences of
complexes of ${\mathcal{C}}$-contramodules that at every term of the complexes
are short exact sequences of ${\mathcal{C}}$-contramodules induced from short
exact sequences of cotorsion $A$-modules of finite flat/injective dimension.
###### Proof.
The argument proceeds along the lines of the proof of [50, Theorem 5.5] (see
also [50, Question 5.4] and [51, Sections 3.9–3.10]).
Part (a): the relative cobar-resolution, totalized by taking infinite direct
sums along the diagonals, provides a closed morphism with a coacyclic cone
from any complex of left ${\mathcal{C}}$-comodules into a complex of coinduced
${\mathcal{C}}$-comodules. The construction from the proof of [50, Theorem
5.5] provides a closed morphism from any complex of coinduced left
${\mathcal{C}}$-comodules into a complex of ${\mathcal{C}}$-comodules termwise
coinduced from injective $A$-modules such that this morphism is coinduced from
an injective morphism of $A$-modules at every term of the complexes.
This allows to obtain a closed morphism with a coacyclic cone from any complex
of coinduced left ${\mathcal{C}}$-comodules into a complex of
${\mathcal{C}}$-comodules termwise coinduced from injective $A$-modules (using
the assumption that countable direct sums of injective $A$-modules have finite
injective dimensions). The same construction from [50] can be also used to
obtain a closed morphism from any complex of left ${\mathcal{C}}$-comodules
with the terms coinduced from cotorsion $A$-modules of finite injective
dimension into a complex of ${\mathcal{C}}$-comodules termwise coinduced from
injective $A$-modules such that the cone is homotopy equivalent to a complex
obtained from short exact sequences of complexes of ${\mathcal{C}}$-comodules
termwise coinduced from short exact sequences of cotorsion $A$-modules of
finite injective dimension using the operation of cone repeatedly.
The assertion of part (a) follows from these observations by a semi-orthogonal
decomposition argument from the proofs of [50, Theorems 5.4–5.5].
Part (b): the contramodule relative bar-resolution, totalized by taking
infinite products along the diagonals, provides a closed morphism with a
contraacyclic cone onto any complex of $A$-cotorsion left
${\mathcal{C}}$-contramodules from a complex of ${\mathcal{C}}$-contramodules
termwise induced from cotorsion $A$-modules. The construction dual to the one
elaborated in the proof of [50, Theorem 5.5] provides a closed morphism onto
any complex of ${\mathcal{C}}$-contramodules termwise induced from cotorsion
$A$-modules from a complex of ${\mathcal{C}}$-contramodules termwise induced
from flat cotorsion $A$-modules such that this morphism is induced from an
admissible epimorphism of cotorsion $A$-modules at every term of the
complexes.
This allows to obtain a closed morphism with a contraacyclic cone onto any
complex of left ${\mathcal{C}}$-contramodules termwise induced from cotorsion
$A$-modules from a complex of ${\mathcal{C}}$-contramodules termwise induced
from flat cotorsion $A$-modules (since an infinite product of flat cotorsion
$A$-modules, being a cotorsion $A$-module of finite injective dimension, has
finite flat dimension). The same construction can be also used to obtain a
closed morphism onto any complex of left ${\mathcal{C}}$-contramodules with
the terms induced from cotorsion $A$-modules of finite flat dimension from a
complex of ${\mathcal{C}}$-contramodules termwise induced from flat cotorsion
$A$-modules such that the cone is homotopy equivalent to a complex obtained
from short exact sequences of complexes of ${\mathcal{C}}$-contramodules
termwise induced from short exact sequences of cotorsion $A$-modules of finite
flat dimension using the operation of cone repeatedly.
Now the dual version of the same semi-orthogonal decomposition argument from
[50] implies part (b). ∎
###### Corollary B.3.2.
Assume that the coring ${\mathcal{C}}$ is a flat left and right $A$-module.
Then the coderived category of left ${\mathcal{C}}$-comodules
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
and the contraderived category of $A$-cotorsion left
${\mathcal{C}}$-contramodules
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
are naturally equivalent. The equivalence is provided by the derived functors
of co-contra correspondence ${\mathbb{R}}\Psi_{\mathcal{C}}$ and
${\mathbb{R}}\Phi_{\mathcal{C}}$ constructed in terms of right resolutions by
complexes of ${\mathcal{C}}$-comodules termwise coinduced from cotorsion
$A$-modules of finite injective/flat dimension and left resolutions by
complexes of ${\mathcal{C}}$-contramodules termwise induced from cotorsion
$A$-modules of finite flat/injective dimension.
###### Proof.
Follows from Theorem B.3.1 and the remarks in Section B.1. ∎
### B.4. Corings with dualizing complexes
Let $A$ and $B$ be associative rings. We call a finite complex of
$A$-$B$-bimodules $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ a
_dualizing complex_ for $A$ and $B$ [12] if
1. (i)
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is simultaneously a complex
of injective left $A$-modules and a complex of injective right $B$-modules (in
the one-sided module structures obtained by forgetting the other module
structure on the other side);
2. (ii)
as a complex of left $A$-modules,
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is quasi-isomorphic to a
bounded above complex of finitely generated projective $A$-modules, and
similarly, as a complex of right $B$-modules,
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is quasi-isomorphic to a
bounded above complex of finitely generated projective $B$-modules;
3. (iii)
the “homothety” maps
$A\longrightarrow\operatorname{Hom}_{B^{\mathrm{op}}}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and
$B\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
are quasi-isomorphisms of complexes.
###### Lemma B.4.1.
Let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a dualizing complex
for associative rings $A$ and $B$. Then
(a) if the ring $A$ is left Noetherian and $F$ is a flat left $B$-module, then
the natural homomorphism of finite complexes of left $B$-modules
$F\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)$
is a quasi-isomorphism;
(b) if the ring $B$ is right coherent and $J$ is an injective left $A$-module,
then the natural homomorphism of finite complexes of left $A$-modules
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)\longrightarrow
J$ is a quasi-isomorphism.
###### Proof.
Part (a): first of all,
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F$ is a complex of
injective left $A$-modules by Lemma 1.6.1(a). Let ${}^{\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a quasi-isomorphism of
complexes of left $A$-modules between a bounded above complex of finitely
generated projective $A$-modules ${}^{\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the complex
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Then it suffices to show
that the induced morphism of complexes of abelian groups
$F\longrightarrow\operatorname{Hom}_{A}({}^{\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)$
is a quasi-isomorphism. Now the complex
$\operatorname{Hom}_{A}({}^{\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)$
is isomorphic to $\operatorname{Hom}_{A}({}^{\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}})\otimes_{B}F$,
and it remains to use the condition (iii) for the morphism
$B\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
together with the flatness condition on the $B$-module $F$.
Part (b): the complex
$\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)$
is a complex of flat left $B$-modules by Lemma 1.6.1(b). Let
${}^{\prime\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a quasi-isomorphism of
complexes of right $B$-modules between a bounded above complex of finitely
generated projective $B$-modules ${}^{\prime\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the complex
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. It suffices to show that
the induced morphims of complexes of abelian groups
${}^{\prime\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)\longrightarrow
J$ is a quasi-isomorphism. The complex ${}^{\prime\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)$
being isomorphic to
$\operatorname{Hom}_{A}(\operatorname{Hom}_{B^{\mathrm{op}}}({}^{\prime\prime}\\!\mskip
1.5muD^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}}),J)$,
it remains to use the condition (iii) for the morphism
$A\longrightarrow\operatorname{Hom}_{B^{\mathrm{op}}}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
together with the injectivity condition on the $A$-module $J$. ∎
The following result is due to Christensen, Frankild, and Holm [12,
Proposition 1.5].
###### Corollary B.4.2.
Let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a dualizing complex
for associative rings $A$ and $B$. Assume that the ring $A$ is left
Noetherian. Then the projective dimension of any flat left $B$-module does not
exceed the length of $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
###### Proof.
Assume that the complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
concentrated in the cohomological degrees from $i$ to $i+d$. It suffices to
show that $\operatorname{Ext}^{d+1}_{B}(F,G)=0$ for any flat left $B$-modules
$F$ and $G$. Let $P_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
projective left resolution of the $B$-module $F$. By Lemma B.4.1(a), the
natural map of complexes of abelian groups
$\operatorname{Hom}_{B}(P_{\text{\smaller\smaller$\scriptstyle\bullet$}},G)\longrightarrow\operatorname{Hom}_{B}(P_{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}G))$
is a quasi-isomorphism. The right-hand side is isomorphic to the complex
$\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}P_{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}G)$,
which is quasi-isomorphic to
$\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F,\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}G)$,
since $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}G$ is a
finite complex of injective $A$-modules. The corollary is proven. Notice that
we have only used “a half of” the conditions (i-iii) imposed on a dualizing
complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. ∎
Let ${\mathcal{C}}$ be a coring over an associative ring $A$ and
${\mathcal{E}}$ be a coring over an associative ring $B$. We assume
${\mathcal{C}}$ to be a flat right $A$-module and ${\mathcal{E}}$ to be a flat
left $B$-module. A _dualizing complex_ for ${\mathcal{C}}$ and ${\mathcal{E}}$
is defined as a triple consisting of a finite complex of
${\mathcal{C}}$-${\mathcal{E}}$-bicomodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, a finite
complex of $A$-$B$-bimodules
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and a morphism of
complexes of $A$-$B$-bimodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the following
properties:
1. (iv)
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing complex for
the rings $A$ and $B$;
2. (v)
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of
injective left ${\mathcal{C}}$-comodules (forgetting the right
${\mathcal{E}}$-comodule structure) and a complex of injective right
${\mathcal{E}}$-comodules (forgetting the left ${\mathcal{C}}$-comodule
structure);
3. (vi)
the morphism of complexes of left ${\mathcal{C}}$-comodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{C}}\otimes_{A}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
induced by the morphism of complexes of left $A$-modules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a quasi-isomorphism;
4. (vii)
the morphism of complexes of right ${\mathcal{E}}$-comodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}{\mathcal{E}}$
induced by the morphism of complexes of right $B$-modules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a quasi-isomorphism.
###### Lemma B.4.3.
(a) Suppose that the ring $A$ is left Noetherian. Then for any
${\mathcal{C}}$-injective ${\mathcal{C}}$-${\mathcal{E}}$-bicomodule
${\mathcal{K}}$ and any left ${\mathcal{E}}$-contramodule ${\mathfrak{F}}$
which is a projective object of
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$,
the left ${\mathcal{C}}$-comodule
${\mathcal{K}}\odot_{\mathcal{E}}{\mathfrak{F}}$ is injective.
(b) Suppose that the ring $B$ is right coherent. Then for any
${\mathcal{E}}$-injective ${\mathcal{C}}$-${\mathcal{E}}$-bicomodule
${\mathcal{K}}$ and any injective left ${\mathcal{C}}$-comodule
${\mathcal{J}}$, the left ${\mathcal{E}}$-contramodule
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{J}})$ is a
projective object of
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
###### Proof.
Part (a): one can assume the left ${\mathcal{E}}$-contramodule
${\mathfrak{F}}$ to be induced from a flat (cotorsion) $B$-module $F$; then
${\mathcal{K}}\odot_{\mathcal{E}}{\mathfrak{F}}\simeq{\mathcal{K}}\otimes_{B}F$.
Hence it suffices to check, e. g., that the class of injective left
${\mathcal{C}}$-comodules is preserved by filtered inductive limits. Let us
show that ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ is a locally
Noetherian Grothendieck abelian category.
The following is a standard argument (cf. [10]). Let ${\mathcal{L}}$ be a left
${\mathcal{C}}$-comodule. For any finitely generated $A$-submodule
$U\subset{\mathcal{L}}$, the full preimage ${\mathcal{L}}_{U}$ of
${\mathcal{C}}\otimes_{A}U\subset{\mathcal{C}}\otimes_{A}{\mathcal{L}}$ with
respect to the ${\mathcal{C}}$-coaction map
${\mathcal{L}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{L}}$ is a
${\mathcal{C}}$-subcomodule in ${\mathcal{L}}$ contained in $U$. Since the
left ${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}{\mathcal{L}}$ is a
filtered inductive limit of its ${\mathcal{C}}$-subcomodules
${\mathcal{C}}\otimes_{A}U$, it follows that the ${\mathcal{C}}$-comodule
${\mathcal{L}}$ is a filtered inductive limit of its
${\mathcal{C}}$-subcomodules ${\mathcal{L}}_{U}$. Given that $A$ is a left
Noetherian ring, we can conclude that any left ${\mathcal{C}}$-comodule is the
union of its $A$-finitely generated ${\mathcal{C}}$-subcomodules.
Part (b): one can assume the left ${\mathcal{C}}$-comodule ${\mathcal{J}}$ to
be coinduced from a left $A$-module $J$, and the $A$-module $J$ to have the
form $\operatorname{Hom}_{\mathbb{Z}}(A,I)$ for a certain injective abelian
group $X$. Then we have
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{K}},{\mathcal{J}})\simeq\operatorname{Hom}_{A}({\mathcal{K}},J)\simeq\operatorname{Hom}_{\mathbb{Z}}({\mathcal{K}},X)$.
The ${\mathcal{E}}$-contramodule
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{K}},X)$ only depends on the right
${\mathcal{E}}$-comodule structure on ${\mathcal{K}}$, so one can assume
${\mathcal{K}}$ to be the right ${\mathcal{E}}$-comodule coinduced from an
injective right $B$-module $I$. Now
$\operatorname{Hom}_{\mathbb{Z}}(I\otimes_{B}{\mathcal{E}},\>X)\simeq\operatorname{Hom}_{B}({\mathcal{E}},\operatorname{Hom}_{\mathbb{Z}}(I,X))$
is the left ${\mathcal{E}}$-contramodule induced from the left $B$-module
$\operatorname{Hom}_{\mathbb{Z}}(I,X)$. The latter is a flat cotorsion
$B$-module by Lemmas 1.3.3(b) and 1.6.1(b). ∎
###### Lemma B.4.4.
Let
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a dualizing complex for
corings ${\mathcal{C}}$ and ${\mathcal{E}}$. Then
(a) assuming that the ring $A$ is left Noetherian, for any left
${\mathcal{E}}$-contramodule ${\mathfrak{F}}$ that is a projective object of
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
the adjunction morphism
${\mathfrak{F}}\longrightarrow\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}{\mathfrak{F}})$
is a quasi-isomorphism of finite complexes over the exact category
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$;
(b) assuming that the ring $B$ is right coherent, for any injective left
${\mathcal{C}}$-comodule ${\mathcal{J}}$ the adjunction morphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\longrightarrow{\mathcal{J}}$
is a quasi-isomorphism of finite complexes over the abelian category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$.
###### Proof.
Part (a): one can assume the ${\mathcal{E}}$-contramodule ${\mathfrak{F}}$ to
be induced from a flat (cotorsion) left $B$-module $F$; then
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}{\mathfrak{F}}\simeq{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F$.
It follows from (the proof of) Lemma B.4.3(a) that the morphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F\longrightarrow{\mathcal{C}}\otimes_{A}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F$
induced by the quasi-isomorphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{C}}\otimes_{A}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
from (vi) is a homotopy equivalence of complexes of injective left
${\mathcal{C}}$-comodules. Hence we have a quasi-isomorphism
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)\longrightarrow\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)$.
Furthermore, by (vii) there is a quasi-isomorphism
$\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}{\mathcal{E}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)\longrightarrow\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)$
and a natural isomorphism
$\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}{\mathcal{E}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F)\simeq\operatorname{Hom}_{B}({\mathcal{E}},\>\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F))$.
Finally, the natural morphism
$\operatorname{Hom}_{B}({\mathcal{E}},F)\longrightarrow\operatorname{Hom}_{B}({\mathcal{E}},\>\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}F))$
is a quasi-isomorphism by Lemmas 1.3.3(b) and B.4.1(a).
Part (b): one can assume the ${\mathcal{C}}$-comodule ${\mathcal{J}}$ to be
induced from an injective left $A$-module $J$; then
$\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\simeq\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)$.
It follows from (the proof of) Lemma B.4.3(b) that the morphism
$\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}{\mathcal{E}},\>J)\longrightarrow\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>J)$
induced by the quasi-isomorphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}{\mathcal{E}}$ from
(vii) is a homotopy equivalence of complexes of projective objects in
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
Hence we have a quasi-isomorphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)\longleftarrow{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}{\mathcal{E}},\>J)\simeq{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}\operatorname{Hom}_{B}({\mathcal{E}},\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J))\simeq{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)$.
Furthermore, by (vi) and Lemma 1.6.1(b) there is a quasi-isomorphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)\longrightarrow{\mathcal{C}}\otimes_{A}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},J)$.
Now it remains to apply Lemma B.4.1(b). ∎
The following theorem does not depend on the existence of any dualizing
complexes.
###### Theorem B.4.5.
(a) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; asssume that
${\mathcal{C}}$ is a flat right $A$-module and $A$ is a left Noetherian ring.
Then the coderived category
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
of the abelian category of left ${\mathcal{C}}$-comodules is equivalent to the
homotopy category of complexes of injective left ${\mathcal{C}}$-comodules.
(b) Let ${\mathcal{E}}$ be a coring over an associative ring $B$; assume that
${\mathcal{E}}$ is a flat left $B$-module and $B$ is a right coherent ring.
Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
of the exact category of $B$-cotorsion left ${\mathcal{E}}$-contramodules is
equivalent to the homotopy category of complexes of projective objects in
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
###### Proof.
Part (a) holds, since there are enough injective objects in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and the class of injectives
is closed under infinite direct sums (the class of injective left $A$-modules
being closed under infinite direct sums). Part (b) is true, because there are
enough projective objects in
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
and the class of projectives is preserved by infinite products (the class of
flat left $B$-modules being preserved by infinite products). See Proposition
A.3.1(b) or [51, Sections 3.7 and 3.8] for further details. ∎
More generally, the assertion of part (a) holds if ${\mathcal{C}}$ is a flat
right $A$-module and countable direct sums of injective left $A$-modules have
finite injective dimensions. Similarly, the assertion of (b) is true if
${\mathcal{E}}$ is a flat left $B$-module and countable products of flat
cotorsion left $B$-modules have finite flat dimensions (see Corollary A.6.2).
###### Corollary B.4.6.
Let ${\mathcal{C}}$ be a coring over an associative ring $A$ and
${\mathcal{E}}$ be a coring over an associative ring $B$. Assume that
${\mathcal{C}}$ is a flat right $A$-module, ${\mathcal{E}}$ is a flat left
$B$-module, the ring $A$ is left Noetherian, and the ring $B$ is right
coherent. Then the data of a dualizing complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the corings
${\mathcal{C}}$ and ${\mathcal{E}}$ induces an equivalence of triangulated
categories
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}^{\mathbb{L}}\nobreak{-}$.
###### Proof.
Follows from Lemmas B.4.3–B.4.4 and Theorem B.4.5. ∎
Now let us suppose that a coring ${\mathcal{E}}$ over an associative ring $B$
is a projective left $B$-module. Then the category
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}$ is abelian with enough
projectives; the latter are the direct summands of the left
${\mathcal{E}}$-contramodules induced from projective left $B$-modules.
###### Lemma B.4.7.
Assume that any left $B$-module has finite cotorsion dimension (or
equivalently, any flat left $B$-module has finite projective dimension). Then
(a) any left ${\mathcal{E}}$-contramodule can be embedded into a $B$-cotorsion
left ${\mathcal{E}}$-contramodule in such a way that the cokernel is a
finitely iterated extension of ${\mathcal{E}}$-contramodules induced from flat
left $B$-modules;
(b) any left ${\mathcal{E}}$-contramodule can be presented as the quotient
contramodule of a finitely iterated extension of ${\mathcal{E}}$-contramodules
induced from flat left $B$-modules by a $B$-cotorsion left
${\mathcal{E}}$-contramodule.
###### Proof.
The proof of part (a) is similar to that of [50, Lemma 3.1.3(b)] and uses
Theorem 1.3.1(a). The proof of part (b) is similar to that of Lemma B.2.3 and
based on part (a) and the fact that there are enough contramodules induced
from flat (and even projective) $B$-modules in
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}$ (cf. Lemma 4.3.3). ∎
Let us call a left ${\mathcal{E}}$-contramodule ${\mathfrak{F}}$ _strongly
contraflat_ if the functor
$\operatorname{Hom}^{\mathcal{E}}({\mathfrak{F}},{-})$ takes short exact
sequences of $B$-cotorsion ${\mathcal{E}}$-contramodules to short exact
sequences of abelian groups (cf. Section 4.3 and [50, Section 5.1.6 and
Question 5.3]).
Assuming that any flat $B$-module has finite projective dimension, it follows
from Lemma B.4.7(a) that the Ext groups computed in the exact category
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
and in the abelian category ${\mathcal{E}}{\operatorname{\mathsf{--contra}}}$
agree. We denote these by $\operatorname{Ext}^{{\mathcal{E}},*}({-},{-})$.
###### Corollary B.4.8.
Assume that any flat left $B$-module has finite projective dimension. Then
(a) A left ${\mathcal{E}}$-contramodule ${\mathfrak{F}}$ is strongly
contraflat if and only if
$\operatorname{Ext}^{{\mathcal{E}},>0}({\mathfrak{F}},{\mathfrak{Q}})=0$ for
any $B$-cotorsion left ${\mathcal{E}}$-contramodule ${\mathfrak{Q}}$. The
class of strongly contraflat ${\mathcal{E}}$-contramodules is closed under
extensions and the passage to kernels of surjective morphisms in
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}$.
(b) A left ${\mathcal{E}}$-contramodule is strongly contraflat if and only if
it is a direct summand of a finitely iterated extension of
${\mathcal{E}}$-contramodules induced from flat $B$-modules.
###### Proof.
Part (a) follows from (a weak version of) Lemma B.4.7(a). Part (b) follows
from Lemma B.4.7(b) and part (a). ∎
###### Theorem B.4.9.
(a) Assuming that any flat left $B$-module has finite projective dimension,
for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$ the triangulated
functor
${\mathsf{D}}^{\star}({\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\star}({\mathcal{E}}{\operatorname{\mathsf{--contra}}})$
induced by the embedding of exact categories
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}\longrightarrow{\mathcal{E}}{\operatorname{\mathsf{--contra}}}$
is an equivalence of triangulated categories.
(b) Assuming that countable products of projective left $B$-modules have
finite projective dimensions (in particular, if the ring $B$ is right coherent
and flat left $B$-modules have finite projective dimensions), the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{E}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathcal{E}}$-contramodules is equivalent to
the homotopy category of complexes of projective left
${\mathcal{E}}$-contramodules.
(c) Assuming that countable products of flat left $B$-modules have finite
projective dimensions, the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{E}}{\operatorname{\mathsf{--contra}}})$
is equivalent to the absolute derived category of the exact category of
strongly contraflat left ${\mathcal{E}}$-contramodules.
###### Proof.
Part (a) follows from Lemma B.4.7(a) and the dual version of Proposition
A.5.6. For part (b), see Corollary A.6.2 or [51, Section 3.8 and Remark 3.7].
Part (c) can be deduced from part (b) together with the fact that strongly
contraflat left ${\mathcal{E}}$-contramodules have finite projective
dimensions in ${\mathcal{E}}{\operatorname{\mathsf{--contra}}}$ and
Proposition A.5.6. Alternatively, notice that countable products of strongly
contraflat ${\mathcal{E}}$-contramodules have finite projective dimensions by
(the proof of) Corollary B.4.8(b), so Corollary A.6.2 applies. ∎
###### Corollary B.4.10.
In the situation of Corollary B.4.6, assume additionally that the coring
${\mathcal{E}}$ is a projective left $B$-module. Then the data of a dualizing
complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the corings
${\mathcal{C}}$ and ${\mathcal{E}}$ induces an equivalence of triangulated
categories
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{E}}{\operatorname{\mathsf{--contra}}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}^{\mathbb{L}}\nobreak{-}$.
###### Proof.
In addition to what has been said in Corollaries B.4.2, B.4.6 and Theorem
B.4.9, we point out that in our present assumptions the assertions of Lemmas
B.4.3(a) and B.4.4(a) apply to any strongly contraflat left
${\mathcal{E}}$-contramodule ${\mathfrak{F}}$. Indeed, in view of Corollary
B.4.8(b) one only has to check that the functor
${\mathcal{N}}\odot_{\mathcal{E}}{-}$ takes short exact sequences of strongly
contraflat left ${\mathcal{E}}$-contramodules to short exact sequences of
abelian groups for any right ${\mathcal{E}}$-comodule ${\mathcal{N}}$. This
follows from part (a) of the same Corollary, as
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{N}}\odot_{\mathcal{E}}{\mathfrak{F}},\>{\mathbb{Q}}/{\mathbb{Z}})\simeq\operatorname{Hom}^{\mathcal{E}}({\mathfrak{F}},\operatorname{Hom}_{\mathbb{Z}}({\mathcal{N}},{\mathbb{Q}}/{\mathbb{Z}}))$
and the left ${\mathcal{E}}$-contramodule
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{N}},{\mathbb{Q}}/{\mathbb{Z}})$ is
$B$-cotorsion by Lemma 1.3.3(b). Therefore, one can construct the derived
functor
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}^{\mathbb{L}}{-}$
using strongly contraflat resolutions. ∎
### B.5. Base ring change
Let ${\mathcal{C}}$ be a coring over an associative ring $A$. Given a right
${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left ${\mathcal{C}}$-comodule
${\mathcal{M}}$, their _cotensor product_
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}$
is an abelian group constructed as the kernel of the natural pair of maps
${\mathcal{N}}\otimes_{A}{\mathcal{M}}\birarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}$.
Given a left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ and a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, their group of
_cohomomorphisms_
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},{\mathfrak{P}})$ is
constructed as the cokernel of the natural pair of maps
$\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}{\mathcal{M}},\>{\mathfrak{P}})\birarrow\operatorname{Hom}_{A}({\mathcal{M}},{\mathfrak{P}})$
[50, Sections 1.2.1 and 3.2.1].
Let ${\mathcal{C}}$ be a coring over an associative ring $A$ and
${\mathcal{E}}$ be a coring over an associative ring $B$. A _map of corings
${\mathcal{C}}\longrightarrow{\mathcal{E}}$ compatible with a ring
homomorphism $A\longrightarrow B$_ is an $A$-$A$-bimodule morphism such that
the maps $A\longrightarrow B$, ${\mathcal{C}}\longrightarrow{\mathcal{E}}$,
and the induced map
${\mathcal{C}}\otimes_{A}{\mathcal{C}}\longrightarrow{\mathcal{E}}\otimes_{B}{\mathcal{E}}$
form commutative diagrams with the comultiplication and counit maps in
${\mathcal{C}}$ and ${\mathcal{E}}$ [50, Section 7.1.1].
Let ${\mathcal{C}}\longrightarrow{\mathcal{E}}$ be a map of corings compatible
with a ring map $A\longrightarrow B$. Given a left ${\mathcal{C}}$-comodule
${\mathcal{M}}$, one defines a left ${\mathcal{E}}$-comodule
${}_{B}{\mathcal{M}}$ by the rule
${}_{B}{\mathcal{M}}=B\otimes_{A}{\mathcal{M}}$, the coaction map being
constructed as the composition $B\otimes_{A}{\mathcal{M}}\longrightarrow
B\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}\longrightarrow
B\otimes_{A}{\mathcal{E}}\otimes_{A}{\mathcal{M}}\longrightarrow{\mathcal{E}}\otimes_{A}{\mathcal{M}}\simeq{\mathcal{E}}\otimes_{B}(B\otimes_{A}{\mathcal{M}})$.
Similarly, given a right ${\mathcal{C}}$-comodule ${\mathcal{K}}$, there is a
natural right ${\mathcal{E}}$-comodule structure on the tensor product
${\mathcal{K}}_{B}={\mathcal{K}}\otimes_{A}B$. Given a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, one defines a left
${\mathcal{E}}$-contramodule ${}^{B}{\mathfrak{P}}$ by the rule
${}^{B}{\mathfrak{P}}=\operatorname{Hom}_{A}(B,{\mathfrak{P}})$, the
contraaction map being constructed as the composition
$\operatorname{Hom}_{B}({\mathcal{E}},\operatorname{Hom}_{A}(B,{\mathfrak{P}}))\simeq\operatorname{Hom}_{A}({\mathcal{E}},{\mathfrak{P}})\longrightarrow\operatorname{Hom}_{A}({\mathcal{E}}\otimes_{A}B,\>{\mathfrak{P}})\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}B,\>{\mathfrak{P}})\simeq\operatorname{Hom}_{A}(B,\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}}))\longrightarrow\operatorname{Hom}_{A}(B,{\mathfrak{P}})$.
Assuming that ${\mathcal{C}}$ is a flat right $A$-module, the functor
${\mathcal{M}}\longmapsto{}_{B}{\mathcal{M}}\colon{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{E}}{\operatorname{\mathsf{--comod}}}$
has a right adjoint functor, which is denoted by
${\mathcal{N}}\longmapsto{}_{\mathcal{C}}{\mathcal{N}}\colon{\mathcal{E}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and constructed by the rule
${}_{\mathcal{C}}{\mathcal{N}}={\mathcal{C}}_{B}\mathbin{\text{\smaller$\square$}}_{\mathcal{E}}{\mathcal{M}}$.
In particular, the functor
${\mathcal{N}}\longmapsto{}_{\mathcal{C}}{\mathcal{N}}$ takes a coinduced left
${\mathcal{E}}$-comodule ${\mathcal{E}}\otimes_{B}U$ into the coinduced left
${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}U$ (where $U$ is an
arbitrary left $B$-module) [50, Section 7.1.2].
Given a coring ${\mathcal{C}}$ over an associative ring $A$ and an associative
ring homomorphism $A\longrightarrow B$, one can define a coring
${}_{B}{\mathcal{C}}_{B}$ over the ring $B$ by the rule
${}_{B}{\mathcal{C}}_{B}=B\otimes_{A}{\mathcal{C}}\otimes_{A}B$. The counit in
${}_{B}{\mathcal{C}}_{B}$ is constructed as the composition
$B\otimes_{A}{\mathcal{C}}\otimes_{A}B\longrightarrow
B\otimes_{A}A\otimes_{A}B\longrightarrow B$, and the comultiplication is
provided by the composition
$B\otimes_{A}{\mathcal{C}}\otimes_{A}B\longrightarrow
B\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{C}}\otimes_{A}B\simeq
B\otimes_{A}{\mathcal{C}}\otimes_{A}A\otimes_{A}{\mathcal{C}}\otimes_{A}B\longrightarrow
B\otimes_{A}{\mathcal{C}}\otimes_{A}B\otimes_{A}{\mathcal{C}}\otimes_{A}B\simeq(B\otimes_{A}{\mathcal{C}}\otimes_{A}B)\otimes_{B}(B\otimes_{A}{\mathcal{C}}\otimes_{A}B)$.
There is a natural map of corings
${\mathcal{C}}\longrightarrow{}_{B}{\mathcal{C}}_{B}$ compatible with the ring
map $A\longrightarrow B$.
Assuming that ${\mathcal{C}}$ is a flat right $A$-module and $B$ is a
faithfully flat right $A$-module, the functors
${\mathcal{M}}\longrightarrow{}_{B}{\mathcal{M}}$ and
${\mathcal{N}}\longmapsto{}_{\mathcal{C}}{\mathcal{N}}$ are mutually inverse
equivalences between the abelian categories of left ${\mathcal{C}}$-comodules
and left ${}_{B}{\mathcal{C}}_{B}$-comodules. One proves this by applying the
Barr–Beck monadicity theorem to the conservative exact functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
B{\operatorname{\mathsf{--mod}}}$ taking a left ${\mathcal{C}}$-comodule
${\mathcal{M}}$ to the left $B$-module $B\otimes_{A}{\mathcal{M}}$ (see [36,
Section 2.1.3] or [50, Section 7.4.1]).
Now let us assume that ${\mathcal{C}}$ is a flat left $A$-module,
${\mathcal{E}}$ is a flat left $B$-module, and $B$ is a flat left $A$-module.
Then the functor ${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}$ takes
$A$-cotorsion left ${\mathcal{C}}$-contramodules to $B$-cotorsion left
${\mathcal{E}}$-contramodules (by Lemma 1.3.3(a)) and induces an exact functor
between the exact categories
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}\longrightarrow{\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
The left adjoint functor
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}$ to this exact
functor may not be everywhere defined, but one can easily see that it is
defined on the full subcategory of left ${\mathcal{E}}$-contramodules induced
from cotorsion left $B$-modules in the exact category
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$,
taking an induced contramodule
${\mathfrak{Q}}=\operatorname{Hom}_{B}({\mathcal{E}},V)$ over ${\mathcal{E}}$
to the induced contramodule
${}^{\mathcal{C}}{\mathfrak{Q}}=\operatorname{Hom}_{A}({\mathcal{C}},V)$ over
${\mathcal{C}}$ (where $V$ is an arbitrary cotorsion left $B$-module; see
Lemma 1.3.4(a)).
Finally, we assume that ${\mathcal{C}}$ is flat left $A$-module and $B$ is a
faithfully flat left $A$-module. It is clear from Example 3.2.1 that the
functor
${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}\allowbreak\longrightarrow{}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
is _not_ an equivalence of exact categories in general (and not even in the
case when ${\mathcal{C}}=A$). The aim of this section is to prove the
following slightly weaker result (cf. Corollaries 4.6.3–4.6.5 and 5.3.3).
###### Theorem B.5.1.
Let ${\mathcal{C}}$ be a coassociative coring over an associative ring $A$ and
$A\longrightarrow B$ be an associative ring homomorphism. Assume that
${\mathcal{C}}$ is a flat left $A$-module and $B$ is a faithfully flat left
$A$-module. Then the triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
induced by the exact functor ${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}$
is an equivalence of triangulated categories.
###### Corollary B.5.2.
In the assumptions of the previous Theorem, the exact functor
${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}\longrightarrow{}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
is fully faithful and induces isomorphisms between the Ext groups computed in
the exact categories
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
and
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
Any short sequence (or, more generally, bounded above complex) in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
which this functor transforms into an exact sequence in
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
is exact in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$.
###### Proof.
See [50, Section 4.1]. ∎
The proof of Theorem B.5.1 is based on the following technical result, which
provides adjusted resolutions for the construction of the left derived functor
of the partial left adjoint functor
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}$ to the exact functor
${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}$. For completeness, we
formulate three versions of the (co)induced resolution theorem; it is the
third one that will be used in the argument below.
###### Theorem B.5.3.
(a) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; assume that
${\mathcal{C}}$ is a flat right $A$-module. Then the coderived category
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
of the abelian category of left ${\mathcal{C}}$-comodules is equivalent to the
quotient category of the homotopy category of complexes of coinduced
${\mathcal{C}}$-comodules by its minimal triangulated subcategory containing
the total complexes of the short exact sequences of complexes of
${\mathcal{C}}$-comodules termwise coinduced from short exact sequences of
$A$-modules and closed under infinite direct sums.
(b) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; assume that
${\mathcal{C}}$ is a projective left $A$-module. Then the contraderived
category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathcal{C}}$-contramodules is equivalent to
the quotient category of the homotopy category of complexes of induced
${\mathcal{C}}$-contramodules by its minimal triangulated subcategory
containing the total complexes of the short exact sequences of complexes of
${\mathcal{C}}$-contramodules termwise induced from short exact sequences of
$A$-modules and closed under infinite products.
(c) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; assume that
${\mathcal{C}}$ is a flat left $A$-module. Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
of the exact category of $A$-cotorsion left ${\mathcal{C}}$-contramodules is
equivalent to the quotient category of the homotopy category of complexes of
${\mathcal{C}}$-contramodules termwise induced from cotorsion $A$-modules by
its minimal triangulated subcategory containing the total complexes of the
short exact sequences of complexes of ${\mathcal{C}}$-contramodules termwise
induced from short exact sequences of cotorsion $A$-modules and closed under
infinite products.
###### Proof.
This is yet another version of the results of [53, Proposition 1.5 and Remark
1.5], [54, Theorem 4.2.1], the above Propositions A.2.1 and A.3.1(b), etc. The
difference is that the categories of coinduced comodules or induced
contramodules are not even exact. This does not change much, however. Let us
sketch a proof of part (c); the proofs of parts (a-b) are similar (but
simpler).
The contramodule relative bar-resolution, totalized by taking infinite
products along the diagonals, provides a closed morphism with a contraacyclic
cone onto any complex of $A$-cotorsion left ${\mathcal{C}}$-contramodules from
a complex of ${\mathcal{C}}$-contramodules termwise induced from cotorsion
$A$-modules. This proves that the natural functor from the homotopy category
of complexes of left ${\mathcal{C}}$-contramodules termwise induced from
cotorsion $A$-modules to the contraderived category of $A$-cotorsion left
${\mathcal{C}}$-contramodules is a Verdier localization functor [51, Lemma
1.6].
In order to prove that the natural functor from the quotient category of the
homotopy category of termwise induced complexes in the formulation of the
theorem to the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
is fully faithful, one shows that any morphism in
$\mathsf{Hot}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
from a complex of contramodules termwise induced from cotorsion $A$-modules to
a contraacyclic complex over
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
factorizes through an object of the minimal triangulated subcategory
containing the total complexes of the short exact sequences of complexes of
${\mathcal{C}}$-contramodules termwise induced from short exact sequences of
cotorsion $A$-modules and closed under infinite products.
Let us only explain the key step, as the argument is mostly similar to the
ones spelled out in the references above. Let
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be the total
complex of a short exact sequence
${\mathfrak{U}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{V}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{W}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of complexes of $A$-cotorsion ${\mathcal{C}}$-contramodules, and let
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex
of ${\mathcal{C}}$-contramodules termwise induced from cotorsion $A$-modules.
The construction dual to the one explained in [50, Theorem 5.5] provides a
closed morphism onto the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ from a complex
of ${\mathcal{C}}$-contramodules
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ termwise
induced from flat cotorsion $A$-modules such that this morphism is induced
from an admissible epimorphism of cotorsion $A$-modules at every term of the
complexes (cf. the proof of Theorem B.3.1(b)).
Let ${\mathfrak{R}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ denote the
kernel of the morphism of complexes
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then the cone ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of the admissible monomorphism
${\mathfrak{R}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
maps naturally onto
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the cone
isomorphic to the total complex of a short exact sequence of complexes of
${\mathcal{C}}$-contramodules termwise induced from a short exact sequence of
cotorsion $A$-modules. As a morphism of graded ${\mathcal{C}}$-contramodules,
the composition
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
factorizes through the graded ${\mathcal{C}}$-contramodule
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The latter
being a projective graded object of
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$,
it follows that the composition
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is homotopic to zero by Lemma A.2.2. ∎
###### Proof of Theorem B.5.1.
First of all, we notice that the functor
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}$ acting between the
full subcategories of left ${}_{B}{\mathcal{C}}_{B}$-contramodules induced
from cotorsion $B$-modules and left ${\mathcal{C}}$-contramodules induced from
cotorsion $A$-modules in
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
is fully faithful. Indeed, one has
$\displaystyle\operatorname{Hom}^{{}_{B}{\mathcal{C}}_{B}}(\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B},U),\>\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B},V))\simeq\operatorname{Hom}_{B}(U,\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B},V))$
$\displaystyle\simeq\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B}\otimes_{B}U,\>V)\simeq\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}U,\>V)\simeq\operatorname{Hom}^{\mathcal{C}}(\operatorname{Hom}_{A}({\mathcal{C}},U),\operatorname{Hom}_{A}({\mathcal{C}},V)).$
Furthermore, the composition of functors
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}\longmapsto{}^{B}({}^{\mathcal{C}}{\mathfrak{Q}})$
is naturally isomorphic to the identity endofunctor on the full subcategory of
${}_{B}{\mathcal{C}}_{B}$-contramodules induced from cotorsion $B$-modules in
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$,
as
$\operatorname{Hom}_{A}(B,\operatorname{Hom}_{A}({\mathcal{C}},V))\simeq\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}B,\>V)\simeq\operatorname{Hom}_{B}(B\otimes_{A}{\mathcal{C}}\otimes_{A}B,\>V)$.
Clearly, the functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
acting between the homotopy categories of complexes of
${}_{B}{\mathcal{C}}_{B}$-contramodules termwise induced from complexes of
cotorsion $B$-modules and complexes of ${\mathcal{C}}$-contramodules termwise
induced from cotorsion $A$-modules, is “partially left adjoint” to the functor
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon\mathsf{Hot}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\allowbreak\longrightarrow\mathsf{Hot}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$.
By Theorem B.5.3(c), the former functor can be used to construct a left
derived functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\nobreak{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$.
It is easy to see that the functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is left adjoint to the functor
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\nobreak{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$.
According to the above, the composition of the adjoint triangulated functors
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}({}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\colon{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
is isomorphic to the identity functor. Hence the functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
is fully faithful, while the functor
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
is a Verdier localization functor.
It remains to show that the triangulated functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is essentially surjective. It is straightforward from the constructions that
both functors
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
preserve infinite products. So it suffices to prove that the contraderived
category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
coincides with its minimal triangulated subcategory containing all complexes
of ${\mathcal{C}}$-contramodules termwise induced from left $A$-modules
obtained by restriction of scalars from cotorsion left $B$-modules and closed
under infinite products. In view of the argument of Lemmas A.3.3–A.3.4, we
only need to check that there is an admissible epimorphism onto any object of
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
from an induced left ${\mathcal{C}}$-contramodule of the desired type.
The contraaction morphism
$\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
being an admissible epimorphism in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
for any $A$-cotorsion left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, the
problem reduces to constructing an admissible epimorphism in
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ onto any cotorsion left
$A$-module from an $A$-module obtained by restriction of scalars from a
cotorsion left $B$-module. Here we are finally using the assumption that $B$
is a _faithfully_ flat left $A$-module. According to [50, Section 7.4.1], the
latter is equivalent to the ring homomorphism $A\longrightarrow B$ being
injective and its cokernel being a flat left $A$-module. Assuming that, the
morphism
$\operatorname{Hom}_{A}(B,P)\longrightarrow\operatorname{Hom}_{A}(A,P)=P$
induced by the map $A\longrightarrow B$ is an admissible epimorphism in
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ for any cotorsion left
$A$-module $P$ by Lemma 1.3.3(a). It remains to say that the left $B$-module
$\operatorname{Hom}_{A}(B,P)$ is cotorsion by Lemma 1.3.5(a). ∎
## Appendix C Affine Noetherian Formal Schemes
This appendix is a continuation of [54, Appendix B]. Its goal is to construct
the derived co-contra correspondence between quasi-coherent torsion sheaves
and contraherent cosheaves of contramodules on an affine Noetherian formal
scheme with a dualizing complex. The affineness condition allows to speak of
(co)sheaves in terms of (contra)modules over a ring, simplifying the
exposition.
### C.1. Torsion modules and contramodules
Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Given
an $R$-module $E$, we denote by ${}_{(n)}E\subset E$ its $R$-submodule
consisting of all the elements annihilated by $I^{n}$. An $R$-module
${\mathcal{M}}$ is said to be _$I$ -torsion_ if ${\mathcal{M}}=\bigcup_{n\ge
1}\,{}_{(n)}{\mathcal{M}}$, i. e., for any element $m\in{\mathcal{M}}$ there
exists an integer $n\ge 1$ such that $I^{n}m=0$ in ${\mathcal{M}}$. An
$R$-module ${\mathfrak{P}}$ is called an _$(R,I)$ -contramodule_ if
$\operatorname{Ext}^{*}_{R}(R[s^{-1}],{\mathfrak{P}})=0$ for all $s\in I$, or
equivalently, the system of equations $q_{n}=p_{n}+sq_{n+1}$, $n\ge 0$, is
uniquely solvable in $q_{n}\in{\mathfrak{P}}$ for any fixed sequence
$p_{n}\in{\mathfrak{P}}$ (cf. the beginning of Section 1.1).
Just as the category of $I$-torsion $R$-modules, the category of
$(R,I)$-contramodules is abelian and depends only on the $I$-adic completion
of the ring $R$ [54, Theorem B.1.1] (cf. the ending part of Section 1.3
above). We denote the category of $I$-torsion $R$-modules by
$(R,I){\operatorname{\mathsf{--tors}}}$ and the category of
$(R,I)$-contramodules by $(R,I){\operatorname{\mathsf{--contra}}}$. Both
categories are full subcategories in the abelian category of $R$-modules
$R{\operatorname{\mathsf{--mod}}}$, closed under the kernels and cokernels;
the former subcategory is also closed under infinite direct sums, while the
latter one is closed under infinite products.
Given a sequence of elements $r_{n}\in R$ converging to zero in the $I$-adic
topology of $R$ and a sequence of elements $p_{n}\in{\mathfrak{P}}$ of an
$(R,I)$-contramodule ${\mathfrak{P}}$, the result of the “infinite summation
operation” $\sum_{n}r_{n}p_{n}$ is well-defined as an element of
${\mathfrak{P}}$ [54, Section 1.2]. One can define the _contratensor product_
${\mathfrak{P}}\odot_{(R,I)}{\mathcal{M}}$ of an $(R,I)$-contramodule
${\mathfrak{P}}$ and an $I$-torsion $R$-module ${\mathcal{M}}$ as the quotient
$R$-module of the tensor product
${\mathfrak{P}}\otimes_{\mathbb{Z}}{\mathcal{M}}$ by the relations
$(\sum_{n}r_{n}p_{n})\otimes m=\sum_{n}p_{n}\otimes r_{n}m$ for any sequence
$r_{n}$ converging to zero in $R$, any sequence of elements
$p_{n}\in{\mathfrak{P}}$, and any element $m\in{\mathcal{M}}$. Here all but a
finite number of summands in the right-hand side vanish due to the conditions
imposed on the sequence $r_{n}$ and the module ${\mathcal{M}}$.
In fact, however, for any $(R,I)$-contramodule ${\mathfrak{P}}$ and
$I$-torsion $R$-module ${\mathcal{M}}$ there is a natural isomorphism
${\mathfrak{P}}\odot_{(R,I)}{\mathcal{M}}\simeq{\mathfrak{P}}\otimes_{R}{\mathcal{M}}$.
This can be deduced from the fact that the functor
$(R,I){\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful, or alternatively, follows
directly from the observation that any sequence of elements of $I^{n}\subset
R$ converging to zero in the $I$-adic topology of $R$ is a linear combination
of a finite number of sequences converging to zero in $R$ with the
coefficients belonging to $I^{n}$ (cf. [50, proof of Proposition B.9.1]).
Clearly, the tensor product $E\otimes_{R}{\mathcal{M}}$ of any $R$-module $E$
and any $I$-torsion $R$-module ${\mathcal{M}}$ is an $I$-torsion $R$-module.
Similarly, the $R$-module $\operatorname{Hom}_{R}({\mathcal{M}},E)$ of
$R$-linear maps from an $I$-torsion $R$-module ${\mathcal{M}}$ into any
$R$-module $E$ is an $(R,I)$-contramodule. Finally, the $R$-module
$\operatorname{Hom}_{R}(E,{\mathfrak{P}})$ of homomorphisms from any
$R$-module $E$ into an $(R,I)$-contramodule ${\mathfrak{P}}$ is an
$(R,I)$-contramodule [54, Sections 1.5 and B.2].
An $(R,I)$-contramodule ${\mathfrak{F}}$ is said to be _$(R,I)$ -contraflat_
if the functor
${\mathcal{M}}\longmapsto{\mathfrak{F}}\odot_{(R,I)}{\mathcal{M}}$ is exact on
the abelian category of $I$-torsion $R$-modules. According to the above, this
is equivalent to the functor
${\mathcal{M}}\longmapsto{\mathfrak{F}}\otimes_{R}{\mathcal{M}}$ being exact
on the full abelian subcategory $(R,I){\operatorname{\mathsf{--tors}}}\subset
R{\operatorname{\mathsf{--mod}}}$.
By the Artin–Rees lemma, an object
${\mathcal{K}}\in(R,I){\operatorname{\mathsf{--tors}}}$ is injective if and
only if it is injective in $R{\operatorname{\mathsf{--mod}}}$, and if and only
if its submodules ${}_{(n)}{\mathcal{K}}$ of elements annihilated by $I^{n}$
are injective $R/I^{n}$-modules for all $n\ge 1$. By [54, Lemma B.9.2], an
$(R,I)$-contramodule ${\mathfrak{F}}$ is $(R,I)$-contraflat if and only if it
is a flat $R$-module, and if and only if its reductions
${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are flat $R/I^{n}$-modules for all $n\ge
1$. Furthermore, an object
${\mathfrak{F}}\in(R,I){\operatorname{\mathsf{--contra}}}$ is projective if
and only if it is $(R,I)$-contraflat and its reduction
${\mathfrak{F}}/I{\mathfrak{F}}$ is a projective $R/I$-module, and if and only
if all the reductions ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are projective
$R/I^{n}$-modules [54, Corollary B.8.2].
###### Theorem C.1.1.
(a) The coderived category
${\mathsf{D}}^{\mathsf{co}}((R,I){\operatorname{\mathsf{--tors}}})$ of the
abelian category of $I$-torsion $R$-modules is equivalent to the homotopy
category of complexes of injective $I$-torsion $R$-modules.
(b) The contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$ of the
abelian category of $(R,I)$-contramodules is equivalent to the contraderived
category of the exact category of $R$-flat (i. e., $(R,I)$-contraflat)
$(R,I)$-contramodules.
(c) Assume that the Noetherian ring $R/I$ has finite Krull dimension. Then the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$ is
equivalent to the absolute derived category of the exact category of $R$-flat
$(R,I)$-contramodules.
(d) Assume that the Noetherian ring $R/I$ has finite Krull dimension. Then the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$ is
equivalent to the homotopy category of complexes of projective
$(R,I)$-contramodules.
###### Proof.
Part (a) holds, since there are enough injectives in the abelian category
$(R,I){\operatorname{\mathsf{--tors}}}$ and the class of injective objects is
closed under infinite direct sums. Part (b) is true, because there are enough
$(R,I)$-contraflat (and even projective) objects in
$(R,I){\operatorname{\mathsf{--contra}}}$ and the class of $(R,I)$-contraflat
$(R,I)$-contramodules (or even flat $R$-modules) is closed under infinite
products (see Proposition A.3.1(b)).
Parts (c-d) hold, since in their assumptions any $(R,I)$-contraflat
$(R,I)$-contramodule ${\mathfrak{F}}$ has finite projective dimension in
$(R,I){\operatorname{\mathsf{--contra}}}$. Indeed, consider a projective
resolution ${\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
${\mathfrak{F}}$ in $(R,I){\operatorname{\mathsf{--contra}}}$ and apply the
functor $R/I\otimes_{R}{-}$ to it. Being a bounded above exact complex of flat
$R$-modules, the complex
${\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{F}}$
will remain exact after taking the tensor product. By Theorem 1.5.6, the
$R/I$-modules of cycles in the complex
$R/I\otimes_{R}{\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are eventually projective, and it follows that the $(R,I)$-contramodules of
cycles in ${\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
eventually projective, too. Now it remains to apply Corollary A.6.2,
Proposition A.5.6, and/or [50, Remark 2.1]. ∎
For noncommutative generalizations of the next two lemmas (as well as the
theorem following them), see Sections C.5 and D.2.
###### Lemma C.1.2.
(a) For any $R$-module $M$, injective $R$-module $J$, and $n\ge 1$, there is a
natural isomorphism of $R/I^{n}$-modules
$\operatorname{Hom}_{R}(M,J)/I^{n}\operatorname{Hom}_{R}(M,J)\simeq\operatorname{Hom}_{R/I^{n}}({}_{(n)}M,\>{}_{(n)}J)$.
(b) For any $R$-module $M$, flat $R$-module $F$, and $n\ge 1$, there is a
natural isomorphism of $R/I^{n}$-modules
${}_{(n)}(M\otimes_{R}F)\simeq\,{}_{(n)}M\otimes_{R/I^{n}}F/I^{n}F$.
###### Proof.
For any finitely generated $R$-module $E$ there are natural isomorphisms
$\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(E,M),J)\simeq
E\otimes_{R}\operatorname{Hom}_{R}(M,J)$ and
$\operatorname{Hom}_{R}(E,\>M\otimes_{R}F)\simeq\operatorname{Hom}_{R}(E,M)\allowbreak\otimes_{R}F$.
It remains to take $E=R/I^{n}$. ∎
A finite complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of injective
$I$-torsion $R$-modules is said to be a _dualizing complex_ for the pair
$(R,I)$ if for every integer $n\ge 1$ the complex of $R/I^{n}$-modules
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the commutative Noetherian ring $R/I^{n}$. If a finite
complex of injective $R$-modules
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing complex for
the ring $R$, then its subcomplex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\bigcup_{n\ge
0}\,{}_{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ consisting of
all the elements annihilated by some powers of $I\subset R$ is a dualizing
complex for $(R,I)$.
###### Lemma C.1.3.
Let ${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a finite
complex of injective $I$-torsion $R$-modules such that the complex of
$R/I$-modules
${}_{(1)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the ring $R/I$. Then
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing
complex for the pair $(R,I)$.
###### Proof.
Clearly,
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
finite complex of injective $R/I^{n}$-modules. Since any $R/I^{n}$-module $K$
with finitely generated $R/I$-module ${}_{(1)}K$ is finitely generated itself,
and the functors $\operatorname{Ext}^{i}_{R/I^{n}}(R/I,{-})$ take finitely
generated $R/I^{n}$-modules to finitely generated $R/I$-modules, one can check
by induction that the $R/I^{n}$-modules of cohomology of the complex
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
finitely generated.
It remains to show that the natural map
$R/I^{n}\longrightarrow\operatorname{Hom}_{R/I^{n}}({}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism. Both the left-hand and the right-hand sides are finite
complexes of flat (or, equivalently, projective) $R/I^{n}$-modules. By Lemma
C.1.2(a), the functor $P\longmapsto P/IP$ transforms this morphism into the
morphism
$R/I\longrightarrow\operatorname{Hom}_{R/I}({}_{(1)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}_{(1)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
which is a quasi-isomorphism by assumption. Hence the desired assertion
follows by Nakayama’s lemma. ∎
###### Theorem C.1.4.
The data of a dualizing complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for a pair
$(R,I)$ with a Noetherian commutative ring $R$ and an ideal $I\subset R$
induces an equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}((R,I){\operatorname{\mathsf{--tors}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}^{\mathbb{L}}{-}$.
###### Proof.
The constructions of the derived functors are based on Theorem C.1.1(a,c).
Applying the functor
$\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
to a complex of injective $I$-torsion $R$-modules produces a complex of
$R$-flat $(R,I)$-contramodules. Applying the functor
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}{-}$
to a complex of $R$-flat $(R,I)$-contramodules produces a complex of injective
$I$-torsion $R$-modules, and for any (absolutely) acyclic complex of flat
$R$-modules $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the complex of
injective ($I$-torsion) $R$-modules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contractible.
It remains to show that the morphism of finite complexes
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\longrightarrow{\mathcal{J}}$
is a quasi-isomorphism for any $I$-torsion $R$-module ${\mathcal{J}}$, and the
morphism of finite complexes
${\mathfrak{F}}\longrightarrow\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}{\mathfrak{F}})$
is a quasi-isomorphism for any $(R,I)$-contramodule ${\mathfrak{F}}$. Both
assertions follow from Lemma C.1.2, which implies that the morphisms become
quasi-isomorphisms after applying the functors
${\mathcal{K}}\longmapsto{}_{(n)}{\mathcal{K}}$ and
${\mathfrak{P}}\longmapsto{\mathfrak{P}}/I^{n}{\mathfrak{P}}$. In the latter
situation, one also has to use the assertion that the natural morphism
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/I^{n}{\mathfrak{P}}$
is an isomorphism for any $R$-flat $(R,I)$-contramodule ${\mathfrak{P}}$ (see
[54, proof of Lemma B.9.2]). ∎
### C.2. Contraadjusted and cotorsion contramodules
Let $R$ be a Noetherian commutative ring and $s\in R$ be an element. In the
spirit of the definitions from Sections 1.1 and C.1, let us say that an
$R$-module $P$ is _$s$ -contraadjusted_ if
$\operatorname{Ext}_{R}^{1}(R[s^{-1}],P)=0$, and that $P$ is an _$s$
-contramodule_ if $\operatorname{Ext}_{R}^{*}(R[s^{-1}],P)=0$. The property of
an $R$-module $P$ to be $s$-contraadjusted or an $s$-contramodule only depends
on the abelian group $P$ with the operator $s\colon P\longrightarrow P$.
An $R$-module $P$ is an $s$-contramodule if and only if it is an
$(R,(s))$-contramodule, where $(s)\subset R$ denotes the principal ideal
generated by $s\in R$. More generally, given an ideal $I\subset R$, an
$R$-module ${\mathfrak{P}}$ is an $(R,I)$-contramodule if and only if it is an
$s$-contramodule for every $s\in I$, and it suffices to check this condition
for any given set of generators of the ideal $I$ [54, proof of Theorem B.1.1].
Recall that any quotient $R$-module of an $s$-contraadjusted $R$-module is
$s$-contraadjusted. Given an $R$-module $L$, let us denote by $L(s)\subset L$
the image of the morphism $\operatorname{Hom}_{R}(R[s^{-1}],L)\longrightarrow
L$ induced by the localization map $R\longrightarrow R[s^{-1}]$. Equivalently,
the $R$-submodule $L(s)\subset L$ can be defined as the maximal $s$-divisible
$R$-submodule in $R$, i. e., the sum of all $R$-submodules (or even all
$s$-invariant abelian subgroups) in $L$ in which the element $s$ acts
surjectively.
Therefore, if $P=L/L(s)$ denotes the corresponding quotient $R$-module, then
one has $P(s)=0$. Notice also that one has
$\operatorname{Hom}_{R}(R[s^{-1}],P)=0$ for any $R$-module $P$ for which
$P(s)=0$. It follows that the $R$-quotient module $L/L(s)$ is an
$s$-contramodule whenever an $R$-module $L$ is $s$-contraadjusted.
Now let $s$, $t\in R$ be two elements; suppose that an $R$-module $L$ is a
$t$-contramodule. Then the $R$-module $\operatorname{Hom}_{R}(R[s^{-1}],L)$ is
also a $t$-contramodule [54, Section B.2], as is the image $L(s)$ of the
morphism of $t$-contramodules
$\operatorname{Hom}_{R}(R[s^{-1}],L)\longrightarrow L$. Hence the quotient
$R$-module $L/L(s)$ is also a $t$-contramodule. Assuming additionally that the
$R$-module $L$ was $s$-contraadjusted, the quotient module $L/L(s)$ is both a
$t$\- and an $s$-contramodule (i. e., it is an $(R,(t,s))$-contramodule).
Recall that an $R$-module $L$ is said to be _contraadjusted_ if it is
$s$-contraadjusted for every element $s\in R$. Given a contraadjusted
$R$-module $L$ and an ideal $I\subset R$, one can apply the above construction
of a quotient $R$-module successively for all the generators of the ideal $I$.
Proceeding in this way, one in a finite number of steps (equal to the number
of generators of $I$) obtains the unique maximal quotient $R$-module
${\mathfrak{P}}$ of the $R$-module $L$ such that ${\mathfrak{P}}$ is an
$s$-contramodule for every $s\in I$, i. e., ${\mathfrak{P}}$ is an
$(R,I)$-contramodule. We denote this quotient module by
${\mathfrak{P}}=L/L(I)$.
###### Lemma C.2.1.
Let $R$ be a Noetherian commutative ring, $s\in R$ be an element, $F$ be a
flat $R$-module, and $M$ be a finitely generated $R$-module. Then the natural
$R$-module homomorphism
$\operatorname{Hom}_{R}(R[s^{-1}],\>M\otimes_{R}F)\longrightarrow
M\otimes_{R}F$ is injective. In other words, one has
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>M\otimes_{R}F)=0$.
###### Proof.
Consider the sequence of annihilator submodules of the elements $s^{m}\in R$,
$m\ge 1$, in the $R$-module $M$. This is an increasing sequence of
$R$-submodules in $M$. Let $N$ be the maximal submodule in this sequence and
$n$ be a positive integer such that $N$ is the annihilator of $s^{n}$ in $M$.
Then any element of the $R$-module $M\otimes_{R}F$ annihilated by $s^{n+1}$ is
also annihilated by $s^{n}$.
Indeed, the element $s$ acts by an injective endomorphism of an $R$-module
$M/N$; hence so does the element $s^{n+1}$. Since $F$ is flat, it follows that
$s^{n+1}$ must act injectively in the tensor product
$(M/N)\otimes_{R}F\simeq(M\otimes_{R}F)/(N\otimes_{R}F)$. Hence any element of
$M\otimes_{R}F$ annihilated by $s^{n+1}$ belongs to $N\otimes_{R}F$ and is
therefore annihilated by $s^{n}$. ∎
###### Lemma C.2.2.
Let $R$ be a Noetherian commutative ring, $s\in R$ be an element, and
$I\subset R$ be an ideal. Then
(a) whenever an $R$-module $L$ is $s$-contraadjusted, the $R$-modules $L(s)$
and $L/L(s)$ are also $s$-contraadjusted (and $L/L(s)$ is even an
$s$-contramodule);
(b) whenever an $R$-module $L$ is contraadjusted, the $R$-modules $L(s)$ and
$L/L(s)$ are also contraadjusted;
(c) whenever an $R$-module $L$ is contraadjusted, the $R$-modules $L(I)$ and
$L/L(I)$ are also contraadjusted (and $L/L(I)$ is in addition an
$(R,I)$-contramodule);
(d) whenever an $R$-module $L$ is cotorsion and
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>L)=0$, the $R$-modules $L(s)$ and
$L/L(s)$ are also cotorsion.
###### Proof.
The parenthesized assertions in (a) and (c) have been explained above. Since
the class of cotorsion $R$-modules is closed under cokernes of injective
morphisms, while the classes of contraadjusted and $s$-contraadjusted
$R$-modules are even closed under quotients, it suffices to check the
assertions related to the $R$-modules $L(s)$ and $L(I)$. Furthermore, there is
a natural surjective morphism of $R$-modules
$\operatorname{Hom}_{R}(R[s^{-1}],L)\longrightarrow L(s)$, which in the
assumptions of (d) is an isomorphism.
Now part (d) is a particular case of Lemma 1.3.2(a), part (b) is provided by
Lemma 1.2.1(b), and part (a) follows from the similar claim that the
$R$-module $\operatorname{Hom}_{R}(R[s^{-1}],P)$ is $s$-contraadjusted for any
$s$-contraadjusted $R$-module $P$. Finally, (b) implies (c) in view of the
above recursive construction of the $R$-module $L(I)$ and the fact that the
class of constraadjusted $R$-modules is closed under extensions. ∎
###### Corollary C.2.3.
Let $R$ be a Noetherian commutative ring, $s\in R$ be an element, and
$I\subset R$ be an ideal. In this setting
(a) if $F$ is an $s$-contraadjusted flat $R$-module, then the $R$-modules
$F(s)$ and $F/F(s)$ are also flat and $s$-contraadjusted (and $F/F(s)$ is even
an $s$-contramodule);
(b) if $F$ is a contraadjusted flat $R$-module, then $F(I)$ is a
contraadjusted flat $R$-module and $F/F(I)$ is an $R$-flat $R$-contraadjusted
$(R,I)$-contramodule;
(c) if $F$ is a flat cotorsion $R$-module, then $F(I)$ is a flat cotorsion
$R$-module and $F/F(I)$ is an $R$-flat $R$-cotorsion $(R,I)$-contramodule.
###### Proof.
Part (a): in order to prove that the $R$-module $F/F(s)$ is flat, let us check
that the map $M\otimes_{R}F(s)\longrightarrow M\otimes_{R}F$ induced by the
natural embedding $F(s)\longrightarrow F$ is injective for any finitely
generated $R$-module $M$. By Lemma C.2.1, one has
$F(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],F)\subset F$ and
$(M\otimes_{R}F)(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],\>M\otimes_{R}F)\subset
M\otimes_{R}F$. It remains to point out that the natural morphism
$M\otimes_{R}\operatorname{Hom}_{R}(R[s^{-1}],F)\longrightarrow\operatorname{Hom}_{R}(R[s^{-1}],\>M\otimes_{R}F)$
is an isomorphism by Lemma 1.6.2.
Part (b): the above recursive construction of the $(R,I)$-contramodule
$F/F(I)$ together with part (a) imply the assertion that $F/F(I)$ is a flat
$R$-module. Now it remains to use Lemma C.2.2(c).
Part (c) can be proven in the way similar to part (b), using the fact that the
class of cotorsion $R$-modules is closed under extensions and Lemma C.2.2(d).
Alternatively, (c) can be deduced from Theorem 1.3.8. Indeed, if
$F\simeq\prod_{\mathfrak{p}}{\mathfrak{F}}_{\mathfrak{p}}$, where
${\mathfrak{p}}\subset R$ are prime ideals and ${\mathfrak{F}}_{\mathfrak{p}}$
are $(R,{\mathfrak{p}})$-contramodules, then one easily checks that $F/F(I)$
is the product of ${\mathfrak{F}}_{\mathfrak{p}}$ over the prime ideals
${\mathfrak{p}}$ containing $I$, while $F(I)$ is the product of
${\mathfrak{F}}_{\mathfrak{q}}$ over all the other prime ideals
${\mathfrak{q}}$. ∎
###### Lemma C.2.4.
(a) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an $s$-contramodule,
$K$ is a contraadjusted $R$-module, and $F$ is a flat $R$-module. Then
$0\longrightarrow P\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a
contraadjusted $R$-module, $F/F(s)$ is a flat $R$-module, and all the three
modules are $s$-contramodules.
(b) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an $s$-contramodule,
$K$ is a cotorsion $R$-module, and $F$ is a flat $R$-module. Then
$0\longrightarrow P\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a cotorsion
$R$-module, $F/F(s)$ is a flat $R$-module, and all the three modules are
$s$-contramodules.
###### Proof.
Part (a): first of all let us show that the morphism of $R$-modules
$K\longrightarrow F$ restricts to an isomorphism of their submodules
$K(s)\longrightarrow F(s)$. Indeed, we have
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>F)=0$ by Lemma C.2.1 and
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>P)\subset\operatorname{Hom}_{R}(R[s^{-1}],P)=0$
by the definition of an $s$-contramodule, hence
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>\allowbreak K)=0$. Therefore, there are
isomorphisms $K(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],K)$ and
$F(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],F)$. Using the condition that $P$
is an $s$-contramodule, that is $\operatorname{Ext}^{*}(R[s^{-1}],P)=0$ again,
we conclude that $K(s)\simeq F(s)$.
It follows that the sequence $0\longrightarrow P\longrightarrow
K/K(s)\longrightarrow F/F(s)\longrightarrow 0$ is exact. The $R$-module $K$
being contraadjusted by assumption, its quotient $R$-module $F$ is
contraadjusted, too. Now the quotient module $K/K(s)$ is contraadjusted by
Lemma C.2.2(b), the quotient module $F/F(s)$ is flat by Corollary C.2.3(a),
and both are $s$-contramodules by Lemma C.2.2(a).
In part (b), it only remains to prove that $K/K(s)$ is a cotorsion $R$-module.
This follows from the above argument and Lemma C.2.2(d). ∎
###### Corollary C.2.5.
(a) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an
$(R,I)$-contramodule, $K$ is a contraadjusted $R$-module, and $F$ is a flat
$R$-module. Then $0\longrightarrow P\longrightarrow K/K(I)\longrightarrow
F/F(I)\longrightarrow 0$ is a short exact sequence of $(R,I)$-contramodules in
which $K/K(I)$ is a contraadjusted $R$-module and $F/F(I)$ is a flat
$R$-module.
(b) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an
$(R,I)$-contramodule, $K$ is a cotorsion $R$-module, and $F$ is a flat
$R$-module. Then $0\longrightarrow P\longrightarrow K/K(I)\longrightarrow
F/F(I)\longrightarrow 0$ is a short exact sequence of $(R,I)$-contramodules in
which $K/K(I)$ is a cotorsion $R$-module and $F/F(I)$ is a flat $R$-module.
###### Proof.
Follows by recursion from Lemma C.2.4. ∎
###### Lemma C.2.6.
(a) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a contraadjusted
$R$-module, $F$ is a flat $R$-module, and $P$ is an $s$-contramodule. Then
$0\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a
contraadjusted $R$-module, $F/F(s)$ is a flat $R$-module, and all the three
modules are $s$-contramodules.
(b) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a cotorsion
$R$-module, $F$ is a flat $R$-module, and $P$ is an $s$-contramodule. Then
$0\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a cotorsion
$R$-module, $F/F(s)$ is a flat $R$-module, and all the three modules are
$s$-contramodules.
###### Proof.
Part (a): first we prove that the morphism of $R$-modules $K\longrightarrow F$
restricts to an isomorphism of their submodules $K(s)\longrightarrow F(s)$.
Indeed,
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>K)\subset\operatorname{Hom}_{R}(R[s^{-1}]/R,\>F)=0$
by Lemma C.2.1, hence $K(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],K)$ and
$F(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],F)$. Since
$\operatorname{Hom}_{R}(R[s^{-1}],P)=0$ by assumption, we conclude that
$K(s)\simeq F(s)$, and it follows that the sequence $0\longrightarrow
K/K(s)\longrightarrow F/F(s)\longrightarrow P\longrightarrow 0$ is exact.
The $R$-module $F$, being an extension of $s$-contraadjusted $R$-modules $K$
and $P$, is $s$-contraadjusted, too. The rest of the argument coincides with
the respective part of the proof of Lemma C.2.4, and so does the proof of part
(b). ∎
###### Corollary C.2.7.
(a) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a contraadjusted
$R$-module, $F$ is a flat $R$-module, and $P$ is an $(R,I)$-contramodule. Then
$0\longrightarrow K/K(I)\longrightarrow F/F(I)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $(R,I)$-contramodules in which $K/K(I)$ is a
contraadjusted $R$-module and $F/F(I)$ is a flat $R$-module.
(b) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a cotorsion
$R$-module, $F$ is a flat $R$-module, and $P$ is an $(R,I)$-contramodule. Then
$0\longrightarrow K/K(I)\longrightarrow F/F(I)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $(R,I)$-contramodules in which $K/K(I)$ is a
cotorsion $R$-module and $F/F(I)$ is a flat $R$-module.
###### Proof.
Follows by recursion from Lemma C.2.6. ∎
Recall that, according to [54, Theorem B.8.1], the $\operatorname{Ext}$
groups/modules computed in the abelian categories
$R{\operatorname{\mathsf{--mod}}}$ and
$(R,I){\operatorname{\mathsf{--contra}}}$ agree.
###### Corollary C.2.8.
Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Then
(a) any $(R,I)$-contramodule can be embedded into an $R$-cotorsion
$(R,I)$-contramodule in such a way that the quotient $(R,I)$-contramodule is
$R$-flat;
(b) any $(R,I)$-contramodule admits a surjective morphism onto it from an
$R$-flat $(R,I)$-contramodule such that the kernel is an $R$-cotorsion
$(R,I)$-contramodule;
(c) an $(R,I)$-contramodule ${\mathfrak{Q}}$ is $R$-cotorsion if and only if
one has $\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any
$R$-flat $(R,I)$-contramodule ${\mathfrak{F}}$;
(d) an $(R,I)$-contramodule ${\mathfrak{F}}$ is $R$-flat if and only if one
has $\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any
$R$-cotorsion $(R,I)$-contramodule ${\mathfrak{Q}}$.
###### Proof.
Parts (a-b) follow from Theorem 1.3.1 together with Corollaries C.2.5(b) and
C.2.7(b). Part (c) is deduced from (a) and part (d) deduced from (b) easily. ∎
Let us denote by $(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}$ the
full exact subcategory of $R$-contraadjusted $(R,I)$-contramodules and by
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}$ the full exact
subcategory of $R$-cotorsion $(R,I)$-contramodules in the abelian category
$(R,I){\operatorname{\mathsf{--contra}}}$.
Notice that the full subcategory
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\subset(R,I){\operatorname{\mathsf{--contra}}}$
depends only on the $I$-adic completion of the ring $R$, as one can see from
Corollary C.2.8(c). The similar assertion for the full subcategory
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\subset(R,I){\operatorname{\mathsf{--contra}}}$
will follow from the results of Section C.3 below.
###### Theorem C.2.9.
(a) Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal.
Then for any symbol $\star={\mathsf{b}}$, $+$, $-$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}})\longrightarrow{\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}})$
induced by the embedding of exact categories
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$
is an equivalence of triangulated categories.
(b) Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal such
that the quotient ring $R/I$ has finite Krull dimension. Then for any symbol
$\star={\mathsf{b}}$, $+$, $-$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}})\longrightarrow{\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}})$
induced by the embedding of exact categories
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$
is an equivalence of triangulated categories.
###### Proof.
Part (a) follows from Corollary C.2.5(a) or C.2.8(a) together with the
opposite version of Proposition A.5.6. To prove part (b) in the similar way,
one needs to know that in its assumptions any $(R,I)$-contramodule has finite
right homological dimension with respect to the exact subcategory
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\subset(R,I){\operatorname{\mathsf{--contra}}}$.
In view of Corollary C.2.8(c), this follows from the fact that any $R$-flat
$(R,I)$-contramodule has finite projective dimension in
$(R,I){\operatorname{\mathsf{--contra}}}$, which was established in the proof
of Theorem C.1.1 (using [54, Corollary B.8.2]). ∎
###### Remark C.2.10.
Let $X=\operatorname{Spec}R$ be the Noetherian affine scheme corresponding to
the ring $R$ and $U=X\setminus Z$ be the open complement to the closed
subscheme $Z=\operatorname{Spec}R/I\subset X$. Denote by $j\colon
U\longrightarrow X$ the natural open embedding morphism. Then the exact
category $(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}$ is
equivalent to the full exact subcategory in the exact category
$X{\operatorname{\mathsf{--ctrh}}}$ of contraherent cosheaves on $X$
consisting of all the contraherent cosheaves ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}$ with vanishing restrictions
$j^{!}{\mathfrak{P}}={\mathfrak{P}}|_{U}$ to the open subscheme $U$.
Similarly, the exact category
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}$ is equivalent to the
full exact subcategory in the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ consisting of all the
(locally) cotorsion contraherent cosheaves ${\mathfrak{P}}$ on $X$ for which
$j^{!}{\mathfrak{P}}=0$. Indeed, a contraadjusted $R$-module $P$ is an
$s$-contramodule if and only if the corresponding contraherent cosheaf
${\mathfrak{P}}=\widecheck{P}$ on $X$ vanishes in the restriction to
$\operatorname{Spec}R[s^{-1}]\subset\operatorname{Spec}R$ (see Sections
2.2–2.4 for the definitions and notation). In particular, when the ring $R/I$
is Artinian, the exact category $\ker(j^{!}\colon
X{\operatorname{\mathsf{--ctrh}}}\to
U{\operatorname{\mathsf{--ctrh}}})=\ker(j^{!}\colon
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\to
U{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ is abelian and equivalent
to the abelian category
$(R,I){\operatorname{\mathsf{--contra}}}=(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}=(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}$.
### C.3. Very flat contramodules
Unlike the flatness, cotorsion, and contraadjustness properties of
$(R,I)$-contramodules, their very flatness property is _not_ defined in terms
of the similar property of $R$-modules. Instead, it is described in terms of
the reductions modulo $I^{n}$ and the very flatness properties of
$R/I^{n}$-modules.
For a (straightforward) generalization of the next lemma, see Lemma D.1.7.
###### Lemma C.3.1.
(a) Let $F$ be a flat $R$-module and ${\mathfrak{Q}}$ be an
$(R,I)$-contramodule such that the $R/I$-module $F/IF$ is very flat, the
natural $(R,I)$-contramodule morphism
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$
is an isomorphism, and the $R/I$-module ${\mathfrak{Q}}/I{\mathfrak{Q}}$ is
contraadjusted. Then one has
$\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{Q}})=0$.
(b) Let $F$ be a flat $R$-module and ${\mathfrak{Q}}$ be an $R$-flat
$(R,I)$-contramodule such that the $R/I$-module
${\mathfrak{Q}}/I{\mathfrak{Q}}$ is cotorsion. Then one has
$\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{Q}})=0$.
###### Proof.
Part (a): by (the proof of) Lemma 1.6.8(a), the $R/I^{n}$-modules
${\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ and the $R/I$-modules
$I^{n-1}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ are contraadjusted, while by part
(b) of the same lemma the $R/I^{n}$-modules $F/I^{n}F$ are very flat. Now we
follow the argument in the proof of [54, Proposition B.10.1]. The $R$-module
$F$ being flat by assumption, one has
$\operatorname{Ext}^{i}_{R}(F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})\simeq\operatorname{Ext}^{i}_{R/I^{n}}(F/I^{n}F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})=0$
for all $i>0$ and any $n\ge 1$. The natural map
$\operatorname{Hom}_{R}(F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})\longrightarrow\operatorname{Hom}_{R}(F,\>{\mathfrak{Q}}/I^{n-1}{\mathfrak{Q}})$
is surjective, since
$\operatorname{Ext}^{1}_{R/I^{n}}(F/I^{n}F,\>I^{n-1}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})=0$.
By [54, Lemma B.10.3], we have
$\operatorname{Ext}_{R}^{i}(F,\>\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})\simeq\varprojlim_{n}^{i}\operatorname{Hom}_{R}(F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})=0$
for all $i>0$.
Part (b): by (the proof of) Lemma 1.6.8(c), the $R/I^{n}$-modules
${\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ and the $R/I$-modules
$I^{n-1}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ are cotorsion, while by [54, proof
of Lemma B.9.2] the natural map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$
is an isomorphism. The argument continues exactly the same as in part (a). ∎
For generalizations of respective parts of the following corollary, see Lemma
D.3.3 and Corollary D.4.5.
###### Corollary C.3.2.
(a) For any contraadjusted $R$-module $Q$, the $R/I$-module $Q/IQ$ is
contraadjusted. If the natural map $Q\longrightarrow\varprojlim_{n}Q/I^{n}Q$
is an isomorphism for an $R$-module $Q$ and the $R/I$-module $Q/IQ$ is
contraadjusted, then the $R$-module $Q$ is contraadjusted.
(b) A flat $(R,I)$-contramodule ${\mathfrak{Q}}$ is $R$-cotorsion if and only
if the $R/I$-module ${\mathfrak{Q}}/I{\mathfrak{Q}}$ is cotorsion.
###### Proof.
Part (a): the first assertion is Lemma 1.6.6(b). Since the $R/I$-module $F/IF$
is very flat for any very flat $R$-module $F$ (see Lemma 1.2.2(b)), the second
assertion follows from Lemma C.3.1(a). Part (b) is similarly a consequence of
Lemma 1.6.7(a) and Lemma C.3.1(b). ∎
For a generalization of the next corollary, see Corollary D.3.8.
###### Corollary C.3.3.
Let $F$ be a flat $R$-module for which the $R/I$-module $F/IF$ is very flat.
Then $\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{P}})=0$ for any
$R$-contraadjusted $(R,I)$-contramodule ${\mathfrak{P}}$.
###### Proof.
By Corollary C.2.7(a) or C.2.8(b), there exists a short exact sequence of
$(R,I)$-contramodules
$0\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$, where the $R$-module ${\mathfrak{Q}}$ is contraadjusted and the $R$-module
${\mathfrak{G}}$ is flat. Then the $R$-module ${\mathfrak{G}}$ is also
contraadjusted. By Corollary C.3.2(a), it follows that so are the
$R/I$-modules ${\mathfrak{Q}}/I{\mathfrak{Q}}$ and
${\mathfrak{G}}/I{\mathfrak{G}}$. Furthermore, according to the proof of [54,
Lemma B.9.2], one has
${\mathfrak{G}}\simeq\varprojlim_{n}{\mathfrak{G}}/I^{n}{\mathfrak{G}}$. The
$(R,I)$-contramodule ${\mathfrak{Q}}$ being a subcontramodule of
${\mathfrak{G}}$, it follows that $\bigcap_{n}I^{n}{\mathfrak{Q}}=0$, hence
also ${\mathfrak{Q}}\simeq\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$.
Now it remains to apply Lemma C.3.1(a) to (the $R$-module $F$ and) the
$(R,I)$-contramodules ${\mathfrak{Q}}$ and ${\mathfrak{G}}$. ∎
###### Lemma C.3.4.
(a) Any $(R,I)$-contramodule ${\mathfrak{M}}$ can be included in a short exact
sequence of $(R,I)$-contramodules
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$, where the $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are very
flat, while the $R$-module ${\mathfrak{P}}$ is contraadjusted.
(b) Any $(R,I)$-contramodule ${\mathfrak{M}}$ can be included in a short exact
sequence of $(R,I)$-contramodules
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where the $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are very
flat, while the $R$-module ${\mathfrak{P}}$ is contraadjusted.
###### Proof.
First of all, let us show that any $R$-flat $(R,I)$-contramodule
${\mathfrak{G}}$ can be included in a short exact sequence of
$(R,I)$-contramodules
$0\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{G}}\longrightarrow
0$, where the $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are very
flat, while the $R$-module ${\mathfrak{E}}$ is (flat and) contraadjusted. By
Theorem 1.1.1(b), there exists a short exact sequence of $R$-modules
$0\longrightarrow E\longrightarrow
F\longrightarrow{\mathfrak{G}}\longrightarrow 0$ such that the $R$-module $F$
is very flat, while the $R$-module $E$ is contraadjusted. Then the $R$-module
$E$ is also flat.
The short sequence of $R/I^{n}$-modules $0\longrightarrow
E/I^{n}E\longrightarrow
F/I^{n}F\longrightarrow{\mathfrak{G}}/I^{n}{\mathfrak{G}}\longrightarrow 0$ is
exact for every $n\ge 1$. Set ${\mathfrak{E}}=\varprojlim_{n}E/I^{n}E$ and
${\mathfrak{F}}=\varprojlim_{n}F/I^{n}F$; recall that the natural map
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/I^{n}{\mathfrak{G}}$
is an isomorphism according to the proof of [54, Lemma B.9.2]. Passing to the
projective limit, we obtain a short exact sequence of $(R,I)$-contramodules
$0\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{G}}\longrightarrow
0$. The $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}\simeq F/I^{n}F$
are very flat by Lemma 1.2.2(b), and the $R$-module ${\mathfrak{E}}$ is
contraadjusted by Corollary C.3.2(a). Now it is easy to obtain part (a) from
Corollary C.2.8(a) and part (b) from Corollary C.2.8(b). ∎
Let us call an $(R,I)$-contramodule ${\mathfrak{F}}$ _very flat_ if the
functor $\operatorname{Hom}_{R}({\mathfrak{F}},{-})$ takes short exact
sequences of $R$-contraadjusted $(R,I)$-contramodules to short exact sequences
of abelian groups. The next corollary says that this definition is equivalent
to the more familiar formulation is terms of the $\operatorname{Ext}^{1}$
vanishing (cf. the definitions of very flat modules and contraadjusted sheaves
in Sections 1.1 and 2.5).
###### Corollary C.3.5.
Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Then
(a) any $(R,I)$-contramodule can be embedded into an $R$-contraadjusted
$(R,I)$-contramodule in such a way that the quotient $(R,I)$-contramodule is
very flat;
(b) any $(R,I)$-contramodule admits a surjective morphism onto it from an very
flat $(R,I)$-contramodule such that the kernel is an $R$-contraadjusted
$(R,I)$-contramodule;
(c) an $(R,I)$-contramodule ${\mathfrak{Q}}$ is $R$-contraadjusted if and only
if $\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any very
flat $(R,I)$-contramodule ${\mathfrak{F}}$;
(d) an $(R,I)$-contramodule ${\mathfrak{F}}$ is very flat if and only if
$\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any
$R$-contraadjusted $(R,I)$-contramodule ${\mathfrak{Q}}$;
(e) an $(R,I)$-contramodule ${\mathfrak{F}}$ is very flat if and only if the
$R/I^{n}$-module ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ is very flat for every
$n\ge 1$.
###### Proof.
The “if” assertion in part (e) follows Corollary C.3.3 and [54, Lemma B.9.2].
Parts (a-b) are provided by the respective parts of Lemma C.3.4 together with
the “if” assertion in (e). The “if” assertions in parts (c) and (d) are
deduced from parts (a) and (b), respectively; and to prove the “only if”, one
only needs to know that any $(R,I)$-contramodule can be embedded into an
$R$-contraadjusted one. Finally, the “only if” assertion in part (e) follows
from the construction in Lemma C.3.4 on which the proof of part (b) is based.
∎
### C.4. Affine geometry of $(R,I)$-contramodules
All rings in this section are presumed to be commutative and Noetherian. Let
$f\colon R\longrightarrow S$ be a ring homomorphism, $I$ be an ideal in $R$,
and $J$ be an ideal in $S$. We denote by $Sf(I)$ the extension of the ideal
$I\subset R$ in the ring $S$.
###### Lemma C.4.1.
(a) If $f(I)\subset J$, then any $(S,J)$-contramodule is also an
$(R,I)$-contramodule in the $R$-module structure obtained by restriction of
scalars via $f$.
(b) If $J\subset Sf(I)$, then an $S$-module is an $(S,J)$-contramodule
whenever it is an $(R,I)$-contramodule in the $R$-module structure obtained by
restriction of scalars via $f$.
###### Proof.
Part (a) holds, since an $S$-module $Q$ is an $(S,J)$-contramodule if and only
if the system of equations $q_{n}=p_{n}+tq_{n+1}$, $n\ge 0$, is uniquesly
solvable in $q_{n}\in Q$ for any fixed sequence $p_{n}\in Q$ and any $t\in J$.
Part (b) is true, because it suffices to check the previous condition for the
elements $t$ belonging to any given set of generators of the ideal $J\subset
S$ only [54, Sections B.1 and B.7]. ∎
###### Lemma C.4.2.
Assume that $J\subset Sf(I)$, and let ${\mathfrak{P}}$ be an
$(R,I)$-contramodule. Then
(a) the $S$-module $\operatorname{Hom}_{R}(S,{\mathfrak{P}})$ is an
$(S,J)$-contramodule;
(b) the $S$-module $S\otimes_{R}{\mathfrak{P}}$ is an $(S,J)$-contramodule
whenever $f$ is a finite morphism.
###### Proof.
Both assertions follow from Lemma C.4.1(b). Indeed, the $R$-module
$\operatorname{Hom}_{R}(M,{\mathfrak{P}})$ is an $(R,I)$-contramodule for any
$R$-module $M$, because the contramodule infinite summation operations can be
defined on it (see the beginning of Section C.1), while the $R$-module
$M\otimes_{R}{\mathfrak{P}}$ is an $(R,I)$-contramodule for any finitely
generated $R$-module $M$, being the cokernel of a morphism between two finite
direct sums of copies of the $(R,I)$-contramodule ${\mathfrak{P}}$. ∎
Denote by ${\mathfrak{R}}$ the $I$-adic completion of the ring $R$ and by
${\mathfrak{S}}$ the $J$-adic completion of the ring $S$, both viewed as
topological rings. By [54, Theorem B.1.1], the full subcategories
$(R,I){\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ and
$(S,J){\operatorname{\mathsf{--contra}}}\subset
S{\operatorname{\mathsf{--mod}}}$ in the categories of $R$\- and $S$-modules
are equivalent to the categories
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ and
${\mathfrak{S}}{\operatorname{\mathsf{--contra}}}$ of contramodules over the
topological rings ${\mathfrak{R}}$ and ${\mathfrak{S}}$.
Assume that $f(I)\subset J$; then the ring homomorphism $f\colon
R\longrightarrow S$ induces a continuous homomorphism of topological rings
$\phi\colon{\mathfrak{R}}\longrightarrow{\mathfrak{S}}$. According to [54,
Section 1.8], there is a pair of adjoint functors of “contrarestriction” and
“contraextension” of scalars
$R^{\phi}\colon{\mathfrak{S}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$
and $E^{\phi}\colon
R{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{S}}{\operatorname{\mathsf{--contra}}}$.
While the functor $R^{\phi}$ is easily identified with the functor of
restriction of scalars from Lemma C.4.1, the functor $E^{\phi}$, which is left
adjoint to $R^{\phi}$, is defined by the rules that $E^{\phi}$ is right exact
and takes the free ${\mathfrak{R}}$-contramodule
${\mathfrak{R}}[[X]]=\varprojlim_{n}R/I^{n}[X]$ to the free
${\mathfrak{S}}$-contramodule ${\mathfrak{S}}[[X]]=\varprojlim_{n}S/J^{n}[X]$
for any set $X$.
When $f$ is a finite morphism and $J=Sf(I)$, the functor $E^{\phi}$ is simply
the functor of extension of scalars from Lemma C.4.2(b).
###### Lemma C.4.3.
For any $R$-flat $(R,I)$-contramodule ${\mathfrak{F}}$ there is a natural
isomorphism of $(S,J)$-contramodules
$E^{\phi}({\mathfrak{F}})\simeq\varprojlim_{n}(S\otimes_{R}{\mathfrak{F}})/\allowbreak
J^{n}(S\otimes_{R}{\mathfrak{F}})$. In particular, the functor $E^{\phi}$
takes $R$-flat $(R,I)$-contramodules to $S$-flat $(S,J)$-contramodules and
very flat $(R,I)$-contramodules to very flat $(S,J)$-contramodules.
###### Proof.
Since
$F\longmapsto(S\otimes_{R}F)/J^{n}(S\otimes_{R}F)=(S\otimes_{R}F)/(J^{n}\otimes_{R}F)=S/J^{n}\otimes_{R}F$
is an exact functor on the category of flat $R$-modules $F$, so is the functor
$F\longmapsto\varprojlim_{n}(S\otimes_{R}F)/J^{n}(S\otimes_{R}F)$. Hence it
suffices to compute this functor for free ${\mathfrak{R}}$-contramodules, and
indeed we have
$S/J^{n}\otimes_{R}{\mathfrak{R}}[[X]]=S/J^{n}\otimes_{R/I^{n}}(R/I^{n}\otimes_{R}{\mathfrak{R}}[[X]])=S/J^{n}\otimes_{R/I^{n}}R/I^{n}[X]=S/J^{n}[X]$
and
$\varprojlim_{n}S/J^{n}\otimes_{R}{\mathfrak{R}}[[X]]=\varprojlim_{n}S/J^{n}[X]={\mathfrak{S}}[[X]]$,
as desired. The remaining assertions follow by the way of [54, Lemma B.9.2]
and the above Corollary C.3.5(e) (see also Lemma D.1.2 below). ∎
###### Lemma C.4.4.
(a) If the map $\bar{f}\colon R/I\longrightarrow S/J$ is surjective, then the
functor $E^{\phi}$ takes $R$-contraadjusted $(R,I)$-contramodules to
$S$-contraadjusted $(S,J)$-contramodules.
(b) If the morphism $\bar{f}\colon R/I\longrightarrow S/J$ is finite, then the
functor $E^{\phi}$ takes $R$-flat $R$-cotorsion $(R,I)$-contramodules to
$S$-flat $S$-cotorsion $(S,J)$-contramodules.
###### Proof.
Part (a): by Corollary C.2.8(b) or Corollary C.3.5(b), any $R$-contraadjusted
$(R,I)$-contramodule is a quotient $(R,I)$-contramodule of an $R$-flat
$R$-contraadjusted $(R,I)$-contramodule. The functor $E^{\phi}$ being right
exact and the class of contraadjusted $S$-modules being closed under quotents,
it suffices to show that the $S$-module $E^{\phi}({\mathfrak{P}})$ is
contraadjusted for any $R$-flat $R$-contraadjusted $(R,I)$-contramodule
${\mathfrak{P}}$. Then, by Lemma C.4.3, one has
$E^{\phi}({\mathfrak{P}})=\varprojlim_{n}E^{\phi}({\mathfrak{P}})/J^{n}E^{\phi}({\mathfrak{P}})$
and
$E^{\phi}({\mathfrak{P}})/JE^{\phi}({\mathfrak{P}})\allowbreak=S/J\otimes_{R}{\mathfrak{P}}$,
so it remains to apply Corollary C.3.2(a) together with Lemma 1.6.6(b). Part
(b) follows directly from Lemma C.4.3, Corollary C.3.2(b), and Lemma 1.6.7(a)
in the similar way. ∎
We keep assuming that $f(I)\subset J$. Let $g\colon R\longrightarrow T$ be
another ring homomorphism and $K=Tg(I)\subset T$ be the extension of the ideal
$I\subset R$ in the ring $T$. Suppose that the commutative ring
$H=S\otimes_{R}T$ is Noetherian, denote by $f^{\prime}\colon T\longrightarrow
H$ and $g^{\prime}\colon S\longrightarrow H$ the related ring homomophisms,
and set $L=Hg^{\prime}(J)\subset H$.
Let ${\mathfrak{T}}$ and ${\mathfrak{H}}$ denote the adic completions of the
rings $T$ and $H$ with respect to the ideals $K$ and $L$, and let
$\psi\colon{\mathfrak{R}}\longrightarrow{\mathfrak{T}}$,
$\phi^{\prime}\colon{\mathfrak{T}}\longrightarrow{\mathfrak{H}}$, and
$\psi^{\prime}\colon{\mathfrak{S}}\longrightarrow{\mathfrak{H}}$ be the
induced homomorphisms of topological rings.
###### Lemma C.4.5.
(a) For any $(T,K)$-contramodule ${\mathfrak{N}}$ there is a natural
isomorphism of $(S,J)$-contramodules $E^{\phi}R^{\psi}({\mathfrak{N}})\simeq
R^{\psi^{\prime}}E^{\phi^{\prime}}({\mathfrak{N}})$.
(b) Assume that the ring homomorphism $g\colon R\longrightarrow T$ induces an
open embedding of affine schemes
$\operatorname{Spec}T\longrightarrow\operatorname{Spec}R$, while the morphism
$\bar{f}\colon R/I\longrightarrow S/J$ is finite. Then for any $R$-flat
$R$-contraadjusted $(R,I)$-contramodule ${\mathfrak{P}}$ there is a natural
isomorphism of $(H,L)$-contramodules
$E^{\phi^{\prime}}(\operatorname{Hom}_{R}(T,{\mathfrak{P}}))\simeq\operatorname{Hom}_{S}(H,E^{\phi}({\mathfrak{P}}))$.
###### Proof.
Part (a): the functor of contraextension of scalars
$E^{\phi}\colon(R,I){\operatorname{\mathsf{--contra}}}\longrightarrow(S,J){\operatorname{\mathsf{--contra}}}$
is left adjoint to the functor of (contra)restriction of scalars
$R^{\phi}\colon(S,J)\allowbreak{\operatorname{\mathsf{--contra}}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$,
while the functor
$R^{\psi}\colon(T,K){\operatorname{\mathsf{--contra}}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$
is left adjoint to the functor of coextension of scalars taking an
$(R,I)$-contramodule ${\mathfrak{P}}$ to the $(T,K)$-contramodule
$\operatorname{Hom}_{R}(T,{\mathfrak{P}})$. To obtain the desired isomorphism
of functors, one can start with the obvious functorial isomorphism of
$(T,K)$-contramodules
$\operatorname{Hom}_{R}(T,R^{\phi}({\mathfrak{Q}}))\simeq
R^{\phi^{\prime}}\operatorname{Hom}_{S}(H,{\mathfrak{Q}})$ for any
$(S,J)$-contramodule ${\mathfrak{Q}}$, and then pass to the left adjoint
functors.
Part (b): the $T$-module $\operatorname{Hom}_{R}(T,{\mathfrak{P}})$ being flat
by Corollary 1.6.5(a), one has
$E^{\phi^{\prime}}(\operatorname{Hom}_{R}(T,{\mathfrak{P}}))\simeq\varprojlim_{n}\mskip
1.5muH/L^{n}\otimes_{T}\operatorname{Hom}_{R}(T,{\mathfrak{P}})\simeq\varprojlim_{n}\mskip
1.5muS/J^{n}\otimes_{R}\operatorname{Hom}_{R}(T,{\mathfrak{P}})$ and
$\operatorname{Hom}_{S}(H,E^{\phi}({\mathfrak{P}}))\simeq\operatorname{Hom}_{R}(T,E^{\phi}({\mathfrak{P}}))\simeq\operatorname{Hom}_{R}(T,\>\varprojlim_{n}\mskip
1.5muS/J^{n}\otimes_{R}{\mathfrak{P}})\simeq\varprojlim_{n}\operatorname{Hom}_{R}(T,\>S/J^{n}\otimes_{R}{\mathfrak{P}})$.
Finally, one has
$S/J^{n}\otimes_{R}\operatorname{Hom}_{R}(T,{\mathfrak{P}})\simeq\operatorname{Hom}_{R}(T,\>S/J^{n}\allowbreak\otimes_{R}{\mathfrak{P}})$
by Corollary 1.6.3(c), since the $R$-modules $S/J^{n}$ are finitely generated.
∎
###### Lemma C.4.6.
Let $R\longrightarrow S_{\alpha}$ be a collection of morphisms of commutative
rings for which the corresponding collection of morphisms of affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Let $I$ be an ideal in $R$ and $J_{\alpha}$ be its extensions
in $S_{\alpha}$. Then a contraadjusted $R$-module $P$ is an
$(R,I)$-contramodule if and only if the $S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},P)$ are
$(S_{\alpha},J_{\alpha})$-contramodules for all $\alpha$.
###### Proof.
The “only if” is a particular case of Lemma C.4.2(a), and to prove the “if”
one can use the Čech sequence (2) together with Lemma C.4.1(a) and the facts
that the class of $(R,I)$-contramodules is preserved by the functors
$\operatorname{Hom}_{R}$ from any $R$-module as well as the passages to the
cokernels of $R$-module morphisms (and, actually, kernels and extensions,
too). ∎
### C.5. Noncommutative Noetherian rings
The aim of this section is to generalize the main results of [54, Appendix B]
and the above Section C.1 to noncommutative rings. Our exposition is somewhat
sketchy with details of the arguments omitted when they are essentially the
same as in the commutative case. For a further generalization, see Sections
D.1–D.2 below.
Let $R$ be a right Noetherian associative ring, and let $m\subset R$ be an
ideal generated by central elements in $R$. Denote by
${\mathfrak{R}}=\varprojlim_{n}R/m^{n}$ the $m$-adic completion of the ring
$R$, viewed as a topological ring in the projective limit ($=$ $m$-adic)
topology. We refer to [50, Remark A.3] and [54, Section 1.2] for the
definitions of the abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ of _left
${\mathfrak{R}}$-contramodules_ and the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ (see also the beginning of Section D.1).
###### Theorem C.5.1.
The forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful. If $s_{1}$, …, $s_{k}$
are some set of central generators of the ideal $m\subset R$, then the image
of the forgetful functor consists precisely of those left $R$-modules $P$ for
which one has $\operatorname{Ext}^{*}_{R}(R[s_{j}^{-1}],P)=0$ for all $1\le
j\le k$.
In other words, extending the terminology of Section C.2 to the noncommutative
situation, one can say that a left $R$-module is an
${\mathfrak{R}}$-contramodule if and only if it is an $s_{j}$-contramodule for
every $j$.
###### Proof.
The argument follows the proof of [54, Theorem B.1.1]. To show that
$\operatorname{Ext}_{R}^{*}(R[s^{-1}],{\mathfrak{P}})=0$ for any left
${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, one can simply
(contra)restrict the scalars to the subring $K\subset R$ generated by $s$ over
${\mathbb{Z}}$ in $R$, completed adically at the ideal $m\cap K$ or
$(s)\subset K$, reducing to the commutative case, where the quoted theorem
from [54, Appendix B] is directly applicable (cf. [54, Section B.2]).
To prove the fully faithfullness and the sufficiency of condition on an
$R$-module $P$, consider the ring of formal power series
${\mathfrak{T}}=R[[t_{1},\dotsc,t_{k}]]$ with central variables $t_{j}$ and
endow it with the adic topology of the ideal generated by $t_{1}$ …, $t_{k}$.
There is a natural surjective open homomorphism of topological rings
${\mathfrak{T}}\longrightarrow{\mathfrak{R}}$ forming a commutative diagram
with the ring homomorphisms $R\longrightarrow{\mathfrak{T}}$ and
$R\longrightarrow{\mathfrak{R}}$ and taking $t_{j}$ to $s_{j}$. Consider also
the polynomial ring $T=R[t_{1},\dotsc,t_{k}]$; there are natural ring
homomorphisms $R\longrightarrow T\longrightarrow{\mathfrak{T}}$. The argument
is based on two lemmas.
###### Lemma C.5.2.
The kernel ${\mathfrak{J}}$ of the ring homomorphism
${\mathfrak{T}}\longrightarrow{\mathfrak{R}}$ is generated by the central
elements $t_{j}-s_{j}$ as an ideal in an abstract, nontopological ring
${\mathfrak{T}}$. Moreover, any family of elements in ${\mathfrak{J}}$
converging to zero in the topology of ${\mathfrak{T}}$ can be presented as a
linear combination of $k$ families of elements in ${\mathfrak{T}}$, converging
to zero in the topology of ${\mathfrak{T}}$, with the coefficients
$t_{j}-s_{j}$.
###### Proof.
The proof is similar to that in [54, Sections B.3–B.4]; the only difference is
that one has to use the noncommutative versions of Hilbert basis theorem [23,
Theorem 1.9 and Exercise 1ZA(c)] and Artin–Rees lemma [23, Theorem 13.3]. ∎
###### Lemma C.5.3.
The forgetful functor
${\mathfrak{T}}{\operatorname{\mathsf{--contra}}}\longrightarrow
T{\operatorname{\mathsf{--mod}}}$ identifies the category of left
contramodules over the topological ring ${\mathfrak{T}}$ with the full
subcategory in the category of left $T$-modules consisting of all those
modules which are $t_{j}$-contramodules for all $j$.
###### Proof.
This assertion is true for any associative ring $R$; the argument is the same
as in [54, Sections B.5–B.6]. ∎
The proof of the theorem finishes similarly to [54, Section B.7]. The category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ is identified with the full
subcategory in ${\mathfrak{T}}{\operatorname{\mathsf{--contra}}}$ consisting
of those left ${\mathfrak{T}}$-contramodules ${\mathfrak{P}}$ in which the
elements $t_{j}-s_{j}$ act by zero. The latter category coincides with the
category of $T$-modules in which the variables $t_{j}$ act the same as the
elements $s_{j}$ and which are also $t_{j}$-contramodules for all $j$. ∎
###### Proposition C.5.4.
A left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ is a flat $R$-module if
and only if the $R/m^{n}$-module ${\mathfrak{P}}/m^{n}{\mathfrak{P}}$ is flat
for every $n\ge 1$. The natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/m^{n}{\mathfrak{P}}$
is an isomorphism in this case.
###### Proof.
The same as in [54, Lemma B.9.2]. One computes the functor $M\longmapsto
M\otimes_{R}{\mathfrak{P}}$ on the category of finitely generated right
$R$-modules $M$ and uses (the noncommutative version of) the Artin–Rees lemma
for such modules $M$. ∎
###### Proposition C.5.5.
For any flat $R$-module $F$ such that the $R/m$-module $F/mF$ is projective,
and any $R$-contramodule ${\mathfrak{Q}}$, one has
$\operatorname{Ext}_{R}^{>0}(F,{\mathfrak{Q}})=0$.
###### Proof.
This assertion, provable in the same way as [54, Proposition B.10.1] (see also
Corollary D.1.8(b) below), does not depend on any Noetherianity assumptions. ∎
###### Corollary C.5.6.
(a) A left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is projective if and
only if it is a flat $R$-module and the $R/m$-module
${\mathfrak{F}}/m{\mathfrak{F}}$ is projective.
(b) The forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces isomorphisms of all the Ext groups.
###### Proof.
To prove the “only if” assertion in part (a), it suffices to notice the
isomorphism of $R/m^{n}$-modules
${\mathfrak{R}}[[X]]/m^{n}{\mathfrak{R}}[[X]]\simeq{\mathfrak{R}}/m^{n}[X]$,
which holds for any set $X$ and any $n\ge 1$, and use Proposition C.5.4. The
“if” follows from the fully faithfulness assertion in Theorem C.5.1 and
Proposition C.5.5, and part (b) follows from the same together with part (a).
(Cf. [54, Section B.8] and Section D.1 below.) ∎
###### Corollary C.5.7.
(a) The contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathfrak{R}}$-contramodules is equivalent
to the contraderived category of the exact category of $R$-flat left
${\mathfrak{R}}$-contramodules.
(b) Assume that all flat left $R/m$-modules have finite projective dimensions.
Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
is equivalent to the absolute derived category of the exact category of
$R$-flat left ${\mathfrak{R}}$-contramodules.
(c) Assume that all flat left $R/m$-modules have finite projective dimensions.
Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
is equivalent to the homotopy category of complexes of projective left/
${\mathfrak{R}}$-contramodules.
###### Proof.
Clearly, the class of flat left $R$-modules is closed under infinite products
in our assumptions, so it remains to notice that projective left
${\mathfrak{R}}$-contramdodules are $R$-flat and, in the assumption of parts
(b-c), any $R$-flat left ${\mathfrak{R}}$-contramodule has finite projective
dimension in ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$. The latter
assertions follow from Corollary C.5.6(a). (Cf. Theorems C.1.1 and D.2.1.) ∎
Let $K$ be a commutative ring, $I\subset K$ be a finitely generated ideal, and
$A$ and $B$ be associative $K$-algebras. For any $K$-module $L$, we will
denote by ${}_{I}L\subset L$ the submodule of elements annihilated by $I$ in
$L$.
###### Lemma C.5.8.
(a) For any left $A$-module $M$ and injective left $A$-module $J$ there is a
natural isomorphism of $K/I$-modules
$\operatorname{Hom}_{A}(M,J)/I\operatorname{Hom}_{A}(M,J)\simeq\operatorname{Hom}_{A/IA}({}_{I}M,{}_{I}J)$.
(b) For any right $B$-module $N$ and flat left $B$-module $F$ there is a
natural isomorphism of $K/I$-modules
${}_{I}(N\otimes_{B}F)\simeq{}_{I}N\otimes_{B/IB}F/IF$.
###### Proof.
For any finitely presented $K$-module $E$ there are natural isomorphisms
$\operatorname{Hom}_{A}(\operatorname{Hom}_{K}(E,M),J)\simeq
E\otimes_{K}\operatorname{Hom}_{A}(M,J)$ and
$\operatorname{Hom}_{K}(E,\>N\otimes_{B}F)\simeq\operatorname{Hom}_{K}(E,\allowbreak
N)\otimes_{B}F$. So it remains to take $E=K/I$. (Cf. Lemmas C.1.2 and D.2.3.)
∎
Let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a finite complex of
$A$-injective and $B$-injective $A$-$B$-bimodules over $K$ (i. e., it is
presumed that the left action of $A$ and the right action of $B$ restrict to
one and the same action of $K$ in
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$). We are using the
definition of a dualizing complex for a pair of noncommutative rings from
Section B.4.
###### Lemma C.5.9.
(a) Assume that the ring $A$ is left coherent and the ring $B$ is right
coherent. Then ${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the rings $A/IA$ and $B/IB$ whenever
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is dualizing complex the
rings $A$ and $B$.
(b) Assume that the ideal $I$ is nilpotent, the ring $A$ is left Noetherian,
and the ring $B$ is right Noetherian. Then
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing complex for
the rings $A$ and $B$ whenever
${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing
complex for the rings $A/IA$ and $B/IB$.
###### Proof.
Part (a): clearly, ${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
a complex of injective left $A/IA$-modules. The homothety map
$B\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism of finite complexes of flat left $B$-modules (see Lemma
1.6.1(b)) and therefore remains a quasi-isomorphism after taking the tensor
product with $B/IB$ over $B$ on the left, i. e., reducing modulo $I$. By Lemma
C.5.8(a), it follows that the map
$B/IB\longrightarrow\operatorname{Hom}_{A/IA}({}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism.
A bounded above complex of left modules over a left coherent ring is quasi-
isomorphic to a bounded above complex of finitely generated projective modules
if and only if its cohomology modules are finitely presented. To show that the
$A/IA$-modules of cohomology of the complex
${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are finitely
presented, we notice that the complex of left $A/IA$-modules
${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is quasi-isomorphic
to the complex
$\operatorname{Hom}_{B^{\mathrm{op}}}(L^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of right $B$-module homomorphisms from a left resolution
$L^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the right $B$-module
$B/IB$ by finitely generated projective $B$-modules into the complex
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Locally in the
cohomological grading, the complex
$\operatorname{Hom}_{B^{\mathrm{op}}}(L^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a finitely iterated cone of morphisms between shifts of copies of the
complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, so its $A$-modules
of cohomology are finitely presented. It remains to point out that an
$A/IA$-module is finitely presented if and only if it is finitely presented as
an $A$-module.
Part (b): whenever the ideal $I$ is nilpotent, a morphism of finite complexes
of flat $B$-modules is a quasi-isomorphism if and only if it becomes one after
taking the tensor products with $B/IB$ over $B$. This, together with the above
argument, proves that the homothety map
$B\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism. To show that the $A$-modules of cohomology of the
complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are finitely
generated, one can consider the spectral sequence
$E_{2}^{pq}=\operatorname{Ext}^{p}_{B^{\mathrm{op}}}(B/IB,H^{q}D^{\text{\smaller\smaller$\scriptstyle\bullet$}})\Longrightarrow
H^{p+q}\operatorname{Hom}_{B^{\mathrm{op}}}(B/IB,D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
converging from the finite direct sums of copies of the $A/I$-modules
${}_{I}H^{q}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ (on the page
$E_{1}$) to the cohomology $A/IA$-modules
$H^{n}({}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$. Then one
argues by increasing induction in $q$, using the fact that an $A$-module $M$
is finitely generated provided that so is the $A/IA$-module ${}_{I}M$. (Cf.
Lemmas C.1.3 and D.2.4.) ∎
Let $K$ be a commutative Noetherian ring, $I\subset K$ be an ideal, and $A$
and $B$ be associative $K$-algebras such that the ring $A$ is left Noetherian
and the ring $B$ is right Noetherian. A finite complex of $A$-$B$-bimodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over $K$ is
said to be a _dualizing complex for $A$ and $B$ over $(K,I)$_ if
1. (i)
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of
injective left $A$-modules and a complex of injective right $B$-modules;
2. (ii)
any element in ${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is annihilated by some power of the ideal $I$;
3. (iii)
for any (or, equivalently, some) $n\ge 1$, the complex
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={}_{I^{n}}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a dualizing complex for the rings $A/I^{n}A$ and $B/I^{n}B$ (cf. Lemma
C.5.9).
A $K$-module (or $A$-module) is said to be _$I$ -torsion_ if every its element
is annihilated by some power of the ideal $I\subset K$. A _left
$(B,I)$-contramodule_ is a left contramodule over the $I$-adic completion of
the ring $B$, or, equivalently, a left $B$-module that is a
$(K,I)$-contramodule in the $K$-module structure obtained by restriction of
scalars (see Theorem C.5.1 and Section C.1).
We denote the abelian category of $I$-torsion left $A$-modules by
$(A,I){\operatorname{\mathsf{--tors}}}$ and the abelian category of left
$(B,I)$-contramodules by $(B,I){\operatorname{\mathsf{--contra}}}$.
###### Theorem C.5.10.
The choice of a dualizing complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the rings
$A$ and $B$ over $(K,I)$ induces an equivalence between the coderived category
of $I$-torsion left $A$-modules
${\mathsf{D}}^{\mathsf{co}}((A,I){\operatorname{\mathsf{--tors}}})$ and the
contraderived category of left $(B,I)$-contramodules
${\mathsf{D}}^{\mathsf{ctr}}((B,I){\operatorname{\mathsf{--contra}}})$. The
equivalence is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}^{\mathbb{L}}{-}$.
###### Proof.
Notice first of all that an object
${\mathcal{J}}\in(A,I){\operatorname{\mathsf{--tors}}}$ is injective if and
only if the $A/I^{n}A$ modules ${}_{(n)}{\mathcal{J}}={}_{I^{n}}{\mathcal{J}}$
are injective for all $n$ (obviously), and if and only if it is an injective
$A$-module (by the Artin–Rees lemma for centrally generated ideals in
noncommutative Noetherian rings [23, Theorem 13.3]). The dual result for
$B$-flat $(B,I)$-contramodules is provided by Proposition C.5.4.
Furthermore, we identify the coderived category
${\mathsf{D}}^{\mathsf{co}}((A,I){\operatorname{\mathsf{--tors}}})$ with the
homotopy category of (complexes of) injective $I$-torsion $A$-modules (based
on the facts that there are enough injectives and the class of injectives is
closed with respect to infinite direct sums in
$(A,I){\operatorname{\mathsf{--tors}}}$) and the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((B,I){\operatorname{\mathsf{--contra}}})$ with
the absolute derived category of $B$-flat $(B,I)$-contramodules (based on
Corollary C.5.7(b) and Lemma B.4.2). The rest of the argument is no different
from the proof of Theorem C.1.4 (see also Lemma B.4.1) and based on Lemma
C.5.8. ∎
## Appendix D Ind-Affine Ind-Schemes
The aim of this appendix is to lay some bits of preparatory groundwork for the
definition of contraherent cosheaves of contramodules on ind-schemes. We
construct enough very flat and contraadjusted contramodules on an ind-affine
ind-scheme represented by a sequence of affine schemes and their closed
embeddings with finitely generated defining ideals, and also enough cotorsion
contramodules on an ind-Noetherian ind-affine ind-scheme of totally finite
Krull dimension. A version of co-contra correspondence for ind-Noetherian ind-
affine ind-schemes with dualizing complexes (and their noncommutative
generalizations) is also worked out.
### D.1. Flat and projective contramodules
Let $R_{0}\longleftarrow R_{1}\longleftarrow R_{2}\longleftarrow
R_{3}\longleftarrow\dotsb$ be a projective system of associative rings,
indexed by the ordered set of positive integers, with surjective morphisms
between them. Denote by ${\mathfrak{R}}$ the projective limit
$\varprojlim_{n}R_{n}$, viewed as a topological ring in the topology of
projective limit of discrete rings $R_{n}$. Clearly, the ring homomorphisms
${\mathfrak{R}}\longrightarrow R_{n}$ are surjective; let
${\mathfrak{I}}_{n}\subset{\mathfrak{R}}$ denote their kernels. Then the open
ideals ${\mathfrak{I}}_{n}$ form a base of neighborhoods of zero in the
topological ring ${\mathfrak{R}}$.
We are interested in left contramodules over the topological ring
${\mathfrak{R}}$ (see [50, Remark A.3] and [54, Section 1.2] for the
definition). They form an abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ having enough projective
objects and endowed with an exact and faithful forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
preserving infinite products. Here
${\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$ denotes the abelian category
of left modules over the ring ${\mathfrak{R}}$ (viewed as an abstract ring
without any topology).
The projective ${\mathfrak{R}}$-contramodules are precisely the direct
summands of the free ${\mathfrak{R}}$-contramodules ${\mathfrak{R}}[[X]]$.
Here $X$ is an arbitrary set of generators, and
${\mathfrak{R}}[[X]]=\varprojlim_{n}R_{n}[X]$ is the set of all maps
$X\longrightarrow{\mathfrak{R}}$ converging to zero in the topology of
${\mathfrak{R}}$.
For any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ and any closed
two-sided ideal ${\mathfrak{J}}\subset{\mathfrak{R}}$, we will denote by
${\mathfrak{J}}{\times}{\mathfrak{P}}\subset{\mathfrak{P}}$ the image of the
contraaction map
${\mathfrak{J}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$. As usually,
for any left module $M$ over a ring $R$ and any ideal $J\subset R$ the
notation $JM$ (or, as it may be sometimes convenient, $J\mskip 1.5mu{\cdot}M$)
stands for the image of the action map $J\otimes_{R}M\longrightarrow M$. For
an ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, there is a (generally
speaking, proper) inclusion ${\mathfrak{J}}{\mathfrak{P}}={\mathfrak{J}}\mskip
1.5mu{\cdot}\mskip
1.5mu{\mathfrak{P}}\subset{\mathfrak{J}}{\times}{\mathfrak{P}}$. Of course,
${\mathfrak{J}}{\times}{\mathfrak{P}}$ is an ${\mathfrak{R}}$-subcontramodule
in ${\mathfrak{P}}$, while ${\mathfrak{J}}{\mathfrak{P}}$ is in general only
an ${\mathfrak{R}}$-submodule.
###### Lemma D.1.1.
For any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, the natural map
to the projective limit
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/({\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$
is surjective.
###### Proof.
This assertion is true for any topological ring ${\mathfrak{R}}$ with a base
of topology consisting of open right ideals, and any decreasing sequence of
closed abelian subgroups
${\mathfrak{R}}\supset{\mathfrak{I}}_{0}\supset{\mathfrak{I}}_{1}\supset{\mathfrak{I}}_{2}\supset\dotsb$
converging to zero in the topology of ${\mathfrak{R}}$ (meaning that for any
neighborhood of zero ${\mathfrak{U}}\subset{\mathfrak{R}}$ there exists $n\ge
0$ such that ${\mathfrak{I}}_{n}\subset{\mathfrak{U}}$). Indeed, it suffices
to show that for any sequence of elements
$p_{n}\in{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$, $n\ge 0$, there exists an
element $p\in{\mathfrak{P}}$ such that
$p-(p_{0}+\dotsb+p_{n})\in{\mathfrak{I}}_{n+1}{\times}{\mathfrak{P}}$ for all
$n\ge 0$.
Now each element $p_{n}$ can be obtained as an infinite linear combination of
elements of ${\mathfrak{P}}$ with the family of coefficients converging to
zero in ${\mathfrak{I}}_{n}$. The countable sum of such expressions over all
$n\ge 0$ is again an infinite linear combination of elements of
${\mathfrak{P}}$ with the coefficients converging to zero in ${\mathfrak{R}}$.
The value of the latter, understood in the sense of the contramodule infinite
summation operations in ${\mathfrak{P}}$, provides the desired element
$p\in{\mathfrak{P}}$. Another (and perhaps more illuminating) argument can be
found in [50, Lemma A.2.3] (while counterexamples showing that the map in
question may not be injective are provided in [50, Section A.1]). ∎
###### Lemma D.1.2.
Let $P_{0}\longleftarrow P_{1}\longleftarrow P_{2}\longleftarrow\dotsb$ be a
projective system of left $R_{n}$-modules in which the morphism
$P_{n+1}\longrightarrow P_{n}$ identifies $P_{n}$ with
$R_{n}\otimes_{R_{n+1}}P_{n+1}$ for every $n\ge 0$. Let ${\mathfrak{P}}$
denote the ${\mathfrak{R}}$-contramodule $\varprojlim_{n}P_{n}$. Then the
natural map ${\mathfrak{P}}\longrightarrow P_{n}$ identifies $P_{n}$ with
${\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$. Conversely, for any
left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ the projective system
$P_{n}={\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$ satisfies the
above condition.
###### Proof.
Since $P_{n}$ is an $R_{n}$-module and ${\mathfrak{P}}\longrightarrow P_{n}$
is a morphism of ${\mathfrak{R}}$-contramodules, the kernel of this morphism
contains ${\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$. To prove the inverse
inclusion, consider an element $p\in{\mathfrak{P}}$ belonging to the kernel of
the morphism ${\mathfrak{P}}\longrightarrow P_{n}$. The image of the element
$p$ in $P_{n+1}$ belongs to $({\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1})P_{n+1}$
(where ${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$ is an ideal in $R_{n+1}$).
Let us decompose this image accordingy into a finite linear combination of
elements of $P_{n+1}$ with coefficients from
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$, lift all the entering elements of
$P_{n+1}$ to ${\mathfrak{P}}$ and all the elements of
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$ to ${\mathfrak{I}}_{n}$, and
subtract from $p$ the corresponding finite linear combination of elements of
${\mathfrak{P}}$ with coefficients in ${\mathfrak{I}}_{n}$.
The image of the resulting element $p^{\prime}\in{\mathfrak{P}}$ in $P_{n+1}$
vanishes, so its image in $P_{n+2}$ belongs to
$({\mathfrak{I}}_{n+1}/{\mathfrak{I}}_{n+2})P_{n+2}$. Continuing this process
indefinitely, we obtain an expression of the original element $p$ in the form
of a countable linear combination of elements from ${\mathfrak{P}}$ with the
coefficient sequence converging to zero in ${\mathfrak{I}}_{n}$. This proves
the first assertion; the second one is straightforward. ∎
We will say that a left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is
_flat_ if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is an isomorphism and the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are flat for all
$n$.
###### Lemma D.1.3.
If ${\mathfrak{G}}\longrightarrow{\mathfrak{F}}$ is a surjective morphism of
flat ${\mathfrak{R}}$-contramodules then its kernel ${\mathfrak{H}}$ is also a
flat ${\mathfrak{R}}$-contramodule. Moreover, the sequences of $R_{n}$-modules
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ are exact.
###### Proof.
Clearly, for any short exact sequence of ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ there are short exact sequences of $R_{n}$-modules
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{H}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{G}})\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ (because the map
${\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
induced by a surjective map ${\mathfrak{G}}\longrightarrow{\mathfrak{F}}$ is
surjective). If the $R_{n}$-module
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ is flat, then the
tensor product with $R_{n-1}$ over $R_{n}$ transforms this sequence into the
similar sequence corresponding to the ideal
${\mathfrak{I}}_{n-1}\subset{\mathfrak{R}}$.
On the other hand, if the maps
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
and
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$
are isomorphisms, then the passage to the projective limit of the above short
exact sequences allows to conclude that the map
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{H}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{G}})$
is an isomorphism. By Lemma D.1.2, it follows from these observations that
${\mathfrak{H}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{G}})={\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$,
so the sequences
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ are exact. Finally, now if the $R_{n}$-modules
${\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$ are also flat, then
so are the $R_{n}$-modules
${\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$. ∎
###### Lemma D.1.4.
If
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ is a short exact sequence of ${\mathfrak{R}}$-contramodules and the
${\mathfrak{R}}$-contramodules ${\mathfrak{H}}$ and ${\mathfrak{F}}$ are flat,
then so is the ${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$.
###### Proof.
In view of the proof of Lemma D.1.3, we only need to show that the map
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$
is an isomorphism. Choose a termwise surjective map onto the short exact
sequence
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ from a short exact sequence of free ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{W}}\longrightarrow{\mathfrak{V}}\longrightarrow{\mathfrak{U}}\longrightarrow
0$ (e. g., ${\mathfrak{W}}={\mathfrak{R}}[[{\mathfrak{H}}]]$,
${\mathfrak{U}}={\mathfrak{R}}[[{\mathfrak{F}}]]$ or
${\mathfrak{R}}[[{\mathfrak{G}}]]$, and
${\mathfrak{V}}={\mathfrak{W}}\oplus{\mathfrak{U}}$). Let
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{K}}\longrightarrow
0$ be the corresponding short exact sequence of kernels.
Passing to the projective limit of short exact sequences
$0\longrightarrow{\mathfrak{L}}/{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{V}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow
0$, we obtain a short exact sequence
$0\longrightarrow\varprojlim_{n}{\mathfrak{L}}/{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow\varprojlim_{n}{\mathfrak{V}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow\varprojlim{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow
0$. The ${\mathfrak{R}}$-contramodule ${\mathfrak{V}}$ being free, the
intersection
$\bigcap_{n}{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\subset{\mathfrak{V}}$
vanishes; so both maps
${\mathfrak{V}}\longrightarrow\varprojlim_{n}{\mathfrak{V}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}$
and
${\mathfrak{L}}\longrightarrow\varprojlim_{n}{\mathfrak{L}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{L}}$
are isomorphisms, while the map
${\mathfrak{L}}\longrightarrow\varprojlim_{n}{\mathfrak{L}}/{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})$
is, at least, injective. It remains to show that the latter map is surjective;
equivalently, it means vanishing of the derived projective limit
$\varprojlim^{1}_{n}{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})$.
Similarly, the assumptions that the maps
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$
and
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
are isomorphisms are equivalently expressed as the vanishing of derived
projective limits
$\varprojlim^{1}_{n}{\mathfrak{M}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{W}})$
and
$\varprojlim^{1}_{n}{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})$.
The short sequences
$0\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{W}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{U}}\longrightarrow
0$ are exact, because the short sequence
$0\longrightarrow{\mathfrak{W}}\longrightarrow{\mathfrak{V}}\longrightarrow{\mathfrak{U}}\longrightarrow
0$ splits. Passing to the fibered product of two short exact sequences
$0\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{W}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{U}}\longrightarrow
0$ and
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{K}}\longrightarrow
0$ over the short exact sequence
$0\longrightarrow{\mathfrak{W}}\longrightarrow{\mathfrak{V}}\longrightarrow{\mathfrak{U}}\longrightarrow
0$ (into which both of them are embedded), we obtain an exact sequence
$0\longrightarrow{\mathfrak{M}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{W}})\longrightarrow{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})$.
The ${\mathfrak{R}}$-contramodules ${\mathfrak{U}}$ and ${\mathfrak{F}}$ being
flat, by Lemma D.1.3 we have
${\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})={\mathfrak{I}}_{n}{\times}{\mathfrak{K}}$.
It follows that the map
${\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})$
is surjective, so the whole sequence
$0\longrightarrow{\mathfrak{M}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{W}})\longrightarrow{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})\longrightarrow
0$ is exact. Applying the right exact functor $\varinjlim^{1}_{n}$ (which is,
in particular, exact in the middle), we obtain the desired vanishing. ∎
###### Lemma D.1.5.
Any projective ${\mathfrak{R}}$-contramodule is flat.
###### Proof.
It suffices to consider the case of a free left ${\mathfrak{R}}$-contramodule
${\mathfrak{R}}[[X]]$. For any set $X$ and any closed ideal
${\mathfrak{J}}\subset{\mathfrak{R}}$ one has
${\mathfrak{J}}{\times}({\mathfrak{R}}[[X]])={\mathfrak{J}}[[X]]$ and
${\mathfrak{R}}[[X]]/{\mathfrak{J}}[[X]]\simeq({\mathfrak{R}}/{\mathfrak{J}})[[X]]$.
So in particular
${\mathfrak{R}}[[X]]/({\mathfrak{I}}_{n}{\times}{\mathfrak{R}}[[X]])=R_{n}[X]$
is a flat (and even free and projective) left $R_{n}$-module and the natural
${\mathfrak{R}}$-contramodule morphism
${\mathfrak{R}}[[X]]\longrightarrow\varprojlim_{n}{\mathfrak{R}}[[X]]/({\mathfrak{I}}_{n}{\times}{\mathfrak{R}}[[X]])$
is an isomorphism. ∎
Given two ${\mathfrak{R}}$-contramodules ${\mathfrak{P}}$ and
${\mathfrak{Q}}$, we will denote by
$\operatorname{Hom}^{\mathfrak{R}}({\mathfrak{P}},{\mathfrak{Q}})$ and
$\operatorname{Ext}^{{\mathfrak{R}},*}({\mathfrak{P}},{\mathfrak{Q}})$ the
$\operatorname{Hom}$ and $\operatorname{Ext}$ groups in the abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$.
###### Lemma D.1.6.
Let ${\mathfrak{F}}$ be a flat left ${\mathfrak{R}}$-contramodule and
${\mathfrak{P}}$ be a left ${\mathfrak{R}}$-contramodule for which the natural
map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism. Set
$F_{n}={\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ and
$P_{n}={\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$, and assume
that one has
$\operatorname{Ext}^{1}_{R_{0}}(F_{0},P_{0})=0=\operatorname{Ext}^{1}_{R_{n+1}}(F_{n+1},\>\allowbreak\ker(P_{n+1}\to
P_{n}))$ for all $n\ge 0$. Then
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{P}})=0$.
###### Proof.
Let ${\mathfrak{G}}\longrightarrow{\mathfrak{F}}$ be a surjective morphism
onto ${\mathfrak{F}}$ from a projective ${\mathfrak{R}}$-contramodule
${\mathfrak{G}}$ with the kernel ${\mathfrak{H}}$. Set
$G_{n}={\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$ and similarly
for $H_{n}$. By Lemmas D.1.5 and D.1.3, the ${\mathfrak{R}}$-contramodules
${\mathfrak{G}}$ and ${\mathfrak{H}}$ are flat, and the short sequences
$0\longrightarrow H_{n}\longrightarrow G_{n}\longrightarrow
F_{n}\longrightarrow 0$ are exact. Let us show that any
${\mathfrak{R}}$-contramodule morphism
${\mathfrak{H}}\longrightarrow{\mathfrak{P}}$ can be extended to an
${\mathfrak{R}}$-contramodule morphism
${\mathfrak{G}}\longrightarrow{\mathfrak{P}}$.
The data of a morphism of ${\mathfrak{R}}$-contramodules
${\mathfrak{H}}\longrightarrow{\mathfrak{P}}$ is equivalent to that of a
morphism of projective systems of $R_{n}$-modules $H_{n}\longrightarrow
P_{n}$. Let us construct by induction an extension of this morphism to a
morphism of projective systems $G_{n}\longrightarrow P_{n}$ for $n\ge 0$.
Since $\operatorname{Ext}_{R_{0}}^{1}(F_{0},P_{0})=0$, the case of $n=0$ is
clear. Assuming that the morphism $G_{n}\longrightarrow P_{n}$ has been
obtained already, we will proceed to construct a compatible morphism
$G_{n+1}\longrightarrow P_{n+1}$.
Since $G_{n+1}$ is a projective $R_{n+1}$-module, the composition
$G_{n+1}\longrightarrow G_{n}\longrightarrow P_{n}$ can be lifted to an
$R_{n+1}$-module morphism $G_{n+1}\longrightarrow P_{n+1}$. (Notice that what
is actually used here is the vanishing of
$\operatorname{Ext}^{1}_{R_{n+1}}(G_{n+1},\>\ker(P_{n+1}\to P_{n}))$.) The
composition of a morphism so obtained with the embedding
$H_{n+1}\longrightarrow G_{n+1}$ differs from the given map
$H_{n+1}\longrightarrow P_{n+1}$ by an $R_{n+1}$-module morphism
$H_{n+1}\longrightarrow\ker(P_{n+1}\to P_{n})$. Given that
$\operatorname{Ext}^{1}_{R_{n+1}}(F_{n+1},\>\ker(P_{n+1}\to P_{n}))=0$, the
latter map can be extended from $H_{n+1}$ to $G_{n+1}$ and added to the
previously constructed map $G_{n+1}\longrightarrow P_{n+1}$. ∎
Suppose $R$ is an associative ring endowed with a descending sequence of two-
sided ideals $R\supset I_{0}\supset I_{1}\supset I_{2}\supset\dotsb$ such that
the projective system of quotient rings $R/I_{n}$ is isomorphic to our
original projective system $R_{0}\longleftarrow R_{1}\longleftarrow
R_{2}\longleftarrow\dotsb$. For example, one can always take
$R={\mathfrak{R}}$ and $I_{n}={\mathfrak{I}}_{n}$; sometimes there may be
other suitable choices of a ring $R$ with the ideals $I_{n}$ as well.
Then there is a natural ring homomorphism $R\longrightarrow{\mathfrak{R}}$
inducing the isomorphisms $R/I_{n}\simeq{\mathfrak{R}}/{\mathfrak{I}}_{n}$.
The restriction of scalars provides a forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$, which is exact and faithful, and preserves
infinite products. Given two $R$-modules $L$ and $M$, we denote, as usually,
by $\operatorname{Hom}_{R}(L,M)$ and $\operatorname{Ext}_{R}^{*}(L,M)$ the Hom
and Ext groups in the abelian category of $R$-modules.
###### Lemma D.1.7.
Let $F$ be a flat left $R$-module and ${\mathfrak{P}}$ be a left
${\mathfrak{R}}$-contramodule for which the natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism. Set
$P_{n}={\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$, and assume
that that one has
$\operatorname{Ext}^{1}_{R_{0}}(F/I_{0}F,\>P_{0})=0=\operatorname{Ext}^{1}_{R_{n+1}}(F/I_{n+1}F,\>\allowbreak\ker(P_{n+1}\to\nobreak
P_{n}))$ for all $n\ge 0$. Then
$\operatorname{Ext}^{1}_{R}(F,{\mathfrak{P}})=0$.
###### Proof.
The proof is the same as in Lemma C.3.1. ∎
###### Corollary D.1.8.
(a) A left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is projective if and
only if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is an isomorphism and the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are projective for
all $n\ge 0$.
(b) For any flat left $R$-module $F$ such that the $R_{n}$-modules $F/I_{n}F$
are projective for all $n\ge 0$, and any left ${\mathfrak{R}}$-contramodule
${\mathfrak{Q}}$, one has
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{Q}})\simeq\operatorname{Hom}_{R}(F,{\mathfrak{Q}})$
and
$\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{Q}})=\operatorname{Ext}_{R}^{>0}(F,{\mathfrak{Q}})=0$.
###### Proof.
Part (a): the “only if” assertion holds by (the proof of) Lemma D.1.5. To
prove the “if”, consider a short exact sequence of
${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ with a projective ${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$. The
natural map
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$
being an isomorphism because the map
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$
is (or by Lemma D.1.3), one has
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{H}})=0$ by
Lemma D.1.6, hence the short exact sequence splits and the
${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is a direct summand of the
${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$.
Part (b): a natural map
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{Q}})\longrightarrow\operatorname{Hom}_{R}(F,{\mathfrak{Q}})$
for any $R$-module $F$ and ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ is
induced by the $R$-module morphism $F\longrightarrow\varprojlim_{n}F/I_{n}F$.
For an ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ such that the map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism, one has
$\operatorname{Hom}_{R}(F,{\mathfrak{P}})\simeq\varprojlim_{n}\operatorname{Hom}_{R_{n}}(F/I_{n}F,\>{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$,
which is isomorphic to
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})$
by Lemma D.1.2.
When $F$ is also a flat $R$-module with projective $R_{n}$-modules $F/I_{n}F$,
one has $\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{P}})=0$ by Lemma D.1.7 and
$\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})=0$
by part (a). Now to prove the assertion of part (b) in the general case, it
suffices to present an ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ as the
cokernel of an injective morphism of ${\mathfrak{R}}$-contramodules
${\mathfrak{K}}\longrightarrow{\mathfrak{P}}$ with
${\mathfrak{P}}=\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
(and, consequently, the same for ${\mathfrak{K}}$). ∎
When the ideals ${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\subset R_{n+1}$ are
nilpotent (i. e., for every $n\ge 0$ there exists $N_{n}\ge 1$ such that
$({\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1})^{N_{n}}=0$), it follows from
Corollary D.1.8(a) and [54, Lemma B.10.2] that a left
${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is projective if and only if it
is flat and the $R_{0}$-module
${\mathfrak{F}}/{\mathfrak{I}}_{0}{\times}{\mathfrak{F}}$ is projective.
Similarly, it suffices to require that the $R_{0}$-module $F/I_{0}F$ be
projective in Corollary D.1.8(b) in this case.
### D.2. Co-contra correspondence
In this section we consider a pair of projective systems $R_{0}\longleftarrow
R_{1}\longleftarrow R_{2}\longleftarrow\dotsb$ and $S_{0}\longleftarrow
S_{1}\longleftarrow S_{2}\longleftarrow\dotsb$ of associative rings and
surjective morphisms between them. We assume that the rings $S_{n}$ are left
Noetherian, the rings $R_{n}$ are right coherent, and the kernels
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\subset R_{n+1}$ of the ring
homomorphisms $R_{n+1}\longrightarrow R_{n}$ are finitely generated as right
ideals.
Set ${\mathfrak{S}}=\varprojlim_{n}S_{n}$, and denote by
${\mathfrak{J}}_{n}\subset{\mathfrak{S}}$ the kernels of the natural
surjective ring homomorphisms ${\mathfrak{S}}\longrightarrow S_{n}$. Given a
left ${\mathfrak{S}}$-module $M$, we denote by ${}_{S_{n}}\\!M\subset M$ its
submodule consisting of all the elements annihilated by ${\mathfrak{J}}_{n}$;
so ${}_{S_{n}}\\!M$ is the maximal left $S_{n}$-submodule in $M$. A left
${\mathfrak{S}}$-module ${\mathcal{M}}$ is said to be _discrete_ if every its
element is annihilated by the ideal ${\mathfrak{J}}_{n}$ for $n$ large enough.
In other words, a left ${\mathfrak{S}}$-module ${\mathcal{M}}$ is discrete if
its increasing filtration
${}_{S_{0}}{\mathcal{M}}\subset{}_{S_{1}}{\mathcal{M}}\subset{}_{S_{2}}{\mathcal{M}}\subset\dotsb$
is exhaustive, or equivalently, if the left action map
${\mathfrak{S}}\times{\mathcal{M}}\longrightarrow{\mathcal{M}}$ is continuous
in the projective limit topology of ${\mathfrak{S}}$ and the discrete topology
of ${\mathcal{M}}$. We denote the full abelian subcategory of discrete left
${\mathfrak{S}}$-modules by
${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}\subset{\mathfrak{S}}{\operatorname{\mathsf{--mod}}}$.
###### Theorem D.2.1.
(a) The coderived category
${\mathsf{D}}^{\mathsf{co}}({\mathfrak{S}}{\operatorname{\mathsf{--discr}}})$
of the abelian category of discrete left ${\mathfrak{S}}$-modules is
equivalent to the homotopy category of complexes of injective discrete left
${\mathfrak{S}}$-modules.
(b) The contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathfrak{R}}$-contramodules is equivalent
to the contraderived category of the exact category of flat left
${\mathfrak{R}}$-contramodules.
###### Proof.
Part (a) holds, because the category
${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}$, being a locally Noetherian
Grothendieck abelian category, has enough injective objects, and injectivity
in ${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}$ is preserved by infinite
direct sums.
Since any left ${\mathfrak{R}}$-contramodule is a quotient contramodule of a
flat (and even projective) one, and the class of flat left
${\mathfrak{R}}$-contramodules is closed under extensions and the passage to
the kernels of surjective morphisms (see Lemmas D.1.3–D.1.5), in order to
prove the assertion (b) it only remains to show that the class of flat left
${\mathfrak{R}}$-contramodules is preserved by infinite products (see
Proposition A.3.1(b)). The latter follows from the definition of flatness for
left ${\mathfrak{R}}$-contramodules, the coherence condition on the rings
$R_{n}$, and the next lemma. ∎
###### Lemma D.2.2.
For any family of left ${\mathfrak{R}}$-contramodules
${\mathfrak{P}}_{\alpha}$ and any $n\ge 0$, the two
${\mathfrak{R}}$-subcontramodules
${\mathfrak{I}}_{n}{\times}\prod_{\alpha}{\mathfrak{P}}_{\alpha}$ and
$\prod_{\alpha}{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}_{\alpha}$ coincide in
$\prod_{\alpha}{\mathfrak{P}}_{\alpha}$.
###### Proof.
The former subcontramodule is obviously contained in the latter one; we have
to prove the converse inclusion. For every $m\ge 0$, pick a finite set of
generators $\bar{r}_{m}^{\gamma}$ of the right ideal
${\mathfrak{I}}_{m}/{\mathfrak{I}}_{m+1}\subset{\mathfrak{R}}/{\mathfrak{I}}_{m+1}$,
and lift them to elements $r_{m}^{\gamma}\in{\mathfrak{R}}$. Then, for any
left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, any element of the
subcontramodule
${\mathfrak{I}}_{n}{\times}{\mathfrak{P}}\subset{\mathfrak{P}}$ can be
expressed in the form $\sum_{m\ge n}^{\gamma}r_{m}^{\gamma}p_{m}^{\gamma}$
with some $p_{m}^{\gamma}\in{\mathfrak{P}}$. In particular, any element $p$ of
the product $\prod_{\alpha}{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}_{\alpha}$
can be presented in the form $p=\big{(}\sum_{m\ge
n}^{\gamma}r_{m}^{\gamma}p_{m}^{\gamma,\alpha}\big{)}_{\alpha}$ with some
elements $p_{m}^{\gamma,\alpha}\in{\mathfrak{P}}_{\alpha}$. Now one has
$\big{(}\sum_{m\ge
n}^{\gamma}r_{m}^{\gamma}p_{m}^{\gamma,\alpha}\big{)}_{\alpha}=\sum_{m\ge
n}^{\gamma}r_{m}^{\gamma}((p_{m}^{\gamma,\alpha})_{\alpha})$, which is an
infinite sum of elements of $\prod_{\alpha}{\mathfrak{P}}_{\alpha}$ with the
coefficient family still converging to zero in
${\mathfrak{I}}_{n}\subset{\mathfrak{R}}$, proving that $p$ belongs to
${\mathfrak{I}}_{n}{\times}\prod_{\alpha}{\mathfrak{P}}_{\alpha}$. (Cf. [54,
Lemmas 1.3.6–1.3.7].) ∎
A right ${\mathfrak{R}}$-module ${\mathcal{N}}$ is said to be discrete if
every its element is annihilated by the ideal
${\mathfrak{I}}_{n}=\ker({\mathfrak{R}}\to R_{n})$ for $n\gg 0$, i. e., in the
notation similar to the above, if
${\mathcal{N}}=\bigcup_{n}{\mathcal{N}}_{R_{n}}$. The abelian category of
discrete right ${\mathfrak{R}}$-modules is denoted by
${\operatorname{\mathsf{discr--}}}{\mathfrak{R}}$. For any associative ring
$S$, any ${\mathfrak{R}}$-discrete $S$-${\mathfrak{R}}$-bimodule
${\mathcal{K}}$, and any left $S$-module $U$, the abelian group
$\operatorname{Hom}_{S}({\mathcal{N}},U)$ is naturally endowed with a left
${\mathfrak{R}}$-contramodule structure as the projective limit of the
sequence of left $R_{n}$-modules
$\operatorname{Hom}_{S}({\mathcal{N}},U)=\varprojlim_{n}\operatorname{Hom}_{S}({\mathcal{N}}_{R_{n}},U)$.
For any discrete right ${\mathfrak{R}}$-module ${\mathcal{N}}$ and any left
${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, their _contratensor product_
${\mathcal{N}}\odot_{\mathfrak{R}}{\mathfrak{P}}$ is defined as the inductive
limit of the sequence of abelian groups
$\varinjlim_{n}{\mathcal{N}}_{R_{n}}\otimes_{R_{n}}{\mathfrak{P}}/({\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$.
To give a more fancy definition, the contratensor product
${\mathcal{N}}\odot_{\mathfrak{R}}{\mathfrak{P}}$ is the cokernel (of the
difference) of the natural pair of abelian group homomorphisms
${\mathcal{N}}\otimes_{\mathbb{Z}}{\mathfrak{R}}[[{\mathfrak{P}}]]\birarrow{\mathcal{N}}\otimes_{\mathbb{Z}}{\mathfrak{P}}$.
Here one map is induced by the left contraaction map
${\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$ and the other
one is the composition
${\mathcal{N}}\otimes_{\mathbb{Z}}{\mathfrak{R}}[[{\mathfrak{P}}]]\longrightarrow{\mathcal{N}}[{\mathfrak{P}}]\longrightarrow{\mathcal{N}}\otimes_{\mathbb{Z}}{\mathfrak{P}}$,
where ${\mathcal{N}}[{\mathfrak{P}}]$ is the group of all finite formal linear
combinations of elements of ${\mathfrak{P}}$ with the coefficients in
${\mathcal{N}}$, the former map to be composed is induced by the discrete
right action map
${\mathcal{N}}\times{\mathfrak{R}}\longrightarrow{\mathcal{N}}$, and the
latter map is just the obvious one. The contratensor product is a right exact
functor of two arguments
$\odot_{\mathfrak{R}}\colon{\operatorname{\mathsf{discr--}}}{\mathfrak{R}}\times{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathbb{Z}}{\operatorname{\mathsf{--mod}}}$.
For any ${\mathfrak{R}}$-discrete $S$-${\mathfrak{R}}$-bimodule
${\mathcal{K}}$, any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, and
any left $S$-module $U$, there are natural isomorphisms of abelian groups
$\operatorname{Hom}_{S}({\mathcal{K}}\odot_{\mathfrak{R}}{\mathfrak{P}},\>U)\simeq\varprojlim_{n}\operatorname{Hom}_{S}({\mathcal{K}}_{R_{n}}\otimes_{R_{n}}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}},\>U)\simeq\varprojlim_{n}\operatorname{Hom}_{R_{n}}({\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}},\>\operatorname{Hom}_{S}({\mathcal{K}}_{R_{n}},U))\simeq\operatorname{Hom}^{\mathfrak{R}}({\mathfrak{P}},\operatorname{Hom}_{S}({\mathcal{K}},U))$
(cf. [50, Sections 3.1.2 and 5.1.1]).
Given surjective ring homomorphisms $g\colon S\longrightarrow{}^{\prime}\\!S$
and $f\colon R\longrightarrow{}^{\prime}\\!R$, a left $S$-module $M$, and a
right $R$-module $N$, we denote by ${}_{{}^{\prime}\\!S}M$ the submodule of
all elements annihilated by the left action of $\ker(g)$ in $M$ and by
$N_{\mskip 1.5mu{}^{\prime}\\!R}$ the submodule of all elements annihilated by
the right action of $\ker(f)$ in $N$. So ${}_{{}^{\prime}\\!S}M$ is the
maximal left ${}^{\prime}\\!S$-submodule in $M$ and $N_{\mskip
1.5mu{}^{\prime}\\!R}$ is the maximal right ${}^{\prime}\\!R$-submodule in
$N$. Similarly, for any left $R$-module $P$ we denote by
${}^{{}^{\prime}\\!R\\!}P$ its maximal quotient left ${}^{\prime}\\!R$-module
$P/\ker(f)P$.
###### Lemma D.2.3.
Assume that $\ker(g)$ is finitely generated as a left ideal in $S$ and
$\ker(f)$ is finitely generated as a right ideal in $R$. Let $K$ be an
$S$-$R$-bimodule such that its submodules ${}_{{}^{\prime}\\!S}K$ and
$K_{\mskip 1.5mu{}^{\prime}\\!R}$ coincide; denote this
${}^{\prime}\\!S$-${}^{\prime}\\!R$-subbimodule in $K$ by ${}^{\prime}\\!K$.
Then
(a) for any injective left $S$-module $J$ there is a natural isomorphism of
left ${}^{\prime}\\!R$-modules
${}^{{}^{\prime}\\!R\\!}\operatorname{Hom}_{S}(K,J)\simeq\operatorname{Hom}_{\mskip
1.5mu{}^{\prime}\\!S}({}^{\prime}\\!K,\mskip 1.5mu{}_{{}^{\prime}\\!S}J)$;
(b) for any flat left $R$-module $F$ there is a natural isomorphism of left
${}^{\prime}\\!S$-modules
${}_{{}^{\prime}\\!S}(K\otimes_{R}F)\simeq{}^{\prime}\\!K\otimes_{\,{}^{\prime}\\!R}{}^{{}^{\prime}\\!R\\!}F$.
###### Proof.
Part (a): for any finitely presented right $R$-module $E$, there is a natural
isomorphism
$\operatorname{Hom}_{S}(\operatorname{Hom}_{R^{\mathrm{op}}}(E,K),J)\simeq
E\otimes_{R}\operatorname{Hom}_{S}(K,J)$. Taking $E={}^{\prime}\\!R$, we get
$\operatorname{Hom}_{\mskip 1.5mu{}^{\prime}\\!S}({}_{{}^{\prime}\\!S}K,\mskip
1.5mu{}_{{}^{\prime}\\!S}J)\simeq\operatorname{Hom}_{S}({}_{{}^{\prime}\\!S}K,J)=\operatorname{Hom}_{S}(K_{\mskip
1.5mu{}^{\prime}\\!R},\mskip
1.5muJ)\simeq{}^{{}^{\prime}\\!R\\!}\operatorname{Hom}_{S}(K,J)$.
Part (b): for any finitely presented left $S$-module $E$ there is a natural
isomorphism
$\operatorname{Hom}_{S}(E,\>K\otimes_{R}F)\simeq\operatorname{Hom}_{S}(E,K)\otimes_{R}F$.
Taking $E={}^{\prime}\\!S$, we get $K_{\mskip
1.5mu{}^{\prime}\\!R}\otimes_{\,{}^{\prime}\\!R}{}^{{}^{\prime}\\!R\\!}F\simeq
K_{\mskip
1.5mu{}^{\prime}\\!R}\otimes_{R}F={}_{{}^{\prime}\\!S}K\otimes_{R}F\simeq{}_{{}^{\prime}\\!S}(K\otimes_{R}F)$.
∎
We recall the definition of a dualizing complex for a pair of noncommutative
rings from Section B.4.
###### Lemma D.2.4.
Let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a finite complex of
$S$-injective and $R$-injective $S$-$R$-bimodules. Suppose that the
subcomplexes
${}_{{}^{\prime}\\!S}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\mskip
1.5mu{}^{\prime}\\!R}$ coincide in
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and denote this complex of
${}^{\prime}\\!S$-${}^{\prime}\\!R$-bimodules by
${}^{\prime}\\!D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
(a) Assume that the ring $S$ is left coherent, the ring $R$ is right coherent,
$\ker(g)$ is finitely generated as a left ideal in $S$, and $\ker(f)$ is
finitely generated as a right ideal in $R$. Then
${}^{\prime}\\!D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the rings ${}^{\prime}\\!S$ and ${}^{\prime}\\!R$
whenever $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is dualizing
complex for the rings $S$ and $R$.
(b) Assume that the ring $S$ is left Noetherian, the ring $R$ is right
Noetherian, and the ideals $\ker(g)\subset S$ and $\ker(f)\subset R$ are
nilpotent. Then $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the rings $S$ and $R$ whenever
${}^{\prime}\\!D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the rings ${}^{\prime}\\!S$ and ${}^{\prime}\\!R$.
###### Proof.
Argue as in the proof of Lemma C.5.9, using Lemma D.2.3 in place of Lemma
C.5.8. ∎
A complex of ${\mathfrak{S}}$-${\mathfrak{R}}$-bimodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is called a
_dualizing complex_ for the projective systems of rings $(S_{n})$ and
$(R_{n})$ if
1. (i)
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of
injective discrete left ${\mathfrak{S}}$-modules and a complex of injective
discrete right ${\mathfrak{R}}$-modules;
2. (ii)
for every $n\ge 0$, the two subcomplexes
${}_{S_{n}}\\!{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and ${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{R_{n}}$
coincide in ${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$;
and
3. (iii)
the latter subcomplex, denoted by
${}_{S_{n}}\\!{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=D^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}={\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{R_{n}}\subset{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
is a dualizing complex for the rings $S_{n}$ and $R_{n}$.
The whole complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ does not have
to be bounded from either side, but its subcomplexes
$D_{n}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, by the definition,
must be finite.
###### Theorem D.2.5.
The choice of a dualizing complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the
projective systems of rings $(S_{n})$ and $(R_{n})$ induces an equivalence
between the coderived category of discrete left ${\mathfrak{S}}$-modules
${\mathsf{D}}^{\mathsf{co}}({\mathfrak{S}}{\operatorname{\mathsf{--discr}}})$
and the contraderived category of left ${\mathfrak{R}}$-contramodules
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$.
The equivalence is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{\mathfrak{S}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathfrak{R}}^{\mathbb{L}}{-}$
of the functor of discrete ${\mathfrak{S}}$-module homomorphisms from
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the functor
of contratensor product with
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
${\mathfrak{R}}$.
###### Proof.
The constructions of the derived functors are based on Theorem D.2.1. Clearly,
a discrete left ${\mathfrak{S}}$-module ${\mathcal{J}}$ is an injective object
in ${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}$ if and only if all the
left $S_{n}$-modules ${}_{S_{n}}{\mathcal{J}}$ are injective. By Lemmas
D.2.3(a), D.1.2, and 1.6.1(b), applying the functor
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{Hom}_{\mathfrak{S}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\varprojlim_{n}\operatorname{Hom}_{S_{n}}({}_{S_{n}}\\!{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\mskip
1.5mu{}_{S_{n}}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
to a complex of injective discrete left ${\mathfrak{S}}$-modules
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ produces a
complex of flat left ${\mathfrak{R}}$-contramodules.
By Lemmas D.2.3(b) and 1.6.1(a), applying the functor
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathfrak{R}}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq\varinjlim_{n}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{R_{n}}\otimes_{R_{n}}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}/({\mathfrak{I}}_{n}{\times}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
to a complex of flat left ${\mathfrak{R}}$-contramodules
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ produces a
complex of injective discrete left ${\mathfrak{S}}$-modules. Let us show that
the complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathfrak{R}}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contractible whenever the complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
contraacyclic.
Any coacyclic complex of injective objects in
${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}$ being contractible, and
coacyclicity being preserved by inductive limits of sequences, it suffices to
show that the complexes of left $S_{n}$-modules
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}\otimes_{R_{n}}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are coacyclic. The functor
${\mathfrak{F}}\longmapsto{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
preserves infinite products of left ${\mathfrak{R}}$-contramodules by Lemma
D.2.2 and short exact sequences of flat left ${\mathfrak{R}}$-contramodules by
Lemma D.1.3. Hence it takes contraacyclic complexes of flat left
${\mathfrak{R}}$-contramodules
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to
contraacyclic complexes of flat left $R_{n}$-modules
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Now any contraacyclic complex of flat left $R_{n}$-modules is absolutely
acyclic by Corollary B.4.2 and [50, Remark 2.1] (see also Section A.6), and
the functor $D_{n}\otimes_{R_{n}}\nobreak{-}$ clearly takes absolutely acyclic
complexes of flat left $R_{n}$-modules to absolutely acyclic complexes of left
$S_{n}$-modules. (Concerning the apparent ambiguity of our terminology, notice
that the classes of complexes of flat objects contraacyclic or absolutely
acyclic with respect to the abelian categories of arbitrary objects coincide
with those of flat objects contraacyclic or absolutly acyclic with respect to
the exact categories of flat objects by Proposition A.2.1.)
Finally, for any complex of injective discrete left ${\mathfrak{S}}$-modules
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the adjunction
morphism
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathfrak{R}}\operatorname{Hom}_{\mathfrak{S}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
has a coacyclic (in fact, contractible) cone by Lemma B.4.1(b) and because the
class of coacyclic complexes is closed with respect to inductive limits of
sequences. For any complex of flat left ${\mathfrak{R}}$-contramodules
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the
adjunction morphism
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{Hom}_{\mathfrak{S}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathfrak{R}}{\mathfrak{F}})$
has a contraacyclic cone by Lemma B.4.1(a) and because the class of
contraacyclic complexes is closed with respect to projective limits of
sequences of surjective morphisms. ∎
### D.3. Very flat and contraadjusted contramodules
For the rest of the appendix we stick to a single projective system of
associative rings and their surjective morphisms $R_{0}\longleftarrow
R_{1}\longleftarrow R_{2}\longleftarrow\dotsb$ with the projective limit
${\mathfrak{R}}=\varprojlim_{n}R_{n}$. In this section we assume that the
rings $R_{n}$ are commutative.
Then an ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is said to be _very
flat_ if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is an isomorphism and the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are very flat for
all $n$ (cf. Section C.3). Clearly, any very flat
${\mathfrak{R}}$-contramodule is flat. In the case when the ideals
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\subset R_{n+1}$ are nilpotent and
finitely generated, it follows from Lemma 1.6.8(b) that a flat
${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is very flat if and only if the
$R_{0}$-module ${\mathfrak{F}}/{\mathfrak{I}}_{0}{\times}{\mathfrak{F}}$ is
very flat.
###### Corollary D.3.1.
The class of very flat ${\mathfrak{R}}$-contramodules contains the projective
${\mathfrak{R}}$-contramodules and is closed under extensions and the passage
to the kernels of surjective morphisms in ${\mathfrak{R}}$-contra. The
projective dimension of any very flat ${\mathfrak{R}}$-contramodule (as an
object of ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$) does not exceed
$1$.
###### Proof.
Follows from Lemmas D.1.3–D.1.4 and Corollary D.1.8(a). ∎
An ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ is said to be
_contraadjusted_ if the functor
$\operatorname{Hom}^{\mathfrak{R}}({-},{\mathfrak{Q}})$ takes short exact
sequences of very flat ${\mathfrak{R}}$-contramodules to short exact sequences
of abelian groups (or, equivalently, of ${\mathfrak{R}}$-modules, or of
${\mathfrak{R}}$-contramodules). It is clear from the first assertion of
Corollary D.3.1 that an ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ is
contraadjusted if and only if
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for
any very flat ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$, and if and only
if $\operatorname{Ext}^{{\mathfrak{R}},>0}({\mathfrak{F}},{\mathfrak{Q}})=0$
for any very flat ${\mathfrak{F}}$.
It follows that the class of contraadjusted ${\mathfrak{R}}$-contramodules is
closed under extensions and the passages to the cokernels of injective
morphisms. Moreover, the second assertion of Corollary D.3.1 implies that any
quotient ${\mathfrak{R}}$-contramodule of a contraadjusted
${\mathfrak{R}}$-contramodule is contraadjusted.
For the rest of this section we assume that (the rings $R_{n}$ are commutative
and) the ideals ${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}=\ker(R_{n+1}\to
R_{n})\subset R_{n+1}$ are finitely generated.
###### Lemma D.3.2.
Let $R$ be a commutative ring and $I\subset R$ be a finitely generated ideal.
Then the $R$-module $IQ$ is contraadjusted for any contraadjusted $R$-module
$Q$.
###### Proof.
The $R$-module $IQ$ is a quotient module of a finite direct sum of copies of
the $R$-module $Q$, and the class of contraadjusted $R$-modules is preserved
by the passages to finite direct sums and quotients. ∎
###### Lemma D.3.3.
Let ${\mathfrak{P}}$ be an ${\mathfrak{R}}$-contramodule for which the natural
map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism. Then the ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ is
contraadjusted whenever the $R_{n}$-modules
${\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$ are contraadjusted
for all $n$.
###### Proof.
Applying Lemma D.3.2 to the ring $R=R_{n+1}$ with the ideal
$I={\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$ and the $R$-module
$Q=P_{n+1}={\mathfrak{P}}/{\mathfrak{I}}_{n+1}{\times}{\mathfrak{P}}$, we
conclude that the $R_{n+1}$-module $\ker(P_{n+1}\to
P_{n})=({\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1})P_{n+1}$ is contraadjusted.
Then it remains to make use of Lemma D.1.6. ∎
When the ideals ${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$ are nilpotent, in
view of Lemma 1.6.8(a) it suffices to require that the $R_{0}$-module
${\mathfrak{P}}/{\mathfrak{I}}_{0}{\times}{\mathfrak{P}}$ be contraadjusted in
Lemma D.3.3.
Recall the above discussion of a ring $R$ with ideals $I_{n}\subset R$ in
Section D.1, and assume the ring $R$ to be also commutative. The following
construction plays a key role in our approach.
###### Lemma D.3.4.
Let ${\mathfrak{P}}$ be an ${\mathfrak{R}}$-contramodule for which the natural
map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism. Then the ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ can
be embedded into a contraadjusted ${\mathfrak{R}}$-contramodule
${\mathfrak{Q}}$ in such a way that the quotient ${\mathfrak{R}}$-contramodule
${\mathfrak{Q}}/{\mathfrak{P}}$ is very flat.
###### Proof.
Let us consider ${\mathfrak{P}}$ as an $R$-module and embed it into a
contraadjusted $R$-module $K$ in such a way that the quotient $R$-module
$F=K/{\mathfrak{P}}$ is very flat. Then, the $R$-module $F$ being, in
particular, flat, there are exact sequences of $R_{n}$-modules
$0\longrightarrow{\mathfrak{P}}/I_{n}{\mathfrak{P}}\longrightarrow
K/I_{n}K\longrightarrow F/I_{n}F\longrightarrow 0$. Furthermore, we have
surjective morphisms of $R_{n}$-modules
${\mathfrak{P}}/I_{n}{\mathfrak{P}}\longrightarrow{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
and the induced short exact sequences
$0\longrightarrow{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}\longrightarrow
K/(I_{n}K+{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})\longrightarrow
F/I_{n}F\longrightarrow 0$.
Passing to the projective limits of these systems of short exact sequences, we
obtain a natural morphism from the short exact sequence
$0\longrightarrow\varprojlim_{n}{\mathfrak{P}}/I_{n}{\mathfrak{P}}\longrightarrow\varprojlim_{n}K/I_{n}K\longrightarrow\varprojlim_{n}F/I_{n}F\longrightarrow
0$ to the short exact sequence
$0\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}\allowbreak\longrightarrow\varprojlim_{n}K/(I_{n}K+{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})\longrightarrow\varprojlim_{n}F/I_{n}F\longrightarrow
0$. The map
$\varprojlim_{n}{\mathfrak{P}}/I_{n}{\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is always surjective, since a surjective map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
(see Lemma D.1.1) factorizes through it. Hence the map
$\varprojlim_{n}K/I_{n}K\longrightarrow\varprojlim_{n}K/(I_{n}K+{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$
is also surjective. Furthermore, the projective system
$K/(I_{n}K+{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$ satisfies the condition
of Lemma D.1.2, because
${\mathfrak{I}}_{n}{\times}{\mathfrak{P}}=I_{n}{\mathfrak{P}}+{\mathfrak{I}}_{n+1}{\times}{\mathfrak{P}}$.
Now put ${\mathfrak{L}}=\varprojlim_{n}K/I_{n}K$ and
${\mathfrak{Q}}=\varprojlim_{n}K/(I_{n}K+{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$,
and also ${\mathfrak{G}}=\varprojlim_{n}F/I_{n}F$. By Lemma D.1.2, we have
${\mathfrak{L}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{L}}=K/I_{n}K$ and
${\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}=F/I_{n}F$. Therefore,
the $R$-contramodule ${\mathfrak{G}}$ is very flat. Besides,
${\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}=K/(I_{n}K+{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$,
and the natural map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}$
is an isomorphism. In addition, the $R_{n}$-modules $K/I_{n}K$ are
contraadjusted by Lemma 1.6.6(b). Finally, the $R_{n}$-module
${\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}$ is a quotient module
of a contraadjusted $R_{n}$-module
${\mathfrak{L}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{L}}=K/I_{n}K$, and
consequently is also contraadjusted. It remains to use Lemma D.3.3. ∎
###### Corollary D.3.5.
Any ${\mathfrak{R}}$-contramodule can be presented as the quotient
contramodule of a very flat ${\mathfrak{R}}$-contramodule by a contraadjusted
${\mathfrak{R}}$-subcontramodule.
###### Proof.
Present an arbitrary ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ as the
quotient contramodule of a free ${\mathfrak{R}}$-contramodule
${\mathfrak{H}}$; apply Lemma D.3.4 to embed the kernel ${\mathfrak{K}}$ of
the morphism ${\mathfrak{H}}\longrightarrow{\mathfrak{P}}$ into a
contraadjusted ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ so that the
quotient contramodule ${\mathfrak{F}}={\mathfrak{Q}}/{\mathfrak{K}}$ is very
flat; consider the induced extensions
$0\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$ and
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$, and use Lemma D.1.4. It is important here that the map
${\mathfrak{K}}\longrightarrow\varprojlim_{n}{\mathfrak{K}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{K}}$
is an isomorphism, since so is the map
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$.
(This is the same construction as in Lemma 1.1.3.) ∎
###### Corollary D.3.6.
Any ${\mathfrak{R}}$-contramodule can be embedded into a contraadjusted
${\mathfrak{R}}$-contramodule in such a way that the quotient contramodule is
very flat.
###### Proof.
Use Corollary D.3.5 to present an arbitrary ${\mathfrak{R}}$-contramodule
${\mathfrak{P}}$ as the quotient ${\mathfrak{R}}$-contramodule of a very flat
${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$ by a contraadjusted
${\mathfrak{R}}$-contramodule ${\mathfrak{K}}$; apply Lemma D.3.4 to embed the
${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$ into a contraadjusted
${\mathfrak{R}}$-contramodule ${\mathfrak{L}}$ so that the quotient
contramodule ${\mathfrak{F}}={\mathfrak{L}}/{\mathfrak{G}}$ is very flat; and
denote by ${\mathfrak{N}}$ the cokernel of the composition of injective
morphisms of ${\mathfrak{R}}$-contramodules
${\mathfrak{K}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{L}}$.
Now the ${\mathfrak{R}}$-contramodule ${\mathfrak{N}}$ is contraadjusted as
the quotient contramodule of a contraadjusted ${\mathfrak{R}}$-contramodule
(by a contraadjusted ${\mathfrak{R}}$-subcontramodule), and the
${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ is embedded into
${\mathfrak{N}}$ with the cokernel ${\mathfrak{F}}$. (This construction comes
from [50, proof of Lemma 9.1.2(a)].) ∎
###### Lemma D.3.7.
Let $F$ be a flat $R$-module and ${\mathfrak{Q}}$ be an
${\mathfrak{R}}$-contramodule such that the $R_{n}$-modules $F/I_{n}F$ are
very flat, the map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}$
is an isomorphism, and the $R_{n}$-modules
${\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}$ are contraadjusted.
Then one has $\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{Q}})=0$.
###### Proof.
This is a particular case of Lemma D.1.7. To show that its conditions are
satisfied, one only has to recall that the ideals
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\subset R_{n+1}$ are finitely
generated by our assumptions, and use Lemma D.3.2. ∎
###### Corollary D.3.8.
An ${\mathfrak{R}}$-contramodule is contraadjusted if and only if it is a
contraadjusted $R$-module. In particular, the $R_{n}$-modules
${\mathfrak{P}}/I_{n}{\mathfrak{P}}$ and
${\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$ are contraadjusted
for any contraadjusted ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$.
Furthermore, one has
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})\simeq\operatorname{Hom}_{R}(F,{\mathfrak{P}})$
and $\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/\allowbreak
I_{n}F,\>{\mathfrak{P}})=\operatorname{Ext}_{R}^{>0}(F,{\mathfrak{P}})=0$ for
any contraadjusted ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ and any flat
$R$-module $F$ for which the $R_{n}$-modules $F/I_{n}F$ are very flat.
###### Proof.
According to the constructions of Lemma D.3.4 and Corollaries D.3.5–D.3.6, any
contraadjusted ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ can be obtained
from the ${\mathfrak{R}}$-contramodules ${\mathfrak{Q}}$ satisfying the
conditions of Lemma D.3.7 using the operations of the passage to the cokernel
of an injective morphism and the passage to a direct summand. This proves the
equation $\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{Q}})=0$, and consequently
also the “only if” part of the first assertion; then the second assertion
follows.
Conversely, suppose that an ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ is
a contraadjusted $R$-module. Let us present ${\mathfrak{P}}$ as the quotient
contramodule of a (very) flat ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$
by a contraadjusted ${\mathfrak{R}}$-subcontramodule ${\mathfrak{Q}}$. As we
have proven, ${\mathfrak{Q}}$ is a contraadjusted $R$-module, and therefore so
is the $R$-module ${\mathfrak{F}}$. Hence the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are contraadjusted,
and by Lemma D.3.3 it follows that ${\mathfrak{F}}$ is a contraadjusted
${\mathfrak{R}}$-contramodule. Now the ${\mathfrak{R}}$-contramodule
${\mathfrak{P}}$ is contraadjusted as the quotient
${\mathfrak{R}}$-contramodule of a contraadjusted one.
The equation
$\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})=0$
holds since the ${\mathfrak{R}}$-contramodule in the first argument is very
flat, and to prove the isomorphism
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})\simeq\operatorname{Hom}_{R}(F,{\mathfrak{P}})$
one argues in the same way as in Corollary D.1.8(b). ∎
### D.4. Cotorsion contramodules
A left ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ is said to be
_cotorsion_ if the functor
$\operatorname{Hom}^{\mathfrak{R}}({-},{\mathfrak{Q}})$ takes short exact
sequences of flat left ${\mathfrak{R}}$-contramodules to short exact sequences
of abelian groups. It follows from Lemmas D.1.3–D.1.5 that a left
${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ is cotorsion if and only if
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for
any flat left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$, and if and only
if $\operatorname{Ext}^{{\mathfrak{R}},>0}({\mathfrak{F}},{\mathfrak{Q}})=0$
for any flat ${\mathfrak{F}}$. Hence the class of cotorsion left
${\mathfrak{R}}$-contramodules is closed under extensions and the passages to
the cokernels of injective morphisms and direct summands.
The _cotorsion dimension_ of a left ${\mathfrak{R}}$-contramodule
${\mathfrak{P}}$ is defined as the supremum of the set of all integers $d$ for
which there exists a flat left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$
such that
$\operatorname{Ext}^{{\mathfrak{R}},d}({\mathfrak{F}},{\mathfrak{P}})\neq 0$.
Clearly, a nonzero ${\mathfrak{R}}$-contramodule is cotorsion if and only if
its cotorsion dimension is equal to zero. Given a short exact sequence of
${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$ of the cotorsion dimensions $k$, $l$, and $m$, respectively, one has
$l\le\max(k,m)$, $k\le\max(l,m+1)$, and $m\le\max(l,k-1)$.
###### Lemma D.4.1.
Let ${\mathfrak{Q}}$ be a left ${\mathfrak{R}}$-contramodule for which the
natural map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}$
is an isomorphism. Set
$Q_{n}={\mathfrak{Q}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{Q}}$, and assume
that the left $R_{0}$-module $Q_{0}$ is cotorsion and the left $R_{n}$-modules
$\ker(Q_{n+1}\to Q_{n})$ are cotorsion for all $n\ge 0$. Then ${\mathfrak{Q}}$
is a cotorsion ${\mathfrak{R}}$-contramodule and a cotorsion left $R$-module.
###### Proof.
The equations
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{Q}})=0=\operatorname{Ext}^{1}_{R}(F,{\mathfrak{Q}})$
for any flat left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ and any flat
left $R$-module $F$ follow from Lemmas D.1.6 and D.1.7, respectively. ∎
For the rest of this section we assume that the rings $R_{n}$ are commutative
and Noetherian. The ring $R$ is also presumed to be commutative.
###### Corollary D.4.2.
Let ${\mathfrak{E}}$ be a flat ${\mathfrak{R}}$-contramodule. Assume that the
flat $R_{n}$-modules
$E_{n}={\mathfrak{E}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{E}}$ are cotorsion
for all $n\ge 0$. Then ${\mathfrak{E}}$ is a flat cotorsion
${\mathfrak{R}}$-contramodule and a cotorsion $R$-module.
###### Proof.
Since $\ker(E_{n+1}\to
E_{n})=({\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1})E_{n+1}\simeq{\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\otimes_{R_{n+1}}E_{n+1}$,
the assertions follow from Lemma D.4.1 in view of Lemma 1.6.4(a). ∎
###### Proposition D.4.3.
Any flat ${\mathfrak{R}}$-contramodule can be embedded into a flat cotorsion
${\mathfrak{R}}$-contramodule in such a way that the quotient
${\mathfrak{R}}$-contramodule is flat.
###### Proof.
In fact, we will even present a functorial construction of such an embedding
of flat ${\mathfrak{R}}$-contramodules. It is based on the functorial
construction of an embedding
$G\longrightarrow\operatorname{\mathrm{FC}}_{S}(G)$ of a flat module $G$ over
a commutative Noetherian ring $S$ into a flat cotorsion module
$\operatorname{\mathrm{FC}}_{S}(G)$ with a flat quotient $S$-module
$\operatorname{\mathrm{FC}}_{S}(G)/G$, which was explained in Lemma 1.3.9. In
addition, we will need the following lemma.
###### Lemma D.4.4.
Let $T$ be a commutative Noetherian ring, $S=T/I$ be its quotient ring by an
ideal $I\subset T$, and $H$ be a flat $T$-module. Then there is a natural
isomorphism of $S$-modules
$\operatorname{\mathrm{FC}}_{T}(H)/I\operatorname{\mathrm{FC}}_{T}(H)\simeq\operatorname{\mathrm{FC}}_{S}(H/IH)$
compatible with the natural morphisms
$H\longrightarrow\operatorname{\mathrm{FC}}_{T}(H)$ and
$H/IH\longrightarrow\operatorname{\mathrm{FC}}_{S}(H/IH)$.
###### Proof.
Since the ideal $I\subset T$ is finitely generated, the reduction functor
$S\otimes_{T}\nobreak{-}$ preserves infinite direct products. For any prime
ideal ${\mathfrak{q}}\subset T$ that does not contain $I$ and any
$T_{\mathfrak{q}}$-module $Q$ one has $Q/IQ=0$; in particular, this applies to
any $T_{\mathfrak{q}}$-contramodule. Now consider a prime ideal
${\mathfrak{q}}\supset I$ in $T$, and let ${\mathfrak{p}}={\mathfrak{q}}/I$ be
the related prime ideal in $S$. It remains to construct a natural isomorphism
of $S$-modules
$\widehat{H}_{\mathfrak{q}}/I\widehat{H}_{q}\simeq\widehat{(H/IH)}_{\mathfrak{p}}$.
Indeed, according to [54, proof of Lemma B.9.2] applied to the Noetherian ring
$T$ with the ideal ${\mathfrak{q}}\subset T$, the complete ring
${\mathfrak{T}}=\widehat{T}_{\mathfrak{q}}$, and the finitely generated
$T$-module $S$, one has
$S\otimes_{T}\widehat{H}_{\mathfrak{q}}\simeq\varprojlim_{n}S\otimes_{T}(H_{\mathfrak{q}}/{\mathfrak{q}}^{n}H_{q})\simeq\varprojlim_{n}(S\otimes_{T}H)_{\mathfrak{p}}/{\mathfrak{p}}^{n}(S\otimes_{T}H)_{\mathfrak{p}}$,
as desired. ∎
So let ${\mathfrak{G}}$ be a flat ${\mathfrak{R}}$-contramodule; set
$G_{n}={\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$. According to
Lemma D.4.4, the flat cotorsion $R_{n}$-modules
$\operatorname{\mathrm{FC}}_{R_{n}}(G_{n})$ naturally form a projective system
satisfying the condition of Lemma D.1.2. Set
${\mathfrak{E}}=\varprojlim_{n}\operatorname{\mathrm{FC}}_{R_{n}}(G_{n})$; by
Corollary D.4.2, the flat ${\mathfrak{R}}$-contramodule ${\mathfrak{E}}$ is
cotorsion. The projective system of $R_{n}$-modules
$\operatorname{\mathrm{FC}}_{R_{n}}(G_{n})/G_{n}$ also satisfies the condition
of Lemma D.4.4, being the cokernel of a morphism of two projective systems
that do; in view of the second assertion of Lemma 1.3.9, it follows that the
${\mathfrak{R}}$-contramodule
${\mathfrak{F}}=\varprojlim_{n}\operatorname{\mathrm{FC}}_{R_{n}}(G_{n})/G_{n}$
is flat. Finally, the short sequence
$0\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ is exact, since $\varprojlim_{n}^{1}G_{n}=0$. ∎
###### Corollary D.4.5.
A flat ${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$ is cotorsion if and only
if the $R_{n}$-modules
${\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$ are cotorsion for
all $n\ge 0$. Any flat cotorsion ${\mathfrak{R}}$-contramodule is a cotorsion
$R$-module.
###### Proof.
In view of Corollary D.4.2, we only have to show that the $R_{n}$-modules
${\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$ are cotorsion for
any flat cotorsion ${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$. For this
purpose, it suffices to apply the construction of Proposition D.4.3 to obtain
a short exact sequence of ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{G}}\longrightarrow
0$, where the ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is flat and the
${\mathfrak{R}}$-contramodule ${\mathfrak{E}}$, by construction, has the
desired property. Now ${\mathfrak{G}}$ is an ${\mathfrak{R}}$-contramodule
direct summand of ${\mathfrak{E}}$, since
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{G}},{\mathfrak{F}})=0$ by
assumption. ∎
From this point on and until the end of this section we assume that the Krull
dimensions of the Noetherian commutative rings $R_{n}$ are uniformly bounded
by a constant $D$. Then it follows from the results of Section D.1 and Theorem
1.5.6 that the projective dimension of any flat ${\mathfrak{R}}$-contramodule
does not exceed $D$. Therefore, the cotorsion dimension of any
${\mathfrak{R}}$-contramodule also cannot exceed $D$.
###### Lemma D.4.6.
(a) Any ${\mathfrak{R}}$-contramodule can be embedded into a cotorsion
${\mathfrak{R}}$-contramodule in such a way that the quotient
${\mathfrak{R}}$-contramodule is flat.
(b Any ${\mathfrak{R}}$-contramodule admits a surjective map onto it from a
flat ${\mathfrak{R}}$-contramodule with the kernel being a cotorsion
${\mathfrak{R}}$-contramodule.
###### Proof.
Part (a): we will argue by decreasing induction in $0\le d\le D$, showing that
any ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ can be embedded into an
${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ of cotorsion dimension $\le d$
in such a way that the quotient ${\mathfrak{R}}$-contramodule
${\mathfrak{Q}}/{\mathfrak{P}}$ is flat.
Assume that we already know this for the cotorsion dimension $d+1$. Let
${\mathfrak{P}}$ be an ${\mathfrak{R}}$-contramodule; pick a surjective
morphism ${\mathfrak{L}}\longrightarrow{\mathfrak{P}}$ onto ${\mathfrak{P}}$
from a projective ${\mathfrak{R}}$-contramodule ${\mathfrak{L}}$. Denote the
kernel of the morphism ${\mathfrak{L}}\longrightarrow{\mathfrak{P}}$ by
${\mathfrak{K}}$ and use the induction assumption to embed it into an
${\mathfrak{R}}$-contramodule ${\mathfrak{N}}$ of cotorsion dimension $\le
d+1$ so that the cokernel ${\mathfrak{H}}={\mathfrak{N}}/{\mathfrak{K}}$ is
flat.
Let ${\mathfrak{G}}$ be the fibered coproduct
${\mathfrak{N}}\sqcup_{\mathfrak{K}}{\mathfrak{L}}$; then the
${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$ is flat as an extension of two
flat ${\mathfrak{R}}$-contramodules ${\mathfrak{H}}$ and ${\mathfrak{L}}$. Now
we use Proposition D.4.3 to embed ${\mathfrak{G}}$ into a flat cotorsion
${\mathfrak{R}}$-contramodule ${\mathfrak{E}}$ so that the cokernel
${\mathfrak{F}}={\mathfrak{E}}/{\mathfrak{G}}$ is a flat
${\mathfrak{R}}$-contramodule. Then the cokernel ${\mathfrak{Q}}$ of the
composition
${\mathfrak{N}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{E}}$ has
cotorsion dimension $\le d$. The ${\mathfrak{R}}$-contramodule
${\mathfrak{P}}$ embeds naturally into ${\mathfrak{Q}}$ with the cokernel
${\mathfrak{F}}$, so the desired assertion is proven.
We have also proven part (b) along the way (it suffices to notice that there
is a natural surjective morphism ${\mathfrak{G}}\longrightarrow{\mathfrak{P}}$
with the kernel ${\mathfrak{N}}$); see also, e. g., the proof of Lemma 1.1.3.
∎
###### Corollary D.4.7.
Any cotorsion ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ is also a
cotorsion $R$-module. Besides, for any flat $R$-module $F$ one has
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})\simeq\operatorname{Hom}_{R}(F,{\mathfrak{P}})$
and
$\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})=\operatorname{Ext}_{R}^{>0}(F,{\mathfrak{P}})=0$.
###### Proof.
Applying the construction of Lemma D.4.6(a) to the
${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, we obtain a short exact
sequence of ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ with a flat ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ and an
${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ obtained from flat cotorsion
${\mathfrak{R}}$-contramodules using the operation of passage to the cokernel
of an injective morphism at most $D$ times. Then ${\mathfrak{P}}$ is a direct
summand of ${\mathfrak{Q}}$, and since the class of cotorsion $R$-modules is
preserved by the passages to the cokernels of injective morphisms and direct
summands, the first assertion follows from Corollary D.4.5. The Ext-vanishing
assertions now hold by the definitions, and the Hom isomorphism is obtained in
the same way as in Corollaries D.1.8 and D.3.8. ∎
## References
* [1] L. Alonso Tarrío, A. Jeremías López, M. J. Souto Salorio. Localization in categories of complexes and unbounded resolutions. _Canadian Journ. of Math._ 52, #2, p. 225–247, 2000.
* [2] D. Arinkin, R. Bezrukavnikov. Perverse coherent sheaves. Moscow Math. Journ. 10, #1, p. 3–29, 2010. arXiv:0902.0349 [math.AG]
* [3] M. F. Atiyah, I. G. MacDonald. Introduction to commutative algebra. Addison-Wesley, 1969–94.
* [4] M. Barr. Coequalizers and free triples. Math. Zeitschrift 116, p. 307–322, 1970.
* [5] H. Becker. Models for singularity categories. Electronic preprint arXiv:1205.4473 [math.CT] .
* [6] L. Bican, R. El Bashir, E. Enochs. All modules have flat covers. Bulletin of the London Math. Society 33, #4, p. 385–390, 2001.
* [7] G. Böhm, T. Brzeziński, R. Wisbauer. Monads and comonads in module categories. Journ. of Algebra 233, #5, p. 1719–1747, 2009. arXiv:0804.1460 [math.RA]
* [8] M. Boekstedt, A. Neeman. Homotopy limits in triangulated categories. Compositio Math. 86, #2, p. 209–234, 1993.
* [9] G. E. Bredon. Sheaf theory. Graduate Texts in Mathematics, 170. Second Edition, Springer, 1997. (See also the Russian translation, 1988, of the first English edition, with the translation editor’s comments by E. G. Sklyarenko.)
* [10] T. Brzezinski, R. Wisbauer. Corings and comodules. London Mathematical Society Lecture Note Series, 309. Cambridge University Press, Cambridge, 2003.
* [11] M. Bunge, J. Funk. Singular coverings of toposes. Lecture Notes in Math. 1890, Springer, 2006.
* [12] L. W. Christensen, A. Frankild, H. Holm. On Gorenstein projective, injective, and flat dimensions—A functorial description with applications. Journ. of Algebra 302, #1, p. 231–279, 2006. arXiv:math.AC/0403156
* [13] J. Curry. Sheaves, cosheaves and applications. Electronic preprint arXiv:1303.3255 [math.AT] .
* [14] P. Deligne. Cohomologie à support propre et construction du functeur $f^{!}$. Appendix to the book [29], pp. 404–421.
* [15] S. Eilenberg, J. C. Moore. Limits and spectral sequences. Topology 1, p. 1–23, 1962.
* [16] S. Eilenberg, J. C. Moore. Foundations of relative homological algebra. Memoirs of the American Math. Society 55, 1965.
* [17] P. C. Eklof, J. Trlifaj. How to make Ext vanish. Bulletin of the London Math. Society 33, #1, p. 41–51, 2001.
* [18] E. Enochs. Flat covers and flat cotorsion modules. Proceedings of the American Math. Society 92, #2, p. 179–184, 1984.
* [19] E. E. Enochs, O. M. G. Jenda. Relative homological algebra. De Gruyter Expositions in Mathematics, 30. De Gruyter, Berlin–New York, 2000.
* [20] E. Enochs, S. Estrada. Relative homological algebra in the category of quasi-coherent sheaves. Advances in Math. 194, #2, p. 284–295, 2005.
* [21] D. Gaitsgory. Ind-coherent sheaves. Electronic preprint arXiv:1105.4857 [math.AG] .
* [22] R. Godement. Topologie algébrique et théorie des faisceaux. Paris, Hermann, 1958–73.
* [23] K. Goodearl, R. Warfield. An introduction to noncommutative Noetherian rings. London Mathematical Society Student Texts, 16. Cambridge University Press, 1989–2004.
* [24] A. Grothendieck. Sur quelques points d’algèbre homologique. Tohoku Math. Journ. 9, #2, p. 119–221, 1957.
* [25] A. Grothendieck, J. Dieudonné. Éléments de géométrie algébrique I. Le langage des schémas. Publications Mathematiques de l’IHÉS 4, p. 5–228, 1960.
* [26] A. Grothendieck, J. Dieudonné. Éléments de géométrie algébrique II. Étude globale élémentaire de quelques classes de morphismes. Publications Mathematiques de l’IHÉS 8, p. 5–222, 1961.
* [27] A. Grothendieck, J. Dieudonné. Éléments de géométrie algébrique III. Étude cohomologique des faisceaux coherents, Première partie. Publications Mathematiques de l’IHÉS 11, p. 5–167, 1961.
* [28] A. Grothendieck, J. Dieudonné. Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes des schémas, Seconde partie. Publications Mathematiques de l’IHÉS 24, p. 5–231, 1965.
* [29] R. Hartshorne. Residues and duality. Lecture Notes in Math. 20, Springer, 1966.
* [30] V. Hinich. DG coalgebras as formal stacks. Journ. of Pure and Applied Algebra 162, #2–3, p. 209–250, 2001. arXiv:math.AG/9812034
* [31] D. Husemoller, J. C. Moore, J. Stasheff. Differential homological algebra and homogeneous spaces. Journ. of Pure and Applied Algebra 5, p. 113–185, 1974\.
* [32] S. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, U. Walther. Twenty-four hours of local cohomology. Graduate Studies in Mathematics 87, American Math. Society, 2007.
* [33] S. Iyengar, H. Krause. Acyclicity versus total acyclicity for complexes over noetherian rings. Documenta Math. 11, p. 207–240, 2006.
* [34] P. Jørgensen. The homotopy category of complexes of projective modules. Advances in Math. 193, #1, p. 223–232, 2005. arXiv:math.RA/0312088
* [35] B. Keller. Derived categories and their uses. In: M. Hazewinkel, Ed., Handbook of algebra, vol. 1, 1996, p. 671–701.
* [36] M. Kontsevich, A. Rosenberg. Noncommutative smooth spaces. The Gelfand Mathematical Seminars 1996–1999, p. 85–108, Birkhaeuser Boston, Boston, MA, 2000. arXiv:math.AG/9812158
* [37] H. Krause. The stable derived category of a Noetherian scheme. Compositio Math. 141, #5, p. 1128–1162, 2005. arXiv:math.AG/0403526
* [38] H. Krause. Approximations and adjoints in homotopy categories. Mathematische Annalen 353, #3, p. 765–781, 2012. arXiv:1005.0209 [math.CT]
* [39] H. W. Lenstra. Galois theory for schemes. Electronic third edition, 2008.
* [40] H. Matsumura. Commutative ring theory. Translated by M. Reid. Cambridge University Press, 1986–2006.
* [41] D. Murfet. Derived categories of quasi-coherent sheaves. Notes, October 2006. Available from http://www.therisingsea.org/notes .
* [42] D. Murfet. The mock homotopy category of projectives and Grothendieck duality. Ph. D. Thesis, Australian National University, September 2007. Available from http://www.therisingsea. org/thesis.pdf .
* [43] J. P. Murre. Lectures on an introduction to Grothendieck’s theory of the fundamental group. Tata Institute of Fundamental Research, Bombay, 1967.
* [44] A. Neeman. The derived category of an exact category. Journ. of Algebra 135, #2, p. 388–394, 1990.
* [45] A. Neeman. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. Journ. of the American Math. Society 9, p. 205–236, 1996.
* [46] A. Neeman. Triangulated categories. Annals of Math. Studies 148, Princeton University Press, 2001.
* [47] A. Neeman. The homotopy category of flat modules, and Grothendieck duality. Inventiones Math. 174, p. 225–308, 2008.
* [48] A. Neeman. Some adjoints in homotopy categories. Annals of Math. 171, #3, p. 2143–2155, 2010.
* [49] A. Polishchuk, L. Positselski. Quadratic algebras. University Lecture Series 37, American Math. Society, Providence, RI, 2005.
* [50] L. Positselski. Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne vol. 70, Birkhäuser/Springer Basel, 2010. xxiv+349 pp. arXiv:0708.3398 [math.CT]
* [51] L. Positselski. Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Memoirs of the American Math. Society 212, #996, 2011. vi+133 pp. arXiv:0905.2621 [math.CT]
* [52] L. Positselski. Mixed Artin–Tate motives with finite coefficients. Moscow Math. Journal 11, #2, p. 317–402, 2011\. arXiv:1006.4343 [math.KT]
* [53] L. Positselski. Coherent analogues of matrix factorizations and relative singularity categories. Electronic preprint arXiv:1102.0261 [math.CT] .
* [54] L. Positselski. Weakly curved A∞-algebras over a topological local ring. Electronic preprint arXiv:1202.2697 [math.CT] .
* [55] M. Raynaud, L. Gruson. Critères de platitude et de projectivité: Techniques de “platification” d’un module. Inventiones Math. 13, #1–2, p. 1–89, 1971.
* [56] R. Rouquier. Dimensions of triangulated categories. Journ. of K-theory 1, #2, p. 193–256, 2008. Erratum: same issue, p. 257–258. arXiv:math.CT/0310134
* [57] S. Rybakov. DG-modules over de Rham DG-algebra. Electronic preprint arXiv:1311.7503 [math.AG] .
* [58] M. Saorín, J. Št’ovíček. On exact categories and applications to triangulated adjoints and model structures. Advances in Math. 228, #2, p. 968–1007, 2011. arXiv:1005.3248 [math.CT]
* [59] J.-P. Schneiders. Quasi-abelian categories and sheaves. Mémoires de la Société Mathématique de France, 2e série 76, 1999. vi+140 pp.
* [60] N. Spaltenstein. Resolutions of unbounded complexes. Compositio Math. 65, #2, p.121–154, 1988.
* [61] R. Thomason, T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift vol. 3, p. 247–435, Birkhäuser, 1990.
* [62] C. A. Weibel. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, 1994.
* [63] J. Xu. Flat covers of modules. Lecture Notes in Math. 1634, Springer, 1996.
|
arxiv-papers
| 2012-09-13T19:17:16 |
2024-09-04T02:49:35.091167
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Leonid Positselski",
"submitter": "Leonid Positselski",
"url": "https://arxiv.org/abs/1209.2995"
}
|
1209.3126
|
11institutetext: Laboratoire Informatique d’Avignon,
BP 91228 84911, Avignon, Cedex 09, France
11email: [email protected] 22institutetext: École
Polytechnique de Montréal,
CP. 6128 succursale Centre-ville, Montréal, Québec, Canada
# Beyond Stemming and Lemmatization:
Ultra-stemming to Improve
Automatic Text Summarization
Juan-Manuel Torres-Moreno, 1122
###### Abstract
In Automatic Text Summarization, preprocessing is an important phase to reduce
the space of textual representation. Classically, stemming and lemmatization
have been widely used for normalizing words. However, even using normalization
on large texts, the curse of dimensionality can disturb the performance of
summarizers. This paper describes a new method for normalization of words to
further reduce the space of representation. We propose to reduce each word to
its initial letters, as a form of Ultra-stemming. The results show that Ultra-
stemming not only preserve the content of summaries produced by this
representation, but often the performances of the systems can be dramatically
improved. Summaries on trilingual corpora were evaluated automatically with
Fresa. Results confirm an increase in the performance, regardless of
summarizer system used.
Keywords: Automatic Text Summarization, Lemmatization, Stemming, Ultra-
Stemming
## 0.1 Introduction
In Natural Language Processing (NLP), pre-processing aims to reduce the
complexity of the vocabulary of the documents. Pre-processing eliminates the
punctuation, filters the function words and normalizes the morphological
variants. In particular, the lemmatization and stemming are two commonly used
techniques to normalize morphological variants.
The lexeme or word-root is the part that does not change and contains its
meaning. The morpheme or variable part is added to the lexeme to form new
words. Morphological analysis is a very important phase of pre-processing of
NLP systems because it allows to reduce the dimension of the vector space
representation in systems of Information Retrieval [3, 32]. Several
applications such as Automatic Summarization, Document Indexing, Textual
Classification and Question-Answering systems among others[3], utilize this
reduction. However, the realization of this analysis may require the use of
external resources (dictionaries, parsers, rules, etc.) which can be expensive
and difficult to build, depending on language or specific domain [32]. Some
algorithms are capable to detect statistically morphological families (posed
as a classification problem), avoiding the utilization of external resources
or a priori knowledge of a language.
Automatic Text Summarization (ATS) is the process to automatically generate a
compressed version of a source document [41]. Query-oriented summaries focus
on a user’s request, and extract the information related to the specified
topic given explicitly in the form of a query [11]. Generic mono-document
summarization tries to cover as much as possible the information content.
Multi-document summarization is a task oriented to creating a summary from a
heterogeneous set of documents on a focused topic. Over the past years,
extensive experiments on query-oriented multi-document summarization have been
carried out. Extractive Summarization produces summaries choosing a subset of
representative sentences from original documents. Sentences are ordered, then
assembled according to their relevance to generate the final summary [31].
This article introduces a new method of normalization of words that reduces
the textual representation space, in order to improve the efficiency of
Automatic Text Summarizers based on Vector Space Model (VSM). We propose
Ultra-stemming which reduces every word(s) to its initial(s) letter(s).
Results show that Ultra-stemming not only preserves the content of the
summaries generated using this new representation, but often, surprisingly the
performance can be dramatically improved. To our knowledge, in summary tasks
no automatic stemming method has explored this extreme possibility. Ultra-
stemming could be an interesting alternative for ATS of documents in languages
$\pi$, where electronic linguistic resources are rare. In these languages,
there are a notable absence of lemmatizers, stemmers, parsers, dictionaries,
corpora and language resources in general (such as Nahuatl and other American
Indian languages). Our tests on trilingual corpora evaluated by the Fresa
algorithm confirm the increase of performance regardless of summarizer used
and a big reduction of complexity in space and time required to generate
summaries. Related work is given in Section 0.2. Section 0.3 presents our
Ultra-stemming strategies coupled with methods of Automatic Text
Summarization. Experiments are presented in Section 0.5, followed by a
discussion and the conclusions in Section 0.6.
## 0.2 Related works
There are several morphological analysis methods [20, 21]. Examples of these
algorithms are the Comparison of Graphs [19], the use of $n$-grams [16, 32],
the search for analogies [27], the surface models based on rules [25, 37], the
probabilistic models [10], the segmentation by optimization [9, 18], the
unsupervised learning of morphological families by ascending hierarchical
classification [4], the lemmatization using Levenshtein distances [12] or
identifying suffixes through entropy [44]. These methods are distinguished by
the type of results obtained, by the identification of lemmas, stems or
suffixes. Flemm111Flemm is available in web site: http://www.univ-
nancy2.fr/pers/namer/Telecharger_Flemm.htm [14] is an analyzer for French
which requires a text previously labeled by WinBrill222WinBrill is available
in web site:
http://www.atilf.fr/scripts/mep.exe?HTML=mep_winbrill.txt;OUVRIR_MENU=1 or by
TreeTagger333Treetagger is available in web site: http://www.ims.uni-
stuttgart.de/projekte/corplex/TreeTagger/DecisionTreeTagger.html. Flemm
produces, among other results, the lemma of each word of the input text.
Treetagger [22] is a multilingual tool that allows to annotate texts with
information of Parts-Of-Speech (POS) 444The types of words are, for example,
nouns, verbs, infinitives and particles. and with information of
lemmatization. TreeTagger uses supervised machine learning and probabilistic
methods [7, 38]. It can be adapted to other languages as long as the lexical
resources and manually labeled corpora are available. FreeLing is another
example of a popular multilingual lemmatizer555FreeLing is available in web
site: http://www.lsi.upc.edu/~nlp/freeling/.
Stemming transforms the variants of words into truncated forms. Two popular
stemming algorithms are the Porter stemming algorithm [37] and the Paice
algorithm [35]. The methods of stemming and lemmatization can be applied when
the terms are morphologically similar. Otherwise when the similarity is
semantic, lexical search methods must be used. To reduce semantic variation,
some systems use long dictionaries. Another systems use thesauri to associate
words to entirely different morphological forms [36]. Both methods are
complementary since the stemming verifies similarities in the spelling level
to infer lexical proximity, while the lexical algorithms use terminographic
data with links to synonyms. [24]. [17] presents an unsupervised genetic
algorithm for stemming inflectional languages. [46] proposes using
morphological merged families into a single term to reduce the linguistic
variety of Spanish indexed texts.
Lexematization [34] seeks morphological rearrangement of words belonging to
the same family using automatic acquisition of morphological knowledge
directly from the texts. Although the constitution on morphological families
may be interesting in itself, its main interest lies in the benefits it
produces for use as normalization mechanism (instead or in addition to
stemming or lemmatization) in specific application domains. Probably the most
common application domain is indexing terms in systems of Information
Retrieval (IR). In recent years there have been numerous articles analyzing in
different languages the efficiency of stemming/lemmatization in IR. In
addition, significant progress has been made in IR in European languages other
than English. In particular, [23] have evaluated corpora of CLEF evaluation
campaigns 666Cross-Language Evaluation Forum, http://www.clef-campaign.org/
(eight European languages). Their results show that morphological
normalization techniques increase the efficiency of the IR systems and it can
be used independently of the language. Reduction algorithms using syntactic
and morphosyntactic variations have shown a significant reduction of storage
costs and management by storing lexemes rather than terms [45]. [1] works on
the impacts of compound words and standardization in IR, finding no
significant performance differences between stemming and lemmatization.
However, the reality is that the linguistic resources necessary to establish
morphological relationships without pre-defined rules are not available for
all languages and all domains, without mention the constant creation of
neologisms [8]. The proposed solution for the specific task of automatic
summarization is the Ultra-stemming of letters.
Research in ATS was introduced by H.P. Luhn in 1958 [30]. In the strategy
proposed by Luhn, the sentences are scored for their component word values as
determined by tf*idf-like weights. Scored sentences are then ranked and
selected from the top until some summary length threshold is reached. Finally,
the summary is generated by assembling the selected sentences in original
source order. Although fairly simple, this extractive methodology is still
used in current approaches. Later on, [13] extended this work by adding simple
heuristic features of sentences such as their position in the text or some key
phrases indicating the importance of the sentences. As the range of possible
features for source characterization widened, choosing appropriate features,
feature weights and feature combinations have became a central issue. A
natural way to tackle this problem is to consider sentence extraction as a
classification task. To this end, several machine learning approaches that
uses document-summary pairs have been proposed [26, 40].
## 0.3 Pre-processing and Ultra-stemming
The following subsections present formally the details of the corpora studied
and the proposed text pre-processing method.
### 0.3.1 Summarization Corpora Description
To study the impact of Ultra-stemming in automatic summary tasks, we used
corpora in three languages: English, Spanish and French. The corpora are
heterogeneous, and different tasks are representive of Automatic
Summarization: generic multi-document summary and mono-document guided by a
subject.
* •
Corpus in English. Piloted by NIST in Document Understanding
Conference777http://duc.nist.gov (DUC) the Task 2 of DUC’04888http://www-
nlpir.nist.gov/projects/duc/guidelines/2004.html, aims to produce a short
summary of a cluster of related documents. We studied generic multi-document-
summarization in English using data from DUC’04. This corpus with 300K words
is compound of 50 clusters, 10 documents each.
* •
Corpus in Spanish. Generic single-document summarization using a corpus from
the journal Medicina
Clínica999http://www.elsevier.es/revistas/ctl_servlet?_f=7032&revistaid=2,
which is composed of 50 medical articles in Spanish, each one with its
corresponding author abstract. This corpus contains 125K words.
* •
Corpus in French. We have studied generic single-document summarization using
the Canadian French Sociological Articles corpus, generated from the journal
Perspectives interdisciplinaires sur le travail et la santé
(Pistes)101010http://www.pistes.uqam.ca/. It contains 50 sociological articles
in French, each one with its corresponding author abstract. This corpus
contains near 400K words.
Table 1 presents the basic statistics on tokens, types and characters of the
three summarization corpora studied.
Corpus | Language | Tokens | Types | Letters
---|---|---|---|---
DUC’04 | English | 294 236 | 17 780 | 1 834 167
Medicina Clínica | Spanish | 125 024 | 9 657 | 793 937
Pistes | French | 380 992 | 18 887 | 2 590 623
Table 1: Basic Statistics for the three Summarization corpora.
Additionally, three large and heterogeneous corpora (generated from novels,
newspaper articles and news on the Internet) were created to measure
statistics of each language. These corpora contains several million tokens in
English, Spanish and French. Table 2 presents basic statistics on tokens and
characters of the three generic corpora.
Generic Corpus | Tokens | Letters
---|---|---
English | 29 346 289 | 177 717 720
Spanish | 21 445 694 | 134 461 092
French | 17 734 663 | 111 169 782
Table 2: Basic Statistics for the three Language Generic corpora.
### 0.3.2 Ultra-stemming
The first step to represent documents in a suitable space is the pre-
processing. As we use extractive summarization as task, documents have to be
chunked into cohesive textual segments that will be assembled to produce the
summary. Pre-processing is very important because the selection of segments is
based on words or bigrams of words. The choice was made to split documents
into full sentences, in this way obtaining textual segments that are likely to
be grammatically correct. Afterwards, sentences pass through several basic
normalization steps in order to reduce computational complexity. An example of
document pre-processing is given in Table 3. The process is composed by the
following steps:
1. 1.
Sentence splitting: a simple rule-based method is used for sentence splitting.
Documents are chunked at the dot, exclamation and question mark signs.
2. 2.
Sentence filtering: words are converted to lowercase and cleared up from
sloppy punctuation. Words with less than 2 occurrences ($f<2$) are eliminated
(Hapax legomenon presents once in a document). Words that do not carry meaning
such as functional or very common words are removed. Small stop-lists
(depending of language) are used in this step.
3. 3.
Word normalization: remaining words are replaced by their canonical form using
lemmatization, stemming, Ultra-stemming or none of them (raw text).
4. 4.
Text Vectorization: Documents are vectorized in a matrix $S_{[P\times N]}$ of
$P$ sentences and $N$ columns, that represent the occurrences of a letter
(Ultra-stemming) or a word (Lemmatization/Stemming/Raw) $j$, $j=1,2,...,N$ in
the sentence $i$, $i=1,2,...,P$.
5. 5.
Summary generation: each summary is generated by a summarizer based on VSM.
For Ultra-stemming using $n=1$ (Fix1), the maximum dimension $N$ may be up to
32 letters. This generates very compact and efficient matrices, as discussed
in 0.3.4.
Original | A federal judge Monday found President Clinton in civil contempt of court for lying in a deposition about the nature of his sexual relationship with former White House intern Monica S. Lewinsky. Clinton, in a January 1998 deposition in the Paula Jones sexual harassment case, swore that he did not have a sexual relationship with Lewinsky. Clinton later explained that he did not believe he had lied in the case because the type of sex he had with Lewinsky did not fall under the definition of sexual relations used in the case.
---|---
Splitted | s0/A federal judge Monday found President Clinton in civil contempt of court for lying in a deposition about the nature of his sexual relationship with former White House intern Monica S. Lewinsky.
s1/Clinton, in a January 1998 deposition in the Paula Jones sexual harassment
case, swore that he did not have a sexual relationship with Lewinsky.
s2/Clinton later explained that he did not believe he had lied in the case
because the type of sex he had with Lewinsky did not fall under the definition
of sexual relations used in the case.
Stemming | s0/feder judg monday found presid clinton civil contempt court lying in deposit natur sexual relationship former white hous intern monica lewinski
s1/clinton januari deposit paula jone sexual harass case swore sexual
relationship lewinski
s2/clinton explain believ lie case type sex lewinski fall denit sexual relat
case
Fix1 | s0/f j m f p c c c c l d n s r f w h i m l
s1/c j d p j s h c s s r l
s2/c l e b l c t s l f d s r u c
Matrix | letter:c d e f h i j l m n p r s u w
s0: 4 0 0 0 1 1 1 2 2 1 1 1 1 0 1
s1: 2 1 0 0 1 0 2 1 0 0 1 1 2 0 0
s2: 3 1 1 0 0 0 0 3 0 0 0 1 2 1 0
Table 3: Example of some pre-processings (Stemming, Ultra-stemming and matrix
generation) applied to the document NYT19990412.0403 from DUC 2006. Document
is split in sentences; punctuation and case are removed; words are normalized.
For comparison, four methods of normalization were applied after filtering:
* •
Lemmatization by simple dictionary of morphological families: 1.32M words-
entries in Spanish, 208K words in English and 316K in French.
* •
Porter’s Stemming, available at Snowball site:
http://snowball.tartarus.org/texts/stemmersoverview.html) for English,
Spanish, French among other languages.
* •
Raw text without normalization.
* •
Ultra-stemming: the $n$ first letters of each word. For example, in the case
of Ultra-stemming of $n=1$ (Fix1), inflected verbs “sing”, “song”, “sings”,
“singing”… or proper names “smith”, “snowboard”, “sex”,… are all replaced by
letter “s”.
### 0.3.3 Why ultra-stemming could work?
Although this technique could be considered a brutal destruction of the
lexicon, Ultra-stemming is, in fact, an extreme stemming. That is, this
truncation represents with minimum information, what we call the stem of the
stem. In the case of Ultra-stemming with $n=1$, the construction of the
vectors-phrases is performed in a space of $j=1,2,...$ 32 classes, which
produces a dense vector representation.
Of course, classes are not equally populated. Figures 1 to 3 show the ranking
of letters of three corpora in English, Spanish and French. The numbers and
function words were previously removed.
In an automatic extractive summarizer, the weight of phrases is represented in
a suitable vector space. However, if the representation is too large, the
resulting representation is very sparse, which can difficult the weighting of
the sentences. Two hypotheses are the basic ideas for using Ultra-stemming in
automatic summarization task.
Figure 1: Scatter plot of first letter ranking for the English corpus. There
are 16.72M of types, after filtering of functional words and punctuation.
Figure 2: Scatter plot of first letter ranking for the Spanish corpus. There
are 4.53M of types, after filtering of functional words and punctuation.
Figure 3: Scatter plot of first letter ranking for the French corpus. There
are 8.53M of types, after filtering of functional words and punctuation.
The first hypothesis is that a more condensed, but retaining important
information of the original representation, would enable a more effective
weighting for phrases extraction. Ultra-stemming produces an extremely compact
representation of documents, in a Vector Space that can reach only thirty
letters, using the representation of one letter per word. One way of
evaluating the efficacity of a vector representation can be by calculating the
density of the resulting matrix. This point will be discussed in detail in the
next section. The other way is to show that two matrices $A$ and $B$ are
equivalent in the sense that they contain a number of similar informations. If
$A<B$, and $A$ and $B$ represent approximately the same information, then it
may be preferable to use the representation given by $A$ instead of $B$.
Now, how does one know that two matrices contain about the same information?
The second hypothesis is that if the matrices $A$ and $B$ are correlated, then
they probably represent similar information. This point will be proved in
Section 0.4 by the Mantel statistic test.
### 0.3.4 Matrix density
Pre-processing and vectorization of documents will produce very sparse
matrices. However the density of matrices generated is directly dependent on
pre-processing algorithm used. Intuitively, the density of matrices generated
by Ultra-stemming must be much greater than those generated by classical
normalizations. We have calculated the density $\delta$ of a matrix
$S_{[P\times N]}$, of $P$ phrases and a vocabulary of $N$ words as a fraction
of occurrences $C_{w}$ of the word $w$ (elements other than 0), divided by the
size of the matrix $\rho=P\times N$. The equation 1 calculates the density of
$S$.
(1) $\delta(S)=\frac{C_{w}>0}{\rho}$
This density can be an indicator of the amount of information in relation to
the volume of the matrix: lower density implies a greater amount of
computation for ranking sentences. As shown in table 4, the matrix produced by
Ultra-stemming of letters produces a higher average density on the studied
corpora. The matrices generated by Ultra-stemming are filled approximately 50%
(56% for English, 64% for Spanish and 47% for French). The volume of the
matrix generated by each pre-processing method in relation to the size of the
matrix in plain text, is given by:
(2) $V=\frac{\rho(\bullet)}{\rho(\textsc{Raw})}$
This volume represents a small fraction (between 5% and 13% depending on the
language) of the matrix equivalent of plain text.
In case of the corpus Medicine Clínica the standard matrices (lemm $\approx$
101%, stem $\approx$ 103%) are slightly larger than the matrix produced by the
plain text (raw). This can be explained by the presence of Hapax legomenon. In
the case of plain text, a large number of Hapax ($f=1$) is eliminated and it
can produce matrices slightly smaller.
DUC’04 | | $\langle P\rangle=238.0$ | Size | Volume V
---|---|---|---|---
Pre-processing | Density $\delta$ | $\langle N\rangle$ | $\rho=\langle P\rangle\times\langle N\rangle$ | Raw=100%
Lemmatization | 2.6% | 405.5 | 96 509.0 | 96.0%
Stemming | 2.4% | 418.2 | 99 531.6 | 99.0%
Raw | 2.3% | 424.3 | 100 983.4 | 100.0%
fix1 | 55.6% | 25.6 | 6 092.8 | 6.0%
Medicina Clínica | | $\langle P\rangle=88.6$ | Size | Volume V
Pre-processing | Density $\delta$ | $\langle N\rangle$ | $\rho=\langle P\rangle\times\langle N\rangle$ | Raw=100%
Lemmatization | 5.9% | 177.0 | 15 682.2 | 101.3%
Stemming | 5.7% | 179.3 | 15 886.0 | 102.6%
Raw | 5.1% | 174.7 | 15 478.4 | 100.0%
fix1 | 63.7% | 22.2 | 1 966.9 | 12.7%
Pistes | | $\langle P\rangle=319.7$ | Size | Volume V
Pre-processing | Density $\delta$ | $\langle N\rangle$ | $\rho=\langle P\rangle\times\langle N\rangle$ | Raw=100%
Lemmatization | 2.0% | 457.7 | 146 326,7 | 90.0%
Stemming | 1.9% | 474.5 | 151 697.7 | 93.0%
Raw | 1.6% | 508.5 | 162 567.5 | 100.0%
fix1 | 46.8% | 25.0 | 7 992.5 | 4.9%
Table 4: Matrix density for three corpora data. The mean dimension of matrix
$S$, $\rho=\langle P\rangle\times\langle N\rangle$. Density $\delta(S)$ is
calculated by equation 1 and Volume by equation 2.
Statistics for summarization DUC’04 English, Medicina Clínica Spanish and
Pistes French corpora, after removing stop-words, Hapax legomenon and
punctuation, are shown in table 5. The mode of letters per word is 5, 6 and 7,
and 6 respectively for each language.
Corpus | Words | Letters | Mean of letters | Mode on generic
---|---|---|---|---
| | | per word | corpus
DUC’04 | | 11 956 sentences | | English
Lemmatization | 137 454 | 800 723 | 5.83 | $\bullet$
Stemming | 137 101 | 764 015 | 5.57 | $\bullet$
Raw | 136 582 | 902 914 | 6.61 | 5
fix1 | 137 461 | 137 461 | 1.00 | $\bullet$
Medicina Clínica | | 4 480 sentences | | Spanish
Lemmatization | 56 063 | 484 281 | 8.64 | $\bullet$
Stemming | 56 067 | 410 048 | 7.31 | $\bullet$
Raw | 56 115 | 526 660 | 9.38 | 6-7
fix1 | 56 347 | 56 347 | 1.00 | $\bullet$
Pistes | | 16 037 Sentences | | French
Lemmatization | 167 056 | 1 505 169 | 9.01 | $\bullet$
Stemming | 167 231 | 1 264 774 | 7.56 | $\bullet$
Raw | 167 677 | 1 589 190 | 9.48 | 6
fix1 | 168 329 | 168 329 | 1.00 | $\bullet$
Table 5: Statistics for three summarization corpora after filtering and
removing punctuation.
Figures 4, 5 and 6 show the average distribution of letters per word by the
three summary corpora, after the filtering described in 0.3.2. Curves are
shown normalized between $[0,1]$ for the large generic and representative of
the language corpora (cf Section 0.3.1) and the corpora used in each of the
summaries experiments.
Figure 4: Scatter plot of mean length of words for two English corpora
(heterogeneous and summarization raw corpora after filtering). Figure 5:
Scatter plot of mean length of words for two Spanish corpora (heterogeneous
and summarization raw corpora after filtering). Figure 6: Scatter plot of mean
length of words for two French corpora (heterogeneous and summarization raw
corpora after filtering).
## 0.4 Matrix test correlation: the test of Mantel
Different methods of data analysis as ranking are based on distance matrices.
[6] indicates: "In some cases, researchers may wish to compare several
distance matrices with one another in order to test a hypothesis concerning a
possible relationship between these matrices. However, this is not always
evident. Usually, values in distance matrices are, in some way, correlated and
therefore the usual assumption of independence between objects is violated in
the classical tests approach. Furthermore, often, spurious correlations can be
observed when comparing two distances matrices."
As [6] shows, in the Mantel test [33], the null hypothesis is that distances
in a matrix $A$ are independent of the distances, for the same objects, in
another matrix $B$. In other words, we are testing the hypothesis that the
process that has generated the data is or is not the same in the two sets.
Then, testing of the null hypothesis is done by a randomization procedure in
which the original value of the statistic is compared with the distribution
found by randomly reallocating the order of the elements in one of the
matrices. The measure used for the correlation between $A$ and $B$ is the
Pearson correlation coefficient:
(3)
$r(A,B)=\frac{1}{P-1}\sum_{i=1}^{P}\sum_{j=1}^{P}\left[\frac{A_{i,j}-\langle
A\rangle}{\sigma_{A}}\right]\left[\frac{B_{i,j}-\langle
B\rangle}{\sigma_{B}}\right]$
where $P$ is the number of elements in the lower (upper) triangular part of
the matrix, $\langle A\rangle$ is mean for $A$ elements and $\sigma_{A}$ is
the standard deviation of $A$ elements.
Coefficient $r>0$ measures the linear correlation and hence is subject to the
same statistical assumptions. Consequently, if non-linear relationships
between matrices exist, they will be degraded or lost ($r<0$). The testing
procedure for the simple Mantel test goes is the same of [6], and it is as
follows: Assume two symmetric dissimilarity matrices $A$ and $B$ of size
$[P\times P]$. The rows and columns correspond to the same objects.
1. 1.
Compute the Pearson correlation coefficient $r(A,B)$ between the corresponding
elements of the lower-triangular part of the $A$ and $B$, using equation 3.
2. 2.
Permute randomly rows and the corresponding columns of the matrix $A$,
creating a new matrix A’.
3. 3.
Compute $r(A^{\prime},B)$ between matrices A’ and B.
4. 4.
Repeat steps 2 and 3 a great number of times. This will constitute the
reference distribution under the null hypothesis.
The calculation of the correlation between the matrix generated by the Ultra-
stemming and others normalization methods is not straightforward, because the
matrices are not square. In general, the matrix produced by the Ultra-stemming
have a smaller number of columns than the other ones. Then, to calculate a
correlation between matrices of different number of columns, each matrix must
be converted in a symmetric matrix.
Let $S^{\prime}_{[P\times N^{\prime}]}$ of $P$ rows and $N^{\prime}$ columns
be a matrix produced by Ultra-stemming, and let $S_{[P\times N]}$ of $P$ rows
and $N$ columns, be a matrix produced by a classic method of normalization
such that stemming, lemmatization, etc. We have the condition that:
$N^{\prime}\leq N$. Let the new matrices be $A_{[P\times P]}=S\times S^{\prime
T}$ and $B_{[P\times P]}=S\times S^{\prime T}$. They are square symmetrical. A
standard Mantel test can indicate the degree of similarity between $A$ and
$B$. If the similarity is high ($r>0$) with a high confidence value
($p\rightarrow 0$), means that the information of the matrix $A$ is
substantially the same as that contained in the matrix $B$. In other words, we
could replace $S^{\prime}$ for $S$, for purposes of a vector representation of
documents.
Tables 6, 7 and 8 show the correlation of the Mantel test for the three
summary corpora studied. The correlation was calculated between the matrices
$S$ generated by lemmatization (Lemm), stemming (Stem), plain text (Raw) and
the matrix $S^{\prime}$ generated by Ultra-stemming Fix1 using the initial
letter. In all cases the correlation is positive with $p$-value $<0.001$,
which is significant. The calculations were performed with the zt program
written in $C$, of Eric Bonnet and Yves Van de Peer111111zt: a software tool
for simple and partial Mantel tests. This program can be downloaded from the
Web site http://bioinformatics.psb.ugent.be/software/details/ZT [5].
DUC’04 | Lemm | Stem | Raw | fix1
---|---|---|---|---
Lemm | $\bullet$ | 0.96149 | 0.91287 | 0.51904
Stem | 0.96149 | $\bullet$ | 0.94492 | 0.53800
Raw | 0.91287 | 0.94492 | $\bullet$ | 0.46914
fix1 | 0.51904 | 0.53800 | 0.46914 | $\bullet$
Table 6: Mantel test correlation for corpus DUC’04 data (English, $p$-value=0.001). Medicina Clínica | Lemm | Stem | Raw | fix1
---|---|---|---|---
Lemm | $\bullet$ | 0.96725 | 0,91174 | 0.58541
Stem | 0.96725 | $\bullet$ | 0,91942 | 0,49614
Raw | 0,91174 | 0,91942 | $\bullet$ | 0,51503
fix1 | 0,58541 | 0,49614 | 0,51503 | $\bullet$
Table 7: Mantel test correlation for corpus Medicina Clínica data (Spanish, $p$-value=0.001). Pistes | Lemm | Stem | Raw | fix1
---|---|---|---|---
Lemm | $\bullet$ | 0.93016 | 0.85708 | 0.53801
Stem | 0.93016 | $\bullet$ | 0,89499 | 0.51641
Raw | 0.85708 | 0,89499 | $\bullet$ | 0.45156
fix1 | 0,53801 | 0.51641 | 0.45156 | $\bullet$
Table 8: Mantel test correlation for corpus Pistes data (French,
$p$-value=0.001).
According to these correlations, in DUC’04 English corpus, the method Fix1 is
more correlated with Stemming normalization. In Spanish and French corpora,
Fix1 seems slightly correlated with the model lemmatization. This is
intuitively correct and according to the reduced variability of English in
relation to Spanish or French.
## 0.5 Experiments
Ultra-stemming method described in the previous section has been implemented
and evaluated in several corpora in English, Spanish and French languages. The
following subsections present details of the different experiments.
### 0.5.1 Summarizers
Three summarization systems were used in our experiments: Cortex, Enertex and
Artex. All systems have used the same text representation based on Vector
Space Model, described in Section 0.3.2.
* •
Cortex is a single-document summarization system using several metrics and an
optimal decision algorithm [43, 41].
* •
Enertex is summarization system based in Textual Energy concept [15]: text is
represented as a spin system where spins $\uparrow$ represents words that
their occurrences are $f>0$ (spins $\downarrow$ if the word is not present).
* •
Artex (AnotheR TEXt summarizer) is a single-document summarization system that
computes the score of a sentence by calculating a dot product between a
sentence vector and a frequencies vector, multiply by lexical used.
We have conducted our experimentation with the following languages,
summarization tasks, summarizers and data sets: 1) Generic multi-document-
summarization in English with the corpus DUC’04; 2) Generic single-document
summarization in Spanish with the corpus Medicina Clínica and 3) Generic
single document summarization in French with the corpus Pistes.
Then, we have applied the summarization algorithms following the pre-
processing algorithm and finally, results have been evaluated using Fresa.
### 0.5.2 Summaries Evaluation
To evaluate the quality of a summary is not an easy task, and remains an open
question. DUC conferences have introduced the ROUGE evaluation [28], wich
measures the overlap of $n$-grams between a candidate summary and reference
summaries written by humans. In other hand, several metrics without references
have been defined and experimented at DUC and TAC121212www.nist.gov/tac
workshops. Fresa measure [42] is similar to Rouge evaluation but it does not
uses reference summaries. It calculates the divergence of probabilities
between the candidate summary and the document source. Among these metrics,
Kullback-Leibler (KL) and Jensen-Shannon (JS) divergences have been used [29,
42] to evaluate the informativeness of summaries. In this paper, we use Fresa,
based in KL divergence with Dirichlet smoothing, like in the 2010 and 2011
INEX edition [39], to evaluate the informative content of summaries by
comparing their $n$-gram distributions with those from source documents.
Fresa only considered absolute log-diff between frequencies. Let $T$ be the
set of terms in the source. For every $t\in T$ , we denote by $C_{t}^{T}$ its
occurrences in the source and by $C_{t}^{S}$ its occurrences in the summary.
The Fresa package computed the divergence between the source and the summaries
as:
(4) ${\mathcal{D}}(T||S)=\sum_{t\in
T}\left|\log\left(\frac{C_{t}^{T}}{|T|}+1\right)-\log\left(\frac{C_{t}^{S}}{|S|}+1\right)\right|$
To evaluate the quality of generated summaries, several automatic measures
were computed:
* •
Fresa1: Unigrams of single stems after removing stop-words.
* •
Fresa2: Bigrams of pairs of consecutive stems (in the same sentence).
* •
FresaSU4: Bigrams with 2-gaps also made of pairs of consecutive stems but
allowing the insertion between them of a maximum of two stems.
* •
$\langle\textsc{Fresa}\rangle=\frac{\textsc{Fresa}_{1}+\textsc{Fresa}_{2}+\textsc{Fresa}_{SU4}}{3}$
is the mean of Fresa values, and represents the final score in our
experiments.
The scores of Fresa are normalized between 0 and 1. High values mean less
divergence regarding the source document summary, reflecting a greater amount
of information content. All summaries produced by systems were evaluated
automatically using Fresa package.
### 0.5.3 Results
Below we present separate results for the three languages. In this way, we
have analyzed linguistic phenomena specific to each language.
#### English corpus
Results in figure 7 show that Ultra-steming improves the score of the three
automatic summarizer systems. This result is remarkable for Fix1, whose
average matrix represents only 6% of the matrix volume in plain text.
Figure 7: Histogram plot of content evaluation for corpus DUC 2004 Task 2,
with $\langle$Fresa$\rangle$ measures, for each summarizer and each
normalization. Figure 8: Scatter plot of $\langle$Fresa$\rangle$ mean of
Ultra-stemming using $n$ first letters (corpus DUC 2004 Task 2, Cortex
summarizer).
As shown in Figure 7, the performance of the three summarizers is improved
using the Ultra-stemming in relation to other normalizations. So, in
particular, using lemmatization (the best score between the two classic
normalizations), the summarizer Artex, goes from 0.02451 to 0.02697 using
normalization Fix1, i.e. an increase of 10%. Cortex increases of 0.02435 to
0.02682, an augmentation of 10.1% and summarizer Enertex increases of 0.02141
to 0.02626, an augmentation of 22.7%.
A detailed analysis for a particular summarizer is shown in Figure 8. This
figure shows the average score Fresa obtained on DUC’04 English corpus, in
function of Ultra-stemming used, of $n=1,2,...,14$ letters, for the automatic
summarizer Cortex. By comparison, the values Fresa for lemmatization (lemm),
stemming (stem) and plain text (raw) are shown in the graph.
#### Spanish corpus
Spanish is a language with greater variability than English. Results in figure
9 shown that ultra-steming improves the score of the three systems of
automatic summarization utilized. In the case of summarizers Cortex and Artex,
stemming and lemmatization substantially obtains the same scores, which does
not occur with Enertex. However, comparing Ultra-stemming against stemming
Fix1, the three summarizers are benefiting of an increased score (Artex 5%,
Enertex 5.25% and Cortex 7.11 %).
Figure 9: Histogram plot of content evaluation for Spanish corpus Medicina
Clínica with $\langle$Fresa$\rangle$ scores for each summarizer.
Figure 10 shows the mean score $\langle$Fresa$\rangle$ on the Spanish corpus
Medicine Clínica, based on the ultra-stemming ($n=1,2,...14$ letters) using
automatic summarizer Cortex. Values Fresa for lemmatization (Lemm), stemming
(Stem) and plain text (Raw) are also shown.
Figure 10: Scatter plot of $\langle$Fresa$\rangle$ mean vs. Ultra-stemming
using $n$ first letters (corpus Medicina Clínica, Cortex summarizer).
#### French corpus
Figure 11: Histogram plot of content evaluation for French corpus Pistes with
Fresa scores for each summarizer. Figure 12: Scatter plot of
$\langle$Fresa$\rangle$ mean vs. Ultra-stemming using $n$ first letters
(corpus Pistes, Cortex summarizer).
Results in figure 11 show that Ultra-stemming improves the score of the three
automatic summarization systems used. In particular, the summarizer Enertex
using a stemming representation obtains a score Fresa of 0.25928, and using
Fix1 representation a score of 0.29311, i.e., an increase of more than 13%.
Finally, Figure 12 shows the detailed mean score $\langle$Fresa$\rangle$ on
French corpus Pistes, as function of $n=1,2,...,14$ letters, using the
automatic summarizer Cortex. As well, it shows the values Fresa for
lemmatization (Lemm), stemming (Stem) and plain text (Raw).
Overall for the three languages, beyond a certain number of letters (5 for
English, 7 for the Spanish and 6 for French) Ultra-stemming loses its
effectiveness and lemmatization score is higher. A view to the table 5 shows
that this limit has a relationship with the mean, rather than the mode of
letters per word in each language. Apparently, using Ultra-stemming is
interesting when using a number of characters less than the mode of the
language in question.
## 0.6 Discussion and conclusion
In this paper we have introduced and tested a simple pre-processing method
suitable for automatic summarization text. Ultra-stemming is fast and simple.
It reduces the size of the matrix representation, but it retains the
information and charateristics of the document. An important aspect of our
approach is that it does not requires linguistic knowledge or resources which
makes it a simple and efficient pre-processing method to tackle the issue of
Automatic Text Summarization.
And what about times ?
In general, the processing times of Ultra-stemming Fix1 are shorter compared
to all others methods. Of course, processing time depends of summarizer
algorithm and pre-processing algorithm. In general, processing time $\tau$ is
function of:
$\tau=\textrm{time(filtering)+time(normalization)+time(summarizer)}$
In our experiments, time(filtering) is independent of the summarizers and
generally, filtering algorithm is very fast. The time(normalization) depends
on algorithm used (stemming, lemmatizaton) and/or extern resource (dictionary
of lemmatization). The time(summarizer) is intrinsic to each summarizer
system.
By example, Cortex is a very fast summarizer with $O(\log\rho^{2})$ (where
$\rho=P\times N$), and processing times for Stemming, Raw and Fix1 are close.
In other hand, Enertex summarizer has a complexity of $O(\rho^{2})$, then it
needs more time to process the same corpus. In this case, Ultra-stemming is a
very interesting alternative to summarize long corpora. Table 9 shows
processing times for each corpus, following the normalization method for
Cortex, Artex and Enertex summarizers. All times are measured in a 7.8 GB of
RAM computer, Core i7-2640M CPU @ 2.80GHz $\times$ 4 processor, running under
32 bits GNU/Linux (Ubuntu Version 12.04).
Summarizer | | Corpus | | Time
---|---|---|---|---
Cortex | DUC’04 | Medicina Clínica | Pistes | Mean (All)
Lemmatization | 0.80’ | 2.88’ | 1.13’ | 1.60’
Stemming | 0.40’ | 0.26’ | 0.53’ | 0.54’
Raw | 0.33’ | 0.26’ | 0.41’ | 0.40’
fix1 | 0.31’ | 0.26’ | 0.38’ | 0.32’
Artex | DUC’04 | Medicina Clínica | Pistes | Mean (All)
Lemmatization | 1.71’ | 3.10’ | 2.70’ | 2.50’
Stemming | 1.35’ | 0.40’ | 2.11’ | 1.29’
Raw | 1.30’ | 0.38’ | 2.13’ | 1.27’
fix1 | 0.41’ | 0.28’ | 0.51’ | 0.40’
Enertex | DUC’04 | Medicina Clínica | Pistes | Mean (All)
Lemmatization | 9.25’ | 3.38’ | 18.63’ | 10.42’
Stemming | 9.28’ | 0.75’ | 18.38’ | 9.47’
Raw | 9.16’ | 0.73’ | 20.76’ | 10.22’
fix1 | 3.93’ | 0.46’ | 8.35’ | 4.25’
Table 9: Statistics of processing times (in minutes) of three summarizers over
three corpora.
Clearly, the lemmatization of a large dictionary is the most time-consuming
strategy. This is notable in the Spanish corpus, using a 1.3M dictionary
entries. Lemmatization is at the same time, the strategy that produces the
best results after the Ultra-stemming (Fixn with $n=1...4$ letters). In the
case of Artex summarizer, the gain in time is dramatic, going from 2.50’ using
lemmatization to 0.40 using Fix1, i.e. a gain of 625%. This gain is 500% for
Cortex and 245% for Enertex.
From our point of view, the Ultra-stemming of $n$ letters has three important
advantages:
1. 1.
A reduction of the space and the calculation time of automatic summarization
algorithms based on the vector space model.
2. 2.
Improving of summary content, when using $n<$ mode in letters per word of each
language.
3. 3.
Applications on resource sparse languages. Typically $\pi$ languages where no
lemmatizers, stemmers or parsers, neither corpora nor native linguist
available, the Ultra-stemming can be an attractive alternative for automatic
document summarizers.
Summarization using the Ultra-stemming representation for sentence scoring,
improve the identification of most relevant sentences from documents. The
results obtained on corpora in English, Spanish and French prove that Ultra-
stemming can achieve good results for content quality. Tests with other
corpora (DUC evaluation campaigns, TAC, INEX, etc.) in mono-and multi-document
guided by a subject, and $\pi$ languages (Nahuatl, Maya, Somali, Interlingua,
etc.) using content evaluation with or without reference summaries still in
progress.
## References
* [1] E. Airio. Word normalization and decompounding in mono- and bilingual IR. Information Retrieval, 9(3):249–271, 2006.
* [2] J. Atserias, B. Casas, E. Comelles, M. González, L. Padró, and M. Padró. FreeLing 1.3: Syntactic and semantic services in an open-source NLP library. In fifth International Conference on Language Resources and Evaluation (LREC’06), ELRA, 2006.
* [3] R. Baeza-Yates and B. Ribeiro-Neto. Modern Information Retrieval. Addison Wesley, 1999.
* [4] D. Bernhard. Apprentissage non supervisé de familles morphologiques par classification ascendante hiérarchique. In TALN’07, volume 1, pages 367–376, 2006.
* [5] Eric Bonnet and Yves Van de Peer. zt: a software tool for simple and partial Mantel tests. Journal of Statistical software, 7(10):1–12, 2002.
* [6] Eric Bonnet and Yves Van de Peer. zt: A sofware tool for simple and partial mantel tests. Journal of Statistical Software, 7(10):1–12, 10 2002.
* [7] L. Breiman, J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Wadsworth and Brooks, Monterey, CA, 1984.
* [8] M.T Cabré Castellví. Typology of neologisms: a complex task. Alfa (São Paulo), 50(2):229–250, 2006.
* [9] M. Creutz and K. Lagus. Unsupervised Discovery of Morphemes. In 6th Workshop of the ACL Special Interest Group in Computational Phonology (SIGPHON), pages 21–30, 2002.
* [10] M. Creutz and K. Lagus. Unsupervised morpheme segmentation and morphology induction from text corpora using Morfessor 1.0. Technical Report A81, Publications in Computer and Information Science, Helsinki University of Technology, 2005.
* [11] Harold Daumé III. Practical structured learning techniques for natural language processing. PhD thesis, Los Angeles, CA, 2006.
* [12] D.P. Lyras and K.N. Sgarbas and N.D. Fakotakis. Using the Levenshtein Edit Distance for Automatic Lemmatization: A Case Study for Modern Greek and English. In 19th IEEE International Conference on Tools with Artificial Intelligence - (ICTAI’07), volume 2, pages 428–435, 2007.
* [13] H. P. Edmundson. New Methods in Automatic Extraction. Journal of the Association for Computing Machinery, 16(2):264–285, 1969.
* [14] F. Namer. Flemm: Un analyseur Flexionnel de Français à base de règles. In Christian Jacquemin, editor, Traitement automatique des Langues pour la recherche d’information, pages 523–547. Hermes, 2000.
* [15] Silvia Fernández, Eric SanJuan, and Juan-Manuel Torres-Moreno. Textual Energy of Associative Memories: performants applications of Enertex algorithm in text summarization and topic segmentation. In Proceedings of the Mexican International Conference on Artificial Intelligence (MICAI’07), pages 861–871, Aguascalientes, Mexique, 2007. Springer-Verlag.
* [16] C.G. Figuerola, R. Gómez Díaz, and E. López de San Román. Stemming and n-grams in Spanish: An evaluation of their impact on information retrieval. Journal of Information Science, 26(6):461–467, 2000.
* [17] A.F. Gelbukh, M. Alexandrov, and S.-Y. Han. Detecting inflection patterns in natural language by minimization of morphological model. In Sanfeliu A., Trinidad J.F.M., and Carrasco-Ochoa J.A., editors, 9th Iberoamerican Congress on Pattern Recognition (CIARP’04), Progress in Pattern Recognition, Image Analysis and Applications, volume 3287, pages 432–438. Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2004.
* [18] J.A. Goldsmith. Unsupervised Learning of the Morphology of a Natural Language. Computational Linguistics, 27(2):153–198, 2001.
* [19] N. Grabar and P. Zweigenbaum. Acquisition automatique de connaissances morphologiques sur le vocabulaire médical. In TALN’99, pages 175–184. Pascal Amsili, Ed., 1999.
* [20] C. Hammarström. Unsupervised Learning of Morphology: Survey, Model, Algorithm and Experiments. Master’s thesis, Department of Computer Science and Engineering, Chalmers University, 2007.
* [21] H. Hammarström. A Naive Theory of Morphology and an Algorithm for Extraction. In R. Wicentowski and G. Kondrak, editors, SIGPHON 2006: ACL Special Interest Group on Computational Phonology, pages 79–88, 2006.
* [22] S. Helmut. Probabilistic part-of-speech tagging using decision trees. In International Conference on New Methods in Language Processing, September 1994.
* [23] V. Hollink, J. Kamps, C. Monz, and M. De Rijke. Monolingual Document Retrieval for European Languages. Information Retrieval, 7(1-2):33–52, January 2004.
* [24] C. Jacquemin and E. Tzoukermann. NLP for term variant extraction: synergy between morphology, lexicon and syntax. In Tomek Strzalkowski, editor, Natural Language Information Retrieval, volume 7 of Text, Speech and Language Technology, pages 25–74. Kluwer Academic Publishers, Dordrecht/Boston/London, 1999.
* [25] T. Korenius, J. Laurikkala, K. Jarvelin, and M. Juhola. Stemming and lemmatization in the clustering of finnish text documents. In CIKM’04: Thirteenth ACM Conference on Information and Knowledge Management, pages 625–633. ACM Press, 2004.
* [26] J. Kupiec, J. Pedersen, and F. Chen. A trainable document summarizer. In Proceedings of the 18th Conference ACM Special Interest Group on Information Retrieval (SIGIR’95), pages 68–73, Seattle, WA, Etats-Unis, 1995\. ACM Press, New York.
* [27] Y. Lepage. Solving analogies on words: an algorithm. In COLING-ACL’98, pages 728–735, 1998.
* [28] Chin-Yew Lin. ROUGE: A Package for Automatic Evaluation of Summaries. In Marie-Francine Moens and Stan Szpakowicz, editors, Proceedings of the Workshop Text Summarization Branches Out (ACL’04), pages 74–81, Barcelone, Espagne, july 2004. ACL.
* [29] Annie Louis and Ani Nenkova. Automatic Summary Evaluation without Human Models. In First Text Analysis Conference (TAC’08), Gaithersburg, MD, Etats-Unis, 17-19 November 2008.
* [30] H.P. Luhn. The Automatic Creation of Literature Abstracts. IBM Journal of Research and Development, 2(2):159–165, 1958.
* [31] I. Mani and M. Mayburi. Advances in Automatic Text Summarization. MIT Press, Cambridge, 1999.
* [32] C.D. Manning and H. Schütze. Foundations of Statistical Natural Language Processing. The MIT Press, 1999.
* [33] Nathan Mantel and Ranchhodbhai S. Valand. A Technique of Nonparametric Multivariate Analysis. Biometrics, 26(3):547–558, Sep., 1970.
* [34] Juan Manuel Torres Moreno. Reagrupamiento en familias y lexematización automática independientes del idioma. Revista Iberoamericana de Inteligencia Artificial, 14(47):38–53, 2010.
* [35] C.D. Paice. Another stemmer. SIGIR Forum, 24(3):56–61, 1990.
* [36] C.D. Paice. Method for Evaluation of Stemming Algorithms Based on Error Counting. Journal of the American Society for Information Science, 47(8):632–649, 1996.
* [37] M.F. Porter. An algorithm for suffix stripping. Program, 40(3):211–218, 2006.
* [38] J. Ross Quinlan. C4.5: Programs for Machine Learning (Morgan Kaufmann Series in Machine Learning). Morgan Kaufmann, 1 edition, 1993.
* [39] Eric SanJuan, Patrice Bellot, Véronique Moriceau, and Xavier Tannier. Overview of the inex 2010 question answering track (qa@inex). In Shlomo Geva, Jaap Kamps, Ralf Schenkel, and Andrew Trotman, editors, Comparative Evaluation of Focused Retrieval, volume 6932 of Lecture Notes in Computer Science, pages 269–281. Springer Berlin / Heidelberg, 2011.
* [40] Simone Teufel and Marc Moens. Sentence extraction as a classification task. In I. Mani and M. Maybury, editors, Proceedings of the ACL/EACL’97 Workshop on Intelligent Scalable Text Summarization, Madrid, Espagne, 11 juillet 1997.
* [41] Juan-Manuel Torres-Moreno. Résumé automatique de documents: une approche statistique. Hermès-Lavoisier, Paris, 2011.
* [42] Juan-Manuel Torres-Moreno, Horacio Saggion, Iria da Cunha, and Eric SanJuan. Summary Evaluation With and Without References. Polibits: Research journal on Computer science and computer engineering with applications, 42:13–19, 2010.
* [43] Juan-Manuel Torres-Moreno, Patricia Velázquez-Morales, and Jean-Guy Meunier. Cortex : un algorithme pour la condensation automatique des textes. In Proceedings of the Conference de l’Association pour la Recherche Cognitive, volume 2, pages 365–366, Lyon, France, 2001.
* [44] A. Medina Urrea. Automatic Discovery of Affixes by means of a Corpus: A Catalog of Spanish Affixes. Journal of Quantitative Linguistics, 7(2):97–114, 2000.
* [45] J. Vilares, M. A. Alonso, and M. Vilares. Extraction of complex index terms in non-English IR: A shallow parsing based approach. Information Processing and Management, 44(4):1517–1537, 2008.
* [46] J. Vilares, D. Cabrero, and M. A. Alonso. Applying productive derivational morphology to term indexing of Spanish texts. In Alexander Gelbukh, editor, Computational Linguistics and Intelligent Text Processing (CICLing’01), volume 2004 of Lecture Notes in Computer Science, pages 336–348. Springer-Verlag, Berlin-Heidelberg-New York, 2001.
|
arxiv-papers
| 2012-09-14T08:45:26 |
2024-09-04T02:49:35.143958
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan-Manuel Torres-Moreno",
"submitter": "Juan Manuel Torres Moreno",
"url": "https://arxiv.org/abs/1209.3126"
}
|
1209.3152
|
# $C\\!P$ violation in $D$ decays
V. Vagnoni Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Italy
###### Abstract
First evidence for $C\\!P$ violation in two-body singly Cabibbo-suppressed
decays of $D^{0}$ mesons reported by LHCb has recently aroused great interest
in charm physics. In this document the latest measurements of $C\\!P$
violation in the charm sector are discussed. LHCb and CDF results on time-
integrated $C\\!P$ asymmetries in $D^{0}\to\pi^{-}\pi^{+}$ and $D^{0}\to
K^{-}K^{+}$ decays are presented in some detail. A search for $C\\!P$
violation performed by Belle in other two-body decays, namely $D^{0}\to
K^{0}_{S}\pi^{0}$, $D^{0}\to K^{0}_{S}\eta^{(\prime)}$,
$D^{+}_{(s)}\to\phi\pi^{+}$ and $D^{+}\to\pi^{+}\eta^{(\prime)}$, is also
presented. Finally, results obtained by CDF with $D^{0}\to
K^{0}_{S}\pi^{+}\pi^{-}$ decays, as well as by LHCb and BaBar with other
multi-body $D$ decays, are shown.
## I Introduction
The violation of $C\\!P$ symmetry is well established in the $K^{0}$ and
$B^{0}$ meson systems (1; 2; 3; 4), and first evidence in the $B^{0}_{s}$
system has been recently reported (5). Experimental measurements of $C\\!P$
violation (CPV) in the quark flavour sector performed so far are generally
well described by the Cabibbo-Kobayashi-Maskawa mechanism (6; 7) of the
Standard Model (SM). However, it is believed that the size of CPV in the SM is
not sufficient to account for the asymmetry between matter and antimatter in
the Universe (8), hence additional sources of $C\\!P$ symmetry breaking are
being searched for as manifestations of physics beyond the SM.
The charm sector is a promising place to probe for the effects of physics
beyond the SM. There has been a renaissance of interest in the past few years
since evidence for $D^{0}$ mixing was first seen (9; 10). Mixing is now well
established (11) at a level which is consistent with expectations (12). The
recent evidence for CPV in singly Cabibbo-suppressed decays of $D^{0}$ mesons
to two-body final states, reported by LHCb (13), has definitely heightened the
theoretical interest in charm physics. Prior to this measurement, $C\\!P$
asymmetries in these decays were expected to be very small in the SM (14; 15;
16; 17), with naïve predictions of up to $\mathcal{O}(10^{-3})$. For this
reason, the asymmetry measured by LHCb, characterized by a central value of
$\mathcal{O}(10^{-2})$, came as big surprise.
Unfortunately, precise theoretical predictions of CPV in this sector are very
difficult to achieve, as the charm quark is too heavy for chiral perturbation
and too light for heavy-quark effective theory to be applied reliably. This
fact means we cannot conclude that the observed effect is a clear sign of
physics beyond the SM (18; 19; 20; 21). In order to investigate these
promising hints further and clarify the picture, carrying out complementary
measurements in other charm decays is of paramount importance.
In the next sections, after a description of the LHCb and CDF measurements
with two-body $D^{0}$ decays, we discuss other recent measurements by the
BaBar, Belle, CDF and LHCb collaborations, involving two-body as well as
three- and four-body $D$ decays.
## II $D^{0}\to\pi^{-}\pi^{+}$ and $D^{0}\to K^{-}K^{+}$
The time-dependent $C\\!P$ asymmetry $A_{C\\!P}(f;\,t)$ for $D^{0}$ decays to
a self-conjugate $C\\!P$ eigenstate $f$ (with $f=\bar{f}$) is defined as
$A_{C\\!P}(f;\,t)=\frac{\Gamma(D^{0}(t)\to f)-\Gamma(\bar{D}^{0}(t)\to
f)}{\Gamma(D^{0}(t)\to f)+\Gamma(\bar{D}^{0}(t)\to f)},$
where $\Gamma$ is the decay rate for the process indicated. In general
$A_{C\\!P}(f;\,t)$ depends on $f$. For $f=K^{-}K^{+}$ and $f=\pi^{-}\pi^{+}$,
$A_{C\\!P}(f;\,t)$ can be expressed in terms of two contributions: a direct
component associated with CPV in the decay amplitudes, and an indirect
component associated with CPV in the mixing or in the interference between
mixing and decay.
The asymmetry $A_{C\\!P}(f;\,t)$ may be written to first order as (22)
$A_{C\\!P}(f;\,t)=a^{\mathrm{dir}}_{C\\!P}(f)\,+\,\frac{t}{\tau}a^{\mathrm{ind}}_{C\\!P},$
where $a^{\mathrm{dir}}_{C\\!P}(f)$ is the direct $C\\!P$ asymmetry, $\tau$ is
the $D^{0}$ lifetime, and $a^{\mathrm{ind}}_{C\\!P}$ is the indirect $C\\!P$
asymmetry, which is universal to a good approximation in the SM (23). The
time-integrated asymmetry measured by an experiment, $A_{C\\!P}(f)$, depends
upon the time-acceptance of that experiment. It can be written as
$A_{C\\!P}(f)=a^{\mathrm{dir}}_{C\\!P}(f)\,+\,\frac{\langle
t\rangle}{\tau}a^{\mathrm{ind}}_{C\\!P},$
where $\langle t\rangle$ is the average decay time in the reconstructed
sample. Denoting by $\Delta$ the differences between quantities for $D^{0}\to
K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$ it is then possible to write
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle=$ $\displaystyle
A_{C\\!P}(K^{-}K^{+})\,-\,A_{C\\!P}(\pi^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle\left[a^{\mathrm{dir}}_{C\\!P}(K^{-}K^{+})\,-\,a^{\mathrm{dir}}_{C\\!P}(\pi^{-}\pi^{+})\right]\,+\,\frac{\Delta\langle
t\rangle}{\tau}a^{\mathrm{ind}}_{C\\!P}.$
In the limit of vanishing $\Delta\langle t\rangle$, $\Delta A_{C\\!P}$ becomes
equal to the difference in the direct $C\\!P$ asymmetry between the two
decays. However, if the time-acceptance is different for the $K^{-}K^{+}$ and
$\pi^{-}\pi^{+}$ final states, a contribution from indirect CPV remains.
The LHCb collaboration performed a measurement of the difference in time-
integrated $C\\!P$ asymmetries between $D^{0}\to K^{-}K^{+}$ and
$D^{0}\to\pi^{-}\pi^{+}$ using 0.62 fb-1 of data (13). The flavour of the
initial state ($D^{0}$ or $\bar{D}^{0}$) is tagged by looking for the charge
of the slow pion ($\pi_{s}^{+}$) in the decay chain $D^{*+}\to
D^{0}\pi_{s}^{+}$.
The raw asymmetry for tagged $D^{0}$ decays to a final state $f$ is given by
$A_{\mathrm{raw}}(f)$, defined as
$A_{\mathrm{raw}}(f)=\frac{N(D^{*+}\to
D^{0}(f)\pi_{s}^{+})\,-\,N(D^{*-}\to\bar{D}^{0}(f)\pi_{s}^{-})}{N(D^{*+}\to
D^{0}(f)\pi_{s}^{+})\,+\,N(D^{*-}\to\bar{D}^{0}(f)\pi_{s}^{-})},$
where $N(X)$ refers to the number of reconstructed events of decay $X$ after
background subtraction.
To first order the raw asymmetries may be written as a sum of four components,
due to physics and detector effects:
$A_{\mathrm{raw}}(f)=A_{C\\!P}(f)\,+\,A_{\mathrm{D}}(f)\,+\,A_{\mathrm{D}}(\pi_{s}^{+})\,+\,A_{\mathrm{P}}(D^{*+}).$
Here, $A_{\mathrm{D}}(f)$ is the asymmetry in efficiency for the $D^{0}$ decay
into the final state $f$, $A_{\mathrm{D}}(\pi_{s}^{+})$ is the asymmetry in
efficiency for the slow pion from the $D^{*+}$ decay chain, and
$A_{\mathrm{P}}(D^{*+})$ is the production asymmetry for $D^{*+}$ mesons. The
first-order expansion is valid since the individual asymmetries are small.
For a two-body decay of a spin-0 particle to a self-conjugate final state
there can be no $D^{0}$ detection asymmetry, i.e.
$A_{\mathrm{D}}(K^{-}K^{+})=A_{\mathrm{D}}(\pi^{-}\pi^{+})=0.$ Moreover,
$A_{\mathrm{D}}(\pi_{s}^{+})$ and $A_{\mathrm{P}}(D^{*+})$ are independent of
$f$ and thus those terms cancel in the difference
$A_{\mathrm{raw}}(K^{-}K^{+})\,-\,A_{\mathrm{raw}}(\pi^{-}\pi^{+})$, resulting
in
$\Delta
A_{C\\!P}=A_{\mathrm{raw}}(K^{-}K^{+})\,-\,A_{\mathrm{raw}}(\pi^{-}\pi^{+}).$
Note that the production asymmetry $A_{\mathrm{P}}(D^{*+})$ can be neglected
in the case of the CDF experiment. This is because, in contrast with LHCb
which is a forward spectrometer and employs flavour-asymmetric $pp$
collisions, CDF was a symmetric detector in pseudorapidity ($\eta$) and
operated at a flavour-symmetric $p\bar{p}$ collider. Hence, integrating the
measurement over a symmetric $\eta$ range, the possible presence of a
production asymmetry is removed by construction, i.e.
$A_{\mathrm{P}}(D^{*+})=0$.
The mass difference ($\delta m$) spectra of candidates selected by LHCb as in
Ref. (13), where $\delta m=m(h^{-}h^{+}\pi_{s}^{+})-m(h^{-}h^{+})-m(\pi^{+})$
for $h=K,\pi$, are shown in Figure 1. The $D^{*+}$ signal yields are
approximately $1.44\times 10^{6}$ in the $K^{-}K^{+}$ sample, and $0.38\times
10^{6}$ in the $\pi^{-}\pi^{+}$ sample.
Figure 1: LHCb fits to the $\delta m$ spectra, where the $D^{0}$ is
reconstructed in the final states (a) $K^{-}K^{+}$ and (b) $\pi^{-}\pi^{+}$.
Fits are performed on the samples in order to determine
$A_{\mathrm{raw}}(K^{-}K^{+})$ and $A_{\mathrm{raw}}(\pi^{-}\pi^{+})$. The
production and detection asymmetries can vary with $p_{\mathrm{T}}$ and
pseudorapidity $\eta$, and so can the detection efficiency of the two
different $D^{0}$ decays, in particular through the effects of the particle
identification requirements. For this reason, since the different masses of
the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ final states lead to differences in the
kinematic distributions of accepted signal events in the two cases, LHCb
performs the analysis in several kinematic bins defined by the
$p_{\mathrm{T}}$ and $\eta$ of the $D^{*+}$ candidates, the momentum of the
slow pion, and the sign of $p_{x}$ of the slow pion at the $D^{*+}$ vertex.
In each bin, one-dimensional unbinned maximum likelihood fits to the $\delta
m$ spectra are performed. A value of $\Delta A_{C\\!P}$ is determined in each
measurement bin as the difference between $A_{\mathrm{raw}}(K^{-}K^{+})$ and
$A_{\mathrm{raw}}(\pi^{-}\pi^{+})$. A weighted average is performed to yield
the result $\Delta A_{C\\!P}=(-0.82\pm 0.21)\%$, where the uncertainty is
statistical only. Including the systematic uncertainty, LHCb measures the
time-integrated difference in $C\\!P$ asymmetry between $D^{0}\to K^{-}K^{+}$
and $D^{0}\to\pi^{-}\pi^{+}$ decays to be
$\Delta A_{C\\!P}^{\mathrm{LHCb}}=\left[-0.82\pm 0.21(\mathrm{stat.})\pm
0.11(\mathrm{syst.})\right]\%.$
Dividing the central value by the sum in quadrature of the statistical and
systematic uncertainties, the significance of the measured deviation from zero
is $3.5\sigma$. This is the first evidence for $C\\!P$ violation in the charm
sector.
A similar measurement has been performed by CDF (24). Figure 2 shows the
invariant $D^{0}\pi_{s}$ mass for $D^{0}$ decays to $\pi^{-}\pi^{+}$ and
$K^{-}K^{+}$, corresponding to an integrated luminosity of 9.7 fb-1. The
$D^{*+}$ signal yields are approximately $1.2\times 10^{6}$ in the
$K^{-}K^{+}$ sample, and $0.55\times 10^{6}$ in the $\pi^{-}\pi^{+}$ sample.
These yields are very close to those of LHCb, despite a difference in
integrated luminosity by a factor 15.
Figure 2: CDF distributions of $D^{0}\pi^{+}$ mass with fit results overlaid
for (a) $D^{0}\to\pi^{-}\pi^{+}$, (b) $\bar{D}^{0}\to\pi^{-}\pi^{+}$, (c)
$D^{0}\to K^{-}K^{+}$ and (d) $\bar{D}^{0}\to K^{-}K^{+}$.
CDF uses a different approach from LHCb to take into account differences in
kinematics between the two decay modes. The kinematic distributions are
equalized by means of an event-by-event reweighting technique, and then a
single fit is performed on the reweighted sample integrated in phase space.
The final result obtained by CDF is
$\Delta A_{C\\!P}^{\mathrm{CDF}}=\left[-0.62\pm 0.21(\mathrm{stat.})\pm
0.10(\mathrm{syst.})\right]\%,$
which deviates from zero by $2.7\sigma$. This result is compatible with the
LHCb measurement, with comparable accuracy and less than $1\sigma$ difference
between the central values.
CDF has also provided individual measurements of $A_{C\\!P}(D^{0}\to
K^{-}K^{+})$ and $A_{C\\!P}(D^{0}\to\pi^{-}\pi^{+})$ (25), using a carefully
constructed combination of raw asymmetries measured from tagged $D^{*+}\to
D^{0}(K^{-}\pi^{+})\pi^{+}$, $D^{*+}\to D^{0}(\pi^{-}\pi^{+})\pi^{+}$,
$D^{*+}\to D^{0}(K^{-}K^{+})\pi^{+}$ and untagged $D^{0}\to K^{-}\pi^{+}$
decays. The calculation assumes that the production asymmetry of $D^{*+}$ is
negligible. This holds at CDF but not at LHCb, where this kind of measurement
is considerably more involved and has not been performed yet. Using part of
the full data sample, corresponding to about 6 fb-1, CDF obtains:
$\displaystyle A_{C\\!P}^{D^{0}\to K^{-}K^{+}}$ $\displaystyle=$
$\displaystyle\left[-0.24\pm 0.22(\mathrm{stat.})\pm
0.09(\mathrm{syst.})\right]\%,$ $\displaystyle
A_{C\\!P}^{D^{0}\to\pi^{-}\pi^{+}}$ $\displaystyle=$
$\displaystyle\left[0.22\pm 0.24(\mathrm{stat.})\pm
0.11(\mathrm{syst.})\right]\%.$
## III Other two-body decays
The Belle collaboration has searched for CPV in the decay $D^{0}\to
K^{0}_{S}P^{0}$ (26), where $P^{0}$ denotes a neutral pseudoscalar meson that
is either a $\pi^{0}$, $\eta$, or $\eta^{\prime}$, using an integrated
luminosity of 791 fb-1. The observed $K^{0}_{S}P^{0}$ final states are
mixtures of $D^{0}\to\bar{K}^{0}P^{0}$ and $D^{0}\to K^{0}P^{0}$ decays where
the former are Cabibbo-favored and the latter are doubly Cabibbo-suppressed.
SM $K^{0}-\bar{K}^{0}$ mixing leads to a small $C\\!P$ asymmetry in final
states containing a neutral kaon, even if no $C\\!P$ violating phase exists in
the charm decay. Figure 3 shows the distributions of the mass difference
$M(D^{*})-M(D)$ for the various decay modes. No evidence for CPV in these
decays is observed, as Belle measures:
$\displaystyle A_{C\\!P}^{D^{0}\to K^{0}_{S}\pi^{0}}$ $\displaystyle=$
$\displaystyle\left[-0.28\pm 0.19(\mathrm{stat.})\pm
0.10(\mathrm{syst.})\right]\%,$ $\displaystyle A_{C\\!P}^{D^{0}\to
K^{0}_{S}\eta}$ $\displaystyle=$ $\displaystyle\left[+0.54\pm
0.51(\mathrm{stat.})\pm 0.16(\mathrm{syst.})\right]\%,$ $\displaystyle
A_{C\\!P}^{D^{0}\to K^{0}_{S}\eta^{\prime}}$ $\displaystyle=$
$\displaystyle\left[+0.98\pm 0.67(\mathrm{stat.})\pm
0.14(\mathrm{syst.})\right]\%.$
Figure 3: Distributions of the mass difference $M(D^{*})-M(D)$ for the
$D^{0}\to K^{0}_{S}P^{0}$ decay modes studied by Belle. Left plots show the
mass difference between $D^{*+}$ and $D^{0}$ and right plots show that between
$D^{*-}$ and $D^{0}$. The top plots are for the $K_{S}^{0}\pi^{0}$ final
state, the middle plots for $K_{S}^{0}\eta$, and the bottom plots for
$K_{S}^{0}\eta^{\prime}$. The points with error bars are the data and the
histograms show the results of the parameterizations of the data.
Belle has also searched for CPV in $D^{+}_{(s)}\to\phi\pi^{+}$ (27) decays.
For the $\phi\pi^{+}$, this is achieved by measuring the $C\\!P$ violating
asymmetries for the Cabibbo-suppressed decays $D^{+}\to K^{+}K^{-}\pi^{+}$ and
the Cabibbo-favored decays $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ in the
$K^{+}K^{-}$ mass region of the $\phi$ resonance, using 955 fb-1 of data. The
mass distributions are shown in Figure 4. Belle finds about $0.237\times
10^{6}$ $D^{\pm}$ and $0.723\times 10^{6}$ $D^{\pm}_{s}$ decays. Assuming
negligible CPV in Cabibbo-favoured decays, Belle measures
$A_{C\\!P}^{D^{+}\to\phi\pi^{+}}=\left[+0.51\pm 0.28(\mathrm{stat.})\pm
0.05(\mathrm{syst.})\right]\%.$
The result shows no evidence for CPV and agrees with SM predictions.
Figure 4: Invariant mass distributions with the fitted functions superimposed
for (a) $\phi\pi^{+}$ and (b) $\phi\pi^{-}$, observed by Belle.
Another relevant result has been obtained by Belle using 791 fb-1 of data,
with the most sensitive search for CPV in the decays $D^{+}\to\pi^{+}\eta$ and
$D^{+}\to\pi^{+}\eta^{\prime}$ (28). Figure 5 shows the $\pi^{+}\eta$ and
$\pi^{+}\eta^{\prime}$ invariant mass distributions. The final results
obtained by Belle for CPV in these decay modes are:
$\displaystyle A_{C\\!P}^{D^{+}\to\pi^{+}\eta}$ $\displaystyle=$
$\displaystyle\left[+1.74\pm 1.13(\mathrm{stat.})\pm
0.19(\mathrm{syst.})\right]\%,$ $\displaystyle
A_{C\\!P}^{D^{+}\to\pi^{+}\eta^{\prime}}$ $\displaystyle=$
$\displaystyle\left[-0.12\pm 1.12(\mathrm{stat.})\pm
0.17(\mathrm{syst.})\right]\%.$
Again, no evidence for CPV is found with these modes.
Figure 5: Invariant mass distributions for (left) $\pi^{+}\eta$ and (right)
$\pi^{+}\eta^{\prime}$ final states, observed by Belle. Points with error bars
and histograms correspond to the data and the fit, respectively.
## IV $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$
With the same trigger used to collect $D^{0}\to h^{+}h^{-}$ decays, and using
an offline selection based on a neural network, CDF is able to reconstruct
about $0.35\times 10^{6}$ decays to resonances of $D^{*}$ tagged decays, using
6 fb-1 of integrated luminosity (29). Figure 6 shows the
$K^{0}_{S}\pi^{+}\pi^{-}$ mass spectrum and the $D^{0}\to
K^{0}_{S}\pi^{+}\pi^{-}$ Dalitz plot obtained by CDF. The analysis is made
both in a model-independent way, by binning the Dalitz plot and looking for
bin-by-bin asymmetries, and by fitting the population of each resonance in
$D^{0}$ and $\bar{D}^{0}$ decays using an isobar model.
Figure 6: Invariant $K^{0}_{S}\pi^{+}\pi^{-}$ mass distribution (top) and
$D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ Daltiz plot (bottom) obtained by CDF.
The search for CPV in bin-by-bin asymmetries shows no evidence of deviation
from a Gaussian distribution with zero mean and unit width. The full Dalitz
fit includes a parameterization of the efficiency over the Dalitz plane. An
event-by-event reweighting is applied in order to equalize $D^{*+}$ and
$D^{*-}$ kinematics. Again, no evidence of CPV is found in any of the
considered resonant modes. CDF measures the following asymmetry integrated
over the Dalitz plane:
$A_{C\\!P}=\left[-0.05\pm 0.57(\mathrm{stat.})\pm
0.54(\mathrm{syst.})\right]\%.$
## V Other multi-body $D$ decays
LHCb has demonstrated the ability to select large samples of three-body
decays. For example, about $0.37\times 10^{6}$ signal $D^{+}\to
K^{-}K^{+}\pi^{+}$ decays with high purity have been selected using 35 pb-1 of
integrated luminosity with data taken during 2010 (30). The reconstructed
$K^{-}K^{+}\pi^{+}$ mass distribution and the $D^{+}\to K^{-}K^{+}\pi^{+}$
Dalitz plot are shown in Figure 7. The charge asymmetries in the control modes
$D^{+}\to K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\to K^{-}K^{+}\pi^{+}$ are
investigated to eliminate the possibility of observing detector asymmetries as
signals of CPV. No significant charge asymmetries are observed, indicating
that such systematic effects are negligible at this level of precision. The
Dalitz plot of the signal mode is then binned according to four binning
schemes, two of which account for the resonant structure of the decay and two
of which do not. The distributions of bin-by-bin asymmetries are consistent
with no CPV in each of the binning schemes.
Figure 7: Invariant $K^{-}K^{+}\pi^{+}$ mass distribution (top) and $D^{+}\to
K^{-}K^{+}\pi^{+}$ Daltiz plot (bottom) obtained by LHCb.
With four-body decays the phase space becomes five-dimensional, and the
difficulty of the analysis grows considerably. However, in four-body decays
the method of $T$-odd moments becomes available. This has been applied e.g. by
the BaBar collaboration searching for CPV in $D^{+}\to
K^{0}_{S}K^{+}\pi^{+}\pi^{-}$ decays (31). The procedure consists in defining
a triple product of the momenta of three of the particles, i.e.
$C_{T}=\vec{p}_{K^{+}}\cdot(\vec{p}_{\pi^{+}}\times\vec{p}_{\pi^{-}})$. It is
then possible to define the quantity $A_{T}$ for $D^{+}$ decays (and its
analogue $\bar{A}_{T}$ for $D^{-}$ decays) as:
$A_{T}=\frac{\Gamma(C_{T}>0)-\Gamma(C_{T}<0)}{\Gamma(C_{T}>0)+\Gamma(C_{T}<0)}.$
Then, if the quantity $\mathcal{A}_{T}=\frac{1}{2}(A_{T}-\bar{A}_{T})$ differs
from zero, this is a sign of CPV. No evidence for CPV is found, as BaBar
obtains:
$\mathcal{A}_{T}=\left[-1.20\pm 1.00(\mathrm{stat.})\pm
0.46(\mathrm{syst.})\right]\%.$
## VI Conclusions
We have summarized some of the latest results involving two-, three- and four-
body $D$ decays at BaBar, Belle, CDF and LHCb. In the past months, first
evidence of CPV in charm decays has been first obtained by LHCb with a
$3.5\sigma$ significance, then followed by CDF with $2.7\sigma$. This has
triggered a big effort by the theory community, in order to understand whether
any effect of physics beyond the SM was manifesting itself or whether the
result could be explained in the framework of the SM. Precise theoretical
predictions in this sector are very difficult to achieve unfortunately, so it
is not yet possible to draw firm conclusions. Nevertheless, if the result is
confirmed at more than $5\sigma$ with increased statistics by LHCb, one
expects CPV to show up in other charm decays as well, and the pattern of
asymmetries will help theorists to decode the overall picture and determine
whether it is truly a SM effect. For the moment, all the efforts looking for
CPV in other $D$ decays besides $D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to
K^{+}K^{-}$ have been frustrated. But the search has just started.
## References
* (1) J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. Lett. 13 (1964) 138.
* (2) BaBar collaboration, B. Aubert et al., Phys. Rev. Lett. 87 (2001) 091801.
* (3) Belle collaboration, K. Abe et al., Phys. Rev. Lett. 87 (2001) 091802.
* (4) Particle Data Group, K. Nakamura et al., J. Phys. G37 (2010) 075021.
* (5) LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 108 (2012) 201601.
* (6) N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.
* (7) M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
* (8) W.-S. Hou, Chin. J. Phys. 47 (2009) 134.
* (9) BaBar collaboration, B. Aubert et al., Phys. Rev. Lett. 98 (2007) 211802.
* (10) Belle collaboration, M. Staric et al., Phys. Rev. Lett. 98 (2007) 211803.
* (11) Heavy Flavor Averaging Group, D. Asner et al., arXiv:1010.1589 [hep-ex].
* (12) A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir, and A. A. Petrov, Phys. Rev. D 69 (2004) 114021.
* (13) LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 108 (2012) 111602.
* (14) S. Bianco, F. L. Fabbri, D. Benson, and I. Bigi, Riv. Nuovo Cim. 26N7 (2003) 1.
* (15) M. Bobrowski, A. Lenz, J. Riedl, and J. Rohrwild, JHEP 1003 (2010) 009.
* (16) Y. Grossman, A. L. Kagan, and Y. Nir, Phys. Rev. D 75 (2007) 036008.
* (17) A. A. Petrov, PoS BEAUTY2009 (2009) 024.
* (18) E. Franco, S. Mishima and L. Silvestrini, JHEP 1205 (2012) 140.
* (19) G. F. Giudice, G. Isidori and P. Paradisi, JHEP 1204 (2012) 060.
* (20) Y. Grossman, A. L. Kagan and J. Zupan, Phys. Rev. D 85 (2012) 114036.
* (21) B. Bhattacharya, M. Gronau and J. L. Rosner, arXiv:1207.0761 [hep-ph].
* (22) I. I. Bigi, A. Paul, and S. Recksiegel, JHEP 1106 (2011) 089.
* (23) A. L. Kagan and M. D. Sokoloff, Phys. Rev. D 80 (2009) 076008.
* (24) CDF collaboration, T. Aaltonen et al., arXiv:1207.2158 [hep-ex].
* (25) CDF collaboration, T. Aaltonen et al., Phys. Rev. D 85 (2012) 012009.
* (26) Belle collaboration, B. R. Ko et al., Phys. Rev. Lett. 106 (2011) 211801.
* (27) Belle collaboration, M. Staric et al., Phys. Rev. Lett. 108 (2012) 071801.
* (28) Belle collaboration, E. Won et al., Phys. Rev. Lett. 107 (2011) 221801.
* (29) CDF collaboration, T. Aaltonen et al., Phys. Rev. D 86 (2012) 032007.
* (30) LHCb collaboration, R. Aaij et al., Phys. Rev. D 84 (2011) 112008.
* (31) BaBar collaboration, J. P. Lees et al., Phys. Rev. D 84 (2011) 031103.
|
arxiv-papers
| 2012-09-14T10:59:49 |
2024-09-04T02:49:35.151017
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vincenzo Vagnoni",
"submitter": "Vincenzo Maria Vagnoni",
"url": "https://arxiv.org/abs/1209.3152"
}
|
1209.3254
|
# Nonplanar Periodic Solutions for Spatial Restricted 3-Body and 4-Body
Problems ***Supported by National Natural Science Foundation of China.
Xiaoxiao Zhao and Shiqing Zhang
Yangtze Center of Mathematics and College of Mathematics, Sichuan University,
Chengdu 610064, People’s Republic of China
> Abstract: In this paper, we study the existence of non-planar periodic
> solutions for the following spatial restricted 3-body and 4-body problems:
> for $N=2\ \mbox{or}\ 3$, given any positive masses $m_{1},\cdots,m_{N}$, the
> mass points of $m_{1},\cdots,m_{N}$ move in the plane of $N$ circular obits
> centered at the center of masses, the sufficiently small mass moves on the
> perpendicular axis passing the center of masses. Using variational
> minimizing methods, we establish the existence of the minimizers of the
> Lagrangian action on anti-T/2 or odd symmetric loop spaces. Moreover, we
> prove these minimizers are non-planar periodic solutions by using the
> Jacobi’s Necessary Condition for local minimizers.
>
> Keywords: Restricted 3-body problems; Restricted 4-body problems; nonplanar
> periodic solutions; variational minimizers; Jacobi’s Necessary Conditions.
>
> 2000 AMS Subject Classification 34C15, 34C25, 58F.
## 1 Introduction and Main Results
In this paper, we study the spatial circular restricted 3-body and 4-body
problems. For $N=2\ \mbox{or}\ 3$, suppose points of positive masses
$m_{1},\cdots,m_{N}$ move in the plane of their circular orbits
$q_{1}(t),\cdots,q_{N}(t)$ with the radius $r_{1},\cdots,r_{N}>0$ and the
center of masses is at the origin; suppose the sufficiently small mass point
does not influence the motion of $m_{1},\cdots,m_{N}$, and moves on the
vertical axis of the moving plane for the given masses $m_{1},\cdots,m_{N}$,
here the vertical axis passes through the center of masses.
It is known that $q_{1}(t),\cdots,q_{N}(t)(N=2\ \mbox{or}\ 3)$ satisfy the
Newtonian equations:
$m_{i}\ddot{q_{i}}=\frac{\partial U}{\partial q_{i}},\ \ \ \ i=1,\cdots,N,$
(1.1)
where
$U=\sum\limits_{1\leq i<j\leq N}\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}.$ (1.2)
The orbit $q(t)=(0,0,z(t))\in R^{3}$ for sufficiently small mass is governed
by the gravitational forces of $m_{1},\cdots,m_{N}(N=2\ \mbox{or}\ 3)$ and
therefore it satisfies the following equation
$\ddot{q}=\sum\limits_{i=1}^{N}\frac{m_{i}(q_{i}-q)}{|q_{i}-q|^{3}},\ N=2\
\mbox{or}\ 3.$ (1.3)
For $N\geq 2$, there are many papers concerned with the restricted N-body
problem, see [3,4,6,8-10] and the references therein. In [8], Sitnikov
considered the following model: two mass points of equal mass
$m_{1}=m_{2}=\frac{1}{2}$ move in the plane of their elliptic orbits and the
center of masses is at rest, the third mass point which does not influence the
motion of the first two moves on the line perpendicular to the plane
containing the first two mass points and goes through the center of mass, and
he used geometrical methods to prove the existence of the oscillatory
parabolic orbit of
$\ddot{z}(t)=\frac{-z(t)}{(|z(t)|^{2}+|r(t)|^{2})^{3/2}},$ (1.4)
where $r(t)=r(t+2\pi)>0$ is the distance from the center of mass to one of the
first two mass points. McGehee [6] used the stable and unstable manifolds to
study the homoclinic orbits (parabolic orbits) of (1.4). In[4], Mathlouthi
studied the periodic solutions for the spatial circular restricted 3-body
problems by minimax variational methods. Recently, Li, Zhang and Zhao[3] used
variational minimizing methods to study spatial circular restricted N+1-body
problem with a zero mass moving on the vertical axis of the moving plane for N
equal mass.
Motivated by [3], we use the Jacobi’s Necessary Condition for local minimizers
to further study the spatial circular restricted 3-body and 4-body problems
with a sufficiently small mass moving on the perpendicular axis of the
circular orbits plane for any given masses $m_{1},\cdots,m_{N}(N=2\ \mbox{or}\
3)$.
Define
$W^{1,2}(R/TZ,R)=\bigg{\\{}u(t)\Big{|}u(t),u^{\prime}(t)\in L^{2}(R,R),\
u(t+T)=u(t)\bigg{\\}}.$
The inner product and the norm of $W^{1,2}(R/TZ,R)$ are
$<u,v>=\int_{0}^{T}(uv+u^{\prime}\cdot v^{\prime})dt,\ \ \ \ \ \ \ \ \ \ $
(1.5)
$\|u\|=\Big{[}\int_{0}^{T}|u|^{2}dt\Big{]}^{\frac{1}{2}}+\Big{[}\int_{0}^{T}|u^{\prime}|^{2}dt\Big{]}^{\frac{1}{2}}.$
(1.6)
The functional corresponding to the equation (1.3) is
$\displaystyle f(q)$
$\displaystyle=\int_{0}^{T}\Big{[}\frac{1}{2}|\dot{q}|^{2}+\sum\limits_{i=1}^{N}\frac{m_{i}}{|q-q_{i}|}\Big{]}dt,\
\ \ \ q\in\Lambda_{j},j=1,2$ (1.7)
$\displaystyle=\int_{0}^{T}\Big{[}\frac{1}{2}|z^{\prime}|^{2}+\frac{m_{1}}{\sqrt{r_{1}^{2}+z^{2}}}+\cdots+\frac{m_{N}}{\sqrt{r_{N}^{2}+z^{2}}}\Big{]}dt\triangleq
f(z),\ N=2\ \mbox{or}\ 3,$
where
$\Lambda_{1}=\left\\{q(t)=(0,0,z(t))\Big{|}z(t)\in W^{1,2}(R/TZ,R),\
z(t+\frac{T}{2})=-z(t)\right\\},$
and
$\Lambda_{2}=\bigg{\\{}q(t)=(0,0,z(t))\Big{|}z(t)\in W^{1,2}(R/TZ,R),\
z(-t)=-z(t)\bigg{\\}}.\ \ \ \ \ $
Our main results are the following:
Theorem 1.1 For $N=2$, the minimizer of $f(q)$ on the closure
$\overline{\Lambda}_{i}$ of $\Lambda_{i}(i=1,2)$ is a nonplanar and
noncollision periodic solution.
Theorem 1.2 For $N=3$, the minimizer of $f(q)$ on the closure
$\overline{\Lambda}_{i}$ of $\Lambda_{i}(i=1,2)$ is a nonplanar and
noncollision periodic solution.
## 2 Preliminaries
In this section, we will list some basic Lemmas and inequality for proving our
Theorems 1.1 and 1.2.
Lemma 2.1(Palais’s Symmetry Principle([7])) Let $\sigma$ be an orthogonal
representation of a finite or compact group $G$, $H$ be a real Hilbert space,
$f:H\rightarrow R$ satisfies $f(\sigma\cdot x)=f(x),\forall\sigma\in G,\forall
x\in H$.
Set $F=\\{x\in H|\sigma\cdot x=x,\ \forall\sigma\in G\\}$. Then the critical
point of $f$ in $F$ is also a critical point of $f$ in $H$.
Remark 2.1 By Palais’s Symmetry Principle, we know that the critical point of
$f(q)$ in $\overline{\Lambda}_{i}=\Lambda_{i}(i=1,2)$ is a periodic solution
of Newtonian equation (1.3).
Lemma 2.2(Tonelli[1]) Let $X$ be a reflexive Banach space, $S$ be a weakly
closed subset of $X$, $f:S\rightarrow R\cup+\infty$. If $f\not\equiv+\infty$
is weakly lower semi-continuous and coercive($f(x)\rightarrow+\infty$ as
$\|x\|\rightarrow+\infty$), then $f$ attains its infimum on $S$.
Lemma 2.3(Poincare-Wirtinger Inequality[5]) Let $q\in W^{1,2}(R/TZ,R^{K})$ and
$\int_{0}^{T}q(t)dt=0$, then
$\int_{0}^{T}|q(t)|^{2}dt\leq\frac{T^{2}}{4\pi^{2}}\int_{0}^{T}|\dot{q}(t)|^{2}dt.$
(2.1)
Lemma 2.4 $f(q)$ in (1.7) attains its infimum on
$\bar{\Lambda}_{i}=\Lambda_{i}(i=1,2)$.
Proof. By using Lemma 2.3, for $\forall z\in\Lambda_{i},\ i=1,2$, the
equivalent norm of (1.6) in $\Lambda_{i}(i=1,2)$ is
$\|z\|\cong\Big{[}\int_{0}^{T}|z^{\prime}|^{2}dt\Big{]}^{\frac{1}{2}}.$ (2.2)
Hence by the definitions of $f(q)$, it is easy to see that $f$ is $C^{1}$ and
coercive on $\Lambda_{i}(i=1,2)$. In order to get Lemma 2.4, we only need to
prove that $f$ is weakly lower semi-continuous on $\Lambda_{i}(i=1,2)$. In
fact, for $\forall z_{n}\in\Lambda_{i}$, if $z_{n}\rightharpoonup z$ weakly,
by compact embedding theorem, we have the uniformly convergence:
$\max\limits_{0\leq t\leq T}|z_{n}(t)-z(t)|\rightarrow 0,\ \ \ \
n\rightarrow\infty,$ (2.3)
which implies
$\int_{0}^{T}\frac{m_{1}}{\sqrt{r_{1}^{2}+z_{n}^{2}}}+\cdots+\frac{m_{N}}{\sqrt{r_{N}^{2}+z_{n}^{2}}}dt\rightarrow\int_{0}^{T}\frac{m_{1}}{\sqrt{r_{1}^{2}+z^{2}}}+\cdots+\frac{m_{N}}{\sqrt{r_{N}^{2}+z^{2}}}dt,\
N=2\ \mbox{or}\ 3.$ (2.4)
It is well-known that the norm and its square are weakly lower semi-
continuous. Therefore, combined with (2.4), one has
$\liminf\limits_{n\rightarrow\infty}f(z_{n})\geq f(z),$
that is, $f$ is weakly lower semi-continuous on $\Lambda_{i}(i=1,2)$. By lemma
2.2, we can get that $f(q)$ in (1.7) attains its infimum on
$\bar{\Lambda}_{i}=\Lambda_{i}(i=1,2)$. $\Box$
Lemma 2.5(Jacobi’s Necessary Condition[2]) Let $F\in C^{3}([a,b]\times R\times
R,R)$. If the critical point $y=\tilde{y}(x)$ corresponds to a minimum of the
functional $\int_{a}^{b}F(x,y(x),y^{\prime}(x))dx$ on $M=\\{y\in
W^{1,2}([a,b],R)|y(a)=A,y(b)=B\\}$ and if $F_{y^{\prime}y^{\prime}}>0$ along
this critical point, then the open interval $(a,b)$ contains no points
conjugate to $a$, that is, for $\forall c\in(a,b)$, the following problem:
$\left\\{\begin{array}[]{ll}-\frac{d}{dx}(Ph^{\prime})+Qh=0,&\\\ h(a)=0,\ \
h(c)=0,\end{array}\right.$ (2.5)
has only the trivial solution $h(x)\equiv 0,\ \forall x\in(a,c)$, where
$P=\frac{1}{2}F_{y^{\prime}y^{\prime}}|_{y=\tilde{y}},\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ $ (2.6)
$Q=\frac{1}{2}\Big{(}F_{yy}-\frac{d}{dx}F_{yy^{\prime}}\Big{)}\Big{|}_{y=\tilde{y}}.$
(2.7)
Remark 2.2 It is easy to see that Lemma 2.5 is suitable for the fixed end
problem. In this paper, we consider the periodic solutions of (1.3) on
$\overline{\Lambda}_{i}=\Lambda_{i}(i=1,2)$, hence we need to establish a
similar conclusion as Lemma 2.5 for the periodic boundary problem.
Lemma 2.6 Let $F\in C^{3}(R\times R\times R,R)$. Assume that $u=\tilde{u}(t)$
is a critical point of the functional $\int_{0}^{T}F(t,u(t),u^{\prime}(t))dt$
on $W^{1,2}(R/TZ,R)$ and $F_{u^{\prime}u^{\prime}}|_{u=\tilde{u}}>0$. If the
open interval $(0,T)$ contains a point $c$ conjugate to $0$, then
$u=\tilde{u}(t)$ is not a minimum of the functional
$\int_{0}^{T}F(t,u(t),u^{\prime}(t))dt$.
Proof. Suppose $u=\tilde{u}(t)$ is a minimum of the functional
$\int_{0}^{T}F(t,u(t),u^{\prime}(t))dt$. The second variation of
$\int_{0}^{T}F(t,u(t),u^{\prime}(t))dt$ is
$\int_{0}^{T}(Ph^{\prime 2}+Qh^{2})dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
(2.8)
where
$P=\frac{1}{2}F_{u^{\prime}u^{\prime}}|_{u=\tilde{u}},\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ $ (2.9)
$Q=\frac{1}{2}\Big{(}F_{uu}-\frac{d}{dt}F_{uu^{\prime}}\Big{)}\Big{|}_{u=\tilde{u}}.$
(2.10)
Set
$Q_{\tilde{u}}(h)=\int_{0}^{T}(Ph^{\prime 2}+Qh^{2})dt.$ (2.11)
For $\forall h\in C_{0}^{1}([0,T],R)$, it is easy to see that
$Q_{\tilde{u}}(h)\geq 0$. Then by $Q_{\tilde{u}}(\theta)=0$, $\theta$ is a
minimum of $Q_{\tilde{u}}(h)$. The Euler-Lagrange equation which is called the
Jacobi equation of (2.11) is
$-\frac{d}{dt}(Ph^{\prime})+Qh=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (2.12)
Since the interval $(0,T)$ contains a point $c$ conjugate to $0$, there exists
a nonzero Jacobi field $h_{0}\in C^{2}([0,T],R)$ satisfying
$\left\\{\begin{array}[]{ll}-\frac{d}{dt}(Ph_{0}^{\prime})+Qh_{0}=0,&\\\
h_{0}(0)=0,\ \ h_{0}(c)=0.\end{array}\right.$ (2.13)
Let
$\hat{h}(t)=\left\\{\begin{array}[]{ll}h_{0}(t)&\ \ \ \ t\in[0,c],\\\ 0&\ \ \
\ t\in(c,T],\end{array}\right.$ (2.14)
we have $\hat{h}\in C^{2}([0,T]\backslash\\{c\\},R)$,
$\hat{h}(0)=\hat{h}(c)=\hat{h}(T)=0$ and
$Q_{\tilde{u}}(\hat{h})=\int_{0}^{T}(P\hat{h}^{\prime
2}+Q\hat{h}^{2})dt=\int_{0}^{c}(Ph_{0}^{\prime 2}+Qh_{0}^{2})dt=0.$ (2.15)
Notice that we can extend $\hat{h}$ periodically when we take T as the period,
so $\hat{h}\in W_{0}^{1,2}(R/TZ,R)$. For $\forall h\in C_{0}^{1}([0,T],R)$, it
is easy to check that $Q_{\tilde{u}}(h)\geq 0$. Then by (2.15), one has
$\hat{h}\in C^{2}([0,T]\backslash\\{c\\},R)\cap W_{0}^{1,2}(R/TZ,R)$ is a
minimum of $Q_{\tilde{u}}(h)$. Hence we get
$-\frac{d}{dt}(P\hat{h}^{\prime})+Q\hat{h}=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ $ (2.16)
Combine with $\hat{h}(0)=\hat{h}(c)=0$, by the uniqueness of initial value
problems for second order differential equation, we have $\hat{h}(t)\equiv 0$
on [0,c], which contradicts the definition of $\hat{h}$. Therefore, Lemma 2.6
holds. $\Box$
## 3 Proof of Theorem 1.1
In this section, we consider the spatial circular restricted 3-body problem
with a sufficiently small mass moving on the vertical axis of the moving plane
for arbitrary given positive masses $m_{1},m_{2}$. Suppose the planar circular
orbits are
$q_{1}(t)=r_{1}e^{\sqrt{-1}\frac{2\pi}{T}t},\ \ \ \
q_{2}(t)=-r_{2}e^{\sqrt{-1}\frac{2\pi}{T}t},$ (3.1)
here the radius $r_{1},r_{2}$ are positive constants depending on
$m_{i}(i=1,2)$ and $T$ (see Lemma 3.1). We also assume that
$m_{1}q_{1}(t)+m_{2}q_{2}(t)=0.$ (3.2)
The functional corresponding to the equation (1.3) is
$\displaystyle f(q)$
$\displaystyle=\int_{0}^{T}\Big{[}\frac{1}{2}|\dot{q}|^{2}+\frac{m_{1}}{|q-q_{1}|}+\frac{m_{2}}{|q-q_{2}|}\Big{]}dt,\
\ \ \ q\in\Lambda_{i},\ i=1,2$ (3.3)
$\displaystyle=\int_{0}^{T}\Big{[}\frac{1}{2}|z^{\prime}|^{2}+\frac{m_{1}}{\sqrt{r_{1}^{2}+z^{2}}}+\frac{m_{2}}{\sqrt{r_{2}^{2}+z^{2}}}\Big{]}dt\triangleq
f(z).$
Lemma 3.1 The radius $r_{1},r_{2}$ of the planar circular orbits for the
masses $m_{1},m_{2}$ are
$r_{1}=\Big{(}\frac{T}{2\pi(m_{1}+m_{2})}\Big{)}^{\frac{2}{3}}m_{2},\ \ \ \
r_{2}=\Big{(}\frac{T}{2\pi(m_{1}+m_{2})}\Big{)}^{\frac{2}{3}}m_{1}.$
Proof. Substituting (3.1) into (3.2), it is easy to get
$r_{2}=\frac{m_{1}}{m_{2}}r_{1}.$ (3.4)
It follows from (1.1) and (1.2) that
$\ddot{q_{1}}=m_{2}\frac{q_{2}-q_{1}}{|q_{2}-q_{1}|^{3}}.$ (3.5)
Then by (3.1) and (3.4), we have
$-\frac{4\pi^{2}}{T^{2}}q_{1}=m_{2}\frac{(-\frac{m_{1}}{m_{2}}-1)q_{1}}{r_{1}^{3}|-\frac{m_{1}}{m_{2}}-1|^{3}},$
(3.6)
which implies
$r_{1}=\Big{(}\frac{T}{2\pi(m_{1}+m_{2})}\Big{)}^{\frac{2}{3}}m_{2}.\ \ \ \ \
$ (3.7)
Hence by (3.4), one has
$r_{2}=\Big{(}\frac{T}{2\pi(m_{1}+m_{2})}\Big{)}^{\frac{2}{3}}m_{1}.\ \ \
\Box$ (3.8)
Proof of Theorem 1.1 Clearly, $q(t)=(0,0,0)$ is a critical point of $f(q)$ on
$\bar{\Lambda}_{i}=\Lambda_{i}(i=1,2)$. For the functional (3.3), let
$F(z,z^{\prime})=\frac{1}{2}|z^{\prime}|^{2}+\frac{m_{1}}{\sqrt{r_{1}^{2}+z^{2}}}+\frac{m_{2}}{\sqrt{r_{2}^{2}+z^{2}}}.$
Then the second variation of (3.3) in the neighborhood of $z=0$ is given by
$\int_{0}^{T}(Ph^{\prime 2}+Qh^{2})dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
(3.9)
where
$P=\frac{1}{2}F_{z^{\prime}z^{\prime}}|_{z=0}=\frac{1}{2},\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3.10)
$Q=\frac{1}{2}\Big{(}F_{zz}-\frac{d}{dt}F_{zz^{\prime}}\Big{)}\Big{|}_{z=0}=-\Big{(}\frac{m_{1}}{2r_{1}^{3}}+\frac{m_{2}}{2r_{2}^{3}}\Big{)}.$
(3.11)
The Euler equation of (3.9) is called the Jacobi equation of the original
functional (3.3), which is
$-\frac{d}{dt}(Ph^{\prime})+Qh=0,\ \ \ \ \ \ \ \ \ \ \ \ \ $ (3.12)
that is,
$h^{\prime\prime}+\Big{(}\frac{m_{1}}{r_{1}^{3}}+\frac{m_{2}}{r_{2}^{3}}\Big{)}h=0.\
\ \ \ \ \ \ \ \ \ \ \ \ $ (3.13)
Next, we study the solution of (3.13) with initial values $h(0)=0,\
h^{\prime}(0)=1$. It is easy to get
$h(t)=\sqrt{\frac{r_{1}^{3}r_{2}^{3}}{m_{2}r_{1}^{3}+m_{1}r_{2}^{3}}}\cdot
sin\sqrt{\frac{m_{1}}{r_{1}^{3}}+\frac{m_{2}}{r_{2}^{3}}}t.$ (3.14)
It follows from (3.7) and (3.8)that
$\sqrt{\frac{m_{1}}{r_{1}^{3}}+\frac{m_{2}}{r_{2}^{3}}}=\sqrt{\frac{m_{1}^{4}+m_{2}^{4}}{m_{1}^{3}m_{2}^{3}}}(m_{1}+m_{2})\cdot\frac{2\pi}{T}.$
(3.15)
Hence
$h(t)=\frac{\sqrt{m_{1}^{3}m_{2}^{3}}T}{2\pi\sqrt{m_{1}^{4}+m_{2}^{4}}(m_{1}+m_{2})}\cdot
sin\Bigg{(}\sqrt{\frac{m_{1}^{4}+m_{2}^{4}}{m_{1}^{3}m_{2}^{3}}}(m_{1}+m_{2})\cdot\frac{2\pi}{T}t\Bigg{)},$
(3.16)
which is not identically zero on
$[0,\frac{\sqrt{m_{1}^{3}m_{2}^{3}}T}{\sqrt{m_{1}^{4}+m_{2}^{4}}(m_{1}+m_{2})}]$.
Since
$m_{1}^{6}+m_{2}^{6}\geq 2\sqrt{m_{1}^{6}\cdot
m_{2}^{6}}=2m_{1}^{3}m_{2}^{3}>m_{1}^{3}m_{2}^{3},$ (3.17)
one has
$\displaystyle(m_{1}^{4}+m_{2}^{4})(m_{1}+m_{2})^{2}$
$\displaystyle>m_{1}^{6}+m_{2}^{6}$ (3.18) $\displaystyle>m_{1}^{3}m_{2}^{3},$
which implies
$\frac{\sqrt{m_{1}^{3}m_{2}^{3}}}{\sqrt{m_{1}^{4}+m_{2}^{4}}(m_{1}+m_{2})}<1.$
(3.19)
Therefore
$\frac{\sqrt{m_{1}^{3}m_{2}^{3}}T}{\sqrt{m_{1}^{4}+m_{2}^{4}}(m_{1}+m_{2})}<T.$
(3.20)
Choose
$0<c=\frac{\sqrt{m_{1}^{3}m_{2}^{3}}T}{2\sqrt{m_{1}^{4}+m_{2}^{4}}(m_{1}+m_{2})}<\frac{T}{2}$,
we have
$h(c)=\frac{\sqrt{m_{1}^{3}m_{2}^{3}}T}{2\pi\sqrt{m_{1}^{4}+m_{2}^{4}}(m_{1}+m_{2})}\cdot
sin\pi=0.$ (3.21)
Case 1: Minimizing $f(q)$ on $\bar{\Lambda}_{1}=\Lambda_{1}$.
Let
$\tilde{h}(t)=\left\\{\begin{array}[]{ll}h(t)&\ \ \ \ t\in[0,c],\\\ 0&\ \ \ \
t\in(c,\frac{T}{2}],\\\ -h(t-\frac{T}{2})&\ \ \ \
t\in(\frac{T}{2},\frac{T}{2}+c],\\\ 0&\ \ \ \
t\in(\frac{T}{2}+c,T].\end{array}\right.$ (3.22)
It is easy to check that $\tilde{h}(t)\in
C^{2}([0,T]\backslash\\{c,\frac{T}{2},\frac{T}{2}+c\\},R)\cap W^{1,2}(R,R)$,
$\tilde{h}(t+\frac{T}{2})=-\tilde{h}(t)$, $\tilde{h}(0)=h(0)=0$,
$\tilde{h}(c)=h(c)=0$ and $\tilde{h}$ is a nonzero solution of (3.12). Notice
that we can extend $\tilde{h}$ periodically when we take T as the period, so
$\tilde{h}\in\Lambda_{1}$. Then by Lemma 2.6, $q(t)=(0,0,0)$ is not a local
minimum for $f(q)$ on $\Lambda_{1}$. Hence the minimizers of $f(q)$ on
$\Lambda_{1}$ are not always at the center of masses, they must oscillate
periodically on the vertical axis, that is, the minimizers are not always co-
planar, therefore, we get the non-planar periodic solutions.
Case 2: Minimizing $f(q)$ on $\bar{\Lambda}_{2}=\Lambda_{2}$.
Let
$\bar{h}(t)=\left\\{\begin{array}[]{ll}h(t)&\ \ \ \ t\in[0,c],\\\ 0&\ \ \ \
t\in(c,T-c],\\\ -h(T-t)&\ \ \ \ t\in(T-c,T].\end{array}\right.$ (3.23)
It is easy to check that $\bar{h}(t)\in
C^{2}([0,T]\backslash\\{c,T-c\\},R)\cap W^{1,2}(R,R)$,
$\bar{h}(-t)=-\bar{h}(t)$, $\bar{h}(0)=h(0)=0$, $\bar{h}(c)=h(c)=0$ and
$\bar{h}$ is a nonzero solution of (3.12). Notice that we can extend $\bar{h}$
periodically when we take T as the period, so $\bar{h}\in\Lambda_{2}$. Then by
Lemma 2.6, $q(t)=(0,0,0)$ is not a local minimum for $f(q)$ on $\Lambda_{2}$.
Hence the minimizers of $f(q)$ on $\Lambda_{2}$ are not always at the center
of masses, they must oscillate periodically on the vertical axis, that is, the
minimizers are not always co-planar, therefore, we get the non-planar periodic
solutions.
Combined with Lemma 2.4, the proof is completed. $\Box$
## 4 Proof of Theorem 1.2
In this section, we consider the spatial circular restricted 4-body problem
with a sufficiently small mass moving on the vertical axis of the moving plane
for arbitrary positive masses $m_{1},m_{2},m_{3}$. Suppose there exists
$\theta_{1},\theta_{2},\theta_{3}\in(0,2\pi)$ such that the planar circular
orbits are
$q_{1}(t)=r_{1}e^{\sqrt{-1}\frac{2\pi}{T}t}e^{\sqrt{-1}\theta_{1}},\ \
q_{2}(t)=r_{2}e^{\sqrt{-1}\frac{2\pi}{T}t}e^{\sqrt{-1}\theta_{2}},\ \
q_{3}(t)=r_{3}e^{\sqrt{-1}\frac{2\pi}{T}t}e^{\sqrt{-1}\theta_{3}},$ (4.1)
here the radius $r_{1},r_{2},r_{3}$ are positive constants depending on
$m_{i}(i=1,2,3)$ and $T$ (see Lemma 4.2). We also assume that
$m_{1}q_{1}(t)+m_{2}q_{2}(t)+m_{3}q_{3}(t)=0$ (4.2)
and
$|q_{i}-q_{j}|=l,\ \ 1\leq i\neq j\leq 3,$ (4.3)
where the constant $l>0$ depends on $m_{i}(i=1,2,3)$ and $T$ (see Lemma 4.1).
The functional corresponding to the equation (1.3) is
$\displaystyle f(q)$
$\displaystyle=\int_{0}^{T}\Big{[}\frac{1}{2}|\dot{q}|^{2}+\frac{m_{1}}{|q-q_{1}|}+\frac{m_{2}}{|q-q_{2}|}+\frac{m_{3}}{|q-q_{3}|}\Big{]}dt,\
\ \ \ q\in\Lambda_{i},\ i=1,2$ (4.4)
$\displaystyle=\int_{0}^{T}\Big{[}\frac{1}{2}|z^{\prime}|^{2}+\frac{m_{1}}{\sqrt{r_{1}^{2}+z^{2}}}+\frac{m_{2}}{\sqrt{r_{2}^{2}+z^{2}}}++\frac{m_{3}}{\sqrt{r_{3}^{2}+z^{2}}}\Big{]}dt\triangleq
f(z).$
In order to get Theorem 1.2, we firstly prove Lemmas 4.1 and 4.2 as follows.
Lemma 4.1 Let $M=m_{1}+m_{2}+m_{3}$, we have
$l=\sqrt[3]{\frac{MT^{2}}{4\pi^{2}}}$.
Proof. It follows from (1.1) and (1.2) that
$\ddot{q_{1}}=m_{2}\frac{q_{2}-q_{1}}{|q_{2}-q_{1}|^{3}}+m_{3}\frac{q_{3}-q_{1}}{|q_{3}-q_{1}|^{3}}.$
(4.5)
Then by (4.1)-(4.3), one has
$\displaystyle-\frac{4\pi^{2}}{T^{2}}q_{1}$
$\displaystyle=\frac{1}{l^{3}}(m_{2}q_{2}+m_{3}q_{3}-m_{2}q_{1}-m_{3}q_{1})$
(4.6) $\displaystyle=\frac{1}{l^{3}}(-m_{1}q_{1}-m_{2}q_{1}-m_{3}q_{1}),$
which implies
$l^{3}=\frac{MT^{2}}{4\pi^{2}},\ \ \ \ \ \ $ (4.7)
that is,
$l=\sqrt[3]{\frac{MT^{2}}{4\pi^{2}}}.\ \ \ \Box$ (4.8)
Lemma 4.2 The radius $r_{1},r_{2},r_{3}$ of the planar circular orbits for the
masses $m_{1},m_{2},m_{3}$ are
$r_{1}=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ \ $
$r_{2}=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ \ $
$r_{3}=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}l.\ \ \ \ \ \ $
Proof. Choose the geometrical center of the initial configuration
($q_{1}(0),q_{2}(0),q_{3}(0)$) as the origin of the coordinate (x,y). Without
loss of generality, by (4.3), we suppose the location coordinates of
$q_{1}(0),q_{2}(0),q_{3}(0)$ are
$A_{1}(\frac{\sqrt{3}l}{3},0),A_{2}(-\frac{\sqrt{3}l}{6},\frac{l}{2}),A_{3}(-\frac{\sqrt{3}l}{6},-\frac{l}{2})$.
Then we can get the coordinate of the center of masses $m_{1},m_{2},m_{3}$ is
$C(\frac{\frac{\sqrt{3}}{3}m_{1}l-\frac{\sqrt{3}}{6}m_{2}l-\frac{\sqrt{3}}{6}m_{3}l}{M},\frac{\frac{m_{2}}{2}l-\frac{m_{3}}{2}l}{M})$.
To make sure the Assumption (4.2) holds, we introduce the new coordinate
$\left\\{\begin{array}[]{ll}X=x-\frac{\frac{\sqrt{3}}{3}m_{1}l-\frac{\sqrt{3}}{6}m_{2}l-\frac{\sqrt{3}}{6}m_{3}l}{M},\\\
Y=y-\frac{\frac{m_{2}}{2}l-\frac{m_{3}}{2}l}{M}.\end{array}\right.$
Hence in the new coordinate (X,Y), the location coordinates of
$q_{1}(0),q_{2}(0),q_{3}(0)$ are
$A_{1}(\frac{\frac{\sqrt{3}}{2}m_{2}l+\frac{\sqrt{3}}{2}m_{3}l}{M},$
$\frac{-\frac{m_{2}}{2}l+\frac{m_{3}}{2}l}{M}),$
$A_{2}(-\frac{\frac{\sqrt{3}}{2}m_{1}l}{M},\frac{\frac{m_{1}}{2}l+m_{3}l}{M}),A_{3}(-\frac{\frac{\sqrt{3}}{2}m_{1}l}{M},-\frac{\frac{m_{1}}{2}l+m_{2}l}{M})$
and the center of masses $m_{1},m_{2},m_{3}$ is at the origin $O(0,0)$. Then
compared with (4.1), we have
$r_{1}=|A_{1}O|=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ \ $
(4.9) $r_{2}=|A_{2}O|=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \
\ \ $ (4.10)
$r_{3}=|A_{3}O|=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}l,\ \ \ \ \ \ $
(4.11)
and
$\tan\theta_{1}=\frac{\sqrt{3}(-m_{2}+m_{3})}{3(m_{2}+m_{3})},\ \ \ \
\tan\theta_{2}=-\frac{\sqrt{3}(m_{1}+2m_{3})}{3m_{1}},\ \ \ \
\tan\theta_{3}=\frac{\sqrt{3}(m_{1}+2m_{2})}{3m_{1}}.\ \ \ \Box$ (4.12)
Proof of Theorem 1.2 Clearly, $q(t)=(0,0,0)$ is a critical point of $f(q)$ on
$\bar{\Lambda}_{i}=\Lambda_{i}(i=1,2)$. For the functional (4.4), let
$F(z,z^{\prime})=\frac{1}{2}|z^{\prime}|^{2}+\frac{m_{1}}{\sqrt{r_{1}^{2}+z^{2}}}+\frac{m_{2}}{\sqrt{r_{2}^{2}+z^{2}}}+\frac{m_{3}}{\sqrt{r_{3}^{2}+z^{3}}}.$
Then the second variation of (4.4) in the neighborhood of $z=0$ is given by
$\int_{0}^{T}(Ph^{\prime 2}+Qh^{2})dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
(4.13)
where
$P=\frac{1}{2}F_{z^{\prime}z^{\prime}}|_{z=0}=1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (4.14)
$Q=\frac{1}{2}\Big{(}F_{zz}-\frac{d}{dt}F_{zz^{\prime}}\Big{)}\Big{|}_{z=0}=-\Big{(}\frac{m_{1}}{2r_{1}^{3}}+\frac{m_{2}}{2r_{2}^{3}}+\frac{m_{3}}{2r_{3}^{3}}\Big{)}.$
(4.15)
The Euler equation of (4.13) is called the Jacobi equation of the original
functional (4.4), which is
$-\frac{d}{dt}(Ph^{\prime})+Qh=0,\ \ \ \ \ \ \ \ \ \ \ \ \ $ (4.16)
that is,
$h^{\prime\prime}+\Big{(}\frac{m_{1}}{r_{1}^{3}}+\frac{m_{2}}{r_{2}^{3}}+\frac{m_{3}}{r_{3}^{3}}\Big{)}h=0.\
\ \ \ \ \ \ \ \ \ \ \ \ $ (4.17)
Next, we study the solution of (4.17) with initial values $h(0)=0,\
h^{\prime}(0)=1$. It is easy to get
$h(t)=\sqrt{\frac{r_{1}^{3}r_{2}^{3}r_{3}^{3}}{m_{3}r_{1}^{3}r_{2}^{3}+m_{2}r_{1}^{3}r_{3}^{3}+m_{1}r_{2}^{3}r_{3}^{3}}}\cdot
sin\sqrt{\frac{m_{1}}{r_{1}^{3}}+\frac{m_{2}}{r_{2}^{3}}+\frac{m_{3}}{r_{3}^{3}}}t.$
(4.18)
Let
$A=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M},\ \ \ \ \ \ \ \ \ $
$B=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M},\ \ \ \ \ \ \ \ \ $
$C=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}.\ \ \ \ \ \ \ \ \ $
It follows from (4.8)-(4.11) that
$\displaystyle\sqrt{\frac{m_{1}}{r_{1}^{3}}+\frac{m_{2}}{r_{2}^{3}}+\frac{m_{3}}{r_{3}^{3}}}$
$\displaystyle=\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}\sqrt{\frac{1}{l^{3}}}$
(4.19)
$\displaystyle=\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}\cdot\frac{2\pi}{\sqrt{M}T}.$
Hence
$h(t)=\frac{\sqrt{M}T}{2\pi\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}}\cdot
sin\Bigg{(}\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}\cdot\frac{2\pi}{\sqrt{M}T}t\Bigg{)},$
(4.20)
which is not identically zero on
$[0,\frac{\sqrt{M}T}{\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}}]$.
It is easy to check that
$\displaystyle M^{2}$ $\displaystyle>m^{2}_{1}+m_{1}m_{2}+m^{2}_{2},\ \ \ \ \
\ \ \ \ $ (4.21) $\displaystyle M^{2}$
$\displaystyle>m^{2}_{1}+m_{1}m_{3}+m^{2}_{3},\ \ \ \ \ \ \ \ \ $
$\displaystyle M^{2}$ $\displaystyle>m^{2}_{2}+m_{2}m_{3}+m^{2}_{3},\ \ \ \ \
\ \ \ \ $
which implies
$\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}>m_{1}+m_{2}+m_{3}=M.$
(4.22)
Therefore
$\frac{\sqrt{M}T}{\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}}<T.$
(4.23)
Choose
$0<c=\frac{\sqrt{M}T}{2\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}}<\frac{T}{2}$,
we have
$h(c)=\frac{\sqrt{M}T}{2\pi\sqrt{\frac{m_{1}}{A^{3}}+\frac{m_{2}}{B^{3}}+\frac{m_{3}}{C^{3}}}}\cdot
sin\pi=0.$ (4.24)
Case 1: Minimizing $f(q)$ on $\bar{\Lambda}_{1}=\Lambda_{1}$.
Let
$\tilde{h}(t)=\left\\{\begin{array}[]{ll}h(t)&\ \ \ \ t\in[0,c],\\\ 0&\ \ \ \
t\in(c,\frac{T}{2}],\\\ -h(t-\frac{T}{2})&\ \ \ \
t\in(\frac{T}{2},\frac{T}{2}+c],\\\ 0&\ \ \ \
t\in(\frac{T}{2}+c,T].\end{array}\right.$ (4.25)
It is easy to check that $\tilde{h}(t)\in
C^{2}([0,T]\backslash\\{c,\frac{T}{2},\frac{T}{2}+c\\},R)\cap W^{1,2}(R,R)$,
$\tilde{h}(t+\frac{T}{2})=-\tilde{h}(t)$, $\tilde{h}(0)=h(0)=0$,
$\tilde{h}(c)=h(c)=0$ and $\tilde{h}$ is a nonzero solution of (4.16). Notice
that we can extend $\tilde{h}$ periodically when we take T as the period, so
$\tilde{h}\in\Lambda_{1}$. Then by Lemma 2.6, $q(t)=(0,0,0)$ is not a local
minimum for $f(q)$ on $\Lambda_{1}$. Hence the minimizers of $f(q)$ on
$\Lambda_{1}$ are not always at the center of masses, they must oscillate
periodically on the vertical axis, that is, the minimizers are not always co-
planar, therefore, we get the non-planar periodic solutions.
Case 2: Minimizing $f(q)$ on $\bar{\Lambda}_{2}=\Lambda_{2}$.
Let
$\bar{h}(t)=\left\\{\begin{array}[]{ll}h(t)&\ \ \ \ t\in[0,c],\\\ 0&\ \ \ \
t\in(c,T-c],\\\ -h(T-t)&\ \ \ \ t\in(T-c,T].\end{array}\right.$ (4.26)
It is easy to check that $\bar{h}(t)\in
C^{2}([0,T]\backslash\\{c,T-c\\},R)\cap W^{1,2}(R,R)$,
$\bar{h}(-t)=-\bar{h}(t)$, $\bar{h}(0)=h(0)=0$, $\bar{h}(c)=h(c)=0$ and
$\bar{h}$ is a nonzero solution of (4.16). Notice that we can extend $\bar{h}$
periodically when we take T as the period, so $\bar{h}\in\Lambda_{2}$. Then by
Lemma 2.6, $q(t)=(0,0,0)$ is not a local minimum for $f(q)$ on $\Lambda_{2}$.
Hence the minimizers of $f(q)$ on $\Lambda_{2}$ are not always at the center
of masses, they must oscillate periodically on the vertical axis, that is, the
minimizers are not always co-planar, therefore, we get the non-planar periodic
solutions.
Combined with Lemma 2.4, the proof is completed. $\Box$
## References
* [1] Ambrosetti. A, Coti Zelati. V, Periodic Solutions of Singular Lagrangian Systems, Birkhäuser, Basel, 1993.
* [2] Gelfand. I, Formin. S, Calculus of Variations, Nauka, Moscow, English edition, Prentice-Hall, Englewood Cliffs, NJ, 1965.
* [3] Li. F. Y, Zhang. S. Q, Zhao. X. X, Periodic solutions for spatial restricted N+1-body problems, arXiv:1209.1304V1.
* [4] Mathlouthi. S, Periodic oribits of the restricted three-body problem, Trans. Amer. Math. Soc, 350(1998) 2265-2276.
* [5] Mawhin. J, Willem. M, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.
* [6] McGehee. R, Parabolic Orbits of the Restricted Three-body Problem, Academic Press, New York, London, 1973.
* [7] Palais. R, The principle of symmetric criticality, Comm. Math. Phys, 69(1)(1979) 19-30.
* [8] Sitnikov, K, Existence of oscillating motions for the three-body problem, Dokl. Akad. Nauk, USSR, 133(2)(1960) 303-306.
* [9] Souissi. C, Existence of parabolic orbits for the restricted three-body problem, Annals of University of Craiova, Math. Comp. Sci. Ser, 31(2004) 85-93.
* [10] Zhang. S. Q, Variational minimizing parabolic and hyperbolic orbits for the restricted 3-body problems, Sci. China. Math, 55(4) 721-725.
|
arxiv-papers
| 2012-09-14T16:46:00 |
2024-09-04T02:49:35.156895
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoxiao Zhao and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1209.3254"
}
|
1209.3354
|
# Cavity Optomechanics with a Bose-Einstein Condensate: Normal Mode Splitting
Muhammad Asjad [email protected] Department of Electronics, Quaid-i-Azam
University, 45320 Islamabad, Pakistan.
###### Abstract
We study the normal mode splitting in a system consisting of a Bose-Einstein
condensates (BEC) trapped inside a Fabry-Pérot cavity driven by single mode
laser field. We analyze the variations in frequency and damping rate of
collective density excitation of BEC imparted by optical field. We study the
occurrence of normal mode splitting which appears as consequences of
hybridization of the fluctuations of intracavity field and condensate mode. It
is shown that normal mode splitting vanishes for weak coupling between
condensate mode and intracavity field. Moreover, we investigate the normal
mode splitting in the transmission spectrum of cavity field.
Bose-Einstein condensate, Optomechanics, Normal mode splitting
## I introduction
Nano optomechanical systems that couple optical degree of freedom to the
mechanical motion of a cantilever have been extensively investigated in recent
past Kippenberg . In such systems, coupling is obtained via radiation pressure
inside a cavity Braginsky ; Mancini ; Zhang , and indirectly via quantum dots
Tian or ions Naik . Significant progress has been made in the investigation
of optomechanics, such as squeezing Fabre ; Tombesi , ultrahigh precision
displacement detection Caves ; LaHaye ; Rugar , mass detection Ekinci ,
gravational wave detection Caves1980 ; V. Braginsky , and the transition
between classical and quantum behaviour of a mechanical system Marshall .
Moreover, entangling the electromagnetic field with motional degree of freedom
of mechanical systems have been explored in various approaches Vitali ;
Paternostro . The entanglement generated in such systems is significant both
philosophically as well as technically in relation with quantum informatics
Nielsen . Further advances in experimental techniques make possible to couple
the mechanical resonators with the statistical ensemble of atoms. In such
systems, the interaction is mediated by the field inside the cavity that
couples the mechanical resonators to the internal and motional degrees of
freedom of the atoms Ian ; genes ; Meiser ; Ritsch .
###
The normal mode splitting (NMS) is the coupling of two degenerate modes with
energy exchange taking place on a time scale faster then the decoherence rate.
Moreover, the NMS is a phenomena ubiquitous in both the classical and quantum
physics. The NMS, in the transmission spectrum of the cavity filed, has been
observed when atoms are coupled to cavity field Thompson . Moreover, the NMS
has also been studied with artificial atoms in circuit quantum electrodynamics
(QED) Wallraff as well as in single quantum dot cavity QED Reithmaier . In
this paper, we study the occurrence of the NMS through a new optomechanical
system, i.e, a coupled BEC-cavity configuration Esteve ; Brennecke ; Ritter ;
Colombe ; Purdy ; Baumann ; Kónya ; Nagy ; asjad . Briefly, the system
consists of Fabry-Pérot cavity containing a condensate that effectively
behaves like a vibrating mirror Ritter . In such systems, the collective
density excitation of the BEC serves the analogy of a moving mirror coupled to
cavity field via radiation pressure force. The strong coupling of the
quantized cavity field with the collective oscillations of the BEC has been
experimentally analyzed with all the atoms are being in the same motional
state Brennecke ; Ritter . Our study reveals that the frequency and the
decoherence rate are sensitive to radiation pressure force. We further analyze
the occurrence of NMS in the position spectrum of the condensate mode. It is
observed that the position spectrum of the condensate mode splits into two
peaks when coupling between condensate mode and intracavity field is
considered. In addition, we also discuss the NMS in the fluctuations spectrum
of the light field emitted by the cavity. We show that the distance between
two peaks increases linearly with BEC-cavity field interaction. The paper is
structured as follows: In Section 2, we give the theoretical model of the
system and its coupling with the environment by using the Heisenberg-Langevin
equations. In section 3, we solve the dynamics in frequency domain and discuss
the results. Finally, we provide concluding remarks in section 4.
## II Model and Hamiltonian of the system
We consider an ensemble of $N$ two level bosonic atoms with resonant frequency
$\omega_{tr}$ is trapped inside a Fabry-Pérot cavity and interact with
standing-wave light field as shown in Fig.1. We assume the atom-field detuning
$\Delta_{a}$ is very large, therefore, one can adiabatically eliminate the
excited atomic level. In the rotating frame at the driving field frequency
$\omega_{P}$, the Hamiltonian of the Bose-Einstein condensate in the limit of
small atom-atom interaction and small value of the external potential can be
written as follows Ritter :
$H=\hbar\,\Delta_{c}c^{\dagger}c+\hbar\,\omega_{a}a^{\dagger}a+\dfrac{1}{\sqrt{2}}\hbar
g\,(a^{\dagger}+a)c^{\dagger}c-i\hbar E(c-c^{\dagger})$ (1)
where $\omega_{a}=4\omega_{r}$, $\Delta_{c}=\omega_{c}-\omega_{P}+NU_{o}/2$,
$U_{o}=g^{2}_{o}/\Delta_{a}$, $\Delta_{a}=\omega_{P}-\omega_{tr}$, and
$g=\sqrt{N}U_{o}/2$. Here, $N$ is the number of BEC atoms, $\omega_{tr}$ is
the transition frequency and $g_{o}$ accounts for the coupling strength
between the single intracavity photon and single condensate atom. The first
term in the Eq.(1) stands for the energy of the intracavity mode with creation
(annihilation) operator $c\,(c^{\dagger})$ and frequency $\omega_{c}$.
Moreover, the empty cavity resonance frequency $\omega_{c}$ is shifted due to
the presence of the BEC inside the cavity by an amount of $NU_{o}/2$. The
second term describes the energy of the Bogoliubov mode of the collective
oscillations of the BEC. Furthermore, $a(a^{\dagger})$ and $\omega_{a}$ are,
respectively, the creation (annihilation) operators and frequency of the
condensate mode. The dynamics of the BEC can be explained as follows: The zero
momentum state is only coupled to the symmetric momentum states, $\pm 2\hbar
k$, due to the absorption and stimulated emission of the cavity photons Ritter
. This can be explained as the condensate mode oscillates at frequency
$\omega_{a}=4\omega_{r}=\hbar k^{2}/2m$, where $m$ is the mass of the atom and
$\omega_{r}$ is recoil frequency. The third term describes the interaction
between intracavity field with condensate mode. Moreover, $g$ accounts for the
strength of interaction between field and BEC and it is clear from the
expression of the $g$ that the single atom-photon coupling is increased by the
square root of the number of condensate atoms. The last term corresponds to
the coupling between cavity mode and input laser field with coupling strength
E and it is related to the input power $P$ with $|E|=\sqrt{2\kappa
P/\hbar\omega_{p}}$, where $\kappa$ is the decay rate of the cavity field.
Figure 1: (Color online) A sample of two-level bosonic atoms with resonant
frequency $\omega_{tr}$ trapped inside a Fabry-Pérot cavity of length $L$ is
interacting with standing laser field. Here, the left-end mirror is
transmissive while the right-end mirror is perfectly reflecting. Moreover, the
cavity is being driven by a laser of frequency $\omega_{p}$.
###
In order to describe the complete dynamics of the subsystems involved in this
problem, an adequate choice is to use the formalism of the quantum Langevin
equations. According to the Heisenberg-Langevin equation of motion, the
commutation relations $[a,a^{\dagger}]=1$ and $[c,c^{\dagger}]=1$, the time
evolution of the $c$ and $a$, can be obtained. We derive the Heisenberg-
Langevin equations for canonical variables and introduce the noise operators
$c_{in}(t)$ and $a_{in}$ weighted with the rates $\kappa$ and $\gamma$ which
describe the dissipation of the intracavity field and condensate mode of
collective oscillations of the BEC respectively. Therefore, Heisenberg-
Langevin equations for condensate mode and intracavity field are given by
$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-(i\omega_{a}+\gamma)a-i\dfrac{g}{\sqrt{2}}\,c^{\dagger}c+\sqrt{2\gamma}\,a_{in},$
$\displaystyle\dot{c}$ $\displaystyle=$
$\displaystyle-(\kappa+i\Delta_{c})\,c-i\dfrac{g}{\sqrt{2}}\,(a+a^{\dagger})c+E$
(2) $\displaystyle+\sqrt{2\kappa}c_{in}.$
We assume that the noise associated with the light field is uncorrelated with
the noise accounts for the condensate mode. For laser field, the noise and
damping are due to the vacuum noise, losses from the cavity and fluctuations
of the laser field. Moreover, the noise and damping accounts for condensate
mode are due to the condensate atoms with additional nearby non-condensed
atoms. In addition, $c_{in}$ and $a_{in}$ are the non-commuting noise
operators associated with optical field and condensate mode respectively. They
have zero mean values and nonzero correlation functions cw :
$\displaystyle\langle\partial a^{\dagger}_{in}(t)\partial
a_{in}(t^{\prime})\rangle$ $\displaystyle=$ $\displaystyle
n_{a}\delta(t-t^{\prime}),$ $\displaystyle\langle\partial a_{in}(t)\partial
a^{\dagger}_{in}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle(n_{a}+1)\delta(t-t^{\prime}),$ $\displaystyle\langle\partial
c^{\dagger}_{in}(t)\partial c_{in}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle n_{c}\delta(t-t^{\prime}),$ $\displaystyle\langle\partial
c_{in}(t)\partial c^{\dagger}_{in}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle(n_{c}+1)\delta(t-t^{\prime}),$
where $n_{c}$ and $n_{a}$ are the occupation numbers of the optical and
condensate modes, respectively. Moreover, all other correlations are zero. As
we are in optical regime and a BEC at a temperature of at most a few
$\mu\mathrm{K}$, therefore one can take $n_{c,a}\rightarrow 0$.
## III Dynamics of small fluctuations: Normal mode splitting
In the following we linearized the operators in Eq.(2) around the steady state
values, $a=\left<a\right>_{ss}+\partial a$,
$c=\left<c\right>_{ss}+\partial\mathrm{c}$. Here, we have assumed that the
fluctuation operators $\partial a$ and $\partial c$ have zero mean. The steady
state value of the intracavity mode is
$\left<c\right>_{ss}=\mathrm{E}/(\kappa+i\,\Delta)$, where the total effective
detuning is
$\Delta=\Delta_{c}-\dfrac{\,\omega_{a}g^{2}_{ac}}{\omega^{2}_{a}+\gamma^{2}}\,\left<c\right>_{\mathrm{ss}}.$
(4)
For the sake of simplicity we assume that the field is real positive and this
can be achieved by adjusting the phase of the laser field. Similarly, the
steady state value of the condensate mode is
$\left<a\,\right>_{ss}=[-i\,g_{ac}/\sqrt{2}\,(\gamma+i\omega_{a}]\left<c\,\right>_{ss}$.
We linearize the Langevin equations of motion given in Eq.(2), and assume that
pump field is intense and keep terms only up to first order in the fluctuation
operators. We rewrite each Heisenberg operator in Eq.(2) as a sum of steady
state value and fluctuation operator with zero mean value. Therefore, the
linearized Heisenberg-Langevin equations are,
$\displaystyle\partial\dot{a}$ $\displaystyle=-(i\omega_{a}+\gamma)\partial
a-i\dfrac{G}{2}\left(\partial
c+\partial{c}^{\dagger}\right)+\sqrt{2\gamma}\,a_{in},$
$\displaystyle\partial\dot{c}$
$\displaystyle=-\left(i\,\Delta+\kappa\right)\partial
c-i\dfrac{G}{2}\left(\partial\mathrm{a}+\partial\mathrm{a}^{\dagger}\right)+\sqrt{2\,\kappa}\,c_{in}.$
(5)
The linearized quantum Langevin equations show the fluctuations of the
Bogoliubov mode as the collective oscillation of the BEC. The condensate mode
is now coupled to the cavity field quadrature fluctuations by the effective
couplings $G=\sqrt{2}\,gc_{s}$ which can be made very large by increasing the
amplitude $c_{s}$ of the intracavity field. Linearized quantum Langevin
equations (5) and their corresponding Hermitian conjugate form a system of
four first-order coupled operator equations, for which the Ruth-Hurwitz
criteria Hurwitz implies that the system will be stable only if the following
stability conditions are satisfied,
$\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle
2\kappa\gamma\\{[\kappa^{2}+(\omega_{a}-\Delta)^{2}][\kappa^{2}+(\omega_{a}+\Delta)^{2}]+\gamma[(\gamma+2\kappa)$
$\displaystyle\times(\kappa^{2}+\Delta^{2})+2\kappa\omega^{2}_{a}]\\}+\Delta\omega_{a}G^{2}(\gamma+2\kappa)^{2}>0$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\omega_{a}(\kappa^{2}+\Delta^{2})-G^{2}\Delta>0.$ (6)
Figure 2: (Color online)(a) Plot normalized effective frequency
$\omega_{eff}/\omega_{a}$ of the condensate mode as a function of the
normalized frequency $\omega/\omega_{a}$. Parameters values are
$\omega_{a}=2\pi\,4\times 3.8\,\rm{kHz}$, $\gamma=2\pi\times 0.4\,\rm{kHz}$,
$\Delta=\omega_{a}$, $\kappa=0.1\,\omega_{a}$ and $G=0.1\,\omega_{a}$ (solid
blue curve), $G=0.2\,\omega_{a}$ (dashed green curve), and $G=0.3\,\omega_{a}$
(red dotted line). (b) Plot of normalized effective damping rate
$\gamma_{eff}/\omega_{a}$ of the condensate versus normalized frequency
$\omega/\omega_{a}$ for three different values of the BEC-field coupling,
$G=0.1\,\omega_{a}$ (solid blue line), $G=0.2\,\omega_{a}$ (dashed green
line), and $G=0.3\,\omega_{a}$ (red dotted line).
Figure 3: (Color online) The displacement spectrum of the Bogoliubov mode
$S_{x_{a}}(\omega)$ as a function of the normalized detuning
$\Delta/\omega_{a}$ and normalized frequency $\omega/\omega_{a}$ for
$\kappa=0.1\,\omega_{a}$ and (a) $G=0.1\,\omega_{a}$ (b) $G=0.2\,\omega_{a}$.
The other parameters values are the same as in Fig.2.
We choose a parameter regime such that the above stability conditions are
satisfied to ensure stability of the system. The Heisenberg-Langevin equations
(5) are linear in creation and annihilation operator. We define the
quadratures, i.e., $\partial x_{a}=(\partial a+\partial
a^{\dagger})/\sqrt{2}$, $\partial p_{a}=(\partial a-\partial
a^{\dagger})/i\sqrt{2}$, $\partial x_{c}=(\partial c+\partial
c^{\dagger})/\sqrt{2}$ and $\partial p_{c}=(\partial c-\partial
c^{\dagger})/i\sqrt{2}$, and solve the Langevin equations in Fourier space.
Here, we introduce the displacement spectrum of Bogoliubov mode of collective
oscillation of the BEC which is obtained from the two-frequency auto-
correlation function $\langle\partial x_{a}(\omega)\partial
x_{a}(\omega^{\prime})\rangle=S_{x_{a}}(\omega)\delta(\omega+\omega^{\prime})$.
Therefore, the displacement spectrum $S_{x_{a}}(\omega)=(1/2\pi)\int
e^{-it(\omega-\omega^{\prime})}\langle\partial x_{a}(\omega)\partial
x_{a}(\omega^{\prime})\rangle d\omega^{\prime}$ of the condensate in Fourier
space is given by
$S_{x_{a}}(\omega)=\dfrac{\omega^{2}_{a}}{|d(\omega)|^{2}}\left[\dfrac{2G^{2}\kappa(\kappa^{2}+\omega^{2}_{a}+\Delta^{2})}{(\kappa^{2}+\Delta^{2}-\omega^{2})^{2}+4\kappa^{2}\omega^{2}}+2\gamma\right],$
(7)
where $\omega$ stands for frequency and
$d(\omega)=\omega^{2}_{a}-\omega^{2}-i\omega\gamma-\dfrac{\omega_{a}\Delta\,G^{2}}{(\kappa-i\omega)^{2}+\Delta^{2}}$
is the modified susceptibility of the condensate mode due to radiation
pressure. The effective frequency ($\omega_{eff}$) of collective oscillation
of the BEC and effective damping rate ($\gamma_{eff}$) of collective
oscillation of the BEC are given by
$\displaystyle\omega_{eff}$ $\displaystyle=$
$\displaystyle\left[\omega^{2}_{a}-\dfrac{\omega_{a}\Delta
G^{2}(\kappa^{2}-\omega^{2}+\Delta^{2})}{(\kappa^{2}-\omega^{2}+\Delta^{2})^{2}+(2\kappa\omega)^{2}}\right]^{\dfrac{1}{2}},$
(8) $\displaystyle\gamma_{eff}$ $\displaystyle=$
$\displaystyle\left[\gamma+\dfrac{2\omega_{a}\kappa\Delta
G^{2}}{(\kappa^{2}-\omega^{2}+\Delta^{2})^{2}+(2\kappa\omega)^{2}}\right].$
(9)
The frequency of the condensate mode is modified due to radiation pressure as
shown in Eq.(8) and this is equivalent to the optical spring effect in case of
opto-mechanical system with moving mirror. The spectrum of condensate mode is
described by the effective susceptibility $d(\omega)$ and displacement
spectrum of collective oscillation of the BEC which consists of two terms, the
first term is proportional to quantum fluctuation of the radiation pressure
and, second term arises from quantum noises associated with the matter waves.
Therefore, the position spectrum of the condensate mode is determined by
radiation pressure and quantum noise.
###
For our numerical calculations, we choose the parameters very close to the
BEC-cavity system Ritter ; Baumann . The recoil frequency
$\omega_{r}=2\pi\times 3.8\,\rm{KHz}$, decay rate $\gamma=2\pi\times
0.4\,\rm{KHz}$, atom-field detuning $\Delta_{a}=2\pi\times 32\,\rm{GHz}$ and
single atom-photon coupling is $g_{o}=2\pi\times 10.9\,\rm{MHz}$. The optical
spring effect leads to the shift in the frequency of the condensate mode and
this shift is small for low-frequency oscillators Corbitt and shift is
significant for high-frequency oscillators Gigan . In Fig.2, we plot
normalized effective frequency $\omega_{eff}/\omega_{a}$ of Bogoliubov mode of
collective oscillation of the BEC as a function of the normalized frequency
$\omega/\omega_{a}$ for three different values of the BEC-intracavity field
interaction, $G=0.1\,\omega_{a}$ (solid blue curve), $G=0.2\,\omega_{a}$
(dashed green curve), and $G=0.3\,\omega_{a}$ (red dotted curve). It is noted
that the deviation in the bare frequency of the condensate mode $\omega_{a}$
is increased as the coupling between BEC and intracavity field is increased.
In Fig.2, we plot the normalized effective damping rate
$\gamma_{eff}/\omega_{a}$ of the condensate mode versus normalized frequency
$\omega/\omega_{a}$ for different values of the BEC-field interaction. The
effective damping rate of the condensate mode is significantly increased as
the BEC-field interaction is increased as shown in Fig.2. Therefore, strong
atom-field interaction causes higher atomic loss and this atomic loss is
maximum near the resonance. The back action of the light causes the heating of
the atoms, and as a consequence, atomic loss was observed in Murch . The
increase in the effective damping is at the basis of cooling of Bogoliubov
mode of the BEC. The BEC couples to the intracavity field due to radiation
pressure force and radiation pressure behaves as an thermal reservoir for the
BEC resonator. The effective temperature of the Bogoliubov mode of the
condensate takes the value between initial thermal reservoir temperature and
optical reservoir. Therefore, when $G>\gamma$ one achieves the ground state
cooling of the Bogoliubov mode.
###
In Fig.3, we plot the displacement spectrum of the BEC as a function of the
normalized detuning $\Delta/\omega_{a}$ and normalized frequency
$\omega/\omega_{a}$ for two different values of the BEC field interaction,
i.e, $G=0.1\,\omega_{a}$, Fig.3, and $G=0.2\,\omega_{a}$, Fig.3. For small
value of interaction between condensate mode and cavity field, i.e
$G=0.1\,\omega_{a}$, the fluctuations spectrum of the Bogoliubov mode barely
shows the normal mode splitting. As we increase the coupling strength, the
normal mode splitting becomes observable. For $G=0.2\omega_{a}$, the normal
mode splitting is clearly seen in Fig.3. It is observed that the normal mode
splits in the presence of the large coupling between BEC and field. The normal
mode splitting is due to the hybridization of the two oscillators: the
electromagnetic field and collective oscillations of the BEC mode. The normal
modes splitting results from the mixing of the fluctuations of the cavity
field with the fluctuations of the condensate. The normal mode occurs at
frequencies
$\omega^{2}_{\pm}=(\Delta^{2}+\omega^{2}_{a}\pm\sqrt{(\Delta^{2}-\omega^{2}_{a})^{2}+4G^{2}\omega_{a}\Delta})/2$.
In this expression, we neglect the cavity decay rate $\kappa$ and damping rate
of the Bogoliubov mode. The splitting is directly proportional to the coupling
rate between Bogoliubov mode and intracavity field, i.e,
$\omega_{+}-\omega_{-}\propto G$.
###
The atom-field system is probed by measuring the optical field transmitted by
the cavity, particularly, its spectrum. The power spectrum of the light
emitted by the cavity is determined from Eq.(5). In frequency space, the
fluctuating $\partial c(\omega)$ of the intracavity field can be obtained by
using Heisenberg Langevin equations (5). With the quantum input-output
relation gardiner $\partial c_{out}=\sqrt{2\kappa}\partial c(\omega)-\partial
c_{in}(\omega)$, the power spectral density of the cavity output field is
defined by taking the Fourier transform of its autocorrelation function
$\langle
c^{\dagger}_{out}(\omega^{\prime})\,c_{out}(\omega)\rangle=S_{cout}(\omega)\delta(\omega^{\prime}+\omega)$
as $S_{cout}(\omega)=(1/2\pi)\int
e^{-it(\omega-\omega^{\prime})}\langle\partial c_{cout}(\omega)\partial
c_{cout}(\omega^{\prime})\rangle d\omega$. Therefore, the power spectra of the
light emitted by the cavity is given by,
$S_{cout}(\omega)=\dfrac{1}{|d_{c}(\omega)|^{2}}\left[4\kappa^{2}|\alpha(\omega)|^{2}+2\gamma|\beta(\omega)|^{2}\right],$
(10)
where,
$\displaystyle\alpha(\omega)$ $\displaystyle=$ $\displaystyle
i\dfrac{\omega_{a}G^{2}}{\sqrt{2}(\omega^{2}_{a}-\omega^{2}-i\omega\gamma)},$
$\displaystyle\beta(\omega)$ $\displaystyle=$
$\displaystyle-i\sqrt{2\kappa}\omega_{a}G\dfrac{\kappa+i\Delta+i\omega}{\sqrt{2}(\omega^{2}_{a}-\omega^{2}-i\omega\gamma)},$
$\displaystyle d_{c}(\omega)$ $\displaystyle=$
$\displaystyle(\kappa-i\omega)^{2}+\Delta^{2}-\dfrac{\sqrt{2}\Delta
G^{2}}{\omega^{2}_{a}-\omega^{2}-i\omega\gamma}.$
Figure 4: (Color online) The emission spectrum $S_{cout}(\omega)$ as a
function of the normalized frequency $\omega/\omega_{a}$ for different values
of the BEC-field coupling strength, $G=0.1\,\omega_{a}$ (solid blue curve),
$G=0.2\,\omega_{a}$ (dotted red curve) and $G=0.3\,\omega_{a}$ (green dashed
curve). Moreover, the cavity detuning $\Delta=\omega_{a}$ and the other
parameters are the same as in Fig.(2).
It is clear from Eq.(10) that the spectrum of the output field consists of two
terms: the first term describes the coupling of intracavity field with the
Bogoliubov mode of collective density oscillations of the BEC and second term
accounts for the vacuum noise associated with condensate mode.
###
In Fig.4, we plot the fluctuations spectrum of the output field as a function
of the normalized frequency $\omega/\omega_{a}$ for different values of
coupling strength between intracavity field with BEC. For small coupling
between condensate mode and cavity field $G=0.1\,\omega_{a}$, one can see that
only single peak (solid Blue curve) is appeared in the fluctuations spectrum
of the output field. However, as the interaction of the intracavity field with
BEC is $G=0.2\,\omega_{a}$ increased, the spectrum splits into two peaks (Red
dotted curve). For $G=0.3\,\omega_{a}$, we observe $S_{cout}(\omega)$ further
splits into two sideband peaks. This splitting is pure result of the coupling
between condensate mode and cavity mode. It is also observed that the
separation between two peaks in the $S_{cout}(\omega)$ is directly
proportional to the BEC-field interaction strength. This splitting depends
linearly on the coupling parameter $G$ and goes to zero in the absence of BEC-
field coupling.
## IV conclusion
In conclusion, we theoretically analyze the normal mode splitting in a system
which consists of Bose-Einstein condensate trapped inside a Fabry-Pérot cavity
driven by laser field. We observe that the frequency and damping rate of the
condensate mode display shift due to the radiation pressure force. The
hybridization of the Bogoliubov mode with the fluctuations of the cavity field
leads to normal mode splitting. The results show that normal mode splitting in
fluctuations spectrum of the condensate mode depends linearly on the coupling
of the BEC with optical field. We also analyze the coupled dynamics of the BEC
and cavity field by probing the emission spectrum of the cavity. Normal mode
splitting is also observed in the transmission spectra by cavity and this
splitting is directly proportional to the BEC-field coupling strength.
Therefore, the coupling rate between Bogoliubov mode and interactivity field
can be determined by measuring the distance between two splitting peaks.
## References
* (1) T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
* (2) V. B. Braginsky, Measurement of Weak Forces in Physics Experiments (University of Chicago Press, Chicago, 1977).
* (3) S. Mancini, V. Giovannetti, D. Vitali and P. Tombesi, Phys. Rev. Lett. 88, 120401 (2002).
* (4) J. Zhang et al., Phys. Rev. A 68, 013808 (2003).
* (5) L. Tian and P. Zoller, Phys. Rev. Lett. 93, 266403 (2004).
* (6) A. Naik et al., Nature (London) 443, 193 (2006).
* (7) C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, Phys. Rev. A 49, 1337 (1994).
* (8) S. Mancini and P. Tombesi, Phys. Rev. A 49 4055 (1994).
* (9) C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg and M. Zimmermann, Rev. Mod. Phys. 52 341 (1980).
* (10) M. D. LaHaye, O. Buu, B. Camarota and K. C. Schwab, Science 304 74 (2004).
* (11) D. Rugar et al., Nature (London) 430, 329 (2004).
* (12) K. L. Ekinci , Y. T. Yang and M. L. Roukes, J. Appl. Phys. 95 2682 (2004).
* (13) C. M. Caves, Phys. Rev. Lett. 45 75 (1980).
* (14) V. Braginsky and S. P. Vyatchanin, Phys. Lett. A 293, 228 (2002).
* (15) W. Marshall, C. Simon, R. Penrose and D. Bouwmeester 2003 Phys. Rev. Lett. 91 130401 (2003).
* (16) M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, Phys. Rev. Lett. 99, 250401 (2007).
* (17) D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Phys. Rev. Lett. 98, 030405 (2007).
* (18) M. A Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
* (19) H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 78, 013824 (2008).
* (20) C. Genes, D. Vitali, and P. Tombesi, Phys. Rev. A 77, 050307(R) (2008).
* (21) D. Meiser and P. Meystre, Phys. Rev. A 73, 033417 (2006).
* (22) C. Genes, H. Ritsch, and D. Vitali, Phys. Rev. A 80, 061803(R) (2009).
* (23) R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
* (24) A. Wallraff et al., Nature (London) 431, 162 (2004).
* (25) J. P. Reithmaier et al., Nature (London) 432, 197 (2004); T. Yoshie et al., ibid. 432, 200 (2004); K. Hennessy et al., ibid. 445, 896 (2007).
* (26) J. Esteve et al., Nature (London) 455, 1216 (2008).
* (27) F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, Nature (London) 450, 268 (2007).
* (28) F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science 322, 235 (2008).
* (29) Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, Nature (London) 450, 272 (2007).
* (30) T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z. Y. Ma, and D. M. Stamper-Kurn, Phys. Rev. Lett. 105, 133602 (2010).
* (31) S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, Appl. Phys. B: Lasers Opt. 95, 213 (2009).
* (32) D. Nagy, G. Kónya, G. Szirmai, and P. Domokos, Phys. Rev. Lett. 104, 130401 (2010).
* (33) D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, Eur. Phys. J. D 55, 659 (2009).
* (34) Muhammad Asjad and Farhan Saif, Phys. Rev. A 84, 033606 (2011).
* (35) C. W. Gardiner, Phys. Rev. Lett. 56, 1917 (1986).
* (36) A. Hurwitz, in Selected Papers on Mathematical Trends in Control Theory, edited R. Bellman and R. Kalaba (New York, Dover, 1964).
* (37) T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, Phys. Rev. Lett. 98, 150802 (2007); T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, ibid. 99, 160801 (2007).
* (38) S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, Nature (London) 444, 67 (2006); O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006); A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97, 243905 (2006).
* (39) K. W. Murch et al Nature Phys. 4 461 (2008).
* (40) C. W. Gardiner, Quantum Noise (Berlin: Springer 1991).
|
arxiv-papers
| 2012-09-15T04:00:34 |
2024-09-04T02:49:35.162750
|
{
"license": "Public Domain",
"authors": "Muhammad Asjad",
"submitter": "Muhammad Asjad Mr.",
"url": "https://arxiv.org/abs/1209.3354"
}
|
1209.3419
|
11institutetext: Oxford University, University of Calabria
[email protected], [email protected], [email protected]
# Tractable Optimization Problems through
Hypergraph-Based Structural Restrictions††thanks: G.Gottlob works at the
Computing Laboratory and at the Oxford Man Institute of Quantitative Finance,
Oxford University. This work was done in the context of the EPSRC grant
EP/G055114/1 “Constraint Satisfaction for Configuration: Logical
Fundamentals,Algorithms, and Complexity” and of Gottlob’s Royal Society
Wolfson Research Merit Award.
Georg Gottlob 11 Gianluigi Greco 22 Francesco Scarcello 221122
###### Abstract
Several variants of the Constraint Satisfaction Problem have been proposed and
investigated in the literature for modelling those scenarios where solutions
are associated with some given costs. Within these frameworks computing an
optimal solution is an NP-hard problem in general; yet, when restricted over
classes of instances whose constraint interactions can be modelled via
(nearly-)acyclic graphs, this problem is known to be solvable in polynomial
time. In this paper, larger classes of tractable instances are singled out, by
discussing solution approaches based on exploiting hypergraph acyclicity and,
more generally, structural decomposition methods, such as (hyper)tree
decompositions.
## 1 Introduction
The Constraint Satisfaction Problem (CSP) is a well-known framework [11] for
modelling and solving search problems, which received considerably attention
in the literature due to its applicability in various areas. Informally, a CSP
instance is defined by singling out the variables of interest, and by listing
the allowed combinations of values for groups of them, according to the
constraints arising in the application at hand. The solutions for this
instance are the assignments of domain values to variables that satisfy all
such constraints. Many apparently unrelated problems from disparate areas
actually turn out to be equivalent to the CSP and can be accommodated within
the CSP framework. Examples are puzzles, conjunctive queries over relational
databases, graph colorability, and checking whether there is a homomorphism
between two finite structures.
###### Example 1
Figure 1 shows a combinatorial crossword puzzle (taken from [15]). A set of
legal words is associated with each horizontal or vertical array of white
boxes delimited by black boxes. A solution to the puzzle is an assignment of a
letter to each white box such that to each white array is assigned a word from
its set of legal words. This problem can be recast in a CSP by associating a
variable with each white box, and by defining a constraint for each array of
white boxes prescribing the legal words that are associated with it. $\lhd$
When assignments are associated with some given cost, however, computing an
arbitrary solution might not be enough. For instance, the crossword puzzle in
Figure 1 may admit more than one solution, and expert solvers may be asked to
single out the most difficult ones, such as those solutions that minimize the
total number of vowels occurring in the used words. In these cases, one is
usually interested in the corresponding _optimization problem_ of computing
the solution of minimum cost, whose modeling is accounted for in several
variants of the basic CSP framework, such as fuzzy, probabilistic, weighted,
lexicographic, valued, and semiring-based CSPs (see [25, 4] and the references
therein).
Figure 1: A crossword puzzle, its associated hypergraph ${\cal H}_{cp}$, and a
hypertree decomposition of width 2 for ${\cal H}_{cp}$.
Since solving CSPs—and the above extensions—is an NP-hard problem, much
research has been spent to identify restricted classes over which solutions
can efficiently be computed. In this paper, structural decomposition methods
are considered [15], which identify tractable classes by exploiting the
structure of constraint scopes as it can be formalized either as a hypergraph
(whose nodes correspond to the variables and where each group of variables
occurring in some constraint induce a hyperedge) or as a suitable binary
encoding of such hypergraph. In particular, we focus on the structural methods
based on the notions of (generalized) hypertree width [18, 19] and treewidth
[28]. In both cases, the underlying idea is that solutions to CSP instances
that are associated with acyclic (or nearly-acyclic) structures can
efficiently be computed via dynamic programming, by incrementally processing
the structure according to some of its topological orderings.
As a matter of fact, however, while in the case of classical CSPs deep and
useful results have been achieved for both graph and hypergraph
representations, in the case of CSP extensions tailored for optimization
problems attention was mainly focused on binary encodings and, in particular,
on the _primal graph_ representation, where nodes correspond to variables and
an edge between two variables indicates that they are related by some
constraint. Discussing whether (and how) hypergraph-based structural
decomposition techniques in the literature can be lifted to such optimization
frameworks is the main goal of this paper. In particular, we consider three
CSP extensions:
(1)
First, we consider optimization problems where every mapping variable-value is
associated with a cost, so that the aim is to find an assignment satisfying
all the constraints and having the minimum total cost.
(2)
Second, we consider the case where costs are associated with the allowed
combinations of simultaneous values for the variables occurring in the
constraint, rather than to individual values. Again, within this setting, we
consider the problem of computing a solution having minimum total cost.
(3)
Finally, we consider a scenario where the CSP instance at hand might not admit
a solution at all, and where the problem is hence to find the assignment
minimizing the total number of violated constraints (and, more generally,
whenever a cost is assigned to each constraint, the assignment minimizing the
total cost of violated constraints).
For each of the above settings, the complexity of computing the optimal
solution is analyzed in this paper, by overviewing some relevant recent
research and by providing novel results. In particular:
* $\blacktriangleright$
We show that optimization problems of kind (1) can be solved in polynomial
time on instances having bounded (generalized) hypertree-width hypergraphs.
This result is based on an algorithm recently designed and analyzed in the
context of _combinatorial auctions_ [13].
* $\blacktriangleright$
We show that even optimization problems of kind (2) are tractable on instances
having bounded (generalized) hypertree-width hypergraphs. Indeed, we describe
how to transform this kind of instances into equivalent instances of kind (1),
by preserving their structural properties.
* $\blacktriangleright$
We observe that optimization problems of kind (3) remain NP-hard even over
instances having an associated acyclic hypergraph. However, there is also good
news: they are shown to be tractable on instances having bounded treewidth
incidence graph encoding. The latter is a binary encoding of the constraint
hypergraph with usually better structural features than the primal graph
encoding (see, e.g., [15, 22]). Again, proof is via a mapping to case (1).
Organization. The rest of the paper is organized as follows. Section 2
discusses preliminaries on CSPs and structural restrictions, and Section 3
provides an overview of the structural decomposition methods based on
treewidth and (generalized) hypertree width. Results for optimization problems
of kind (1) and (2) are discussed in Section 4, whereas problems of kind (3)
are discussed in Section 5. Finally, Section 6 draws our conclusions.
## 2 CSPs, Acyclic Instances, and their Desirable Properties
An instance of a constraint satisfaction problem [11] is a triple
$\mathcal{I}=\langle{\it Var},U,\mathcal{C}\rangle$, where ${\it Var}$ is a
finite set of variables, $U$ is a finite domain of values, and
$\mathcal{C}=\\{C_{1},C_{2},\ldots,C_{q}\\}$ is a finite set of constraints.
Each constraint $C_{v}$, for $1\leq v\leq q$, is a pair $(S_{v},r_{v})$, where
$S_{v}\subseteq{\it Var}$ is a set of variables called the constraint scope,
and $r_{v}$ is a set of substitutions (also called _tuples_) from variables in
$S_{v}$ to values in $U$ indicating the allowed combinations of simultaneous
values for the variables in $S_{v}$. Any substitution from a set of variables
$V\subseteq{\it Var}$ to $U$ is extensively denoted as the set of pairs of the
form $X/u$, where $u\in U$ is the value to which $X\in V$ is mapped. Then, a
solution to $\mathcal{I}$ is a substitution $\theta:{\it Var}\mapsto U$ for
which $q$-tuples $t_{1}\in r_{1},...,t_{q}\in r_{q}$ exist such that
$\theta=t_{1}\cup...\cup t_{q}$.
###### Example 2
In the crossword puzzle of Figure 1, $\it Var$ coincides with the letters of
the alphabet, and a variable $X_{i}$ (denoted by its index $i$) is associated
with each white box. An example of constraint is
$C_{1H}=((1,2,3,4,5),r_{1H})$, and a possible instance for $r_{1H}$ is
$\\{\langle h,o,u,s,e\rangle,\langle c,o,i,n,s\rangle,\langle
b,l,o,c,k\rangle\\}$—in the various constraint names, subscripts $H$ and $V$
stand for “Horizontal” and “Vertical,” respectively, resembling the usual
naming of definitions in crossword puzzles. $\lhd$
The structure of a CSP instance $\mathcal{I}$ is best represented by its
associated hypergraph ${\cal H}(\mathcal{I})=(V,H)$, where $V={\it Var}$ and
$H=\\{S\mid(S,r)\in{\mathcal{C}}\\}$—in the following, $V$ and $H$ will be
denoted by ${\mathcal{N}}({\cal H})$ and ${\mathcal{E}}({\cal H})$,
respectively. As an example, the hypergraph associated with the crossword
puzzle formalized above is illustrated in the central part of Figure 1.
A hypergraph ${\cal H}$ is acyclic iff it has a join tree [3]. A join tree
$J\\!T({\cal H})$ for a hypergraph ${\cal H}$ is a tree whose vertices are the
hyperedges of ${\cal H}$ such that, whenever the same node $X\in V$ occurs in
two hyperedges $h_{1}$ and $h_{2}$ of ${\cal H}$, then $X$ occurs in each
vertex on the unique path linking $h_{1}$ and $h_{2}$ in $J\\!T({\cal H})$.
The notion of acyclicity we use here is the most general one known in the
literature, coinciding with $\alpha$-acyclicity according to Fagin [9]. Note
that the hypergraph ${\cal H}_{cp}$ of Figure 1 is not acyclic. An acyclic
hypergraph is discussed below.
Figure 2: A hypergraph ${\cal H}_{1}$, a join tree $J\\!T({\cal H}_{1})$, the
primal graph ${\it G}({\cal H}_{1}$), and the incidence graph ${\it inc}({\cal
H}_{1})$.
###### Example 3
Consider the hypergraph ${\cal H}_{1}$ shown on the left of Figure 2, which is
associated with a CSP instance over the set of variables $\\{A,...,M\\}$. In
particular, six constraints are defined over the instance whose scopes
precisely correspond to the hyperedges in ${\mathcal{E}}({\cal H}_{1})$; for
instance, $\\{A,B,C\\}$ is an example of constraint scope. Note also that
${\cal H}_{1}$ is acyclic. Indeed, a join tree $J\\!T({\cal H}_{1})$ for it is
reported in the same figure to the right of ${\cal H}_{1}$. $\lhd$
An important property of acyclic instances is that they can efficiently be
processed by dynamic programming. Indeed, according to Yannakakis’ algorithm
[34] (originally conceived in the equivalent context of evaluating acyclic
Boolean conjunctive queries), they can be evaluated by processing any of their
join trees bottom-up, by performing upward semijoins between the constraint
relations, thus keeping the size of the intermediate result small. At the end,
if the constraint relation associated with the root atom of the join tree is
not empty, then the CSP instance does admit a solution. Therefore, the whole
procedure is feasible in $O(n\times r_{max}\times\log r_{max})$, where $n$ is
the number of constraints and $r_{max}$ denotes the size of the largest
constraint relation.
In addition to the polynomial time algorithm for deciding whether a CSP admits
a solution, acyclic instances enjoy further desirable properties:
Acyclicity is efficiently recognizable:
Deciding whether a hypergraph is acyclic is feasible in linear time [31] and
belongs to the class ${\rm L}$ (deterministic logspace). Indeed, this follows
from the fact that hypergraph acyclicity belongs to SL [16], and that SL is
equal to ${\rm L}$ [27].
Acyclic instances can be efficiently solved:
After the bottom-up step described above, one can perform the reverse top-down
step by filtering each child vertex from those tuples that do not match with
its parent tuples. The relations obtained after the top-down step enjoy the
global consistency property, i.e., they contain only tuples whose values are
part of some solution of the CSP. Then, all solutions can be computed with a
backtrack-free procedure, and thus in total polynomial time, i.e., in time
polynomial in the input plus the output [34] (and actually also with
polynomial delay). Alternatively, one may enforce pairwise consistency by
taking the semijoins between all pairs of relations until a fixpoint is
reached. Indeed, acyclic instances that fulfil this property also fulfil the
global consistency property [2].
Acyclic instances are parallelizable:
It has been shown that solving acyclic CSP instances is highly parallelizable,
as this problem (actually, deciding the existence of a solution) is complete
for the low complexity class LOGCFL [16]. Efficient parallel algorithms are
discussed in [16] and [17].
We conclude this section by recalling that the above desirable properties of
acyclic CSP instances have profitably been exploited in various application
scenarios. Indeed, besides their application in the context of Database
Theory, they found applications in Game Theory [14, 8], Knowledge
Representation and Reasoning [21], and Electronic Commerce [13], just to name
a few.
## 3 Generalizing acyclicity
Many attempts have been made in the literature for extending the good results
about acyclic instances to relevant classes of nearly acyclic structures. We
call these techniques structural decomposition methods, because they are based
on the “acyclicization” of cyclic (hyper)graphs. We refer the interested
reader to [29] for a detailed description of how these techniques may be
useful for constraint satisfaction problems and to [22] for further results
about graph-based techniques, when relational structures are represented
according to various graph representations (primal graph, dual graph,
incidence-graph encoding). We also want to mention recent methods such as
Spread-cuts [7] and fractional hypertree decompositions [23].
A survey of most of these techniques is currently available in Wikipedia (look
for “decomposition method”, at http://www.wikipedia.org). In the sequel, we
shall briefly overview the tree and hypertree decomposition methods.
### 3.1 Tree Decompositions
For classes of instances having only binary constraints or, more generally,
constraints whose scopes have a fixed maximum arity, the most powerful
structural method is based on the notion of treewidth.
###### Definition 1 ([28])
A tree decomposition of a graph $G=(V,E)$ is a pair $\langle T,\chi\rangle$,
where $T=(N,F)$ is a tree, and $\chi$ is a labelling function assigning with
each vertex $p\in N$ a set of vertices $\chi(p)\subseteq V$ such that the
following conditions are satisfied: (1) for each node $b$ of $G$, there exists
$p\in N$ such that $b\in\chi(p)$; (2) for each edge $(b,d)\in E$, there exists
$p\in N$ such that $\\{b,d\\}\subseteq\chi(p)$; and, (3) for each node $b$ of
$G$, the set $\\{p\in N\mid b\in\chi(p)\\}$ induces a connected subtree of $T$
(_connectedness condition_). The width of $\langle T,\chi\rangle$ is the
number $\max_{p\in N}(|\chi(p)|-1)$. The treewidth of $G$, denoted by $tw(G)$,
is the minimum width over all its tree decompositions. $\Box$
It is well-known that a graph $G$ is acyclic if and only if $tw(G)=1$.
Moreover, for any fixed natural number $k>0$, deciding whether $tw(G)\leq k$
is feasible in linear time [5].
Any CSP with primal graph $G$ such that $tw(G)\leq k$ can be (efficiently)
turned into an equivalent CSP whose primal graph is acyclic. Let
$\mathcal{I}=\langle{\it Var},U,\mathcal{C}\rangle$ be a CSP instance, let $G$
be the primal graph of ${\cal H}(\mathcal{I})$, and let $\langle
T,\chi\rangle$ be a tree decomposition of $G$ having width $k$. We may build a
new acyclic CSP instance $\mathcal{I}^{\prime}=\langle{\it
Var},U,\mathcal{C}^{\prime}\rangle$ over the same variables and universe as
$\mathcal{I}$, but with a different set of constraints $\mathcal{C}^{\prime}$,
as follows. Firstly, for each vertex $v$ of $T$, we create a constraint
$(S_{v},r_{v})$, where $S_{v}=\chi(v)$ and $r_{v}=U^{|\chi(v)|}$. Then, for
every constraint $(S,r)\in\mathcal{C}$ of the original problem such that
$S\subseteq\chi(v)$, we eliminate from $r_{v}$ all those tuples that do not
match with $r$. The resulting constraint is then added to
$\mathcal{C}^{\prime}$. It can be shown that $\mathcal{I}^{\prime}$ has the
same solutions as $\mathcal{I}$, and that it is acyclic. In fact, observe
that, by construction, $\langle T,\chi\rangle$ is a join tree of the
hypergraph ${\cal H}(\mathcal{I}^{\prime})$ associated with
$\mathcal{I}^{\prime}$, because of the connectedness condition of tree
decompositions. Furthermore, building $\mathcal{I}^{\prime}$ from
$\mathcal{I}$ is feasible in $O(n\times|U|^{k+1})$ where $n$ is the number of
vertices in $T$, and where the size of the largest constraint relation in the
resulting instance is $|U|^{k+1}$. Since one can always consider only tree
decompositions whose number of vertices is bounded by the number of variables
of the problem (i.e., the nodes of the graph), it follows that deciding
whether $\mathcal{I}^{\prime}$ (and hence $\mathcal{I}$) is satisfiable is
feasible in $O(|Var|\times|U|^{k+1}\times\log|U|^{k+1})$. In fact, as for
acyclic instances, even in this case we may compute also solutions for
$\mathcal{I}$ with a backtrack-free search, after the preprocessing of the
instance performed according to the given tree decomposition (i.e., according
to the join tree of the equivalent acyclic instance). As a consequence, all
classes of CSP instances (with primal graphs) having bounded treewidth may be
solved in polynomial time, even if with an exponential dependency on the
treewidth.
Clearly enough, this technique is not very useful for CSP instances with large
constraint scopes. In particular, the class of CSP instances whose associated
constraint hypergraphs are acyclic are not tractable according to tree
decompositions, because acyclic hypergraphs may have unbounded treewidth.
Intuitively, in the primal graph all variables occurring in the same
constraint scope are connected to each other, and thus they lead to a clique
in the graph. It follows that CSP instances having constraint scopes with
large arities have large treewidths, too, because the treewidth of a clique of
$n$ nodes is $n-1$. As an example, Figure 2 reports the graph ${\it G}({\cal
H}_{1})$ associated with the acyclic hypergraph ${\cal H}_{1}$, where one may
notice how the hyperedge $\\{A,C,D,E,F,G,H\\}$ is flattened into a clique over
all its variables.
### 3.2 Hypertree Decompositions
Let us now turn our attention to hypergraph based decompositions. Such
decompositions are similar to tree decompositions, but they use an additional
covering of each set $\chi(p)$ with as few as possible hyperedges. The width
is then no longer defined as the maximum cardinality of $\chi(p)$ over all
decomposition nodes $p$, but as the maximum number of hyperedges used to cover
$\chi(p)$. Intuitively, this notion of width is better, because it will allow
us to expresses more accurately the computational effort needed to transform
an instance into an acyclic one.
###### Definition 2 ([19])
A generalized hypertree decomposition of a hypergraph ${\cal H}$ is a triple
$H\\!D=\langle T,\chi,\lambda\rangle$, where $\langle T,\chi\rangle$ is a tree
decomposition of the primal graph of ${\cal H}$, and $\lambda$ is a labelling
of the tree $T$ by sets of hyperedges of ${\cal H}$ such that, for each vertex
$p\in vertices(T)$, $\chi(p)\subseteq\bigcup_{h\in\lambda(v)}h$. That is, all
variables in the $\chi$ labeling are covered by hyperedges (scopes) in the
$\lambda$ labeling. The width of $H\\!D$ is the number $\max_{p\in
vertices(T)}(|\lambda(p)|)$. The generalized hypertree width of ${\cal H}$,
denoted by $ghw({\cal H})$, is the minimum width over all its generalized
hypertree decompositions. If $I$ is a CSP instance then $ghw(I):=ghw({{\cal
H}}(I))$. $\Box$
Clearly, for each CSP instance $I$, $ghw(I)\leq tw(I)$. Moreover, there are
classes of CSPs having unbounded treewidth whose generalized hypertree width
is bounded[19].
Finding a suitable tree decomposition whose sets $\chi(p)$ may each be covered
with a few hyperedges seems to be quite a hard task even in case we have some
fixed upper bound $k$. Indeed, it has been shown that deciding whether
$ghw({\cal H})\leq k$ is NP-complete (for any fixed $k\geq 3$) [20].
Fortunately, since its first proposal in [18], this notion comes with a
tractable variant, called hypertree decomposition, whose associated width is
at most 3 times (+1) larger than the generalized hypertree width [1]. As a
consequence, it can be shown that every class of CSPs that is tractable
according to generalized hypertree width is tractable according to hypertree
width, as well.
###### Definition 3 ([18])
A hypertree decomposition of a hypergraph ${\cal H}$ is a generalized
hypertree decomposition $H\\!D=\langle T,\chi,\lambda\rangle$ that satisfies
the following additional condition, called Descendant Condition or also
special condition: $\forall p\in vertices(T)$, $\forall h\in\lambda(p)$,
$h\cap\chi(T_{p})\;\subseteq\;\chi(p)$, where $T_{p}$ denotes the subtree of
$T$ rooted at $p$, and $\chi(T_{p})$ the set of all variables occurring in the
$\chi$ labeling of this subtree.
The hypertree width $hw({\cal H})$ of ${\cal H}$ is the minimum width over all
its hypertree decompositions. $\Box$
As an example, on the right part of Figure 1 a hypertree decomposition of the
hypergraph ${\cal H}_{cp}$ in Example 1 is reported. Note that this
decomposition has width 2.
We refer the interested reader to [18, 29] for more details about this notion
and in particular about the descendant condition. Here, we just observe that
the notions of hypertree width and generalized hypertree width are true
generalizations of acyclicity, as the acyclic hypergraphs are precisely those
hypergraphs having hypertree width and generalized hypertree width one. In
particular, the classes of CSP instances having bounded (generalized)
hypertree width have the same desirable computational properties as acyclic
CSPs [16]. Indeed, from a CSP instance $\mathcal{I}=\langle{\it
Var},U,\mathcal{C}\rangle$ and a (generalized) hypertree decomposition $H\\!D$
of ${\cal H}(\mathcal{I})$ of width $k$, we may build an acyclic CSP instance
$\mathcal{I}^{\prime}=\langle{\it Var},U,\mathcal{C}^{\prime}\rangle$ with the
same solutions as $\mathcal{I}$. The overall cost of deciding whether
$\mathcal{I}$ is satisfiable is in this case $O((m-1)\times
r_{max}^{k}\times\log r_{max}^{k})$, where $r_{max}$ denotes the size of the
largest constraint relation and $m$ is the number of vertices of the
decomposition tree, with $m\leq|{\it Var}|$ (in that we may always find
decompositions in a suitable normal form without redundancies, so that the
number of vertices in the tree cannot exceed the number of variables of the
given instance). To be complete, if the input consists of $\mathcal{I}$ only,
we have to compute the decomposition, too. This can be done with a guaranteed
polynomial-time upper bound in the case of hypertree decompositions [18].
In the following two sections, we provide some tractability results for
optimization problems. For the sake of presentation, we give algorithms for
the acyclic case, provided that these results may be clearly extended to any
class of instances having bounded (generalized) hypertree width, after the
above mentioned polynomial-time transformation.
## 4 Optimization Problems over CSP Solutions
In this section, we consider optimization problems where an assignment has to
be singled out that satisfies all the constraints of the underlying CSP
instance and that has minimum total cost; in other words, we look for a “best”
solution among all the possible solutions. In particular, below, we shall
firstly address the case where each possible variable-value mapping is
associated with a cost (also called constraint satisfaction optimization
problem); then we shall consider the case where costs are defined over the
constraints tuples (weighted CSP).
### 4.1 Constraint Satisfaction Optimization Problems
An instance of a constraint satisfaction optimization problem (CSOP) consists
of a pair $\langle\mathcal{I},{w}\rangle$, where $\mathcal{I}=\langle{\it
Var},U,\mathcal{C}\rangle$ is a CSP instance and where ${w}:{\it Var}\times
U\mapsto\mathbb{Q}$ is a function mapping substitutions for individual
variables to rational numbers. For a substitution
$\\{X_{1}/u_{1},...,X_{n}/u_{n}\\}$, we denote by
${w}(\\{X_{1}/u_{1},...,X_{n}/u_{n}\\})$ the value
$\sum_{i=1}^{n}{w}(X_{i},u_{i})$. Then, a solution to a CSOP instance
$\langle\mathcal{I},{w}\rangle$ is a solution $\theta$ to $\mathcal{I}$ such
that ${w}(\theta)\leq{w}(\theta^{\prime})$, for each solution
$\theta^{\prime}$ to $\mathcal{I}$. Details on this framework can be found,
e.g., in [32].
Input: An acyclic CSOP instance $\langle\mathcal{I},{w}\rangle$ with
$\mathcal{I}=\langle{\it Var},U,\mathcal{C}\rangle$,
$\mathcal{C}=\\{(S_{1},r_{1}),...,(S_{q},r_{q})\\}$, and a join tree $T=(N,E)$
of the hypergraph ${\cal H}(\mathcal{I})$; Output: A solution to
$\langle\mathcal{I},{w}\rangle$; var ${t}^{*}:{\it Var}\mapsto U$;
$\ell_{{{t}_{v}}}^{v}:\mbox{rational number, for each tuple }{t}_{v}\in
r_{v}$; ${t}_{{t}_{v},c}:\mbox{tuple in }r_{c}$, for each tuple ${t}_{v}\in
r_{v}$, and for each $(v,c)\in E$; ————————————————————————————————————————–
Procedure $BottomUp$; begin $Done:=$ the set of all the leaves of $T$; while
$\exists v\in T$ such that (i) $v\not\in Done$, and (ii) $\\{c\mid c\mbox{ is
child of }v\\}\subseteq Done$ do $r_{v}:=r_{v}-\\{{t}_{v}\mid\exists(v,c)\in
E\mbox{ such that }\forall{t}_{c}\in\theta_{c},\ {t}_{v}\not\approx\
{t}_{c}\\}$; if $r_{v}=\emptyset$ then EXIT; (* $\mathcal{I}$ is not
satisfiable *) for each ${t}_{v}\in r_{v}$ do
$\ell_{{t}_{v}}^{v}:=w({t}_{v})$; for each $c$ such that $(v,c)\in E$ do
$\bar{t}_{c}:=\arg\min_{{t}_{c}\in r_{c}\mid{t}_{v}\approx\
{t}_{c}}\left(\ell_{{t}_{c}}^{c}-w({t}_{c}\cap{t}_{v})\right);$
${t}_{{t}_{v},c}:=\bar{t}_{c}$; (* set best solution *)
$\ell_{{t}_{v}}^{v}:=\ell_{{t}_{v}}^{v}+\ell_{\bar{t}_{c}}^{c}-w(\bar{t}_{c}\cap{t}_{v})$;
end for end for $Done:=Done\cup\\{v\\}$; end while end;
————————————————————————————————————————– begin (* MAIN *) $BottomUp$; let $r$
be the root of $T$; $\bar{t}_{r}:=\arg\min_{{t}_{r}\in
r_{r}}\ell_{{t}_{r}}^{r}$; ${t}^{*}:=\bar{t}_{r}$; (* include solution *)
$TopDown(r,\bar{t}_{r})$; return ${t}^{*}$; end. Procedure
$TopDown(v:\mbox{vertex of }N$, ${t}_{v}\in r_{v}$); begin for each $c\in N$
s.t. $(v,c)\in E$ do $\bar{t}_{c}:={t}_{{t}_{v},c}$;
${t}^{*}:={t}^{*}\cup\bar{t}_{c}$; (* include solution *)
$TopDown(c,\bar{t}_{c}$); end for end;
Figure 3: Algorithm ComputeOptimalSolution.
Constraint satisfaction optimization problems naturally arise in various
application contexts. As an example they have recently been used in the
context of combinatorial auctions [13], in order to model and solve the
_winner determination problem_ of determining the allocation of the items
among the bidders that maximizes the sum of the accepted bid prices. In
particular, in [13], it has been observed that CSOPs and, in particular, the
winner determination problem, can be solved in polynomial time on some classes
of acyclic instances via a dynamic programming algorithm founded on the ideas
of [34]. This algorithm, named ComputeOptimalSolution, is reported in Figure 3
and will be briefly illustrated in the following.
The algorithm receives in input the instance $\langle\mathcal{I},{w}\rangle$
and a join tree $T=(N,E)$ for ${\cal H}(\mathcal{I})$. Recall that each vertex
$v\in N$ corresponds to a hyperedge of ${\cal H}(\mathcal{I})$ and, in its
turn, to a constraint in $\mathcal{C}$; hence, we shall simply denote by
$(S_{v},r_{v})$ the constraint in $\mathcal{C}$ univocally associated with
vertex $v$.
Based on $\langle\mathcal{I},{w}\rangle$ and $T$, ComputeOptimalSolution
computes an optimal solution (or checks that there is no solution) by looking
for the “conformance” of the tuples in each relation $r_{v}$ with the tuples
in $r_{c}$, for each child $c$ of $v$ in $T$, where ${t}_{v}\in r_{v}$ is said
to _conform_ with ${t}_{c}\in r_{c}$, denoted by ${t}_{v}\approx\ {t}_{c}$, if
for each $X\in S_{v}\cap S_{c}$, $X/u\in{t}_{v}\Leftrightarrow X/u\in{t}_{c}$.
In more detail, ComputeOptimalSolution solves $\langle\mathcal{I},{w}\rangle$
by traversing $T$ in two phases. First, vertices of $T$ are processed from the
leaves to the root $r$, by means of the procedure $BottomUp$ that updates the
weight $\ell_{{{t}_{v}}}^{v}$ of the current vertex $v$. Intuitively,
$\ell_{{{t}_{v}}}^{v}$ stores the cost of the best partial solution for
$\mathcal{I}$ computed by using only the variables occurring in the subtree
rooted at $v$. Indeed, if $v$ is a leaf, then
$\ell_{{{t}_{v}}}^{v}=w({t}_{v})$. Otherwise, for each child $c$ of $v$ in
$T$, $\ell_{{{t}_{v}}}^{v}$ is updated by adding the minimum value
$\ell_{{t}_{c}}^{c}-w({t}_{c}\cap{t}_{v})$ over all tuples ${t}_{c}$
conforming with ${t}_{v}$. The tuple $\bar{t}_{c}$ for which this minimum is
achieved is stored in the variable ${t}_{{t}_{v},c}$ (resolving ties
arbitrarily). Note that if this process cannot be completed, because there is
no tuple in $r_{v}$ conforming with some tuple in each relation associated
with the children of $v$, then we may conclude that $\mathcal{I}$ is not does
not admit any solution. Otherwise, after the root $r\in N$ is reached, this
part ends, and the top-down phase may start.
In this second phase, the tree $T$ is processed starting from the root.
Firstly, the assignment ${t}^{*}$ is defined as the tuple in $r_{r}$ with the
minimum cost over all the tuples in $r_{r}$ (again, resolving ties
arbitrarily). Then, procedure $TopDown$ extends ${t}^{*}$ with a tuple for
each vertex of $T$: at each vertex $v$ and for each child $c$ of $v$,
${t}^{*}$ is extended with the tuple ${t}_{{t}_{v},c}$ resulting from the
bottom-up phase.
Being based on a standard dynamic programming scheme, correctness of
ComputeOptimalSolution can be shown by structural induction on the subtrees of
$T$ [13]. Moreover, by analyzing its running time, one may note that dealing
with cost functions does not (asymptotically) provide any overhead w.r.t.
Yannakakis’s algorithm [34] for plain CSPs. Following [13], the following can
be shown for the more general case of CSOP instances having bounded
generalized hypertree-width hypergraphs.111In all complexity results, we
assume the weighting function ${w}$ be explicitly listed in the input
(otherwise, just add the cost of computing through ${w}$ all cost values for
the variable assignments of the given input instance).
###### Theorem 4.1
Let $\langle\mathcal{I},{w}\rangle$ be a CSOP instance and $H\\!D$ a
(generalized) hypertree decomposition of ${\cal H}(\mathcal{I})$. Moreover,
let $k$ be the width of $H\\!D$ and $m$ be the number of vertices in its
decomposition tree. Then, a solution to $\langle\mathcal{I},{w}\rangle$ can be
computed (or it is discovered that no solution exists) in time $O((m-1)\times
r_{max}^{k}\times\log r_{max}^{k})$, where $r_{max}$ is the size of the
largest constraint relation in $\mathcal{I}$.
### 4.2 Weighted CSPs: Costs over Tuples
Let us now turn to study a slight variation of the above scenario, where costs
are associated with each tuple of the constraint relations, rather than with
substitutions for individual variables. In fact, this is the setting of
weighted CSPs, a well-known specialization of the more general _valued_ CSP
framework [30].
Formally, a weighted CSP (WCSP) instance consists of a tuple
$\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle$, where
$\mathcal{I}=\langle{\it Var},U,\mathcal{C}\rangle$ with
$\mathcal{C}=\\{C_{1},C_{2},\ldots,C_{q}\\}$ is a CSP instance, and where, for
each tuple $t_{v}\in r_{v}$, ${w}_{v}(t_{v})\in\mathbb{Q}$ denotes the cost
associated with $t_{v}$. For a solution $\theta=t_{1}\cup...\cup t_{q}$ to
$\mathcal{I}$, we define ${w}(\theta)=\sum_{v=1}^{q}{w}_{v}(t_{v})$ as its
associated cost. Then, a solution to
$\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle$ is a solution $\theta$ to
$\mathcal{I}$ such that ${w}(\theta)\leq{w}(\theta^{\prime})$, for each
solution $\theta^{\prime}$ to $\mathcal{I}$.
A few tractability results for WCSPs (actually, for valued CSPs) are known in
the literature when structural restrictions are considered over binary
encodings of the constraint hypergraphs. Indeed, it has been observed that
WCSPs are tractable when restricted on classes of instances whose associated
primal graphs are acyclic or nearly-acyclic (see, e.g., [33, 10, 26]).
However, the primal graph obscures much of the structure of the underlying
hypergraph since, for instance, each hyperedge is turned into a clique
there—see the discussion in Section 3.
Therefore, whenever constraints have large arities, tractability results for
primal graphs are useless, and it becomes then natural to ask whether
polynomial-time solvability still holds when moving from (nearly-)acyclic
primal graphs to acyclic hypergraphs, possibly associated with very intricate
primal graphs. Next, we shall positively answer this question, by simply
recasting weighted CSPs as constraint optimization problems, and by
subsequently solving them via the algorithm ComputeOptimalSolution. To this
end, given a WCSP instance $\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle$, we
define its associated CSOP instance, denoted by
${\mbox{CSOP}}(\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle)$, as the pair
$\langle\mathcal{I}^{\prime},w^{\prime}\rangle$ with
$\mathcal{I}^{\prime}=\langle{\it
Var}^{\prime},U^{\prime},\mathcal{C}^{\prime}\rangle$ such that:
* $\bullet$
${\it Var}^{\prime}={{\it Var}\cup\\{D_{1},...,D_{q}\\}}$, where each $D_{v}$
is a fresh auxiliary variable in $\mathcal{I}^{\prime}$;
* $\bullet$
$U^{\prime}=U\cup\bigcup_{v=1}^{q}\bigcup_{t_{v}\in r_{v}}\\{u_{t_{v}}\\}$,
i.e., for each constraint $(S_{v},r_{v})\in\mathcal{C}$, $U^{\prime}$ contains
a fresh value for each tuple in $r_{v}$—intuitively, mapping the variable
$D_{v}$ to $u_{t_{v}}$ encodes that the tuple $t_{v}$ is going to contribute
to a solution for $\mathcal{I}$;
* $\bullet$
$\mathcal{C}^{\prime}=\\{(S_{v}\cup\\{D_{v}\\},r_{v}^{\prime})\mid(S_{v},r_{v})\in\mathcal{C}\\}$,
where $r_{v}^{\prime}=\\{t_{v}\cup\\{D_{v}/u_{t_{v}}\\}\mid t_{v}\in
r_{v}\\}$;
* $\bullet$
$w^{\prime}(X/u)=w_{v}(t_{v})$ if $X=D_{v}$ and $u=u_{t_{v}}$, for some tuple
$t_{v}\in r_{v}$; otherwise, $w^{\prime}(X/u)=0$. That is, the whole cost of
each tuple is determined by the mapping of its associated fresh variable
$D_{v}$.
It is immediate to check that the above transformation is feasible in linear
time. In addition, the transformation enjoys two relevant preservation
properties: Firstly, it preserves the structural properties of the WCSP
instance in that ${\cal H}(\mathcal{I}^{\prime})$ is acyclic if and only if
${\cal H}(P)$ is acyclic; and secondly, it preserves its solutions, in that
$\theta^{\prime}=t_{1}^{\prime}\cup...\cup t_{q}^{\prime}$ is a solution to
$\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle$ if and only if
$\theta=t_{1}\cup...\cup t_{q}$ is a solution to
$\langle\mathcal{I}^{\prime},w^{\prime}\rangle$, where
$t_{v}^{\prime}=t_{v}\cup\\{D_{v}/u_{t_{v}}\\}$ for each $1\leq v\leq q$. By
exploiting these observations and Theorem 4.1, the following can be
established.
###### Theorem 4.2
Let $\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle$ be a WCSP instance and
$H\\!D$ a (generalized) hypertree decomposition of ${\cal H}(\mathcal{I})$.
Moreover, let $k$ be the width of $H\\!D$ and $m$ be the number of vertices in
its decomposition tree. Then, a solution to
$\langle\mathcal{I},{w}_{1},...,{w}_{q}\rangle$ can be computed (or we may
state that there is no solution) in time $O((m-1)\times r_{max}^{k}\times\log
r_{max}^{k})$, where $r_{max}$ is the size of the largest constraint relation
in $\mathcal{I}$.
## 5 Minimizing the Number of Violated Constraints
In this section, we shall complete our picture by considering those scenarios
where problems might possibly be overconstrained and where, hence, the focus
is on finding assignment minimizing the total number of violated constraints.
These kinds of problems are usually referred to in the literature as Max-CSPs
[12], which similarly as WCSPs are specializations of valued CSPs.
Formally, let $\theta:{\it Var}\mapsto U$ be an assignment for a CSP instance
$\mathcal{I}=\langle{\it Var},U,\mathcal{C}\rangle$. We say that the violation
degree of $\theta$, denoted by $\delta(\theta)$, is the number of relations
$r_{v}$ such that there is no tuple $t_{v}\in r_{v}$ with
$t_{v}\subseteq\theta$. An assignment $\theta:{\it Var}\mapsto U$ is a
solution to the Max-CSP instance (associated with $\mathcal{I}$) if
$\delta(\theta)\leq\delta(\theta^{\prime})$, for each assignment
$\theta^{\prime}:{\it Var}\mapsto U$. Note that Max-CSPs instances, by
definition, do always have a solution.
### 5.1 Acyclic Instances Remain Intractable
After the tractability results established in Section 4.2 for WCSPs, one may
expect good news for Max-CSPs, too. Surprisingly, this is not the case.
###### Theorem 5.1
Solving Max-CSPs is $\rm NP$-hard, even when restricted over classes of
instances with acyclic constraint hypergraphs.
###### Proof
Consider any class $\mathcal{T}$ of CSPs instances having an NP-hard
satisfiability problem. Then, let $\mathcal{T}^{\prime}$ be a new class of
Max-CSP instances such that, for each $\mathcal{I}=\langle{\it
Var},U,\mathcal{C}\rangle\in\mathcal{T}$, $\mathcal{T}^{\prime}$ contains an
instance $\mathcal{I}^{\prime}=\langle{\it Var},U,\mathcal{C}^{\prime}\rangle$
with $\mathcal{C}^{\prime}=\mathcal{C}\cup\\{({\it Var},\emptyset)\\}$. That
is, any instance $\mathcal{I}^{\prime}\in\mathcal{T}^{\prime}$ has a
constraint over all variables with an empty constraint relation, and thus it
is not satisfiable. Moreover, because of the big hyperedge associated with
such a constraint, its hypergraph ${\cal H}(\mathcal{I}^{\prime})$ is
trivially acyclic. Also, by construction, there is an assignment for
$\mathcal{I}^{\prime}$ violating only one constraint if and only if
$\mathcal{I}$ is satisfiable. It follows that finding an assignment minimizing
the total number of violated constraints is NP-hard on the class of acyclic
instances $\mathcal{T}^{\prime}$. $\Box$
### 5.2 Incidence graphs and Tractable Cases
Given that hypergraph acyclicity and hence its generalizations are not
sufficient for guaranteeing the tractability of Max-CSPs, it makes sense to
explore acyclicity properties related to suitable graph representations. In
fact, as observed in Section 4.2, it is well-known that valued CSPs (and,
hence, Max-CSPs) are tractable over acyclic primal graphs (e.g., [33, 10,
26]). More precisely, tractability has been observed in the literature to hold
over primal graphs having bounded _treewidth_ (see Section 3). Our main result
in this section is precisely to show that tractability still holds in case the
_incidence graph_ of ${\cal H}(\mathcal{I})$ has bounded treewidth, which is a
more general condition than the bounded treewidth of primal graphs and which
can be used to establish better complexity bounds and to enlarge the class of
tractable instances [22]. The fact that the standard CSP is tractable for
instances whose incidence graphs have bounded treewidth was already shown in
[6]. We here extend this tractability result to Max-CSPs.
Recall that the incidence encoding of a hypergraph ${\cal H}$, denoted by
${\it inc}({\cal H})=(N,E)$, is the bipartite graph where
$N={\mathcal{E}}({\cal H})\cup{\mathcal{N}}({\cal H})$ and $E=\\{\
\\{h,a\\}\mid h\in{\mathcal{E}}({\cal H})\mbox{ and }a\in h)\\}$, i.e. it
contains an edge between $h$ and $a$ if and only if the variable $a$ occurs in
the hyperedge $h$. As an example, Figure 2 reports on the rightmost part the
incidence graph ${\it inc}({\cal H}_{1})$, where nodes associated with
hyperedges in ${\mathcal{E}}({\cal H}_{1})$ are depicted as black circles.
Note that the treewidth of ${\it inc}({\cal H}_{1})$ is 2, which is much
smaller than the treewidth of ${\it G}({\cal H}_{1})$. This does not happen by
chance since, for each hypergraph ${\cal H}$, it holds that $tw({\it
inc}({\cal H}))\leq tw({\it G}({\cal H}))$; in addition, there are also
classes of hypergraphs with incidence encodings of bounded treewidth and
primal encodings of unbounded treewdith (see, e.g., [22]).
While enlarging the class of instances having bounded treewidth, the incidence
encoding still conveys all the information needed to solve Max-CSP instances.
Again, the solution algorithm consists of a transformation into a suitable
CSOP instance. Formally, let $\mathcal{I}=\langle{\it
Var},U,\mathcal{C}\rangle$ be a Max-CSP instance with
$\mathcal{C}=\\{(S_{1},r_{1}),...,(S_{q},r_{q})\\}$, and let $\langle
T,\chi\rangle$ be a $k$-width tree decomposition of ${\it inc}({\cal
H}(\mathcal{I}))$—recall that for each vertex $v\in T$, $\chi(v)$ is a set of
variables (i.e., nodes of ${\mathcal{N}}({\cal H}(\mathcal{I}))$) and
constraint scopes (i..e, edges in ${\mathcal{E}}({\cal H}(\mathcal{I}))$).
Then, the constraint satisfaction optimization problem instance
${\mbox{CSOP}}(\mathcal{I},\langle T,\chi\rangle)$ is the pair
$\langle\mathcal{I}^{\prime},w^{\prime}\rangle$, where
$\mathcal{I}^{\prime}=\langle{\it
Var}^{\prime},U^{\prime},\mathcal{C}^{\prime}\rangle$ and such that:
* $\bullet$
${\it Var}^{\prime}={\it Var}\cup\\{S_{1},...,S_{q}\\}$, that is, also the
constraint scopes of $\mathcal{C}$ belong to the variables of the new problem;
* $\bullet$
$U^{\prime}=U\cup\\{\mathit{unsat}\\}\cup\\{u_{t}\mid t\in r_{i},\mbox{ for
}1\leq i\leq q\\}$;
* $\bullet$
$\mathcal{C}^{\prime}=\\{(\chi(v),r^{\prime}_{v})\mid v\in T\\}$ where the
constraint relation $r^{\prime}_{v}$ is defined as follows. Let
$\mu=|\chi(v)\cap{\it Var}|$, and let $U^{\mu}$ denote the set of all possible
tuples over the $\mu$ variables in $\chi(v)\cap{\it Var}$. Let also
$S_{i_{1}},...S_{i_{h}}$ be the scope-variables in $\chi(v)$. Then, for each
tuple $\theta\in U^{\mu}$, the relation $r^{\prime}_{v}$ contains all tuples
$\theta\cup\\{S_{i_{1}}/v_{i_{1}}\\}\cup\cdots\cup\\{S_{i_{h}}/v_{i_{h}}\\}$,
where $v_{i_{j}}\in U$ ($1\leq j\leq h$) is a value for the scope-variable
$S_{i_{j}}$ such that: $v_{i_{j}}=u_{t}$ if there is a tuple $t\in r_{i_{j}}$
conforming with $\theta$; and $v_{i_{j}}=\mathit{unsat}$, if no such a tuple
exists in $r_{i_{j}}$.
* $\bullet$
$w^{\prime}(X/u)=0$ if $u\neq\mathit{unsat}$; otherwise $w^{\prime}(X/u)=1$,
that is, each constraint of $\mathcal{C}$ that is not satisfied increases the
cost of a solution by a unitary factor.
Note that this transformation is feasible in time exponential in the width of
$\langle T,\chi\rangle$ only. Moreover, solutions of $\mathcal{I}^{\prime}$
with minimum total cost precisely correspond to assignments over $\mathcal{I}$
minimizing the total number of violated constraints. In fact, the following
can be established.
###### Theorem 5.2
Let $\mathcal{I}=\langle{\it Var},U,\mathcal{C}\rangle$ be a Max-CSP instance
with $tw({\it inc}({\cal H}(\mathcal{I})))=k$. Then, a solution to
$\mathcal{I}$ can be computed in time $O(|{\it
Var}|\times|U|^{k+1}\times\log|U|^{k+1})$.
## 6 Conclusion and Discussion
In this paper, classes of tractable CSOP, WCSP, and Max-CSP instances are
singled out by overviewing and proposing solution approaches applicable to
instances whose hypergraphs have bounded (generalized) hypertree width, or
whose incidence graphs have bounded treewidth. The techniques described in
this paper are mainly based on Algorithm ComputeOptimalSolution, which has
been designed to optimize costs expressed as rational numbers and combined via
the summation operation. However, it is easily seen that it remains correct if
costs are specified over an arbitrary totally ordered monoid structure, where
some binary operation $\oplus$ (in place of standard summation) is used in
order to combine costs, provided it is commutative, associative, closed, and
that it verifies identity and monotonicity. It follows that all tractable
classes of CSOP, WCSP, and Max-CSP instances identified in this paper remain
tractable in such extended scenarios, which indeed emerge with valued CSPs
(see, e.g., [4]).
## References
* [1] I. Adler, G. Gottlob, and M. Grohe. Hypertree-Width and Related Hypergraph Invariants. European Journal of Combinatorics, 28, pp. 2167–2181, 2007.
* [2] C. Beeri, R. Fagin, D. Maier, and M. Yannakakis. On the desirability of acyclic database schemes. Journal of the ACM, 30(3), pp. 479–513, 1983.
* [3] P.A. Bernstein and N. Goodman. The power of natural semijoins. SIAM Journal on Computing, 10(4), pp. 751–771, 1981.
* [4] S.Bistarelli, U. Montanari, F. Rossi, T. Schiex, G. Verfaillie, and H. Fargier. Semiring-Based CSPs and Valued CSPs: Frameworks, Properties,and Comparison. _Constraints_ , 4(3), pp. 199–240, 1999.
* [5] H.L. Bodlaender and F.V. Fomin. A Linear-Time Algorithm for Finding Tree Decompositions of Small Treewidth. SIAM Journal on Computing, 25(6), pp. 1305-1317, 1996.
* [6] Ch. Chekuri and A. Rajaraman. Conjunctive Query Containment Revisited. Theoretical Computer Science Vol. 239, Issue 2, 28 May 2000, pp. 211-229, 2000. (Preliminary version in Proc. ICDT’07.) In Proc. International Conference on Database Theory 1997 (ICDT’97), Delphi, Greece, Jan. 1997, Springer LNCS, Vol. 1186, pp.56–70, 1997. Full version:
* [7] D.A. Cohen, P.G. Jeavons, and M. Gyssens. A unified theory of structural tractability for constraint satisfaction problems. Journal of Computer and System Sciences, 74(5):721–743, 2008.
* [8] C. Daskalakis and C.H. Papadimitriou. Computing pure nash equilibria in graphical games via markov random fields. In Proc. of ACM EC’06, pp. 91–99, 2006.
* [9] R. Fagin. Degrees of acyclicity for hypergraphs and relational database schemes. J. ACM, 30(3):514–550, 1983.
* [10] S. de Givry, T. Schiex, and G. Verfaillie. Exploiting Tree Decomposition and Soft Local Consistency In Weighted CSP. In Proc. of AAAI’06, 2006.
* [11] R. Dechter. Constraint Processing. Morgan Kaufmann, 2003.
* [12] E.C. Freuder and R.J. Wallace. Partial Constraint Satisfaction. Artificial Intelligence, 58(1-3), pp. 21–70, 1992.
* [13] G. Gottlob and G. Greco. On the complexity of combinatorial auctions: structured item graphs and hypertree decomposition. In Proc. EC’07, pp. 152–161, 2007. [full version currently available as Technical Report, University of Calabria]
* [14] G. Gottlob, G. Greco, and F. Scarcello. Pure Nash Equilibria: Hard and Easy Games. Journal of Artificial Intelligence Research, 24, pp. 357–406, 2005.
* [15] G. Gottlob, N. Leone, and F. Scarcello. A comparison of structural CSP decomposition methods. Artificial Intelligence, 124(2), pp. 243–282, 2000.
* [16] G. Gottlob, N. Leone, and F. Scarcello. The complexity of acyclic conjunctive queries. Journal of the ACM, 48(3), pp. 431–498, 2001.
* [17] G. Gottlob, N. Leone, and F. Scarcello. Advanced parallel algorithms for processing acyclic conjunctive queries, rules, and constraints. In Proc. of SEKE’00, pp. 167–176, 2000.
* [18] G. Gottlob, N. Leone, and F. Scarcello. Hypertree decompositions and tractable queries. J. of Computer and System Sciences, 64(3), pp. 579–627, 2002.
* [19] G. Gottlob, N. Leone, and F. Scarcello. Robbers, marshals, and guards: game theoretic and logical characterizations of hypertree width. J. of Computer and System Sciences, 66(4), pp. 775–808, 2003.
* [20] G. Gottlob, Z. Miklós, and T. Schwentick. Generalized hypertree decompositions: NP-hardness and tractable variants. In Proc. of PODS’07, pp. 13-22, 2007.
* [21] G. Gottlob, R. Pichler, and F. Wei. Bounded Treewidth as a Key to Tractability of Knowledge Representation and Reasoning. In Proc. of AAAI’06.
* [22] G. Greco and F. Scarcello. Non-Binary Constraints and Optimal Dual-Graph Representations. In Proc. of IJCAI’03, pp. 227-232, 2003.
* [23] M. Grohe and D. Marx. Constraint solving via fractional edge covers. In Proc. of SODA’06, pp. 289–298, Miami, Florida, USA, 2006.
* [24] K. Kask, R. Dechter, J. Larrosa, and A. Dechter. Unifying tree decompositions for reasoning in graphical models. Artificial Intelligence, 166(1-2), pp. 165–193, 2005.
* [25] P. Meseguer, F. Rossi and T. Schiex. Soft Constraints. Handbook of Constraint Programming, Elsevier, 2006.
* [26] S. Ndiaye, P. J gou, and C. Terrioux. Extending to Soft and Preference Constraints a Framework for Solving Efficiently Structured Problems. In Proc. of ICTAI’08, pp. 299–306, 2008.
* [27] O. Reingold. Undirected ST-connectivity in log-space, Journal of the ACM, 55(4), 2008.
* [28] N. Robertson and P.D. Seymour. Graph minors III: Planar tree-width. Journal of Combinatorial Theory, Series B, 36, pp. 49- 64, 1984.
* [29] F. Scarcello, G. Gottlob, and G. Greco. Uniform Constraint Satisfaction Problems and Database Theory. In Complexity of Constraints, LNCS 5250, pp. 156- 195, Springer-Verlag, 2008.
* [30] T. Schiex, H. Fargier, and G. Verfaillie. Valued Constraint Satisfaction Problems: Hard and Easy Problems. In Proc. of IJCAI’95, pp. 631–639, 1995.
* [31] R.E. Tarjan, and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13(3):566-579, 1984.
* [32] E. Tsang. Foundations of Constraint Satisfaction. Academic Press, 1993.
* [33] C. Terrioux and P. J’egou. Bounded Backtracking for the Valued Constraint Satisfaction Problems. In Proc. of CP’03, pp. 709–723, 2003.
* [34] M. Yannakakis. Algorithms for acyclic database schemes. In Proc. of VLDB’81, pp. 82–94, 1981.
|
arxiv-papers
| 2012-09-15T16:40:19 |
2024-09-04T02:49:35.168682
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Georg Gottlob and Gianluigi Greco and Francesco Scarcello",
"submitter": "Francesco Scarcello",
"url": "https://arxiv.org/abs/1209.3419"
}
|
1209.3562
|
# Knots, Braids and First Order Logic
Siddhartha Gadgil Department of Mathematics,
Indian Institute of Science,
Bangalore 560012, India [email protected] and Prathamesh, T . V . H
###### Abstract.
Determining when two _knots_ are equivalent (more precisely _isotopic_) is a
fundamental problem in topology. Here we formulate this problem in terms of
_Predicate Calculus_ , using the formulation of knots in terms of braids and
some basic topological results.
Concretely, Knot theory is formulated in terms of a language with signature
$(\cdot,T,\equiv,1,\sigma,\bar{\sigma})$, with $\cdot$ a $2$-function, $T$ a
$1$-function, $\equiv$ a $2$-predicate and $1$, $\sigma$ and $\bar{\sigma}$
constants. We describe a _finite_ set of axioms making the language into a
(first order) theory. We show that every knot can be represented by a term $b$
in $1$, $\sigma$, $\bar{\sigma}$ and $T$, and knots represented by terms
$b_{1}$ and $b_{2}$ are equivalent if and only if $b_{1}\equiv b_{2}$.
Our formulation gives a rich class of problems in First Order Logic that are
important in Mathematics.
###### 1991 Mathematics Subject Classification:
Primary 57M99; Secondary 17B99
## 1\. Introduction
A (tame) _knot_ is a _smooth embedding_ of the circle $S^{1}$ into
$3$-dimensional Euclidean space $\mathbb{R}^{3}$, or equivalently the
$3$-sphere $S^{3}$ (which is viewed as $\mathbb{R}^{3}$ with an additional
point at infinity). We say that two knots are _isotopic_ if one can be
deformed into the other through smooth embeddings (we give a more precise
definition in Section 2. Determining when two knots are equivalent is a
fundamental problem in topology.
The goal of this paper is to translate this topological problem into a problem
in predicate calculus. We shall in fact formulate the problem of _stable
equivalence of links_ , which generalises knot equivalence, in terms of
predicate calculus. Our formulation is based on the representation of knots in
terms of braids.
We give the definitions on knots, links and stable equivalence in Section 2.
We then state the axiom system for stable equivalence of links in Section 3.
We then recall the formulation of knot theory in algebraic terms via Braids in
Section 4. We reformulate this to give a concise description of stable
equivalence of links in Section 5. This allows us to prove that our axiom
system describes knot theory in Section 6. Finally, we give the topological
background concerning braids and knots in an Appendix (the paper can be read
without this).
## 2\. Knots, Links and stable equivalence
We begin by recalling some basic definitions. We shall assume that all knots
and links are smooth to exclude _wild knots_.
###### Definition 2.1.
A knot K is defined as the image of a smooth, injective map
$h:S^{1}\rightarrow S^{3}$ so that $h^{\prime}(\theta)\neq 0$ for all
$\theta\in S^{1}$.
###### Definition 2.2.
A link $L\subset S^{3}$ is a smooth 1-dimensional submanifold of $S^{3}$ such
that each component of $L$ is a knot and there are only finitely many
components.
We shall regard two links as the same if there is an _ambient isotopy_ between
them, which is defined as follows.
###### Definition 2.3 (Ambient Isotopy).
Two links $L_{1}$ and $L_{2}$ in $S^{3}$ are said to be ambient isotopic if
there exists a smooth map $F:S^{3}\times[0,1]\rightarrow S^{3}$ such that
1. (1)
$F|_{{S^{3}}\times\\{0\\}}$ = $id|_{S^{3}}:S^{3}\rightarrow S^{3}$.
2. (2)
$F|_{S^{3}\times\\{1\\}}(L_{1})=L_{2}$.
3. (3)
$F|_{S^{3}\times\\{1\\}}$ is a diffeomorphism $\forall s\in[0,1]$.
4. (4)
F is smooth.
This gives an equivalence relation by the following well-known theorem.
###### Theorem 2.4.
Ambient Isotopy induces an equivalent relation on the set of all links.
To give a description of knot theory in terms of predicate calculus, we
introduce another equivalence relation on links which we call _stable
equivalence_.
###### Definition 2.5.
A link $L^{\prime}$ is said to be a stabilisation of a link $L$ if the
following conditions hold.
1. (1)
$L^{\prime}=L\cup L^{\prime\prime}$ with $L^{\prime\prime}$ disjoint from $L$.
2. (2)
There is a collection $\\{D_{1},D_{2},\dots,D_{n}\\}$ of disjoint, smoothly
embedded discs in $S^{3}\setminus L$ with
$L^{\prime\prime}=\bigcup\limits_{i=1}^{n}\partial D_{i}$.
###### Definition 2.6.
Two links $L_{1}$ and $L_{2}$ are said to be stably equivalent, denoted
$L_{1}\equiv L_{2}$, if there are stabilisations $L^{\prime}_{1}$ and
$L^{\prime}_{2}$ of $L_{1}$ and $L_{2}$, respectively, that are ambient
isotopic.
It is easy to see that $\equiv$ is an equivalence relation. The following
result follows from the prime decomposition theorem of Knesser-Milnor (see,
for example, [3]).
###### Theorem 2.7.
If $K_{1}$ and $K_{2}$ are knots (regarded as links), then $K_{1}\equiv K_{2}$
if and only if $K_{1}$ is ambient isotopic to $K_{2}$.
We shall denote by $\mathcal{L}$ the set of equivalence classes of links up to
stable equivalence and by $\widetilde{\mathcal{L}}$ the set of ambient isotopy
classes of links. Thus $\mathcal{L}$ is a quotient of
$\widetilde{\mathcal{L}}$.
## 3\. Axioms in First Order Logic
In this section, we list a set of axioms which enable us to describe stable
equivalence of links in terms of first order logic with equality. In later
sections we will further substantiate on why these axioms suffice. Consider a
language with signature $(\cdot,T,\equiv,1,\sigma,\bar{\sigma})$ such that
$\cdot$ is a $2$-function , T is a $1$-function, $\equiv$ is a $2$-predicate,
while $1$, $\sigma$ and $\bar{\sigma}$ are constants. The system of axioms for
the infinite braid group in this language are the following.
* •
Group Axioms (for the set of closed terms)
1. (1)
$\forall x,y,z\quad(x\cdot(y\cdot z)=((x\cdot y)\cdot z)$
2. (2)
$\forall x\quad 1\cdot x=x$
3. (3)
$\forall x\quad x\cdot 1=x$
4. (4)
$\sigma\cdot\bar{\sigma}=\bar{\sigma}\cdot\sigma=1$
* •
Shift operation
1. (1)
$\forall x,y\quad T(x\cdot y)=T(x)\cdot T(y)$
2. (2)
$T(e)=e$
* •
Braid axioms
1. (1)
$\sigma\cdot T(\sigma)\cdot\sigma=T(\sigma)\cdot\sigma\cdot T(\sigma)$
2. (2)
$\forall b\quad\sigma\cdot T^{2}(b)=T^{2}(b)\cdot\sigma$
* •
Equivalence relation
1. (1)
$\forall x\quad x\equiv x$
2. (2)
$\forall x,y\quad x\equiv y\implies y\equiv x$
3. (3)
$\forall x,y,z\quad x\equiv y\wedge y\equiv z\implies x\equiv z$
* •
Markov moves
1. (1)
$\forall x,y,z\quad y\cdot z=1\implies x\equiv y\cdot x\cdot z$
2. (2)
$\forall x\quad x\equiv\sigma\cdot T(x)$
3. (3)
$\forall x\quad x\equiv\bar{\sigma}\cdot T(x)$
Any model of these axioms will be referred to as a link model and the axioms
will be referred to as link axioms.
## 4\. Algebraic Formulation of knot theory
In this section, we recall the algebraic formulation of knots in terms of
Braids. We first recall the definition of braid groups.
###### Definition 4.1.
The braid group $B_{n}$ is the group generated by
$\sigma_{1},\sigma_{2},\dots,\sigma_{n-1}$ with the relations
1. (1)
$\sigma_{i}\cdot\sigma_{j}=\sigma_{j}\cdot\sigma_{i}$, where $1\leq i,j\leq
n-1$, $i\geq j+2$.
2. (2)
$\sigma_{i}\cdot\sigma_{i+1}\cdot\sigma_{i}=\sigma_{i+1}\cdot\sigma_{i}\cdot\sigma_{i+1}$,
where $1\leq i\leq n-2$.
Note that for $m<n$, there is a natural inclusion homomorphism $i_{m}$ from
$B_{m}$ to $B_{n}$ mapping a generator $\sigma_{i}$ of $B_{m}$ to the
generator $\sigma_{i}$ of $B_{n}$. We can thus identify elements in $B_{m}$
with elements in $B_{n}$. From now we shall refer to elements of
$\bigcup_{n\in\mathbb{N}\setminus{\\{1\\}}}B_{n}$ as braids.
Given an integer $m>1$, we associate a diagram to every generator of the braid
group $B_{m}$, as in the following diagram.
One can thus associate a diagram to every element of $B_{m}$ by defining the
diagram associated to the product of two elements as in the figure below
Given an integer $m>1$ and a braid $b\in B_{m}$, we can associate to the braid
a link $\lambda(b,m)$ by closing up the diagram associated to a braid and
smoothening the sharp edges. A more rigorous construction is provided in the
Appendix Appendix: Topological background
This gives a function $\lambda:\mathcal{B}\to\widetilde{\mathcal{L}}$ from the
set
$\mathcal{B}=\\{(b,m):b\in B_{m}\\}$
to the set of links. The following result says that all links are obtained by
this construction.
###### Theorem 4.2 (Alexander).
For every link $L$, there is an integer $m>1$ and a braid $B\in B_{m}$ so that
$L$ is ambient isotopic to $\lambda(b,m)$.
###### Remark 4.3.
For $n>m$ and $b\in B_{m}\subset B_{n}$, $\lambda(b,m)$ is not ambient
isotopic to $\lambda(b,n)$. However it is easy to see that the links are
stably equivalent.
The following lemma is immediate from the fact that a braid
$\prod_{k=1}^{m}\sigma_{i_{k}}^{\epsilon_{k}}\in B_{n}$, where $\epsilon_{k}$
is $1$ or $-1$ and $i_{k}\in\mathbb{N}$, when viewed from the other side of
the plane in which the braid lies is represented by
$\prod_{k=1}^{m}\sigma_{n-i_{k}}^{\epsilon_{k}}$.
###### Lemma 4.4.
The braid elements $\prod_{k=1}^{m}\sigma_{i_{k}}^{\epsilon_{k}}$ and
$\prod_{k=1}^{m}\sigma_{n-i_{k}}^{\epsilon_{k}}$ are associated to the same
link.
To formulate knot theory in terms of braids, we also need to know when two
braids correspond to the same link. To state the result of Markov giving such
a characterisation, consider the equivalence relation $\sim$ on the set
$\mathcal{B}$ generated by
* •
For $a,b\in B_{m}$, $(b,m)\sim(aba^{-1},m)$.
* •
For $b\in B_{m}$, $(b,m)\sim(\sigma_{m}b,m+1)$.
* •
For $b\in B_{m}$, $(b,m)\sim(\sigma^{-1}_{m}b,m+1)$.
The relation $\sim$ on the set $\mathcal{B}$ is called the Markov equivalence.
The first move corresponds to inserting a braid and its inverse below and
above an existing braid respectively. The second corresponds to adding a
strand to the right of existing braid in such a way that it crosses the strand
previously at the extreme right, below the braid. When viewed from the other
side of the plane of the braid it corresponds to shifting the braid to the
right by a position, inserting a strand in the first position in such a way
that it crosses the strand in second position below the braid.
Thus the second and third Markov moves, could be rewritten to give us the
following theorem.
###### Theorem 4.5.
The equivalence relation $\cong$ generated by the relations
1. (1)
For $a,b\in B_{m}$, $m>1$, $(b,m)\cong(aba^{-1},m)$.
2. (2)
For $i_{k}\leq m-1$,
$(\prod_{k=1}^{m}\sigma_{i_{k}}^{\epsilon_{k}},m)\cong(\sigma_{1}\prod_{k=1}^{m}\sigma_{{i_{k}}+1}^{\epsilon_{k}},m+1)$.
3. (3)
For $i_{k}\leq m-1$,
$(\prod_{k=1}^{m}\sigma_{i_{k}}^{\epsilon_{k}},m)\cong(\sigma^{-1}_{1}\prod_{k=1}^{m}\sigma_{{i_{k}}+1}^{\epsilon_{k}},m+1)$.
is the Markov equivalence
###### Theorem 4.6 (Markov).
For $i=1,2$, let $m_{i}>1$ be integers and $b_{i}\in B_{m_{i}}$. Then the
links $\lambda(b_{1},m_{1})$ and $\lambda(b_{2},m_{2})$ are isotopic if and
only if $(b_{1},m_{1})\sim(b_{2},m_{2})$.
To state and equivalent condition for stable equivalence of links, consider
the equivalence relation $\approx$ on $\mathcal{B}$ generated by
* •
For any $\beta_{1},\beta_{2}\in\mathcal{B}$ such that $\beta_{1}\sim\beta_{2}$
, $\beta_{1}\approx\beta_{2}$.
* •
For $m_{1},m_{2}\in\mathbb{N}$ such that $b\in B_{m_{1}},B_{m_{2}}$,
$(b,m_{1})\approx(b,m_{2})$
###### Lemma 4.7.
Two links are stably equivalent if and only if given
$\lambda(b_{1},m_{1})=l_{1}$ and $\lambda(b_{2},m_{2})=l_{2}$, then
$(b_{1},m_{1})\approx(b_{2},m_{2})$.
###### Proof.
The fact that $(b_{1},m_{1})\approx(b_{2},m_{2})$ implies
$\lambda(b_{1},m_{1})\equiv\lambda(b_{2},m_{2})$ follows from the Markov’s
Theorem and Remark 4.3. For the converse, let $l_{1}$ and $l_{2}$ be two links
stably equivalent to each other, such that $\lambda(b_{1},m_{1})=l_{1}$ and
$\lambda(b_{2},m_{2})=l_{2}$. It is easy to see that, for some
$k_{1},k_{2}\geq 0$, $\lambda(b_{1},m_{1}+k_{1})$ is isotopic to
$\lambda(b_{2},m_{2}+k_{2})$ (we can in fact take one of $k_{1}$ and $k_{2}$
to be $0$). Then by Markov’s theorem, $(b_{1},m_{1}+k)\sim(b_{2},m_{2})$,
hence $(b_{1},m_{1})\approx(b_{2},m_{2})$. ∎
## 5\. Stable links and Infinite braids
In the previous section, we recalled the well known formulation of knot theory
in terms of braid groups. However, to formulate in terms of predicate
calculus, we shall reformulate this in terms of a single braid group
$B_{\infty}$. We shall do this by replacing the usual formulation of Markov’s
theorem by its mirror, which is Theorem 4.5 above.
###### Definition 5.1.
The braid group $B_{\infty}$ is the group generated by the set
$\\{\sigma_{i}\\}_{i\in\mathbb{N}}$ with the relations
1. (1)
$\sigma_{i}\cdot\sigma_{j}=\sigma_{j}\cdot\sigma_{i}$, where
$i,j\in\mathbb{N},i\geq j+2$
2. (2)
$\sigma_{i}\cdot\sigma_{i+1}\cdot\sigma_{i}=\sigma_{i}\cdot\sigma_{i+1}\cdot\sigma_{i}$,
where $i\in\mathbb{N}$.
Note that each braid group $B_{n}$ can be regarded as a subset of
$B_{\infty}$. Observe that there is an injective homomorphism $T:B_{\infty}\to
B_{\infty}$, which we call the _shift_ , determined by
$T(\sigma_{i})=\sigma_{i+1}.$
Let $\sigma=\sigma_{1}$ and $\bar{\sigma}=\sigma_{1}^{-1}$. Then observe that
$\sigma_{i}=T^{i-1}(\sigma)$. Thus, the translates of $\sigma$ by the semi-
group generated by $T$ generate $B_{\infty}$.
Consider the equivalence relation $\equiv$ on the group $B_{\infty}$ generated
by
1. (1)
For $a,b\in B_{\infty}$, $aba^{-1}\equiv b$ .
2. (2)
For $b\in B_{\infty}$, $b\equiv\sigma T(b)$ .
3. (3)
For $b\in B_{\infty}$, $b\equiv\bar{\sigma}T(b)$.
We formulate stable equivalence of links in terms of $B_{\infty}$.
###### Theorem 5.2.
There is a surjective function $\Lambda:B_{\infty}\to\mathcal{L}$ so that, for
braids $b_{1},b_{2}\in B_{\infty}$, $\Lambda(b_{1})=\Lambda(b_{2})$ if and
only if $b_{1}\equiv b_{2}$.
###### Proof.
We begin by constructing the function $\Lambda$. For each $n>1$, there is a
homomorphism $f_{n}:B_{n}\to B_{\infty}$ determined by
$f_{n}(\sigma_{i})=\sigma_{i}$. The function is well defined because the braid
relations are preserved in $B_{\infty}$. These functions determine a function
$f:\mathcal{B}\to B_{\infty}$
given by
$f(b,m)=f_{m}(b)$
Observe that
$f^{-1}(b)=\\{(b,m)|b\in\ B_{m}\\}$
By lemma 4.7 , the links $\lambda(b,m_{1})$ and $\lambda(b,m_{2})$ are stably
equivalent for $m_{1}$ and $m_{2}$ such that $b\in B_{m_{1}}$ and $b\in
B_{m_{2}}$. Thus $\lambda$ and $f$ induce a function
$\Lambda:B_{\infty}\to\mathcal{L}$
given by
$\Lambda(b)=\lambda(f^{-1}(b))$
The fact that the map is surjective follows from Alexander’s Theorem. Now we
prove that $\Lambda(b_{1})=\Lambda(b_{2})\Leftrightarrow b_{1}\equiv b_{2}$.
Consider $b_{1},b_{2}\in B_{\infty}$ such that
$\Lambda(b_{1})=\Lambda(b_{2})$. Lemma 4.7 implies that for any $x\in
f^{-1}(b_{1})$ and $y\in f^{-1}(b_{2})$, $x\approx y$. In order to prove that
$\Lambda(b_{1})=\Lambda(b_{2})\implies b_{1}\equiv b_{2}$, it suffices to
prove that for $x,y\in\mathcal{B}$, $x\approx y\implies f(x)\equiv f(y)$. In
terms of the generating set of the relation ($\approx$), it translates to
proving the following
1. (1)
For $b\in B_{m_{1}},B_{m_{2}}$, $f(b,m_{1})\equiv f(b,m_{2})$
2. (2)
For $a,b\in B_{m}$, $f(b,m)\equiv f(aba^{-1},m)$.
3. (3)
For $b\in B_{m}$, $f(b,m)\equiv f(\sigma_{1}b,m+1)$
4. (4)
For $b\in B_{m}$, $f(b,m)\equiv f(\sigma_{1}^{-1}b,m+1)$.
The relation $f(b,m_{1})\equiv f(b,m_{2})$ follows from the fact that
$f(b_{1},m_{1})=f(b_{1},m_{2})$ and $\equiv$ is an equivalence relation.
$f(b,m)\equiv f(aba^{-1},m)$ follows from the first axiom of generating set of
$\equiv$. The third and fourth conditions are easy to verify.
In order to prove that $b_{1}\equiv
b_{2}\implies\Lambda(b_{1})=\Lambda(b_{2})$, from Theorem 4.5 it suffices to
check that for some $a_{1}\in f^{-1}(b_{1})$ and $a_{2}\in f^{-1}(b_{2})$,
$a_{1}\approx a_{2}$. Since Markov moves generate the equivalence relation
$\equiv$ on $B_{\infty}$ , it suffices to check that if two elements are
related to each other by a Markov move then the elements in their inverse
images are $\approx$ equivalent to each other.
Consider the first Markov move for $B_{\infty}$, $b\equiv aba^{-1}$. For an
appropriate $m$ such that $b,aba^{-1}\in B_{m}$, $(aba^{-1},m)\sim(b,m)$. Thus
from Theorem 4.5, $\lambda(aba^{-1})=\lambda(b)$. Similarly,
$(b,m)\sim(\sigma_{1}^{\pm 1}b,m+1)$. Thus $b_{1}\equiv
b_{2}\implies\Lambda(b_{1})=\Lambda(b_{2})$. ∎
## 6\. Knots as a canonical model
From the above we know that equivalence between links can be established by
checking if the corresponding elements in the braid group are related by a
sequence of moves on $B_{\infty}$ induced by the equivalence relation
$\equiv$. Now we prove that the braid group is the canonical model for the
link axioms and thus any model for these axioms can be used to distinguish
knots.
###### Theorem 6.1.
$(B_{\infty},T,\cdot,\equiv,\sigma_{1},\sigma_{1}^{-1})$ is a link model.
###### Proof.
We begin by proving that
$(B_{\infty},T,\cdot,\equiv,\sigma_{1},\sigma_{1}^{-1})$ satisfies the second
braid axiom. From the definitions of $B_{\infty}$ and $T$, it is easy to
derive that $(B_{\infty},T,\cdot,\equiv,\sigma_{1},\sigma_{1}^{-1})$ satisfies
the other link axioms. Consider an arbitrarily chosen element $b\in
B_{\infty}$. Since $B_{\infty}$ is generated by the set
$\\{\sigma_{i}\\}_{i\in\mathbb{N}}$, $b$ can be represented in terms of
generators as $\prod_{k=1}^{n}{\sigma_{i_{k}}^{\epsilon_{k}}}$, where
$i_{k}\in\mathbb{N}$ and $\epsilon_{k}$ is 1 or -1. This leads us to the
following equality
$\sigma_{1}\cdot
T^{2}(b)=\sigma_{1}\cdot\prod_{k=1}^{n}{\sigma_{i_{k}+2}^{\epsilon_{k}}}$
From the definition of $B_{\infty}$, we know that $\sigma_{i}$ and
$\sigma_{j}$ commute with respect to the product operation if $|i-j|\geq 2$.
This implies that for $i_{k}\in\mathbb{N}$,
$\sigma_{1}\cdot{\sigma_{{i_{k}}+2}^{\epsilon_{k}}}={\sigma_{i_{k}+2}^{\epsilon_{k}}}\cdot\sigma_{1}$
which further implies that
$\sigma_{1}\cdot\prod_{k=1}^{n}{\sigma_{{i_{k}}+2}^{\epsilon_{k}}}=\prod_{k=1}^{n}{\sigma_{i_{k}+2}^{\epsilon_{k}}}\cdot\sigma_{1}$
. Since $b$ was an arbitrarily chosen element of $B_{\infty}$, it follows that
$\sigma\cdot T^{2}(b)=T^{2}(b)\cdot\sigma\quad\forall b\in B_{\infty}$
. ∎
###### Definition 6.2 (Canonical Model).
For any signature $L$ and a set of sentences $\mathbb{T}$ in the language $L$,
a structure $A$ is said to be a _canonical model_ if
* •
$A\models\mathbb{T}$
* •
Every element of $A$ is of the form $t^{A}$, where $t$ is a closed term of
$L$.
* •
If $B$ is an $L$-structure and $B\models\mathbb{T}$, there is a unique
homomorphism of structures $f:A\rightarrow B$.
Now we prove that $(B_{\infty},T,\cdot,\equiv,1,\sigma_{1},\sigma_{1}^{-1})$
is a canonical model for the link axioms. In order to do so, consider a model
of the link axioms
$(S,T_{1},*,\equiv^{\prime},1^{\prime},\sigma_{1}^{\prime},\bar{\sigma}_{1}^{\prime})$
and a map $f:B_{\infty}\rightarrow S$ such that
$\displaystyle f(1)$ $\displaystyle=1^{\prime}$ $\displaystyle f(\sigma_{1})$
$\displaystyle=\sigma_{1}^{\prime}$ $\displaystyle f(\sigma_{i})$
$\displaystyle=T_{1}^{i-1}(\sigma_{1}^{\prime})$ $\displaystyle f(b_{1}\cdot
b_{2})$ $\displaystyle=f(b_{1})*f(b_{2})$
The last condition suffices to extend the function to $B_{\infty}$ using the
image on the generating set. However it remains to be proved that it is well
defined.
###### Lemma 6.3.
The map $f:B_{\infty}\rightarrow S$ is well defined.
###### Proof.
In order to prove that the map is well defined, it suffices to prove that
1. (1)
$T_{1}^{i}(\sigma_{1}^{\prime})*T_{1}^{j}(\sigma_{1}^{\prime})=T_{1}^{j}(\sigma_{1}^{\prime})*T_{1}^{i}(\sigma_{1}^{\prime})$,
For $i,j$ such that $j\geq i+2$
2. (2)
$T_{1}^{i}(\sigma_{1}^{\prime})*T_{1}^{i+1}(\sigma_{1}^{\prime})*T_{1}^{i}(\sigma_{1}^{\prime})$
=
$T_{1}^{i+1}(\sigma_{1}^{\prime})*T_{1}^{i}(\sigma_{1}^{\prime})*T_{1}^{i+1}(\sigma_{1}^{\prime})$
Since
$(S,T_{1},*,\equiv^{\prime},1^{\prime}\sigma_{1}^{\prime},\bar{\sigma}_{1}^{\prime})$
is a model of the link axioms,
$\sigma_{1}^{\prime}\cdot
T^{2}(b)=T^{2}(b)\cdot\sigma_{1}^{\prime},\quad\forall b\in S$
Consider two arbitrarily chosen natural numbers $i$ and $j$ such that
$j-i-2\geq 0$. If we substitute $T_{1}^{j-i-2}(\sigma_{1}^{\prime})$ for $b$
we get,
$\sigma_{1}^{\prime}\cdot
T_{1}^{2}(T_{1}^{j-i-2}(\sigma_{1}^{\prime}))=T_{1}^{2}(T_{1}^{j-i-2}(\sigma_{1}^{\prime}))\cdot\sigma_{1}^{\prime}$
Which further leads to the equality,
$T_{1}^{i}(\sigma_{1}^{\prime}\cdot(T_{1}^{j-i}(\sigma_{1}^{\prime}))=T_{1}^{i}((T_{1}^{j-i}(\sigma_{1}^{\prime}))\cdot\sigma_{1}^{\prime})$
From the homomorphism action we get,
$T_{1}^{i}(\sigma_{1}^{\prime})\cdot
T_{1}^{i}(T_{1}^{j-i}(\sigma_{1}^{\prime}))=T_{1}^{i}(T_{1}^{j-i}(\sigma_{1}^{\prime}))\cdot
T_{1}^{i}(\sigma_{1}^{\prime})$ $T_{1}^{i}(\sigma_{1}^{\prime})\cdot
T_{1}^{j}(\sigma_{1}^{\prime})=T_{1}^{j}(\sigma_{1}^{\prime})\cdot
T_{1}^{i}(\sigma_{1}^{\prime})$
Thus condition (1) holds. From the first braid axiom it follows that,
$\sigma_{1}^{\prime}\cdot
T_{1}(\sigma_{1}^{\prime})\cdot\sigma_{1}^{\prime}=T_{1}(\sigma_{1}^{\prime})\cdot\sigma_{1}^{\prime}\cdot
T_{1}(\sigma_{1}^{\prime})$
Given $i\in\mathbb{N}$, $T_{1}^{i}$ is a homomorphism. By applying $T_{1}^{i}$
to the both the sides of the above equation,
$T_{1}^{i}(\sigma_{1}^{\prime})*T_{1}^{i+1}(\sigma_{1}^{\prime})*T_{1}^{i}(\sigma_{1}^{\prime})=T_{1}^{i+1}(\sigma_{1}^{\prime})*T_{1}^{i}(\sigma_{1}^{\prime})*T_{1}^{i+1}(\sigma_{1}^{\prime})$
Thus $f$ is well defined. ∎
###### Theorem 6.4.
$(B_{\infty},T,\cdot,\equiv,1,\sigma_{1},\sigma_{1}^{-1})$ is a canonical
model for link axioms.
###### Proof.
From Theorem 6.1, we know that
$(B_{\infty},T,\cdot,\equiv,\sigma_{1},\sigma_{1}^{-1})$ is a model of the
link axioms and thus the first axiom in the definition of a canonical model
holds true. Since every element in $B_{\infty}$ is of the form
$\prod_{k=1}^{n}T^{i_{k}}(\sigma_{1}^{\epsilon_{k}})$, where
$i_{k}\in\mathbb{N}\cup\\{0\\}$ and $\epsilon_{k}$ is 1 or -1, it follows that
every element is a closed term of $L$. Let
$\prod_{k=1}^{n}{\sigma_{i_{k}}^{\epsilon_{k}}}$ be an arbitrarily chosen
element of $B_{\infty}$. From the definition of $T$ it follows that
$\displaystyle f\circ T(\prod_{k=1}^{n}{\sigma_{i_{k}}^{\epsilon_{k}}})$
$\displaystyle=f(\prod_{k=1}^{n}{\sigma_{i_{k}+1}^{\epsilon_{k}}})$
Since $f$ is a group homomorphism,
$\displaystyle f(\prod_{k=1}^{n}{\sigma_{i_{k}+1}^{\epsilon_{k}}})$
$\displaystyle=\prod_{k=1}^{n}f({\sigma_{i_{k}+1}^{\epsilon_{k}}})$
$\displaystyle=\prod_{k=1}^{n}T^{{i_{k}}+1}_{1}(\sigma_{1}^{\prime\epsilon_{k}})$
$\displaystyle=\prod_{k=1}^{n}T_{1}\circ
T_{1}^{i_{k}}({\sigma_{1}^{\prime\epsilon_{k}}})$ $\displaystyle=T_{1}\circ
f(\prod_{k=1}^{n}{\sigma_{i_{k}}^{\epsilon_{k}}})$
Which implies that
$\displaystyle f\circ T$ $\displaystyle=T_{1}\circ f$
In order to prove that $f$ is a homomorphism of models, it now suffices to
prove that if two elements are related by any of the generating relations of
the stable equivalence, then their images under $f$ are related to each other.
Since
$(S,T_{1},*,\equiv^{\prime},1,\sigma^{\prime}_{1},\bar{\sigma}_{1}^{\prime})$
satisfies the Markov move axioms and $f$ preserves multiplication and
identity,
$x,y,z,\in B_{\infty},((y\cdot z=1)\implies(f(y\cdot x\cdot
z)\equiv^{\prime}f(x)))$
From the equality it follows that $f\circ T=T_{1}\circ f$,
$\sigma^{\prime}_{1}*T_{1}(f(x))=f(\sigma_{1}\cdot T(f(x)))$
and
$\bar{\sigma}^{\prime}_{1}*T_{1}(f(x))=f(\sigma_{1}^{-1}\cdot T(f(x)))$
hence it follows that
$f(x)\equiv^{\prime}f(\sigma_{1}\cdot T(f(x)))$
$f(x)\equiv^{\prime}f(\sigma_{1}^{-1}\cdot T(f(x)))$
The uniqueness of $f$ remains to be proved. Any homomorphism $g$ between the
given models maps $\sigma_{1}$ to $\sigma_{1}^{\prime}$ and has to satisfy the
condition $g\circ T$ = $T_{1}\circ g$. By applying induction, it follows that
$g\circ T^{i}(\sigma_{1})=T^{i}(g(\sigma_{1}))=T^{i}(f(\sigma_{1}))$ for every
$i\in\mathbb{N}$. Thus for any given homomorphism $g$ and any given
$i\in\mathbb{N}$, image of $g(\sigma_{i})$ is $T_{1}^{i}(f(\sigma_{1}))$.
Since the elements of the set $\\{\sigma_{i}\\}_{i\in\mathbb{N}}$ generate
$B_{\infty}$, it follows that $g=f$. ∎
## Appendix: Topological background
### 6.1. Knots and Braids
In order to understand how the above axioms constitute a model of the knots,
we introduce the following conceptual apparatus through which knots can be
reduced to braid groups.
###### Definition 6.5 (Braid Diagrams).
A braid diagram is a set $D\subset\mathbb{R}\times I$ split into topological
intervals called the strands of D such that
1. (1)
The projection $\mathbb{R}\times I\rightarrow I$ maps each strand
homeomorphically onto I.
2. (2)
Every point of $\mathbb{N}\times\\{0,1\\}$ is an end point of a unique strand.
3. (3)
Every point of $\mathbb{R}\times I$ belongs to at most two strands. At each
intersection point of two strands, these strands meet transversally. At every
intersection, one of the intersecting strands is labelled ’undergoing’ and the
other strand is labelled ’overgoing’.
###### Remark 6.6.
Transversality in condition 3 means that in a neighbourhood of a crossing, up
to homeomorphism, D is like the set $\\{(x,y)|xy=0\\}$ . Compactness of
strands and condition(iii) ensures that the number of double points are
finite.
Two braid diagrams $D$ and $D^{\prime}$ are said to be isotopic if there is a
continuous map $F:D\times I\rightarrow\mathbb{R}\times I$ such that for each
$s\in I$, $D_{s}=F(D\times\\{s\\})\subset\mathbb{R}\times I$ is a braid
diagram preserving $D_{0}=D$ and $D_{1}=D^{\prime}$. It is understood that F
maps the crossings of D to crossings of $D_{s}$ while preserving the
information about when a strand goes under or over the other strand.An example
of braid diagram isotopy is given in the figure below.
The term braid diagram shall be here on used to denote the isotopy class of a
braid diagram. A product $*$ on the isotopy classes of braid diagrams is
defined by associating the diagram $D_{1}*D_{2}$ to the diagram obtained by
placing $D_{1}$ on top of $D_{2}$ and ’squeezing’ the resultant diagram into
$\mathbb{R}\times I$, i.e., (x, t)$\rightarrow$(x, t/2). It is easy to see
that the product is well defined and the braid diagram
$e=\cup_{k=1}^{n}\\{(k,t)|t\in[0,1]\\}$, is the identity. The product is
associative because any element of the form $(a*b)*c$ can be isotoped to
$a*(b*c)$ through a continuous map defined as follows.
$F(x,t,\theta)=\begin{cases}(1-\theta)(x,t)+\theta(x,2t),&\quad
t\in[0,1/4],\\\ (1-\theta)(x,t)+\theta(x,t+1/4),&\quad t\in[1/4,1/2],\\\
(1-\theta)(x,t)+\theta(x,(t+1)/2),&\quad t\in[1/2,1].\\\ \end{cases}$
To define a map from Braid Group $B_{n}$ to the set of n-braid diagrams
$\mathtt{B_{n}}$ upto isotopy, we define an equivalence relation on
$\mathtt{B_{n}}$ generated by a set of moves called the reidemeister moves.
###### Definition 6.7 (Reidemeister Moves).
The transformations of the braid diagrams $\Omega_{2},\Omega_{3}$ as shown in
the figure below and their inverses $\Omega_{2}^{-1},\Omega_{3}^{-1}$ are
called Reidemeister moves.
Two distinct braid diagrams are said to be R-equivalent if they are related to
each other by a sequence of Reidemeister moves.
Observe that if there exist braid diagrams a, b, c and d such that a is R-
equivalent to c and b is R-equivalent to d, $a*b$ is R-equivalent to $c*d$. It
follows that the product $*$ on $\mathtt{B_{n}}$, the isotopy classes of braid
diagrams upto R-equivalence, is well defined and $\mathtt{B}_{n}$ is a monoid.
Consider the braid diagrams $\overline{\sigma}_{i}^{+}$ and
$\overline{\sigma}_{i}^{-}$ as in the figure below.
The following result implies that the elementary braid diagrams
$\overline{\sigma}_{i}^{+}$ and $\overline{\sigma}_{i}^{-}$ generate
$\mathtt{B}_{n}$ and $(\mathtt{B}_{n},*)$ is a group.
###### Lemma 6.8.
R-equivalent classes of elementary braid diagrams $\overline{\sigma}_{i}^{+}$
and $\overline{\sigma}_{i}^{-}$ generate $\mathtt{B}_{n}$ as a monoid. Further
each element $\beta\in\mathtt{B}_{n}$ has a two sided inverse $\beta^{-1}$ in
$\mathtt{B}_{n}$.
The maps $\phi_{1_{n_{\epsilon}}}:B_{n}\to\mathtt{B}_{n}$ such that
$\phi_{1_{n_{\epsilon}}}(\sigma_{i})=\overline{\sigma}^{\epsilon}_{i}$, are
well defined because the first axiom of braid groups corresponds to isotopy of
braid diagrams and the second axiom corresponds to the second Reidemeister
move. The following result says that it is an isomorphism of groups.
###### Theorem 6.9.
$\phi_{1_{n_{\epsilon}}}$ is an isomorphism of groups.
To construct Links from braid diagrams, we define the following set of objects
in $\mathbb{R}^{2}\times I$ called Geometric Braids.
###### Definition 6.10 (Geometric Braids).
A geometric braid on $n\geq 2$ strings is a set $b\subset\mathbb{R}^{2}\times
I$ formed by the $n$ disjoint topological intervals called the strings of $b$,
such that
1. (1)
$\pi_{3}:\mathbb{R}^{2}\times I\rightarrow I$ ,i.e., the projection onto I,
maps each string homeomorphically onto I.
2. (2)
$b\cap(\mathbb{R}^{2}\times\\{0\\})=\\{(k,0,0)\\}_{k=1}^{n}$
3. (3)
$b\cap(\mathbb{R}^{2}\times\\{1\\})=\\{(k,0,1)\\}_{k=1}^{n}$
Two geometric braids $b$ and $b^{\prime}$ are said to be isotopic if there is
a continuous map, $F:b\times I\rightarrow R^{2}\times I$ such that
1. (1)
$F_{s}:b\rightarrow\mathbb{R}^{2}\times I$ sending $x\in b$ to F(x, s) is an
embedding, whose image is a geometric braid on n strings.
2. (2)
$F_{0}=id_{0}:b\rightarrow b$
3. (3)
$F_{1}(b)=b^{\prime}$
###### Theorem 6.11.
F extends to an isotopy of $\mathbb{R}^{2}\times I$ which is identity on
boundaries.
The map $i:\mathbb{R}\times I-\mathbb{R}^{2}\times I$ given by
$i(x,t)=(x,0,t)$ embeds braid diagrams in $R^{2}\times I$. Pushing the
undergoing strand in a small neighbourhood of every crossing into
$\mathbb{R}\times(0,\infty)\times I$ by appropriately increasing the second
co-ordinate of the strand while keeping the first and third constant, one
obtains a geometric n-braid. Observe that the geometric braids constructed
from isotopic braid diagrams are isotopic.
The above construction induces a map $\phi_{2_{n}}$ from the isotopy classes
of n-braid diagrams to $\mathbb{B}_{n}$, the set of geometric n-braids upto
isotopy. Projecting a geometric braid onto its first and third co-ordinates
and marking the intersecting strand with greater value of the second co-
ordinate in a neighbourhood of intersection as undercrossing and the other as
overcrossing in the neighbourhood, we obtain a braid diagram. The image of
this braid diagram under $\phi_{2_{n}}$ is isotopic to the original geometric
braid. Thus the map $\phi_{2_{n}}$ is surjective.
The following theorem makes explicit the relationship between braid diagrams
and geometric braids.
###### Theorem 6.12.
$\phi_{2_{n}}(b_{1})=\phi_{2_{n}}(b_{2})$ if and only if $b_{1}$ is
R-equivalent to $b_{2}$.
###### Corollary 6.13.
$\phi_{2_{n}}\circ\phi_{1_{n_{\epsilon}}}:B_{n}\rightarrow\mathbb{B}_{n}$ is
bijective.
If $f:(0,\infty)\to(0,1)$ is an order preserving homeomorphism, then the map
$\phi_{3_{n}}:R^{2}\times I\to D\times S^{1}$ where
$\phi_{3_{n}}(r(cos(\theta),sin(\theta)),t)=(f(r)(cos(\theta),sin(\theta)),e^{it})$
induces a well defined map from geometric $n$-braids to isotopic classes of
Links.
The map $\phi:\mathcal{B}\to L$ obtained by composition of the above maps,
$\phi(b,n)=\phi_{3_{n}}\circ\phi_{2_{n}}\circ\phi_{1_{n_{+}}}(b)$, assigns an
isotopy class of links to each braid which could be labelled as ’closing the
braid’. Markov’s Theorem and Alexander’s Theorem can thus be reformulated in
terms of $\phi$.
###### Theorem 6.14 (Alexander).
$\phi$ is surjective.
###### Theorem 6.15 (Markov).
$\phi(b_{1},m_{1})=\phi(b_{2},m_{2})$ if and only if
$(b_{1},m_{1})\sim(b_{2},m_{2})$.
## References
* [1] W. Hodges. Model theory, volume 42. Cambridge Univ Pr, 1993.
* [2] C. Kassel and V.G. Turaev. Braid groups. Springer Verlag, 2008.
* [3] L.H. Kauffman. On knots, volume 115. Princeton Univ Pr, 1987.
|
arxiv-papers
| 2012-09-17T06:49:35 |
2024-09-04T02:49:35.178548
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Siddhartha Gadgil and T. V. H. Prathamesh",
"submitter": "Siddhartha Gadgil",
"url": "https://arxiv.org/abs/1209.3562"
}
|
1209.3793
|
Polynomial Path Orders: A Maximal Model]Polynomial Path Orders: A Maximal Model
]Martin Avanzini
]Georg Moser
Institute of Computer Science
University of Innsbruck
The first author partially supported by a grant of the University of Innsbruck.
The second author is partially supported by FWF (Austrian Science Fund) project I-608-N18
This paper is concerned with the automated complexity analysis of
term rewrite systems (TRSs for short)
and the ramification of these in implicit computational complexity
theory (ICC for short).
We introduce a novel path order with multiset status, the polynomial path order $\gpop$.
Essentially relying on the principle of
predicative recursion as proposed by Bellantoni and Cook,
its distinct feature is the tight control of resources on compatible TRSs:
The (innermost) runtime complexity of compatible TRSs is polynomially bounded.
We have implemented the technique, as underpinned by our experimental evidence
our approach to the automated runtime complexity analysis is not only feasible,
but compared to existing methods incredibly fast.
As an application in the context of ICC we provide an order-theoretic
characterisation of the polytime computable functions.
To be precise, the polytime computable functions are exactly the functions
computable by an orthogonal constructor TRS compatible with .
§ INTRODUCTION
As a special form of equational logic, term rewriting has found many applications in
automated deduction and verification.
Term rewriting is a conceptually simple but powerful abstract model of
computation that underlies much of declarative programming, and
the automated time complexity analysis of term rewrite systems (TRSs for short) is of particular interest.
A natural way to measure the time complexity of a TRS $\RS$ is to measure the
length $\ell$ of derivations
f(\seq{v}) \rew[\RS] s_1 \to s_2 \cdots \rew[\RS] s_\ell = w
in terms of the sizes of the initial arguments $\seq{v}$.
Maybe surprisingly, this unitary cost model is polynomially invariant <cit.>:
the result $w$ of $f(\seq{v})$ can be computed on a conventional model of computation
in time polynomial in $\ell$.
Runtime complexity analysis is an active research area
in rewriting.
See <cit.> for a broad overview in this research field.
Since the feasible functions are often associated with the
polytime computable functions, estimating polynomial bounds is of particular interest.
Virtually all methods developed in this field go back to termination techniques.
Termination of rewrite systems has been studied extensively,
and majored to a state where it has become practical
to study the termination of real world programs by translations to rewrite systems.
Source languages cover not only functional programs (see for instance <cit.> that studies Haskell),
but also logic (c.f. <cit.> or <cit.> for Prolog programs) and
imperative programs (for Java${}^\text{\texttrademark}$ bytecode in <cit.> and recently <cit.>).
This trend is also reflected in the annual termination competition (TERMCOMP)
[<http://termcomp.uibk.ac.at/>.] that features dedicated categories for all mentioned programming languages.
Verifying that such translations are complexity preserving,
rewriting can provide a unified backend for complexity analysis of programs,
written in different languages and different paradigms.
It is clear that reduction orders, for instance polynomial interpretations
and recursive path orders not only verify termination but also bind the length of reductions.
For instance, the longest possible rewrite sequence in polynomial terminating TRSs is double-exponentially bounded in the size
of the initial term, cf. <cit.>. Similar, multiset path orders (MPO for short) induce primitive recursive
complexity <cit.>, the induced bound for the Knuth-Bendix order is two-recursive <cit.>
and for lexicographic path orders it is even multiply recursive <cit.>.
In a modern termination prover, these orders play a fundamental role in their combination with transformation
techniques like semantic labeling <cit.> and the dependency pair method <cit.>.
Based on a careful analysis of the induced derivational complexity <cit.>,
Schnabl conjectures
[t]he derivational complexity of any rewrite system that can be proven
terminating using a recent termination prover is bounded
by a multiply recursive function.
With our tool , the Tyrolean complexity tool
[$\TCT$ is open source and available from <http://cl-informatik.uibk.ac.at/software/tct>.],
we have demonstrated that a termination prover, employing only suitable miniaturised termination techniques,
can form a powerful complexity analyser. puts special focus on
proving polynomial bounds on the runtime (respectively derivational) complexity of TRSs.
However, it is worth emphasising that
the most powerful techniques for polynomial runtime complexity analysis currently available,
basically employ semantic considerations on the rewrite systems, which are notoriously
We just mention very recently work on a miniaturisation of matrix interpretations due to Middeldorp et al. <cit.>.
Recent breakthroughs in complexity analysis have also been achieved with
the development of variations of dependency pairs <cit.> as well as
modularity results <cit.>.
§.§ Motivation and Contributions
To overcome the notorious inefficiency of semantic techniques in
runtime complexity analysis we aim at a syntactic method to analyse
polynomial runtime complexity of rewrite systems.
A suitable starting point for such an analysis is given by the multiset path order $\MPO$.
$\MPO$ not only induces primitive recursive bounds on the length of derivations, it
even characterises the primitive recursive functions <cit.>:
any function computed by an $\MPO$-terminating TRS is primitive recursive, vice versa,
any primitive recursive function can be stated as an $\MPO$-terminating TRS.
It is well known that the principles of data tiering
introduced by Simmons <cit.> and Leivant <cit.>
can be used to characterise small complexity classes like $\FP$ in
a purely syntactic manner.
In particular Bellantoni and Cook <cit.> embodies the principle
of predicative recursion, a form of tiering, on the definition of the primitive recursive
functions, resulting in a recursion theoretic characterisation of $\FP$.
The here proposed polynomial path order ( for short), embodies
the principle of predicative recursion onto $\MPO$, with the
distinctive feature that induces polynomial bounds
on the length of derivations. To motivate this order,
let us first recapitulate central ideas of <cit.>.
For each function $f$, the arguments to $f$ are separated into normal
and safe ones.
To highlight this separation, we write $f(\svec{x}{y})$ where arguments to the left
of the semicolon are normal, the remaining ones are safe.
Bellantoni and Cooks define a class $\B$,
consisting of a small set of initial functions
and that is closed under safe composition and safe recursion on notation (safe recursion for brevity).
The crucial ingredient in $\B$ is that
a new function $f$ is defined via safe recursion by the equations
\begin{equation}\label{scheme:srn} \tag{\ensuremath{\mathsf{SRN}}}
\begin{array}{r@{\;}c@{\;}l}
f(\sn{0,\vec{x}}{\vec{y}}) & = & g(\sn{\vec{x}}{\vec{y}}) \\
\qquad\quad f(\sn{2z + i,\vec{x}}{\vec{y}}) & = &
h_i(\sn{z,\vec{x}}{\vec{y},f(\sn{z,\vec{x}}{\vec{y}})})\quad i \in \set{1,2}
\end{array}
\end{equation}
for functions $g,h_1$ and $h_2$ already defined in $\B$.
Unlike primitive recursive functions, the
stepping functions $h_i$ cannot perform recursion on the impredicative value $f(\sn{z,\vec{x}}{\vec{y}})$.
This is a consequence of data tiering. Recursion is performed on normal,
and recursively computed result are substituted into safe argument position.
To maintain the separation, safe composition restricts the usual composition operator
so that safe arguments are not substituted into normal argument position.
Precisely, for functions $h$, $\vec{r}$ and $\vec{s}$ already defined in $\B$,
a function $f$ is defined by safe composition using the equation
\begin{equation}\label{scheme:sc} \tag{\ensuremath{\mathsf{SC}}}
f(\sn{\vec{x}}{\vec{y}}) = h(\sn{\vec{r}(\sn{\vec{x}}{})}{\vec{s}(\sn{\vec{x}}{\vec{y}})}) \tpkt
\end{equation}
Crucially, the safe arguments $\vec{y}$ are absent in normal arguments to $h$.
The main result from <cit.> states that $\B = \FP$.
Polynomial path orders enforce safe recursion on compatible TRSs.
In order to employ the separation of normal and safe arguments,
we fix for each defined symbol
a partitioning of argument positions into normal and safe positions.
For constructors we fix that all argument positions are safe.
Moreover restricts recursion to normal argument.
Dual only safe argument positions allow the substitution of recursive calls.
Via the order constraints we can also guarantee that functions
are composed in a safe manner.
This syntactic account of predicative recursion delineates a class of rewrite systems:
a rewrite system $\RS$ is called predicative recursive if $\RS$ is
compatible with $\POPSTAR$.
For motivation consider the TRS $\RSsat$ given in Example <ref> the that encodes
the function problem $\FSAT$ associated to the well-known
satisfiability problem $\SAT$.
Notably $\FSAT$ is complete for the class of function problems over $\NP$ ($\FNP$ for short),
compare <cit.>.
The TRS $\RSsat$ is defined as follows.
A conjunctive normal form is encoded as a list of non-empty clauses,
clauses being lists of literals, in the obvious way.
Lists are constructed as usual from the constant $\nil$ and the binary constructor ($\cons$).
Literals are encoded as binary strings (build from the $\varepsilon$, $\mZ$ and $\mO$) with
the most significant bit reserved for its plurality.
The TRS $\RSsat$ contains a conditional
\begin{align*}
\mif(\sn{}{\mtrue,t,e}) & \to t & \mif(\sn{}{\mfalse,t,e}) & \to e
\end{align*}
and defines negation
\begin{align*}
\mneg(\sn{}{\mO(x)}) & \to \mZ(x) & \mneg(\sn{}{\mZ(x)}) & \to \mO(x)
\end{align*}
as well as equality:
\begin{align*}
\meq(\sn{\mZ(x)}{\mZ(y)}) & \to \meq(\sn{x}{y})
& \meq(\sn{\mZ(x)}{\mO(y)}) & \to \mfalse &
\meq(\sn{\varepsilon}{\varepsilon}) & \to \mtrue\\
\meq(\sn{\mO(x)}{\mO(y)}) & \to \meq(\sn{x}{y})
& \meq(\sn{\mO(x)}{\mZ(y)}) & \to \mfalse \tpkt
\end{align*}
A list of literals is consistent if an atom does not
occur positively and negatively.
\begin{align*}
\verify(\sn{\nil}{}) & \to \mtrue & \verify(\sn{l \cons ls}{}) & \to \mif(\sn{}{\member(\sn{\mneg(\sn{}{l}),ls}{})}, \mfalse, \verify(\sn{ls}{})) \\
\member(\sn{x,\nil}{}) & \to \mfalse & \member(\sn{x,y \cons ys}{}) & \to \mif(\sn{}{\meq(\sn{x}{y}), \mtrue, \member(\sn{x,ys}{})})
\end{align*}
The computed assignment will be a consistent list of literals.
Note that a satisfying assignment necessarily contains a literal for every clause $c$.
The following rules guess such an assignment and verify whether it is consistent.
\begin{align*}
\issat(\sn{c}{}) & \to \issat'(\sn{\guess(\sn{c}{})}{}) &
\issat'(\sn{as}{}) & \to \mif(\sn{}{\verify(\sn{as}{}),as,\unsat}) \\
\guess(\sn{\nil}{}) & \to \nil &
\guess(\sn{c \cons cs}{}) & \to \choice(\sn{c}{}) \cons \guess(\sn{cs}{}) \tpkt
\end{align*}
Here $\choice$ given by the rules
\begin{align*}
\choice(\sn{a \cons \nil}{}) & \to a &
\choice(\sn{a \cons b \cons bs}{}) & \to a &
\choice(\sn{a \cons b \cons bs}{}) & \to \choice(\sn{b \cons bs}{})
\end{align*}
selects nondeterministically an literal from a clause.
This concludes the definition of $\RSsat$.
It can be verified that $\RSsat$ is compatible with the multiset path order $\gmpo$
with underlying precedence $\qp$ satisfying
, ',
, , , , , ' , ,
Using the separation of argument positions as indicated in the rules,
where in the spirit of $\B$ constructors admit only safe arguments,
we can even prove compatibility with $\gpop$ based on the same precedence, i.e.,
$\RSsat$ is predicative recursive.
Note that $\RSsat$ does not rigidly follow
safe recursion (<ref>) and safe composition (<ref>).
Notably values are formed from an arbitrary algebra and are not restricted to words.
Also $\gpop$ allows in principle arbitrary deep right-hand sides.
Still the main principle, namely prohibition of recursion on impredicative values,
remains reflected.
In total, we establish following results.
Automated Runtime Complexity Analysis of TRSs
We establish that for predicative recursive TRSs $\RS$,
the (innermost) runtime complexity function is polynomially bounded.
To the best of our knowledge, the polynomial path order is the first purely syntactic
approach that establishes feasible bound on the runtime complexity of TRSs.
We have implemented the here proposed techniques in .
The experimental evidence obtained indicates the viability of the method.
For the predicative recursive TRS $\RSsat$ from Example <ref>
this result implies that the number of rewrite steps starting from $\issat(\sn{c}{})$ is polynomially bounded in
the size of the CNF $c$. This can even be automatically verified[
To our best knowledge $\TCT$ is currently the only complexity
tool that can provide a complexity certificate for the TRS $\RSsat$,
compare <http://termcomp.uibk.ac.at>.].
Due to the polynomial invariance theorem <cit.> we can thus
that $\FSAT$ belongs to $\FNP$.
Resource free characterisation of $\FP$
The class of predicative recursive rewrite systems
entail new order-theoretic characterisation of $\FP$, the polytime computable functions.
This bridges the gap to implicit computational complexity (ICC for short) theory.
is sound for $\FP$, i.e., (confluent and) predicative recursive TRSs
compute only polytime computable functions.
Moreover we can also prove that predicative recursive TRSs are complete for $\FP$,
in the sense that every polytime computable function $f$
is defined by a (orthogonal and) predicative recursive TRS $\RS_f$.
Parameter Substitution
We extend upon by proposing
a generalisation , admitting the same properties as outlined above,
but that allows to handle more general recursion schemes that make
use of parameter substitution.
As a corollary to this and the fact that the runtime complexity
of a TRS forms an invariant cost model we conclude a non-trivial
closure property of Bellantoni and Cooks definition of the feasible functions.
The present article collects our ongoing work on polynomial path orders.
The order $\POPSTAR$ has been introduced first in <cit.>, extended to
quasi-precedences in <cit.> and the extension $\POPSTARP$
appeared first in the Workshop on Termination of 2009 <cit.>.
Apart from the usual corrections of technicalities, we make here the following new contributions:
* The presented definition of $\POPSTAR$ is more liberal and captures predicative recursion more precisely,
compare <cit.> and Definition <ref> from Section <ref>.
* To show that $\POPSTAR$ is sound for $\FP$, we relied in <cit.>
on a certain typing of constructors that guaranteed that sizes of values are polynomial in
their depth. In particular, the typing prohibited tree structures a priori.
Our new soundness result (c.f Theorem <ref> from Section <ref>) is more general and permits arbitrary values.
* The propositional encoding used in our automation of polynomial path orders (c.f. Section <ref>)
has been considerably overhauled.
§.§ Related Work
There are several accounts of predicative analysis of recursion in the (ICC) literature.
We mention only those related works which are directly comparable to our work. See <cit.> for an
overview on ICC.
The mental predecessor of $\POPSTAR$ is the path order for $\FP$
as put forward in <cit.>.
Our main motivation lies in the observation that this order is directly only applicable to a handful of simple TRSs.
This order only gains power if addition transformations are performed.
But unfortunately powerful transformations are difficult to find automatically.
Notable the clearest connection of our work is to Marion's light multiset path order (LMPO for short) <cit.>.
This path order forms a strict extension of the here
proposed order $\POPSTAR$. Similar to $\POPSTAR$ it characterises $\FP$.
As exemplified below however,
compatible TRSs do not admit polynomially bounded runtime complexity in general.
This renders $\LMPO$ non-usable in our complexity analyser $\TCT$.
The definition of $\POPSTAR$ has been calibrated with some effort to prevent such behaviour.
The TRS $\RSbin$ is given by the following rules:
\begin{align*}
\bin(\sn{x,\Null}{}) & \to \ms(\Null) &
\bin(\sn{\Null,\ms(y)}{}) & \to \Null &
\bin(\sn{\ms(x),\ms(y)}{}) & \to \mP(\sn{}{\bin(\sn{x,\ms(y)}{}),\bin(\sn{x,y}{})})
\end{align*}
For a precedence $\qp$ that fulfils $\bin \sp \ms$ and $\bin \sp \mP$ we obtain
that $\RSbin$ is compatible with $\LMPO$.
However it is straightforward to verify that the family of terms
$\bin(\ms^n(\Null),\ms^m(\Null))$ admits (innermost) derivations whose length grows exponentially in $n$.
Still the underlying function can be proven polynomial, essentially relying on memoisation techniques, c.f. <cit.>.
The result of our main theorem can also be obtained
if one considers polynomial interpretations, where the interpretations of
constructor symbols is restricted. Such restricted polynomial interpretations
are called additive in <cit.>. Note that additive polynomial interpretations
also characterise the functions computable in polytime, cf. <cit.>.
Although incomparable to our technique,
unarguably such semantic techniques admit a better intensionality,
but are difficult to implement efficiently in an automated setting.
In our complexity tool , we see as a fruitful and fast extension that handles
systems in a fraction of a second.
We also want to mention recent approaches for the automated analysis of
resource usage in programs.
Notably, Hoffmann et al. <cit.> provide an automatic multivariate amortised
cost analysis exploiting typing, which extends earlier results on amortised
cost analysis.
To indicate the applicability of our method we have employed a straightforward (and complexity preserving)
transformation of the RAML programs considered
in <cit.> into TRSs. Equipped with our complexity
analyser can handle all examples from <cit.>.
Finally Albert et al. <cit.> present an automated complexity tool
for Java${}^\text{\texttrademark}$ Bytecode programs and
Gulwani et al. <cit.> as well as Zuleger et al. <cit.>
provide an automated complexity tool for C programs.
§.§ Outline
The remainder of this paper is organised as follows.
In the next section we recall basic notions and starting points of
this paper.
In Section <ref> we introduce polynomial path orders along with our main result.
In the subsequent Sections <ref>–<ref> we show that the (innermost)
runtime-complexity of predicative recursive TRSs is polynomially bounded:
in Section <ref> we set the stage by introducing a notion of predicative interpretation;
in Section <ref> we present an extended version of the aforementioned path order on sequences <cit.>,
and we show that our extension is still sound (c.f. Corollary <ref>);
section <ref> finally shows that predicative interpretations embed derivations into the order on sequences,
establishing our central argument.
In Section <ref> we then present our ramification of polynomial path orders in ICC.
Parameter substitution is incorporated in Section <ref>.
Our implementation is detailed in Section <ref> and ample experimental evidence
is provided in Section <ref>.
Finally, we conclude and present future work in Section <ref>.
§ PRELIMINARIES
We denote by $\N$ the set of natural numbers $\{0,1,2,\dots\}$.
Let $R$ be a binary relation.
The transitive closure of $R$ is denoted by $R^+$ and
its transitive and reflexive closure by $R^{\ast}$.
For $n\in \N$ we denote by $R^n$ the $n$-fold composition of $R$.
The binary relation $R$ is well-founded if
there exists no infinite chain $a_0, a_1, \dots$ with $a_i \mathrel{R} a_{i+1}$
for all $i \in \N$. Moreover, we say that $R$ is well-founded on a set $A$ if
there exists no such infinite chain with $a_0 \in A$.
The relation $R$ is finitely branching if for all elements $a$, the set $\{b \mid a \mathrel{R} b\}$ is finite.
A proper order is an irreflexive and transitive binary relation.
A preorder is a reflexive and transitive binary relation.
An equivalence relation is reflexive, symmetric and transitive.
For a preorder $\succcurlyeq$, we denote the induced equivalence relation by $\eqi$ and induced proper order
by $\succ$.
A multiset is a collection in which elements are allowed to
occur more than once. We denote by $\msetover(A)$ the set of multisets over $A$
and write $\mset{a_1,\dots,a_n}$ to denote multisets with elements $a_1, \dots,a_n$.
We use $m_1 \uplus m_2$ for the summation and $m_1 \backslash m_2$ for difference on multisets $m_1$ and $m_2$.
The multiset extension $\mextension{R}$ of a relation $R$ on $A$ is the
relation on $\msetover(A)$ such that $M_1 \mextension{R} M_2$ if there exists
multisets $X,Y \in \msetover(A)$ satisfying
* $M_2 = (M_1 \backslash X) \uplus Y$,
* $\varnothing \not= X \subseteq M_1$ and
* for all $y \in Y$ there exists an element $x \in X$ such that $x \mathrel{R} y$.
In order to cleanly extend this definition to preorders and equivalences, we follow <cit.>.
Let $\eqi$ denote an equivalence relation over the set $A$ and let ${\succcurlyeq} = {\succ} \cup {\eqi}$ be a binary
relation over $A$ so that $\succ$ and $\eqi$ are compatible in the following sense:
${{\eqi} \cdot {\succ} \cdot {\eqi}} \subseteq {{\succ}}$.
Let $\eclass{a}$ denotes the equivalence class of $a \in A$ with respect to $\eqi$.
By the compatibility requirement,
the extension $\sqsupset$ of $\succ$ to equivalence classes
such that $\eclass{a}\sqsupset\eclass{b}$ if and only if $a \succ b$,
is well defined.
We define the strict multiset extension $\mextension{\succ}$ of $\succcurlyeq$ as
${M_1} \mextension{\succ} {M_2}$ if and only if ${\eclass{M_1}} \mextension{\sqsupset} {\eclass{M_2}}$.
Further, the weak multiset extension $\mextension{\succcurlyeq}$ of $\succcurlyeq$ is given
by ${M_1} \mextension{\succcurlyeq} {M_2}$ if and only if ${\eclass{M_1}} \mextension{\sqsupset} {\eclass{M_2}}$
or ${\eclass{M_1}} = {\eclass{M_2}}$ holds.
Note that if ${\succcurlyeq}$ is a preorder (on $A$) then
$\mextension{\succ}$ is a proper order
and $\mextension{\succcurlyeq}$ a preorder on $\msetover{A}$, cf. <cit.>.
Also $\mextension{\succ}$ is well-founded if $\succ$ is well-founded.
§.§ Complexity Theory
We assume modest familiarity in complexity theory, notations
are taken from <cit.>.
The functional problem $F_R$ associated with an binary relation $R$
is: given $x$ find some $y$ such that $(x,y) \in R$ holds if $y$ exists,
otherwise return $\m{no}$.
A binary relation $R$ on words is called polynomial balanced
if for all $(x,y) \in R$, the size of $y$ is polynomially bounded in $x$.
The relation $R$ is polytime decidable if $(x,y) \in R$
is decided by a deterministic Turing machine (TM for short) $M$ operating in polynomial time.
The class $\NP$ is the class of languages $L$ admitting polynomially balanced, polytime decidable
relations $R_L$ <cit.>: $L = \{x \mid (x,y) \in R_L \text{ for some $y$}\}$.
The class $\FNP$ is the class of function problems associated with $\NP$ in the above way.
The class of polynomial time computable functions $\FP$ (polytime computable for short)
is the subclass resulting if we only consider function problems in $\FNP$ that
can be solved in polynomial time <cit.>.
We say that a function problem $F$ reduces to a function problem $G$ if there
exist functions $s$ and $r$, both computable in logarithmic space, such that
for all $x,y$ with $F$ computing $y$ on input $x$,
$G$ computes on input $s(x)$ the output $z$ with $r(z) = y$.
Note that both $\FNP$ and $\FP$ are closed under reductions.
We also note that nondeterministic Turing machines running in polynomial time compute function
problems from $\FNP$.
Let $N$ be a nondeterministic Turing machine that computes the function problem $F$ in polynomial time.
Then $F \in \FNP$.
Define the following relation $R$: $(x,y) \in R$ if and only if $y$ is the encoding of an accepting
computation of $N$ on input $x$.
Since $N$ operates in polynomial time, $R$ is polynomially balanced, as
it can be checked in linear time that $y$ encodes an accepting run of $N$ on input $x$,
$R$ is polytime decidable.
Hence the functional problem $F_R$ that computes an accepting runs $y$ of $N$ on input $x$ is in $\FNP$.
Finally notice that $F$ reduces to $F_R$.
To see this, employ following reduction:
the function $r$ is simply the identity function; the logspace computable function $s$ extracts the result of $N$ on input $x$
from the accepting run $y$ computed by $F_R$ on input $x$.
We conclude the lemma since $\FNP$ is closed under reductions.
§.§ Term Rewriting
We assume at least nodding acquaintance with the basics of term rewriting <cit.>.
We fix the bare essential of
notions and notation that are used in the paper.
Throughout the paper, we fix a countably infinite set of variables $\VS$
and a finite signature $\FS$ of function symbols.
The signature $\FS$ is partitioned into defined symbols $\DS$ and
constructors $\CS$.
The set of values, basic terms and
terms is defined according to the grammar
∋v x |c(v)
∋s x |f(v)
∋t x |c(t) |f(t)
where $x \in \VS$, $c \in \CS$, and $f \in \DS$.
The arity of a function symbol $f \in \FS$ is denoted by $\ar(f)$.
We write $s \superterm t$ if $t$ is a subterm of $t$, the
strict part of $\superterm$ is denoted by $\supertermstrict$.
The size of a term $t$ is denoted by $\size{t}$ and refers
to the number of variables and function symbols contained in $t$.
We denote by $\depth(t)$ the depth of $t$ which is
defined as $\depth(t) = 1$ if $t \in \VS$ and $\depth(f(\seq{t})) = 1 + \max\{\depth(t_i) \mid i = 1,\dots n\}$.
Here we employ the convention that the maximum of an empty set is equal to $0$.
Let $\qp$ be a preorder on the signature $\FS$, called quasi-precedence
or simply precedence. We always write $\sp$ for the
induced proper order and
$\ep$ for the induced equivalence on $\FS$.
We lift the equivalence ${\ep}$ to terms
modulo argument permutation:
$s \eqi t$ if either $s = t$ or
$s = f(\seq{s})$ and $t = g(\seq{t})$ where $f \ep g$
and for some permutation $\pi$,
$s_i \eqi t_{\pi(i)}$ for all $i \in \{1,\dots,n\}$.
Further we write $s \esuperterm t$ if $t$ is a subterm
of $s$ modulo $\eqi$, i.e., $s~{\superterm} \cdot {\eqi}~{t}$.
We denote by $\sigbelow{f}{\FS} \defsym \{g \mid f \sp g\}$ the set of function symbols
below $f$ in the precedence $\qp$.
This notion is extended to sets $F \subseteq \FS$ by
$\sigbelow{F}{\FS} \defsym \bigcup_{f \in F} \sigbelow{f}{\FS}$.
The rank of a function symbol is inductively defined by
$\rk(f) = \max\{1 + \rk(g) \mid f \sp g\}$.
A rewrite rule is a pair $(l,r)$ of terms, in notation $l \to r$,
such that $l$ is not a variable and all variables in $r$ occur also in $l$.
Here $l$ is called the left-hand, and $r$ the right-hand side of $l \to r$.
A term rewrite system (TRS for short) $\RS$ over
$\TERMS$ is a set of rewrite rules.
In the following, $\RS$ always denotes a TRS.
If not mentioned otherwise, we assume $\RS$ is finite.
A relation on $\TERMS$ is a rewrite relation if it is
closed under contexts and closed under substitutions.
The smallest rewrite relation that contains $\RS$ is denoted by
A term $s \in \TERMS$ is called a normal form if there is no
$t \in \TERMS$ such that $s \rew t$.
With $\NF(\RS)$ we denote the set of all normal forms of a TRS $\RS$.
Whenever $t$ is a normal form of $\RS$ we write $s \rsn t$ for $s \rss t$.
The innermost rewrite relation, denoted as $\irew$, is the restriction
of $\rew$ where arguments are normal forms.
The TRS $\RS$ is terminating if no infinite rewrite sequence exists.
The TRS $\RS$ has unique normal forms if for all
$s, t_1, t_2 \in \TERMS$ with $s \rsn t_1$ and
$s \rsn t_2$ we have $t_1 = t_2$.
The TRS $\RS$ is called confluent if for all $s, t_1, t_2 \in \TERMS$
with $s \rss t_1$ and $s \rss t_2$ there exists a term $u$ such that
$t_1 \rss u$ and $t_2 \rss u$. An orthogonal TRS is
a left-linear and non-overlapping TRS <cit.>. Note that
every orthogonal TRS is confluent.
The TRS $\RS$ is a constructor TRS if all left-hand sides are basic terms.
A defined function symbol is completely defined (with respect to $\RS$)
if it does not occur in any term in normal form, i.e.,
functions are reducible on all terms.
The TRS $\RS$ is completely defined if each defined symbol
is completely defined.
§.§ Rewriting as Computational Model
We fix call-by-value semantics and only consider
constructor TRSs $\RS$. Input and output are taken from the set of
values $\Val$, and defined symbols $f \in \DS$ denote computed functions.
More precise, a (finite) computation of $f\in \DS$ on input $\seq{v} \in \Val$
is given by innermost reductions
f(\seq{v}) = t_0 \irew t_1 \irew \cdots \irew t_\ell = w \tpkt
If the above computation ends in a value, i.e., $w \in \Val$,
we also say that $f$ computes on input $\seq{v}$ in $\ell$ steps the value $w$.
To also account for nondeterministic computation,
we capture semantics of $\RS$ by assigning to each $n$-ary defined symbol
$f \in \DS$ an $n+1$-ary relation $\sem{f}$ that maps
input arguments $\seq{v}$ computed values $w$.
A finite set $\NA$ of non-accepting patterns is used to distinguish meaningful outputs
$w$ from outputs that should not be considered part of the computation.
A value $w$ is accepting with respect to $\NA$
if there exists no $p \in \NA$ and substitution $\sigma$ such that $p\sigma = w$ holds.
A typical example of a meaningful value that should not be accepted is the constant $\unsat$
appearing in the TRS $\RSsat$ from Example <ref>.
Below functional problem are extended to $n+1$-ary relations in the obvious way.
Let $\NA$ be a set of non-accepting patterns.
For each $n$-ary symbol, $f \in \DS$ the TRS $\RS$
the relation $\sem{f} \subseteq {\Val^{n+1}}$ defined by $f$
is given by
\begin{equation*}
{(\seq{v},w) \in \sem{f}} \quad\defiff\quad {\m{f}(\seq{v}) \irsn[\RS] w} \text{ and $w$ is accepting}\tpkt
\end{equation*}
We say that $\RS$ computes the functional problems associated with $\sem{f}$.
Note that if $\RS$ is confluent, then $\sem{f}$ is in fact a (partial) function.
Following <cit.> we adopt an unitary cost model for rewriting,
where each reduction step accounts for one time unit.
Reductions are of course measured in the size of the input.
The (innermost) runtime complexity function $\ofdom{\rc[\RS]}{\N \to \N}$
relates sizes of basic terms $f(\seq{v}) \in \BASICS$ to the maximal
length of computation. Formally
\rc[\RS](n) \defsym
\max\{\ell \mid \exists s \in \BASICS, \size{s} \leqslant n \text{ and } f(\seq{v}) = t_0 \irew t_1 \irew \dots \irew t_\ell\} \tpkt
The restriction $s \in \BASICS$ accounts for the fact that computations start only from basic terms.
We sometimes use $\dheight(s) \defsym \max\{\ell \mid \exists t.~s \irsl{\ell} t\}$
to refer to the derivation height of a single term $s$.
Note that the runtime complexity function is well-defined if $\RS$ is terminating,
i.e., $\irew$ is well-founded.
If $\rc[\RS]$ is asymptotically bounded from above by a linear, quadratic,…, polynomial function,
we simply say that the runtime of $\RS$ is linear, quadratic,…, or respectively polynomial.
By folklore it is known that rewriting can be implemented with only polynomial overhead
if terms grow only polynomial during reductions.
In <cit.> we have shown that the unitary cost model is reasonable
for full rewriting (the deterministic case was proven independently in <cit.> using essentially the same approach).
It is not difficult to see that the central Lemma <cit.> that estimates the implementation
cost of a single rewrite step can be specialised to innermost rewriting.
We obtain following proposition by specialising <cit.> to innermost rewriting.
Let $\RS$ be a TRS whose is at least linear.
There exists a polynomial $p_\RS$ such that for any $f(\seq{v}) \in \BASICS$ of
size up to $n$,
* any normal form of $f(\seq{v})$ can be computed on a Turing machine in nondeterministic time
$p_\RS(\rc[\RS](n))$; and
* some normal form of $f(\seq{v})$ is computable on a Turing machine in deterministic time
Hence there are no surprises here.
By Proposition <ref> and Proposition <ref> we obtain:
Let $\RS$ be a rewrite system with polynomial runtime.
Then the functional problems associated with $\sem{f}$ defined by $\RS$ are contained in $\FNP$.
If $\RS$ is confluent, i.e. deterministic, then $\sem{f}$ is a (partial) function contained in $\FP$.
Our choice of adopting call-by-value semantics is rested
in the observation that the unitary cost model of unrestricted rewriting
often overestimates the runtime complexity of computed functions.
This has to do with the unnecessary duplication of redexes.
Consider the constructor TRS $\RSdup$ given by the following rules.
1: btree() →leaf
2: btree((n)) →dup(btree(n))
3: dup(t) →node(t,t)
Then for $n \in \N$, $\m{btree}(\ms^n(\mZ))$ computes a binary tree of height $n$
in a linear number of steps.
On the other hand, $\RSdup$ gives also rise to a non-innermost reduction
\m{btree}(\sn{\ms^{n}(\mZ)}{})
\rew \m{dup}(\m{btree}(\sn{\ms^{n-1}(\mZ)}{}))
\rew \m{node}(\m{btree}(\sn{\ms^{n-1}(\mZ)}{}), \m{btree}(\sn{\ms^{n-1}(\mZ)}{}))
\rew \dots
obtained by preferring $\m{dup}$ over $\m{btree}$.
The length of the derivation is however exponential in $n$.
By Proposition <ref> we obtain $\sem{\m{btree}} \in \FP$.
As indicated later, our analysis can automatically classify the function $\sem{\m{btree}}$ as feasible.
§ THE POLYNOMIAL PATH ORDER
We arrive at the formal definition of polynomial path order ( for short).
Variants of the here presented definition have been presented in earlier conference publications,
see <cit.>.
The order essentially embodies the predicative analysis of recursion
set forth by Bellantoni and Cook <cit.>.
In , the separation of argument positions is taken into account in the notion of safe mapping.
A safe mapping $\safe$ is a function
$\ofdom{\safe}{\FS \to 2^\N}$ that associates
with every $n$-ary function symbol $f$ the set of safe argument positions
$\{i_1, \dots , i_m\} \subseteq \{1,\dots,n\}$.
Argument positions included in $\safe(f)$ are called safe,
those not included are
called normal and collected in $\normal(f)$.
For $n$-ary constructors $c$
we require that all argument positions are safe,
i.e., $\safe(c) = \{1,\dots,n\}$.
We refine term equivalence so that the safe mapping is taken into account.
Let ${\qp}$ denote a precedence and $\safe$ a safe mapping.
We define safe equivalence $\eqis$ for terms $s,t \in TERMS$
inductively as follows:
$s \eqis t$ if either $s = t$ or
$s = f(\seq{s})$, $t = g(\seq{t})$, $f \ep g$
and there exists a permutation $\pi$ such that for all $i \in \{1,\dots,n\}$,
$s_i \eqis t_{\pi(i)}$ and $i \in \safe(f)$ if and only if $\pi(i) \in \safe(g)$.
To avoid notational overhead, we suppose that for each $k+l$ ary function symbol
$f$, the first $k$ argument positions are normal, and the remaining
argument positions are safe, i.e., $\safe(f) = \{k+1,\dots,k+l\}$.
This allows use to write $f(\pseq{s})$
where the separation of safe from normal arguments is directly indicated in terms.
Let $\qp$ denote a quasi-precedence. We require that the precedence adheres the partitioning of
$\FS$ into defined symbols and constructors in the following sense.
Then in particular $\eqis$ preserves values, i.e.,
if $s \in \Val$ and $s \eqis t$ then also $t \in \Val$.
A precedence $\qp$ is admissible (for ) if $f \ep g$ implies
that either both $f$ and $g$ are defined symbols, or both are constructors.
The following definition introduces an auxiliary order $\gsq$,
the full order $\gpop$ is then presented in Definition <ref>.
Let ${\qp}$ denote a precedence and $\safe$ a safe mapping.
Consider terms $s, t \in \TERMS$ such that $s = f(\pseq[k][l]{s})$.
Then $s \gsq t$ if one of the following alternatives holds:
* $s_i \geqsq t$ for some $i \in \{1,\dots,k+l\}$ and,
if $f \in \DS$ then $i$ is a normal argument position ($i \in \{1,\dots,k\}$);
* $f \in \DS$, $t = g(\pseq[m][n]{t})$ where $f \sp g$
and $s \gsq t_i$ for all $i = 1,\dots,m+n$.
Here we set ${\geqsq} \defsym {\gsq} \cup {\eqis}$.
Consider a function $f$ defined by safe composition from $r$ and $s$, compare scheme (<ref>).
The purpose of this auxiliary order is to embody safe composition in the full order $\gpop$.
Note that the auxiliary order can orient $f(\sn{\vec{x}}{\vec{y}}) \gsq r(\sn{\vec{x}}{})$ for defined symbol $f$
with $f \sp r$.
On the other hand, $f(\sn{\vec{x}}{\vec{y}})$ and safe arguments $y_i$ are incomparable,
and consequently the orientation of $f(\sn{\vec{x}}{\vec{y}})$ and $s(\sn{\vec{x}}{\vec{y}})$ fails.
Let ${\qp}$ denote a precedence and $\safe$ a safe mapping.
Consider terms $s, t \in \TERMS$ such that $s = f(\pseq[k][l]{s})$.
Then $s \gpop t$ if one of the following alternatives holds:
* $s_i \geqpop t$ for some $i \in \{1,\dots,k+l\}$, or
* $f \in \DS$, $t = g(\pseq[m][n]{t})$ where $f \sp g$
and the following conditions hold:
* $s \gsq t_j$ for all normal argument positions $j = 1,\dots,m$;
* $s \gpop t_j$ for all safe argument positions $j = m+1,\dots,m+n$;
* $t_j \not\in \TA(\sigbelow{\Fun(s)}{\FS},\VS)$ for at most one safe argument position $j \in \{m+1,\dots,m+n\}$;
* $f \in \DS$, $t = g(\pseq[m][n]{t})$ where $f \ep g$
and the following conditions hold:
* $\mset{\seq[k]{s}} \gpopmul \mset{\seq[m]{t}}$;
* $\mset{\seq[k+l][k+1]{s}} \geqpopmul \mset{\seq[m+n][m+1]{t}}$.
Here ${\geqpop} \defsym {\gpop \cup \eqis}$.
We say a constructor TRS $\RS$ is predicative recursive if
$\RS$ is compatible with an instance $\gpop$ with underlying admissible precedence.
We use the notation $\caseref{\gpop}{i}$ and respectively $\caseref{\gsq}{i}$
to refer to the $i$ case in Definition <ref> respectively Definition <ref>.
We emphasise that $\gpop$ is blind on constructors, in particular $\gpop$ collapses
to the subterm relation (modulo equivalence) on values.
Suppose the precedence underlying $\gpop$ is admissible.
If $s \gpop t$ and $s \in \Val$ then $s \esupertermstrict t$,
in particular $t \in \Val$.
The case ia accounts for definitions by safe composition (<ref>).
The final restriction put onto ia is used to prevent multiple recursive calls
as indicated in Example <ref>.
We remark that restrictions put onto ia are weaker compared to the corresponding clause given
in <cit.>
[The early definition from <cit.>,
used the full order $\gpop$ only on one argument of the right-hand side (the one that possibly holds the recursive call),
the remaining arguments were all oriented with the auxiliary order $\gsq$.
To retain completeness, in <cit.> we allowed also the
admittedly ad hoc use of a subterm comparison on safe arguments.].
The case ep restricts the corresponding case in
by taking the separation of normal and safe argument positions into account.
Note that here normal arguments need to decrease. This reflects that as in (<ref>)
recursion is performed on normal argument positions.
We arrive at the central theorem of this paper.
Let $\RS$ be predicative recursive TRS.
Then the innermost derivation height of any basic term
$f(\svec{u}{v})$ is bounded by a polynomial in the
maximal depth of normal arguments $\vec{u}$.
The polynomial depends only on $\RS$ and the signature $\FS$.
In particular, the runtime complexity of $\RS$ is polynomial.
The proof of Theorem <ref> is rather involved, and outlined at the end of this
section. The formal proof is then presented in the subsequent Sections <ref>–<ref>.
We clarify first Definition <ref> on several examples.
Consider the TRS $\RSmul$ expressing multiplication in Peano arithmetic.
1: +(y) →y
3: ×(,y) →
2: +((x)y) →(+(xy))
4: ×((x),y) →+(y×(x,y))
The TRS $\RSmul$ is predicative recursive, using the precedence
${\times}~\sp~{+}~\sp~{\ms}$ and the safe mapping as indicated in the rules:
The rules $\rlbl{1}$ and $\rlbl{3}$ are oriented by st.
The rule $\rlbl{3}$ is oriented by $\cpop{ia}$ using ${+}~\sp~{\ms}$
and $+(\sn{\ms(\sn{}{x})}{y}) \cpop{ep} +(\sn{x}{y})$.
Note that the latter enforces that the first argument to $+$ is normal.
Similar, the final rule $\rlbl{4}$ is oriented by ia, employing
${\times}\sp{+}$ together with $\times(\sn{\ms(\sn{}{x}),y}{}) \csq{st} y$
and $\times(\sn{\ms(\sn{}{x}),y}{}) \cpop{ep} \times(\sn{x,y}{})$.
Note that the latter two inequalities require that
the both argument positions of $\times$ are normal, i.e., are
used for recursion.
Now consider the extension of $\RSmul$ from Example <ref> by the two rules
5: exp(,y) →()
6: exp((x),y) →×(y,exp(x,y))
that express exponentiation $y^x$ in an exponential number of steps.
The definition of $\m{exp}$ does not follow predicative recursion,
in particular since $\times$ admits no safe argument positions it
cannot serve as stepping function. Independent on the safe mapping for $\m{exp}$,
rule $\rlbl{6}$ cannot be oriented using polynomial path orders.
Finally, for a negative example consider $\RSmul$ from Example <ref> where the rule $\rlbl{4}$ is replaced by the rule
$$\rlbl{4a}:~\times(\sn{\ms(\sn{}{x}),y}{}) \to +(\sn{\times(\sn{x,y}{})}{y})\tpkt$$
The resulting system admits polynomial runtime complexity, but does not follow the rigid
scheme of predicative recursion. For this reason, it cannot be handled by .
terms $\times(\sn{\ms(\sn{}{x}),y}{})$ and $\times(\sn{x,y}{})$
is incomparable with respect to $\gsq$ independent on the precedence, and
consequently also orientation of left- and right-hand side with
ia fails.
Finally, we stress that the restriction to innermost reductions
is essential for the correctness of Theorem <ref>.
This has to do with unnecessary duplication of redexes as pointed out in Example <ref>.
Reconsider the TRS $\RSdup$ from Example <ref>.
Then $\RSdup \subseteq {\gpop}$ with any admissible precedence
satisfying $\m{btree} \sp \m{dup}$. Theorem <ref>
thus implies that the (innermost) runtime complexity of $\RSdup$ is
polynomial. On the other hand, we already observed that $\RSdup$ admits exponentially long
outermost reductions.
*Proof Outline
The proof of Theorem <ref> requires a variety of ingredients.
In Section <ref>, we define predicative interpretations $\ints$
that flatten terms to sequences of terms, essentially separating
safe from normal arguments. This allows us to analyse terms independent from safe arguments.
In Section <ref> we introduce an order $\gpopv[][]$ on sequences of terms,
that is simpler compared to $\gpop$ and does not rely on the separation
of argument positions.
In Section <ref> we show that predicative interpretations
embeds innermost rewrite steps into $\gpopv[][]$:
in 1,..., ;
at (,0) $\dots$;
at (,-) $\dots$;
in 1,..., ;
In Theorem <ref> we show that the length of $\gpopv[][]$ descending sequences
starting from basic terms can be bound appropriately.
§ PREDICATIVE INTERPRETATIONS
Fix a safe mapping $\safe$ on the signature $\FS$.
In this section, we define the predicative interpretation
that guided by $\safe$ interpret terms as sequences.
For this, define the normalised signature $\FSn$ be given as
\begin{equation*}
\FSn \defsym \bigl\{ \fn \mid f \in \FS, \normal(f) = \{i_1,\dots,i_k\} \text{ and } \ar(\fn) = k \}\bigr\}
\end{equation*}
The predicative interpretation of a term $f(\pseq{s})$ results in a sequence
$\lst{\fn(\seq[k]{a})} \append a_{k+1} \append \cdots \append a_{k+l}$,
where $\append$ denotes concatenation of sequences and the
sequences $a_i$ are predicative interpretations of the corresponding arguments $s_i$ ($i = 1,\dots,k+l$).
To denote sequences, we use an auxiliary variadic function symbol
$\listsym$. Here variadic means that the arity of
$\listsym$ is finite but arbitrary. We always write $\lseq{t}$
for $\listsym(\seq{t})$, in particular if we write $f(\seq{t})$ then $f \not=\listsym$.
Note that in the interpretations, terms have sequences as arguments.
We reflect this in the next definition.
The set of terms with sequence arguments
$\TS \subseteq \TA(\FSn \uplus \set{\listsym},\VS)$
and the set of sequences $\LS \subseteq \TA(\FSn \uplus \set{\listsym},\VS)$ is inductively defined as follows:
* $\VS \subseteq \TS$, and
* if $\seq{t} \in \TS$ then $\lseq{t} \in \LS$, and
* if $\seq{a} \in \LS$ and $f \in \FSn$ then $f(\seq{a}) \in \TS$.
We always write $a,b, \dots$, possibly extended by
subscripts, for elements from $\TS$ and $\LS$.
The restriction of $\TS$ and $\LS$ to ground terms is denoted
by $\GTS$ and $\GLS$ respectively.
When no confusion can arise from this we call terms with sequence arguments simply terms.
Further, we sometimes abuse set notation and write $b \in \lseq{a}$ if
$b = a_i$ for some $i \in \{1,\dots,n\}$.
We denote by $a \append b$ the concatenation of $a \in \TLS$ and $b \in \TLS$.
To avoid notational overhead we identify terms with singleton sequences.
Let $\tolst(a) \defsym \lst{a}$ if $a \in \TS$ and $\tolst(a) \defsym a$ if $a \in \LS$.
We set $a \append b \defsym \lst{a_1~\cdots~a_n~b_1~\cdots~b_m}$
where $\tolst(a) = \lseq{a}$ and $\tolst(b) = \lseq[m]{b}$.
We define the length over $\TLS$ as $\len(a) \defsym n$ where $\tolst(a) = \lseq{a}$.
The sequence width $\width$ (or width for short)
of an element $a \in \TLS$ is given recursively by
\begin{equation*}
\width(a) \defsym
\begin{cases}
1 & \text{if $a$ is a variable, }\\
\max \{1,\width(a_1),\dots,\width(a_n)\} & \text{if $a = f(\seq{a})$, and}\\
\sum_{i=1}^n \width(a_i)
& \text{if $a = \lseq{a}$.}
\end{cases}
\end{equation*}
We will tacitly employ $\len(a) \leqslant \width(a)$ and $\width(a \append b) = \width(a) + \width(b)$ for all $a,b \in \TLS$.
We definite the norm of $t \in \TERMS$ in correspondence to the depth of $t$, but
disregard normal argument positions.
\begin{equation*}
\norm{t} =
\begin{cases}
1 & \text{$t$ is a variable} \\
1+ \max\{\norm{t_{k+1}},\dots,\norm{t_{k+l}}\} & \text{$t=f(\pseq{t})$}
\end{cases}
\end{equation*}
Note that since all argument positions of constructors are safe,
the norm $\norm{\cdot}$ and depth $\depth(\cdot)$ coincides on values.
Predicative interpretations are given by two mappings $\ints$ and $\intn$:
the interpretation $\ints$ is applied on safe arguments and removes values;
the mapping $\intn$ is applied to normal arguments and additionally encodes
the norm of the given term as tally sequence. Consequently we keep
track of the maximal depth of values at normal argument positions.
Let $\theconst \not \in \FSn$ be a fresh constant.
To encode natural numbers $n\in \N$,
define its tally sequence representation $\natToSeq{n}$
as the sequence containing $n$ occurrences of this fresh constant:
$\natToSeq{0} = \nil$ and $\natToSeq{n+1} = \theconst \append \natToSeq{n}$.
A predicative interpretation
is a pair $(\ints,\intn)$ of mappings $\ofdom{\ints,\intn}{\TERMS \to \TAL^{\!\ast}\!(\FS \cup \{\theconst\})}$
defined as follows:
\begin{align*}
\ints(t) & \defsym
\begin{cases}
\nil & \text{ if $t$ is a value} \\
\lst{\fn(\intn(t_1), \dots, \intn(t_k))} \append \ints(t_{k+1}) \append \cdots \append \ints(t_{k+l}) & \text{ otherwise where ($\star$)}
\end{cases}\\
\intn(t) & \defsym \ints(t) \append \NM{t} \tpkt
\end{align*}
Here ($\star$) stands for $t = f(\pseq{t})$.
In the next section we introduce the order $\gpopv[][]$ on
sequences $\GTLS$.
In the subsequent section we then embed innermost $\RS$-steps into this order,
and use $\gpopv[][]$ to estimate the length of reductions accordingly.
Since for basic terms $s = f(\pseq{u})$ in particular
\ints(s) = \lst{\fn(\intn(u_1),\dots,\intn(u_k))} \append \ints(u_{k+1}) \append \cdots \append \ints(u_{k+l})
= \lst{\fn(\natToSeq{\depth(u_1)},\dots,\natToSeq{\depth(u_k)})}
the so obtained bound will depend on depths of normal arguments only.
To get the reader prepared for the definition of $\gpopv[][]$,
we exemplify Definition <ref> on a predicative recursive TRS.
Consider following predicative recursive TRS $\RS_f$ where we suppose that besides $f$, also $g$ and $h$ are defined symbols:
\begin{align*}
\rlbl{1}:~f(\sn{\mZ}{y}) & \to y
& \rlbl{2}:~f(\sn{\ms(x)}{y}) & \to g(\sn{h(\sn{x}{})}{f(\sn{x}{y})})
\end{align*}
Consider a substitution $\ofdom{\sigma}{\Var \to \Val}$.
Using that $\intn(v) = \natToSeq{\depth(v)}$ for all values $v$,
the embedding $\ints(l\sigma) \gpopv[][] \ints(r\sigma)$ of root steps ($l \to r \in \RS_f$)
results in the following order constraints.
\begin{align*}
\lst{\fn(\natToSeq{1})} & \gpopv[][] \nil && \text{from rule \rlbl{1}}\\
\mparbox[r]{45mm}{\lst{\fn(\natToSeq{\depth(x\sigma) + 1})}} & \gpopv[][] \mparbox[l]{65mm}{\lst{\gn(\intn(h(\sn{x\sigma}{})))~\fn(\natToSeq{\depth(x\sigma)})}} && \text{from rule \rlbl{2}}.
% & = \lst{\gn(\lst{\hn(\natToSeq{\depth(x\sigma)})~\theconst})~\fn(\natToSeq{\depth(x\sigma) + 1})}
% & \ints(y\sigma) & = \nil
% \\
% \intn(f(\sn{\mZ}{y})\sigma) & = \fn(\natToSeq{1}) \append \natToSeq{\depth(y\sigma) + 1}
% & \intn(y\sigma) & = \natToSeq{\depth(y\sigma)}
% \\[3mm]
% \ints(f(\sn{\ms(x)}{y})) & =\lst{\fn(\natToSeq{\depth(x\sigma) + 1})}
% & \ints(g(\sn{h(\sn{x}{})}{f(\sn{x}{y})})) & =
% \\
% \intn(f(\sn{\ms(x)}{y})) & = \fn(\natToSeq{\depth(x\sigma) + 1}) \append \natToSeq{\depth(y\sigma) + 1}
\end{align*}
where $\intn(h(\sn{x\sigma}{})) = \lst{\hn(\intn(x\sigma))}\append \NM{h(\sn{x\sigma}{})} = \lst{\hn(\natToSeq{\depth(x\sigma)})~\theconst}$.
Closure under context follows using standard inductive reasoning.
To deal with steps below normal argument positions, it is also necessary to
orient images of $\intn$. On the TRS $\RS_f$ this results additionally in following constraints:
\begin{align*}
\lst{\fn(\natToSeq{1})} \append \natToSeq{\depth(y\sigma) + 1} & \gpopv[][] \natToSeq{\depth(y\sigma)} && \text{from rule \rlbl{1}}\\
\mparbox[r]{45mm}{\lst{\fn(\natToSeq{\depth(x\sigma) + 1})}\append \natToSeq{\depth(y\sigma) + 1}} & \gpopv[][] \mparbox[l]{65mm}{\lst{\gn(\intn(h(\sn{x}{})))~\fn(\natToSeq{\depth(x\sigma)})} \append \natToSeq{\depth(y\sigma) + 1}}
&& \text{from rule \rlbl{2}}.
\end{align*}
To get a polynomial bound on $\gpopv[][]$ descending sequences, we need to control
the length of right-hand sides appropriately. Precisely we will require
that for a global constant $k \in \N$, $\len(b) \leqslant \width(a) + k$ whenever $a \gpopv[][] b$ holds.
In particular $k$ will be more than twice the maximal size of a right-hand side in the analysed TRS $\RS$.
Note that due to the following lemma, if $l\sigma \irew r\sigma$ with $\ofdom{\sigma}{\VS \to \Val}$ is a root step of a predicative TRS $\RS$,
then $\len(\intq(r\sigma)) \leqslant \width(\intq(l\sigma)) + k$ for $\intq \in \{\ints,\intn\}$.
Let $s = f(\pseq{s}) \in \Tb$, $t \in \TA$, $\ofdom{\sigma}{\VS \to \Val}$ and define $k \defsym 2\cdot\size{t}$. Then
* $\len(\ints(t\sigma)) \leqslant \size{t}$; and
* if $s \gsq t$ then $\len(\intn(t\sigma)) \leqslant \max \{ \norm{s_1\sigma}, \dots, \norm{s_k\sigma} \} + k$; and
* if $s \gpop t$ then $\len(\intn(t\sigma)) \leqslant \max \{ \norm{s_1\sigma}, \dots, \norm{s_k\sigma},\norm{s\sigma}\} + k$.
The first property follows by induction on $t$, employing that $\ints(x\sigma) = \nil$.
A standard induction on $\gpop$ (respectively $\gsq$) proves the second and third properties.
For the cases $s \cpop{st} t$ (respectively $s \csq{st} t$) and $s \cpop{ep} t$, we use
Lemma <ref>, the remaining cases follow from induction hypothesis directly.
§ THE POLYNOMIAL PATH ORDER ON SEQUENCES
The polynomial path order on sequences ( for short), denoted by $\gpopv[][]$,
constitutes a generalisation of the path order for $\FP$ as put forward in <cit.>.
Whereas we previously uses the notion of safe mapping to
dictate predicative recursion on compatible TRSs,
the order on sequences relies on the explicit separation of safe
arguments as given by predicative interpretations.
Following Buchholz <cit.>, we present finite approximations
$\gpopv[k][l]$ of $\gpopv[][]$.
The parameters $k \in \N$ and $l \in \N$ are used to controls the width and depth
of right-hand sides.
Fix a precedence $\qp$ on the normalised signature $\FSn$.
We extend term equivalence with respect to $\qp$ to sequences by
disregarding the order on elements.
We define $a \eqi b$ if $a = b$ or there exists a permutation $\pi$
such that $a_i \eqi b_{\pi(i)}$ for all $i = 1,\dots,n$,
where either
(i) $a = \lseq{a}$, $b = \lseq{b}$, or
(ii) $a = f(\seq{a})$, $b = g(\seq{b})$ and $f \ep g$.
In correspondence to $\gpop$, the order $\gpopv[k][l]$
is based on an auxiliary order $\gppv[k][l]$
defined next. The full order is then introduced in Definition <ref>.
Let $k,l \geqslant 1$.
We define $\gppv[k][l]$ with respect to the precedence $\qp$ inductively as follows:
$f(\seq{a}) \gppv[k][l] b$ if $a_i \geqppv[k][l] b$ for some $i \in \{1,\dots,n\}$;
$f(\seq{a}) \gppv[k][l] g(\seq[m]{b})$ if $f \sp g$ and the following conditions are satisfied:
* $f(\seq{a}) \gppv[k][l-1] b_j$ for all $j = 1,\dots,m$;
* $m \leqslant k$;
$f(\seq{a}) \gppv[k][l] \lseq[m]{b}$ if the following conditions are satisfied:
* $f(\seq{a}) \gppv[k][l-1] b_j$ for all $j = 1,\dots,m$;
* $m \leqslant \width(f(\seq{a})) + k$;
$\lseq{a} \gppv[k][l] \lseq[m]{b}$ if the following conditions are satisfied:
* $\lseq[m]{b} \eqi c_1 \append \cdots \append c_n$;
* $a_i \geqppv[k][l] c_i$ for all $i = 1, \dots, n$;
* $a_{i_0} \gppv[k][l] c_{i_0}$ for at least one $i_0 \in \{1, \dots, n\}$;
* $m \leqslant \width(\lseq{a}) + k$;
Here ${\geqppv[k][l]}$ denotes ${\gppv[k][l]} \cup {\eqi}$. We write
$\gppv[k]$ to abbreviate $\gppv[k][k]$.
Recall that the auxiliary order $\gsq$ underlying $\POPSTAR$ is used to orient normal arguments in right-hand sides.
Similar, the auxiliary order $\gppv[k][l]$ is to orient the predicative interpretations of this
normal arguments. We exemplify the order $\gppv[k][l]$ on the Example <ref>.
Reconsider rule $\rlbl{2}$ from Example <ref> where
in particular $f(\sn{\ms(x)}{y}) \gsq h(\sn{x}{})$.
Define the precedence $\fn \sp \hn \sp \theconst$.
First recall that by definition of the operator $\append$ we have
$$\natToSeq{n} = \lst{\theconst~\cdots~\theconst} = \theconst\append\cdots\append \theconst \append \nil \append \cdots \append \nil$$
with $n$ occurrences of $\theconst$ and $m$ occurrences of $\nil$.
Using $n$ times $\theconst \eqi \theconst$ and $m$ times $\theconst \cppv{ialst}[k][l] \nil$
we can thus prove $\natToSeq{n+m} \cppv{ms}[k][l] \natToSeq{n}$ for all $m \geqslant 1$ whenever $l \geqslant 2$.
Let $k \geqslant 12$ be at least twice the size of the right-hand sides,
and consider a substitution $\ofdom{\sigma}{\Var \to \Val}$.
To show $\intq(f(\sn{\ms(x\sigma)}{y\sigma})) \gppv[k] \intq(h(\sn{\ms(x\sigma)}{}))$ for $\intq \in \{\ints,\intn\}$,
we can even show the stronger property $\fn(\natToSeq{\depth(x\sigma)} + 1) \gppv[k][10] \intq(h(\sn{\ms(x\sigma)}{}))$
\begin{align*}
\rlbl{1}: && \natToSeq{\depth(x\sigma) + 1}
& \cppv{ms}[k][7] \natToSeq{\depth(x\sigma)}
&& \text{as $\depth(x\sigma) + 1 > \depth(x\sigma)$}
\\
\rlbl{2}: && \fn(\natToSeq{\depth(x\sigma)} + 1)
& \cppv{ia}[k][8] \hn(\natToSeq{\depth(x\sigma)}) = \ints(h(\sn{x\sigma}{}))
&& \text{using $\fn \sp \hn$ and \rlbl{1}}
\\
\rlbl{3}: && \fn(\natToSeq{\depth(x\sigma)} + 1)
& \cppv{ialst}[k][9] \lst{\hn(\depth(x\sigma))~\theconst} = \intn(h(\sn{x\sigma}{}))
&& \text{by \rlbl{2} and $\fn(\dots) \cppv{ia}[k][8] \theconst$}
\end{align*}
We arrive at the definition of the full order $\gpopv[k][l]$.
Let $k,l \geqslant 1$.
We define $\gpopv[k][l]$ inductively as the least extension of $\gppv[k][l]$ such that:
$f(\seq{a}) \gpopv[k][l] b$ if $a_i \geqpopv[k][l] b$ for some $i \in \{1,\dots,n\}$;
$f(\seq{a}) \gpopv[k][l] g(\seq[m]{b})$ if $f \ep g$
and following conditions are satisfied:
* $\mset{\seq{a}}~\mextension{\gpopv[k][l]}~\mset{\seq[m]{b}}$;
* $m \leqslant k$;
$f(\seq{a}) \gpopv[k][l] \lseq[m]{b}$
and following conditions are satisfied:
* $f(\seq{a}) \gpopv[k][l-1] b_{j_0}$ for at most one $j_0 \in\{1,\dots,m\}$;
* $f(\seq{a}) \gppv[k][l-1] b_j$ for all $j \neq j_0$;
* $m \leqslant \width(f(\seq{a})) + k$;
$\lseq{a} \gpopv[k][l] \lseq[m]{b}$
and following conditions are satisfied:
* $\lseq[m]{b} \eqi c_1 \append \cdots \append c_n$;
* $a_i \geqpopv[k][l] c_i$ for all $i = 1,\dots, n$;
* $a_{i_0} \gpopv[k][l] c_{i_0}$ for at least one $i_0 \in \{1,\dots, n\}$;
* $m \leqslant \width(\lseq{a}) + k$;
Here ${\geqpopv[k][l]}$ denotes ${\gpopv[k][l]} \cup {\eqi}$.
We write $\gpopv[k]$ to abbreviate $\gpopv[k][k]$.
Reconsider the rules from Example <ref>,
and let $k \geqslant 12$.
We consider only substitutions $\ofdom{\sigma}{\VS \to \Val}$.
First consider a rewrite step $f(\sn{\mZ}{y\sigma}) \irew y\sigma$ due to rule 1. Exploiting the shape of $\sigma$, we
$\ints(f(\sn{\mZ}{y\sigma})) = \fn(\natToSeq{1}) \cpopv{ialst}[k] \nil = \ints(y\sigma)$
and similar
\intn(f(\sn{\mZ}{y\sigma}))
= \lst{\fn(\natToSeq{1})} \append \natToSeq{\depth(y\sigma) + 1}
\cpopv{ms} \natToSeq{\depth(y\sigma)}
= \intn(y\sigma) \tpkt
Finally consider a rewrite step $f(\sn{\ms(x\sigma)}{y\sigma}) \irew g(\sn{h(\sn{x\sigma}{})}{f(\sn{x\sigma}{y\sigma})})$ caused by rule 2.
This case is slightly more involved. Essentially we use $\cpopv{ep}[k][l]$ to orient the recursive call (proof step 5),
and $\cppv{ia}[k][l]$ for the remaining elements not containing $\fn$ (proof step 6).
\begin{align*}
\rlbl{4}: && \natToSeq{\depth(x\sigma) + 1}
& \cpopv{ms}[k][9] \natToSeq{\depth(x\sigma)}
% && \text{as $\depth(x\sigma) + 1 > \depth(x\sigma)$}
\\
\rlbl{5}: && \fn(\natToSeq{\depth(x\sigma)} + 1)
& \cpopv{ep}[k][9] \fn(\natToSeq{\depth(x\sigma)})
&& \text{by \rlbl{4}}
\\
\rlbl{6}: && \fn(\natToSeq{\depth(x\sigma)} + 1)
& \cppv{ia}[k][10] \gn(\intn(h(\sn{x\sigma}{})))
&& \text{using $\fn \sp \gn$ and \rlbl{3}}
\\
\rlbl{7}: && \fn(\natToSeq{\depth(x\sigma)} + 1)
& \cpopv{ialst}[k][11] \lst{\gn(\intn(h(\sn{x\sigma}{})))~\fn(\natToSeq{\depth(x\sigma)})}
&& \text{using \rlbl{5} and \rlbl{6}}
\\
&&& = \ints(g(\sn{h(\sn{x\sigma}{})}{f(\sn{x\sigma}{y\sigma})}))
\\
\rlbl{8}: && \intn(f(\sn{\ms(x\sigma)}{y\sigma})) & =
\lst{\fn(\natToSeq{\depth(x\sigma) + 1})}\append \natToSeq{\depth(y\sigma) + 1} \\
&&& \cpopv{ialst}[k][k] \lst{\gn(\intn(h(\sn{x\sigma}{})))~\fn(\natToSeq{\depth(x\sigma)})}\append \natToSeq{\depth(y\sigma) + 1}
&& \text{using \rlbl{7}}
\\
&&& = \intn(g(\sn{h(\sn{x\sigma}{})}{f(\sn{x\sigma}{y\sigma})}))
\end{align*}
The next lemma collects some frequently used properties.
The following properties hold for all $k \geqslant 1$ and $a,b,c_1,c_2 \in \TLS$.
* ${\gppv[l]} \subseteq {\gpopv[l]} \subseteq {\gpopv[k]}$ for all $l \leqslant k$;
* ${\eqi} \cdot {\gpopv[k]} \cdot {\eqi} \subseteq {\gpopv[k]}$;
* $a \gpopv[k] b$ implies ${c_1 \append a \append c_2} \gpopv[k] {c_1 \append b \append c_2}$.
The first two properties follow by standard reasoning.
For the final property on proves $a \append c_2 \cpopv{ms}[k] b \append c_2$
by case analysis on the assumption $a \gpopv[k] b$. Crucially, $\len(b \append c_2)$ is bounded
by $\width(a \append c_2) + k$ as required in $\cpopv{ms}[k]$.
The general property is then an easy consequence from Property <ref>.
Following <cit.> we define $\Slow[k]$ that measures the
$\gpopv[k]$-descending lengths on sequences. To simplify matters,
we restrict the definition of $\Slow[k]$ to ground sequences.
As images of predicative interpretations are always ground, this suffices for our purposes.
We define $\ofdom{\Slow[k]}{\GTLS \to \N}$
\begin{equation*}
\Slow[k](a) \defsym 1 + \max
\{ \Slow[k](b) \mid b \in \GTLS \text{ and } a \gpopv[k] b
\}\tpkt
\end{equation*}
Note that due to Lemma l:approxmodeqi, $\Slow[k](a) = \Slow[k](b)$
whenever $a \eqi b$.
Sequences are intended to act purely as a container,
not contributing to $\Slow[k]$ themselves.
The next lemma confirms our intention,
exploiting that conceptually clause $\cpopv{ms}[k]$ amounts to a product-wise extension of $\gpopv[k]$ to sequences.
For $\lseq{a} \in \GLS$ it holds that $\Slow[k](\lseq{a}) = \sum_{i=1}^n \Slow[k](a_i)$.
Let $a = \lseq{a} \in \GLS$.
We first show $\Slow[k](a) \geqslant \sum_{i=1}^n \Slow[k](a_i)$.
Let $b,c \in \GTLS$ and consider maximal sequences
$b \gpopv[k] b_1 \gpopv[k] \cdots \gpopv[k] b_o$ and
$c \gpopv[k] c_1 \gpopv[k] \cdots \gpopv[k] c_p$.
Using Lemma l:approxsubseq repeatedly we get
$b \append c \gpopv[k] b_1 \append c \gpopv[k] \cdots \gpopv[k] b_o \append c$,
$c \append b_o \gpopv[k] c_1 \append b_o \gpopv[k] \cdots \gpopv[k] c_p \append b_o$.
Since $b_o \append c \eqi c \append b_o$ and employing
Lemma l:approxsubseq we see
$\Slow[k](b \append c) \geqslant \Slow[k](b) + \Slow[k](c)$
for all $b,c \in \GTLS$. We conclude
\Slow[k](a) = \Slow[k](a_1 \append \cdots \append a_n) \geqslant \sum_{i=1}^n \Slow[k](a_i)
with a straight forward induction on $n$.
It remains to verify $\Slow[k](a) \leqslant \sum_{i=1}^n \Slow[k](a_i)$.
For this we show that $a \gpopv[k] b$ implies $\Slow[k](b) < \sum_{i=1}^n \Slow[k](a_i)$
by induction on $\Slow[k](a)$.
Consider the base case $\Slow[k](a) = 0$.
Since $a$ is ground it follows that $a = \nil$, the claim is trivially satisfied.
For the inductive step $\Slow[k](a) > 1$,
let $a \gpopv[k] b$.
Since $a$ is a sequence, $a \cpopv{ms} b$.
Hence $b \eqi b_1 \append \cdots \append b_n$ where $a_i \geqpopv[k] b_i$
and thus $\Slow[k](b_i) \leqslant \Slow[k](a_i)$ for all $i =1,\dots,n$.
Additionally $a_{i_0} \gpopv[k] b_{i_0}$ and hence $\Slow[k](b_{i_0}) < \Slow[k](a_{i_0})$
for at least one $i_0 \in \{1,\dots,n\}$.
As in the first half of the proof, one verifies $\Slow[k](b_i) \leqslant \Slow[k](b)$
for all $i = 1,\dots,n$.
Note $\Slow[k](b) < \Slow[k](a)$ as $a \gpopv[k] b$,
hence induction hypothesis is applicable to $b$ and all $b_i$ ($i = 1,\dots,n$).
It follows that
\begin{align*}
\Slow[k](b) = \sum_{c \in b} \Slow[k](c)
= \sum_{i=1}^n \sum_{c \in b_i} c
= \sum_{i=1}^n \Slow[k](b_i)
< \sum_{i=1}^n \Slow[k](a_i) \tpkt
\end{align*}
This concludes the second half of the proof.
The central theorem of this section states that $\Slow[k](f(a_1, \dots, a_n))$
is polynomial in $\sum_{i}^n \Slow[k](a_i)$,
where the polynomial bound depends only on $k$ and the rank $p$ of $f$.
The proof of this is rather involved.
To cope with the multiset comparison underlying $\cpopv{ep}[k]$,
we introduce as a first step an order-preserving extension $\MSlow{n}{k}$ of $\Slow[k]$ to multisets of sequences,
in the sense that $\MSlow{n}{k}(\seq{a}) > \MSlow{m}{k}(\seq[m]{b})$ holds
whenever $\mset{\seq[n]{a}} \mextension{\gpopv[k]} \mset{\seq[m]{b}}$ (provided $k \geqslant m,n$, c.f. Lemma <ref>).
As the next step toward our goal, we estimate $\Slow[k](f(a_1,\dots,a_n))$ in terms
of $\MSlow{n}{k}(\seq{a})$
whenever $n \leqslant k$ and $\rk(f) \leqslant k$.
Technically we bind following functions by polynomials $q_{k,p}$.
For all $k,p \in \N$ with $k \geqslant 1$
we define $\ofdom{\Fpop{k}{p}}{\N \to \N}$ as
\begin{multline*}
\Fpop{k}{p}(m) \defsym \max\{ \Slow(f(\seq{a})) \mid \\
f(\seq{a}) \in \GTS,
~\rk(f) \leqslant p,~n \leqslant k
\text{ and } \MSlow{n}{k}(a_1,\dots,a_n) \leqslant m \} \tpkt
\end{multline*}
Noting that also $\MSlow{n}{k}(\seq{a})$ is polynomial in $\max_{i=1}^n\Slow[k](a_i)$, say $q_k$, which depends only on $k$,
we obtain $\Slow(f(\seq{a})) \leqslant q_{k,p}(q_{k}(\max_{i=1}^n \Slow[k](a_i)))$ whenever $k \geqslant n$.
The definition of $\MSlow{n}{k}$ is defined in terms of an order-preserving homomorphism from $\msetover(\N)$ to $\N$.
To illustrate the construction carried out below, consider the following example.
Let $k \geqslant 1$ and
let $c > m_1 \geqslant \dots \geqslant m_k$ be natural numbers in descending order, dominated by $c \in \N$.
Consider multisets $\msetover(\N)$ of size $k$.
If we conceive such multisets as base-$c$ representations of numbers using $k$ digits, then we can form
a chain $M_1 \mextension{>} M_2 \mextension{>} \dots$ that can be understood as a decreasing counter
that wrongly wraps from $\{m_1,\dots, m_i+1,0,\dots,0\}$ to $\{m_1,\dots, m_i,m_i,\dots,m_i\}$.
It is not difficult to prove that the maximal length of such a chain is bounded by
\begin{align*}
\sum_{m_2=1}^{m_1} \dots \sum_{m_{k-1}=1}^{m_{k-2}} \sum_{m_k=1}^{m_{k-1}} m_k
~\in~\sum_{m_2=1}^{m_1} \dots \sum_{m_{k-1}=1}^{m_{k-2}} \Omega( m_{k-1}^2 )
~\in~\Omega( m_1^k ) \tpkt
\end{align*}
We now show that this upper bound serves also as a lower bound for
multisets $\msetover(\N)$ of size $n \leqslant k$.
As in the example,
the function $\ofdom{\homo{n}{k}{c}}{\N^l \to \N}$ (where $n \leqslant k$) defined below interprets
multisets $M \in \msetover(\N)$ as natural numbers encoded in base-$c$ with $k$ digits,
where the $i$ largest $m_i \in M$ represents the $i$ most significant digit.
Formally, for $k\geqslant n \in \N$ and $c \in \N$ we define the family of functions $\ofdom{\homo{n}{k}{c}}{\N^l \to \N}$
such that
\begin{align*}
\homo{n}{k}{c}(\seq[n]{m}) = \sum_{i=1}^n \sort{n}(\seq[n]{m},i) \cdot c^{(k - i)} \tpkt
\end{align*}
Here $\sort{n}(\seq[n]{m},i)$ denote the $i$ element of $\seq[n]{m}$ sorted in descending order, i.e.,
$\sort{n}(\seq[m]{n},i) \defsym m_{\pi(i)}$
for $i =1,\dots,n$ and some permutation $\pi$ such that $m_{\pi(i)} \geqslant m_{\pi(i+1)}$ ($i \in \{1,\dots,n-1\}$).
Let $k,n,c \in \N$ such that $k \geqslant 1$ and $k \geqslant n$. Then for all $\seq[l]{n} \in \N$ we have:
* $c > \max\set{\seq[l]{n}}$ implies $c^{n} > \homo{l}{n}{c}(\seq[n]{m})$.
* $\mset{\seq[n]{m}} \mextension{>} \mset{\seq[n']{m'}}$ implies $\homo{n}{k}{c}(\seq[n]{m}) > \homo{n'}{k}{c}(\seq[n']{m'})$
for all $c > \seq[n]{m} \geqslant 1$.
The mapping $\MSlow{n}{k}$ is obtained by extend $\homo{l}{k}{\cdot}$ to multisets over $\GTLS$.
Let $k,n \in \N$ such that $k \geqslant n$.
We define $\ofdom{\MSlow{n}{k}}{\GLS^n \to \N}$ as
\MSlow{n}{k}(\seq[n]{a}) \defsym \homo{n}{k}{c}(\Slow[k](a_1),\dots,\Slow[k](a_n))
where $c = 1 + \max \set{\Slow[k](a_i) \mid i \in \{1,\dots,n\}}$.
By Lemma l:homo4, $\MSlow{l}{k}(\seq[l]{a})$ is polynomially bounded
in $\Slow[k](a_i)$ ($i = 1,\dots,l$).
By Lemma l:homo5 we obtain that
$\MSlow{l}{k}$ is indeed order preserving as outlined above.
Let $\seq{a},\seq[m]{b}\in \GTLS$ and let $k \geqslant m,n$. Then
\mset{\seq{a}} \mextension{\gpopv[k]} \mset{\seq[m]{b}}
\quad \IImp \quad \MSlow{n}{k}(\seq{a}) > \MSlow{m}{k}(\seq[m]{b}) \tpkt
In Theorem <ref> below we prove $\Fpop{k}{p}(m) \leqslant c_{k,p} \cdot {(m+2)}^{d_{k,p}}$,
where the constants $c_{k,p},d_{k,p} \in \N$ are defined as follows:
$d_{k,0} \defsym k+1$ and $d_{k,p+1} \defsym {(d_{k,p})}^{k}+1$;
further we set $c_{k,0} \defsym k^k$ and $c_{k,p+1} \defsym {(c_{k,p} \cdot k)}^e$
where $e = {\sum_{i=0}^k {(k \cdot d_{k,p})}^i}$.
Inevitably the proof of Theorem <ref> is technical,
the reader is advised to skip the formal proof on the first read.
Theorem <ref> is proven by induction on $p$ and $m$.
Consider term $f(\seq{a})$ with $k \geqslant n$ and $\MSlow{n}{k}(\seq{a}) \leqslant m$.
At the heart of the proof, we have to show that
$c_{k,p} \cdot {(m+2)}^{d_{k,p}} > \Slow[k](b)$ for arbitrary $b$ with $f(\seq{a}) \gpopv[k] b$.
The most involved case is $f(\seq{a}) \cpopv{ialst}[k] b$ for $b = \lseq[o]{b}$.
Here it is fundamental to give precise bounds on the elements $b_j$
with $f(\seq{a}) \gppv[k][l] b_j$.
Since $\gppv[k][l]$ constraints $b_j$ to only contain symbols ranked below $\rk(f) = p$ in the precedence,
conceptually $\Slow[k](b_j)$ is bounded by iterated application of the induction hypothesis on $p$.
Since $l$ essentially controls the depth of $b_j$ (compare Example <ref>),
$l$ serves as a bound on the number of iterations.
To properly account for all cases of $\gppv[k][l]$, matters get slightly more involved.
To bind $\Slow[k](b_j)$ sufficiently,
we define for $l \in \N$ a family of auxiliary functions
$\ofdom{g_{l,k,p}}{\N \to \N}$ such that
\begin{align*}
g_{l,k,p}(m) & =
\begin{cases}
k^l \cdot m^l & \text {if $p = 0$ or $l = 0$, and} \\
c_{k,p-1} \cdot (m \cdot g_{l-1,k,p}(m))^{k \cdot d_{k,p-1}} & \text{otherwise.}
\end{cases}
\end{align*}
Having as premise the induction hypothesis (on $p$) of the main proof, the next lemma verifies that
$g_{l,k,\rk(f)}(m+2)$ sufficiently binds $\gppv[k][l]$-descendants of $f(\seq{a})$.
Let $f(\seq{a}) \in \GTS$.
Let $k \geqslant n$ and
$m \geqslant \MSlow{n}{k}(\seq{a})$.
Suppose $\Fpop{k}{p}(m') \leqslant c_{k,p} {(m'+2)}^{d_{k,p}}$ for all $p < \rk(f)$ and $m'$.
Then $f(\seq{a}) \gppv[k][l] b$ implies $\Slow(b) \leqslant g_{k,l,\rk(f)}(m+2)$
for all $b \in \GTLS$.
We prove the claim by induction on $l$
and case analysis on $f(\seq{a}) \gppv[k][l] b$.
First note that $f(\seq{a})\cppv{st}[k][l] b$ implies
that $a_i \gppv[k][l] b$ for some $i \in \{1,\dots,n\}$
and consequently $\Slow(b) \leqslant \Slow(a_i)$.
As by definition $\Slow(a_i) \leqslant m$ the lemma follows trivially.
As in the base case $l = 1$ either $b = \nil$ or $f(\seq{a}) \cppv{st}[k][l] b$,
it suffices to consider only the remaining cases of the inductive step.
Assuming $f(\seq{a}) \gppv[k][l+1] b$ we show
$\Slow(b) \leqslant g_{k,l+1,\rk(f)}(m+2)$.
$f(\seq{a}) \cppv{ia}[k][l+1] b$ where $b = g(\seq[o]{b})}$]
$f(\seq{a}) \gppv[k][l] b_j$ for all $j = 1,\dots,o$, and $f \sp g$.
Set $m' \defsym \MSlow{o}{k}(\seq[o]{b})$. We have
\begin{align*}
m' & < \max\set{\Slow[k](b_j) + 1 \mid j \in \{1,\dots,o\}}^k
&& \text{by definition and Lemma~\eref{l:homo}{4}} \\
& \leqslant {(g_{k,l,\rk(f)}(m+2)+ 1)}^k
&& \text{applying induction hypothesis $o$ times}
\end{align*}
As the assumption also gives $\rk(g) < \rk(f)$
we have
\begin{align*}
\Slow(b)
& \leqslant \Fpop{k}{\rk(g)}(m') && \text{by definition of $\Fpop{k}{\rk(g)}$} \\
& \leqslant c_{k,\rk(g)} \cdot {(m' + 2)}^{d_{k,\rk(g)}} && \text{by assumption} \\
& \leqslant c_{k,\rk(f)-1} \cdot {(m' + 2)}^{d_{k,\rk(f)-1}} && \text{as $\rk(g) < \rk(f)$} \\
% & = \homo{o}{k}{\max\set{\Slow[k](c_i) + 1 \mid i \in \{1,\dots,o\}}}(\Slow[k](c_1),\dots,\Slow[k](c_n)) \\
& < c_{k,\rk(f)-1} \cdot {({(g_{k,l,\rk(f)}(m+2)+ 1)}^k + 2)}^{d_{k,\rk(f)-1}} && \text{substituting bound for $m'$} \\
& \leqslant c_{k,\rk(f)-1} \cdot {(g_{k,l,\rk(f)}(m+2)+ 3)}^{k \cdot d_{k,\rk(f)-1}} && \text{using $1 \leqslant k$}\\
& \leqslant c_{k,\rk(f)-1} \cdot {((m+2) \cdot g_{k,l,\rk(f)}(m+2))}^{k \cdot d_{k,\rk(f)-1}} && \text{using $2 \leqslant g_{k,l,\rk(f)}(m+2)$}\\
& = g_{k,l+1,\rk(f)}(m+2) && \text{using $\rk(f) > 0$}\tpkt
\end{align*}
$f(\seq{a}) \cppv{ialst}[k][l+1] b$ where $b = \lseq[o]{b}}$]
Ordering constraints give $o \leqslant \width(a) + k$
and $f(\seq{a}) \gppv[k][l] b_j$ ($j = 1,\dots,o$).
Exploiting that $a_i$ is ground, a standard induction shows
that $\width(a_i) \leqslant \Slow[k](a_i)$, and consequently
$\width(a_i) \leqslant m$.
\begin{align}
\label{e:bindwidth}
\tag{\dag}
o \leqslant \width(a) + k
= \max\set{1,\width(a_1), \dots,\width(a_n)} + k
\leqslant m + k \leqslant k \cdot (m + 1) \tpkt
\end{align}
If $\rk(f) = 0$ then
we see
\begin{align*}
\Slow[k](b) & = \sum_{j=1}^o \Slow[k](b_i) && \text{using Lemma~\ref{l:slowsum}}\\
& \leqslant \sum_{j=1}^o g_{k,l0}(m+2) && \text{applying induction hypothesis $o$ times}\\
& \leqslant k \cdot (m + 1) \cdot g_{k,l0}(m+2) && \text{using \eqref{e:bindwidth}} \\
& = k \cdot (m + 1) \cdot k^{l} \cdot {(m + 2)}^{l} && \text{by assumption $\rk(f) = 0$} \\
& < k^{l+1} \cdot {(m + 2)}^{l+1}
= g_{k,l+10}(m+2) \tpkt
\end{align*}
Otherwise $\rk(f) > 0$ and we conclude
\begin{align*}
\Slow[k](b) & \leqslant k \cdot (m + 1) \cdot g_{k,l,\rk(f)}(m+2)
&& \text{as in the case $\rk(f) = 0$}\\
& < c_{k,\rk(f)-1} \cdot {((m + 2) \cdot g_{k,l,\rk(f)}(m+2))}^{k \cdot d_{k,\rk(f)-1}}
&& \text{as $k \leqslant c_{k,\rk(f)-1}$ and $1 < k \cdot d_{k,\rk(f)-1}$} \\
& = g_{k,l+1,\rk(f)}(m+2) && \text{by assumption $\rk(f) > 0$}\tpkt
\end{align*}
For all $k,p \in \N$ there exist constants $c,d \in \N$ (depending only on $k$ and
$p$) such that for all $m$: $\Fpop{k}{p}(m) \leqslant c \cdot {(m+2)}^{d}$.
Fix $a = f(\seq[n]{a}) \in \GTS$ such that $\rk(f) = p$, $k \geqslant n$ and $\MSlow{n}{k}(a_1,\dots,a_n) \leqslant m$.
To show the theorem, we prove that for all $b$ with $a \gpopv[k] b$
we have $\Slow[k](b) < c_{k,p} \cdot {(m+2)}^{d_{k,p}}$ by induction on the rank $p$ and side induction on $m$.
$ p = 0 $
The base case of the side induction is trivial, so consider the inductive step $m > 0$.
We first prove $\Slow[k](b) < k^k \cdot {(m+1)}^{k+1} + k^l \cdot {(m + 2)}^l$
by induction on $\gpopv[k][l]$.
$f(\seq{a}) \cpopv{st}[k][l] b}$] Then $a_i \geqpopv[k][l] b$, and
we conclude since $\Slow[k](b) \leqslant \Slow[k](a_i) \leqslant m$ using
the assumptions and Lemma l:homo4.
$f(\seq{a}) \cpopv{ep}[k][l] b$ where $g(\seq[o]{b})$
The ordering constraints give $o \leqslant k$, $f \ep g$ and
$\mset{\seq{a}} \mextension{\gpopv[k][l]} \mset{\seq[o]{b}}$.
Set $m' \defsym \MSlow{o}{k}(\seq[o]{b})$.
Hence $m' < \MSlow{n}{k}(\seq[n]{a}) \leqslant m$
by Lemma <ref> and assumption $n \leqslant k$.
Thus side induction hypothesis gives
$\Fpop{k}{0}(m') \leqslant c_{k,0} \cdot {(m' + 2)}^{c_{k,0}} = k^k {(m' + 2)}^{k+1}$.
As the ordering constraints imply $\rk(g) = \rk(f) = 0$
we conclude
\begin{align*}
\Slow[k](g(\seq[o]{b}))
& \leqslant \Fpop{k}{0}(m')
&& \text{by definition of $\Fpop{k}{0}$} \\
& = c_{k,0} \cdot {(m' + 2)}^{d_{k,0}}
&& \text{by side induction hypothesis}\\
& = k^k \cdot {(m' + 2)}^{k+1}
&& \text{by definition}\\
& < k^k \cdot {(m+1)}^{k+1} + k^{l} \cdot {(m + 2)}^{l}
&& \text{using $m' < m$.}
\end{align*}
$f(\seq{a}) \cpopv{ialst}[k][l]$ where $\lseq[o]{b}$
The ordering constraints give
(i) $a \gpopv[k][l-1] b_{j_0}$ for some $j_0 \in\{1,\dots,o\}$,
(ii) $a \gppv[k][l-1] b_j$ for all $j \neq j_0$, and
(iii) $o \leqslant \width(a) + k$.
By induction hypothesis on (i) we get $\Slow[k](b_{j_0}) < k^k \cdot {(m+1)}^{k+1} + k^{l-1} \cdot {(m + 2)}^{l-1}$,
the preparatory Lemma <ref> on (ii) gives $\Slow[k](b_j) \leqslant k^{l-1} \cdot {(m + 2)}^{l-1}$ for $j \not = j_o$.
Exactly as in Equation (<ref>) we obtain $o \leqslant k \cdot (m + 1) < k \cdot (m + 2)$ from (iii).
Summing up we have
\begin{align*}
\Slow[k](b) & = \sum_{j =1}^o \Slow[k](b_j)
&& \text{by Lemma~\ref{l:slowsum}}\\
& < k^k \cdot {(m+1)}^{k+1} + k^{l-1} \cdot {(m + 2)}^{l-1}
&& \text{by induction hypothesis on (i), and}\\
& \quad + (k \cdot (m+2) - 1) \cdot k^{l-1} \cdot {(m + 2)}^{l-1}
&& \text{using $o < k \cdot (m+2)$ and Lemma~\ref{l:pop:aux} on (ii)}\\
% & = k^k \cdot {(m+1)}^{k+1} + k^{l-1} \cdot {(m + 2)}^{l-1} \\
% & \quad + k^{l} \cdot {(m + 2)}^l - k^{l-1} \cdot {(m + 2)}^{l-1} \\
& = k^k \cdot {(m+1)}^{k+1} + k^{l} \cdot {(m + 2)}^{l} \tpkt
\end{align*}
Since $\gpopv[k] = \gpopv[k][k]$ this preparatory step gives
\begin{align*}
\Slow[k](b) < k^k \cdot {(m+1)}^{k+1} + k^k \cdot {(m + 2)}^k
% & = k^k \cdot ({(m+1)}^{k+1} + {(m + 2)}^k) \\
\leqslant k^k \cdot {(m + 2)}^{k+1}
\end{align*}
and concludes the base case.
By induction hypothesis on $p$ we get
$\Fpop{k}{p}(m) \leqslant c_{k,p} \cdot {(m+2)}^{d_{k,p}}$, side induction hypothesis gives
$\Fpop{k}{p+1}(m') \leqslant c_{k,p+1} \cdot {(m+2)}^{d_{k,p+1}}$ for all $m' < m$.
A standard induction reveals $g_{l,k,p+1}(n) \leqslant c_{k,p}^{\sum_{i=0}^{l-1} {(k \cdot d_{k,p})}^i} \cdot n^{\sum_{i=1}^{l-1} {(k \cdot d_{k,p})}^i}$
for all $n \in \N$.
We continue with the proof of the lemma, and show that for all $l \geqslant 1$, if $f(\seq{a}) \gpopv[k][l] b$ then
\begin{equation}
\label{t:pop:a}
\tag{$\ddag$}
\Slow[k](b) \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m + 2)}^{{(k \cdot d_{k,p})}^l}
\end{equation}
by induction on $l$.
The only interesting case is $a \cpopv{ialst}[k][l+1] b$.
Then $b = \lseq[o]{b}$ with
(i) $a \gpopv[k][l] b_{j_0}$ for some $j_0 \in \{1,\dots,o\}$,
(ii) $a \gppv[k][l] b_j$ for all $j \neq j_0$, and
(iii) $o \leqslant \width(a) + k$.
By induction hypothesis on (i) we get $\Slow[k](b_{j_0}) \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m + 2)}^{{(k \cdot d_{k,p})}^{l}}$,
Lemma <ref> on (ii) gives $\Slow[k](b_j) \leqslant g_{l,k,p+1}(m + 2)$ for $j \not = j_o$ and
(iii) gives $o \leqslant k \cdot (m + 1)$ as in Equation (<ref>).
Summing up we see
\begin{align*}
\Slow[k](b) & = \sum_{j=1}^{o} \Slow[k](b_j) && \text{by Lemma~\ref{l:slowsum}}\\
& \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m + 2)}^{{(k \cdot d_{k,p})}^l} && \text{by induction hypothesis}\\
& \quad + k \cdot (m+1) \cdot g_{l,k,p+1}(m+2) && \text{by Lemma~\ref{l:pop:aux} and bound on $o$}\\
& \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m + 2)}^{{(k \cdot d_{k,p})}^{l}} \\
& \quad + k \cdot (m+1) \cdot c^{\sum_{i=0}^{l-1} {(k \cdot d_{k,p})}^i} \cdot {(m+2)}^{\sum_{i=1}^{l-1} {(k \cdot d_{k,p})}^i} && \text{bound on $g_{l,k,p+1}(m+2)$}\\
& < c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m + 2)}^{{(k \cdot d_{k,p})}^{l}} \\
& \quad + c_{k,p+1} \cdot {(m+2)}^{\sum_{i=0}^{l-1} {(k \cdot d_{k,p})}^i}
&& \text{using $k \cdot c_{k,p}^{\sum_{i=0}^{l-1} {(k \cdot d_{k,p})}^i} \leqslant c_{k,p+1}$}\\
% & \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} \\
% & \quad + c_{k,p+1} \cdot ((m + 2)^{{(k \cdot d_{k,p})}^{l}} \cdot {(m+2)}^{\sum_{i=0}^{l-1} {(k \cdot d_{k,p})}^i})
% && \\
& \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m+2)}^{\sum_{i=0}^{l} {(k \cdot d_{k,p})}^i} \\
& \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m+2)}^{{(k \cdot d_{k,p})}^{l+1}}
\end{align*}
as desired.
Using $\eqref{t:pop:a}$, $\gpopv[k] = \gpopv[k][k]$
and ${(k \cdot d_{k,p})}^k < {(k \cdot d_{k,p})}^k + 1 < d_{k,p+1}$ we finally get
\begin{align*}
\Slow[k](b) & \leqslant c_{k,p+1} \cdot {(m+1)}^{d_{k,p+1}} + c_{k,p+1} \cdot {(m + 2)}^{{(k \cdot d_{k,p})}^k} \\
& = c_{k,p+1} \cdot ({(m+1)}^{d_{k,p+1}} + {(m + 2)}^{{(k \cdot d_{k,p})}^k}) \\
& < c_{k,p+1} \cdot {(m + 2)}^{d_{k,p+1}}
\end{align*}
and conclude the inductive case.
As a consequence, the number of $\gpopv[k]$-descents on
basic terms interpreted with predicative interpretation $\ints$
is polynomial in sum of depths of normal arguments.
Let $f \in \DS$ with at most $k$ normal arguments. There exists a constant $d \in \N$ depending only on $k$ such that:
$$\Slow[k](\ints(f(\sn{\seq{m}{u}}{\vec{v}}))) \in \bigO\bigl({(\max_{i=1}^m \depth(u_i))}^{d}\bigr)$$
for all $\seq[m+n]{u} \in \Val$.
Let $s = f(\sn{\seq{m}{u}}{\vec{v}})$ be as given by the corollary. Recall that
\begin{align*}
\ints(s)
& = \lst{\fn(\intn(u_1), \dots, \intn(t_{u_m}))} \append \ints(u_{m+1}) \append \cdots \append \ints(u_{m+n}) \\
& = \lst{\fn(\natToSeq{\depth(u_1)}, \dots, \natToSeq{\depth(u_m)})}
\end{align*}
As $\Slow[k](\theconst)$ is constant, say $\Slow[k](\theconst) = c$, by Lemma <ref> we see that
$\Slow[k](\natToSeq{\depth(u_i)}) = c \cdot \depth(u_i)$.
Putting things together is tedious but not difficult:
\begin{align*}
\Slow[k](s) =
& \Slow[k](\fn(\natToSeq{\depth(u_1)}, \dots, \natToSeq{\depth(u_m)}))
&& \text{by Lemma~\ref{l:slowsum}}
\\
& \leqslant \Fpop{k}{\rk(f)}\bigl(\MSlow{l}{k}(\natToSeq{\depth(u_1)}, \dots, \natToSeq{\depth(u_m)})\bigr)
\\
& \leqslant \Fpop{k}{\rk(f)}\Bigl({\bigl(1+ \max_{i=1}^m \Slow[k](\NM{u_i})\bigr)}^k\Bigr)
&& \text{by Lemma~\eref{l:homo}{4}}
\\
& \leqslant \Fpop{k}{\rk(f)}\Bigl({\bigl(c \cdot (1 + \max_{i=1}^m \depth(u_i))\bigr)}^k\Bigr)
&& \text{using $\Slow[k](\NM{u_i}) \leqslant c \cdot \depth(u_i)$}
\\
& \in \bigO\Bigl({\bigl(c \cdot (1 + \max_{i=1}^m \depth(u_i))\bigr)}^{k+d'}\Bigr)
&& \text{by Theorem~\ref{t:pop}}
\\
& \in \bigO\bigl({\max_{i=1}^m \depth(u_i)}^{k+d'}\bigr)
\end{align*}
Set $d \defsym k + d'$ and note that $d$ depends only on $k$ and $\rk(f)$ as desired.
§ PREDICATIVE EMBEDDING
Fix a predicative recursive TRS $\RS$ and signature $\FS$,
and let $\gpop$ be the polynomial path order underlying $\RS$
based on the (admissible) precedence $\qp$.
We denote by $\qp$ also the homomorphic precedence on $\FSn$
given by: $\fn \ep \gn$ if $f \ep g$ and $\fn \sp \gn$ if $f \sp g$.
Further, we set $f \sp \theconst$ for all $\fn \in \FSn$.
We denote by $\gpopv[\ell]$ (and respectively $\gppv[\ell]$) the approximation
given in Definition <ref> (respectively Definition <ref>) with underlying precedence $\qp$.
We now establish the embedding of $\irew$ into $\gpopv[\ell]$
for some $\ell$ depending only on $\RS$.
To simplify matters, we suppose for now that $\RS$ is completely defined.
Since then normal forms and values coincide, $s \irew t$
if $s = C[l\sigma]$ and $t = C[r\sigma]$ where
${l \to r} \in \RS$ and all arguments of $l\sigma$ are values.
In particular, this implies that the substitution $\sigma$
maps variables to values.
Lemma <ref> below proves the embedding of root steps
for the case $l \gpop r$. In Lemma <ref> we then show that the embedding is closed under contexts.
The next lemma, exploited in Lemma <ref>,
connects the auxiliary orders $\gsq$ and $\gppv[k][l]$ (compare Example <ref>).
Suppose $s = f(\pseq{s}) \in \BASICS$, $t \in \TERMS$ and $\ofdom{\sigma}{\VS \to \Val}$.
Then for predicative interpretation $\intq \in \{\ints, \intn\}$
we have
s \gsq t \quad \IImp \quad \fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t}] \intq(t\sigma) \tpkt
Let $s = f(\pseq{s}) \in \BASICS$, $t \in \TERMS$.
We continue by induction on the definition of $\gsq$.
$s \csq{st} t$
Then $s_i \geqsq t$ for some normal argument position $i \in \{1,\dots,k\}$.
Note that by assumption $s \in \BASICS$, $s_i \in \Val$
and so Lemma <ref> (employing ${\gsq} \subseteq {\gpop}$)
gives $s_i \esuperterm t$ and $t \in \Val$, consequently
$t\sigma \in \Val$
and furthermore $\norm{s_i\sigma} \geqslant \norm{t\sigma}$.
As $t\sigma \in \Val$, we get
$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{ialst}[2 \cdot \size{t}] \nil = \ints(t\sigma)$ which concludes the case $\intq = \ints$.
For the case $\intq = \intn$,
observe $\intn(s_i\sigma) = \NM{s_i\sigma}$ and $\intn(t\sigma) = \NM{t\sigma}$
since both $s_i\sigma$ and $t\sigma$ are values.
If $\norm{s_i\sigma} = \norm{t\sigma}$ then obviously
$\intn(s_i\sigma) = \intn(t\sigma)$. Otherwise
$\norm{s_i\sigma} > \norm{t\sigma}$ and then
$\intn(s_i\sigma) \cppv{ms}[2\cdot\size{t}] \intn(t\sigma)$,
employing $\theconst \cppv{ialst}[2\cdot\size{t}] \nil$.
Hence overall $\intn(s_i\sigma) \geqppv[2\cdot\size{t}] \intn(t\sigma)$.
Since the position $i$ is normal, $\intn(s_i\sigma)$ is a direct subterm of $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma))$
and we conclude $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{st}[2\cdot\size{t}] \intn(t\sigma)$ as desired.
$s \csq{ia} t$
By the assumption $t = g(\pseq[m][n]{t})$ where $f \sp g$ and $s \gsq t_i$ for all $j \in =1,\dots,m+n$.
We consider the more involved case $t\not\in\Val$.
Let $\ints(t_i\sigma) = \lst{v_{i,1}~\cdots~v_{i,j_i}}$ for all safe argument positions
$i = m+1, \dots m+n$ of $g$, i.e.,
\begin{align*}
\ints(t\sigma)
& = \lst{\gn(\intn(t_1), \dots, \intn(t_m))~v_{m+1,1}~\cdots~v_{m+1,j_{m+1}}\quad\cdots\quad v_{m+n,1}~\cdots~v_{m+n,j_{m+n}}} \tpkt
\end{align*}
By induction hypothesis on $i = 1,\dots, m$
we get $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t_i}] \intn(t_i\sigma)$.
Since $\size{t_i} < \size{t}$, using Lemma l:approxkmon
we have in particular $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma))
\gppv[2\cdot\size{t} - 2] \intn(t_i\sigma)$.
Using this, $\fn \sp \gn$, and $m < \size{t}$ we conclude
\begin{equation}
\label{eq:egsq:gtv}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{ia}[2\cdot\size{t} -1] \gn(\intn(t_1), \dots, \intn(t_m))\tpkt
\end{equation}
By induction hypothesis on safe argument positions of $g$ we get
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t_i}] \lst{v_{i,1}~\cdots~v_{i,j_i}} = \ints(t_i\sigma)
for all $i =m+1,\dots,m+n$.
Using a simple inductive argument
one verifies
\begin{equation}
\label{eq:egsq:safe}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t}-1] v_{i,j} \text{ for all $i = m+1,\dots,m+n$ and $j = 1,\dots,j_i$}
\end{equation}
from this.
Let $\intq \in \{\ints, \intn\}$.
\begin{align*}
\len(\intq(t\sigma))
& \leqslant 2\cdot\size{t} + \max \{ \norm{s_1\sigma}, \dots, \norm{s_k\sigma}\}
&& \text{by Lemma~\eref{l:int:len}{gsq}} \\
& \leqslant 2\cdot\size{t} + \width(\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma))) \tpkt
\end{align*}
Using this, Equations (<ref>), Equations (<ref>)
and possibly $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{ia}[2\cdot\size{t}] \theconst$
we have $u \cppv{ialst}[2\cdot\size{t}] \intq(t\sigma)$ as desired.
We conclude this auxiliary lemma.
Suppose $s = f(\pseq{s}) \in \BASICS$, $t \in \TERMS$ and $\ofdom{\sigma}{\VS \to \Val}$.
Then for predicative interpretation $\intq \in \{\ints, \intn\}$
we have
s \gpop t \quad \IImp \quad \intq(s\sigma) \gpopv[2\cdot\size{t}] \intq(t\sigma) \tpkt
Let $s$, $t$, $\sigma$ be as given in the lemma.
We prove the stronger assertions
* $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gpopv[2\cdot\size{t}] \ints(t\sigma)$,
* $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t}] \ints(t\sigma)$ if $t \in \termsbelow[\Fun(s)]$, and
* $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \append \NM{s\sigma} \gpopv[2\cdot\size{t}] \intn(t\sigma)$.
We continue with the proof of the assertions by induction on $\gpop$.
$s \cpop{st} t$
Exactly as in Lemma <ref> we conclude $s_i\sigma \esuperterm t\sigma$ and $t\sigma \in \Val$.
The latter implies $\ints(t\sigma) = \nil$ and thus
Assertion <ref> and Assertion <ref> are immediate.
For Assertion <ref>, observe that
= \norm{t\sigma} \leqslant \norm{s_i\sigma}
\leqslant \width(\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \append \NM{s\sigma})$
where the latter inequality is obtained by case analysis on $i$.
From this and $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{ia}[2\cdot\size{t} - 1] \theconst$ we get
$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \append \NM{s\sigma} \cpopv{ms}[2\cdot\size{t}] \NM{t\sigma} = \intn(t\sigma)$ as desired.
$s \cpop{ia} t$
The assumption gives $t = g(\pseq[m][n]{t})$ where $f \sp g$
and further $s \gsq t_i$ for all normal argument positions $i = 1,\dots,m$ and
$s \gpopps t_i$ for all safe argument positions $i = m+1,\dots,m+n$ of $g$.
Additionally $t_{i_0} \not\in \termsbelow[\Fun(s)]$ for at most one argument position $i_0$.
We first verify Assertion <ref> and Assertion <ref> for the non-trivial case $t \not \in \Val$.
Set $v \defsym \gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma))$ and
let $\ints(t_i\sigma) = \lst{v_{i,1}~\cdots~v_{i,j_i}}$ for all safe argument positions
$i = m+1,\dots,m+n$, hence
\begin{align*}
\ints(t\sigma) = \lst{\gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma))~v_{m+1,1}~\cdots~v_{m+1,j_{m+1}}\quad\cdots\quad v_{m+n,1}~\cdots~v_{m+n,j_{m+n}}} \tpkt
\end{align*}
Applying Lemma <ref> on all normal arguments of $t$ we see
\begin{align}
\label{e:c3:0}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t} - 1] \gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma))
\end{align}
from the assumptions $\fn \sp \gn$ and $s \gsq t_i$ for all $i = 1,\dots,m$.
Since $s \gpop t_{i_0}$ by assumption, induction hypothesis on $i_0$ gives
\begin{align*}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gpopv[2\cdot \size{t_{i_0}}] \lst{v_{i_0,1}, \dots, v_{i_0,j_{i_0}}} = \ints(t_{i_0}\sigma)\tpkt
\end{align*}
Employing $2\cdot\size{t_{i_0}} \leqslant 2\cdot\size{t} - 1$,
it is not difficult to check that due to the above inequality we have
\begin{align}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) & \gpopv[2\cdot\size{t}-1] {v_{i_0,j_0}}
&& \text{for some $j_0 \in \{1,\dots,j_{i_0}\}$}\label{e:c3:1}\\
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) & \gppv[2\cdot\size{t}-1] {v_{i_0,j}}
&& \text{for all $j = 1,\dots,j_{i_0}$, $j \not=j_0$.}\label{e:c3:2}
\end{align}
Similar induction hypothesis on safe argument positions $i = m+1,\dots,m+n$ ($i \not=i_0$) of $g$,
where in particular $t_i \in \termsbelow[\Fun(s)]$ by assumption,
\begin{align}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) & \gppv[2\cdot\size{t}-1] {v_{i,j}}
&& \text{for all $i = m+1,\dots,m+n$, $i \not=i_0$ and $j = 1,\dots,j_{i}$.} \label{e:c3:3}
\end{align}
Observe $\len(\ints(t\sigma)) \leqslant \size{t}$ by Lemma l:int:lenS.
Summing up, Assertion <ref> follows by $\cpopv{ialst}[2\cdot\size{t}]$
using Equations (<ref>), (<ref>), (<ref>) and (<ref>).
Likewise, Assertion <ref> follows using additionally
$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{ia}[2\cdot\size{t}-1] \theconst$
\begin{align*}
\len(\intn(t\sigma))
& \leqslant 2\cdot\size{t} + \max \{ \norm{s_1\sigma}, \dots, \norm{s_k\sigma}, \norm{s\sigma}\}
&& \text{by Lemma~\eref{l:int:len}{gpop}} \\
& \leqslant 2\cdot\size{t} + \width(\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \append \NM{s\sigma}) \tpkt
\end{align*}
Finally, for Assertion <ref> we proceed exactly as above, but strengthen the inequality (<ref>) to
$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t}-1] {v_{i_0,j_0}}$
which follows as $t_{i_0} \in \termsbelow[\Fun(s)]$ by assumption, and thus
induction hypothesis can be strengthened to
$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\size{t_{i_0}}] \ints(t_{i_0}\sigma)$.
This concludes the case $\cpop{ia}$.
$s \cpop{ep} t$
Then $t = g(\pseq[m][n]{t})$ where $f \ep g$.
Further, the assumption gives
$\mset{\seq[k]{s}} \gpopmul \mset{\seq[m]{t}}$
and $\mset{\seq[k+l][k+1]{s}} \geqpopmul \mset{\seq[m+n][m+1]{t}}$.
Hence $t \not \in \termsbelow[\Fun(s)]$ and Property <ref> is vacuously
We prove Property <ref> and Property <ref>.
Using that $s_i \in \Val$ for all normal argument positions $i = 1,\dots,m$
and employing Lemma <ref> we see exactly as in the
case $s \cpop{st} t$ above that
$\mset{\seq[k]{s}} \gpopmul \mset{\seq[m]{t}}$ implies
$\mset{\intn(s_1\sigma), \dots, \intn(s_k\sigma)} \mextension{\gpopv[2\cdot\size{t}-1]} \mset{\intn(t_1\sigma), \dots, \intn(t_m\sigma)}$.
\begin{equation}
\label{eq:root:ep}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cpopv{ep}[2\cdot\size{t} -1] \gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma))
\end{equation}
follows as by assumption $\fn \ep \gn$ and clearly $m \leqslant \size{t} \leqslant 2\cdot\size{t} - 1$.
Note that the assumption $\mset{\seq[k+l][k+1]{s}} \geqpopmul \mset{\seq[m+n][m+1]{t}}$ together with
$s_i \in \Val$ for all $i = k+1,k+l$ gives
$t_j \in \Val$, and consequently $\ints(t_j\sigma) = \nil$ for all
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cpopv{ialst}[2\cdot\size{t}]
\lst{\gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma))}% \append \ints(t_{m+1}\sigma) \append \cdots \append \ints(t_{m+n}\sigma)
= \ints(t\sigma)
which concludes Assertion <ref>.
To prove Assertion <ref>
we additionally verify $\norm{t\sigma} \leqslant \norm{s\sigma}$
by case analysis on $\norm{t\sigma}$.
Thus $\NM{s\sigma} \geqpopv[2\cdot\size{t}] \NM{t\sigma}$ follows.
Using this and Equation (<ref>) we obtain
$$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \append \NM{s\sigma}
\cpopv{ms}[2\cdot\size{t}] \gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma)) \append \NM{t\sigma} = \intn(t\sigma)$$
by Lemma l:approxkmon and Lemma l:approxsubseq.
We conclude the lemma.
Let $\ell \geqslant \max\{\ar(\fn) \mid \fn \in \FSn \}$ and $s,t \in \TERMS$. Then for $\intq \in \{\intn,\ints\}$,
\intq(s) \gpopv[\ell] \intq(t) \quad \IImp \quad \intq(C[s]) \gpopv[\ell] \intq(C[t]) \tpkt
It suffices to consider the inductive step.
Consider terms $s = f(s_1,\dots,s_i, \dots, s_{k+l})$ and $t = f(s_1,\dots,t_i, \dots, s_{k+l})$.
We show that for $\intq \in \{\intn,\ints\}$, under the assumption $\intq(s_i) \gpopv[\ell] \intq(t_i)$ also
$\intq(f(s_1,\dots,s_i, \dots, s_{k+l})) \gpopv[\ell] \intq(f(s_1,\dots,t_i, \dots, s_{k+l}))$ holds.
$\intq = \ints$
Consider the non-trivial case $t \not \in \Val$.
Without loss of generality, suppose the first $k$
argument positions of $f$ are normal,
and the remaining $l$ positions are safe.
Depending on the position $i$, we distinguish two cases.
If $i \in \{k+1,\dots,k+l\}$ is safe, then by definition
\begin{align*}
\ints(s) & = \lst{\fn(\intn(s_1), \dots, \intn(s_k))}
\append \ints(s_{k+1}) \append \cdots \append \ints(s_i) \append \cdots \append \ints(s_{k+l}) \text{, and} \\
\ints(t) & = \lst{\fn(\intn(s_1), \dots, \intn(s_k))}
\append \ints(s_{k+1}) \append \cdots \append \ints(t_i) \append \cdots \append \ints(s_{k+l})
\end{align*}
If $i$ is a normal argument position, the assumption
$\intn(s_i) \gpopv[\ell] \intn(t_i)$ and $\ell \geqslant k$ gives
\begin{align*}
\fn(\intn(s_1), \dots, \intn(s_i) ,\dots, \intn(s_k))
\cpopv{ms}[\ell] \fn(\intn(s_1), \dots, \intn(t_i) ,\dots, \intn(s_k))
\end{align*}
and the lemma follows again using Lemma l:approxsubseq.
$\intq = \intn$
Recall that $\intn(s) = \ints(s) \append \NM{s}$ and $\intn(t) = \ints(t) \append \NM{t}$.
If $\norm{s} \geqslant \norm{t}$ then $\intn(s) \gpopv[l] \intn(t)$ follows
from $\ints(s) \gpopv[l] \ints(t)$ and Lemma l:approxsubseq.
Hence suppose $\norm{s} < \norm{t}$.
First, consider the more involved case $t \not \in \Val$.
As $\norm{s} < \norm{t}$ implies that $i$ is a safe argument position of $f$, we thus have
\begin{align*}
\intn(s) & = \lst{\fn(\intn(s_1), \dots, \intn(s_k))}
\append \ints(s_{k+1}) \append \cdots \append \ints(s_i) \append \cdots \append \ints(s_{k+l}) \append \NM{s} \text{, and} \\
\intn(t) & = \lst{\fn(\intn(s_1), \dots, \intn(s_k))}
\append \ints(s_{k+1}) \append \cdots \append \ints(t_i) \append \cdots \append \ints(s_{k+l}) \append \NM{t}
\end{align*}
Using Lemma l:approxmodeqi
and Lemma l:approxsubseq
we see that $\intn(s) \gpopv[\ell] \intn(t)$ follows from
$\ints(s_i) \append \NM{s} \gpopv[\ell] \ints(t_i) \append \NM{t}$.
Note $\norm{s_i} < \norm{s}$
and observe that
the assumption $\norm{s} < \norm{t}$ gives $\norm{t} = \norm{t_i} + 1$
by the shape of $s$ and $t$.
Thus using Lemma l:approxsubseq and the assumption $\intn(s_i) \gpopv[\ell] \intn(t_i)$
we can even prove the stronger property
$\ints(s_i) \append \NM{s_i} \append \theconst \gpopv[\ell] \ints(t_i) \append \NM{t_i} \append \theconst = \intn(t)$.
By similar reasoning we can also prove $t \in \Val$ where $\intn(t) = \norm{t}$.
This concludes the case analysis.
We have established the embedding for completely defined TRSs.
Putting things together we obtain:
This allows us to estimate the derivation height in terms of $\Slow[\ell]$.
Let $\RS$ be a completely defined TRS compatible with $\gpop$.
Define $\ell \defsym \max\{\ar(\fn) \mid \fn \in \FSn \} \cup \{2\cdot\size{r} \mid {l \to r} \in \RS \}$
and $\intq \in \{\intn,\ints\}$. Then $\dheight(s, \irew) \leqslant \Slow[\ell](\intq(s))$.
Suppose $\RS$ is completely defined TRS compatible with $\gpop$, and
let $\ell$ be given by the Lemma.
Consider a maximal $\RS$-derivation
\mparbox[c]{1cm}{s}
\mparbox[c]{1cm}{\irew[\RS]} \mparbox[c]{1cm}{s_1}
\mparbox[c]{1cm}{\irew[\RS]} \mparbox[c]{1cm}{s_2}
\mparbox[c]{1cm}{\irew[\RS]} \mparbox[c]{1cm}{\cdots}
\mparbox[c]{1cm}{\irew[\RS]} \mparbox[c]{1cm}{s_m\tkom}
starting from an arbitrary term $s$, i.e., $m = \dheight(s, \irew[\RS])$.
Using Lemma <ref> together with Lemma <ref> $m$-times
we get
\mparbox[c]{1cm}{\intq(s)}
\mparbox[c]{1cm}{\gpopv[\ell]} \mparbox[c]{1cm}{\intq(s_1)}
\mparbox[c]{1cm}{\gpopv[\ell]} \mparbox[c]{1cm}{\intq(s_2)}
\mparbox[c]{1cm}{\gpopv[\ell]} \mparbox[c]{1cm}{\cdots}
\mparbox[c]{1cm}{\gpopv[\ell]} \mparbox[c]{1cm}{\intq(s_m)}
and consequently $m \leqslant \Slow[\ell](\intq(s))$ by definition.
The final proof step is to lift the requirement that $\RS$ is completely defined.
Call a normal form $s$ garbage if its root symbol is defined.
Let $\bot \not \in \FS$ be a fresh constructor symbol.
For each garbage term $s$ we extend $\RS$ by a rule
that replaces $s$ with $\bot$. Although infinite, the resulting
system is completely defined.
Let $\bot$ be a fresh constructor symbol $\bot \not \in \FS$ and $\RS$ a TRS over $\FS$.
We define $\RSS$ over the signature $\FS \cup \{\bot\}$ by
\RSS \defsym \{t \to \bot \mid \text{$t \in \TA(\FS \cup \{\bot\},\VS)$ is a normal form of $\RS$ with defined root symbol} \}\tpkt
We set $\RSbot \defsym \RS \cup \RSS$.
We extend the precedence $\qp$ on $\FS$ to $\FS \cup \{\bot\}$ so that $\bot$ is minimal.
As clearly $s \cpop{ia} \bot$ for each garbage term $s$,
for predicative TRS $\RS$ the TRS $\RSS$ is compatible with $\gpop$.
Note that $\RSS$ is confluent and terminating, in particular every term
$s$ has a unique normal form with respect to $\RSS$, in notation $\normalise{s}$.
Clearly $\normalise{f(\seq{s})} = \normalise{f(\normalise{s_1},\dots, \normalise{s_n})}$.
Exploiting that the additional rules do not interfere with
pattern matching of $\RS$, the TRS $\RSbot$ is able to simulate $\RS$ in the following sense.
Suppose $\RS$ is a constructor TRS. Then
s \irew t \quad\IImp\quad \normalise{s} \irst[\RSbot] \normalise{t}
Suppose $s \irew t$, i.e., $s = C[f(l_1\sigma, \dots, l_n\sigma)]$ and
$t = C[r\sigma]$ for some context $C$, rule ${f(\seq{l}) \to r} \in \RS$
and substitution $\sigma$ where $l_i \sigma \in \NF(\RS)$ for all $i = 1,\dots,n$.
We continue by induction on $C$.
Let $\sigma_{\normalise{}}(x) \defsym \normalise{x\sigma}$ for all $x \in \dom(\sigma)$.
Consider the base case $C = \hole$. Since $\RS$ is by assumption a constructor TRS,
the direct arguments of the left-hand sides of $\RS$ do not contain defined symbols,
consequently $\normalise{l_i}\sigma = \normalise{l_i\sigma_{\normalise{}}} = l_i\sigma_{\normalise{}}$
is a constructor term
for all $i = 1,\dots,n$.
We conclude the inductive step
\normalise{f(l_1\sigma, \dots, l_n\sigma)} = f(l_1\sigma_{\normalise{}}, \dots, l_n\sigma_{\normalise{}})
\irew[\RSbot] r{\sigma_{\normalise{}}}
\irss[\RSbot] \normalise{(r\sigma)} \tpkt
Here in the first equality we employ that $f(l_1\sigma_{\normalise{}}, \dots, l_n\sigma_{\normalise{}})$ is not a normal form of $\RS$.
For the inductive step, let $s = f(s_1, \dots, s_i, \dots, s_n)$ and
$t = f(s_1, \dots, t_i, \dots, s_n)$ where $s_i \irew t_i$. Induction hypothesis
gives $\normalise{s_i} \irew[\RSbot] \normalise{t_i}$.
\normalise{s} = f(\normalise{s_1}, \dots, \normalise{s_i}, \dots, \normalise{s_n})
\irew[\RSbot] f(\normalise{s_1}, \dots, \normalise{t_i}, \dots, \normalise{s_n})
\irss[\RSbot] \normalise{t} \tpkt
For the first equality we employ that
$\normalise{s_i} \not \in \NF(\RS)$. This concludes the proof.
An immediate consequence is the following.
Let $\RS$ be a predicative recursive TRS. Then
$\RSbot$ is a completely defined TRS compatible with $\gpop$.
Further $\dheight(s, \irew) \leqslant \dheight(s, \irew[\RSbot])$
for all basic terms $s$.
We have already observed that $\RSbot$ is compatible with $\gpop$.
Moreover it is completely defined by definition.
As $\RS$ is predicative recursive, it is a constructor TRS.
To prove the second halve of the assertion, consider a maximal derivation
\mparbox[c]{1cm}{s} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{s_1} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{s_2} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{\cdots} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{s_m}
starting from a basic term $s$, i.e., $m = \dheight(s, \irew)$.
If $m = 0$ the lemma is immediate.
For the case $m > 0$,
$m$-times application of Lemma <ref> gives
\mparbox[c]{1cm}{\normalise{s}} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{\normalise{s_1}} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{\normalise{s_2}} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{\cdots} \mparbox[c]{1cm}{\irew[\RS]}
\mparbox[c]{1cm}{\normalise{s_m}} \tpkt
Hence overall, $\dheight(s, \irew) \leqslant \dheight(\normalise{s}, \irew[\RSbot])$.
Since by assumption $s$ is a basic term not in normal form,
we have $\normalise{s} = s$ and the lemma follows.
We arrive at the proof of the main theorem:
Let $\RS$ be a predicative recursive TRS and fix an
arbitrary basic term $s = f(\pseq[m][n]{u})$.
Set $\ell \defsym \max\{\ar(\fn) \mid \fn \in \FSn \} \cup \{2\cdot\size{r} \mid {l \to r} \in \RSbot\}$
and note that $\ell$ is well defined since $\FSn$ and $\RS$ are finite.
Putting things together we see
\begin{align*}
\dheight(s, \irew) & \leqslant \dheight(s, \irew[\RSbot])
&& \text{using Lemma~\ref{l:rss:simul}} \\
& \leqslant \Slow[\ell](\ints(s))
&& \text{using Lemma~\ref{l:embed:bound}}\\
& \in \bigO\bigl((\max_{i=1}^m \depth(u_i))^{d}\bigr)
&& \text{using Corollary~\ref{c:pop}}
\end{align*}
where $d$ depends only on $\ell$.
§ AN ORDER-THEORETIC CHARACTERISATION OF THE POLYTIME FUNCTIONS
We now present the application of polynomial path orders
in the context of implicit computational complexity (ICC).
As by-product of Proposition <ref> and Theorem <ref> we immediately obtain
that $\POPSTAR$ is sound for $\FNP$ respectively $\FP$.
Let $\RS$ be a predicative recursive TRS.
For every relation $\sem{f}$ defined by $\RS$,
the functional problem $F_f$ associated with $\sem{f}$ is in $\FNP$.
Moreover, if $\RS$ is confluent than $\sem{\mf} \in \FP$.
Although it is decidable whether a TRS $\RS$ is predicative recursive (we
present a sound and complete automation in Section <ref>),
confluence is undecidable in general. To get a decidable result for $\FP$,
one can replace by an decidable criteria, for instance orthogonality.
We will now also establish that is complete for $\FP$, that is,
every function $f \in \FP$ is computed by some confluent (even orthogonal) predicative recursive TRS.
For this we use the term rewriting formulation of the
predicative recursive functions from <cit.>.
For each $k,l \in \N$ the set of function symbols $\Fb^{k,l}$
with $k$ normal and $l$ safe argument positions is the
least set of function symbols such that
* $\epsilon \in \Fb^{0,0}$, $\mS_1,\mS_2 \in \Fb^{0,1}$, $\m{P} \in \Fb^{0,1}$, $\m{C} \in \Fb^{0,4}$
and $\m{I}^{k,l}_j, \m{O}^{k,l} \in \Fb^{k,l}$, where $j = 1, \dots, k+l$;
* if $\vec{r} = \seq[m]{r} \in \Fb^{k,0}$, $\vec{s} = \seq[n]{s} \in \Fb^{k,l}$
and $h \in \Fb^{m,n}$ then $\m{SC}[h,\vec{r}, \vec{s}] \in \Fb^{k,l}$;
* if $g \in \Fb^{k,l}$ and $h_1,h_2 \in \Fb^{k+1,l+1}$ then $\m{SRN}[g,h_1,h_2] \in \Fb^{k+1,l}$;
The predicative signature is given by $\Fb \defsym \bigcup_{k,l \in \N} \Fb^{k,l}$.
Only the constant $\epsilon$ and dyadic successors $\mS_1,\mS_2$,
which serve the purpose of encoding natural numbers in binary,
are constructors.
The remaining symbols from $\Fb$ are defined
by the following (infinite) schema of rewrite rules $R_\B$.
Here we $k,l$ range over $\N$ and we abbreviate
$\vec{x} = \seq[k]{x}$ and $\vec{y} = \seq[l]{y}$ for $k$ respectively $l$ distinct variables.
4@lInitial Functions
$\m{P}(\sn{}{\epsilon})$ $\to$ $\epsilon$
$\m{P}(\sn{}{\mS_i(\sn{}{x})})$ $\to$ $x$ for $i=1,2$
$\m{I}^{k,l}_j(\svec{x}{y})$ $\to$ $ x_j$ for all $j = 1,\dots,k$
$\m{I}^{k,l}_j(\svec{x}{y})$ $\to$ $y_{j-k}$ for all $j =k+1, \dots, l+k$
$\m{C}(\sn{}{\epsilon, y, z_1, z_2})$ $\to$ $y$
$\m{C}(\sn{}{\mS_i(\sn{}{x}), y, z_1, z_2})$ $\to$ $z_i$ for $i = 1, 2$
$\m{O}(\svec{x}{y})$ $\to$ $\epsilon$
4@lSafe Composition ($\m{SC}$)
$\m{SC}[h,\vec{r}, \vec{s}](\svec{x}{y})$ $\to$ $h(\sn{\vec{r}(\sn{\vec{x}}{})}{\vec{s}(\svec{x}{y})})$
4@lSafe Recursion on Notation ($\m{SRN}$)
$\m{SRN}[g,h_1,h_2](\sn{\epsilon, \vec x}{\vec y})$ $\to$ $ g(\svec{x}{y})$
$\m{SRN}[g,h_1,h_2](\sn{\mS_i (\sn{}{z}), \vec x}{\vec y})
$ $\to$ $h_i(\sn{z, \vec x}{\vec y, \m{SRN}[g,h_1,h_2](\sn{z, \vec x}{\vec y})})$ for $i = 1, 2$
We emphasise that the above rules are all orthogonal.
Also, we stress that the system $R_\B$ is dupped infeasible in <cit.>.
Indeed $R_\B$ admits an exponential lower bound on the derivation height
which has to do with effects caused by duplicating redexes as
explained already in Example <ref> on page ex:dup.
Therefore $R_\B$ is not (directly) suitable as a term-rewriting characterisation
of the predicative recursive functions.
However this exponential lower-bound is only correct if we consider unrestricted rewriting.
The following proposition verifies that $R_\B$ generates only polytime computable functions.
Let $f \in \FP$. There exists a finite restriction $\RS_f \subsetneq R_\B$
such that $\RS_f$ computes $f$.
We arrive at our completeness result.
For every $f \in \FP$ there exists an orthogonal predicative recursive
TRS $\RS_f$ that computes $f$.
Take the TRS $\RS_f \subsetneq R_\B$ from Proposition <ref> that computes $f$.
Obviously $\RS_f$ is orthogonal hence confluent,
it remains to verify that $\RS_f$ is compatible with some instance $\gpop$.
To define $\gpop$ we use the separation of normal from safe argument positions
as indicated in the rules.
To define the precedence underlying $\gpop$, we
first define a mapping $\lh$ from the signature of $\Fb$
into the natural numbers as follows:
* $\lh(f) \defsym 0$ if $f$ is one of $\epsilon$, $\mS_0$, $\mS_1$, $\m{C}$, $\m{P}$, $\m{I}^{k,l}_j$ or $\m{O}^{k,l}$;
* $\lh(\m{SC}[h,\vec{r}, \vec{s}]) \defsym 1 + \lh(h) + \sum_{r \in \vec{r}} \lh(r) + \sum_{s \in \vec{s}} \lh(s)$;
* $\lh(\m{SRN}[g,h_1,h_2]) \defsym 1 + \lh(g) + \lh(h_1) + \lh(h_2)$.
Finally for each pair of function symbol $f$ and $g$ occurring in $\RS_f$ set
$f \sp g$ if and only if $\lh(f) > \lh(g)$.
Then $\sp$ defines an admissible precedence.
It is straight forward to verify that $\RS_f \subseteq {\gpop}$ where
$\gpop$ is based on the precedence $\sp$ and the safe mapping as indicated in
Definition <ref>.
By Theorem <ref> and Theorem <ref> we thus obtain a
precise characterisation of the class polytime computable functions.
The class of confluent (or orthogonal) predicative recursive TRSs define exactly $\FP$.
§ A NON-TRIVIAL CLOSURE PROPERTY OF THE POLYTIME COMPUTABLE FUNCTIONS
Bellantoni already observed that the class $\B$ is closed under
predicative recursion on notation with parameter substitution (scheme (<ref>)).
Essentially this recursion scheme allows substitution on safe argument positions. More precise,
a new function $f$ is defined by the equations
\begin{equation}\label{scheme:srnps} \tag{\ensuremath{\mathsf{SRN_{PS}}}}
\begin{array}{r@{\;}c@{\;}l}
f(\sn{0,\vec{x}}{\vec{y}}) & = & g(\sn{\vec{x}}{\vec{y}}) \\
\qquad\quad f(\sn{2z + i,\vec{x}}{\vec{y}}) & = &
h_i(\sn{z,\vec{x}}{\vec{y},f(\sn{z,\vec{x}}{\vec{p}(\svec{x}{y})})}),~i \in \set{1,2} \tpkt
\end{array}
\end{equation}
Notably closure of $\B$ under parameter substitution has been proven also been Beckmann and Weiermann <cit.>
based on rewriting techniques.
In the following we introduce a polynomial path order beyond MPO,
the polynomial path order with parameter substitution ( for short).
The next definition introduces $\POPSTARP$.
It is a variant of $\POPSTAR$ where
clause $\cpop{ep}$ has been modified and allows computation at safe argument positions.
Let $s, t \in \TERMS$ such that $s = f(\pseq[k][l]{s})$.
Then $s \gpopps t$ with respect to the precedence $\qp$ and safe mapping $\safe$ if either
* $s_i \geqpopps t$ for some $i \in \{1,\dots,k+l\}$, or
* $f \in \DS$, $t = g(\pseq[m][n]{t})$ where $f \sp g$
and the following conditions hold:
* $s \gsq t_j$ for all normal argument positions $j = 1,\dots,m$;
* $s \gpopps t_j$ for all safe argument positions $j = m+1,\dots,m+n$;
* $t_j \not\in \TA(\sigbelow{\Fun(s)}{\FS},\VS)$ for at most one safe argument position $j \in \{m+1,\dots,m+n\}$;
* $f \in \DS$, $t = g(\pseq[m][n]{t})$ where $f \ep g$
and the following conditions hold:
* $\mset{\seq[k]{s}} \gpopmulps \mset{\seq[m]{t}}$;
* $s \gpopps t_j$ and $t_j \in \TA(\sigbelow{\Fun(s)}{\FS},\VS)$ for all safe argument positions $j = m+1, \dots, m+n$.
Here ${\geqpopps} \defsym {\gpopps \cup \eqis}$.
We adapt the notion of predicative recursive TRS to in the obvious way.
It is not difficult to see that $\POPSTARP$ extends the analytic power of $\POPSTAR$.
For any underlying admissible precedence $\qp$, ${\gpop} \subseteq {\gpopps}$.
Note that is strictly more powerful than , as witnessed by
the following example.
Consider the TRS $\RSrev$ defining the reversal of
lists in a tail recursive fashion:
\begin{align*}
\mrev(\sn{xs}{}) & \to \mrevt(\sn{xs}{\mnil}) &
\mrevt(\sn{\nil}{ys}) & \to ys &
\mrevt(\sn{\mcs(x, xs)}{ys}) & \to \mrevt(\sn{xs}{\mcs(x,ys)}) \tpkt
\end{align*}
Then $\RSrev \subseteq {\gpopps}$ with precedence
$\mrev \sp \mrevt \sp \mnil \ep \mcs$.
Note that orientation of the last rule with $\gpopps$
breaks down to $\mcs(x, xs) \gpopps xs$ and $\mrevt(\sn{\mcs(x, xs)}{ys}) \gpopps \mcs(x,ys)$.
On the other hand, $\gpop$ fails as the corresponding clause $\cpop{ep}$ requires $ys \geqpop \mcs(x,ys)$.
The order is complete for the class of
polytime computable functions.
To show that it is sound, we prove that
induces polynomially bounded runtime complexity in the sense of Theorem <ref>.
The crucial observation is that the embedding of $\irew$ into $\gpopv$ does not break
if we relax compatibility constraints to $\RS \subseteq {\gpopps}$.
Suppose $s = f(\pseq{s}) \in \Tb$, $t \in \TERMS$ and $\ofdom{\sigma}{\VS \to \Val}$.
Then for predicative interpretation $\intq \in \{\ints, \intn\}$
we have
s \gpopps t \quad \IImp \quad \intq(s\sigma) \gpopv[2\cdot\size{t}] \intq(t\sigma) \tpkt
First one verifies that
Lemma <ref> holds even if we replace
$\gpop$ by $\gpopps$.
In particular, the assumptions give
\begin{align}
\label{l:embed:root:ps:len}
\len(\intn(t\sigma)) \leqslant 2\cdot\size{t} + \max \{ \norm{s_1\sigma}, \dots, \norm{s_k\sigma}, \norm{s\sigma}\}
\end{align}
The proof proceeds then in correspondence to
Lemma <ref> by induction on
$\gpopps$. We cover only the new case.
Let $s$, $t$, $\sigma$ be as given in the lemma.
$s \cpopps{ep} t$
Then $t = g(\pseq[m][n]{t})$ where $f \ep g$.
Further, the assumption gives
$\mset{\seq[k]{s}} \gpopmul \mset{\seq[m]{t}}$.
As $t \not \in \termsbelow[\Fun(s)]$
it suffices to verify Property <ref> and Property <ref>
from Lemma <ref>.
Exactly as in the corresponding case of Lemma <ref> we see
\begin{equation}
\label{eq:root:ep:ps}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cpopv{ep}[2\cdot\size{t} -1] \gn(\intn(t_1\sigma), \dots, \intn(t_m\sigma)) \tpkt
\end{equation}
As by assumption $s \gpopps t_j$ and $t_j \in \termsbelow[\Fun(s)]$, induction hypothesis gives
\begin{equation}
\label{eq:ti:ep:ps}
\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \gppv[2\cdot\size{t} -1] \ints(t_j\sigma) \tpkt
\end{equation}
As $\len(\ints(t\sigma)) \leqslant \size{t}$ by Lemma l:int:lenS,
we obtain $\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cpopv{ialst}[2\cdot\size{t}] \ints(t\sigma)$
from Equation (<ref>) and Equation (<ref>).
Likewise, from this Assertion <ref> follows by $\cpopv{ms}$ using additionally
$\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \cppv{ia}[2\cdot\size{t}-1] \theconst$
\begin{align*}
\len(\intn(t\sigma))
& \leqslant 2\cdot\size{t} + \max \{ \norm{s_1\sigma}, \dots, \norm{s_k\sigma}, \norm{s\sigma}\}
&& \text{by Equation~\eqref{l:embed:root:ps:len}} \\
& \leqslant 2\cdot\size{t} + \width(\fn(\intn(s_1\sigma), \dots, \intn(s_k\sigma)) \append \NM{s\sigma}) \tpkt
\end{align*}
Following the Proof of Theorem <ref>, but replacing
Lemma <ref> by Lemma <ref> we obtain:
Let $\RS$ be predicative recursive TRS (in the sense of Definition <ref>).
Then the innermost derivation height of any basic term
$f(\svec{u}{v})$ is bounded by a polynomial in the
maximal depth of normal arguments $\vec{u}$.
The polynomial depends only on $\RS$ and the signature $\FS$.
Using this theorem, Proposition <ref> states
that is sound for the polytime computable functions.
Lemma <ref> together with Theorem <ref>
shows completeness of for the polytime computable functions.
The class of confluent (or orthogonal) predicative recursive TRSs (in the sense of Definition <ref>) define exactly $\FP$.
§ AUTOMATION OF POLYNOMIAL PATH ORDERS
In this section we present an automation of polynomial path orders,
for brevity we restrict our efforts to the order $\gpop$.
Consider a constructor TRS $\RS$. Checking whether $\RS$ is predicative
recursive is equivalent to guessing a precedence $\qp$ and partitioning
of argument positions so that $\RS \subseteq {\gpop}$ holds for the induces
order $\gpop$.
As standard for recursive path orders <cit.>,
this search can be automated by encode the constraints imposed by Definition <ref>
into propositional logic.
To simplify the presentation, we extend language of propositional
logic with truth-constants $\top$ and $\bot$ in the obvious way.
In the constraint presented below we employ the following atoms.
*Propositional Atoms
To encode the separation of normal from safe arguments, we
introduce $f \in \DS$ and $i = 1, \dots, \ar(f)$ the atoms $\esafe{f}{i}$
so that $\esafe{f}{i}$ represents the assertion that the $i$ argument position of $f$ is safe.
Further we set $\esafe{f}{i} \defsym \top$ for $n$-ary $f \in \CS$ and $i = 1,\dots,n$ which
reflects that argument positions of constructors are always safe.
One verifies that predicative recursive
TRSs are even compatible with $\gpop$ as induced
by an admissible precedence where constructors are equivalent,
that is, polynomial path orders are blind on constructors.
This is exploited in the propositional encoding of precedences, where
we encode a precedence $\qp$ on the set of defined symbols $\DS$ only:
For each pair of symbols $f,g \in \DS$, we introduce
propositional atoms $\esp{f}{g}$ and $\eep{f}{g}$ so that $\esp{f}{g}$
represents the assertion $f \sp g$, and likewise
$\eep{f}{g}$ represents the assertion $f \ep g$.
Overall we define for function symbols $f$ and $g$ the propositional formulas
\begin{equation*}
\enc{f \sp g} \defsym
\begin{cases}
\top & \text{if $f\in\DS$ and $g\in \CS$,} \\
\bot & \text{if $f\in\CS$ and $g\in \CS$,} \\
\esp{f}{g} & \text{otherwise.}
\end{cases}
\quad
\enc{f \ep g} \defsym
\begin{cases}
\top & \text{if $f\in\CS$ and $g\in \CS$, } \\
\bot & \text{if $f\in\DS$ and $g \in \DS$,} \\
\eep{f}{g} & \text{otherwise.}
\end{cases}
\end{equation*}
To ensure that the variables $\esp{f}{g}$ and respectively $\eep{f}{g}$ encode a preorder on $\DS$
we encode an order preserving homomorphism into the natural order $>$.
To this extend, to each $f \in \DS$ we associate a natural number $\rk_f$ encoded as binary string
with $\lceil \log_2(\size{\DS}) \rceil$ bits.
It is straight forward to define Boolean formulas $\enc{\rk_f > \rk_g}$ (respectively $\enc{\rk_f = \rk_g}$) that are satisfiable iff
the binary numbers $\rk_f$ and $\rk_g$ are decreasing (respectively equal) in the natural order. Using these we set
\begin{align*}
\vprec(\DS)
\defsym \bigwedge_{f,g \in \DS} (\enc{f \sp g} \imp \enc{\rk_f > \rk_g})
\wedge \bigwedge_{f,g \in \DS} (\enc{f \ep g} \imp \enc{\rk_f = \rk_g})
\end{align*}
We say that a propositional assignment $\mu$ induces the precedence
$\qp$ if $\mu$ satisfies $\enc{f \sp g}$ when $f \sp g$ and $\enc{f \ep g}$ when $f \ep g$.
The next lemma verifies that $\vprec$ serves our needs.
For any valuation $\mu$ that satisfies $\vprec(\DS)$,
$\mu$ induces an admissible precedence on $\FS$.
Vice versa, for any admissible precedence $\qp$ on $\FS$,
any valuation $\mu$, satisfying $\mu(\enc{f \sp g})$ iff $f \sp g$ and
$\mu(\enc{f \sp g})$ iff $f \ep g$, also satisfies
the formula $\vprec(\DS)$.
Order Constraints
For concrete pairs of terms $s = f(\seq{s})$ and $t$, we define the order constraints
\enc{s \gpop t} \defsym \enc{s \cpop{st} t} \vee \enc{s \cpop{ia} t} \vee \enc{s \cpop{ep} t}
which enforces the orientation $f(\seq{s}) \gpop t$ using propositional formulations
of the three clauses in Definition <ref>.
To complete the definition for arbitrary left-hand sides, we set $\enc{x \gpop t} \defsym \bot$ for all $x \in \VS$.
Further weak orientation is given by
\enc{s \geqpop t} \defsym \enc{s \gpop t} \vee \enc{s \eqis t} \tkom
where the constraint $\enc{s \eqis t}$ refers to a formulation of
Definition <ref> in propositional logic, defined as follows.
For $s = t$ we simply set $\enc{s \eqis t} \defsym \top$.
Consider the case $s = f(\seq{s})$ and $t = g(\seq{t})$.
Then $s \eqis t$ if $f \ep g$
and moreover $s_i \eqis t_{\pi(i)}$ for all $i = 1,\dots,n$ and
some permutation $\pi$ on argument positions that takes the separation of normal and safe positions into account.
To encode $\pi(i) = j$, we use fresh atoms $\pi_{i,j}$ for $i,j=1,\dots,n$.
The propositional formula $\vperm(\pi,n) \defsym \bigwedge_{i=1}^{n} \eone(\pi_{i,1}, \dots, \pi_{i,n})$
is used to assert that the atoms $\pi_{i,j}$ reflect a permutation on $\{1,\dots,n\}$.
Here $\eone(\pi_{i,1}, \dots, \pi_{i,n})$ expresses that exactly one of its arguments evaluates to $\top$.
We set
\begin{align*}
\enc{s \eqis t} \defsym
\enc{f \ep g} \wedge \vperm(\pi,n)
\wedge~\bigl({\bigwedge_{j=1}^{n} \pi_{i,j} \imp \enc{s_i \eqis t_j} \wedge (\esafe{f}{i} \iff \esafe{g}{j})}\bigr) \tpkt
\end{align*}
To complete the definition, we set $\enc{s \eqis t} = \bot$ for
the remaining cases.
Suppose $\mu$ induces an admissible precedence $\qp$ and satisfies $\enc{s \eqis t}$.
Then $s \eqis t$ with respect to the precedence $\qp$.
Vice versa, if $s \eqis t$ then $\enc{s \eqis t}$ is satisfiable by assignments $\mu$
that induce the precedence underlying $\eqis$.
We now define the encoding for the different cases underlying the definition of $\gpop$.
Assuming that $\enc{s_i \geqpop t}$ enforces $s_i \gpop t$ clause $\cpop{st}$ is expressible
\begin{align*}
\enc{f(\seq{s}) \cpop{st} t} \defsym \bigor_{i=1}^n \enc{s_i \geqpop t}
\end{align*}
in propositional logic.
For clause $\cpop{ia}$ we use propositional atoms $\alpha_i$ ($i = 1,\dots,m$)
to mark the unique argument position of $t = g(\seq[m]{t})$ that allows
$t_i \not\in \termsbelow[\Fun(s)]$.
The propositional formula $\ezeroone(\seq[m]{\alpha})$ expresses that zero or one
$\alpha_i$ valuates to $\top$.
Further, we introduce the auxiliary constraint
\begin{align*}
\enc{g(\seq[m]{t}) \in \termsbelow[F]} \defsym \bigvee_{f \in F} \enc{f \sp g} \wedge \bigwedge_{j=1}^m \enc{t_j \in \termsbelow[F]}
\end{align*}
and $\enc{x \in \termsbelow[F]} \defsym \top$ for $x \in \VS$.
Using these, clause $\cpop{ia}$ becomes expressible as
\begin{multline*}
\enc{f(\seq{s}) \cpop{ia} g(\seq[m]{t})} \defsym
\enc{f \in \DS}
\wedge \enc{f \sp g} \\
\wedge \bigwedge_{j=1}^m (\esafe{g}{j} \imp \enc{s \gpop t_j})
\wedge \bigwedge_{j=1}^m (\neg \esafe{g}{j} \imp \enc{s \gsq t_j}) \\
\wedge \ezeroone(\seq[m]{\alpha})
\wedge \bigwedge_{j=1}^m (\neg \alpha_j \imp \enc{t_j \in \termsbelow[\Fun(s)]}) \tpkt
\end{multline*}
Here $\enc{f \in \DS} = \top$ if $f \in \DS$ and otherwise $\enc{f \in \DS} = \bot$.
The propositional formula $\enc{s \gsq t}$ expresses the orientation with the $\gsq$ and is given by
\begin{align*}
\enc{f(\seq{s}) \gsq t} \defsym \enc{f(\seq{s}) \csq{st} t} \vee \enc{f(\seq{s}) \csq{ia} t}
\end{align*}
and otherwise $\enc{x \gsq t} = \bot$, where
\begin{align*}
\enc{f(\seq{s}) \csq{st} t} & \defsym \bigor_{i=1}^n ((\enc{s_i \gsq t} \vee \enc{s_i \eqis t}) \wedge (\enc{f \in \DS} \imp \neg \esafe{f}{i})) \\
\enc{f(\seq{s}) \csq{ia} t} & \defsym
\begin{cases}
\enc{f \in \DS} \wedge \enc{f \sp g} & \text{ if $t = g(\seq[m]{t})$} \\
\quad \wedge \bigwedge_{j=1}^m \enc{f(\seq{s}) \gsq t_j} \\
\bot & \text{ if $t \in \VS$}.
\end{cases}
\end{align*}
This concludes the propositional formulation of clause $\cpop{ia}$.
The main challenge in formulating clause $\cpop{ep}$
is to deal with the encoding of multiset-comparisons.
We proceed as in <cit.> and encode the underlying multiset cover.
Let $\succ_\mul$ denote the multiset extension of a binary relation ${\succcurlyeq} = {\succ} \uplus {\eqi}$.
Then a pair of mapping $(\gamma, \varepsilon)$
where $\ofdom{\gamma}{\set{1,\dots,m} \to \set{1,\dots,n}}$
and $\ofdom{\varepsilon}{\set{1,\dots,n} \to \set{\top,\bot}}$
is a multiset cover on multisets $\mset{\seq{a}}$ and $\mset{\seq[m]{b}}$
if the following holds for all $j \in \{1,\dots,m\}$:
* if $\gamma(j) = i$ then $a_i \succcurlyeq b_j$, in this case we say that $a_i$ covers $b_j$;
* if $\varepsilon(j) = \top$ then $s_{\tau(j)} \eqi t_j$ and $\tau$ is invective on $\{j\}$,
i.e., $a_{\tau(j)}$ covers only $b_j$.
The multiset cover $(\gamma, \varepsilon)$ is said to be strict if at least one cover is strict,
i.e., $\varepsilon(j) = \bot$ for some $j \in \{1,\dots,m\}$.
It is straight forward to verify that multiset covers characterise the multiset extension
of $\succ$ in the following sense.
We have $\mset{\seq{a}} \mextension{\succcurlyeq} \mset{\seq[m]{b}}$ if and only if there
exists a multiset cover $(\gamma, \varepsilon)$ on $\mset{\seq{a}}$ and $\mset{\seq[m]{b}}$.
Moreover, $\mset{\seq{a}} \mextension{\succ} \mset{\seq[m]{b}}$ if and only if the cover is strict.
Consider the orientation $f(\seq{s}) \cpop{ep} g(\seq[m]{t})$.
Then normal arguments are strictly, and safe arguments weakly decreasing with
respect to the multiset-extension of $\gpop$.
Since the partitioning of normal and safe argument is not fixed,
in the encoding of $\cpop{ep}$ we formalise a multiset-comparison on all arguments,
where the underlying multiset-cover $(\gamma, \varepsilon)$
will be restricted so that if $s_i$ covers $t_j$, i.e., $\gamma(i) = j$,
then both $s_i$ and $t_j$ are safe or respectively normal.
To this extend, for a specific multiset cover $(\gamma, \varepsilon)$ we introduce variables $\gamma_{i,j}$ and
$\varepsilon_i$, where $\gamma_{i,j} = \top$ represents $\gamma(j) = i$ and
$\varepsilon_i = \top$ denotes $\varepsilon(i) = \top$ ($1 \leqslant i \leqslant n$, $1 \leqslant j \leqslant m$).
We set
\begin{multline*}
\enc{f(\seq{s}) \cpop{ep} g(\seq[m]{t})} \defsym
\enc{f \in \DS}
\wedge \enc{f \sp g} \\
\wedge
\bigwedge_{i=1}^{n} \bigwedge_{j=1}^{m} \Bigl( \gamma_{i,j} \to \bigl( \varepsilon_i \to \enc{s_i \eqis t_j} \bigr)
\wedge \bigl( \neg \varepsilon_i \to \enc{s_i \gpop t_j} \bigr)
\wedge \bigl( \esafe{f}{i} \iff \esafe{g}{j} \bigr)
\Bigr) \\
% jedes normale t_j ist von einem s_i "gecovered"
\wedge \bigwedge_{j=1}^m \eone(\gamma_{1,j},\dots,\gamma_{n,j}) %\bigvee_{i=1}^n \gamma_{i,j}
% wenn s_i fuer "cover by equality" dann wird % von "s_i" genau ein t_j "gecovered"
\wedge \bigwedge_{i=1}^{n} \bigl(\varepsilon_i \to \eone(\gamma_{i,1},\dots,\gamma_{i,m})\bigr)
% mindestens ein strikter abstieg
\wedge \bigvee_{i=1}^n \bigl( \neg \esafe{f}{i} \wedge \neg \varepsilon_i \bigr) \tpkt
\end{multline*}
Here the first line establishes the Condition d:mscover1, where
$\esafe{f}{i} \iff \esafe{g}{j}$ additionally enforces the separation of normal from safe arguments.
The final line formalises
that $\gamma$ maps $\{1,\dots,m\}$ to $\{1,\dots,n\}$, Condition d:mscover2
as well as the strictness condition on normal arguments.
This completes the encoding of $\gpop$.
Suppose $\mu$ induces an admissible precedence $\qp$ and satisfies $\enc{s \gpop t}$.
Then $s \gpop t$ with respect to the precedence $\qp$.
Vice versa, if $s \gpop t$ then $\enc{s \gpop t}$ is satisfiable assignments $\mu$
that induce the precedence underlying $\gpop$.
As a predicative recursive TRS $\RS$ is a constructor TRS compatible with some
polynomial path order $\gpop$, putting the constraints together we get the following theorem.
Let $\RS$ be a constructor TRS.
The propositional formula
\vpredrec(\RS) \defsym \vprec(\DS) \wedge \bigwedge_{{l \to r} \in \RS} \enc{l \gpop r}
is satisfiable if and only if $\RS$ is predicative recursive.
We have implemented this reduction to in our complexity analyser .
As underlying -solver we employ the open source solver <cit.>.
On the example from the introduction, outputs
the following result in a fraction of a second.
Efficiency Considerations
The -solver requires its input in CNF.
For a concise translation of $\vpredrec(\RS)$ to CNF
we use the approach of Plaisted and Greenbaum <cit.> that
gives an equisatisfiable CNF linear in size.
Our implementation also eliminates redundancies resulting from
multiple comparisons of the same pair of term $s, t$ by
replacing subformulas $\enc{s \gpop t}$ with unique
propositional atoms $\delta_{s,t}$. Since $\enc{s \gpop t}$
occurs only in positive contexts, it suffices to
add $\delta_{s,t} \imp \enc{s \gpop t}$, resulting in an equisatisfiable formula.
Also during construction of $\vpredrec(\RS)$ our implementation
performs immediate simplifications under Boolean laws.
§ EXPERIMENTAL ASSESSMENT
In this section we present an empirical evaluation of polynomial path orders.
We selected two testbeds: Testbed constitutes
of 597 terminating constructor TRSs, obtained
by restricting the innermost runtime complexity problemset
from the termination problem database (TPDB for short), version 8.0
to known to be terminating constructor TRSs.
Termination is checked against the full run of the complexity competition from December 2011
Testbed , containing 290 examples, results from restricting Testbed to
orthogonal systems.
Unarguably the TPDB is an imperfect choice as examples were collected primarily to
assess the strength of termination provers, but it is at the moment the only
extensive source of TRSs. Since the creation of the dedicated
complexity categories in 2008 the situation, although slowly, changes to the better.
Experiments were conducted with $\TCT$ version 1.9.1
[Available from <http://cl-informatik.uibk.ac.at/software/tct/projects/tct/archive/>.],
on a laptop with 4Gb of RAM and Intel${}^\text{\textregistered}$ Core${}^\text{\texttrademark}$ i7–2620M CPU (2.7GHz, quad-core).
We assess the strength of $\POPSTAR$ and $\POPSTARP$ in comparison to its predecessors $\MPO$ and $\LMPO$.
The implementation of $MPO$ and $\LMPO$ follows the line of polynomial path orders
as explained in Section <ref>.
We contrast these syntactic techniques to interpretations
as implemented in our complexity tool $\TCT$.
The last column show result of constructor restricted
matrix interpretations <cit.> (dimension $1$ and $3$)
as well as polynomial interpretations <cit.> (degree $2$ and $3$),
run in parallel on the quad-core processor.
We employ interpretations in their default configuration of ,
noteworthy coefficients (respectively entries in coefficients)
range between $0$ and $7$, and we also make use of the usable argument positions
criterion <cit.> that weakens monotonicity constraints.
Table <ref>
[Full evidence available at <http://cl-informatik.uibk.ac.at/software/tct/experiments/popstar>.]
shows totals on systems that can respectively cannot be handled.
To the right of each entry we annotate the average execution time, in seconds.
Empirical Evaluation, comparing syntactic to semantic techniques.
It is immediate that syntactic techniques cannot compete with the expressive
power of interpretations.
In Testbed there are in fact only three examples
compatible with where could not find interpretations.
There are additionally four examples compatible with but not so with interpretations,
including the TRS $\RSbin$ from Example <ref>.
All but one (noteworthy the merge-sort algorithm from Steinbach and Kühlers collection
of these do in fact admit exponential runtime-complexity,
thus a priori they are not compatible to the restricted interpretations.
We emphasise that parameter substitution significantly increases the strength of
, 13 examples are provable by but neither by nor .
could benefit from parameter substitution,
we conjecture that the resulting order is still sound for $\FP$.
On Testbed , containing only orthogonal TRSs, in total 75 systems (26% of the testbed)
can be verified to encode polytime computable functions, 35 (12% of the testbed)
can be verified polytime computable by only syntactic techniques.
It should be noted that not all examples appearing in our collection encode polytime computable
functions, the total amount of such systems is unknown.
Table <ref> clearly illustrates one of our main motivations for investigating
syntactic techniques.
Our complexity analyser recursively decomposes complexity problems using
various complexity preserving transformation techniques,
discarding those problems that can be handled by basic techniques as
contrasted in Table <ref>.
Certificates are only obtained
if finally all subproblems can be discarded,
above all it is crucial that subproblems can be discarded
succeeds on average 14 times faster than polynomial and
matrix interpretations run parallel, it can be safely
preposed to interpretations, speeding up the overall procedure.
Note that the difficulty of implementing interpretations efficiently
is also reflected in the total number of timeouts.
§ CONCLUSION AND FUTURE WORK
We propose a new order, the polynomial path order $\POPSTAR$.
The order $\POPSTAR$ is a syntactical restriction of multiset path orders,
with the distinctive feature that the (innermost) runtime complexity
of compatible TRSs lies in $O(n^d)$ for some $d$.
Based on $\POPSTAR$, we delineate a class of rewrite systems, dubbed
systems of predicative recursion,
so that the class of functions computed by these systems
corresponds to $\FP$, the class of polytime computable functions.
We have shown that an extension of $\POPSTAR$, the order $\POPSTARP$
that also accounts for parameter substitution,
increases the intensionality of $\POPSTAR$.
In contrast to interpretations,
$\POPSTAR$ is partly lacking in intensionality but surpluses
in verification time.
In our complexity prover , we do not intend to replace
semantic techniques, but rather prepose them by , in
order to improve both in analytic power and speed.
With we are in particular interested in obtaining asymptotically
tight bounds.
Although we could estimate the degree of the witnessing
bounding function for and ,
a bound extracted from our proof yields unnecessarily an overestimation,
compare Theorem <ref> and particular the preceding construction of the degree $d_{k,p}$.
Partly this is due to the underlying multiset extension.
Future investigations will certainly include establishing
tighter bounds.
§ ACKNOWLEDGEMENT
We are in particular thankful to Nao Hirokawa for fruitful discussions.
|
arxiv-papers
| 2012-09-17T20:38:54 |
2024-09-04T02:49:35.187906
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Martin Avanzini and Georg Moser",
"submitter": "Martin Avanzini",
"url": "https://arxiv.org/abs/1209.3793"
}
|
1209.4004
|
# Polymorphism and bistability in adherent cells
Shiladitya Banerjee Department of Physics, Syracuse University, Syracuse, New
York 13244, USA Luca Giomi School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA International School
for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
###### Abstract
The optimal shapes attained by contractile cells on adhesive substrates are
determined by the interplay between intracellular forces and adhesion with the
extracellular matrix. We model the cell as a contractile film bounded by an
elastic cortex and connected to the substrate via elastic links. When the
adhesion sites are continuously distributed, optimal cell shape is constrained
by the adhesion geometry, with a spread area sensitively dependent on the
substrate stiffness and contractile tension. For discrete adhesion sites,
equilibrium cell shape is convex at weak contractility, while developing local
concavities at intermediate values of contractility. Increasing contractility
beyond a critical value, controlled by mechanical and geometrical properties
of adhesion, cell boundary undergoes a discontinuous transition to a star-
shaped configuration with cusps and protrusions, accompanied by a region of
bistability and hysteresis.
## I Introduction
Mechanical force generation during cell-matrix adhesion is strongly influenced
by the ability of cells to actively probe the mechanical and geometrical cues
in the extracellular matrix Discher _et al._ (2005). Matrix stiffness plays a
profound role in regulating a variety of cellular processes, from
morphogenesis, motility to cell spreading and cytoskeletal activity. Cells
adhering to softer substrates spread less and prefer to have well rounded
morphologies, while they are more likely to exhibit branched patterns on
stiffer substrates with greater spread area Yeung _et al._ (2005); Chopra
_et al._ (2011). Experiments on micro-patterned adhesive islands revealed that
cell fate, proliferation and spreading sensitively depend on adhesion geometry
Chen _et al._ (1997). However, cellular response to extracellular
determinants is strongly linked to myosin dependent activity of the cell
cytoskeleton Galbraith and Sheetz (1998). While myosin activity can influence
force transmission by regulating the growth of focal adhesions Riveline _et
al._ (2001), it can also drive changes in cell morphology, as seen by
pharmacologically disrupting the cell cytoskeleton Bar-Ziv _et al._ (1999);
Asano _et al._ (2009) or by inhibiting myosin-II activity Théry _et al._
(2006). Traction forces exerted by cells on substrates can now be determined
accurately using traction force microscopy or micropillar arrays Pelham and
Wang (1997); Du Roure _et al._ (2005), but the feedback between cell
morphology and mechanics during adhesion to a matrix requires further
theoretical investigation. In this article we present a minimal mechano-
geometric model for isolated adherent cells that addresses a fundamental
question in cell mechanics and morphogenesis: How intercellular and
extracellular forces cooperate to control the geometry of cell shapes? At time
scales when the cell is fully spread and develops stronger focal adhesions,
the dominant forces in the cell stem from surface tension induced by
actomyosin contractility and elasticity in the actomyosin cortex. These
intracellular forces act in opposition to receptor-mediated adhesive forces in
determining optimal cell shapes Lecuit and Lenne (2007); Mader _et al._
(2007). Although chemical pathways can trigger a feedback between cell
activity and cell-substrate adhesion Buchsbaum (2007), we instead focus on
their mechanical cooperativity in regulating cell shapes. Tuning stiffness of
the matrix and acto-myosin contractility, we discuss how cells can be driven
through a series of morphological transitions - convex, concave, cusps and
protrusions with associated hysteresis. In addition, we provide several
analytical results relating geometrical properties of cells e.g. curvature,
spread radius to mechanical properties such as substrate stiffness and
contractile surface tension, that are amenable to experimental verification
and quantitative comparison.
The paper is organized as follows. In section 2, we introduce a minimal free
energy model describing the optimal shape of the contact line of an adherent
cell. We retain three major contributions to the free energy stemming from :
intracellular contractility, bending elasticity in the cell contour and
adhesion to an extracellular substrate. We then proceed to study optimal
solutions of the free energy for two distinct cases : (1) continuously
distributed adhesion sites and (2) discrete adhesion sites. For continuously
distributed adhesion sites, optimal shape of the cell contact line is always
constrained by the geometry of the adhesion patch. We explicitly provide
solution for the spread area of a cell constrained by a circular adhesion
area, and analyze its dependence on substrate stiffness (Fig. 1). The result
is in excellent qualitative and quantitative agreement with experimental
trends. For discrete adhesion sites, cell contour develops non-uniform
curvatures along its non-adherent segments. For low contractility and softer
adhesions, the cell contour has a convex shape and one can analytically
describe the shape and calculate traction forces transmitted to the substrate.
In section 3 we discuss that by tuning contractility and substrate stiffness
cell contour can be guided through a series of morphological transitions.
Furthermore at intermediate values of substrate stiffness (see Fig. 5),
increasing contractility beyond a critical value leads to formation of cusps
and protrusions at adhesion sites. This transition involves a discontinuous
change of the cell geometry characterized by a jump in the turning number of
the contact line. The transition is further accompanied by a region of
bistability and hysteresis in the dependence of cell perimeter on
contractility. This result indicates how strongly intracellular and
extracellular forces can control geometric properties of an adherent cell.
## II Contractile Film Model for Adherent Cells
We consider a thin film of an adherent cell subject to internal contractile
forces. The shape of the cell contact line is parametrized by the contour
${\bm{r}}(s)$, where $s$ represents arc-length. The total mechanical energy of
the cell can be approximated, on the basis of symmetry arguments, in the form:
$E=\sigma\int dA+\oint ds\,\left(\alpha\kappa^{2}+\lambda\right)+k_{s}\oint
ds\,\rho\,|\bm{r}-\bm{r}_{0}|^{2}\;,$ (1)
where $\sigma$ is the effective surface tension in the cell due to
cytoskeletal contractility, $\kappa$ is the local curvature of the cell
boundary, $\alpha$ the associated bending rigidity and $\lambda$ represents
line tension at the cell boundary. The last term in Eq. (1) represent the
strain energy induced by the cell on a substrate of stiffness $k_{s}$ through
focal adhesions localized at the cell edge Wozniak _et al._ (2004) with
density $\rho(s)$, so that the total number of adhesions is $N_{A}=\oint
ds\rho$. For cells adhering to a thin continuous substrate (see Sec. II.1),
$\bm{r}_{0}$ can be considered as the position of the cell boundary once the
cell is fully spread and forces are predominately contractile, while for cells
cultured on elastomeric pillars (see Sec. II.2), this is simply the pillar’s
rest position at the adhesion points. In the analytical framework presented
here, we will treat the reference shape as an adjustable parameter to
investigate different experimental situations.
The model assumes that the overall effect of acto-myosin contractility, that
pulls the cell contour inwards reducing its contact area with the substrate,
can be described by an effective surface tension $\sigma$. Thus the first term
in Eq. (1) should not be interpreted as the classic hydrostatic tension that
occurs at the interface between two fluids, but as an active normal stress
resulting from the action of the motors. In order to estimate the order of
magnitude of $\sigma$, we assume that the active myosin motors cross-linked
with the cortical F-actin gel of mean thickness $h\simeq 0.1\ \mu$m, are
distributed with an average areal density $\rho_{m}\simeq 10^{4}\mu m^{-2}$,
with effective stiffness $k_{m}\simeq 1$ pN/nm and mean stretch
$\Delta_{m}\simeq 1$ nm Howard (2001). Surface tension $\sigma$ can then be
estimated as $\sigma\simeq h\rho_{m}k_{m}\Delta_{m}\simeq$ 1 nN/$\mu$m. This
estimate comes to the same order of magnitude as reported for endothelial
cells Lemmon _et al._ (2005); Bischofs _et al._ (2009) and epithelial cells
Mertz _et al._ (2012).
The second term in Eq. (1) describes the elasticity of the cell cortex. This
consists of a bending energy density $\alpha\kappa^{2}$, reflecting the
resistance of cortical actin in response of a change in curvature, and an
effective line tension $\lambda$ that, similarly to the bulk tension $\sigma$,
embodies the contractile forces due to the actin fibers lining the cell
periphery Bischofs _et al._ (2008, 2009). The Euler-Lagrange equations for
the shape that minimizes the energy (1) can be derived with standard methods
Mumford (1993); Giomi and Mahadevan (2012). This yields:
$\alpha\left(2\kappa^{\prime\prime}+\kappa^{3}\right)-\lambda\kappa-\sigma+2k_{s}\rho(\bm{r}-\bm{r}_{0})\cdot\bm{n}=0$
(2)
Here prime denotes derivative with respect to arc-length $s$ and
$\bm{n}=\bm{r}^{\prime\prime}/|\bm{r}^{\prime\prime}|$ is the normal vector.
Eq. (2) expresses the balance between the total stress acting on a cross-
section of the cortex and the body force
$\bm{K}=2k_{s}\rho(\bm{r}-\bm{r}_{0})$ due to adhesion:
$\frac{d}{ds}(\bm{F}+\bm{\varSigma}+\bm{\varLambda})+\bm{K}=0$ (3)
where $\bm{F}=\alpha\kappa^{2}\bm{t}+2\alpha\kappa^{\prime}\bm{n}$ (with
$\bm{t}$ the tangent vector) is the elastic stress resultant,
$\bm{\varSigma}=-\sigma[(\bm{r}\cdot\bm{t})\bm{n}-(\bm{r}\cdot\bm{n})\bm{t}]$
is the stress contribution of bulk contractility and
$\bm{\varLambda}=-\lambda\bm{t}$ that of peripheral contractility.
Previous theoretical models Bar-Ziv _et al._ (1999); Bischofs _et al._
(2008, 2009) have analyzed the competition of bulk and peripheral
contractility and ignored the bending elasticity of the actin cortex (i.e.
$\alpha=0$). In analogy with the Laplace law of capillarity, the steady state
cell contour is then described by concave circular arcs of radius
$\lambda/\sigma$ connecting adhesion sites. Here we focus on the opposite
limit and consider the regime in which the force balance is dominated by the
competition between cortex elasticity and bulk contractility, while the effect
of peripheral contractility is negligible (i.e. $\lambda=0$). In this
scenario, the curvature is generally non-uniform, especially in the
neighborhood of adhesion sites. As we will see in the remainder of this
article, incorporating bending elasticity leads to an extremely rich
polymorphism and allows for a transition from purely convex to purely concave
cell shape reminiscent of that observed in experiments on cardiac myocytes
Chopra _et al._ (2011). Alternative models for cellular geometry and
mechanics include a growing class finite element models Deshpande _et al._
(2006); Loosli _et al._ (2010); Farsad and Vernerey (2012), continuum
mechanical models Banerjee and Marchetti (2011); Edwards and Schwarz (2011) or
network models Torres _et al._ (2012).
The Contractile Film Model defined by Eq. (1) is inspired by a classic problem
in mechanics: finding the optimal shape of a capillary film bounded by an
elastic rod. This problem was formulated by Lévy Lévy (1884) in 1884 and for
over a century it drew the attention of many researchers Tadjbakhsh and Odeh
(1967); Flaherty _et al._ (1972); Arreaga _et al._ (2002); Vassilev _et
al._ (2008); Djondjorov _et al._ (2011); Mora _et al._ (2012); Giomi and
Mahadevan (2012) due to its tremendous richness of polymorphic and multi-
stable behaviors. Unlike this simple system consisting of a film spanning an
elastic boundary, however, the model proposed here for adhering cells does not
involve any constraint on the length of the boundary, which is then only
softly constrained by the adhesion with the substrate. This feature,
introduces in the model a number of crucial mechanical properties, including
an adaptive bending stiffness of the cell boundary.
### II.1 Continuous adhesions
Figure 1: Relative cell size $R/R_{0}$ as a function of substrate stiffness
$k_{s}$ (solid black circles) for smooth muscle cells, 4 hours after plating
on continuous elastic gels Engler _et al._ (2004). Cell radius is estimated
from the projected cell area reported in Engler _et al._ (2004) as
$R=\sqrt{\text{area}}/\pi$. Substrate stiffness $k_{s}$ is determined from
substrate Young’s modulus $E_{s}$ as : $k_{s}=aE_{s}$, where $a$ is the
characteristic focal adhesion size, with $a\sim 1\ \mu m$. Solid (red) line
represents the solution to Eq. (4) with with $\sigma=1.05$ nN/$\mu$m and
$\alpha/R_{0}^{3}=0.16$ nN/$\mu$m.
In this case the periphery of the cell forms contact with a single continuous
adhesion site, so that $\rho=1/\mathcal{L}$ with $\mathcal{L}=\oint ds$ the
perimeter of the cell. In presence of a uniform and isotropic substrate, we
can assume the reference configuration to be a circle of radius $R_{0}$ so
that a natural minimizer of the energy (1) would be a circle or radius $R$.
Thus, setting $\lambda=0$, $\kappa=R^{-1}$ and $\rho^{-1}=2\pi R$ in Eq. (2)
yields the following cubic equation:
$(k_{s}+\pi\sigma)R^{3}-k_{s}R_{0}R^{2}-\pi\alpha=0\;,$ (4)
The equation contains two length scales, $R_{0}$ and
$\xi=(\alpha/\sigma)^{1/3}$, and a dimensionless control parameter
$k_{s}/\sigma$ expressing the relative amount of adhesion and contraction. For
very soft anchoring $k_{s}\ll\sigma$ and Eq. (4) admits the solution $R=\xi$.
Thus in non-adherent cell segments, corresponding to the limit $k_{s}=0$,
radius of curvature scales with surface tension as $R\sim\sigma^{-1/3}$. The
same scaling law is also predicted using active cable network models of an
adherent cell Torres _et al._ (2012). If the cell is rigidly pinned at
adhesion sites, $k_{s}\gg\sigma$ and $R\rightarrow R_{0}$. For intermediate
values of $k_{s}/\sigma$ the optimal radius $R$ interpolates between $\xi$ and
$R_{0}$ and is an increasing function of the substrate stiffness $k_{s}$, in
case $\xi<R_{0}$, or a decreasing function if $\xi>R_{0}$. For $\xi=R_{0}$,
the lower and upper bound coincide, and the solution is $R=R_{0}$. In
particular, the case $R_{0}>\xi$ reproduces the experimentally observed trend
that cell projected area increases with increasing substrate stiffness before
reaching a plateau at higher stiffnesses Engler _et al._ (2004); Yeung _et
al._ (2005); Chopra _et al._ (2011). We fit the solution to Eq. (4) to the
measured projected areas of smooth muscle cells (SMCs) adhering to continuous
elastic gels of varying substrate elastic modulus Engler _et al._ (2004), as
shown in Fig. 1. Data for the spread area of SMCs are taken 4 hours after
plating onto the substrate, when they retain rounded morphologies. The fitted
value for surface tension $\sigma=1.05$ nN/$\mu$m comes to the same order of
magnitude as reported for endothelial cells Lemmon _et al._ (2005); Bischofs
_et al._ (2009), epithelial cells Mertz _et al._ (2012) and is consistent
with the numerical estimate provided earlier. The fit also provides a value
for the bending rigidity $\alpha=4.62\times 10^{-16}$ Nm2. The asymptotic
behavior and various limits of the solution are well captured by the
interpolation formula:
$R\approx\frac{k_{s}R_{0}+3\pi\sigma\,\xi}{k_{s}+3\pi\sigma}$ (5)
indicating that larger surface tension, hence larger cell contractility
$\sigma$ leads to lesser spread area, consistent with the experimental
observation that myosin-II activity retards the spreading of cells Wakatsuki
_et al._ (2003). Standard stability analysis of this solution under a small
periodic perturbation in the cell radius shows that the circular shape is
always stable for any values of the parameters $\sigma$, $k_{s}$ and $R_{0}$.
### II.2 Discrete adhesions
Figure 2: Cell anchored onto three pointwise adhesions located at the vertices
of an equilateral triangle. The curvature (a) and the tangent angle (b) as
function of arc-length for $\sigma R_{0}^{3}/\alpha=10$, $k_{s}R_{0}^{3}=50$
and $N_{A}=3$. The circles are obtained from a numerical minimization of a
discrete version of the energy (1), while the solid lines corresponds to our
analytical approximation. (c) The total cell length $\mathcal{L}$ as a
function of adhesion stiffness. For small stiffnesses the cell boundary form a
curve of constant width (lower inset) and $\mathcal{L}=\pi w$, with $w$ the
width of the curve. This property breaks down for larger stiffnesses when
inflection points develops (upper inset). (d) The curvature $\kappa_{0}$ at
the adhesion points as a function of the substrate stiffness for various
contractility values. The points are obtained from numerical simulations while
the solid lines correspond to our analytical approximation. Figure 3: Cell
anchored onto three pointwise adhesions located at the vertices of an
equilateral triangle. (a) $\sigma<\sigma_{c1}$, cell contour is everywhere
convex with constant width. (b) $\sigma=\sigma_{p}$, cell contour is purely
concave with cusps at adhesion points and without protrusions. (c)
$\sigma>\sigma_{c2}$, cusps are connected to the substrate by means of a
protrusion of length $\ell$.
For cells adhering to discrete number of adhesion sites, one can show that the
circular solution for the cell boundary is never stable and there is always a
non-circular configuration with lower energy. For simplicity, we assume that
$N_{A}$ adhesion sites are located at the vertices of a regular polygon of
circumradius $R_{0}$, with density $\rho(s)=\sum_{i=0}^{N_{A}-1}\delta(s-iL)$,
and $L$ the distance between subsequent adhesions. Optimal cell shape is given
by the solution of the equation:
$\alpha\left(2\kappa^{\prime\prime}+\kappa^{3}\right)-\sigma+2k_{s}\sum_{i=0}^{N_{A}-1}\delta(s-iL)\left({\bm{r}}-{\bm{r}}_{0}\right)\cdot{\bm{n}}=0\;.$
(6)
Due to the $N_{A}$-fold symmetry of the adhesion sites, adhesion springs
stretch by an equal amount $\Delta$ in the direction of the normal vector:
$(\bm{r}_{i}-\bm{r}_{0i})\cdot\bm{n}_{i}=\Delta$, $i=1,\,2\ldots N_{A}$. As a
consequence of the localized adhesion forces, the curvature is non-analytical
at the adhesion points. Integrating Eq. (6) along an infinitesimal
neighborhood of a generic adhesion point $i$, one finds the following
condition for the derivative of the curvature at the adhesion points:
$\kappa^{\prime}_{i}=-\frac{k_{s}}{2\alpha}\,\Delta\;.$ (7)
The local curvature of the segment lying between adhesion points is on the
other hand determined by the equation
$\alpha\left(2\kappa^{\prime\prime}+\kappa^{3}\right)-\sigma=0$, with the
boundary conditions : $\kappa(iL)=\kappa((i+1)L)=\kappa_{0}$. Without loss of
generality we consider a segment located in $s\in[0,L]$. Although an exact
analytic solution this nonlinear equation is available (see Ref. Vassilev _et
al._ (2008) and Appendix C), an excellent approximation can be obtained by
neglecting the cubic nonlinearity (Fig. 2a-b). With this simplification, Eq.
(6) admits a simple solution of the form:
$\kappa(s)=\kappa_{0}+\frac{\sigma}{4\alpha}\,s(s-L)\;.$ (8)
Eqs. (8) and (7) immediately allow us to derive a condition on the cell
perimeter: $L=2k_{s}\Delta/\sigma$. Furthermore, the latter condition leads to
a linear relation between traction force $T=2k_{s}\Delta$, and cell size :
$T=\sigma L\;,$ (9)
which is indeed observed in traction force measurements on large epithelial
cells Mertz _et al._ (2012).
To determine the end-point curvature $\kappa_{0}$, we use the turning tangents
theorem for a simple closed curve Gray (1997), which requires
$\int_{0}^{L}ds\,\kappa=2\pi/N_{A}$. This leads to following relation between
local curvature and segment length, or equivalently traction force, at the
adhesion sites :
$\kappa_{0}=\frac{\sigma
L^{2}}{24\alpha}+\frac{2\pi}{N_{A}L}=\frac{T^{2}}{24\,\alpha\sigma}+\frac{2\pi\sigma}{N_{A}T}\;.$
(10)
A plot of $\kappa_{0}$ as a function of the substrate stiffness is shown in
Fig. 2c.
Finally, to determine the optimal length of the cell segment $L$, we are going
to make use of a remarkable geometrical property of the curve obtained from
the solution of Eq. (6) with discrete adhesions: the fact of being a curve of
constant width Gray (1997). The width of a curve is the distance between the
uppermost and lowermost points on the curve (see lower inset of Fig. 2d). In
general, such a distance depends on how the curve is oriented. There is
however a special class of curves, where the width is the same regardless of
their orientation. The simplest example of a curve of constant width is
clearly a circle, in which case the width coincides with the diameter. A
fundamental property of curves of constant width is given by the Barbier’s
theorem Gray (1997), which asserts that the perimeter $\mathcal{L}$ of any
curve of constant width is equal to width $w$ multiplied by $\pi$:
$\mathcal{L}=\pi w$. As illustrated in Fig. 2d, this is confirmed by numerical
simulations for low to intermediate values for contractility and stiffness.
With our setting, the cell width is given by:
$w=(R_{0}-\Delta)(1+\cos\pi/N_{A})+h(L/2)\;,$ (11)
where $h(s)=\int_{0}^{s}ds^{\prime}\,\sin\theta(s^{\prime})$ is the height of
the curve above a straight line connecting subsequent adhesions and
$\theta(s)=\int_{0}^{s}ds^{\prime}\,\kappa(s^{\prime})=\theta_{0}+\kappa_{0}s+\frac{\sigma}{24\alpha}\,s^{2}(2s-3L)$
(12)
the angle formed by the tangent vector with the $x-$axis of a suitable
oriented Cartesian frame (Fig. 3a). For small angles $h$ can be approximated
as : $h(s)\approx s(L-s)\left[\pi/(N_{A}L)-(\sigma/48\alpha)\,s(L-s)\right]$.
Using this together with Eq. (11) and the Barbier’s theorem with
$\mathcal{L}=N_{A}L$ allow us to obtain a quartic equation for the cell length
and the traction force, whose approximate solution is given by:
$T\simeq\frac{\sigma
R_{0}}{\left(g_{0}+\frac{\sigma}{2k_{s}}\right)\left[1+\frac{7\sigma
R_{0}^{3}}{\alpha
g_{1}}\left(g_{0}+\frac{\sigma}{2k_{s}}\right)^{-4}\right]^{1/7}}\;,$ (13)
where, $g_{0}=(4N_{A}^{2}-\pi^{2})/\left[4\pi N_{A}(1+\cos\pi/N_{A})\right]$
and $g_{1}=768(1+\cos\pi/N_{A})$. Eq. (13) supports the experimental trend
that traction force increases monotonically with substrate stiffness $k_{s}$
before plateauing to a finite value for higher stiffnesses Ghibaudo _et al._
(2008); Mitrossilis _et al._ (2009). The plateau value increases with
increasing contractility (Fig. 4a). Traction force grows linearly with
increasing contractility for $\sigma R_{0}^{3}/\alpha\ll 1$, before saturating
to the value $2k_{s}R_{0}$ at large contractility $\sigma R_{0}^{3}/\alpha\gg
1$, as shown in Fig. 4b. Eq. (13) is also consistent with experimentally
observed trend that reducing contractility by increasing the dosage of myosin
inhibitor Blebbistatin, leads to monotonic drop in traction forces Mitrossilis
_et al._ (2009).
In the calculation presented in this section we have neglected the
contribution of peripheral contractility embodied in the effective line
tension $\lambda$. From the point of view of force balance, increasing
$\lambda$ has the effect of rotating the stress resultant toward the
tangential direction. This creates a boundary layer between the adhesion
points, where the curvature $\kappa_{0}$ is dictated by the balance between
adhesion and bending, and the central region, where the curvature
$\kappa\approx\sigma/\lambda$ is dominated by the balance between normal and
tangential contractility. The size of the boundary layer is approximatively
$\sqrt{\alpha/\lambda}$.
## III Inflections, cusps and protrusions
For low to intermediate values of $\sigma$ and $k_{s}$, cell shape is convex
and has constant width. Upon increasing $\sigma$ above a $k_{s}-$dependent
threshold $\sigma_{c1}$, however, the cell boundary becomes inflected (see
Fig. 5 and upper inset of Fig. 2d). Initially a region of negative curvature
develops in proximity of the mid point between two adhesions, but as the
surface tension is further increased, the size of this region grows until
positive curvature is preserved only in a small neighborhood of the adhesion
points. Due to the presence of local concavities, the cell boundary is no
longer a curve of constant width. Convex and concave regions are separated by
inflection points, given by the solution to $\kappa=0$, or explicitly:
$s^{2}-Ls+4\alpha\kappa_{0}/\sigma=0$. In order for this equation to have real
solutions one needs $\sigma L^{3}>96\pi\alpha/N_{A}$. Fig. 4c shows
$\sigma_{c1}$ as a function of $k_{s}$. Prior experimental studies Théry _et
al._ (2006); James _et al._ (2008) have indicated that lamellipodia formation
is predominant along convex edges or sharp corners, whereas contractile stress
fibers assemble along concave regions. Lamellipodia formation implies greater
motile activity along those corners. On softer substrates, with weak adhesion,
cell is more motile on average than on stiff substrates. This is because
increasing substrate stiffness promotes formation of concave arcs along non-
adherent sites, thus reducing the total area spanned by convex regions. As
such, lamellipodia distribution is controlled by the geometry of adhesion
sites for both continuous and discrete cases.
Figure 4: Traction force as a function of substrate stiffness (a) and
contractility (b) obtained from a numerical minimization of a discrete analog
of Eq. (1). Solid curves denote the approximate traction values obtained from
Eq. (13). (c) Boundary length $\mathcal{L}$ obtained by increasing (squares)
and then decreasing (triangles) the contractility for substrate stiffnesses
$k_{s}R_{0}^{3}/\alpha=100$ (green squares, black triangles) and
$k_{s}R_{0}^{3}/\alpha=120$ (red squares, blue triangles). The diagram shows
bistability in the range $\sigma_{p}<\sigma<\sigma_{c2}$. (d) The critical
contractility $\sigma_{c1}$ and $\sigma_{c2}$ as functions of substrate
stiffness.
Upon increasing $\sigma$ above a further threshold value $\sigma_{c2}$, the
inflected shape collapses giving rise to the star-shaped configurations shown
in upper right corner of Fig. 5. These purely concave configurations are made
by arcs whose ends meet in a cusp. The cusp is then connected to the substrate
by a protrusion consisting of a straight segment of length $\ell$ that extends
until the adhesion point rest position, so that $\Delta\approx 0$ (Fig. 3c)
(see Appendix B). The cell boundary becomes pinned at adhesion sites as a
result of having to satisfy force-balance, Eq. (6), and adhesion-induced
boundary condition, Eq. (7), while accommodating large contractile tensions at
its neighbourhood. This results in spontaneous expansion in the cell
perimeter. Unlike the previous transition from convex to non-convex shapes,
this second transition occurs discontinuously and is accompanied by a region
of bistability in the range $\sigma_{p}<\sigma<\sigma_{c2}$, where
$\sigma_{p}$ is the value of $\sigma$ at which the protrusions have zero
length and the shape of the cell is that sketched in Fig. 3b. This is clearly
visible in the hysteresis diagram in Fig. 4d showing the optimal length
obtained by numerically minimizing a discrete analog of Eq. (1) in a cycle and
using as initial configuration the output of the previous minimization. The
onset of bistability is regulated by substrate stiffness as shown in Fig. 4c,
with stiffer substrates promoting transition to cusps at lower $\sigma_{c2}$.
Away from the protrusion, the curvature has still the form given in Eq. (8),
with $\kappa_{0}=0$ so that the boundary is everywhere concave or flat and the
bending moment $\bm{M}=2\alpha\kappa\bm{\hat{z}}$ does not experience any
unphysical discontinuity at the protrusion’s origin.
From the shape of the cell at $\sigma=\sigma_{p}$ we can construct all the
shapes at $\sigma>\sigma_{p}$ by mean of a similarity transformation. To see
this let us set $\ell=0$ at $\sigma=\sigma_{p}$ so that the shape of the cell
will be of the kind illustrated in Fig. 3b. In the following we will refer to
this as the reference shape. The approximated expression for the curvature is
the same given in Eq. (8), but with $\kappa_{0}=\Delta=0$ and
$\kappa^{\prime}$ unconstrained since the last term in Eq. (6) vanishes
identically. The quantities $\sigma_{p}$ and the length $L_{p}$ of the
reference shape are left to determine. To achieve this, a first condition can
be obtained by observing that: $x(L_{p}/2)=R_{0}\sin\pi/N_{A}$, where $x(s)$
is the projection of the curve on the edge of the circumscribed polygon (see
Fig. 3b). A second condition is given by the theorem of turning tangents for a
simple closed curve with $N_{A}$ cusps:
$\int_{0}^{L_{p}}ds\,\kappa=\pi(2-N_{A})/N_{A}$ (see Appendix B). In the case
$N_{A}=3$, for instance, the right-hand side is equal to $-\pi/3$,
corresponding to the fact that the tangent vector rotates clockwise by
$60^{\circ}$ as we move counterclockwise along the curve from one cusp to the
next. These allow us to approximate:
$\displaystyle
L_{p}\approx\frac{2N_{A}R_{0}}{\pi(N_{A}-2)}\,\sin\frac{\pi}{N_{A}}\;,$ (14a)
$\displaystyle\sigma_{p}\approx\frac{3\alpha\pi^{4}}{R_{0}^{3}\sin^{3}\frac{\pi}{N_{A}}}\left(\frac{N_{A}-2}{N_{A}}\right)^{4}\;,$
(14b)
which define the reference shape shown in Fig. 3b.
Figure 5: Phase diagram in $\sigma$-$k_{s}$ plane showing optimal
configuration obtained by numerical minimization of the energy (1) for
$N_{A}=3$.
Next, following Ref. Flaherty _et al._ (1972); Djondjorov _et al._ (2011),
we notice that the force balance equation
$2\kappa^{\prime\prime}+\kappa^{3}-\sigma/\alpha=0$ is invariant under the
scaling transformation:
$(s,\,\kappa,\,\sigma)\rightarrow\left(\Lambda\,s,\,\frac{\kappa}{\Lambda},\,\frac{\sigma}{\Lambda^{3}}\right)\;.$
(15)
Consequently, the equilibrium shape obtained for a given value of
$\sigma>\sigma_{p}$ are similar to the reference shape with a scaling factor
$\Lambda=(\sigma_{p}/\sigma)^{1/3}<1$. Accordingly, the closed curve is
rescaled so that $L=\Lambda L_{p}$ and $A=\Lambda^{2}A_{p}$ with $A_{p}$ the
area of the reference shape. This beautiful geometric property immediately
translates into the following algorithm to construct shapes with protrusion
(Fig. 3c): 1) Given the surface tension $\sigma>\sigma_{p}$ we calculate the
scaling factor $\Lambda$. 2) We rescale the reference curve so that $L=\Lambda
L_{p}$. 3) Finally, we fill the distance between the adhesion points and the
cusps with straight segments of length $\ell=R_{0}(1-\Lambda)$ (since $R_{0}$
is the circumradius of the reference shape and $\Lambda\,R_{0}$ that of the
rescaled shape). This latter step, ultimately allows us to formulate a scaling
law for the length of protrusions that can be tested in experiments:
$\ell/R_{0}=1-(\sigma_{p}/\sigma)^{1/3}\;.$ (16)
It should be stressed that our knowledge of the convex/concave transition is
still very preliminary. This instability is different from the classical Euler
buckling Love (1927), which originates from the trade-off between compression
and bending and is a supercritical pitchfork bifurcation. The appearance of
cusps is reminiscent, to some extent, of the sulcification instability in neo-
Hookean solids Biot (1965); Hohlfeld and Mahadevan (2011, 2012); Tallinen _et
al._ (2013), but there is far from being a precise mapping. One of the
fundamental aspect that distinguishes our model form classical elasticity
relies on the fact that the perimeter is not hardly constrained, but only
subject to a soft constraint by mean of the adhesion springs. The length of an
elastic object affects its overall flexibility (i.e. long filaments are floppy
and easy to bend, while short filaments are stiff), thus, when the effective
surface tension is increased, the whole cell boundary becomes shorter and
stiffer. Because stiff materials are difficult to bend, but easy to break, a
possible interpretation could be the following. For sufficiently large
adhesion, increasing the surface tension has the effect of bending and
stiffening the cell boundary in proximity of the adhesion sites, until, above
a certain surface tension, the cell boundary is too stiff to continue bending
and fractures. The cracks are localized at the adhesion points, where the
curvature initially focuses, giving rise to the cusps observed in the
simulations. However, a thorough understanding of this phenomenon remains a
challenge for the future.
## IV Discussion
The Contractile Film Model describes equilibrium cell shapes and does not
account for the dynamics associated with adhesion remodeling and actin
filament turnovers. Adhesion sites are static, with controllable density and
spatial distribution, as can be best realized using micropatterning techniques
Théry _et al._ (2006). The model provides a quantitative framework to
describe how polymorphic cell shapes arise by tuning substrate stiffness,
adhesion geometry and cell activity. The presence of bending deformations in
the cell periphery naturally allows for optimal cell shapes with non-uniform
boundary curvatures, a feature not included in previous theoretical works on
cell shapes Bar-Ziv _et al._ (1999); Bischofs _et al._ (2008, 2009). Bending
in the cell boundary can also arise due to splay deformations in the Arp2/3
regulated actin array in the lamellipodium. Although our model relies on local
mechanotransduction through adhesions localized at the cell edge, in reality
traction stresses penetrate inside the cell up to a characteristic depth
controlled by cellular and substrate stiffness Mertz _et al._ (2012). Our
model is thus applicable to cell sizes much larger than traction penetration
depth and predicts the same trend on the dependence of traction forces on
substrate stiffness as derived using long-range elastic models Banerjee and
Marchetti (2012). Although, local mechanosensing at cell periphery coupled
with global surface tension due to cytoskeletal contractility can accurately
capture experimental trends for cell size and traction forces, the effect of
non-local interactions of the cytoskeleton with the substrate cannot be
neglected at actin remodeling time scales.
An important consequence of increasing surface tension is the loss of
stability of smooth shapes and a discontinuous transition to cusps and
protrusion (Fig. 4 and 5). The transition is favored on stiffer substrates
(see Fig. 5) and leads to spontaneous expansion in the cell perimeter and
relaxation of localized adhesion springs. Such a transition could also
possibly occur on cellular timescales via chemo-mechanical instabilities
induced by coupling of motor activity with ligand-receptor kinetics at
adhesion sites. Instead here it emerges as a consequence of the cell boundary
satisfying of energy minimization and global geometrical constraint imposed by
the theorem of turning tangents. To our knowledge no experimental evidence has
yet been put forward of such instabilities. The result can be tested in cell
traction assays by varying motor activity in a cycle.
Finally, recent experiments on multicellular systems Mertz _et al._ (2012)
demonstrated that cohesive cell colonies behave like single cells in their
distribution of traction stresses and presence of supracellular actin cables
localized to the colony periphery. The Contractile Film Model can be
conveniently used to study shapes of strongly coupled cell colonies, where
colony surface tension stems from actomyosin contractility as well as strength
of cadherins mediating cell-cell adhesions.
###### Acknowledgements.
We thank Cristina Marchetti for many useful discussions. SB acknowledges
support from National Science Foundation through awards DMR-0806511 and
DMR-1004789. LG is supported by the NSF Harvard MRSEC, the Harvard Kavli
Institute for Nanobio Science & Technology and the Wyss Institute.
## Appendix A $-$ Numerical Simulations
The data shown in Figs. 1 and 4a,b,d have been obtained by numerically
minimizing the following discrete version of the energy (1):
$E_{1}=\frac{\sigma}{2}\sum_{i=1}^{N-1}(x_{i}y_{i+1}-x_{i+1}y_{i})+\alpha\sum_{i=1}^{N}\langle
s_{i}\rangle\kappa_{i}^{2}+k_{s}\sum_{i=1}^{N_{A}}|\bm{r}_{i}-\bm{r}_{0i}|^{2}$
(17)
where the first term corresponds to the area of the irregular polygon of
vertices $\bm{r}_{i}=(x_{i},y_{i})$, with $i=1\,2\ldots\,N$, and the third sum
represents the energetic contribution of the $N_{A}$ adhesion points.
$\kappa_{i}$ is the unsigned curvature at the vertex $i$:
$\kappa_{i}=|\bm{t}_{i}-\bm{t}_{i-1}|/\langle s_{i}\rangle$ with
$\bm{t}_{i}=(\bm{r}_{i+1}-\bm{r}_{i})/|\bm{r}_{i+1}-\bm{r}_{i}|$ the tangent
vector at $i$ and $\langle s_{i}\rangle=(s_{i}+s_{i-1})/2$, with
$s_{i}=|\bm{r}_{i+1}-\bm{r}_{i}|$. The discrete energy (17) was minimized
using a standard conjugate gradient algorithm. Using (17) allows a direct
comparison between simulations and the analytical results presented in the
previous sections. However, for very large substrate stiffness, the discrete
curve develops self-intersections and the energy becomes ill-defined. In this
regime, it is more convenient to approximate the cell as a simplicial complex
consisting of mesh $M$ of equilateral triangles. The edges of the triangles
can then be treated as elastic springs of zero rest-length, so that the total
energy of the mesh is given by:
$E_{2}=\Sigma\sum_{e\in M}|e|^{2}+\alpha\sum_{v\in\partial M}\langle
s_{v}\rangle\,\kappa_{v}^{2}+k_{s}\sum_{i=1}^{N_{A}}|\bm{r}_{i}-\bm{r}_{0i}|^{2}$
(18)
where $v$ and $e$ represent respectively the vertices and the edges of the
mesh and $\Sigma$ is a spring constant. If the triangles in the mesh are
equilateral, this yields a discrete approximation of the interfacial energy
$\sigma A$, with the spring stiffness proportional to the surface tension:
i.e. $\sigma\approx 4\Sigma\sqrt{3}/(2-B/E)$, where $B/E$ is the ratio between
the number of boundary edges $B$ and the total number of edges $E$ of the
triangular mesh Giomi and Mahadevan (2012).
## Appendix B $-$ Kinks, cusps and protrusions
Figure 6: Example of singular points: kink (left), cusps (center), protrusion
(right). The red dot indicated the adhesion point rest position $\bm{a}$,
while $\bm{\hat{\Delta}}=(\bm{r}-\bm{r}_{0})/|\bm{r}-\bm{r}_{0}|$. For a cusp
$\bm{n}\cdot\bm{\Delta}=0$, while for a protrusion, the normal vector is
undefined at the point of adhesion.
We present here some additional mathematical aspects on the occurrence of
cusps and formation of protrusions in the large contractility and stiffness
regime. In particular we show that the shape consisting of $N_{A}$ cusps that
extend until the adhesion rest point through a set of straight protrusions, is
the only regular convex $N_{A}$-fold star shape to be mechanically stable
within the Contractile Film Model.
A kink is a singular point on a curve where the tangent vector switches
discontinuously between two orientations (Fig. 6, left). The magnitude of the
discontinuity can be measured from the external angle $\phi$. A cusp, is a
kink with $\phi=\pi$, so that the tangent vector switches between equal and
opposite orientation (rotation by a larger angle would give rise to self-
intersections). In the case of a simple closed curve with kinks, the theorem
of turning tangents can be reformulated as follows:
$\oint ds\,\kappa+\sum_{i}\phi_{i}=2\pi\;,$ (19)
where the summation runs over all the kinks. In the case of a convex polygon,
for instance, $\kappa=0$ and (19) asserts that the sum of the external angle
of a polygon is equal to $2\pi$. In a convex $N_{A}$-fold star, the external
angle is bounded in the range $\phi\in[2\pi/N_{A},\pi]$, where
$\phi=2\pi/N_{A}$ corresponds to a regular polygon. As described in the main
text, Euler-Langrange equation for cellular force-balance is given by:
$\alpha(2\kappa^{\prime\prime}+\kappa^{3})-\sigma+2k_{s}\sum_{i=1}^{N_{A}}\delta(s-s_{i})(\bm{r}-\bm{r}_{0})\cdot\bm{n}=0\;.$
(20)
Let $s_{1}=0$ be the position of a generic adhesion point. Then, integrating
Eq. (20) in the range $s\in[-\epsilon,\epsilon]$ and taking the limit
$\epsilon\rightarrow 0$ yields:
$2\alpha\kappa^{\prime}(0)+k_{s}\,\bm{n}(0)\cdot\bm{\Delta}(0)=0\;,$ (21)
which expresses that the elastic restoring force originating in the boundary
must balance the body force $k_{s}\,\bm{n}\cdot\bm{\Delta}$ due to the
adhesion spring. For a kink, as that shown on the left of Fig. 6,
$\bm{n}\cdot\bm{\Delta}=\Delta\cos(\pi-\phi/2)=-\Delta\cos(\phi/2)$, force
balance gives us:
$\kappa^{\prime}(0)=\frac{k_{s}}{2\alpha}\,\Delta\cos(\phi/2)\;,$ (22)
Now, in a configuration consisting of a regular convex $N_{A}$-fold star, the
signed curvature is everywhere negative and has single minimum at the midpoint
between kinks. The latter property implies $\kappa^{\prime}(0)<0$, which
however contradicts Eq. (22) being the right-hand side always positive for any
positive value of $k_{s}$, $\alpha$ and $\Delta$. From this we conclude that
such a configuration cannot be a possible equilibrium shape.
In the case of a cusp, $\bm{n}\cdot\bm{\Delta}=0$ and the adhesion force is
all exerted along the tangent direction, hence $\kappa^{\prime}(0)=0$. In
Appendix C, however, we show that Eq. (20) has no solution with
$\kappa^{\prime}(0)=0$ that satisfies (19) with $\phi=\pi$. The only case left
is then that illustrated on the right of Fig. 6, in which the cusp extends
through a straight protrusion until the adhesion rest position, so that
$\Delta=0$. In this configuration, the adhesion force exerted by the substrate
is zero and so is the elastic force acting in the protrusion, this being
straight. In other words, the cell is pinned at the adhesion rest position
while the elastic force is zero. The case in which the protrusion has zero
length, is a special instance of this scenario from which one can construct
all the shapes having nonzero protrusion length by mean of a similarity
transformation, described in the main text.
## Appendix C $-$ Solution of the nonlinear Elastica equation
In this section we give and exact analytical expression of the general
solution of the equation:
$\alpha(2\kappa^{\prime\prime}+\kappa^{3})-\sigma=0\;,\qquad\qquad\kappa(0)=\kappa(L)=\kappa_{0}\;,$
(23)
and prove what asserted in Appendix B that a cusp as that shown in Fig. 6
(center) cannot exist. As a starting point let us make the equation
dimensionless by taking $t=s/L$ and $\hat{\kappa}=L\kappa$ and
$\hat{\sigma}=\sigma L^{3}/\alpha$. We have then:
$2\hat{\kappa}^{\prime\prime}+\hat{\kappa}^{3}-\hat{\sigma}=0\,,\qquad\qquad\hat{\kappa}(0)=\hat{\kappa}(1)=\hat{\kappa}_{0}$
(24)
where the prime now stands for a differentiation with respect to $t$. Without
loss of generality, we can chose $t=0$ as the point where the derivative of
$\hat{\kappa}$ vanishes. For the previously mentioned cusp, this point will be
in particular a point of adhesion and $t\in[0,1]$. Otherwise, this will be
identified as the mid point between adhesions upon translating $t\rightarrow
t-1/2$. Then, integrating the equation with respect to $\hat{\kappa}$ and
using the fact that $\hat{\kappa}(0)=\hat{\kappa}_{0}$ and
$\hat{\kappa}^{\prime}(0)=0$, we obtain:
$(\hat{\kappa}^{\prime})^{2}+\tfrac{1}{4}(\hat{\kappa}^{4}-\hat{\kappa}_{0}^{4})-\hat{\sigma}(\hat{\kappa}-\hat{\kappa}_{0})=0$
(25)
Introducing the new variable $y=1/(\hat{\kappa}_{0}-\hat{\kappa})$ we can
reduce the order on the nonlinearity by one unit:
$(y^{\prime})^{2}=(\hat{\kappa}_{0}^{3}-\sigma)y^{3}-\tfrac{3}{2}\hat{\kappa}_{0}^{2}y^{2}+\hat{\kappa}_{0}y-\tfrac{1}{4}$
(26)
This equation is of the form:
$(y^{\prime})^{2}=P(y)=h^{2}(y-a)(y-b)(y-c)$ (27)
with $h^{2}=\hat{\kappa}_{0}^{3}-\sigma$ and $a$, $b$ and $c$ the roots of the
cubic polynomial $P(y)$, and is suitable to be solved in terms of elliptic
functions. Now, one can verify that $P(y)$ has always a single real root,
$y=\alpha$ and a pair of complex conjugate roots $y=\beta\pm i\gamma$ for any
physical value of $\hat{\kappa}_{0}$ and $\hat{\sigma}_{0}$. Thus:
$P(y)=h^{2}(y-\alpha)[(y-\beta)^{2}+\gamma^{2}]$ (28)
which allows us to calculate the elliptic integral Greenhill (1892):
$t=\int_{y}^{\infty}\frac{dy}{\sqrt{P(y)}}=\omega^{-1}\operatorname{cn}^{-1}\left(\frac{y-z_{1}}{y-z_{2}},m\right)$
(29)
where $z_{1}$ and $z_{2}$ are the roots of the quadratic equations:
$z^{2}-2\alpha z+2\alpha\beta-(\beta^{2}+\gamma^{2})=0$ (30)
and $\omega$ and $m$ is given by:
$\omega^{2}=\frac{h(z_{1}-z_{2})}{2}\,,\qquad m^{2}=\frac{\beta-
z_{2}}{z_{1}-z_{2}}$ (31)
Finally, solving (29) for $y$ and going back to our original variables, we
have:
$\hat{\kappa}=\hat{\kappa}_{0}-\frac{1-\operatorname{cn}(\omega
t,m)}{z_{1}-z_{2}\operatorname{cn}(\omega t,m)}$ (32)
Figure 7: The curve obtained by solving Eqs. (33) (black) and (34) (red). The
left panel corresponds to the cusp shown in Fig. 6 (left). The absence of
intersections between the curves implies that such a configuration is not
mechanically stable.
In Eqs. (29) and (32) we use the standard notation for Jacobi elliptic
functions Davis (2010). Namely, given the incomplete elliptic integral of the
first kind:
$u=F(\phi,m)=\int_{0}^{\phi}\frac{dt}{\sqrt{1-m^{2}\sin^{2}t}}$
with $0<m^{2}<1$, the elliptic modulus, then $\phi$ is the Jacobi amplitude:
$\phi=\operatorname{am}(u,m)$ and $\operatorname{cn}(u,m)=\cos\phi$.
The expression (32), satisfies by construction the boundary conditions
$\hat{\kappa}(0)=\hat{\kappa}_{0}$ and $\hat{\kappa}^{\prime}(0)=0$. In order
for it to be a legitimate solution of the problem, we further need it to be
periodic, so that $\hat{\kappa}(0)=\hat{\kappa}(1)$ and to satisfy the theorem
of turning tangents (19). Periodicity can be easily implemented by recalling
that $\operatorname{cn}(x+4K(m))=\operatorname{cn}(x)$, where
$K(m)=F(\frac{\pi}{2},m)$ is the complete elliptic integral of first kind.
This results in the following condition for the frequency: $\omega=4K(m)$, or
more explicitly:
$h(z_{1}-z_{2})=32K^{2}(m)$ (33)
From the theorem of turning tangents applied to the case of a simple closed
curve with $N_{A}$ cups, we obtain :
$\hat{\kappa}_{0}-\pi\left(\frac{2-N_{A}}{N_{A}}\right)\\\
=\frac{4}{z_{1}z_{2}}\left[z_{1}K(m)-\frac{z_{1}-z_{2}}{\sqrt{1-m^{2}}}\,\Pi\left(\frac{z_{2}^{2}}{z_{1}^{2}}\,\bigg{|}\,\frac{m^{2}}{m^{2}-1}\right)\right]$
(34)
where $\Pi$ is the complete elliptic integral of third kind:
$\Pi(n\,|\,m)=\int_{0}^{\frac{\pi}{2}}\frac{dt}{(1-n\sin^{2}t)\sqrt{1-m\sin^{2}t}}$
(35)
Solving simultaneously the transcendental equations (33) and (34), the
quadratic equation (30) and its associated cubic, allows to calculate
$\hat{\kappa}_{0}$ and $\hat{\sigma}$, from which one can obtain $\kappa_{0}$
and $L$. The solution is then complete. With this machinery in hand we can now
answer the original question. Is there any solution corresponding to the cusp
in the center of Fig. 6, with $\hat{\kappa}^{\prime}(0)=0$ and, say,
$N_{A}=3$, such that $\int_{0}^{1}dt\,\hat{\kappa}=-\pi/3$? A numerical
solution of Eqs. (33) (black) and (34) (red) is plotted in Fig. 7 as a
function of $\hat{\kappa}_{0}$ and $\hat{\sigma}$. The curves never intersect
in the physical range of $\hat{\kappa}_{0}$ and $\hat{\sigma}$, thus such a
solution does not exist.
## References
* Discher _et al._ (2005) D. Discher, P. Janmey, and Y. Wang, Science, 310, 1139 (2005).
* Yeung _et al._ (2005) T. Yeung, P. Georges, L. Flanagan, B. Marg, M. Ortiz, M. Funaki, N. Zahir, W. Ming, V. Weaver, and P. Janmey, Cell Motil Cytoskel, 60, 24 (2005).
* Chopra _et al._ (2011) A. Chopra, E. Tabdanov, H. Patel, P. Janmey, and J. Kresh, Am J Physiol-Heart C, 300, H1252 (2011).
* Chen _et al._ (1997) C. Chen, M. Mrksich, S. Huang, G. Whitesides, and D. Ingber, Science, 276, 1425 (1997).
* Galbraith and Sheetz (1998) C. Galbraith and M. Sheetz, Curr Opin Cell Biol, 10, 566 (1998).
* Riveline _et al._ (2001) D. Riveline, E. Zamir, N. Balaban, U. Schwarz, T. Ishizaki, S. Narumiya, Z. Kam, B. Geiger, and A. Bershadsky, Science’s STKE, 153, 1175 (2001).
* Bar-Ziv _et al._ (1999) R. Bar-Ziv, T. Tlusty, E. Moses, S. Safran, and A. Bershadsky, Proc Natl Acad Sci USA, 96, 10140 (1999).
* Asano _et al._ (2009) Y. Asano, A. Jiménez-Dalmaroni, T. Liverpool, M. Marchetti, L. Giomi, A. Kiger, T. Duke, and B. Baum, HFSP journal, 3, 194 (2009).
* Théry _et al._ (2006) M. Théry, A. Pépin, E. Dressaire, Y. Chen, and M. Bornens, Cell Motil Cytoskel, 63, 341 (2006).
* Pelham and Wang (1997) R. Pelham and Y. Wang, Proc Natl Acad Sci USA, 94, 13661 (1997).
* Du Roure _et al._ (2005) O. Du Roure, A. Saez, A. Buguin, R. Austin, P. Chavrier, P. Siberzan, and B. Ladoux, Proc Natl Acad Sci USA, 102, 2390 (2005).
* Lecuit and Lenne (2007) T. Lecuit and P. Lenne, Nat Rev Mol Cell Bio, 8, 633 (2007).
* Mader _et al._ (2007) C. Mader, E. Hinchcliffe, and Y. Wang, Soft Matter, 3, 357 (2007).
* Buchsbaum (2007) R. Buchsbaum, J Cell Sci, 120, 1149 (2007).
* Wozniak _et al._ (2004) M. Wozniak, K. Modzelewska, L. Kwong, and P. Keely, BBA-Mol Cell Res, 1692, 103 (2004).
* Howard (2001) J. Howard, _Mechanics of motor proteins and the cytoskeleton_ (Sinauer Associates Sunderland, MA, 2001).
* Lemmon _et al._ (2005) C. Lemmon, N. Sniadecki, S. Ruiz, J. Tan, L. Romer, and C. Chen, Mechanics & chemistry of biosystems: MCB, 2, 1 (2005).
* Bischofs _et al._ (2009) I. Bischofs, S. Schmidt, and U. Schwarz, Phys Rev Lett, 103, 48101 (2009).
* Mertz _et al._ (2012) A. Mertz, S. Banerjee, Y. Che, G. German, Y. Xu, C. Hyland, M. Marchetti, V. Horsley, and E. Dufresne, Phys Rev Lett, 108, 198101 (2012).
* Bischofs _et al._ (2008) I. Bischofs, F. Klein, D. Lehnert, M. Bastmeyer, and U. Schwarz, Biophys J, 95, 3488 (2008).
* Mumford (1993) D. Mumford, in _Algebraic geometry and its applications_ , edited by C. Bajaj (Springer-Verlag, New York, 1993) pp. 507–518.
* Giomi and Mahadevan (2012) L. Giomi and L. Mahadevan, Proc R Soc A, 468, 1851 (2012).
* Deshpande _et al._ (2006) V. S. Deshpande, R. M. McMeeking, and A. G. Evans, Proc Natl Acad Sci USA, 103, 14015 (2006).
* Loosli _et al._ (2010) Y. Loosli, R. Luginbuehl, and J. G. Snedeker, Phil Trans R Soc A, 368, 2629 (2010).
* Farsad and Vernerey (2012) M. Farsad and F. J. Vernerey, Int J Num Meth Eng, 92, 238 (2012).
* Banerjee and Marchetti (2011) S. Banerjee and M. C. Marchetti, EPL (Europhysics Letters), 96, 28003 (2011).
* Edwards and Schwarz (2011) C. M. Edwards and U. S. Schwarz, Physical Review Letters, 107, 128101 (2011).
* Torres _et al._ (2012) P. G. Torres, I. Bischofs, and U. Schwarz, Physical Review E, 85, 011913 (2012).
* Lévy (1884) M. M. Lévy, J Math Pure Appl, 10, 5 (1884).
* Tadjbakhsh and Odeh (1967) I. Tadjbakhsh and F. Odeh, J Math Anal Appl, 18, 59 (1967).
* Flaherty _et al._ (1972) J. Flaherty, J. Keller, and S. Rubinow, SIAM Journal on Applied Mathematics, 23, 446 (1972).
* Arreaga _et al._ (2002) G. Arreaga, R. Capovilla, C. Chryssomalakos, and J. Guven, Phys Rev E, 65, 031801 (2002).
* Vassilev _et al._ (2008) V. Vassilev, P. Djondjorov, and I. Mladenov, J Phys A-Math Gen, 41, 435201 (2008).
* Djondjorov _et al._ (2011) P. Djondjorov, V. Vassilev, and I. Mladenov, Int J Mech Sci, 53, 355 (2011).
* Mora _et al._ (2012) S. Mora, T. Phou, J.-M. Fromental, B. Audoly, and Y. Pomeau, Phys Rev E, 86, 026119 (2012).
* Engler _et al._ (2004) A. Engler, L. Bacakova, C. Newman, A. Hategan, M. Griffin, and D. Discher, Biophysical Journal, 86, 617 (2004).
* Wakatsuki _et al._ (2003) T. Wakatsuki, R. Wysolmerski, and E. Elson, J Cell Sci, 116, 1617 (2003).
* Gray (1997) A. Gray, _Modern differential geometry of curves and surfaces with mathematica_ (CRC-Press, Boca Raton, FL, 1997).
* Ghibaudo _et al._ (2008) M. Ghibaudo, A. Saez, L. Trichet, A. Xayaphoummine, J. Browaeys, P. Silberzan, A. Buguin, and B. Ladoux, Soft Matter, 4, 1836 (2008).
* Mitrossilis _et al._ (2009) D. Mitrossilis, J. Fouchard, A. Guiroy, N. Desprat, N. Rodriguez, B. Fabry, and A. Asnacios, Proc Natl Acad Sci USA, 106, 18243 (2009).
* James _et al._ (2008) J. James, E. D. Goluch, H. Hu, C. Liu, and M. Mrksich, Cell Motil Cytoskel, 65, 841 (2008).
* Love (1927) A. E. H. Love, _A Treatise on the Mathematical Theory of Elasticity_ , 4th ed. (Cambridge University Press, Cambridge, 1927).
* Biot (1965) M. A. Biot, _Mechanics of incremental deformations_ (John Wiley and Sons, New York, 1965).
* Hohlfeld and Mahadevan (2011) E. Hohlfeld and L. Mahadevan, Phys Rev Lett, 106, 105702 (2011).
* Hohlfeld and Mahadevan (2012) E. Hohlfeld and L. Mahadevan, Phys Rev Lett, 109, 025701 (2012).
* Tallinen _et al._ (2013) T. Tallinen, J. S. Biggins, and L. Mahadevan, Phys Rev Lett, 110, 024302 (2013).
* Banerjee and Marchetti (2012) S. Banerjee and M. C. Marchetti, Phys Rev Lett, 109, 108101 (2012).
* Greenhill (1892) A. Greenhill, _The applications of elliptic functions_ (Macmillan and Co., 1892).
* Davis (2010) H. Davis, _Introduction to nonlinear differential and integral equations_ (Dover publications, Mineola NY, 2010).
|
arxiv-papers
| 2012-09-18T16:02:18 |
2024-09-04T02:49:35.247978
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shiladitya Banerjee and Luca Giomi",
"submitter": "Shiladitya Banerjee",
"url": "https://arxiv.org/abs/1209.4004"
}
|
1209.4022
|
# Game Theoretic Formation of a Centrality Based Network
Ryan Tatko Applied Research Laboratory &
Department of Economics
Penn State University
University Park, PA 16802
E-mail: [email protected] Christopher Griffin Applied Research Laboratory &
Department of Mathematics
Penn State University
University Park, PA 16802
E-mail: [email protected]
###### Abstract
We model the formation of networks as a game where players aspire to maximize
their own centrality by increasing the number of other players to which they
are path-wise connected, while simultaneously incurring a cost for each added
adjacent edge. We simulate the interactions between players using an algorithm
that factors in rational strategic behavior based on a common objective
function. The resulting networks exhibit pairwise stability, from which we
derive necessary stable conditions for specific graph topologies. We then
expand the model to simulate non-trivial games with large numbers of players.
We show that using conditions necessary for the stability of star topologies
we can induce the formation of hub players that positively impact the total
welfare of the network.
## I Introduction
In the last fifteen years, the emerging study of network science has produced
results impacting a broad variety of dynamic systems, from biological growth
to human interaction [1, 2]. The recent surge in society’s dependence on
social networks places extra importance in the study of how these networks
form, grow, and eventually stagnate. Although commonly accepted models produce
some accurate network characteristics, they rely almost purely on
probabilistic methods of network growth [3]. Even the most popular stochastic
generative models like Watts and Strogatz’s small-world [4] and Barabási and
Albert’s preferential attachment models [5] fall short of describing real
network behavior. In contrast, we will explain the formation of networks using
game theoretic principles, where individual “players” make rational decisions
based on maximizing an assigned payoff function under a given information set.
Applying game theory allows us to model social behavior and interpret the
resulting network structure using the underlying economic factors that a
player might take into consideration.
In this paper, we study the dynamic formation of complex networks where the
principle source of utility for each player comes from how central he is in
the network. The concept of centrality, as a metric, is commonly used in graph
theory and social network analysis to measure the relative importance of a
single node within a graph [6, 7]. Consequently, centrality can be applied to
networks to gauge how influential an individual is, determine the chain of
command within an organization, or in theory, study how social constructions
such as cities form and develop. Individuals often place importance on being
in the “middle”; those with high centrality in a social network often learn
about new information before most of the rest of the group.
There are numerous measures of centrality commonly used in network analysis;
betweenness, closeness, degree centrality [8], and even Google’s PageRank [9,
10] are each used to explain centrality in different contexts. In the case of
degree centrality, nodes with a higher number of direct connections relative
to other nodes in a graph are more central to the “flow” of information, as
the (unweighted) shortest paths through the graph tend to traverse through
highly central nodes rather than less central ones. While the number of direct
links is a large factor in an node’s centrality, the number of indirect links
to a node is also important in determining the influence of a specific node
within a network. When we consider indirect connectivity, degree centrality is
not sufficient.
Introduced by Leo Katz in the 1950’s [11], Katz centrality is a generalization
of degree centrality that measures the relative influence of a node within a
network. The Katz centrality of a node primarily depends on that node’s
immediate neighbors, but also the nodes connected to these immediate neighbors
and so on. Formally, the Katz centrality of node $i$ is defined as
$C_{K_{i}}=\sum_{k=1}^{\infty}\sum_{j=1}^{n}\alpha^{k}(\mathbf{A}^{k})_{ij}\qquad\alpha\in(0,1)$
(1)
Where $\mathbf{A}$ is the adjacency matrix of the network. Here, the powers of
$k$ measure the presence of links through intermediate nodes. For example, in
the matrix $\mathbf{A}^{2}$, if $a_{13}$ = 1, node $1$ and node $3$ are
connected through one immediate neighbor. Distant links are penalized by a
factor $\alpha\in(0,1)$ that assigns weights to each link based on the
distance between the nodes. For this definition of centrality to be
meaningful, $\alpha$ must be smaller than $1/\lambda_{0}$ where $\lambda_{0}$
is the largest eigenvalue of the adjacency matrix $\mathbf{A}$. Assuming this,
Katz centrality can be calculated as
$\mathbf{C}_{K}=((I-\alpha\mathbf{A}^{T})^{-1}-I)\mathbf{1}_{n}$ (2)
Here $\mathbf{A}^{T}$ is the transposed adjacency matrix of the network,
$\mathbf{I}_{n}$ is the $n\times n$ identity matrix, and $\mathbf{1}_{n}$ is a
ones vector of size $n$. We use this form of centrality in order to apply our
results to directed graphs as well as undirected graphs.
## II Literature review
The study of networks is a relatively new direction in the game theory and
network science literature [12, 13, 14]. Jackson and Wolinski [15] analyzed
the relationship between stability and efficiency of simple economic networks
composed of individuals equipped with a utility function. From this, they were
able to deduce network topologies most likely to form given a specific set of
conditions related to redistributive structures of the network. More recent
work by Goyal and Joshi [16] looked at networks resulting from the formation
of oligopolies between collaborating firms. Their research focused on firms
forming $pairwise$ $stable$ links, a key characteristic of networks that
depend on a mutual benefit to maintain a connection. Additional work on this
is extensive. See [15, 17, 18, 19, 20, 21, 22] and their references. In this
paper we analyze different topologies of pairwise stable networks and the
dynamics of heterogeneous link costs between players.
## III Notational Preliminaries
In this section we introduce conventional network science and game theory
notation. Assume that there is a set of players, $N=\\{1,2,3,\dots,n\\}$ where
$n\in\mathbb{N}$. A graph $G$ is a defined by a set of vertices (players) and
edges connecting them. If we define $G^{N}$ to be the complete set of all
possible adjacent edges, the set of all graphs over $N$ is defined as
$G^{\prime}=\\{G:G\subset G^{N}\\}$. The set of $N$ players within a network
have relationships characterized by binary variables, where
$a_{ij}\in\\{0,1\\}$ represents this relationship between any two players
$i,j$. Let $a_{ij}=1$ if $i$ is directly connected to $j$ and 0 otherwise. The
degree of a player $\eta_{i}$ is the number of direct connections $i$ has in
the graph, defined as $\sum_{j=1}^{n}a_{ij}$. As is common in network science
literature, we ignore the nonsensical possibility of self loops so that
$a_{ii}$=0. The complete graph $K_{n}$ is a graph which is defined by a
specific degree sequence $\eta_{i}=n-1$, $\forall\,i\in N$.
In our network formation game, we define a simple strategy set
$S_{i}=\\{s_{i}^{1},s_{i}^{2}\\}$ that gives each player the ability to form
($s_{i}^{1}$) or veto ($s_{i}^{2}$) a link with another player. We equate the
act of vetoing a link with deleting a pre-existing connection between two
players, if it exists. Let $S=\Pi_{i=1}^{n}S_{i}$ be the set of all possible
strategy profiles. We define a network game $\mathcal{G}(N,S,\pi)$, where
$\pi:S\to\mathbb{R}$ is a payoff function assigned to the set of players $N$.
## IV Theoretical Results
### IV-A Objective Model
In our work, we consider a non-cooperative game structure where a player’s
objective relies on maximizing his relative centrality while simultaneously
increasing the size of the network. In the context of real world networks,
highly centralized individuals often possess the largest “sphere of
influence,” which becomes most powerful when in close proximity to a vast
number of people. Specifically, we will study the case where players attempt
to maximize the probability they are landed on by a random walk within their
connected component. We can capture this objective as:
$\lambda_{i}=R_{i}p_{i}K_{i}$ (3)
Here $\lambda_{i}$ is the benefit function equal to the number of nodes in
$i$’s component (minus itself) $p_{i}$, multiplied by the scaled component-
wise Katz centrality of the node, $K_{i}$. This benefit function captures an
interesting trade off the player faces between maintaining a high centrality
while increasing the size of his component. We then multiply this value by
some arbitrary award $R_{i}$ a player (individuals, store, websites, etc.)
receives when another player (potentially not directly connected) engages in
an interaction with this vertex. The centrality in the payoff function is
defined as:
$K_{i}=\frac{C_{K_{i}}}{\sum_{j\in H(j)}C_{K_{j}}}$ (4)
Where $C_{K_{i}}$ is equal to the Katz centrality of $i$ in the player’s
connected subgraph $H(i)$. Our centrality measurement acts as a probability
mass that maintains a sum of one no matter the size of the network, and so is
consistent with the notion that (all else equal) players lose centrality as
the total size of the connected network grows. To maintain consistency in the
model, we define the component centrality of an isolate node to be equal to
one. We may assume a linear cost associated with establishing out-links:
$\phi_{i}=\sum_{j\neq i}\gamma_{ij}A_{ij}$ (5)
We assume that $\gamma_{ij}=\gamma_{i}$, $\forall j\in N$ and $\gamma_{i}$ is
a bilateral fixed cost for establishing individual links for all players. The
payoff function is derived as
$\pi_{i}=\lambda_{i}-\phi_{i}=R_{i}p_{i}K_{i}-\sum_{j\neq i}\gamma_{i}A_{ij}$
(6)
From this point on, when we denote the link cost as $\gamma$ without subscript
we assume that all players share a link cost, so that $\gamma_{i}=\gamma$ for
$i=1,\dots,N$. Simple marginal analysis shows that a player $i$ will establish
a link with another player $j$ as long as $\Delta\lambda_{i}>\gamma$, or the
increase in the benefit for $i$ is greater than the linear cost associated
with adding an additional link. For an undirected graph, a link will be made
between $i$ and $j$ if and only if $\Delta\pi_{i},\Delta\pi_{j}>0$. In this
sense, players attempt to minimize link maintenance cost while simultaneously
gaining centrality by establishing (or deleting) links with other players.
### IV-B Stability of Complete Graph Topologies
Pairwise stability as described by Jackson [15] is simply defined as a network
where no player benefit from creating a new link, and no two players benefit
from severing an existing link. We assume that both players must bilaterally
agree to the creation of a link, while any player can sever a link. This is
similar to “friending” on Facebook. Under topologically-specific conditions,
complete networks and star networks are pairwise stable using our model. In
the following sections, we construct the Katz centrality in manipulatable
terms to derive pairwise stable conditions for these network structures.
By calculating the Katz centrality as defined in (2) explicitly, it is
possible to express $\mathbf{K}_{i}$ in terms of the parameters $n$ and
$\alpha$. From this expression, pairwise stability can be defined for $any$
complete graph $K_{n}$ and link cost $\gamma$. Lemma 7 summarizes this result.
###### Lemma IV.1.
For a complete graph $K_{n}$, the centrality vector $\mathbf{K}$ is given by:
$K_{i}=\frac{(n-1)\alpha}{1-(n-1)\alpha}\cdot\frac{1-(n-1)\alpha}{n(n-1)\alpha}=\frac{1}{n}$
(7)
∎
The expression in Equation $\eqref{kn}$ defines the Katz centrality of a
vertex in a complete graph and shows that it is equivalent to the sum of the
infinite series $\sum_{k=1}^{\infty}((n-1)\alpha)^{k}$. We observe that the
each vertex is equally central in the complete graph and the relative
centrality of a vertex decreases harmonically as $n$ increases.
To prove stability, we consider the deletion of a link as the one possible
complete graph manipulation. We define a nearly-complete graph as the graph
$K_{n}^{(-1)}$ with degree sequence $\\{n-1,n-1,\ldots,n-2,n-2\\}$. The
centrality of a nearly-complete graph can be derived from $\eqref{vecC}$ by
changing the adjacency matrix so that $a_{ij}$ = 0 for exactly one pair of
$i,j\in N$.
###### Lemma IV.2.
For a complete graph minus one link between two vertices, the centrality
vector $\mathbf{K}_{n}^{(-1)}$ has two distinct values $K_{b},K_{s}$ given by:
$\displaystyle K_{b}=\frac{(2n-2)\alpha+n}{(n-2)(2n\alpha+n+1)}$
$\displaystyle K_{s}=\frac{2\alpha+1}{2n\alpha+n+1}$ (8)
∎
Lemma IV.2 is illustrated in Figure 1, where we show the two possible types of
vertices in $\mathbf{K}_{n}^{(-1)}$.
Figure 1: The complete graph K5 and the nearly complete graph G5,9 s.t.
$a_{1,2}$=0. For high values of $\gamma$, one player benefits from a link
deletion and the complete graph loses stability. This causes a cascading
effect of s2 strategy dominance, resulting in a graph with fewer (and often
zero) links.
Expression $\eqref{ks}$ shows that there are two possible centrality values
for vertices in a nearly-complete graph, where $K_{s}$ is the centrality of
the two vertices with $n-2$ degrees and $K_{b}$ is the centrality for all
other vertices. From the expressions it follows that the deletion of a single
link in a complete graph always results in a lower centrality and
$K_{s}<K_{b}$.
###### Theorem IV.3.
A complete network in which $a_{ij}=1,\forall i\neq j\in N$ will result in a
pairwise stable equilibrium as long as
$R_{i}(n-1)(K_{i}-K_{i}^{(-1)})>\gamma$.
###### Proof:
Consider a complete graph $K_{n}$ where $N$ is the set of $n$ players. By
definition, the graph is a single component so $p_{i}=(n-1)$, $\forall i\in
N$. For the complete graph, the payoff function can be simplified as
$\pi_{i}^{\star}=R_{i}(n-1)K_{i}-(n-1)\gamma$ (9)
Where $\pi_{i}^{\star}$ represents the payoff for each player $i$ in the
complete graph $K_{n}$. If a player deviates away from $\pi_{i}^{\star}$ by
deleting an existing link, that player’s payoff is given by
$\pi_{i}=R_{i}(n-1)K_{i}^{(-1)}-(n-2)\gamma$ (10)
If $\pi_{i}\leq\pi_{i}^{\star}$, than no player in $K_{n}$ will benefit from
deviating from the complete structure. Thus
$\pi_{i}^{\star}-\pi_{i}>0\implies R_{i}(n-1)(K_{i}-K_{i}^{(-1)})>\gamma$ (11)
From $\eqref{kn}$ and $\eqref{ks}$, we get the explicit necessary condition
for stability:
$\frac{n-1}{n(2n\alpha+n+1)}>\frac{\gamma}{R_{i}}$ (12)
As long as this inequality holds, the complete graph is pairwise stable and
any player will receive a non-positive marginal payoff from deleting an
existing link with another player. ∎
Note that if a fixed link cost is chosen and players are subsequently added to
the complete network, the necessary stability inequality is violated and the
complete network will collapse and reform into smaller components.
### IV-C Stability of Star Graph Topologies
When we study real world networks, it is difficult to find “completeness” on a
large scale; rarely does every individual in even the closest communities form
bonds with every other individual. Networks with star topologies are a common
phenomena, appearing often in computer and social systems aiming to optimize
efficiency. In its basic form, a star graph $S_{n}$ is defined by $n$ vertices
and $n$+1 edges, where each “leaf” vertex is connected to only a single “hub”
vertex. We induce the stability in star networks by choosing separate link
costs $\delta$ and $\zeta$ for leaf and hub vertices. These “tax breaks”
simulate a heterogeneous population of players found in nearly every real
world system. These non-trivial networks exhibit interesting properties
explained by their stability conditions.
Like our previous section on complete graphs, we derive star graph centrality
explicitly in terms of parameters $n$ and $\alpha$.
###### Lemma IV.4.
For any star graph $S_{n}$, the centrality vector $\mathbf{K}$ has two
possible values: $K_{b}$, the centrality of the central vertex and $K_{s}$,
the centrality of the leaves, where:
$\displaystyle K_{b}$ $\displaystyle=\frac{\alpha+1}{n\alpha+2}$
$\displaystyle K_{s}$ $\displaystyle=\frac{(n-1)\alpha+1}{(n-1)(n\alpha+2)}$
(13)
From $\eqref{231}$ we see that the centrality of the hub vertex is high
relative to the leaf vertices for low values of $n$. To determine a pairwise
stable condition we consider the result of two possible strategies: (1) a link
is added between two leaf vertices and (2) the a leaf vertex deletes its
single link with the hub vertex.
###### Lemma IV.5.
For any single-degree vertices $i,j$ in a star graph $S_{n}$, if a link is
formed between them, i.e., $a_{ij}$=1, then the centrality $K_{s}^{(+1)}$ of
both vertices is given by
$K_{s}^{(+1)}=\frac{(n-3)\alpha^{2}+(1-n)\alpha-2}{(n-3)(\alpha-1)n\alpha-2n-6\alpha}$
(14)
The action in Lemma 14 are illustrated in Figure 2
Figure 2: The star graph S5 and the graph G${}_{5},5$ s.t. $a_{2,3}$=1. Star
topologies arise often in many network contexts. For lower values of $\gamma$,
s1 strategy dominates as leaf nodes connect with each other. The resulting
stable graph is often the complete graph.
###### Theorem IV.6.
If the central node of a star network $S_{n}$ has a link cost $\zeta$, while
the leaves have link cost $\delta$ so that $\zeta\leq\gamma$, then $S_{n}$ is
pairwise stable if for every leaf $i$ with a link cost $\delta$,
$R_{i}(n-1)(K_{i}^{(+1)}-K_{i})<\delta<R_{i}(n-1)K_{i}$ and for the hub $j$
with a link cost $\zeta$, $R_{j}(n-1)K_{j}-R_{j}(n-2)K_{j}^{(-1)}>\zeta$.
###### Remark IV.7.
Before providing the proof, we remark, we will derive exact conditions:
$R_{i}(n-1)(K_{i}^{(+1)}-K_{i})<\delta<R_{i}(n-1)K_{i}$ and
$R_{j}(n-1)K_{j}-R_{j}(n-2)K_{j}^{(-1)}>\zeta$.
###### Proof:
Assume that there is a payoff function for vertex $i$ defined as
$\pi_{i}=R_{i}p_{i}K_{i}-\sum_{j\neq i}\gamma_{i}A_{ij}$
$\gamma_{i}=\begin{cases}\zeta&\text{if }\eta_{i}=(n-1)\\\
\delta&\text{otherwise}\\\ \end{cases}$
where $R_{i}$ is a constant reward obtained by vertex $i$, $p_{i}$ is the
number of nodes in $i$’s component (minus itself), $K_{i}$ is the centrality
of $i$, and $\eta_{i}$ is the number degrees of vertex $i$. Because there is
only one component for a connected graph, $p_{i}=(n-1)$ for all $i\in N$. Let
$\gamma_{i}$ be the cost for maintaining an existing link between $i$ and $j$.
In order for a star graph to be pairwise stable, the hub must have no
incentive to delete its link between itself and a leaf, or
$\pi_{i}-\pi_{i}^{(-1)}>0$ (15)
We expand this using the centrality terms defined in $\eqref{231}$ and
$\eqref{232}$ to show that a hub will not delete a link if
$\displaystyle\frac{(n-1)(\alpha+1)}{n\alpha+2}-\frac{(n-2)(\alpha+1)}{(n-1)\alpha+2}>\frac{\zeta}{R_{i}}$
$\displaystyle\frac{(\alpha+1)(\alpha+2)}{(n\alpha+2)(n\alpha-\alpha+2)}>\frac{\zeta}{R_{i}}$
(16)
To show that a leaf will neither delete a link or establish a new link, we get
these two necessary inequalities:
$\displaystyle\pi_{i}^{(+1)}-\pi_{i}$ $\displaystyle<0$
$\displaystyle\pi_{i}-\pi_{iso}$ $\displaystyle>0$ (17)
Where $\pi_{i}^{(+1)}$ is the payoff of a leaf linking with another leaf and
$\pi_{iso}$ is the payoff for an isolate node. Because an isolated node has no
neighbors, it’s payoff is zero. From this we show that any leaf $i$ will not
disconnect from the hub if
$\frac{(n-1)^{2}\alpha+n-1}{(n-1)(n\alpha+2)}>\frac{\delta}{R_{i}}$ (18)
and $i$ will not link with another leaf as long as
$\frac{(n-1)((n-3)\alpha^{2}+(1-n)\alpha-2)}{(n-3)(\alpha-1)n\alpha-2n-6\alpha}-\\\
\frac{(n-1)^{2}\alpha+n-1}{(n-1)(n\alpha+2)}<\frac{\delta}{R_{i}}$ (19)
Rewriting these expressions in terms of variables in the payoff function, a
star graph is pairwise stable if for any leaf $i$:
$(n-1)(K_{i}^{(+1)}-K_{i})<\frac{\delta}{R_{i}}<(n-1)K_{i}$ (20)
and for any hub $j$:
$(n-1)K_{j}-(n-2)K_{j}^{(-1)}>\frac{\zeta}{R_{j}}$ (21)
∎
## V Empirical Results
We have shown that star networks are pairwise stable for games with
heterogeneous link costs between players on a small-scale. To illustrate the
complexity of real world networks with numerous players, we have developed an
algorithm that computes possible pairwise stable game solutions given a game
definition and a specific set of parameters. The blueprint of the algorithm is
as follows:
Add $N$ isolate nodes to the null network.
Each isolated node is given an objective function $\pi$ and strategy set
$\\{s_{1},s_{2}\\}$.
Choose two nodes $i,j$ at random from the set $\\{1,2,\ldots,N\\}$. If both
(either) nodes benefit from the addition (deletion) of a link between them,
the aijth entry in the network’s adjacency matrix $\mathbf{A}$ is changed to a
one (zero).
A stable network is achieved when neither the addition nor deletion of a
single link results in a higher payoff for all $i$ in $N$. Note that the
algorithm does not assume players know which links contribute a higher payoff
compared to others, i.e., the interactions are treated as random events
between players with imperfect information. This is an important distinction
between our model and network games proposed in previous literature [23].
Using this algorithm, we analyze the game $\mathcal{G}(100,S,\pi)$ with
network parameters $R_{i}=1$, $\alpha=.0075$, and $\gamma=.25$, where $\pi$ is
the payoff function defined in $\eqref{pi}$. As previously described, the
player’s objective relies on centralizing his position in the network at the
cost of fixed-price connections. The network shown in Fig. 3 is one of many
pairwise stable topologies resulting from these parameters.
Figure 3: A simulated pairwise stable solution for the game with N=100,
$\alpha$=.0075, $\gamma$=.25. Size and color intensity is determined by the
degrees of the node.
Notice that we have multiple connected components with a single giant
component emerging from the network simulation. This is consistent with our
previous remark regarding the stability of increasingly large single component
complete networks. Without further analysis, we infer that either players in
the smaller components cannot afford to be part of a larger network at the
cost of their current position, or (more likely) a player in the giant
component would not not benefit from taking on an additional link with a
“lesser” central player. The pairwise stable graph shown in Fig. 3 has an
average degree of 3.56 and total payoff ($\sum_{i=1}^{N}\pi_{i}$) of 7.50.
Figure 4: A simulated pairwise stable solution for the game with N=100,
$\alpha$=.0075, $\delta$=.25, $\zeta$=.20. Size and color intensity is
determined by the degrees of the node.
Next we consider an altered version of the game defined above, where a select
number of players are offered a discounted link cost. We define the new
network parameters $R_{i}=1$, $\alpha=.0075$, $\delta=.25$, $\zeta=.20$, where
$\zeta$ is the link cost for exactly five incentivized players. One example
pairwise stable network (shown in Fig. 4) exhibits an entirely different
structure than the previously studied network. The five players act as hubs,
accumulating a large portion of the links to form pseudo-star topologies
within the network. Over multiple simulations, we observe that hubs only
accumulate when players are offered lower link costs compared to their peers.
However, a lower link cost does not necessarily guarantee that player becomes
a hub. It appears that the number of hubs a network can sustain exhibits a
positive correlation with the size of the network; intuitively, a network can
only sustain a few “superstar” nodes vying for connections.
Interestingly, the incentivized network sustains an average degree of 3.42 and
a larger total payoff of 18.45. Incorporating incentivized players in the
network not only increases the total welfare of the network, but also a number
of players $without$ discounted link costs. From Fig. 5 we see that players
indirectly benefit from the centrality of neighboring hubs and are thus more
highly centralized in a larger component. Representative runs are shown in
Figure 5.
Figure 5: The payoffs of 100 players with variable link costs, shown on a base
10 logarithmic scale. Note that as the margin between costs is increased, the
payoffs of hub and intermediate players increases as well.
## VI Conclusions and Future Directions
In this paper we have created an $N$-player game that models the strategic
interaction between players vying for centrality within a dynamic system. From
our objective function we derived parameter-specific $(n,\alpha,\gamma)$
conditions for pairwise stability in the some commonly found real world
network topologies and used those to interpret our algorithm’s empirical
results. Interestingly, our research has found that introducing incentivized
players with inherent “advantages” into the model increases the total welfare
of the network and creates a better semblance of communal structure when
compared to a completely homogeneous population of players.
In the future, we would like to analyze the results of more complex parameter
adjustments, such as implementing an evolving cost distribution based on a
player’s current position within the network. The algorithm described in the
paper can further be adjusted to allow for games with perfect information,
where players possess more possible linking strategies to choose from.
In addition, we would like to analyze games that produce more realistic small-
world behavior. These networks are found in numerous social systems [24] and a
complex game theoretic model analyzing the dynamics of small-world phenomena
may be of interest.
## Acknowledgment
Portions of Dr. Griffin’s work were supported by professional development
funding from the Applied Research Laboratory. Portions of Mr. Tatko’s work
were supported by the Applied Research Laboratory undergraduate honors
program.
## References
* [1] S. Bulò and I. Bomze, “Infection and immunization: A new class of evolutionary game dynamics,” _Games and Economic Behavior_ , vol. 71, pp. 193–211, 2011.
* [2] M. Jackson and A. Watts, “The evolution of social and economic networks,” _Journal of Economic Theory_ , vol. 106, no. 2, pp. 265–295, 2002.
* [3] L. Li, D. Alderson, J. Doyle, and W. Willinger, “Towards a theory of scale-free graphs: Definition, properties, and implications,” _Internet Mathematics_ , vol. 2, no. 4, pp. 431–523, 2005.
* [4] D. Watts and S. Strogatz, “Collective dynamics of ’small-world’ networks,” _Nature_ , vol. 393, pp. 440–442, 6684 1999.
* [5] A. Barabási and R. Albert, “Emergence of scaling in random networks,” _Science_ , vol. 286, no. 5439, pp. 509–512, 1999.
* [6] P. J. Carrington, J. Scott, and S. Wasserman, _Models and Methods in Social Network Analysis_. Cambridge University Press, 2005.
* [7] D. Knoke and S. Yang, _Social Network Analysis_ , ser. Quantitative Applications in the Social Sciences. SAGE Publications, 2008, no. 154.
* [8] M. E. J. Newman, _Networks: An Introduction_. Oxford University Press, 2010.
* [9] S. Brin and L. Page, “The anatomy of a large-scale hypertextual web search engine,” in _Seventh International World-Wide Web Conference (WWW 1998)_ , 1998.
* [10] T. Haveliwala, “Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search,” _IEEE Transactions on Knowledge and Data Engineering_ , vol. 15, no. 4, pp. 784–796, 2003.
* [11] L. Katz, “A new status index derived from sociometric index,” _Psychometrika_ , pp. 39–43, 1953.
* [12] C. Griffin and A. Squicciarini, “Toward a game theoretic model of information release in social media with experimental results,” in _Proc. 2nd Workshop on Semantic Computing and Security_ , San Francisco, CA, USA, May 28 2012\.
* [13] A. C. Squicciarini, S. Sundareswaran, and C. Griffin, “A game theoretical perspective of users’ registration in online social platforms,” in _Accepted to Third IEEE International Conference on Privacy, Security, Risk and Trust_ , October 9 - 11 2011.
* [14] A. Squicciarini and C. Griffin, “An informed model of personal information release in social networking sites,” in _ASE/IEEE Conference on Privacy, Security, Risk and Trust_ , Amsterdam, Netherlands, September 3-5 2012\.
* [15] M. Jackson and A. Wolinsky, “A strategic model of social and economic networks,” _Journal of Economic Theory_ , vol. 71, no. 1, p. 1996, 1996.
* [16] S. Goyal and S. Joshi, “Networks of collaboration in oligopoly,” _Games and Economic Behavior_ , vol. 43, no. 1, pp. 57–85, 2003.
* [17] B. Dutta and S. Mutuswami, “Stable networks,” _Journal of Economic Theory_ , vol. 76, no. 2, pp. 322–344, 1997.
* [18] V. Bala and S. Goyal, “A noncooperative model of network formation,” _Econometrica_ , vol. 68, no. 5, pp. 1181–1229, 2000.
* [19] T. Furusawa and H. Konishi, “Free trade networks,” _Journal of International Economics_ , vol. 72, no. 2, pp. 310–335, 2007.
* [20] P. Belleflamme and F. Bloch, “Market sharing agreements and collusive networks,” _International Economic Review_ , vol. 45, no. 2, pp. 387–411, 2004.
* [21] A. Calvó-Armengol and M. Jackson, “Networks in labor markets: Wage and employment dynamics and inequality,” _Journal of Economic Theory_ , vol. 132, no. 1, pp. 27–46, 2007.
* [22] M. Jackson, “A survey of models of network formation: Stability and efficiency,” _Game Theory and Information_ , 2003.
* [23] P. Holme and G. Ghoshal, “Dynamics of networking agents competing for high centrality and low degree,” _Physical Review Letters_ , 2006.
* [24] D. Watts, _Small Worlds_. Princeton University Press, 1999.
|
arxiv-papers
| 2012-09-18T16:38:06 |
2024-09-04T02:49:35.256557
|
{
"license": "Public Domain",
"authors": "Ryan Tatko and Christopher Griffin",
"submitter": "Christopher Griffin",
"url": "https://arxiv.org/abs/1209.4022"
}
|
1209.4029
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-267 LHCb-PAPER-2012-023 18 September 2012
Search for the rare decay $K_{\rm\scriptscriptstyle
S}^{0}\rightarrow\mu^{+}\mu^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
A search for the decay $K_{\rm\scriptscriptstyle
S}^{0}\rightarrow\mu^{+}\mu^{-}$ is performed, based on a data sample of 1.0
fb-1 of $pp$ collisions at $\sqrt{s}$ = 7 TeV collected by the LHCb experiment
at the Large Hadron Collider. The observed number of candidates is consistent
with the background-only hypothesis, yielding an upper limit of
$\mathcal{B}(K_{\rm\scriptscriptstyle
S}^{0}\rightarrow\mu^{+}\mu^{-})<11(9)\times 10^{-9}$ at 95 (90)$\%$
confidence level. This limit is a factor of thirty below the previous
measurement.
Published in the Journal of High Energy Physics
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, C. Gaspar35, R.
Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V.
Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A.
Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado
Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T.
Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51,
P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P.
Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52,
M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N.
Hussain52, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P.
Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F.
Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K.
Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M.
Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D.
Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45, C.
Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M.
Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac
Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O. Maev27,35,
J. Magnin1, M. Maino20, S. Malde52, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, A. Massafferri1, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E.
Maurice6, A. Mazurov16,30,35, J. McCarthy42, G. McGregor51, R. McNulty12, M.
Meissner11, M. Merk38, J. Merkel9, D.A. Milanes13, M.-N. Minard4, J. Molina
Rodriguez54, S. Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38,
F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47,
N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N.
Nikitin29, T. Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53,
A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S.
Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro36, W. Qian3, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N. Rauschmayr35, G.
Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A.
Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35, V.
Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, P. Schaack50, M.
Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B.
Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I.
Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35,
P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L. Shekhtman31, O.
Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K. Sobczak5, F.J.P.
Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, H.
Voss10, C. Voß55, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R.
Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J.
Wicht35, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M.
Williams50,p, F.F. Wilson46, J. Wishahi9, M. Witek23,35, W. Witzeling35, S.A.
Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z.
Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a,
F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The decay $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ is a
Flavour Changing Neutral Current (FCNC) transition that has not yet been
observed. This decay is suppressed in the Standard Model (SM), with an
expected branching fraction [1, 2]
${\cal B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})=(5.0\pm
1.5)\times 10^{-12},$
while the current experimental upper limit is $3.2\times 10^{-7}$ at 90$\%$
confidence level (C.L.) [3].
Although the dimuon decay of the $K^{0}_{\rm\scriptscriptstyle L}$ meson is
known to be ${\cal B}(K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-})=(6.84\pm 0.11)\times 10^{-9}$ [4], in agreement
with the SM, effects of new particles can still be observed in
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ decays. In the most
general case, the decay width of $K^{0}_{\rm\scriptscriptstyle
L,S}\rightarrow\mu^{+}\mu^{-}$ can be written as [5]
$\Gamma(K^{0}_{\rm\scriptscriptstyle
L,S}\rightarrow\mu^{+}\mu^{-})=\frac{m_{K}}{8\pi}\sqrt{1-\left(\frac{2m_{\mu}}{m_{K}}\right)^{2}}\left[|A|^{2}+\left(1-\left(\frac{2m_{\mu}}{m_{K}}\right)^{2}\right)|B|^{2}\right],$
(1)
where $A$ is an S-wave amplitude and $B$ a P-wave amplitude. These two
amplitudes have opposite $C\\!P$ eigenvalues, and in absence of $C\\!P$
violation ($K^{0}_{\rm\scriptscriptstyle S}$ $=K_{1}^{0}$,
$K^{0}_{\rm\scriptscriptstyle L}$ $=K_{2}^{0}$), $K^{0}_{\rm\scriptscriptstyle
L}$ decays would be generated only by $A$ while $K^{0}_{\rm\scriptscriptstyle
S}$ decays would be generated only by $B$. The decay width
$\Gamma(K^{0}_{\rm\scriptscriptstyle L}\rightarrow\mu^{+}\mu^{-})$ receives
long-distance111The long-distance scales correspond to masses below that of
the $c$ quark, while short-distance scales correspond to masses of the $c$
quark and above. contributions to $A$ from intermediate two-photon states, as
well as short distance contributions to the real part of $A$. In any model
with the same basis of effective FCNC operators as the SM, the contributions
from $B$ can be neglected for ${\cal B}(K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-})$. The decay width of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ depends on the
imaginary part of the short-distance contributions to $A$ and on the long-
distance contributions to $B$ generated by intermediate two-photon states.
Therefore, the measurement of ${\cal B}(K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-})$ in agreement with the SM does not necessarily
imply that ${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})$ has to agree with the SM. Contributions up to
one order of magnitude above the SM expectation are allowed [2]; enhancements
of the branching fraction above $10^{-10}$ are less likely. The study of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ has been suggested
as a possible way to look for new light scalars [1].
In addition, bounds on the upper limit of ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ close to
$10^{-11}$ could be very useful to discriminate among scenarios beyond the SM
if other modes, such as $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ (charge
conjugation is implied throughout this paper), were to indicate a non-standard
enhancement of the $s\rightarrow d\ell\bar{\ell}$ transition [2]. The KLOE
collaboration has searched for the related decay $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow e^{+}e^{-}$, which is affected by a larger helicity suppression
than the muonic mode, and set an upper limit on the branching fraction ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow e^{+}e^{-})<9\times 10^{-9}$ at
90% confidence level [6].
The LHC produces $\sim 10^{13}$ $K^{0}_{\rm\scriptscriptstyle S}$ per
$\mbox{\,fb}^{-1}$ inside the LHCb acceptance. In this paper, a search for
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ is presented using
1.0 fb-1 of $pp$ collisions at $\sqrt{s}$ = 7 TeV collected by LHCb in 2011.
Dimuon candidates are classified in bins of a multivariate discriminant, and
compared to background and signal expectations. The background present in the
signal region is a combination of combinatorial background and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays in which
both pions are misidentified as muons. The number of expected signal
candidates for a given branching fraction hypothesis is obtained by
normalising to the measured $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ rate. The results obtained by the measurements in
different bins are combined, and a limit is set using the
$\textrm{CL}_{\textrm{s}}$ method [7, 8]. The data in the signal region were
only analysed once the full analysis strategy was defined, including the
selection, the binning and the evaluation of systematic uncertainties.
The LHCb apparatus, and the aspects of the trigger relevant for this analysis
are presented in Sect. 2. Section 3 is devoted to the full signal selection
and to the definition of the multivariate method used as the main
discriminant. In Sect. 4 the different backgrounds for
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ decay are
described, as well as the expected background in the signal region. The
normalisation, required to convert the number of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ candidates to the branching fraction, is detailed
in Sect. 5. The systematic uncertainties are described in Sect. 6. The limit
setting procedure, together with the corresponding expected and observed
limits, is presented in Sect. 7, and conclusions are drawn in Sect. 8.
## 2 Experimental setup
The LHCb detector [9] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream.
The combined tracking system has a momentum resolution $\Delta p/p$ that
varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (${\rm
IP}$) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$) with respect to the beam direction. Charged hadrons are
identified using two ring-imaging Cherenkov detectors. Photon, electron and
hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. For this analysis, the events are first required to
pass a hardware trigger which selects at least one muon with $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the subsequent software
trigger [10], at least one of the final state tracks is required to be of good
quality and to have $\mbox{$p_{\rm
T}$}>1.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, an ${\rm IP}$ $>0.5$$\rm\,mm$
and the $\chi^{2}$ of the impact parameter (IP $\chi^{2}$) above $200$. The IP
$\chi^{2}$ is defined as the difference between the $\chi^{2}$ of the proton-
proton, $pp$, interaction point (primary vertex, PV) built with and without
the considered track. A prescale factor of two is applied to the lines
triggered by the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates. The $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates responsible for the trigger of both the hardware and software
levels are called TOS (trigger on signal).
Events with a reconstructed $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ candidate can also be triggered independently of
the signal candidate if some other combination of particles in the underlying
event passes the trigger. Such candidates are called TIS (trigger
independently of signal). The TIS and TOS categories are not exclusive as
muons from both the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates and from the underlying event can pass the trigger. There is
overlap between the two, which allows the determination of trigger
efficiencies from the data [11]. Finally, minimum bias candidates triggered by
a dedicated random trigger (MB) provide a negligible amount of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates. Instead
they allow the selection of a sample of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ useful to understand the distributions that the
signal would have in the case of no trigger bias.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [12] with a
specific LHCb configuration [13]. Decays of hadronic particles are described
by EvtGen [14] in which final state radiation is generated using Photos [15].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [16, *Agostinelli:2002hh] as
described in Ref. [18].
## 3 Selection and multivariate classifier
The $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates are
reconstructed requiring two tracks with opposite curvature with hits in the
VELO and in the tracking stations. About 40% of the
$K^{0}_{\rm\scriptscriptstyle S}$ mesons with the two daughter tracks inside
the LHCb acceptance decay in the VELO detector. Those tracks are required to
be of high quality ($\chi^{2}<5$ per degree of freedom), to have an ${\rm IP}$
$\chi^{2}$ greater than 100 and a distance of closest approach of less than
0.3 $\rm\,mm$. The two tracks are required to be identified as muons [19]. The
reconstructed $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates are required to have a proper decay time greater than 8.9
${\rm\,ps}$ and to point to the PV (${\rm IP}(K^{0}_{\rm\scriptscriptstyle
S})<400$ $\,\upmu\rm m$). The secondary vertex, SV, of the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidate is
required to be downstream of the PV. If more than one PV is reconstructed, the
PV associated to the $K^{0}_{\rm\scriptscriptstyle S}$ is the one that
minimises its ${\rm IP}$ $\chi^{2}$. Furthermore, $\Lambda\rightarrow
p\pi^{-}$ decays are vetoed via a requirement in the Armenteros-Podolanski
plane [20], by including cuts on the transverse momentum of the daughter
tracks with respect to the $K^{0}_{\rm\scriptscriptstyle S}$ flight direction
and on their longitudinal momentum asymmetry. The reconstructed
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ mass is required to
be in the range [450,1500] ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay is used
as a control channel and is reconstructed and selected in the same way as the
signal candidates, with the exception of the particle identification
requirements on the daughter tracks and the mass range, which is requested to
be between 400 and 600 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Figure 1 shows the mass spectrum for selected $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ candidates in the MB sample after applying the
set of cuts described above and in the $\pi\pi$ and $\mu\mu$ mass hypotheses:
the two mass peaks are separated by 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This separation, combined with the
LHCb mass resolution of about 4 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
such combinations of tracks, is used to discriminate the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ signal from
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays where both
pions are misidentified as muons.
Figure 1: Mass spectrum for selected $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ candidates in the MB sample. The black points
correspond to the mass reconstructed under the $\pi\pi$ mass hypothesis for
the daughters, while the red triangles correspond to the mass reconstructed
under the $\mu\mu$ mass hypothesis.
In order to further increase the background rejection, a boosted decision tree
(BDT) [21] with the AdaBoost algorithm [22] is used. The variables entering in
the BDT discriminant are:
* •
the decay time of the $K^{0}_{\rm\scriptscriptstyle S}$ candidate, computed
using the distance between the SV and the PV, and the reconstructed momentum
of the $K^{0}_{\rm\scriptscriptstyle S}$ candidate;
* •
the smallest muon IP $\chi^{2}$ of the two daughter tracks with respect to any
of the PVs reconstructed in the event;
* •
the $K^{0}_{\rm\scriptscriptstyle S}$ IP $\chi^{2}$ with respect to the PV;
* •
the distance of closest approach between the two daughter tracks;
* •
the secondary vertex $\chi^{2}$, which adds complementary information with
respect to the distance of closest approach of the tracks, as it uses
information on the uncertainty of the vertex fit;
* •
the angle of the decay plane in the $K^{0}_{\rm\scriptscriptstyle S}$ rest
frame with respect to the $K^{0}_{\rm\scriptscriptstyle S}$ flight direction,
which is isotropic for signal decays, but not necessarily for background
candidates;
* •
variables used to discriminate against material interactions, as further
detailed below.
An important source of background consists of muons resulting from
interactions between the particles produced in the PV and the detector
material in the region of the VELO. The position of the SV of the background
candidates from the $K^{0}_{\rm\scriptscriptstyle S}$ mass sidebands in the
$x-z$ plane is shown in Fig. 2. The structures observed correspond to the
position of the material inside the VELO detector. To discriminate against
this background, two different approaches are used for the TIS and TOS trigger
categories, consisting of two different choices of variables for the BDT.
For the TOS category, two additional variables are included in the BDT, the
$p_{\rm T}$ of the $K^{0}_{\rm\scriptscriptstyle S}$ and a boolean matter veto
that uses the VELO geometry to assess whether a given decay vertex coincides
with a point in the detector material or not. Muons from material interactions
have a harder $p_{\rm T}$ spectrum than muons from other background sources
and hence are more likely to be selected by the trigger. The use of this
variable in the BDT provides 50% less background yield for the same signal
efficiency than simply applying the veto as a selection cut.
For the TIS category, the coordinates of the position of the SV in the
laboratory frame are used to deal with this background. As the simultaneous
use of the lifetime, $p_{\rm T}$ of the $K^{0}_{\rm\scriptscriptstyle S}$
meson, and the SV position allows the BDT to effectively compute the mass of
the candidate, a fake signal peak could be artificially created out of the
combinatorial background. Hence the $p_{\rm T}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$ meson is not used in the TIS analysis. This
second approach provides a factor of two less background yield for the same
signal efficiency than the matter veto (and $K^{0}_{\rm\scriptscriptstyle S}$
$p_{\rm T}$) for the TIS analysis, while, on the contrary, the matter veto
boolean variable gives a factor of four less background yield for the same
signal efficiency than the SV coordinates for the TOS analysis.
Figure 2: Position in the $x-z$ plane of the secondary vertices of the
background candidates found in the high mass sideband for (left) TIS
candidates and (right) TOS candidates. The lighter coloured areas correspond
to higher density of points.
Because of these different approaches and to take into account the biases on
the variable distributions introduced by the trigger, the data sample is split
in two subsamples according to the TIS and TOS categories, for which BDT
discriminants are optimised separately. In the TOS analysis, the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays are required
to have at least one of the daughters with a $p_{\rm T}$ above 1.3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in order to minimise the difference in
the momentum distributions with respect to the triggered
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates. The
candidates that are simultaneously TIS and TOS are analysed only as TIS
candidates to avoid counting them twice. Only one per mille of the TOS
candidates overlap with TIS candidates.
In addition, the BDT discriminants for both trigger categories are defined and
trained on data using $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ candidates as signal sample and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates in the
upper mass sideband as background sample. For the background sample, the
region above 1100 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (above the $\phi$
resonance) is used to define the BDT settings and the region between 504 and
1000 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to train the BDT algorithm
chosen. For the signal sample, the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ TIS events are used to train the BDT for the TIS
category, while $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$
decays with both pions misidentified as muons and passing the same trigger
requirements as the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
signal are used for the TOS category. In order to minimise the differences
between misidentified $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ events and $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ decays, tight muon identification requirements
(including cuts in the quality of the tracks or in the number of muon hits
shared by different tracks) are applied to the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ sample. These tight requirements are chosen such
that the efficiency of the trigger in the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ simulated decays is the same as in the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ simulated decays.
In addition, the TOS and TIS categories are further split in two equal-sized
subsamples, corresponding to the first and second halves of the data taking
period. This procedure prevents possible biases related to the use of the same
events in the mass sidebands both to train the BDT discriminant and to
evaluate the background in the signal region, while making maximal use of the
available data both for BDT training and background evaluation. Thus, in
total, four different samples are defined (two subsamples for the TIS trigger
category and two subsamples for the TOS trigger category) and combined as
described in Sect. 7.
Candidates with low values of the BDT response are not considered because of
the large amount of background in that region. This requirement provides about
$50\%$ signal efficiency and $99\%$ background rejection, depending on the
sample. The rest of the candidates are classified in ten bins of equal signal
efficiency, i.e. a total of forty bins are combined to get the
$\textrm{CL}_{\textrm{s}}$ limit.
## 4 Background
The search region is defined as the mass range
$[492,504]$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The background level is
calibrated by interpolating the observed yield from mass sidebands
($[470,492]$ and $[504,600]$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) to
the signal region. This is done by means of an unbinned maximum likelihood fit
in the sidebands, using a model with two components. The first component is a
power law that describes the tail of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ decays where both pions are misidentified as
muons; this model has been checked to be appropriate using MC simulation. The
second component is an exponential function describing the combinatorial
background. As an illustration, Fig. 3 shows the distribution of candidates
for all BDT bins and for TIS and TOS samples, respectively. The expected total
background yield in the most sensitive BDT bins of both samples ranges from 0
to 1 candidates.
Figure 3: Background model fitted to the data separated along (left) TIS and
(right) TOS trigger categories. The vertical lines delimit the search window.
Other sources of background, such as $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\mu^{-}\bar{\nu}_{\mu}$, $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}\gamma$, $K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-}\gamma$, $K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\pi^{+}\mu^{-}\bar{\nu}_{\mu}$ and $K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-}$ decays, are negligible for the current analysis.
In the case of $K^{0}_{\rm\scriptscriptstyle L}\rightarrow\mu^{+}\mu^{-}$ and
$K^{0}_{\rm\scriptscriptstyle L}\rightarrow\mu^{+}\mu^{-}\gamma$, the
contributions have been evaluated using the ratio of the
$K^{0}_{\rm\scriptscriptstyle S}$ and $K^{0}_{\rm\scriptscriptstyle L}$
lifetimes and the proper time acceptance measured in data with the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays. The
contributions of the other decay modes have been determined using MC simulated
events.
## 5 Normalisation
A normalisation is required to translate the number of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ signal decays into
a branching fraction measurement. Two normalisations are determined
independently for TIS and TOS candidates. The ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ is computed
using
$\frac{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})}{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}=\frac{\epsilon_{\pi\pi}}{\epsilon_{\mu\mu}}\frac{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}},$ (2)
where, in a given BDT bin, $N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}$ is the observed number of signal decays,
$N_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}$ the number of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays, and
$\epsilon_{\pi\pi}/\epsilon_{\mu\mu}$ the ratio of the corresponding
efficiencies. The efficiencies are factorised as
$\epsilon=\epsilon^{\text{SEL}}\epsilon^{\text{PID}}\epsilon^{\text{TRIG/SEL}}$
where:
* •
$\epsilon^{\text{SEL}}$ is the offline selection efficiency. It includes the
geometrical acceptance, reconstruction and selection, i.e, it is the
probability for a $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$
($K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$) decay generated
in a $pp$ collision, to have been reconstructed and selected;
* •
$\epsilon^{\text{PID}}$ is the efficiency of the muon identification for
reconstructed and selected $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ signal decays;
* •
$\epsilon^{\text{TRIG/SEL}}=N^{\text{SEL\&PID\&TRIG}}/N^{\text{SEL\&PID}}$,
where TRIG denotes either the TIS or the TOS categories, is the trigger
efficiency for decays that would be offline selected. Under this definition,
trigger efficiencies can be determined from data using the procedure described
in Ref. [11].
The ratio of reconstruction and selection efficiencies between
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays is evaluated
in bins of $p_{\rm T}$ and rapidity of the $K^{0}_{\rm\scriptscriptstyle S}$
meson using simulated events reweighted in order to reproduce the
$K^{0}_{\rm\scriptscriptstyle S}$ $p_{\rm T}$ and rapidity spectra measured in
data [23]. The reconstruction and selection efficiency for
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays is between
60% and 85% (depending on which point in the phase space a given event is
from) of that of the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ decays due to difference in the material
interactions of the pions compared to muons.
The factor $\epsilon^{\text{PID}}$ is evaluated in bins of the BDT (both for
the TOS and TIS categories) by measuring the muon identification efficiency as
a function of $p$ and $p_{\rm T}$ using calibration muons. The sample of
calibration muons is obtained from a $J/\psi\rightarrow\mu^{+}\mu^{-}$ sample
in which positive muon identification is required for only one of the tracks.
The $p$ and $p_{\rm T}$ spectra of the pions from
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays in a MB
sample is later used to get the efficiency for $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ decays. The $\epsilon^{\text{PID}}$ efficiency is
between $68\%$ and $82\%$ (depending on the BDT bin and the sample). It is
measured with a precision between $1\%$ and $10\%$. For the ratio of trigger
efficiencies, different strategies are considered for the TIS and TOS samples.
For the TIS samples, the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ yield is normalised to the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ TIS yield. In this
case, the trigger efficiencies cancel in the ratio, because the probability to
trigger on the underlying event is independent of the decay mode of the
$K^{0}_{\rm\scriptscriptstyle S}$ meson. This cancellation is verified in
simulation. The normalisation expression for TIS decays reads
$\frac{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})}{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}^{\text{TIS}}}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{TIS}}},$ (3)
where $N^{\text{TIS}}_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}$ and
$N^{\text{TIS}}_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}$
are the number of TIS decays in a given BDT bin for signal and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ modes respectively.
$N^{\text{TIS}}_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}$ is
found to be around 9000 for every BDT bin.
For the TOS sample, the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ yield is normalised to the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ yield from MB
triggers. The normalisation requires in this case an absolute determination of
the TOS trigger efficiency for $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$, $\epsilon^{\text{TOS/SEL}}_{\mu\mu}$, as well as
the knowledge of the average prescale factor of the MB trigger,
$s^{\text{MB}}$. The absolute TOS trigger efficiency for the signal is
computed using muons from $B^{+}\\!\rightarrow
J/\psi(\rightarrow\mu^{+}\mu^{-})K^{+}$ decays.222To avoid bias, it is
required that another object be the origin of the trigger and not the muons
alone, i.e. the muons from this sample are TIS. The $p$ and $p_{\rm T}$
spectra of the $B^{+}\\!\rightarrow J/\psi(\rightarrow\mu^{+}\mu^{-})K^{+}$
muons are reweighted in order to match those of pions from the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays. Trigger
unbiased $p$ and $p_{\rm T}$ spectra of the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ decays can be obtained from the MB sample. The
TOS efficiency is found to be at the level of 20% for all BDT bins. The
normalisation expression for TOS decays reads
$\frac{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})}{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{s^{\text{MB}}}{\epsilon^{\text{TOS/SEL}}_{\mu\mu}}\frac{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}^{\text{TOS}}}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}},$ (4)
$N_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}$
being the number of $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$
decays from the MB trigger and $N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}^{\text{TOS}}$ denoting the number of signal
decays from the TOS category. $N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}$ is found to be around 1000 for every
BDT bin.
The quantities
$\alpha_{\text{TIS}}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{{\cal
B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{TIS}}}$ (5)
and
$\alpha_{\text{TOS}}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{s^{\text{MB}}}{\epsilon^{\text{TOS/SEL}}_{\mu\mu}}\frac{{\cal
B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}}$ (6)
are called normalisation factors and are defined for each of the BDT bins. For
a given number $N$ of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ signal decays, the corresponding value of ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ is then
$\alpha\times N$. Using the value of ${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})$ from Ref. [4], the normalisation factors are in
the range $[6.6,16.2]\times 10^{-8}$ for the TIS category, and
$[0.9,7.8]\times 10^{-8}$ for the TOS category, depending on the BDT bin. From
the normalisation factors, around $2\times 10^{-4}$ ($6\times 10^{-5}$) SM
candidates are expected per BDT bin for the TOS (TIS) analysis.
## 6 Systematic uncertainties
The quantities considered in the determination of the branching fraction that
are affected by systematic uncertainties are listed below.
* •
The background expectations per bin, obtained by comparing the results with
the model described in Sect. 4 to those computed: a) if the combinatorial
background is modelled by a linear function; b) if the mass range over which
the fit is performed is modified; c) repeating the fit excluding (together
with the signal region) the 12${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ left
and right windows neighbouring the search window and comparing the fit
prediction to the yields in those regions; no correlation is considered among
the different bins for this systematic uncertainty.
* •
The ratios of reconstruction and selection efficiencies and absolute muon
identification efficiencies, for which systematic uncertainties are obtained
from the difference between different methods in the data reweighting of the
MC computed ratios and from the comparison to simulation respectively (around
20% for the ratios and 5% for muon identification efficiencies); no
correlation is considered among the different bins.
* •
The branching fraction of the normalisation channel ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-})$ $=(69.20\pm
0.05)\%$ [4]; its uncertainty affects coherently the signal expectations of
the forty bins of the analysis.
* •
The absolute TOS efficiency, for which the systematic uncertainty is obtained
from the comparison to simulation (around 15%, depending on the BDT bin); no
correlation is considered among the different bins.
* •
The effective prescale factor of the MB sample, $s^{\text{MB}}=(2.70\pm
0.76)\times 10^{-6}$. The uncertainty is evaluated from the difference between
the prescale factor as measured in data and the value of the prescale as set
in the trigger system. This systematic uncertainty affects coherently the
signal expectations of the twenty bins of the TOS analysis.
The leading systematic uncertainties are those coming from the absolute TOS
efficiency and $s^{\text{MB}}$ factor for the TOS analysis and from the ratio
of reconstruction and selection efficiencies for the TIS analysis.
## 7 Results
The modified frequentist approach (or $\textrm{CL}_{\textrm{s}}$ method) [7,
8] is used to assess the compatibility of the observation with expectations as
a function of ${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})$.
Test statistics are built from pseudo-experiments for the signal plus
background and background-only hypotheses. For each pseudo-experiment a
product of likelihood ratios is computed depending on the expected number of
signal events for a given branching fraction, $s_{i}$, the expected number of
background events, $b_{i}$ and the observed number of events, $d_{i}$ for bin
$i$. The $\textrm{CL}_{\textrm{s+b}}$ ($\textrm{CL}_{\textrm{b}}$) is defined
as the probability for signal plus background (background only) generated
pseudo-experiments to have a test-statistic value larger than or equal to that
observed in the data. The $\textrm{CL}_{\textrm{s}}$ is defined as the ratio
of confidence levels
$\frac{\textrm{CL}_{\textrm{s+b}}}{\textrm{CL}_{\textrm{b}}}$. This ratio is
used to set the exclusion (upper) limit on the branching fraction, whereas
$1-\textrm{CL}_{\textrm{b}}$ is used as a $p$-value to claim evidence or
observation. A $95(90)\%$ confidence level exclusion corresponds to
$\textrm{CL}_{\textrm{s}}=0.05(0.1)$.
The values of $b_{i}$ are obtained from the fit of the mass sidebands, as
detailed in Sect. 4. The values of $s_{i}$ depend on the assumed branching
fraction, as well as on the normalisation factors computed in Sect. 5. The
uncertainties on the input parameters are taken into account by fluctuating
the signal and background expectations when generating the $b$ and $s+b$
ensembles. These fluctuations are performed via asymmetric Gaussian priors,
following the formula
$x^{\prime}_{i}=x_{i}\left(1+\frac{1}{2}r(s_{+}-s_{-})+\frac{1}{2}r^{2}(s_{+}+s_{-})\right)$
(7)
where $x_{i}$ is the central value of the parameter, $r$ is a random number
generated from a normal distribution and $s_{+}$ and $s_{-}$ are the relative
(signed) errors of $x_{i}$ [24]. Correlations are implemented by using the
same value of $r$ for the parameters that should fluctuate coherently.
Figure 4: $\textrm{CL}_{\textrm{s}}$ curves for (a) TIS, (b) TOS categories and for (c) the combined sample. The solid line corresponds to the observed $\textrm{CL}_{\textrm{s}}$. The dashed line corresponds to the median of the $\textrm{CL}_{\textrm{s}}$ for an ensemble of background-alone experiments. In each plot, two bands are shown. The green (dark) band covers $68\%$ ($1\sigma$) of the $\textrm{CL}_{\textrm{s}}$ curves obtained in the background only pseudo-experiments, while the yellow (light) band covers $95\%$ ($2\sigma$). Table 1: Upper limits on ${\cal B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ for the TIS and the TOS categories separately, and for the combined analysis. The last entry in the table is the $p$-value of the background-only hypothesis. Quantity | TIS | TOS | Combined
---|---|---|---
Expected upper limit at 95 (90)% C.L. $[10^{-9}]$ | $42\,(33)$ | $13\,(10)$ | $11\,(9)$
Observed upper limit at 95 (90)% C.L. $[10^{-9}]$ | $24\,(19)$ | $15\,(12)$ | $11\,(9)$
$p$-value | 0.95 | 0.20 | $0.27$
The observed distribution of events is compatible with background
expectations, giving a $p$-value of $27\%$. In particular, in the last 4 bins
of the BDT output, corresponding to the most significant region of the
analysis, just one candidate is observed in each of the trigger categories, in
agreement with the background expectations. Figure 4 shows the expected and
observed $\textrm{CL}_{\textrm{s}}$ curves for the TIS category and for the
TOS category as well as for the combined measurement. The upper limit found is
11 (9)$\times 10^{-9}$ at 95 (90)% confidence level and is a factor of thirty
below the previous world best limit. Table 1 summarises the limits in the TIS,
TOS categories, and the combined result.
## 8 Conclusions
A search for $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ has
been performed using 1.0 fb-1 of data collected at the LHCb experiment in
2011. This search profits from the $10^{13}$ $K^{0}_{\rm\scriptscriptstyle S}$
produced inside the LHCb acceptance and the powerful discrimination against
the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay in which
both pions are misidentified as muons, achieved thanks to the LHCb mass
resolution for two body decays of the $K^{0}_{\rm\scriptscriptstyle S}$ meson.
The candidates observed are consistent with the expected background, with the
$p$-value for the background only hypothesis being $27\%$. The measured upper
limit
${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})<11(9)\times 10^{-9}$
at 95(90)% confidence level is an improvement of a factor of thirty below the
previous world best limit[3].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] G. Ecker and A. Pich, The longitudinal muon polarization in $K_{L}\rightarrow\mu^{+}\mu^{-}$, Nucl. Phys. B366 (1991) 189
* [2] G. Isidori and R. Unterdorfer, On the short distance constraints from $K_{L,S}\rightarrow\mu^{+}\mu^{-}$, JHEP 01 (2004) 009, arXiv:hep-ph/0311084
* [3] S. Gjesdal et al., Search for the decay $K_{S}^{0}\rightarrow 2\mu$ , Phys. Lett. B44 (1973) 217
* [4] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021
* [5] G. D’Ambrosio, G. Ecker, G. Isidori, and H. Neufeld, Radiative nonleptonic kaon decays, arXiv:hep-ph/9411439
* [6] KLOE Collaboration, F. Ambrosino et al., Search for the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow e^{+}e^{-}$ decay with the KLOE detector, Phys. Lett. B672 (2009) 203, arXiv:0811.1007
* [7] A. L. Read, Presentation of search results: the CLs technique, J. Phys. G28 (2002) 2693
* [8] T. Junk, Confidence level computation for combining searches with small statistics, Nucl. Instrum. Meth. A434 (1999) 435, arXiv:hep-ex/9902006
* [9] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [10] R. Aaij and J. Albrecht, Muon triggers in the High Level Trigger of LHCb, LHCb-PUB-2011-017
* [11] E. Lopez Asamar et al., Measurement of trigger efficiencies and biases, CERN-LHCb-2008-073
* [12] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 Physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [13] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [14] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [15] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [16] GEANT4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [17] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [18] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. of Phys: Conf. Ser. 331 (2011) 032023
* [19] G. Lanfranchi et al., The Muon Identification procedure of the LHCb experiment for the first data, LHCb-PUB-2009-013
* [20] J. Podolanski and R. Armenteros, Analysis of V-events, Phil. Mag. 45 (1954) 13
* [21] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
* [22] Y. Freund and R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Jour. Comp. and Syst. Sc. 55 (1997) 119
* [23] LHCb Collaboration, R. Aaij et al., Measurement of $V^{0}$ production ratios in $pp$ collisions at $\sqrt{s}=0.9$ and 7 TeV, JHEP 08 (2011) 034, arXiv:1107.0882
* [24] R. Barlow, Asymmetric errors, eConf C030908 (2003) , arXiv:physics/0401042
|
arxiv-papers
| 2012-09-18T16:50:32 |
2024-09-04T02:49:35.263040
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, C.\n Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V.\n Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H.\n Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E.\n Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.\n C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S. T.\n Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K.\n Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, D.\n Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain,\n R. S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov,\n R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E.\n Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S.\n Malde, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J.\n Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, P. Schaack, M. Schiller, H.\n Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A.\n Smith, E. Smith, M. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro,\n D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T.\n Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X.\n Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, H. Voss, C. Vo{\\ss}, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M.\n Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Xabier Cid Vidal",
"url": "https://arxiv.org/abs/1209.4029"
}
|
1209.4056
|
# Testing Lipschitz Property over Product Distribution and its Applications to
Statistical Data Privacy
Kashyap Dixit
Pennsylvania State University
[email protected] Madhav Jha
Pennsylvania State University
[email protected] Abhradeep Thakurta
Pennsylvania State University
[email protected]
###### Abstract
Analysis of statistical data privacy has emerged as an important area of
research. In this work we design algorithms to test privacy guarantees of a
given Algorithm $\mathcal{A}$ executing on a data set $\mathcal{D}$ which
contains potentially sensitive information about individuals. We design an
efficient algorithm $\mathcal{A}_{\bm{test}}$ which can verify whether
$\mathcal{A}$ satisfies _generalized differential privacy_ guarantee.
Generalized differential privacy [BBG+11] is a relaxation of the notion of
differential privacy initially proposed by [DMNS06]. By now differential
privacy is the most widely accepted notion of statistical data privacy.
To design Algorithm $\mathcal{A}_{\bm{test}}$, we show a new connection
between the differential privacy guarantee and Lipschitzness property of a
given function. More specifically, we show that an efficient algorithm for
testing of Lipschitz property can be transformed into
$\mathcal{A}_{\bm{test}}$ which can test for generalized differential privacy.
Lipschitz property testing and its variants, first studied by [JR11], has been
explored by many works [JR11, AJMR12b, AJMR12a, CS12] because of its intrinsic
connection to data privacy as highlighted by [JR11]. To develop a Lipschitz
property tester with an explicit application in privacy has been an intriguing
problem since the work of [JR11]. In our work, we present such a direct
application of lipschitz tester to testing privacy . We provide concrete
instantiations of Lipschitz testers (over both the hypercube and the hypergrid
domains) which are used in $\mathcal{A}_{\bm{test}}$ to test for privacy of
Algorithm $\mathcal{A}$ when the underlying data set $\mathcal{D}$ is drawn
from the hypercube and the hypergrid domains respectively.
Apart from showing a direct connection between testing of privacy and
Lipschitzness testing, we generalize the work of [JR11] to the setting of
distribution property testing. We design an efficient Lipschitz testing
algorithm when the distribution over the domain points is not uniform. More
precisely, we design an efficient Lipschitz tester for the case where the
domain points are drawn from hypercube according to some fixed product
distribution. This result is of independent interest to the property testing
community. It is important to note that to the best of our knowledge our
results on Lipschitz testing over product distributions is the only positive
result in property testing literature for non-uniform distributions after
[AC06].
## 1 Introduction
Consider a data sharing platform like _BlueKai_ , _TellApart_ or _Criteo_.
These platforms extensively collect and share user data with third-parties
(e.g., advertisers) to enhance specific user experience (e.g., better
behavioral targeting). Now, the third party applications use these data to
train their machine learning algorithms for better prediction abilities.
Since, the data which gets shared is extremely rich in user information, it
immediately poses privacy concerns over the user information [Kor10, CKN+11].
One way to address the privacy concerns due to the third-party learning
algorithms is to train the third party algorithms “in-house”, i.e., within the
data sharing platform itself thus, making sure that the trained machine
learning model preserves privacy of the underlying training data. In this
paper, we study a theoretical abstraction of the above mentioned problem.
Let $\mathcal{D}$ be a data set where each record corresponds to a particular
user and contains potentially sensitive information about the user (for
example, the click history of the user for a set of advertisements displayed).
Let $\mathcal{A}$ be an algorithm that we would like to execute on the data
set $\mathcal{D}$ (possibly to obtain some global trends about the users in
$\mathcal{D}$) without compromising individual’s privacy. This challenging
problem has recently received a lot of attention in the form of theoretical
investigation in determining the privacy-utility trade-offs for various old
and new algorithms. However, even if an algorithm is provably “safe”, in
practice the algorithm will be implemented in a programming language that may
originate from untrusted third party. This brings its own set of challenges
and has primarily been addressed in the following way: transform the algorithm
$\mathcal{A}$ into a variant which provably satisfies some theoretically sound
notion of data privacy (e.g., _differential privacy_ [DMNS06]) either by
syntactic manipulation (e.g. [McS09, RP10]) or doing so in some
algorithmic/systems framework (eg. [NRS07, JR11, MTS+12, RSK+10]). While each
approach has its own appeal, they all have a few shortcomings. For example,
they suffer from weak utility guarantees [NRS07, MTS+12, RSK+10] or take
prohibitively large running time [JR11] or require use of specialized syntax
[McS09, RP10] making it somewhat nontrivial for a non-privacy expert to
produce an effective transformation.
In this work, we take a new approach to the above problem which we call
privacy testing. Specifically, we initiate the study of testing whether an
input algorithm $\mathcal{A}$ satisfies statistical privacy guarantees. We do
this by formulating the problem in the well-studied framework of property
testing [RS96a, GGR98a].
##### Privacy testing
Before we execute an Algorithm $\mathcal{A}$ which claims to satisfy a pre-
approved notion of privacy, we test for the validity of such a claim. To the
best of our knowledge, ours is the first work to study this approach. More
precisely, in this work we initiate the study of testing an algorithm
$\mathcal{A}$ for differential privacy guarantees. Differential privacy in the
recent past has become a well established notion of privacy [Dwo06, Dwo08,
Dwo09]. Roughly speaking, differential privacy guarantees that the output of
an algorithm $\mathcal{A}$ will not depend “too much” on any particular record
of the underlying data set $\mathcal{D}$. We design testing algorithms to test
whether $\mathcal{A}$ satisfies _generalized_ differential privacy [BBG+11] or
not. _Generalized differential privacy_ is a relaxation of differential
privacy and follows the same principles as differential privacy. Under
specific setting of parameters, generalized differential privacy collapses to
the definition of differential privacy. For a precise definition, see Section
2.1. It seems to us (and we make it more formal later on) that it may not be
possible to design a computationally efficient testing algorithm for testing
the notion of exact differential privacy, since in some sense it is a worst
case notion privacy (see [BBG+11, BD12] for a discussion on this).
##### Testing Lipschitz property under product distribution and its connection
to privacy testing
The goal of testing properties of functions is to distinguish between
functions which satisfy a given property from functions which are “far” from
satisfying the property. The notion of “far” is usually the fraction of points
in the domain of the function on which the function needs to be redefined to
make it satisfy the property.
To test for generalized differential privacy, we show a new connection between
differential privacy and the problem of testing Lipschitz property which was
first studied by [JR11]. A recent line of work [JR11, AJMR12b, AJMR12a] has
sought to explore applications of sublinear algorithms (specifically,
_property testers_ and _reconstructors_) to data privacy. We continue this
line of work and show the first application of property testers (which are
vastly more efficient than property reconstructors) to the setting of data
privacy. Indeed, prior to this work it was not clear if property testers for
Lipschitz property can be used at all in data privacy setting.
Let $\mathcal{T}$ be the universe from which data sets are drawn where each
data set has the same number of records. A function
$f:\mathcal{T}\rightarrow\mathbb{R}$ is $\alpha$-Lipschitz if for all pair of
points $x,x^{\prime}\in\mathcal{T}$ the following condition holds:
$|f(x)-f(x^{\prime})|\leq d_{H}(x,x^{\prime})$ where $d_{H}$ is the Hamming
distance between $x$ and $x^{\prime}$ (that is, $d_{H}(x,x^{\prime})$ is the
number of entries in which $x$ and $x^{\prime}$ differ). To define Lipschitz
tester, we define the notion of distance between functions $f$ and $g$ defined
on the same (finite) domain $\mathcal{T}$ under distribution $\bm{Distr}$ as
follows:
$dist(f,g)\stackrel{{\scriptstyle\mbox{def}}}{{=}}\Pr\limits_{x\sim\bm{Distr}}[f(x)\neq
g(x)]$. A Lipschitz tester gets an oracle access to function $f$, a distance
parameter $\epsilon\in(0,1]$. It accepts Lipschitz functions $f$ and rejects
with high probability functions $f$ which are $\epsilon$-far from Lipschitz
property. Namely, functions $f$ for which $\min dist(f,g)>\epsilon$, where the
minimum is taken over all Lipschitz functions $g$. In this work, we extend the
result of [JR11] to the setting of product distribution.
While $\bm{Distr}$ is usually taken to be the uniform distribution in the
property testing literature, in our setting it will be important to allow
$\bm{Distr}$ to be more general distribution. Taking $\bm{Distr}$ to be
something other than uniform distribution is challenging to investigate even
for the special case of product distributions. Indeed, prior to this work the
only positive result known for the product distribution setting is the work by
[AC06] for monotonicity testing. For the setting where $\bm{Distr}$ is an
arbitrary unknown distribution there are exponential lower bounds on
computational efficiency of the tester are known [HK07]. Above result is
stated for functions with discrete range of the form $\delta\mathbb{Z}$.
In this paper, we show that one can use a Lipschitz property testing algorithm
($\bm{Liptest}$) as a proxy for testing generalized differential privacy. The
tester $\bm{Liptest}$ should be able to sample _efficiently_ the data set
according to a given probability distribution defined over domain of these
data sets (see Definition 2.2). It has been shown that this additional
requirement is sufficient to give strong privacy guarantees for the algorithm
being tested.( For further details see Section 3.) Additionally, for practical
applications, this tester should run efficiently, especially over the large
data set domain.
With the above motivation in mind, we have designed such a Lipschitz tester
with sub-linear time complexity (with respect to the domain size) for the
hypercube domain $\mathcal{T}=\\{0,1\\}^{d}$ with product distribution defined
on data sets in $\mathcal{T}$. (For further details, we refer the reader to
Section 4.) With this construction, we can test the privacy guarantees of an
algorithm in time that is poly-logarithmic in domain size.
### 1.1 Related Work
In the last few years, various notions of data privacy have been proposed.
Some of the most prominent are $k$-anonymity [Swe02], $\ell$-diversity
[MGKV06], differential privacy [DMNS06], noiseless privacy [BBG+11], natural
differential privacy [BD12] and generalized differential privacy [BBG+11].
While ad-hoc notions like $k$-anonymity and $\ell$-diversity being broken
[GKS08], privacy community has pretty much converged to theoretical sound
notions of privacy like differential privacy. In this paper, we work with the
definition of generalized differential privacy (GDP), which is a
generalization of differential privacy, noiseless privacy and natural
differential privacy. The primary difference between GDP and the other related
definitions is that it incorporates both the randomness in the underlying data
set $\mathcal{D}$ and the randomness of the Algorithm $\mathcal{A}$, where as
other notions consider either the randomness of the data or the randomness of
the algorithm.
In this paper, we design algorithms ($\mathcal{A}_{\bm{test}}$) to test
whether a given algorithm $\mathcal{A}$ satisfies GDP. In all our algorithms,
we assume that $\mathcal{A}$ is given as a “white-box”, i.e., complete access
to the source code of $\mathcal{A}$ is provided. In this paper, all the
instantiations of $\mathcal{A}_{\bm{test}}$ are probabilistic and use
Lipschitz property testing algorithms as underlying tool set. On a related
note, in the field of formal verifications there have been recent works [RP10]
using which one can guarantee that a given algorithm $\mathcal{A}$ satisfy
differential privacy. The caveat of these kind of static analysis based
algorithms is that it needs the source code for $\mathcal{A}$ to be written in
a type-safe language which is hard for a non-expert to adapt to.
One of the primary reason for considering the sublinear (with respect to the
domain size) time Lipschitz testers is the large size of domain often
encountered in the study of statistical privacy of databases. The property
testers ([RS96b, GGR98b]) have been extensively studied for various
approximation and decision problems. They are of particular interest because
they usually have sublinear (in input size) running time which is of
particular interests in the problem with large inputs. Some of the ideas and
definitions in this paper have been taken from the work on distribution
testing ([HK07, GS09, AC06]). Lipschitz property testers were introduced in
[JR11] (which gave the explicit tester for the hypercube domain) and have
since then been studied in [AJMR12b, AJMR12a] for the hypergrid domain.
Recently [CS12] have proposed an optimal Lipshcitz tester for the hypercube
domain with the underlying distribution being uniform.
### 1.2 Our Contributions
* •
Formulate testing of data privacy property as Lipschitz property testing: In
this paper we initiate the study of testing privacy properties of a given
candidate algorithm $\mathcal{A}$. The specific privacy property that we test
is _generalized differential privacy_ (GDP) (see Definition 2.2). In order to
design a tester for GDP property, we cast the problem of testing GDP property
as a problem of testing Lipschitzness. (See Theorem 3.1.) The problem of
testing Lipschitzness was initially proposed by [JR11].
* •
Design a generic transformation to convert an Algorithm $\mathcal{A}$ to its
GDP variant: We design a generic transformation to convert a candidate
algorithm $\mathcal{A}$ to its generalized differentially private variant.
(See Theorem 3.5.)
* •
New results for Lipschitz property testing: In order to allow our privacy
tester to be effective for a large class of data generating distributions, we
extend the existing results of Lipschitz property testing to work with product
distributions. We give the first efficient tester for the Lipschitz property
for the hypercube domain which works for arbitrary product distribution. (See
Theorem 4.1.) Previous works (even for other function properties) have mostly
focused on the case of uniform distribution. To the best of our knowledge this
is the only non-trivial positive result in property testing over arbitrary
product distribution apart from the result of [AC06] on monotonicity testing.
* •
Concrete instantiation of privacy testers based on old and new Lipschitz
testers We instantiate privacy tester using Lipschitz tester described in the
previous item to get a concrete instantiation of privacy tester. This also
leads to a concrete instantiation of Item 2 mentioned above. We also
instantiate privacy testers based on known Lipschitz testers in the
literature. This is summarized in Section 5.
### 1.3 Organization of the paper
In Section 2, we introduce the notions of privacy used in this paper, namely,
differential privacy and generalized differential privacy. We also introduce
the concepts of general property testing and the specific instantiation of
Lipschitz property testing. In Section 3, we show the formal connection
between testing of generalized differential privacy (GDP) and Lipschitz
property testing. In Section 4, we state our new results of Lipschitz property
testing over product distributions in the hypercube domain. In Section 5, we
show that Lipschitz testers over the hypergrid domain can be used to test for
GDP when the data sets are drawn uniformly from the hypergrid domain. Lastly,
in Section 6 we conclude with discussions and open problems.
## 2 Preliminaries
### 2.1 Differential Privacy and Generalized Differential Privacy
In the last few years, differential privacy [DMNS06] has become a well-
accepted notion of statistical data privacy in the data privacy community. At
a high-level the definition of differential privacy implies that the output of
a differentially private algorithm will be “almost” the same from an
adversary’s perspective irrespective of an individual’s presence or absence in
the underlying data set. The reason that it is a meaningful notion is because
the presence or absence of an individual in the data set does not affect the
output of the algorithm “too much”. This high-level intuition can be
formalized as below:
###### Definition 2.1 ($(\alpha,\gamma)$-Differential Privacy [DMNS06,
DKMN06]).
A randomized algorithm $\mathcal{A}$ is $(\alpha,\gamma)$-differentially
private if for any two data sets $\mathcal{D}$ and $\mathcal{D}^{\prime}$
drawn from a domain $\mathcal{T}$ with
$|\mathcal{D}\Delta\mathcal{D}^{\prime}|=1$ ($\Delta$ being the symmetric
difference), and for all measurable sets $\mathcal{O}\subseteq
Range(\mathcal{A})$ the following holds:
$\Pr[\mathcal{A}(\mathcal{D})\in\mathcal{O}]\leq
e^{\alpha}\Pr[\mathcal{A}(\mathcal{D}^{\prime})\in\mathcal{O}]+\gamma$
.
In the above definition if $\gamma=0$, we simply call it $\alpha$-differential
privacy. In this paper we intend to test if an algorithm $\mathcal{A}$ is
$\alpha$-differentially private. In order to test the above, we mould the
problem into a problem of testing Lipschitzness over the probability measure
induced by Algorithm $\mathcal{A}$ over a finite set $S$ (see Section 3 for
more discussion on this). Since, we want to test Lipschitzness efficiently
with respect to the size of the set $S$, we will use a relaxed notion of
differential privacy called _generalized differential privacy_ (GDP) [BBG+11].
The main idea behind GDP is that it allows us to incorporate the randomness
over the data generating distribution. This in turn allows us to incorporate
the failure probability of the Lipschitzness testing algorithm (over the
randomness of the data generating distribution). The definition of GDP below
is a slight modification to the definition proposed in [BBG+11] and in most
natural settings is stronger than [BBG+11].
###### Definition 2.2 ($(\alpha,\gamma,\beta)$-Generalized Differential
Privacy).
Let $\bm{Dist}$ be the distribution over the space of all data sets drawn from
domain $\mathcal{T}$. Let $W\subseteq\mathcal{T}$ be a set such that
$\Pr_{\mathcal{D}\sim\bm{Distr}}[\mathcal{D}\in W]\leq\beta$. A randomized
algorithm $\mathcal{A}$ is $(\alpha,\gamma,\beta)$-generalized differentially
private (GDP) if for any pair data sets
$\mathcal{D},\mathcal{D}^{\prime}\in\mathcal{T}\setminus W$ with
$|\mathcal{D}\Delta\mathcal{D}^{\prime}|=1$ ($\Delta$ being the symmetric
difference) and for all measurable sets $\mathcal{O}\subseteq
Range(\mathcal{A})$ the following holds:
$\Pr[\mathcal{A}(\mathcal{D})\in\mathcal{O}]\leq
e^{\alpha}\Pr[\mathcal{A}(\mathcal{D}^{\prime})\in\mathcal{O}]+\gamma$, where
the probability is over the randomness of the Algorithm $\mathcal{A}$.
It is worth mentioning here that the above definition generalizes the
_noiseless privacy_ definition [BBG+11] and _natural differential privacy_
definition [BD12] in the literature. While in both noiseless and natural
differential privacy definitions the randomness is _solely_ over the data
generating distribution $\bm{Dist}$, in GDP the randomness is both over the
data generating distribution and the randomness of the algorithm.
At a high-level what GDP says is that there exists a set $W$ of “bad” data
sets where $(\alpha,\gamma)$-differential privacy condition does not hold. But
the probability of drawing a data set $\mathcal{D}$ (over the data generating
distribution $\bm{Distr}$) from $W$ is at most $\beta$ (which is usually
negligible in the problem parameters). In fact if we set $\beta=0$, then we
recover $(\alpha,\gamma)$-differential privacy definition (see Definition 2.1)
exactly. Similarly, it can be shown that under different choices of
$(\alpha,\gamma,\beta)$ GDP implies both noiseless privacy and natural
differential privacy.
### 2.2 Lipschitz Property Testing
In this work we show that efficiently testing whether an algorithm is
$(\alpha,\beta,\gamma)$-generalized differentially private reduces to the
problem of testing (with high success probability over the probability measure
induced by Algorithm $\mathcal{A}$) if the output is Lipschitz. (For further
details see section, see section 3.)
###### Definition 2.3.
Given a function $f:\mathcal{T}\rightarrow\mathbb{R}$ from a metric space
$(\mathcal{T},d_{\mathcal{T}})$ to $(\mathbb{R},d_{\mathbb{R}})$, where
$d_{D}$ and $d_{R}$ denote the distance function on the domain $D$ and the
range $R$ respectively. The function $f$ is c-Lipschitz if
$d_{\mathbb{R}}(f(x),f(y))\leq c\cdot d_{\mathcal{T}}(x,y)$.
Property testing ([GGR98b],[RS96b]) is a well studied area pertaining to
randomized approximation algorithms for decision problems usually having
sublinear time and query complexity. At one end of the spectrum, most of the
work previously done in this area assume a uniform distribution over domain
elements. The other end is to consider the setting where the distribution over
the domain points is not known ([HK07]).
Here, we assume that the probability measure over domain elements is known and
is not necessarily uniform. Although seemingly important, to the best of our
knowledge, this is the first time that such a setting is explored in the
lipschitz property testing. To state our results, we will need the following
notation.
Let $\mathcal{P}$ (e.g. Lipschitzness in this case) be the property that needs
to be tested over the range of function $f:D\rightarrow R$. We define the
distance of the function $f$ from $\mathcal{P}$ as follows.
###### Definition 2.4.
Let $\mathcal{P}$ and $\mathcal{T}$ be defined as above. The
$\mathcal{P}\textrm{-distance}$ between functions $f,g\in\mathcal{F}$ is
defined by $dist_{\mathcal{P}}(f,g)\stackrel{{\scriptstyle
def}}{{=}}\Pr_{x\sim\mathcal{T}}\\{f(x)\neq g(x)\\}$. The
$\mathcal{P}$-distance of a function $f$ from property $\mathcal{P}$ is
defined as
$dist_{\mathcal{P}}(f,\mathcal{P})=min_{g\in\mathcal{P}}dist_{\mathcal{P}}(f,g)$.
We say that $f$ is $\epsilon$-far from a property $\mathcal{P}$ if
$dist_{\mathcal{P}}(f,\mathcal{P})\geq\epsilon$.
We will need the notion of the image diameter of a function $f$ for explaining
our results, which, roughly speaking, is the difference between maximum and
minimum values taken by $f$ on domain $\mathcal{T}$.
###### Definition 2.5 (Image diameter).
The image diameter of a function $f:\mathcal{T}\rightarrow\mathbb{R}$, denoted
by $ImD(f)$, is the difference between the maximum and the minimum values
attained by $f$, i.e.,
$\max_{x\in\mathcal{T}}f(x)-\min_{x\in\mathcal{T}}f(x)$.
## 3 Test for Generalized Differential Privacy
In this work we initiate the study of testing whether a given algorithm
$\mathcal{A}$ satisfies statistical data privacy guarantees. As a specific
instantiation of the problem, we study the notion of generalized differential
privacy (GDP) (see Definition 2.2). Roughly speaking, GDP guarantee ensures
that the output of Algorithm $\mathcal{A}$ when executed on data set
$\mathcal{D}$ does not depend “too much” on any one entry of $\mathcal{D}$.
The term “too much” is formalized by three parameters $\alpha$, $\gamma$ and
$\beta$, where the first two parameters ($\alpha$ and $\gamma$) depends on the
randomness of the Algorithm $\mathcal{A}$ and the parameter $\beta$ depends on
the randomness of the distribution $\bm{Distr}$ generating the data. We refer
to the guarantee as $(\alpha,\gamma,\beta)$-Generalized Differential Privacy
(or simply $(\alpha,\gamma,\beta)$-GDP).
Given an algorithm $\mathcal{A}$, we design a tester $\mathcal{A}_{\bm{test}}$
with the following property: if the tester outputs $\bm{YES}$, then Algorithm
$\mathcal{A}$ is $(\alpha,\gamma,\beta)$-generalized differentially private
where the parameters $\beta$ and $\gamma$ can be made arbitrarily small (at
the cost of increased running time). If the tester outputs $\bm{NO}$, then the
Algorithm $\mathcal{A}$ is not $\alpha$-differentially private. We state this
formally below.
###### Theorem 3.1 ($(\theta,\alpha,\gamma,\beta)$-Privacy testing).
Let $\bm{Liptest}$ be a $\theta$-approximate Lipschitz tester (see Definition
3.2 below), let $\bm{Distr}$ be a distribution on the domain of datasets
$\mathcal{T}$ and let $\mathcal{A}$ be an algorithm which on input
$\mathcal{D}\sim\bm{Distr}$ outputs a value $\mathcal{A}(\mathcal{D})$ in the
finite set $\Gamma$. Suppose there is an oracle ${\cal O}_{\mathcal{A}}$ which
for every value $o\in\Gamma$ and for every $\mathcal{D}\in\mathcal{T}$ allows
constant time access to the probability measure
$\mu(\mathcal{A}(\mathcal{D})=o)$ (where the measure is over the randomness of
the algorithm $\mathcal{A}$). Then there exists a “testing” algorithm
$\mathcal{A}_{\bm{test}}$ which on input privacy parameters
$\alpha,\beta\in(0,1]$, failure probability parameter $\gamma\in(0,1]$ and
access to ${\cal O}_{\mathcal{A}}$ and $\bm{Distr}$ satisfies the following
guarantee.
* •
(soundness) If Algorithm $\mathcal{A}_{\bm{test}}$ outputs $\bm{NO}$, then the
candidate algorithm $\mathcal{A}$ is not $\alpha$-differentially private.
* •
(completeness) If Algorithm $\mathcal{A}_{\bm{test}}$ outputs $\bm{YES}$, then
with probability at least $1-\gamma$ the candidate algorithm $\mathcal{A}$ is
$(\alpha\theta,0,\beta)$-generalized differentially private.
The algorithm $\mathcal{A}_{\bm{test}}$ uses $\bm{Liptest}$ as a subroutine
and runs in time $O(|\Gamma|\cdot(\text{Run time of }\bm{Liptest}))$.
To prove Theorem 3.1, we show a new connection between testing
$(\alpha,0,\beta)$-GDP and the problem of testing Lipschitz property. The
study of testing Lipschitz property was initiated by [JR11]. We present an
algorithm $\mathcal{A}_{\bm{test}}$ for testing $(\alpha,0,\beta)$-GDP based
on a generalization of Lipschitz tester presented in [JR11]. We formally
define the (generalized) Lipschitz tester below where the definition differs
from the standard property testing definition (example, as used in [JR11]) in
two aspects: (i) we require Lipschitz testers to only distinguish between
Lipschitz functions from functions which are far from $\theta$-Lipschitz
functions for some fixed $\theta\geq 1$ and (ii) we measure distance between
functions (in particular, how “far” the function is from satisfying the
property) with respect to a pre-defined probability measure $\bm{Distr}$ on
the domain.
###### Definition 3.2 ($\theta$-approximate Lipschitz tester).
A $\theta$-approximate Lipschitz tester $\bm{Liptest}$ is a randomized
algorithm that gets as input: (i) oracle access to function
$f:\mathcal{T}\rightarrow\mathbb{R}$; (ii) oracle access to independent
samples from distribution $\bm{Distr}$ on $\mathcal{T}$ and (iii) parameters
$\epsilon,\gamma\in(0,1]$. It outputs a $\bm{YES}$/$\bm{NO}$ value and
provides the following guarantee.
* •
If $\bm{Liptest}$ outputs $\bm{NO}$, then with probability 1, the function $f$
is not Lipschitz.
* •
If $\bm{Liptest}$ outputs $\bm{YES}$, then with probability at least
$1-\gamma$, there exists a set $W\subseteq\mathcal{T}$ such that (i) the input
function $f$ is $\theta$-Lipschitz on the domain $\mathcal{T}\setminus W$ and
(ii) $\Pr_{\mathcal{D}\sim\bm{Distr}}[\mathcal{D}\in W]\leq\epsilon$.
We remark that setting $\theta=1$ and $\bm{Distr}$ to be the uniform
distribution on $\mathcal{T}$ recovers the standard definition of property
tester (in our case, Lipschitz tester as defined in [JR11]).
In Section 3.2, we show that one can extend the connection between GDP and
Lipschitz testing to design an algorithm $\mathcal{A}_{\bm{privGen}}$ which
converts the candidate algorithm $\mathcal{A}$ in to a
$(\alpha,\gamma,\beta)$-generalized differentially private algorithm.
### 3.1 (Generalized) Differential Privacy as Lipschitz Property over a
Probability Measure
Consider the domain of the data sets $\mathcal{T}$ to be a finite set and
assume that (the randomized) Algorithm $\mathcal{A}$, whose privacy property
is to be tested, maps a data set $\mathcal{D}\in\mathcal{T}$ to another finite
set $\Gamma$, i.e. any output of $\mathcal{A}$ is always an element in
$\Gamma$. Now let us look at the privacy guarantee of GDP (see Definition
2.2). Ignoring the parameters $\beta$ and $\gamma$, the privacy guarantee
suggests that for any pair of neighboring data sets
$\mathcal{D},\mathcal{D}^{\prime}\in\mathcal{T}$ (drawn from the distribution
$\bm{Distr}$) and any $o\in\Gamma$, the following is true:
$\displaystyle
e^{-\alpha}\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)\leq\mu(\mathcal{A}(\mathcal{D})=o)\leq
e^{\alpha}\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)$ (1)
The measure $\mu$ is the probability induced by the randomness of the
Algorithm $\mathcal{A}$. Taking logarithm of (1), we get
$\displaystyle|\log\mu(\mathcal{A}(\mathcal{D})=o)-\log\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)|\leq\alpha$
(2)
We will use the following formulation of (2):
$|\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D})=o)-\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)|\leq
d_{H}(\mathcal{D},\mathcal{D}^{\prime})$, where $d_{H}$ is the Hamming metric.
Now, if we view the expression
$\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D})=o)$ as a function
$\lambda_{o}:\mathcal{T}\to\mathbb{R}$ defined by setting
$\lambda_{o}(\mathcal{D})=\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D})=o)$,
then we get the following condition:
$|\lambda_{o}(\mathcal{D})-\lambda_{o}(\mathcal{D}^{\prime})|\leq
d_{H}(\mathcal{D},\mathcal{D}^{\prime})$. This condition is exactly the
Lipschitzness guarantee for $\lambda_{o}$ under the Hamming metric. Using this
observation we state the following meta-algorithm $\mathcal{A}_{\bm{test}}$
(Algorithm 1) to test whether given Algorithm $\mathcal{A}$ is
$(\alpha,0,\beta)$-generalized differentially private. In Algorithm 1
(Algorithm $\mathcal{A}_{\bm{test}}$), we use a black box Lipschitz property
tester $\bm{Liptest}$. Later in the paper we instantiate $\bm{Liptest}$ with a
specific testing algorithms.
Algorithm 1 $\mathcal{A}_{\bm{test}}$: Generalized Differential Privacy (GDP)
tester
0: Algorithm $\mathcal{A}$, data generating distribution $\bm{Distr}$, data
domain $\mathcal{T}$, output range $\Gamma$, privacy parameters
$(\alpha,\beta)$ and failure parameter $\gamma$
1: $flag\leftarrow\bm{FALSE}$
2: Let $\bm{Liptest}$ be a $\theta$-approximate Lipschitz tester defined in
Definition 3.2.
3: for all values $o\in\Gamma$ do
4: Define function $\lambda_{o}:\mathcal{T}\rightarrow\mathbb{R}$ by setting
$\lambda_{o}(\mathcal{D})=\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D})=o)$.
5: Run $\bm{Liptest}$ on $\lambda_{o}$ with proximity parameter
$\frac{\beta}{\Gamma}$ and failure probability parameter
$\frac{\gamma}{|\Gamma|}$.
6: If $\bm{Liptest}$ outputs $\bm{NO}$, then $flag\leftarrow\bm{TRUE}$
7: end for
8: If $flag=\bm{FALSE}$, then output $\bm{YES}$, otherwise output $\bm{NO}$
At a high-level Algorithm $\mathcal{A}_{\bm{test}}$ does the following. For
each possible output $o\in\Gamma$, it defines a function table $\lambda_{o}$
(with the domain $\mathcal{T}$). It then invokes the Lipschitz testing
algorithm $\bm{Liptest}$ to test $\lambda_{o}$ for Lipschitzness property. If
for every output $o\in\Gamma$, $\bm{Liptest}$ outputs $\bm{YES}$, then
$\mathcal{A}_{\bm{test}}$ outputs affirmative, and outputs negative otherwise.
#### 3.1.1 Proof of Theorem 3.1
The claim about the running time of Algorithm $\mathcal{A}_{\bm{test}}$ stated
in Theorem 3.1 follows directly from the definition of Algorithm
$\mathcal{A}_{\bm{test}}$ (Algorithm 1). We state and prove the soundness and
completeness guarantees of Theorem 3.1 separately as Claim 3.3 and Claim 3.4
respectively below.
###### Claim 3.3 (Soundness guarantee).
If Algorithm $\mathcal{A}_{\bm{test}}$ (Algorithm 1) outputs $\bm{NO}$, then
the candidate algorithm $\mathcal{A}$ is not $\alpha$-differentially private.
###### Proof.
If Algorithm $\mathcal{A}_{\bm{test}}$ outputs a $\bm{NO}$, then there exists
an $o\in\Gamma$ such that $\bm{Liptest}$ outputs NO on $\lambda_{o}$. By
defintion of $\bm{Liptest}$ (see Definition 3.2), we get that $\lambda_{o}$ is
not Lipschitz. In other words, we have,
$|\lambda_{o}(\mathcal{D})-\lambda_{o}(\mathcal{D}^{\prime})|=|\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D})=o)-\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)|>1$.
Therefore, either
$\mu(\mathcal{A}(\mathcal{D})=o)>e^{\alpha}\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)$
or
$\mu(\mathcal{A}(\mathcal{D})=o)<e^{-\alpha}\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)$,
as required. ∎
###### Claim 3.4 (Completeness guarantee).
If Algorithm $\mathcal{A}_{\bm{test}}$ (Algorithm 1) outputs $\bm{YES}$, then
with probability at least $1-\gamma$ (over the randomness of $\bm{Liptest}$),
the candidate algorithm $\mathcal{A}$ is $(\alpha\theta,0,\beta)$-generalized
differentially private.
###### Proof.
If Algorithm $\mathcal{A}$ outputs $\bm{YES}$, then by the union bound it
follows that with probability at least $1-{\gamma}$, the following condition
holds for every $o\in\Gamma$: There exists a set $W_{o}\subseteq\mathcal{T}$
such that (i) $\lambda_{o}$ satisfies $\theta$-Lipschitz condition for every
$\mathcal{D},\mathcal{D}^{\prime}\in\mathcal{T}\setminus W_{o}$ and (ii)
$\Pr\limits_{x\sim\bm{Distr}}[x\in W_{o}]<\frac{\beta}{|\Gamma|}$.
Let $W=\displaystyle\bigcup_{o\in\Gamma}W_{o}$. We show that with probability
at least $1-\gamma$ (over the randomness of $\bm{Liptest}$), the following
holds: algorithm $\mathcal{A}$ satisfies $\alpha\theta$-differential privacy
condition on the set $\mathcal{T}\setminus W$ and
$\Pr\limits_{\mathcal{D}\sim\bm{Distr}}[\mathcal{D}\in W]\leq\beta$.
Condition (i) above implies that for every $o\in\Gamma$, $\lambda_{o}$ is
$\theta$-Lipschitz on $\mathcal{T}\setminus W$. Therefore, we get the
following for every neighboring pairs of data sets
$\mathcal{D},\mathcal{D}^{\prime}\in\mathcal{T}\setminus W$.
$\displaystyle|\lambda_{o}(\mathcal{D})-\lambda_{o}(\mathcal{D}^{\prime})|\leq\theta$
$\displaystyle\Rightarrow|\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D})=o)-\frac{1}{\alpha}\log\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)|\leq\theta$
$\displaystyle\Rightarrow
e^{-\alpha\theta}\leq\frac{\mu(\mathcal{A}(\mathcal{D})=o)}{\mu(\mathcal{A}(\mathcal{D}^{\prime})=o)}\leq
e^{\alpha\theta}$
Also, using Condition (ii) and the union bound over all $o\in\Gamma$, we get
the following.
$\Pr\limits_{\mathcal{D}\sim\bm{Distr}}[\mathcal{D}\in
W]\leq\sum_{o\in\Gamma}\Pr\limits_{\mathcal{D}\sim\bm{Distr}}[\mathcal{D}\in
W_{o}]\leq\beta.$
Since Conditions (i) and (ii) both hold with probability at least $1-\gamma$
(over the randomness of $\bm{Liptest}$), we get the desired claim.
∎
### 3.2 Application of GDP tester to ensure privacy for the output of a given
candidate algorithm
In this section we will demonstrate how one can use Algorithm
$\mathcal{A}_{\bm{test}}$ (Algorithm 1) designed in the previous section to
guarantee $(\alpha,\beta,\gamma)$-generalized differential privacy to the
output produced by a candidate Algorithm $\mathcal{A}$. The details are given
in Algorithm 2. The theoretical guarantees for Algorithm 2 are given below.
###### Theorem 3.5 ($(\theta,\alpha,\gamma,\beta)$-generalized differentially
private mechanism).
Let $\bm{Liptest}$ be a $\theta$-approximate Lipschitz tester (see Definition
3.2) used in the testing algorithm $\mathcal{A}_{\bm{test}}$ (Algorithm 1).
Under the same assumptions of Theorem 3.1, following are true for Algorithm
$\mathcal{A}_{\bm{privGen}}$ (Algorithm 2).
* •
(privacy) Algorithm $\mathcal{A}_{\bm{privGen}}$ (Algorithm 2) is
$(\alpha\theta,\beta,\gamma)$-generalized differentially private (GDP).
* •
(utility) If the candidate Algorithm $\mathcal{A}$ is $\alpha$-differentially
private, then Algorithm $\mathcal{A}_{\bm{privGen}}$ (Algorithm 2) always
produces the output $\mathcal{A}(\mathcal{D})$.
Algorithm 2 $\mathcal{A}_{\bm{privGen}}$: Generalized differentially private
mechanism
0: Data set $\mathcal{D}$, candidate algorithm $\mathcal{A}$, testing
algorithm $\mathcal{A}_{\bm{test}}$, data generating distribution
$\bm{Distr}$, data domain $\mathcal{T}$, output set $\Gamma$, privacy
parameters $(\alpha,\beta,\gamma)$
1: Run $\mathcal{A}_{\bm{test}}$ with parameters
$\mathcal{A},\bm{Distr},\mathcal{T},\Gamma$, privacy parameters
$(\alpha,\beta)$, and failure parameter $\gamma$
2: If $\mathcal{A}_{\bm{test}}$ outputs $\bm{YES}$, then output
$\mathcal{A}(\mathcal{D})$, output $\bm{FAILURE}$ otherwise
#### 3.2.1 Proof of Theorem 3.5
The proof of Theorem 3.5 follows from the two claims below.
###### Claim 3.6 (Privacy).
Algorithm $\mathcal{A}_{\bm{privGen}}$ (Algorithm 2) is
$(\alpha\theta,\gamma,\beta)$-generalized differentially private (GDP).
###### Proof.
First note that from Claim 3.4, it follows that if Algorithm
$\mathcal{A}_{\bm{test}}$ (Algorithm 1) outputs $\bm{YES}$, then w.p. $\geq
1-\gamma$, the candidate algorithm $\mathcal{A}$ is
$(\alpha\theta,0,\beta)$-GDP. Now to complete the proof, we provide the
following argument.
* •
Case 1 [Algorithm 2 outputs $\mathcal{A}(D)$]: We define event $Ev$ to be the
following: For every $o\in\Gamma$ there exists a set
$W_{o}\subseteq\mathcal{T}$ such that (i) $\lambda_{o}$ satisfies
$\theta$-Lipschitz condition for every
$\mathcal{D},\mathcal{D}^{\prime}\in\mathcal{T}\setminus W_{o}$ and (ii)
$\Pr\limits_{x\sim\bm{Distr}}[x\in W_{o}]<\beta$. As implied by the GDP
guarantee, event $Ev$ holds with probability $1-\gamma$. Hence, we have the
following for all $o\in\Gamma\cup\\{\bm{FAILURE}\\}$
$\displaystyle\Pr[\mathcal{A}_{\bm{privGen}}(\mathcal{D})=o]$
$\displaystyle\leq\Pr[\mathcal{A}_{\bm{privGen}}(\mathcal{D})=o|Ev]\Pr[Ev]+\Pr[\bar{Ev}]$
$\displaystyle\leq
e^{\alpha\theta}\Pr[\mathcal{A}_{\bm{privGen}}(\mathcal{D}^{\prime})=o|Ev]\Pr[Ev]+\gamma$
$\displaystyle\leq
e^{\alpha\theta}\Pr[\mathcal{A}_{\bm{privGen}}(\mathcal{D}^{\prime})=o\wedge
Ev]+\gamma$ $\displaystyle\leq
e^{\alpha\theta}\Pr[\mathcal{A}_{\bm{privGen}}(\mathcal{D}^{\prime})=o]+\gamma$
* •
Case 2[Algorithm 2 outputs $\bm{FAILURE}$]: In this case, the output is
trivially $(\alpha,\gamma,\beta)$-generalized differentially private since the
output (i.e., $\bm{FAILURE}$) is independent of the data set $\mathcal{D}$.
With this the proof is complete. ∎
###### Claim 3.7 (Utility).
If the candidate Algorithm $\mathcal{A}$ is $\alpha$-differentially private,
then Algorithm $\mathcal{A}_{\bm{privGen}}$ (Algorithm 2) always produces the
output $\mathcal{A}(\mathcal{D})$.
The proof of the above claim follows from the fact that if the candidate
algorithm $\mathcal{A}$ is $\alpha$-differentially private, then
$\mathcal{A}_{\bm{test}}$ will always output $\bm{YES}$.
## 4 Lipschitz Property Testing over Hypercube domain
In this section, we present a $(1+\delta)$-approximate Lipschitz tester (see
Definition 3.2) for functions defined on $\mathcal{T}={\\{0,1\\}}^{d}$ where
the notion of distance is with respect to any product distribution.
Specifically, the points in the data set are distributed according to the
product distribution $\Pi=Ber(p_{1})\times Ber(p_{2})\times...,\times
Ber(p_{d})$ where $Ber(p)$ denotes the Bernoulli distribution with probability
$p$. For any vertex $x=(x_{1},x_{2},...,x_{d})\in\mathcal{T}$, $x_{i}=1$ with
probability $p_{i}$ and $0$ with probability $1-p_{i}$. Each vertex in
$x\in\mathcal{T}$ has an associated probability mass $p_{x}=p_{i_{1}}\cdot
p_{i_{2}}\cdots
p_{i_{k}}\cdot(1-p_{j_{1}})\cdot(1-p_{j_{2}})\cdots(1-p_{j_{d-k}})$ where $k$
is the hamming weight of $x$, also denoted by $H(x)$ and
$i_{1},i_{2},...,i_{k}$ denote the indices of $x$ with bit-value $1$.
In this section, we prove the following theorem which gives a $1$-approximate
Lipschitz tester for $\delta\mathbb{Z}$-valued functions. A function is
$\delta\mathbb{Z}$ valued if it produces outputs in integral multiples of
$\delta$.
###### Theorem 4.1.
Let $\mathcal{T}=\\{0,1\\}^{d}$ be the domain from which the data set are
drawn according to a product probability distribution $\Pi=Ber(p_{1})\times
Ber(p_{2})\times...,\times Ber(p_{d})$. The Lipschitz property of
functions$f:\mathcal{T}\rightarrow\delta\mathbb{Z}$ on these data sets can be
tested non-adaptively and with one sided error probability $\omega$ in
$O(\frac{d\cdot\min\\{d,ImD(f)\\}}{\delta(\epsilon-d^{2}\delta)}\ln(\frac{2}{\omega}))$
time for $\delta\in(0,1]$. Here $ImD$ is the image diameter defined in
Definition 2.5.
Following is an easy corollary of the above giving a $(1+\delta)$-approximate
Lipschitz tester for $\mathbb{R}$-valued functions.
###### Corollary 4.2 (of Theorem 4.1).
Let $\mathcal{T}=\\{0,1\\}^{d}$ be the domain from which the data set are
drawn according to a product probability distribution $\Pi=Ber(p_{1})\times
Ber(p_{2})\times...,\times Ber(p_{d})$. There is an algorithm that on input
parameters $\delta\in(0,1],\epsilon\in(0,1),d$ and oracle access to a function
$f:\\{0,1\\}^{d}\rightarrow\mathbb{R}$ has the following behavior: It accepts
if $f$ is Lipschitz and rejects with probability at least $1-\omega$ if $f$ is
$\epsilon$-far (with respect to the distribution $\Pi$) from
$(1+\delta)$-Lipschitz and runs in
$O(\frac{d\cdot\min\\{d,ImD(f)\\}}{\delta(\epsilon-d^{2}\delta)}\ln(\frac{2}{\omega}))$
time. Here $ImD$ is the image diameter defined in Definition 2.5.
The proof of above theorem and corollary appears in Section 4.1. To state the
proof we need the following technical result.
We define a distribution on edges of the hypercube where the probability mass
of an edge ${\\{x,y\\}}$ is given by $\frac{p_{x}+p_{y}}{d}$. Note that
$\sum_{(x,y)\in E(H_{d})}\frac{(p_{x}+p_{y})}{d}=1$. Thus the probability
distribution (we call it $D_{E}$ henceforth) on the edges defined above is
consistent. Our tester is based on detecting violated edges (that is, edges
which violate Lipschitz property) sampled from distribution $D_{E}$. Our main
technical lemma (Lemma 4.3) gives a lower bound on the probability of sampling
a violated edge according to distribution $D_{E}$ for a function that is
$\epsilon$-far from Lipschitz. (Recall that $\epsilon$-far is measured with
respect to the distribution $\Pi$.)
###### Lemma 4.3.
Let function $f:\\{0,1\\}^{d}\rightarrow\delta\mathbb{Z}$ be $\epsilon$-far
from Lipschitz. Then
$\displaystyle\sum_{(x,y)\in V(f)}{\frac{(p_{x}+p_{y})}{d}}$
$\displaystyle\geq$ $\displaystyle\frac{\delta(\epsilon-d^{2}\delta)}{d\cdot
ImD(f)}$
Here $ImD$ is the image diameter defined in Definition 2.5.
We prove the above lemma in section 4.2.1.
### 4.1 Lipschitz tester
In this section we prove Theorem 4.1 and Corollary 4.2. We first present the
algorithm stated in Theorem 4.1.
Algorithm 3 Lipschitz Tester
0: Data domain $\mathcal{T}=\\{0,1\\}^{d}$, product distribution on data set
$\Pi=Ber(p_{1})\times Ber(p_{2})\times...,\times Ber(p_{d})$, failure
probability parameter $\omega$, $\mathcal{P}$-distance parameter
$\epsilon^{\prime}$, discretization parameter $\delta$
1: Set $\epsilon=\epsilon^{\prime}-d^{2}\delta$.
2: Sample $\left\lceil\frac{2}{\epsilon}\ln(\frac{2}{\omega})\right\rceil$
vertices $z_{1},z_{2},...,z_{t}$ independently from $\mathcal{T}$ according to
the distribution $\Pi$
3: Let $r=\max_{i=1}^{t}f(z_{i})-\min_{i=1}^{t}f(z_{i})$
4: If $r>d$, reject
5: Sample
$\left\lceil\frac{dr}{\delta\epsilon}\ln(\frac{2}{\omega})\right\rceil$ edges
independently with each edge $(x,y)$ picked with probability
$\frac{(p_{x}+p_{y})}{d}$ from the hypercube $\mathcal{T}$
6: If any of the sampled edges are violated, then reject, else accept
###### Proof of Theorem 4.1.
First observe that if input function $f$ is Lipschitz then the Algorithm 3
always accepts. This is because a Lipschitz function $f$ has image diameter
(see Definition 2.5) at most $d$ (and hence cannot be rejected in Step 4.
Moreover, it does not have any violated edges (and hence cannot be rejected in
Step 6). Next consider the case when $f$ is $\epsilon$-far from Lipschitz.
Towards this we first extend Claim 3.1 of [JR11] about sample diameter $r$ to
our setting where the distance (in particular, the notion of $\epsilon$-far)
is measured with respect to product distribution.
###### Claim 4.4.
The steps 1. and 2. of the tester outputs $r\in\delta\mathbb{Z}$ such that
$r\leq ImD(f)$ and with probability at least $1-\frac{\omega}{2}$ (failure
probability at most $\frac{\omega}{2}$), $f$ is $\epsilon$-close to having
diameter that is at most $r$.
###### Proof.
Sort the points in ${\\{0,1\\}}^{d}$ according the function value in non-
decreasing order. Let $L$ be the first $\ell$-points such that their
probability mass sums up to $\frac{\epsilon}{2}$ and $R$ be the set of last
$\ell^{\prime}$ points such that their probability mass sums up to
$\frac{\epsilon}{2}$. The rest of the proof is very similar to the proof of
Claim 3.1 in [JR11], so we omit the details here. ∎
Having established Claim 4.4, rest of the proof is identical to [JR11] and we
omit the details. ∎
###### Proof of Corollary 4.2.
It is identical to the proof of Corollary 1.2 in [JR11] and we omit the
details. ∎
### 4.2 Repair Operator and Proof of Lemma 4.3
We show a transformation of an arbitrary function
$f:\\{0,1\\}^{d}\rightarrow\delta\mathbb{Z}$ into Lipschitz function by
changing $f$ on certain points, whose probability mass is related to the
probability mass (with respect to $D_{E}$) of the violated edges of
$\mathcal{T}$. This is achieved by repairing one dimension of $\mathcal{T}$ at
a time as explained henceforth. To achieve this, we define an asymmetric
version of the basic operator of [JR11]. The operator redefines function
values so that it reduces the gap asymmetrically according to the Hamming
weights (and probability masses in-turn) of the endpoints of the violated
edge. This is the main difference from previous approaches ([JR11], [AJMR12b])
which do not work if applied directly, because of the varying probability
masses of the vertices with respect to the Hamming weight. We first define the
building block of the repair operator which is called the asymmetric basic
operator.
###### Definition 4.5 (Asymmetric basic operator).
Given $f:\\{0,1\\}^{d}\rightarrow\delta\mathbb{Z}$, for each violated edge
$\\{x,y\\}$ along dimension $i$, where $f(x)<f(y)-1$, define $B_{i}$ as
follows.
1. 1.
If $H(x)>H(y)$, then $B_{i}[f](x)=f(x)+(1-p_{i})\delta$ and
$B_{i}[f](y)=f(y)-p_{i}\delta$
2. 2.
If $H(x)<H(y)$, then $B_{i}[f](x)=f(x)+p_{i}\delta$ and
$B_{i}[f](y)=f(y)-(1-p_{i})\delta$
Now we define the repair operator.
###### Definition 4.6 (Repair operator).
Given $f:\\{0,1\\}^{d}\rightarrow\delta\mathbb{Z}$, $A_{i}[f](x)$ is obtained
from $f$ by several applications of the asymmetric basic operator (see
Definition 4.5) $B_{i}$ along dimension $i$ followed by a single application
of the rounding operator. Specifically, let $f^{\prime}$ be the function
obtained from $f$ by applying $B_{i}$ repeatedly until there are no violated
edges along the $i$-th dimension. Then, $A_{i}[f]$ is defined to be
$\mathbf{R}[f^{\prime}]$ where the rounding operator $\mathbf{R}$ rounds the
function values to the closest $\delta\mathbb{Z}$-valued function.
In effect, we have the following picture for the repair operation.
$\displaystyle f=f_{0}\xrightarrow{\mathbf{R}\circ
B^{\lambda_{1}}_{1}}f_{1}\xrightarrow{\mathbf{R}\circ
B^{\lambda_{2}}_{2}}f_{2}\xrightarrow{}\cdots\xrightarrow{}f_{d-1}\xrightarrow{\mathbf{R}\circ
B^{\lambda_{d}}_{d}}f_{d}.$
Now we define a measure called violation score which will be used to show the
progress of repair operation. As shown later, the violation score is
approximately preserved along any dimension $j\neq i$ when we apply the repair
operator to repair the edges along dimension $i$. Note that the violation
score closely resembles the violation score in [JR11] except that it depends
on the function value as well as the probability masses of the end-points of
the edge.
###### Definition 4.7.
The violation score of an edge with respect to function $f$, denoted by
$vs(\\{x,y\\})$, is $\max(0,(p_{x}+p_{y})(|f(x)-f(y)|-1))$. The violation
score along dimension $i$, denoted by $VS^{i}(f)$, is the sum of violation
scores of all edges along dimension $i$
The violation score of an edge $\\{x,y\\}$ is positive iff it is violated and
violation score of a $\delta\mathbb{Z}$ valued function is contained in the
interval $\left[\delta(p_{x}+p_{y}),ImD(f)(p_{x}+p_{y})\right]$. Let
$V^{i}(f)$ denote be the set of edges along dimension $i$ violated by $f$.
Then
$\displaystyle\delta\cdot\sum_{\\{x,y\\}\in V^{i}(f)}(p_{x}+p_{y})\leq
VS^{i}(f)\leq\sum_{\\{x,y\\}\in V^{i}(f)}(p_{x}+p_{y})\cdot ImD(f)$ (3)
Lemma 4.9 shows that $A_{i}$ does not increase the violation score in
dimensions other than $i$ more than the additive value of $\delta$. The lemma
makes use of the following claim.
###### Claim 4.8 (Rounding is safe).
Given $a,b\in\mathbb{R}$ satisfying $|a-b|\leq 1$, let $a^{\prime}$
(respectively, $b^{\prime}$) be the value obtained by rounding $a$
(respectively, $b$) to the closest $\delta\mathbb{Z}$ integer. Then
$|a^{\prime}-b^{\prime}|\leq 1$.
###### Proof.
Assume without loss of generality $a\leq b$. For $x\in\mathbb{R}$, let
$\left\lfloor{x}\right\rfloor_{\delta}$ be the largest value in
$\delta\mathbb{Z}$ not greater than $x$. Observe that
$a^{\prime}\in{\\{\left\lfloor{a}\right\rfloor_{\delta},\left\lfloor{a}\right\rfloor_{\delta}+\delta\\}}$.
Using the fact that $\left\lfloor{a}\right\rfloor_{\delta}\leq
b^{\prime}\leq\left\lfloor{a}\right\rfloor_{\delta}+1+\delta$, we see that if
$a^{\prime}=\left\lfloor{a}\right\rfloor_{\delta}+\delta$ then
$|b^{\prime}-a^{\prime}|\leq 1$ always holds. Therefore, assume
$a^{\prime}=\left\lfloor{a}\right\rfloor_{\delta}$. This can happen only if
$a\leq\left\lfloor{a}\right\rfloor_{\delta}+\delta/2$. The latter implies
$b\leq\left\lfloor{a}\right\rfloor_{\delta}+1+\delta/2$ (using the fact that
$b-a\leq 1$). That is
$b^{\prime}\neq\left\lfloor{a}\right\rfloor_{\delta}+1+\delta$. In other
words, $b^{\prime}\leq\left\lfloor{a}\right\rfloor_{\delta}+1$ again implying
$b^{\prime}-a^{\prime}\leq 1$, as required. ∎
###### Lemma 4.9.
For all $i,j\in[d]$, where $i\neq j$, and every function
$f:\\{0,1\\}^{d}\rightarrow\delta\mathbb{Z}$, the following holds.
* •
(progress) Applying the repair operator $A_{i}$ does not introduce new
violated edges in dimension $j$ if the dimension $j$ is violation free, i.e.
$VS_{j}(f)=0\Rightarrow VS_{j}(A^{i}[f])=0$.
* •
(accounting) Applying the repair operator $A_{i}$ does not increase the
violation score in dimension $j$ by more than $\delta$, i.e.
$VS_{j}(A^{i}[f])\leq VS_{j}(f)+\delta$.
###### Proof.
Let $f^{\prime}$ be the function obtained from $f$ by applying $B_{i}$
repeatedly until there are no violated edges along the $i$-th dimension. We
prove the following stronger claim to prove the lemma.
###### Claim 4.10.
$VS_{j}(f^{\prime})\leq VS_{j}(f).$
We prove the above claim momentarily but first prove the lemma using the above
claim. The function $A_{i}[f]$ is obtained by rounding the values of
$f^{\prime}$ to the closest $\delta\mathbb{Z}$ values. Since rounding can
never create new edge violations by Claim 4.8, we immediately get the first
part of the lemma. The second part follows from the observation that the
rounding step modifies each function value by at most $\delta/2$.
Correspondingly, the violation score of an edge along the $j$-th dimension
changes by at most $2\cdot(\delta/2)\cdot(p_{u}+p_{v})$ where the factor 2
comes because both endpoints of an edge may be rounded. Summing over all edges
in the $j$-th dimension, we get, $\mbox{increase in violation
score}\leq\sum_{\\{u,v\\}}{\delta\cdot(p_{u}+p_{v})}=\delta$ where the last
equality holds because edges along the $j$-th dimension form a perfect
matching and therefore the probabilities $p_{u}+p_{v}$ sum to 1.
###### Proof of Claim 4.10.
Following the proof outline of a similar proof in [JR11], we show that
application of the asymmetric basic operator in dimension $i$ does not
increase the violation score in dimension $j\neq i$. Standard arguments
[GGL+00, DGL+99, JR11, AJMR12b] show that it is enough to analyze the effect
of applying $B_{i}$ on one fixed disjoint square formed by adjacent edges that
cross dimensions $i$ and $j$. (This is because edges along dimensions $i$ and
$j$ form disjoint squares in the hypercube. So having established Claim 4.10
for one fixed square of the hypercube, the full claim follows by summing up
the inequalities over all such squares.)
Consider the two dimensional function
$f:\\{x_{b},x_{t},y_{b},y_{t}\\}\rightarrow\delta\mathbb{Z}$ where
$\\{x_{b},x_{t},y_{b},y_{t}\\}$ are positioned such that
$H(y_{t})=H(x_{t})+1=H(y_{b})+1=H(x_{b})+2$ where $H(x_{b})$ denotes the
hamming weight of $x_{b}$. Assume that the basic operator is applied along the
dimension $i$. We show that the violation score along dimension $j$ does not
increase. Assume that the violation score along edge $\\{x_{b},x_{t}\\}$
increases. First, assume that the $B_{i}[f](x_{t})>B_{i}[f](x_{b})$. (The
other case is very similar and we will prove it later.) Then $B_{i}$ increases
$f(x_{t})$ and/or decreases $f(x_{b})$. Assume that $B_{i}$ increases
$f(x_{t})$. (The other case is symmetrical.) This implies that
$\\{x_{t},y_{t}\\}$ is violated and $f(x_{t})<f(y_{t})$. Let $f_{k}(x)$ (resp.
$f_{k}(y)$) denote the value of $f(x)$ (resp. $f(y)$) after $k$ applications
of $B_{i}$ on an edge $(x,y)$, for an integer $k\geq 0$. If $(x,y)$ is
violated after $k-1$ applications of the basic operator, then
$f_{k}(x)=f_{k-1}(x)+p_{i}\delta$ and $f_{k}(y)=f_{k-1}(y)-(1-p_{i})\delta$
else $f_{k}(x)=f_{k-1}(x)$ and $f_{k}(y)=f_{k-1}(y)$. We will study the effect
of applying $B_{i}$ on $(x_{t},y_{t})$ multiple (say $\lambda\geq 1$) times.
Recall that the repair operator is applied only if the edge is violated. This
means that
$\displaystyle f_{\lambda-1}(x_{t})$ $\displaystyle<$ $\displaystyle
f_{\lambda-1}(y_{t})-1$ $\displaystyle\Rightarrow
f(x_{t})+(\lambda-1)p_{i}\delta$ $\displaystyle<$ $\displaystyle
f(y_{t})-(\lambda-1)(1-p_{i})\delta-1$ $\displaystyle\Rightarrow
f(x_{t})+(\lambda-1)\delta+1$ $\displaystyle<$ $\displaystyle f(y_{t})$
$\displaystyle\Rightarrow f(x_{t})+\lambda\delta+1$ $\displaystyle\leq$
$\displaystyle f(y_{t})$
The second inequality follows from the observation that since the edge is
being corrected in the $\lambda^{th}$ application, it must have been corrected
in all previous applications as well. The last inequality follows from the
fact that $f$ is a $\delta\mathbb{Z}$-valued function and $\frac{1}{\delta}$
is an integer. We subtract $(1-p_{i})(\lambda-1)\delta$ from both sides in the
above inequality and do some rearrangement to achieve the following.
$\displaystyle f(y_{t})-(1-p_{i})(\lambda-1)\delta$ $\displaystyle\geq$
$\displaystyle f(x_{t})+\lambda\delta+1-(1-p_{i})(\lambda-1)\delta$
$\displaystyle\Rightarrow f(y_{t})-(1-p_{i})(\lambda-1)\delta$
$\displaystyle\geq$ $\displaystyle f(x_{t})+(\lambda-1)p_{i}\delta+1+\delta$
$\displaystyle\Rightarrow f_{\lambda-1}(y_{t})$ $\displaystyle\geq$
$\displaystyle f_{\lambda-1}(x_{t})+1+\delta$
The above inequality is crucial for the remaining proof of the lemma 4.3. Now
consider the cases when either the bottom edge is also violated or is not
violated.
If the bottom edge is not violated then we have $f_{\lambda-1}(x_{b})\geq
f_{\lambda-1}(y_{b})-1$ and $f_{\lambda-1}(x_{b})$ and $f_{\lambda-1}(y_{b})$
are not modified by the basic operator. Since $vs(\\{x_{t},x_{b}\\})$
increases, $f_{\lambda-1}(x_{t})>f_{\lambda-1}(x_{b})+1-p_{i}\delta$.
Combining the above inequalities, we get $f_{\lambda-1}(y_{t})\geq
f_{\lambda-1}(x_{t})+1+\delta>f_{\lambda-1}(x_{b})+2+(1-p_{i})\delta\geq
f_{\lambda-1}(y_{b})+1+(1-p_{i})\delta>f_{\lambda-1}(y_{b})+1$. Thus the
violation score increases along $\\{x_{t},x_{b}\\}$ by
$(p_{x_{b}}+p_{x_{t}})p_{i}\delta$ and decreases along $\\{y_{b},y_{t}\\}$ by
$(p_{y_{b}}+p_{y_{t}})(1-p_{i})\delta=(p_{x_{b}}+p_{x_{t}})\left(\frac{p_{i}}{1-p_{i}}\right)(1-p_{i})\delta$
which is same as $(p_{x_{b}}+p_{x_{t}})p_{i}\delta$, keeping the violation
score along the dimension $j$ unchanged.
If the bottom edge is violated, then the increase in $vs(\\{x_{b},x_{t}\\})$
implies that $f_{\lambda-1}(x_{b})$ must decrease (after application of
$B_{i}$) by $p_{i}\delta$ (since $H(x_{b})<H(y_{b})$) implying
$f_{\lambda-1}(y_{b})+1<f_{\lambda-1}(x_{b}))$. Therefore
$f_{\lambda-1}(x_{t})+p_{i}\delta>f_{\lambda-1}(x_{b})+1-p_{i}\delta$ or
$f_{\lambda-1}(x_{t})>f_{\lambda-1}(y_{t})+1-2p_{i}\delta$. Therefore
$f_{\lambda-1}(y_{t})>f_{\lambda-1}(x_{t})+1>f(x_{b})+2-2p_{i}\delta\geq
f(y_{b})+3-2p_{i}\delta+\delta\geq f(y_{b})+1+\delta$. The last inequality is
true since $\delta\leq 1$ and $p_{i}\leq 1$. Thus, $vs(\\{x_{t},x_{b}\\})$
increases by at most $(p_{x_{b}}+p_{x_{t}})2p_{i}\delta$ while
$vs(\\{y_{t},y_{b}\\})$ decreases by
$(p_{y_{t}}+p_{y_{b}})2(1-p_{i})\delta=(p_{x_{b}}+p_{x_{t}})2p_{i}\delta$,
ensuring that violation score along the vertical dimension does not increase.
Now we turn to the case when $B_{i}[f](x_{t})<B_{i}[f](x_{b})$. By the
arguments very similar to the first case, it can be proved that
$f_{\lambda-1}(x_{t})\geq f_{\lambda-1}(y_{t})+1+\delta$ and the application
of basic operator decreases $f(x_{t})$ by $p_{i}\delta$ and increases
$f(y_{t})$ by $(1-p_{i})\delta$.
If the bottom edge is not violated then $f_{\lambda-1}(y_{b})\geq
f_{\lambda-1}(x_{b})-1$ and $f_{\lambda-1}(x_{b})$ and $f_{\lambda-1}(y_{b})$
are not modified by the basic operator. Since $vs(\\{x_{t},x_{b}\\})$
increases, $f_{\lambda-1}(x_{b})>f_{\lambda-1}(x_{t})+1-p_{i}\delta$.
Combining the above inequalities, we get $f_{\lambda-1}(y_{b})\geq
f_{\lambda-1}(x_{b})-1>f(x_{t})-p_{i}\delta\geq f(y_{t})+1+\delta(1-p_{i})$.
Thus the violation score increases along $\\{x_{t},x_{b}\\}$ by
$(p_{x_{b}}+p_{x_{t}})p_{i}\delta$ and decreases along $\\{y_{b},y_{t}\\}$ by
$(p_{y_{b}}+p_{y_{t}})(1-p_{i})\delta=(p_{x_{b}}+p_{x_{t}})\left(\frac{p_{i}}{1-p_{i}}\right)(1-p_{i})\delta$
which is same as $(p_{x_{b}}+p_{x_{t}})p_{i}\delta$, keeping the violation
score along the dimension $j$ unchanged.
If the bottom edge is violated, then the increase in $vs(\\{x_{b},x_{t}\\})$
implies that $f_{\lambda-1}(x_{b})$ must increase implying
$f_{\lambda-1}(y_{b})>f_{\lambda-1}(x_{b})+1$. Therefore, the increase in
$vs\\{x_{b},x_{t}\\}$ implies that
$f_{\lambda-1}(x_{b})+p_{i}\delta>f_{\lambda-1}(x_{t})-p_{i}\delta+1$ or
$f_{\lambda-1}(x_{b})>f_{\lambda-1}(x_{t})-2p_{i}\delta+1$. Combining the
above inequalities, we get
$f_{\lambda-1}(y_{b})>f_{\lambda-1}(x_{b})+1>f_{\lambda-1}(x_{t})-2p_{i}\delta+2\geq
f_{\lambda-1}(y_{t})+3+\delta-2p_{i}\delta\geq f_{\lambda-1}(y_{t})+1+\delta$.
The last inequality is true since $\delta\leq 1$ and $p_{i}\leq 1$. Thus,
$vs(\\{x_{t},x_{b}\\})$ increases by at most
$(p_{x_{b}}+p_{x_{t}})2p_{i}\delta$ while $vs(\\{y_{t},y_{b}\\})$ decreases by
$(p_{y_{t}}+p_{y_{b}})2(1-p_{i})\delta=(p_{x_{b}}+p_{x_{t}})2p_{i}\delta$,
ensuring that violation score along the vertical dimension does not increase.
∎
∎
#### 4.2.1 Proof of Lemma 4.3
Using the arguments very similar to [JR11] as given below, we can get the
following sequence of inequalities
$\displaystyle
Dist(f_{i-1},f_{i})=Dist(f_{i-1},A_{i}(f_{i-1}))\leq\sum_{(x,y)\in
V_{i}(f_{i-1})}(p_{x}+p_{y})$
$\displaystyle\leq\frac{1}{\delta}VS^{i}(f_{i-1})\leq\frac{1}{\delta}VS^{i}(f)+2(d-i)\delta\leq\frac{1}{\delta}\sum_{(x,y)\in
V^{i}(f)}(p_{x}+p_{y})\cdot ImD(f)+2(d-i)\delta$
Here functions $\\{f_{i}\\}_{i=0}^{i=d}$ are defined in the same way as
[JR11]. The first inequality holds because $A_{i}$ modifies $f$ only at the
endpoints points $x$ and $y$ of violated edge $(x,y)$ along dimension $i$,
thus paying $p_{x}+p_{y}$. The second and fourth inequalities follow from
Equation (3) and the third inequality holds because of Lemma 4.9. Therefore,
by triangle inequality, we have
$\displaystyle
Dist(f,f_{d})\leq\sum_{i\in[d]}Dist(f_{i-1},f_{i})\leq\sum_{i\in[d]}\left(\sum_{(x,y)\in
V^{i}f(H)}(p_{x}+p_{y})\cdot\frac{ImD(f)}{\delta}\right)+2(d-i)\delta$
$\displaystyle\leq\left(\sum_{(x,y)\in
V(f))}(p_{x}+p_{y})\cdot\frac{ImD(f)}{\delta}\right)+d^{2}\delta$
For a function which is $\epsilon$-far from Lipschitz, we have
$Dist(f,f_{d})\geq\epsilon$. Therefore, from the above inequality, we have
$\displaystyle\sum_{(x,y)\in V(f)}{\frac{(p_{x}+p_{y})}{d}}$
$\displaystyle\geq$ $\displaystyle\frac{\delta(\epsilon-d^{2}\delta)}{d\cdot
ImD(f)}$
## 5 Instantiation of privacy tester using Lipschitz testers
In this section, we instantiate the privacy tester of Section 3 with both
known Lipschitz testers as well as the Lipschitz tester developed in this
work. This is presented in the table below. The third column gives the
“approximation factor” as defined in Definition 3.2 for the various testers.
The final column gives the privacy tester parameters that each of the tester
achieves. The last row gives the result of Lipschitz tester (Section 4)
developed in this work.
Reference | Functions | Approximation factor ($\theta$) | Distribution | Tester running time | Privacy tester
---|---|---|---|---|---
[JR11] | ${\\{0,1\\}}^{d}\rightarrow\mathbb{R}$ | $1+\delta$ | Uniform | $O(\frac{d\cdot ImD(f)}{\epsilon\delta})$ | $(1+\delta,\alpha,\gamma,\beta)$
[AJMR12b] | ${\\{1,\ldots,n\\}}^{d}\rightarrow\mathbb{R}$ | $1+\delta$ | Uniform | $\tilde{O}\left(\frac{d\min{\\{ImD(f),nd\\}}}{\delta\epsilon}\right)$ | $(1+\delta,\alpha,\gamma,\beta)$
[CS12] | ${\\{0,1\\}}^{d}\rightarrow\mathbb{R}$ | $1$ | Uniform | $O(\frac{d}{\epsilon})$ | $(1,\alpha,\gamma,\beta)$
This work | ${\\{0,1\\}}^{d}\rightarrow\mathbb{R}$ | $1+\delta$ | Product | $O\left(\frac{d\cdot ImD(f)}{(\epsilon-d^{2}\delta)\delta}\right)$ | $(1+\delta,\alpha,\gamma,\beta)$
## 6 Discussions and Open Problems
In this section we discuss about some of the interesting implications of our
current work and some of the new avenues it opens up. Also we state some of
the open problems that remains unresolved in our work.
##### Privacy:
In this work, we took the first step towards designing efficient testing
algorithm for statistical data privacy. Our work indicates that it is indeed
possible to design efficient testing algorithms for some existing notions of
statistical data privacy (e.g., generalized differential privacy). It is
important that the current paper should be treated as an initial study of the
problem and in no way should be interpreted conclusive. It is interesting to
explore other rigorous notions of data privacy, their applications and design
testers for them.
In this paper, we test for generalized differential privacy, which is a
relaxation of differential privacy. It remains an open problem to design a
privacy tester for exact differential privacy. The problem seems to be
challenging because of the fact that if we want to design an efficient tester,
then usually the utility guarantees for the tester allow it to fail with some
probability. Now, differential privacy being a worst case notion, it is not
clear how to incorporate the failure property of the tester and yet make
precise claims about differential privacy.
In the current work, we have designed privacy testers for algorithms where the
domain of the data sets are either hypercube or hypergrid. A natural question
that arises is that if we can extend the current results to design privacy
testers when the data sets are drawn from continuous domain, unlike hypercube
or hypergrid.
##### Lipschitz Testing:
This work presents the first Lipschitz property tester for the setting where
the domain points are sampled from a distribution that is not uniform. Because
of possible applications to statistical data privacy, this work has motivated
the design of such Lipschitz testers for other domains, e.g. hypergrid. Also,
this paper mainly shows the tester for the product distribution over the
hypercube domain, but it still remains open to design testers for other
distributions that may be correlated in some way (e.g., pairwise correlation).
##### Acknowledgements:
We would like to thank Sofya Raskhodnikova and Adam Smith for various
suggestions and comments during the course of this project.
## References
* [AC06] Nir Ailon and Bernard Chazelle. Information theory in property testing and monotonicity testing in higher dimension. Inf. Comput., 204(11):1704–1717, 2006.
* [AJMR12a] Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Limitations of local filters of lipschitz and monotone functions. In Gupta et al. [GJRS12], pages 387–398.
* [AJMR12b] Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Testing lipschitz functions on hypergrid domains. In Gupta et al. [GJRS12], pages 387–398.
* [BBG+11] Raghav Bhaskar, Abhishek Bhowmick, Vipul Goyal, Srivatsan Laxman, and Abhradeep Thakurta. Noiseless database privacy. In Dong Hoon Lee and Xiaoyun Wang, editors, ASIACRYPT, volume 7073 of Lecture Notes in Computer Science, pages 215–232. Springer, 2011\.
* [BD12] Abhishek Bhowmick and Cynthia Dwork. Natural differential privacy. In Personal communication, 2012.
* [CKN+11] Joseph A. Calandrino, Ann Kilzer, Arvind Narayanan, Edward W. Felten, and Vitaly Shmatikov. ”you might also like: ” privacy risks of collaborative filtering. In IEEE Symposium on Security and Privacy, 2011.
* [CS12] Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and lipschitz testing over the hypercube. CoRR, abs/1204.0849, 2012.
* [DGL+99] Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In RANDOM, pages 97–108, 1999.
* [DKMN06] Cynthia Dwork, Krishnaram Kenthapadi, Frank Mcsherry, and Moni Naor. Our data, ourselves: Privacy via distributed noise generation. In EUROCRYPT, pages 486–503. Springer, 2006.
* [DMNS06] Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In TCC, pages 265–284, 2006.
* [Dwo06] Cynthia Dwork. Differential privacy. In ICALP, 2006.
* [Dwo08] Cynthia Dwork. Differential privacy: A survey of results. In TAMC, pages 1–19. Springer, 2008.
* [Dwo09] Cynthia Dwork. The differential privacy frontier. In TCC, pages 496–502. Springer, 2009.
* [GGL+00] Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301–337, 2000.
* [GGR98a] Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653–750, 1998.
* [GGR98b] Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653–750, 1998.
* [GJRS12] Anupam Gupta, Klaus Jansen, José D. P. Rolim, and Rocco A. Servedio, editors. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012\. Proceedings, volume 7408 of Lecture Notes in Computer Science. Springer, 2012.
* [GKS08] Srivatsava Ranjit Ganta, Shiva Prasad Kasiviswanathan, and Adam Smith. Composition attacks and auxiliary information in data privacy. In KDD, pages 265–273, 2008.
* [GS09] Dana Glasner and Rocco A. Servedio. Distribution-free testing lower bound for basic boolean functions. Theory of Computing, 5(1):191–216, 2009.
* [HK07] Shirley Halevy and Eyal Kushilevitz. Distribution-free property-testing. SIAM J. Comput., 37(4):1107–1138, 2007.
* [JR11] Madhav Jha and Sofya Raskhodnikova. Testing and reconstruction of lipschitz functions with applications to data privacy. In Rafail Ostrovsky, editor, FOCS, pages 433–442. IEEE, 2011.
* [Kor10] Aleksandra Korolova. Privacy violations using microtargeted ads: A case study. In ICDMW, 2010.
* [McS09] Frank D. McSherry. Privacy integrated queries: an extensible platform for privacy-preserving data analysis. In SIGMOD, 2009.
* [MGKV06] Ashwin Machanavajjhala, Johannes Gehrke, Daniel Kifer, and Muthuramakrishnan Venkitasubramaniam. l-diversity: Privacy beyond k-anonymity. In ICDE, page 24, 2006.
* [MTS+12] Prashanth Mohan, Abhradeep Thakurta, Elaine Shi, Dawn Song, and David Culler. Gupt: privacy preserving data analysis made easy. In SIGMOD, 2012.
* [NRS07] Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Smooth sensitivity and sampling in private data analysis. In STOC, 2007.
* [RP10] Jason Reed and Benjamin C. Pierce. Distance makes the types grow stronger: a calculus for differential privacy. In ICFP, 2010.
* [RS96a] Ronitt Rubinfeld and Madhu Sudan. Robust characterization of polynomials with applications to program testing. SIAM J. Comput., 25(2):252–271, 1996.
* [RS96b] Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252–271, 1996.
* [RSK+10] Indrajit Roy, Srinath T. V. Setty, Ann Kilzer, Vitaly Shmatikov, and Emmett Witchel. Airavat: Security and privacy for mapreduce. In NSDI, 2010.
* [Swe02] Latanya Sweeney. $k$-anonymity: A model for protecting privacy. International Journal on Uncertainty, Fuzziness and Knowledge-based Systems, 10(5):557–570, 2002.
|
arxiv-papers
| 2012-09-18T18:51:17 |
2024-09-04T02:49:35.271505
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kashyap Dixit and Madhav Jha and Abhradeep Thakurta",
"submitter": "Kashyap Dixit",
"url": "https://arxiv.org/abs/1209.4056"
}
|
1209.4073
|
# Second Order Ergodic Theorem for Self-Similar Tiling Systems
Konstantin Medynets Konstantin Medynets, Department of Mathematics, U.S.
Naval Academy, Annapolis, MA 21402, USA [email protected] and Boris Solomyak
Boris Solomyak, Department of Mathematics, University of Washington, Seattle,
WA 98195-4350, USA [email protected]
###### Abstract.
We consider infinite measure-preserving non-primitive self-similar tiling
systems in Euclidean space $\mathbb{R}^{d}$. We establish the second-order
ergodic theorem for such systems, with exponent equal to the Hausdorff
dimension of a graph-directed self-similar set associated with the
substitution rule.
B. S. is supported in part by NSF grant DMS-0968879.
## 1\. Introduction
Let $\mathbb{X}=(X,\mu,\mathbb{T})$ be a conservative, ergodic, measure
preserving dynamical system on a $\sigma$-finite measure space. If $f\in
L^{1}(X,\mu)$, then Hopf’s ratio ergodic theorem says that the growth of
$S_{n}f(x)=f(x)+\cdots+f(T^{n-1}x)$ is independent of $f$ in the sense that if
$g\in L^{1}(X,\mu)$ with $\int g\,d\mu\neq 0$, then
$\frac{S_{n}f(x)}{S_{n}g(x)}\to\frac{\int f\,d\mu}{\int g\,d\mu}\mbox{ for
}\mu\mbox{-a.e. }x\in X.$
It turns out that due to the measure $\mu$ being infinite, it is impossible to
replace functions $S_{n}g(x)$ by constants $\\{a_{n}\\}$ [A, Theorem 2.4.2].
However, it was observed earlier [Fi1, ADF, LS] that for some systems, the
ratios $S_{n}f(x)/a_{n}$ (for some choice of $a_{n}$) still converge to $\int
f\,d\mu$, though in a weaker sense (second-order averages). The asymptotic
behavior of the sequence $\\{a_{n}\\}$ is an invariant of the dynamical
system.
The main result of the present paper is the following ergodic theorem showing
that for self-similar tilings the sequence $\\{a_{n}\\}$ can be chosen as
$\\{n^{\alpha+1}\\}$ where $\alpha$ is an intrinsic parameter of the system
reflecting self-similarity (the precise statements, with all technical
assumptions, are Theorem 5.5 and Theorem 6.11).
###### Theorem 1.1.
(i) Let $\mathbb{X}=(\Omega,\mu,{{\mathbb{R}}}^{d})$ be a non-primitive self-
similar substitution tiling system preserving an infinite ergodic measure
$\mu$. Assume that the measure $\mu$ is non-zero and finite on some open
subset of $\Omega$. Then there exist positive parameters $\alpha$ and $c$ such
that for every function $f\in L^{1}(\Omega,\mu)$ and $\mu$-almost every tiling
$\mathcal{T}\in\Omega$, we have
(1.1)
$\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{1}^{t}\frac{\int_{B_{R}}f(\mathcal{T}-u)\,du}{c(2R)^{\alpha}}\,\frac{dR}{R}=\int_{\Omega}fd\mu.$
Here $B_{R}$ is the ball of radius $R$ centered at the origin.
(ii) The parameters $\alpha$ and $c$ are invariants of the measure-theoretic
isomorphism of the system $\mathbb{X}$.
(iii) This result is also valid for a large class of one-dimensional symbolic
substitution systems with integrals in the left-hand side replaced by the
corresponding sums.
Our work was originally inspired by A. Fisher’s paper [Fi1], where he obtained
a similar ergodic theorem for a single substitution system $(X_{\sigma},S)$
generated by the map $\sigma(0)=000$ and $\sigma(1)=101$. Iterating the map
$\sigma^{n}(1)$, $n\geq 1$, we get a sequence where the appearances of 1’s and
0’s resemble the process of constructing the middle-thirds Cantor set. Fisher
used this analogy to show that for any function $f\in L^{1}(X_{\sigma},\mu)$
and $\mu$-a.e point $x\in X_{\sigma}$,
$\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{\sum_{i=0}^{k-1}f(S^{i}x)}{ck^{\alpha}}\cdot\frac{1}{k}=\int_{X_{\sigma}}fd\mu,$
where $\mu$ is the unique $S$-invariant measure $\mu$ on $X_{\sigma}$ with
$\mu([1])=1$. Here $\alpha=\log(2)/\log(3)$ is the Hausdorff dimension and $c$
is the average density of the Hausdorff measure on the middle-thirds Cantor
set. Average densities were introduced by Bedford and Fisher [BF] (see
Definition 5.4). The middle-thirds Cantor set arising in the study of the
substitution $\sigma$ is a special case of the self-similar set (in general,
for a graph-directed iterated function system) that one can associate to every
self-similar tiling substitution (Section 3.1). In the proof of Theorem 1.1,
we show the parameters $\alpha$ and $c$ always arise as the Hausdorff
dimension and the appropriate average density of the Hausdorff measure of the
associated graph-directed sets.
It is well-known that primitive substitution dynamical systems (both symbolic
and tiling versions) are uniquely ergodic. Non-primitive substitutions and
their invariant measures have been recently studied in [Y, BKM, BKMS, HY],
where it was shown, in particular, that for a large class of non-primitive
symbolic substitutions infinite ($\sigma$-finite) invariant measures appear
naturally. This has also been extended to the tiling setting in [CS]. The
present work continues this line of research.
We observe that the second-order ergodic theorem is not a universal result.
Its validity depends on intrinsic properties of the dynamical system in
question. We refer the reader to the paper [ADF] of Aaronson-Denker-Fisher for
the discussion of second order ergodic theorems for Markov shifts. We also
mention the paper [LS] of Ledrappier and Sarig establishing the second order
ergodic theorem for certain horocycle flows.
The basic idea of the proof of Theorem 1.1 has roots in the work of Bedford
and Fisher [BF]. In a nutshell, it follows from the Birkhoff ergodic theorem
for the renormalization, or “scaling” map, associated with the substitution,
restricted to a certain fractal subset of the tiling space. The negative part
(going into the past) of this renormalization map turns out to be essentially
the time-1 map of the scenery flow arising from “zooming in” into the fractal.
The structure of the paper is the following. In Section 2 we give the
definition of the substitution tiling system $(\Omega,{{\mathbb{R}}}^{d})$
generated by a tile substitution $\mathcal{G}$. We then recall the
classification of infinite ergodic invariant measures established in [CS]. In
subsection 2.3 we explicitly state technical assumptions on the tiling system
needed for Theorem 1.1.
In Section 3 we show how to associate a graph-directed set to the substitution
rule $\mathcal{G}$. In the following sections we use this fractal to count
frequencies of prototiles in tilings of $\Omega$.
In Section 4, we show that the transverse dynamical system (generated by
iterations of the substitution rule) is measure-theoretically isomorphic to a
Markov chain.
Sections 5 and 6 are devoted to the proof of Theorem 1.1. As a corollary, we
establish that almost every sequence from a substitution space admits a non-
zero “$\alpha$-dimensional frequency,” where the parameter $\alpha$ comes from
Theorem 1.1. Section 7 contains a few examples and open questions.
## 2\. Tiling Dynamical Systems
In this section we fix our notation and present necessary definitions from
tiling dynamics. We mostly follow conventions of [CS], see also [Ro].
### 2.1. Tiling Space
Fix a finite alphabet $\mathcal{W}$ and an integer $d\geq 1$. By a tile in
$\mathbb{R}^{d}$ we mean a pair $T=(F,i)$ of a compact set $F$ that is the
closure of its interior, and a letter (label) $i\in\mathcal{W}$. Two
geometrically identical sets labeled by different letters are treated as
distinct tiles. The set $F$ will be called the support of the tile $T$, in
symbols, $\textrm{supp}(T)=F$. A tiling is a family of tiles $\mathcal{T}$
such that $\mathbb{R}^{d}=\cup\\{\textrm{supp}(T):T\in\mathcal{T}\\}$ and
distinct tiles have disjoint interiors. A patch is a finite set of tiles with
disjoint interiors. The support of a patch $P$ is the set
$\textrm{supp}(P)=\cup\\{\textrm{supp}(T):T\in P\\}$. If $\mathcal{T}$ is a
tiling, its finite subsets are called $\mathcal{T}$-patches.
The translate of a tile $T=(F,i)$ by a vector $u\in\mathbb{R}^{d}$ is the tile
$T+u=(F+u,i)$. Similarly, a translate of a patch $P$ by $u\in\mathbb{R}^{d}$
is the patch $P+u=\\{T+u:T\in P\\}$. We say that two patches $P_{1}$ and
$P_{2}$ are translationally equivalent if $P_{1}=P_{2}+u$ for some vector
$u\in\mathbb{R}^{d}$.
###### Definition 2.1.
Let $\mathcal{A}$ be a finite set of tiles in $\mathbb{R}^{d}$ such that
distinct tiles from $\mathcal{A}$ are not translationally equivalent. Tiles
from the set $\mathcal{A}$ are called prototiles. We will call translations of
prototiles $\mathcal{A}$-tiles.
Denote by $\mathcal{A}^{+}$ the set of patches made of translates of tiles
from $\mathcal{A}$. We assume that every prototile $T\in\mathcal{A}$ is
centered at the origin, in the sense that $\textbf{0}\in{\rm
int}(\textrm{supp}(T))$.
Let $\varphi$ be an expanding linear transformation in ${{\mathbb{R}}}^{d}$. A
map $\mathcal{G}:\mathcal{A}\rightarrow\mathcal{A}^{+}$ is called a tile
substitution with expansion $\varphi$ if
(2.1) $\textrm{supp}(\mathcal{G}(T))=\varphi(\textrm{supp}(T))\mbox{ for every
tile }T\in\mathcal{A}.$
In other words, the substitution $\mathcal{G}$ shows how to subdivide the
inflated tile $\varphi(\textrm{supp}(T))$ into translates of prototiles. The
tile substitution can be written explicitly as follows:
(2.2) ${\mathcal{G}}(T)=\bigcup_{T^{\prime}\in{\mathcal{A}}}\\{T^{\prime}+u:\
u\in{\mathcal{D}}_{T,T^{\prime}}\\}\ \ \mbox{for all}\ \ T\in{\mathcal{A}},$
where ${\mathcal{D}}_{T,T^{\prime}}$ is a finite (possibly empty) subset of
${{\mathbb{R}}}^{d}$, the tiles in the right-hand side have disjoint
interiors, and
(2.3) $\varphi({\rm
supp}(T))=\bigcup_{T^{\prime}\in{\mathcal{A}}}\bigcup_{u\in{\mathcal{D}}_{T,T^{\prime}}}({\rm
supp}(T^{\prime})+u).$
The substitution $\mathcal{G}$ is extended to translates of prototiles by
$\mathcal{G}(T+u)=\mathcal{G}(T)+\varphi(u)$; and to patches by
$\mathcal{G}(P)=\cup\\{\mathcal{G}(T):T\in P\\}$. The linearity of $\varphi$
and the equation (2.1) imply that the patch $\mathcal{G}(P)$ is well-defined.
###### Remark 2.2.
In this paper we restrict ourselves to the self-similar case, i.e.
$\varphi=\lambda\cdot O$, where $O$ is an orthogonal matrix and $\lambda>1$.
We refer to the corresponding ${\mathcal{G}}$ as self-similar tiling
substitution. The more general case of an arbitrary expansion map $\varphi$,
referred to as self-affine, is not covered by our main results.
###### Definition 2.3.
For a given tiling substitution
$\mathcal{G}:\mathcal{A}\rightarrow\mathcal{A}^{+}$, let
$M_{\mathcal{G}}=(m_{A,B})_{A,B\in\mathcal{A}}$ be the matrix with $m_{A,B}$
being the number of translates of the prototile $A$ in the patch
$\mathcal{G}(B)$ (i.e. $m_{A,B}=\\#{\mathcal{D}}_{A,B}$). The matrix
$M_{\mathcal{G}}$ is called the substitution matrix of $\mathcal{G}$.
The substitution is called primitive if some power of the substitution matrix
has only positive entries. We emphasize that our focus is on the non-primitive
case.
The following example will help illustrate the concepts as we go along. We
call it the integer Sierpiński carpet tiling substitution, by analogy with A.
Fisher’s integer Cantor set substitution [Fi1].
###### Example 2.4.
Suppose that the prototile set $\mathcal{A}$ consists of two $1\times 1$
squares on the plane labeled by $0$ and $1$ (we will call them the “0-tile”
and “1-tile”). Consider the following tile substitution ${\mathcal{G}}$:
$\begin{array}[]{|c|}\hline\cr 0\\\
\hline\cr\end{array}\mapsto\begin{array}[]{|c|c|c|}\hline\cr 0&0&0\\\
\hline\cr 0&0&0\\\ \hline\cr 0&0&0\\\ \hline\cr\end{array}\qquad\mbox{ and
}\qquad\begin{array}[]{|c|}\hline\cr 1\\\
\hline\cr\end{array}\mapsto\begin{array}[]{|c|c|c|}\hline\cr 1&1&1\\\
\hline\cr 1&0&1\\\ \hline\cr 1&1&1\\\ \hline\cr\end{array}$
The expansion map is a dilation: $\varphi=3I$. The substitution matrix
$M_{\mathcal{G}}=\left(\begin{array}[]{cc}9&1\\\ 0&8\end{array}\right)$ is
non-primitive.
###### Definition 2.5.
Let $\mathcal{G}:\mathcal{A}\rightarrow\mathcal{A}^{+}$ be a tile
substitution. Denote by $\Omega_{\mathcal{G}}$ the set of all tilings of
$\mathbb{R}^{d}$ by tiles from $\mathcal{A}$ such that
$\mathcal{T}\in\Omega_{\mathcal{G}}$ if every $\mathcal{T}$-patch is a
subpatch of $\mathcal{G}^{n}(T)+u$ for some $T\in\mathcal{A}$,
$u\in\mathbb{R}^{d}$, and $n\geq 1$. The set $\Omega_{\mathcal{G}}$ is called
the tiling space corresponding to the substitution $\mathcal{G}$. The group
$\mathbb{R}^{d}$ has a natural translation action on $\Omega_{\mathcal{G}}$
given by $u:\mathcal{T}\mapsto\mathcal{T}-u$ for every $u\in\mathbb{R}^{d}$
and $\mathcal{T}\in\Omega_{\mathcal{G}}$. The pair
$(\Omega_{\mathcal{G}},\mathbb{R}^{d})$ is called a substitution tiling
system.
Let $\|\cdot\|$ be the Euclidean norm on $\mathbb{R}^{d}$. For
$x\in\mathbb{R}^{d}$ and $R>0$, set
$B_{R}(x)=\\{u\in\mathbb{R}^{d}:\,\|u-x\|\leq R\\}$. We will write $B_{R}$ for
$B_{R}(\textbf{0})$. For a compact set $K$ and a tiling $\mathcal{T}$, denote
by $\mathcal{T}[[K]]$ the set of all $\mathcal{T}$-patches $P$ with
$K\subset\textrm{supp}(P)$. Define a metric $\rho$ on the space
$\Omega_{\mathcal{G}}$ as follows. Given tilings
$\mathcal{T}^{\prime},\mathcal{T}^{\prime\prime}\in\Omega_{\mathcal{G}}$, let
$\rho(\mathcal{T}^{\prime},\mathcal{T}^{\prime\prime})$ be the minimum of
$2^{-1/2}$ and
$\inf\limits\\{r>0:\exists g\in
B_{r},P^{\prime}\in\mathcal{T}^{\prime}[[B_{1/r}]],P^{\prime\prime}\in\mathcal{T}^{\prime\prime}[[B_{1/r}]]\mbox{
such that }P^{\prime}-g=P^{\prime\prime}\\}.$
With respect to the topology generated by $\rho$, two tilings are close to
each other if they agree on a large ball around the origin after a small
translation. The cut-off parameter $2^{-1/2}$ is needed to fulfill the
triangle inequality.
###### Definition 2.6.
(1) We say that the tiling system $\Omega_{\mathcal{G}}$ has finite local
complexity (FLC) if for every tiling $\mathcal{T}\in\Omega_{\mathcal{G}}$ and
$R>0$, there are only finitely many $\mathcal{T}$-patches of diameter less
than $R$ up to translation equivalence. (Note that finite pattern condition
and translational finiteness are sometimes used in the literature synonymously
with FLC).
(2) The tiling substitution $\mathcal{G}$ is called admissible if for every
prototile $T\in\mathcal{A}$ there exists a tile
$\mathcal{T}\in\Omega_{\mathcal{G}}$ such that $T\in\mathcal{T}$.
It is not always trivial to verify these conditions. Of course,
$\Omega_{\mathcal{G}}$ in Example 2.4 has FLC. Less obvious examples of FLC
tile substitutions (with tiles having fractal boundary) are considered e.g. in
[K2, So1], and two of them are discussed in Section 7. There exist primitive
tile substitution systems that do not have the FLC property, see [K1, p.244]
and [D, FR] (the latter ones have polygonal tiles).
Next let us verify that the tiling substitution in Example 2.4 is admissible.
If we put a 1-tile on the plane so that one of its corners is at the origin
and start iterating the substitution, we will obtain an increasing sequence of
patches which agree with each other and tend to a tiling of a quarter-plane.
Since the second iterate
${\mathcal{G}}^{2}\left(\,\begin{array}[]{|c|}\hline\cr 1\\\
\hline\cr\end{array}\,\right)$ contains a patch of the form
$\begin{array}[]{|c|c|}\hline\cr 1&1\\\ \hline\cr 1&1\\\ \hline\cr\end{array}$
, we can start with this “seed” centered at the origin and obtain a tiling of
the entire plane, which is the union of the four quarter-plane tilings. This
tiling is then in $\Omega_{\mathcal{G}}$, which confirms admissibility.
The FLC assumption implies the following result. The proof can be found, for
example, in [RW, Lemma 2].
###### Proposition 2.7.
If the tiling system has the FLC property, then the set $\Omega_{\mathcal{G}}$
is compact with respect to the topology generated by the metric $\rho$. The
action of the group $\mathbb{R}^{d}$ by translations on $\Omega_{\mathcal{G}}$
is continuous.
Along with the translation ${{\mathbb{R}}}^{d}$-action, we have the
substitution action on $\Omega_{\mathcal{G}}$, which is denoted by the same
letter ${\mathcal{G}}:\,\Omega_{\mathcal{G}}\to\Omega_{\mathcal{G}}$. These
two dynamical systems are intertwined by the relation
(2.4)
${\mathcal{G}}({\mathcal{T}}-y)={\mathcal{G}}({\mathcal{T}})-\varphi(y),\
{\mathcal{T}}\in\Omega_{\mathcal{G}},\ y\in{{\mathbb{R}}}^{d}.$
The following result shows that any tiling in $\Omega_{\mathcal{G}}$ has a
preimage under the map $\mathcal{G}$, see [CS, Lemma 2.8].
###### Proposition 2.8.
If $\mathcal{G}$ is admissible, then the map
$\mathcal{G}:\Omega_{\mathcal{G}}\rightarrow\Omega_{\mathcal{G}}$ is a
continuous surjection.
In fact, this is almost immediate. To find a pre-image of
${\mathcal{T}}\in\Omega_{\mathcal{G}}$ under ${\mathcal{G}}$ one needs to
“compose” or “combine” its tiles into patches that are translates of
substituted prototiles (i.e. ${\mathcal{G}}(T),\ T\in{\mathcal{A}}$), so that
the resulting tiling, after rescaling by $\varphi^{-1}$, belongs to
$\Omega_{\mathcal{G}}$. This is always possible locally, by the definition of
$\Omega_{\mathcal{G}}$ (Definition 2.5), and one only has take a subsequential
limit.
One of the important issues in the theory of substitutions is to understand
when the map $\mathcal{G}$ is invertible. This property is sometimes referred
to as recognizability or unique composition. In fact,
${\mathcal{T}}\in\Omega_{\mathcal{G}}$ has a unique pre-image under
${\mathcal{G}}$ whenever the “composition” described above is unique. Global
invertibility of ${\mathcal{G}}$ is equivalent to non-periodicity of the
tiling space for primitive tile substitutions [So2], but the extension to the
non-primitive case is by no means trivial.
###### Definition 2.9.
A tiling $\mathcal{T}\in\Omega_{\mathcal{G}}$ is periodic if there is a non-
zero vector $u\in\mathbb{R}^{d}$ such that $\mathcal{T}=\mathcal{T}+u$. Such a
vector $u$ is called a period of ${\mathcal{T}}$. A tile substitution
$\mathcal{G}$ is called non-periodic if the set $\Omega_{\mathcal{G}}$ has no
periodic tilings.
The set of periods of a tiling in ${{\mathbb{R}}}^{d}$ is a subgroup of
${{\mathbb{R}}}^{d}$. Periodic tilings can be further classified according to
the rank of the group of periods, but this will not concern us in this paper.
A classical argument shows that ${\mathcal{G}}$ cannot be invertible if the
tiling space contains a periodic tiling. Indeed, if
${\mathcal{G}}({\mathcal{S}})={\mathcal{T}}$ and
${\mathcal{T}}={\mathcal{T}}+u$, then
${\mathcal{G}}({\mathcal{S}})={\mathcal{G}}({\mathcal{S}}+\varphi^{-1}u)={\mathcal{T}}$
by (2.4). If ${\mathcal{S}}={\mathcal{S}}+\varphi^{-1}u$, we can repeat this,
obtaining shorter and shorter periods. However, this cannot go on
indefinitely, since the period of a tiling cannot be shorter than the diameter
of the largest ball which is contained in the interior of all the prototiles.
Next we discuss minimal components of our tiling dynamical systems. Recall
that a dynamical system is minimal if it has no proper closed invariant
subsets. It is well-known that tiling dynamical systems arising from primitive
substitutions are minimal (see e.g. [Ro]).
###### Definition 2.10.
A minimal component of the system $(\Omega_{\mathcal{G}},\mathbb{R}^{d})$ is a
closed $\mathbb{R}^{d}$-invariant set that contains no proper closed invariant
subsets. We note that if a minimal component $\Omega$ contains a tiling with
period $u\in\mathbb{R}^{d}$, then every tiling of $\Omega$ has the period $u$.
An easy consequence of Proposition 2.8 is that
$\Omega_{{\mathcal{G}}}=\Omega_{{\mathcal{G}}^{k}}$ for any $k\in{\mathbb{N}}$
(see [CS, Lemma 2.9]), hence one can replace the tiling substitution with its
power, whenever convenient. Reordering the letters in the alphabet
$\mathcal{A}$ and replacing $\mathcal{G}$ with its higher power
$\mathcal{G}^{k}$ if needed, the substitution matrix can be reduced to the
following form:
(2.5)
$M_{\mathcal{G}}=\left(\begin{array}[]{ccccccc}F_{1}&0&\cdots&0&X_{1,s+1}&\cdots&X_{1,m}\\\
0&F_{2}&\cdots&0&X_{2,s+1}&\cdots&X_{2,m}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\cdots&\vdots\\\
0&0&\cdots&F_{s}&X_{s,s+1}&\cdots&X_{s,m}\\\
0&0&\cdots&0&F_{s+1}&\cdots&X_{s+1,m}\\\
\vdots&\vdots&\cdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0&0&\cdots&F_{m}\end{array}\right)$
The square matrices $F_{i}$ on the main diagonal are either zero matrices or
contain only strictly positive entries. For any fixed $j=s+1,...,m$, at least
one of the matrices $X_{k,j}$ is non-zero. The block-triangular form (2.5)
allows us to give an effective description of minimal components of the
system.
For each $i=1,\ldots,m$, denote by $\mathcal{A}_{i}$ the set of prototiles
corresponding to the block $F_{i}$. Denote by $\Omega_{i}$ the set of tilings
$\mathcal{T}\in\Omega_{\mathcal{G}}$ whose patches are subpatches of
$\mathcal{G}^{n}(T)$, $n\geq 0$, with $T\in\mathcal{A}_{i}$. For $i\leq s$ it
follows from the structure of $M_{\mathcal{G}}$ that
${\mathcal{G}}({\mathcal{A}}_{i})\subset{\mathcal{A}}_{i}^{+}$, and then
$\Omega_{i}$ is just the substitution tiling system for the restriction of
${\mathcal{G}}$ to ${\mathcal{A}}_{i}$. Since $F_{i}$ is strictly positive for
$i\leq s$, we see that $(\Omega_{i},{{\mathbb{R}}}^{d})$ is minimal. It is not
hard to show ([CS, Lemma 2.10]) that the sets
$\\{\Omega_{1},\ldots,\Omega_{s}\\}$ are the only minimal components of the
tiling system.
In Example 2.4, the unique minimal component consists of the tilings with
0-tiles only. It is, of course, periodic, with the lattice ${\mathbb{Z}}^{2}$
being the group of periods, so it is topologically conjugate to the
translation action on the 2-torus ${{\mathbb{R}}}^{2}/{\mathbb{Z}}^{2}$.
Next we state a sufficient condition for the invertibility of the map
$\mathcal{G}$, following [CS], for which we need another definition.
###### Definition 2.11.
(1) Denote by $\mathcal{A}_{\textrm{per}}$ the set of all tiles
$T\in\mathcal{A}$ that occur in periodic tilings from minimal components. Set
$\mathcal{A}_{\textrm{nonp}}=\mathcal{A}\setminus\mathcal{A}_{\textrm{per}}$.
We emphasize that prototiles which do not appear in minimal components belong
to $\mathcal{A}_{\textrm{nonp}}$ by default.
(2) A substitution $\mathcal{G}$ is said to satisfy the non-periodic border
condition (NBC) if for every tile $T\in\mathcal{A}_{\textrm{nonp}}$, the
$\mathbb{R}^{d}$-boundary of the patch $\mathcal{G}(T)$ is contained in the
union of $\mathcal{A}_{\textrm{nonp}}$-tiles of $\mathcal{G}(T)$ (or rather,
their boundaries).
For the proof of the following result see [CS, Theorem 4.1 and 4.4].
###### Theorem 2.12.
(1) If the dynamical system $(\Omega_{\mathcal{G}},\mathbb{R}^{d})$ has no
periodic tilings, then the substitution
$\mathcal{G}:\Omega_{\mathcal{G}}\rightarrow\Omega_{\mathcal{G}}$ is a
homeomorphism.
(2) Assume that a substitution $\mathcal{G}$ satisfies the non-periodic border
condition. Then for every tiling $\mathcal{T}\in\Omega_{\mathcal{G}}$ that
contains a tile from $\mathcal{A}_{\textrm{nonp}}$ there exists a unique
tiling $\mathcal{T}^{\prime}$ such that
$\mathcal{G}(\mathcal{T}^{\prime})={\mathcal{T}}$.
###### Remark 2.13.
It is conjectured in [CS, Section 4] that the NBC condition may be dropped in
part (2) of the theorem, that is, ${\mathcal{G}}$ is always invertible on the
set of non-periodic tilings. It is clear that NBC is satisfied in Example 2.4.
For specific examples, even those which fail the NBC, invertibility of
${\mathcal{G}}$ on non-periodic tilings can sometimes be verified by
inspection, by observing that the “composition,” discussed after Proposition
2.8, is unique. In view of Theorem 2.12, for tiling systems with NBC, periodic
tilings can only exist in minimal components. [CS, Example 4.6] shows that the
latter property may fail without the NBC.
### 2.2. Invariant Measures
###### Definition 2.14.
(1) A measure $\mu$ on $\Omega_{\mathcal{G}}$ is called invariant if
$\mu(U-u)=\mu(U)$ for every $u\in\mathbb{R}^{d}$ and every Borel set
$U\subset\Omega_{\mathcal{G}}$. An invariant measure $\mu$ is called ergodic
if whenever a Borel set $X$ is translation-invariant, i.e. $X-u=X$ for every
$u\in\mathbb{R}^{d}$, either $\mu(X)=0$ or $\mu(\Omega_{\mathcal{G}}\setminus
X)=0$.
(2) By the transversal of $\Omega_{\mathcal{G}}$ we mean the family of all
tilings $\mathcal{T}\in\Omega_{\mathcal{G}}$ such that one of the
$\mathcal{T}$-tiles is exactly a prototile from $\mathcal{A}$. Recall that
each prototile contains the origin in the interior of its support. Throughout
the paper, the transversal will be denoted by
$\Gamma\subset\Omega_{\mathcal{G}}$.
(3) A transverse measure is a Borel measure $\nu$ on $\Gamma$ such that
$\nu(U)=\nu(U-u)$ for every Borel subset $U\subset\Gamma$ and
$u\in\mathbb{R}^{d}$ for which $U-u\subset\Gamma$.
###### Proposition 2.15.
There is a one-to-one correspondence between finite (resp. $\sigma$-finite)
transverse measures and finite (resp. $\sigma$-finite) invariant measures [CS,
Section 7].
Consider the transversal $\Gamma$. For a prototile $T\in\mathcal{A}$, set
$\Gamma_{T}=\\{\mathcal{T}\in\Gamma:\,T\in\mathcal{T}\\}.$
Then $\Gamma=\coprod_{T\in\mathcal{A}}\Gamma_{T}$ is a disjoint union. The
following result provides a description of “natural” $\sigma$-finite ergodic
measures, for the proof see Theorems 3.1 and 5.22 in [CS].
###### Theorem 2.16.
(i) Each finite ergodic measure is supported by one of the minimal components
$\\{\Omega_{1},\ldots,\Omega_{s}\\}$.
(ii) Let $i\in\\{s+1,\ldots,m\\}$ be such that the matrix $F_{i}$ is nonzero
and there exist $A\in\mathcal{A}_{i}$ and $n>0$ such that a translate of $A$
appears in the interior of $\mathcal{G}^{n}(A)$. Then there exists a unique
(up to scaling) invariant ergodic $\sigma$-finite measure $\mu$ supported by
$\Omega_{i}$ such that $0<\mu^{\rm tr}(\Gamma_{C})<\infty$ for some (and, in
fact, for all) prototile $C\in\mathcal{A}_{i}$, where $\mu^{\rm tr}$ is the
transverse measure corresponding to $\mu$. Moreover, the vector $(\mu^{\rm
tr}(\Gamma_{C}))_{C\in{\mathcal{A}}_{i}}$ is a right Perron-Frobenius
eigenvector of $F_{i}$.
###### Remark 2.17.
Denote by ${\mathcal{L}}^{d}$ the Lebesgue measure on ${{\mathbb{R}}}^{d}$.
Observe that the substitution matrix $M_{\mathcal{G}}$ has a strictly positive
left eigenvector $({\mathcal{L}}^{d}({\rm supp}(T)))_{T\in{\mathcal{A}}}$,
corresponding to the eigenvalue $\lambda=\rho(A)$. This follows from (2.1) and
the fact that the tile boundaries have zero $d$-dimensional Lebesgue measure.
(The latter is proved e.g. in [P, Prop. 1.2] by B. Praggastis; she does not
assume primitivity there.) Note also that $\rho(A_{i})=\lambda$ for $i\leq s$
and $\rho(A_{i})<\lambda$ for $i=s+1,\ldots,m$. The latter inequality follows
from the existence of strictly positive left eigenvector, see [G, Theorem
III.6].
In view of Theorem 2.16, the study of ergodic measures can be reduced to the
study of the dynamics on one of the sets $\Omega_{i}$. In fact, we can simply
restrict the substitution to the subset of prototiles
$\bigcup_{j=1}^{i}{\mathcal{A}}_{j}$; the substitution matrix will then be
obtained by truncating the matrix in (2.5) so that the diagonal block $F_{i}$
will be in the lower-right corner. This implies that it is enough to consider
substitution tiling systems whose incidence matrices have the following form:
(2.6) $M_{\mathcal{G}}=\left(\begin{array}[]{cc}A&C\\\ 0&B\end{array}\right),$
where $A$ and $B$ are square matrices; $B$ is a primitive matrix; $C$ and $A$
are non-zero matrices. Note that $B$ is uniquely determined (it will be
$F_{i}$ when we consider $\Omega_{i}$); the matrix $A$ does not have to be
primitive and may contain zero diagonal blocks. In view of the discussion
above, we will always have $\rho(B)<\rho(A)$. Note that $B=[8]$, a $1\times 1$
matrix, in Example 2.4.
### 2.3. Technical Assumptions
Here we summarize the assumptions we will be implicitly imposing on the tiling
systems in question. Throughout the paper, the symbols $\mathcal{G}$ and
$\varphi(x)=\lambda\cdot O(x)$, $\Omega_{\mathcal{G}}$ will be reserved for a
self-similar tile substitution, the associated expansion map, and the tiling
space, respectively. The set of prototiles corresponding to the matrix $B$
will be denoted by $\mathcal{B}$. Furthermore, we will always assume that the
tiling substitution $\mathcal{G}:\mathcal{A}\rightarrow\mathcal{A}^{+}$
satisfies the following conditions:
1. (1)
Every prototile is a compact subset of $\mathbb{R}^{d}$ that is the closure of
its interior. Note that this implies that the Hausdorff dimension of the
boundary of every prototile is at least $d-1$, see, for example, Corollary
IV.2 and Theorem VII.3 in [HuW].
2. (2)
The tiling system $\Omega_{\mathcal{G}}$ has finite local complexity.
3. (3)
The tile substitution $\mathcal{G}$ is admissible.
4. (4)
The substitution $\mathcal{G}$ satisfies NBC: the non-periodic border
condition (see Definition 2.11).
5. (5)
The substitution matrix $M_{\mathcal{G}}$ has the form (2.6), with $C$ non-
zero, $B$ primitive, $\rho(B)>1$.
6. (6)
We have $\alpha:=\log(\rho(B))/\log(\lambda)>d-1$. The meaning of $\alpha$
will be clarified in Section 3.1.
###### Remark 2.18.
1\. The admissibility assumption implies that for every prototile
$T\in\mathcal{B}$ there is $n>0$ such that a translate of $T$ occurs in the
interior of $\mathcal{G}^{n}(T)$.
2\. One can show that if the ${{\mathbb{R}}}^{d}$-boundary of the patches
${\mathcal{G}}(T)$, for $T\in\mathcal{B}$, is contained in the union of
$\mathcal{B}$-tiles of ${\mathcal{G}}(T)$ (this implies NBC), then condition
(6) above also holds. We leave this as an exercise.
We summarize dynamical properties of tiling systems, which follow from our
assumptions.
###### Proposition 2.19.
The space $\Omega_{\mathcal{G}}$ is compact. The map
$\mathcal{G}:\Omega_{\mathcal{G}}\rightarrow\Omega_{\mathcal{G}}$ is a
continuous surjection that is invertible on non-periodic tilings. The
dynamical system $(\Omega_{\mathcal{G}},\mathbb{R}^{d})$ has a unique (up to
scaling) infinite $\sigma$-finite measure $\mu$ such that the corresponding
transverse measure is positive and finite on one (equivalently on all) of the
sets $\Gamma_{T}$, $T\in\mathcal{B}$.
### 2.4. Hierarchical Structure
For each $k\in{\mathbb{Z}}$, define prototiles of order $k$ as
$\varphi^{k}(T)=((\varphi^{k}({\rm supp}(T)),i)$, where $T\in{\mathcal{A}}$
and $i$ is the label of $T$. Tiles of order $k$ are defined as translates of
the prototiles of order $k$. We say that they have “type $\mathcal{B}$” if $T$
has type $\mathcal{B}$.
Given a tiling ${\mathcal{T}}\in\Omega_{\mathcal{G}}$, using the surjectivity
of the substitution map $\mathcal{G}$, find a sequence of tilings
$\\{{\mathcal{T}}_{k}\\}_{k\in\mathbb{Z}}$ such that
${\mathcal{T}}_{0}={\mathcal{T}}$ and
$\mathcal{G}({\mathcal{T}}_{k})={\mathcal{T}}_{k+1}$ for every
$k\in\mathbb{Z}$. Denote by ${\mathcal{T}}^{(k)}$ the tiling obtained from
${\mathcal{T}}_{k}$ by replacing each tile with the corresponding tile of
order $k$, i.e. ${\mathcal{T}}^{(k)}=\varphi^{k}({\mathcal{T}}_{k})$. Note
that the tiles of ${\mathcal{T}}^{(k+1)}$ are obtained from tiles of
${\mathcal{T}}^{(k)}$ by “composition,” roughly speaking, by taking
appropriate unions, and the inverse operation is “subdivision,” determined by
the substitution rule. Observe that if ${\mathcal{T}}$ contains a tile of type
$\mathcal{B}$, then these tilings are uniquely defined; in fact,
(2.7) ${\mathcal{T}}^{(k)}=\varphi^{k}{\mathcal{G}}^{-k}({\mathcal{T}})\ \
\mbox{for all}\ k\in{\mathbb{Z}}.$
The tiles of ${\mathcal{T}}^{(k)}$ will be called tiles of order $k$ obtained
from ${\mathcal{T}}$. The tiles of ${\mathcal{T}}^{(k)}$ for $k>0$ will
sometimes be referred to as “supertiles of ${\mathcal{T}}$”.
## 3\. Transverse Dynamics
One of the main technical ingredients in our proof will be the dynamical
system $(\Omega_{\mathcal{G}},\mathcal{G})$, restricted to a certain fractal
subset defined below. Recall that $\varphi=\lambda\cdot O$, $\lambda>1$, and
$O$ is an orthogonal matrix.
### 3.1. Graph-Directed Iterated Function Systems
Consider the directed graph $G=(V,E)$ such that the set of vertices $V$
coincides with the alphabet $\mathcal{B}$ and the multiplicity of the set of
edges from $T$ to $T^{\prime}$ is exactly the number of occurrences of the
(translate of) prototile $T^{\prime}$ in the patch $\mathcal{G}(T)$. It
follows that the transpose matrix $B^{t}$ is exactly the incidence matrix of
the graph $G$. We will denote by $\mathcal{E}_{T,T^{\prime}}$ the set of edges
connecting a vertex $T$ to a vertex $T^{\prime}$. We will use the symbols
$s(e),r(e)$ respectively, to denote the source and range of a directed edge.
For each vertex (prototile) $T\in V$, consider the set $S_{T}={\rm supp}(T)$.
Notice that there is a one-to-one correspondence between edges in
$\mathcal{E}_{T_{1},T_{2}}$ and the set of translates of (the order $(-1)$
tile) $\varphi^{-1}(T_{2})$ in $\varphi^{-1}(\mathcal{G}(T_{1}))$. We shall
fix such a correspondence.
This can be made precise using formula (2.2) for the tile substitution: the
set of edges ${\mathcal{E}}_{T,T^{\prime}}$ corresponds to
${\mathcal{D}}_{T,T^{\prime}}$, for $T,T^{\prime}\in\mathcal{B}$. Given
$e\in{\mathcal{E}}_{T,T^{\prime}}$, we denote the corresponding vector in
${\mathcal{D}}_{T,T^{\prime}}$ by $u_{e}$. Then the similitude
(3.1) $f_{e}:\,x\mapsto\varphi^{-1}(x+u_{e})$
maps the set $S_{T^{\prime}}$ onto the translate of
$\varphi^{-1}(S_{T^{\prime}})$ corresponding to the edge $e$ in the patch
$\varphi^{-1}(\mathcal{G}(T))$ according to (2.3). Observe that distinct edges
define different maps. Then $G=(V,E)$, $\\{f_{e}\\}_{e\in E}$, is a graph-
directed system, and it uniquely defines a family of non-empty compact sets
$\\{K_{T}\\}_{T\in\mathcal{B}}$ of $\mathbb{R}^{d}$ such that
(3.2)
$K_{T}=\bigcup_{T^{\prime}\in\mathcal{B}}\bigcup_{e\in\mathcal{E}_{T,T^{\prime}}}f_{e}(K_{T^{\prime}}),$
see [MW] or [Fa, p.48]. Note that $K_{T}\subset S_{T}$ for every
$T\in\mathcal{B}$. The set $K_{T}$ is obtained from $S_{T}$ by consecutively
removing all “$\varphi$-preimages” of tiles from
$\mathcal{A}\setminus\mathcal{B}$. We note that the union (3.2) need not be
disjoint.
Observe that the contraction coefficient of every map $f_{e}$, $e\in E$, is
exactly $1/\lambda$. Thus, to find the Hausdorff dimension of the sets
$\\{K_{T}\\}_{T\in\mathcal{B}}$, one needs to consider the matrix $D^{(s)}$
with the entries
$D_{T_{1},T_{2}}^{(s)}=\sum_{e\in\mathcal{E}_{T_{1},T_{2}}}\frac{1}{\lambda^{s}}=\frac{1}{\lambda^{s}}m_{T_{2},T_{1}}.$
It follows from [MW], see also [Fa, Corollary 3.5]111This requires the open
set condition which can be verified by setting $U_{T}=\textrm{int}(S_{T})$ and
noting that
$U_{T}\supset\bigcup_{T^{\prime}\in\mathcal{B}}\bigcup_{e\in\mathcal{E}_{T,T^{\prime}}}f_{e}(U_{T^{\prime}})$,
the union being disjoint., that the Hausdorff dimension of each set $K_{T}$,
$T\in\mathcal{B}$, is the unique positive number $\alpha$ satisfying
$1=\rho(D^{(\alpha)})=\frac{1}{\lambda^{\alpha}}\rho(B^{t})=\frac{\rho(B)}{\lambda^{\alpha}}.$
Therefore, the Hausdorff dimension of every set $K_{T}$, $T\in\mathcal{B}$, is
equal to
(3.3) $\alpha=\log(\rho(B))/\log(\lambda).$
###### Remark 3.1.
It is proved in [MW], see also [Fa, Corollary 3.5], that the
$\alpha$-dimensional Hausdorff measure of $K_{T}$, denoted
${\mathcal{H}}^{\alpha}(K_{T})$, is positive and finite. This, together with
(3.2), implies that
(3.4) ${\mathcal{H}}^{\alpha}(K_{T}\cap K_{T^{\prime}})=0\ \ \mbox{for}\ T\neq
T^{\prime},$
and $({\mathcal{H}}^{\alpha}(K_{T}))_{T\in\mathcal{B}}$ is a left Perron-
Frobenius eigenvector of $B$.
We will also need the fact that
$\\{{\mathcal{H}}^{\alpha}|_{K_{T}}\\}_{T\in\mathcal{B}}$ is the list of
natural self-similar graph-directed measures on the attractors. This means
that $\\{{\mathcal{H}}^{\alpha}|_{K_{T}}\\}_{T\in\mathcal{B}}$ is, up to
scaling, the unique list of finite and positive Borel measures $\eta_{{}_{T}}$
on $K_{T}$, $T\in\mathcal{B}$, such that
(3.5)
$\eta_{{}_{T}}=\sum_{T^{\prime}\in\mathcal{B}}\sum_{e\in{\mathcal{E}}_{T,T^{\prime}}}\frac{1}{\rho(B)}(\eta_{{}_{T^{\prime}}}\circ
f_{e}^{-1}),$
see [E, 3.5]. Moreover, these natural measures may be obtained as projections
of appropriate Markov measures on the sequence space, as we are now going to
explain.
Let $T\in\mathcal{B}$. By the definition of graph-directed sets, $x\in K_{T}$
if and only if there is an infinite path $(e_{0},e_{1},\ldots)$ in the graph
$G$ such that $T=s(e_{0})$ and
(3.6) $\\{x\\}=\bigcap\limits_{k=0}^{\infty}K_{(e_{0},\ldots,e_{k})},$
where
(3.7) $K_{(e_{0},\ldots,e_{k})}=f_{e_{0}}\circ f_{e_{1}}\circ\cdots\circ
f_{e_{k}}(K_{r(e_{k})}).$
###### Definition 3.2.
(1) Let $X_{G}$ be the two-sided edge shift space associated with the graph
$G=(V,E)$, i.e.
$X_{G}=\\{(e_{n})\in E^{\mathbb{Z}}:e_{n+1}\mbox{ follows }e_{n}\mbox{ in the
graph }G\mbox{ for every }n\in\mathbb{Z}\\}.$
Formally, “$e_{n+1}\mbox{ follows }e_{n}\mbox{ in the graph }G$” means
$r(e_{n+1})=s(e_{n})$. The left shift on $X_{G}$ is denoted by $S$.
(2) We will refer to $X_{G}$ as the set of infinite two-sided paths
$(e_{n})_{n\in{\mathbb{Z}}}$ in the graph $G$. We will need it in the next
section; for now, let $X_{G}^{+}$ be the set of one-sided infinite paths
$(e_{n})_{n\geq 0}$ in the graph $G$. The natural projection
$\pi_{+}:\,X_{G}^{+}\to{{\mathbb{R}}}^{d}$ is defined by
$\pi_{+}\left((e_{n})_{n\geq 0}\right)=\lim_{k\to\infty}f_{e_{0}}\circ
f_{e_{1}}\circ\cdots\circ f_{e_{k}}(x_{0}),$
which is independent of $x_{0}\in{{\mathbb{R}}}^{d}$. For $T\in\mathcal{B}$
let
(3.8) $X^{+}_{G}(T)=\\{(e_{n})_{n\geq 0}\in X_{G}^{+}:\ s(e_{0})=T\\}$
be the set of infinite paths in $G$ starting at the vertex $T$. Clearly,
$X_{G}^{+}=\coprod_{T\in{\mathcal{B}}}X^{+}_{G}(T)$ is a disjoint union. It
follows from (3.6) that
$K_{T}=\pi_{+}(X^{+}_{G}(T)).$
Using (3.1), the natural projection $\pi_{+}$ can be written explicitly as
follows:
(3.9) $\pi_{+}\left((e_{n})_{n\geq
0}\right)=\sum_{n=0}^{\infty}\varphi^{-n-1}u_{e_{n}}.$
(3) Let $w=(w_{T})_{T\in\mathcal{B}}$ be the right Perron-Frobenius
eigenvector for the matrix $B^{t}$, such that $\sum_{T\in\mathcal{B}}w_{T}=1$.
We consider the Markov measure $\overline{\eta}$ on $X_{G}^{+}$, with initial
probabilities (i.e. probabilities of starting at $T\in\mathcal{B}$) equal to
$w_{T}$ and the probability of moving along an edge $e$ equal to
$\frac{w_{r(e)}}{\rho(B)w_{s(e)}}$. Consistency follows from the fact that $w$
is the right eigenvector of the transition matrix for the graph, with
eigenvalue $\rho(B)$. In view of Remark 3.1, we have
$w_{T}=c_{0}^{-1}{\mathcal{H}}^{\alpha}(T)\ \ \mbox{where}\ \
c_{0}=\sum_{T^{\prime}\in{\mathcal{B}}}{\mathcal{H}}^{\alpha}(T^{\prime}),$
and hence for a cylinder set $[e_{0},\ldots,e_{n}]\subset X_{G}^{+}$ we obtain
(3.10)
$\overline{\eta}([e_{0},\ldots,e_{n}])=\frac{w_{r(e_{n})}}{\rho(B)^{n+1}}=\frac{{\mathcal{H}}^{\alpha}(r(e_{n}))}{c_{0}\rho(B)^{n+1}},.$
The next lemma follows from the theory of graph-directed IFS (see e.g. the
proof of [MW, Theorem 3]).
###### Lemma 3.3.
For $T\in\mathcal{B}$ consider
$\eta_{{}_{T}}:=\overline{\eta}|_{X^{+}_{G}(T)}\circ\pi_{+}^{-1},$
that is, the natural projection of the measure $\overline{\eta}$ restricted to
$X^{+}_{G}(T)$. Then $\\{\eta_{{}_{T}}\\}_{T\in\mathcal{B}}$ is the list of
graph-directed self-similar measures satisfying (3.5), and
$\eta_{{}_{T}}=c_{0}^{-1}{\mathcal{H}}^{\alpha}|_{K_{T}}\ \ \mbox{for}\
T\in\mathcal{B},$
where
$c_{0}=\sum_{T^{\prime}\in{\mathcal{B}}}{\mathcal{H}}^{\alpha}(T^{\prime})$.
### 3.2. “Cantorization” of tilings
Recall that the graph-directed set $K_{T}$ is defined for every prototile
$T\in\mathcal{B}$. If $T^{\prime}=T+x$ is a translate of a prototile
$T\in\mathcal{B}$, then we write $K_{T^{\prime}}$ for the set $K_{T}+x$.
###### Definition 3.4.
(1) The “cantorization” of a tiling $\mathcal{T}\in\Omega_{\mathcal{G}}$ is
the set
${\mathcal{C}}({\mathcal{T}})=\bigcup\\{K_{T}:\ T\in{\mathcal{T}}\ \mbox{and
$T$ is type ${\mathcal{B}}$}\\}.$
(2) Denote
$\Omega_{0}=\\{\mathcal{T}\in\Omega_{\mathcal{G}}:\textbf{0}\in\mathcal{C}(\mathcal{T})\\},$
where 0 stands for the zero vector.
Next we present some equivalent conditions for the property
${\mathcal{T}}\in\Omega_{0}$, which are immediate from the definitions.
###### Remark 3.5.
(1) We have ${\mathcal{T}}\in\Omega_{0}$ if and only if there exist
$T_{0}\in\mathcal{B}$ and $x\in K_{T_{0}}$ such that
$T_{0}-x\in{\mathcal{T}}$.
(2) We have ${\mathcal{T}}\in\Omega_{0}$ if and only if there is a nested
sequence of type ${\mathcal{B}}$ tiles of order $-k$ obtained from
${\mathcal{T}}$, such that the intersection of their supports is the origin
${\bf 0}$. More formally (compare (2.7)), we have that
${\mathcal{T}}\in\Omega_{0}$ if and only if there is a sequence of type
$\mathcal{B}$ tiles $T_{-k}\in\varphi^{-k}{\mathcal{G}}^{k}({\mathcal{T}})$,
$k\geq 1$, such that
${\rm supp}(T_{-k-1})\subset{\rm supp}(T_{-k}),\ k\geq 0,\ \ \mbox{and}\ \
\bigcap_{k=0}^{\infty}{\rm supp}(T_{-k})=\\{{\bf 0}\\}.$
###### Proposition 3.6.
(i) The set $\Omega_{0}$ is compact in the tiling metric.
(ii) The map $\mathcal{G}:\Omega_{0}\rightarrow\Omega_{0}$ is a homeomorphism.
(iii) For every tiling $\mathcal{T}\in\Omega_{0}$, we have that
$\mathcal{C}(\mathcal{G}^{-1}(\mathcal{T}))=\varphi^{-1}(\mathcal{C}(\mathcal{T}))$
and $\mathcal{C}(\mathcal{G}(\mathcal{T}))=\varphi(\mathcal{C}(\mathcal{T}))$.
###### Proof.
(i) We only need to show that the set $\Omega_{0}$ is closed in
$\Omega_{\mathcal{G}}$. Consider a tiling $\mathcal{T}\notin\Omega_{0}$. Then
$\textbf{0}\notin\mathcal{C}({\mathcal{T}})$. Then ${\bf 0}$ does not belong
to $K_{T}$ for any tile of type ${\mathcal{B}}$ containing the origin, which
is an open condition, since $K_{T}$ is compact.
(ii) The continuity of the map
$\mathcal{G}:\Omega_{\mathcal{G}}\rightarrow\Omega_{\mathcal{G}}$ is well-
known and easily follows from the definition of the tiling metric. Note that
tiles of type ${\mathcal{B}}$ belong to ${\mathcal{A}}_{\rm nonp}$, since they
do not appear in minimal components, see Definition 2.11(1). So
${\mathcal{G}}$ is one-to-one on $\Omega_{0}$. Hence, in view of Theorem 2.12,
we only need to show that $\mathcal{G}(\Omega_{0})=\Omega_{0}$.
Remark 3.5(2) implies that if ${\mathcal{T}}\in\Omega_{0}$ then
${\mathcal{G}}({\mathcal{T}})\in\Omega_{0}$. Indeed, using the notation of the
remark, we have $\varphi
T_{-k}\in\varphi^{-(k-1)}{\mathcal{G}}^{k-1}({\mathcal{G}}({\mathcal{T}}))$,
so $\\{\varphi T_{-k}\\}_{k\geq 1}$ is a nested sequence of tiles of order
$-(k-1)$ obtained from ${\mathcal{G}}({\mathcal{T}})$, all of type
$\mathcal{B}$, and clearly the intersection of their supports is $\\{{\bf
0}\\}$.
Since the map $\mathcal{G}$ is invertible on $\Omega_{0}$ by Theorem 2.12, we
can find a (unique) tiling $\mathcal{T}_{-1}$ with
$\mathcal{G}(\mathcal{T}_{-1})=\mathcal{T}$. We claim that
${\mathcal{T}}_{-1}\in\Omega_{0}$. If not, then for some $k>0$ all the tiles
of order $-k$ obtained from ${\mathcal{T}}_{-1}$ containing the origin are of
type ${\mathcal{A}}\setminus{\mathcal{B}}$. But the substitution of
${\mathcal{A}}\setminus{\mathcal{B}}$ tiles contains only
${\mathcal{A}}\setminus{\mathcal{B}}$ tiles, so we get a contradiction with
the assumption that the origin lies in a ${\mathcal{B}}$ tile of
${\mathcal{T}}$. We have proved that $\mathcal{G}|_{\Omega_{0}}$ is a
homeomorphism.
(iii) It follows from Equation (3.2) that
$\varphi^{-1}({\mathcal{C}}({\mathcal{G}}(T)))=K_{T}$ for every prototile
$T\in\mathcal{B}$. Therefore,
$\varphi^{-1}({\mathcal{C}}({\mathcal{G}}({\mathcal{T}})))={\mathcal{C}}({\mathcal{T}})\
\ \mbox{for every tiling}\ {\mathcal{T}}\in\Omega_{0}.$
Ths implies the last statement of the proposition. ∎
## 4\. Transverse Measures
In this section we show that the dynamical system $(\Omega_{0},\mathcal{G})$
is measure-theoretically isomorphic to a mixing Markov chain, namely, the edge
shift on the graph $G$ with the incidence matrix $B^{t}$, equipped with the
measure of maximal entropy. We note that properties of
$(\Omega_{\mathcal{G}},\mathcal{G})$ as a topological dynamical system were
earlier considered in [AP].
Let $\mu$ be the $\mathbb{R}^{d}$-ergodic measure on $\Omega_{\mathcal{G}}$ as
described in Proposition 2.19. It is unique up to scaling; we will normalize
it later. There exists a unique Borel $\sigma$-finite transverse measure
$\mu^{{\rm tr}}$ on the transversal $\Gamma$ such that
(4.1) $\mu(U-\Theta)=\mu^{{\rm tr}}(U)\cdot{\mathcal{L}}^{d}(\Theta),$
where $\mathcal{L}^{d}$ is the $d$-dimensional Lebesgue measure and
$U-\Theta=\\{{\mathcal{T}}-x:\ {\mathcal{T}}\in U,\ x\in\Theta\\},$
for all Borel sets $U\subset\Gamma_{Q}$ and $\Theta\subset{\rm supp}(Q),\
Q\in\mathcal{B}$, see Section 7 in [CS] for the details. (Actually, in [CS]
this is only proved for $U$ contained in a small ball centered at the origin,
but the formula in stated generality follows from shift invariance of the
measure $\mu$.) This means that “locally” the measure $\mu$ behaves as a
product measure.
Following [CS], we give the following definition. Recall that
$\Gamma_{T}=\\{\mathcal{T}\in\Gamma:\,T\in\mathcal{T}\\}$.
###### Definition 4.1.
For every $Q\in\mathcal{B}$ and $n\geq 0$, define
$\mu^{{\rm tr}}_{n,Q}=\mu^{{\rm tr}}(\mathcal{G}^{n}(\Gamma_{Q})-x),$
where $x$ is a vector such that $\mathcal{G}^{n}(\Gamma_{Q})-x\subset\Gamma$.
Since $\mu^{{\rm tr}}$ is a transverse measure, the definition of $\mu^{{\rm
tr}}_{n,Q}$ does not depend on the choice of $x$.
The next result follows from Lemma 5.11 in [CS] and the Perron-Frobenius
theorem for primitive matrices.
###### Lemma 4.2.
There exists a (unique) right Perron-Frobenius eigenvector $\xi$ for the
matrix $B$ such that
(4.2) $\mu^{{\rm tr}}_{n,{Q}}=\frac{\xi_{Q}}{\rho(B)^{n}}\mbox{ for every
}Q\in\mathcal{B}\mbox{ and }n\geq 0.$
Let $G=(V,E)$ be the graph of the iterated function system constructed in
Section 3.1.
###### Definition 4.3.
(1) The itinerary of ${\mathcal{T}}\in\Omega_{0}$ for the
${\mathcal{G}}$-dynamics is a two-sided infinite path
$\beta({\mathcal{T}})=(e_{n})_{n\in{\mathbb{Z}}}\in X_{G}$ (see Definition
3.2), defined as follows: $\beta({\mathcal{T}})=(e_{n})_{n\in{\mathbb{Z}}}$ if
for all $n\in{\mathbb{Z}}$ the tiling ${\mathcal{G}}^{n}({\mathcal{T}})$ has a
tile $T_{n}$ of type $s(e_{n})=r(e_{n-1})\in\mathcal{B}$ containing the origin
and $T_{n+1}$ occurs in ${\mathcal{G}}(T_{n})$ in the position corresponding
to $e_{n}$.
(2) Observe that $\varphi^{n}T_{-n}$ for $n>0$ forms an increasing sequence of
supertiles of the tiling ${\mathcal{T}}$. We will call it the compatible
sequence of supertiles of ${\mathcal{T}}$ containing the origin.
(3) Note that the itinerary need not be unique. Denote by $\Omega_{0}^{*}$ the
set of all tilings $\mathcal{T}\in\Omega_{0}$ for which the itinerary is
unique. The itinerary is non-unique if and only if for some
$n\in{\mathbb{Z}}$, the origin ${\bf 0}$ lies on the common boundary of two
tiles of ${\mathcal{B}}$ type
$T_{n},T_{n}^{\prime}\in{\mathcal{G}}^{n}({\mathcal{T}})$ and, moreover, ${\bf
0}\in K_{T_{n}}\cap K_{T_{n}^{\prime}}$. Note that just being on the boundary
of a tile may not lead to non-uniqueness.
Thus $\beta:\,\Omega_{0}^{*}\to X_{G}$ is a well-defined function, whereas
$\beta$ may be considered as a multi-valued function on all of $\Omega_{0}$.
Observe that $\Omega_{0}^{*}$ is a ${\mathcal{G}}$-invariant Borel subset of
$\Omega_{0}$.
(4) By definition, $\beta\circ{\mathcal{G}}=S\circ\beta$, where $S$ is the
left shift on $X_{G}$. This holds, in an appropriate sense, even when the
itinerary is non-unique.
###### Remark 4.4.
(1) Many properties of the tiling dynamical system can be expressed using the
symbolic dynamics provided by the itinerary $\beta$. In particular, if we fix
the left one-sided sequence $(e_{n})_{n<0}$, this corresponds to the set of
translates of ${\mathcal{T}}\in\Omega_{0}^{*}$, such that the origin stays in
the interior of its original tile of type $r(e_{-1})$. This is a “piece” of
the translation orbit of ${\mathcal{T}}$. On the other hand, fixing the right
half of the symbolic orbit $(e_{n})_{n\geq 0}$ corresponds to the transversal;
more precisely, for all ${\mathcal{T}}\in(\Gamma_{T}+x)\cap\Omega^{*}_{0}$ for
a fixed vector $x$, the sequences $\beta({\mathcal{T}})$ agree in $n\geq 0$.
(2) There are, however, some complications. First, $\beta$ is not well-defined
on $\Omega_{0}\setminus\Omega_{0}^{*}$. Second, $\beta$ need not be one-to-one
and need not be onto (even if extended to $\Omega_{0}$ as a multi-valued
function). The reason is that the sequence $(e_{n})_{n<0}$ determines a
sequence of compatible supertiles whose union need not be the entire space
${{\mathbb{R}}}^{d}$. We will deal with such sequences by showing that they
have zero measure of maximal entropy for $S$. To get an example of such a
sequence, let ${\mathcal{G}}$ be the substitution from Example 2.4. Then the
graph $G$ has a single vertex and eight loops, corresponding to the eight
1-tiles in the substitution of a 1-tile. Taking a constant sequence
$(e_{n})_{n<0}$, corresponding to the lower-left 1-tile, for instance, will
yield the tiling of the 1st quadrant of the plane; see the discussion
following Definition 2.6. More generally, if the sequence $(e_{n})_{n<0}$
eventually consists of edges corresponding to the 1-tiles on one of the sides
of the substituted 1-tile, then the union of compatible supertiles will only
cover a half-plane or a quarter-plane.
###### Definition 4.5.
(1) Define $X^{*}_{G}$ as the set of $(e_{n})\in X_{G}$ such that the
compatible increasing sequence of supertiles, corresponding to
$(e_{n})_{n<0}$, has all of ${{\mathbb{R}}}^{d}$ as the limit (i.e. the union)
of the supports. It is clear that $X^{*}_{G}$ is $S$-invariant.
(2) We define the natural projection map $\pi:\,X^{*}_{G}\to\Omega_{0}$ so
that $(e_{n})$ is an itinerary of ${\mathcal{T}}:=\pi(\overline{e})$. It is
possible to describe ${\mathcal{T}}$ explicitly, as a limit of an increasing
compatible sequence of patches (whose supports are the supports of supertiles
of ${\mathcal{T}}$). The condition $\overline{e}=(e_{n})_{n\in{\mathbb{Z}}}\in
X_{G}$ means, by definition, that
(4.3) $r(e_{n})+u_{e_{n}}\in{\mathcal{G}}(s(e_{n}))\ \ \ \mbox{for all}\
n\in{\mathbb{Z}}$
(recall that the vertices of $G$ are identified with the prototiles in
${\mathcal{B}}$). A tile of ${\mathcal{T}}=\pi(\overline{e})$ containing the
origin (possibly non-unique) must be
(4.4)
$T_{0}-\sum_{n=0}^{\infty}\varphi^{-n-1}u_{e_{n}}=T_{0}-\pi_{+}(\overline{e}_{+}),$
where $T_{0}=s(e_{0})$ and $\overline{e}_{+}=(e_{n})_{n\geq 0}$ (recall that
$\pi_{+}$ was defined in Definition 3.2(2)). Note that this already guarantees
${\mathcal{T}}\in\Omega_{0}$, in view of Remark 3.5(1) and (3.9). Now we let
(4.5)
$\pi(\overline{e})=\lim_{k\to\infty}\left[{\mathcal{G}}^{k}(s(e_{-k}))-\sum_{n=-k}^{\infty}\varphi^{-n-1}u_{e_{n}}\right].$
We claim that these patches are increasing and compatible. Indeed,
${\mathcal{G}}^{k}(s(e_{-k}))-\sum_{n=-k}^{\infty}\varphi^{-n-1}u_{e_{n}}\subset{\mathcal{G}}^{k+1}(s(e_{-k-1}))-\sum_{n=-k-1}^{\infty}\varphi^{-n-1}u_{e_{n}}$
reduces to
$u_{e_{-k-1}}+s(e_{-k})\in{\mathcal{G}}(s(e_{-k-1})),$
which follows from (4.3), keeping in mind that $s(e_{-k})=r(e_{-k-1})$. Thus,
the right-hand side of (4.5) is well-defined, and it is a tiling of the entire
${{\mathbb{R}}}^{d}$ if $\overline{e}\in X_{G}^{*}$.
###### Lemma 4.6.
We have $\pi\circ S={\mathcal{G}}\circ\pi$ on $X^{*}_{G}$ and
$\pi\circ\beta={\textit{I}d}$ on $\beta(\Omega_{0}^{*})$.
###### Proof.
This is an immediate consequence of the definitions. ∎
Next we consider the measure of maximal entropy (the Parry measure)
$\overline{\nu}$ for the shift $S$ on $X_{G}$. Recall that the incidence
matrix for the graph $G$ is $B^{t}$. The Parry measure (of the edge shift) is
a Markov measure, given by
(4.6)
$\overline{\nu}([e_{1},\ldots,e_{n}]_{k})=u_{s(e_{1})}v_{s(e_{1})}\prod_{j=1}^{n}\frac{v_{r(e_{j})}}{\rho(B)v_{s(e_{j})}}=\frac{u_{s(e_{1})}v_{r(e_{n})}}{\rho(B)^{n}},$
where $[e_{1},\ldots,e_{n}]_{k}$ is a cylinder set in $X_{G}$ starting at the
index $k\in{\mathbb{Z}}$, $n\geq 0$, $u=(u_{Q})_{Q\in\mathcal{B}}$ is the left
Perron-Frobenius eigenvector of $B^{t}$, and $v=(v_{Q})_{Q\in\mathcal{B}}$ is
the right Perron-Frobenius eigenvector of $B^{t}$, normalized so that
$\sum_{Q\in\mathcal{B}}u_{Q}v_{Q}=1$. The measure is clearly shift-invariant.
We have
$\overline{\nu}(\\{\overline{e}\in X_{G}:\
s(e_{0})=Q\\})=\sum_{Q^{\prime}\in\mathcal{B}}\sum_{e\in{\mathcal{E}}_{Q,Q^{\prime}}}u_{Q}v_{Q^{\prime}}\cdot\rho(B)^{-1}=u_{Q}v_{Q},$
which implies that $\overline{\nu}$ is a probability measure. For the vector
$u$ we can take the vector $\xi$ from Lemma 4.2, which is a right Perron-
Frobenius eigenvector for $B$, and for the vector $v$ we can take
$({\mathcal{H}}^{\alpha}(K_{Q}))_{Q\in\mathcal{B}}$, which is a left Perron-
Frobenius eigenvector for $B$, see Remark 3.1. Since the measure $\mu$ was
defined up to scaling, we can normalize it (this will also affect the
transverse measure) so that
(4.7)
$\sum_{Q\in\mathcal{B}}\xi_{Q}{\mathcal{H}}^{\alpha}(K_{Q})=\sum_{Q\in\mathcal{B}}\mu^{{\rm
tr}}(\Gamma_{Q}){\mathcal{H}}^{\alpha}(K_{Q})=1.$
Then we have
(4.8)
$\overline{\nu}([e_{1},\ldots,e_{n}]_{k})=\frac{\xi_{s(e_{1})}}{\rho(B)^{n}}\,{\mathcal{H}}^{\alpha}(K_{r(e_{n})})\
\ \mbox{for}\ k\in{\mathbb{Z}},\ n\geq 0.$
It is well-known that the measure-preserving transformation
$(X_{G},S,\overline{\nu})$ is ergodic, where $S$ is the left shift.
###### Lemma 4.7.
We have $\overline{\nu}(X_{G}\setminus X^{*}_{G})=0$.
###### Proof.
Recall that there is an integer $k>0$ such that for each prototile
$T\in\mathcal{B}$ the interior of the patch ${\mathcal{G}}^{k}(T)$ contains a
translate of $T$. We can assume without loss of generality, passing from
${\mathcal{G}}$ to ${\mathcal{G}}^{k}$, that $k=1$. Then for any vertex of the
graph $G$ (i.e. $Q\in\mathcal{B}$) there is an edge $e$, with $s(e)=Q$, which
corresponds to the choice of an interior tile in the patch ${\mathcal{G}}(Q)$.
A one-sided path $(e_{n})_{n<0}$, which includes infinitely many of these
“interior” edges, will necessarily belong to $X_{G}^{*}$. Indeed, choosing an
interior supertile of order $n$ inside the supertile of order $n+1$, for
$n>0$, implies that the union of the compatible sequence of supertiles
contains the ball of radius $\delta\lambda^{n}$ centered at the origin, for
some $\delta>0$. A standard argument shows that this is a full measure set. To
verify this, note that the set of paths, which avoid the selected edges, has a
growth rate equal to the spectral radius of a matrix $B^{\prime}$ having at
least one entry in each row smaller than that of $B$, whence
$\rho(B^{\prime})<\rho(B)$. ∎
###### Definition 4.8.
Define the measure $\nu$ on $\Omega_{0}$ as the “push-forward” of
$\overline{\nu}$ on $X_{G}^{*}$ via the map $\pi$, that is,
$\nu(U)=\overline{\nu}(\pi^{-1}(U))\ \ \ \mbox{for Borel}\
U\subset\Omega_{0}.$
Since $\overline{\nu}$ is $S$-invariant, we obtain from Lemma 4.6 that the
measure $\nu$ is ${\mathcal{G}}$-invariant on $\Omega_{0}$.
###### Lemma 4.9.
We have $\nu(\Omega_{0}\setminus\Omega_{0}^{*})=0$.
###### Proof.
The argument is almost the same as in the proof of Lemma 4.7. It is enough to
prove that the set of tilings ${\mathcal{T}}\in\Omega_{0}$, for which there
exists $n\in{\mathbb{Z}}$ such that ${\bf 0}$ is on the boundary of a tile of
type ${\mathcal{B}}$ in ${\mathcal{G}}^{n}({\mathcal{T}})$, has $\nu$ measure
zero. Considering the itineraries of such tilings, we see that they must
contain only finitely many edges $e_{i}$ corresponding to the tile of type
$r(e_{i})$ in the interior of ${\mathcal{G}}(s(e_{i}))$, for $i\geq 0$. But
the growh rate of such sequences is strictly less than $\rho(B)$, hence their
$\overline{\nu}$ measure equals zero, as desired. ∎
###### Theorem 4.10.
Suppose that the Markov measure $\overline{\nu}$ is defined by (4.8), using
the normalization (4.7), and $\nu=\overline{\nu}\circ\pi^{-1}$. Then the
following hold:
(i) The probability-preserving system $(\Omega_{0},{\mathcal{G}},\nu)$ is
measure-theoretically isomorphic to $(X_{G},S,\overline{\nu})$, hence ergodic.
(ii) For any $Q\in\mathcal{B}$ and all Borel sets $\Theta\subset\Gamma_{Q},\
W\subset K_{Q}$ we have
(4.9) $\nu(\Theta-W)=\mu^{{\rm tr}}(\Theta)\cdot{\mathcal{H}}^{\alpha}(W).$
###### Proof.
(i) This follows from Lemmas 4.6, 4.7, and 4.9.
(ii) The left-hand side of (4.9) is well-defined, since
$\Gamma_{Q}-K_{Q}\subset\Omega_{0}$ by Remark 3.5(1). First let us prove the
equality for $\Theta=\Gamma_{Q}$. Recall that $X_{G}^{+}(Q)$ denotes the set
of one-sided paths in $G$ starting at $Q$. We have
$\nu(\Gamma_{Q}-W)=\overline{\nu}\left(\\{\overline{e}\in X_{G}:\
\overline{e}_{+}\in X^{+}_{G}(Q)\ \mbox{and}\ \pi_{+}(\overline{e}_{+})\in
W\\}\right),$
using the fact that ${\mathcal{T}}\in\Gamma_{Q}-W$, with $W\subset K_{Q}$, has
an itinerary with $s(e_{0})=Q$, and $\nu$ almost every tiling has a unique
itinerary by Lemma 4.9. The measure $\overline{\nu}$ on $X_{G}$ induces a
measure $\overline{\nu}_{+}$ on $X_{G}^{+}$ via the projection
$\overline{e}\mapsto\overline{e}_{+}$. Comparing (4.6) with (3.10) we see that
$\overline{\nu}_{+}|_{X^{+}_{G}(Q)}=c_{0}^{-1}\xi_{Q}\cdot\overline{\eta}|_{X^{+}_{G}(Q)}.$
Thus,
$\displaystyle\nu(\Gamma_{Q}-W)$ $\displaystyle=$
$\displaystyle\overline{\nu}_{+}|_{X^{+}_{G}(Q)}(\pi_{+}^{-1}W)$
$\displaystyle=$ $\displaystyle
c_{0}^{-1}\xi_{Q}\cdot\overline{\eta}|_{X^{+}_{G}(Q)}(\pi_{+}^{-1}W)$
$\displaystyle=$ $\displaystyle c_{0}^{-1}\xi_{Q}\cdot\eta_{Q}(W)$
$\displaystyle=$ $\displaystyle\xi_{Q}{\mathcal{H}}^{\alpha}(W),$
where we used Lemma 3.3 in the last step.
Now let us verify (4.9) for an arbitrary Borel $\Theta\subset\Gamma_{Q}$. The
transversal $\Gamma_{Q}$ is topologically a Cantor set, in which the Borel
$\sigma$-algebra is generated by the sets of the form
${\mathcal{G}}^{n}(\Gamma_{Q^{\prime}}-x)$, $Q^{\prime}\in{\mathcal{A}}$,
where $x$ is such that $Q+x\in{\mathcal{G}}^{n}(Q^{\prime})$. We have
$\nu({\mathcal{G}}^{n}(\Gamma_{Q^{\prime}}-x-W)=\nu({\mathcal{G}}^{n}(\Gamma_{Q^{\prime}}-\varphi^{-n}(x+W)))=\nu(\Gamma_{Q^{\prime}}-\varphi^{-n}(x+W)),$
using the fact that $\nu$ is ${\mathcal{G}}$-invariant. Note that $W\subset
K_{Q}$ and $Q+x\in{\mathcal{G}}^{n}(Q^{\prime})$ imply
$W+x\subset{\mathcal{C}}({\mathcal{G}}^{n}(Q^{\prime}))=\varphi^{n}K_{Q^{\prime}}$,
hence $\varphi^{-n}(x+W)\subset K_{Q^{\prime}}$, and by the case of (4.9)
already proved,
$\displaystyle\nu(\Gamma_{Q^{\prime}}-\varphi^{-n}(x+W))$ $\displaystyle=$
$\displaystyle\xi_{Q^{\prime}}\cdot{\mathcal{H}}^{\alpha}(\varphi^{-n}W)$
$\displaystyle=$
$\displaystyle\frac{\xi_{Q^{\prime}}\cdot{\mathcal{H}}^{\alpha}(W)}{\lambda^{nd}}$
$\displaystyle=$
$\displaystyle\frac{\xi_{Q^{\prime}}\cdot{\mathcal{H}}^{\alpha}(W)}{\rho(B)^{n+1}}$
$\displaystyle=$ $\displaystyle\mu^{{\rm
tr}}(\Gamma_{Q^{\prime}}-x)\cdot{\mathcal{H}}^{\alpha}(W),$
by Lemma 4.2 and Definition 4.1. The proof is complete.
∎
## 5\. Second Order Ergodic Theorem
In this section we establish the second order theorem for tiling substitution
systems. We begin with Lemma 5.3 saying that the second order ergodic theorem
can be established by checking the convergence of second order averages for
one (any) function only. This lemma was originally proved in [Fi1, Theorem 4]
for the discrete case. We include the proof for the reader’s convenience. The
proof is based on the following generalization of Hopf’s ratio ergodic
theorem. Recall that a group action is free if the identity is the only group
element for which there exists a fixed point. Our tiling translation action is
free in the measure-theoretic sense, since tilings containing at least one
tile of type ${\mathcal{B}}$ are non-periodic [CS, Corollary 4.5], and these
tilings form an invariant set of full $\mu$ measure. Recall that $B_{R}$
denotes the closed Euclidean ball. We note that the dynamical system in the
following theorem need not be conservative as the ergodic sums get averaged
over symmetric balls versus $[0,n]^{d}$.
###### Theorem 5.1 (M. Hochman [H]).
Let $\\{T^{u}\\}_{u\in\mathbb{R}^{d}}$ be a free ergodic measure preserving
action on a standard $\sigma$-finite measure space $X$. Then for $\mu$-a.e.
$x\in X$ and every $f,g\in L^{1}(X,\mu)$ with $\int_{X}gd\mu\neq 0$, we have
$\frac{\int_{B_{R}}f(T^{u}(x))du}{\int_{B_{R}}g(T^{u}(x))du}\to\frac{\int_{X}fd\mu}{\int_{X}gd\mu}\
\ \mbox{as }R\to\infty.$
###### Remark 5.2.
In fact, [H] considers non-singular free ergodic actions of ${\mathbb{Z}}^{d}$
or ${{\mathbb{R}}}^{d}$, which includes measure-preserving actions, and
averaging is over balls in any norm. We note that for our purposes it would
suffice to use an older ratio ergodic theorem of M. Becker [Beck], but it
would require a little additional argument, so we chose to quote the recent
more general result of M. Hochman.
###### Lemma 5.3.
Let $\\{T^{u}\\}_{u\in\mathbb{R}^{d}}$ be a free ergodic measure preserving
action on a standard $\sigma$-finite measure space $(X,\mu)$. Assume that
there exists $\alpha>d-1$ such that for some function $g\in
L^{\infty}(X,\mu)\cap L^{1}(X,\mu)$ with $\int_{X}g\,d\mu\neq 0$, the limit
$\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{1}^{t}\frac{\int_{B_{R}}g(T^{u}(x))\,du}{(2R)^{\alpha}}\,\frac{dR}{R}$
exists and is finite for $\mu$ a.e. $x\in X$. Then
(i) this limit is constant almost everywhere;
(ii) writing this limit as $c\cdot\int_{X}gd\mu$ for some constant $c$, we get
that for every function $f\in L^{1}(X,\mu)$,
$\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{1}^{t}\frac{\int_{B_{R}}f(T^{u}(x))\,du}{c(2R)^{\alpha}}\,\frac{dR}{R}=\int_{X}f\,d\mu$
for $\mu$-a.e. $x\in X$.
###### Proof.
(i) Set
$S_{R}^{g}(x)=\int_{B_{R}}g(T^{u}(x))\,du\ \ \mbox{and}\ \
\overline{g}(x)=\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{1}^{t}\frac{S_{R}^{g}(x)}{(2R)^{\alpha}}\,\frac{dR}{R}.$
By assumption, the function $\overline{g}(x)$ is finite for $\mu$-a.e. $x\in
X$, and it is straightforward to check that $\overline{g}$ is measurable. We
claim that $\overline{g}(T^{v}(x))=\overline{g}(x)$ for every
$v\in\mathbb{R}^{d}$ and $\mu$-a.e. $x\in X$. Indeed,
$\int_{B_{R}}g(T^{u}(x))\,du-\int_{B_{R}}g(T^{u+v}(x))\,du=O(R^{d-1}\|g\|_{\infty})\
\ \mbox{as}\ R\to\infty,$
and the assumption $\alpha>d-1$ yields the claim. Since the measure $\mu$ is
ergodic, we get that the function $\overline{g}$ is constant almost
everywhere.
(ii) Applying Theorem 5.1, we obtain that
$\frac{S_{R}^{f}(x)}{S_{R}^{g}(x)}\to\frac{\int f\,d\mu}{\int g\,d\mu}\ \mbox{
as }R\to\infty.$
Assume for definiteness that $\int g\,d\mu>0$ and $\int f\,d\mu\geq 0$. It
follows that for any $\varepsilon>0$ there is $R_{0}>0$ such that for all
$R>R_{0}$,
$S_{R}^{g}(x)\frac{\int f\,d\mu}{\int g\,d\mu}\,(1-\varepsilon)\leq
S_{R}^{f}(x)\leq S_{R}^{g}(x)\frac{\int fd\mu}{\int
g\,d\mu}\,(1+\varepsilon).$
Dividing the inequalities by $2^{\alpha}R^{\alpha+1}$ and integrating with
respect to $R$ yields
$(1-\varepsilon)\int g\,d\mu\,\frac{\int f\,d\mu}{\int
g\,d\mu}\leq\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{1}^{t}\frac{S_{R}^{f}(x)}{(2R)^{\alpha}}\,\frac{dR}{R}\leq(1+\varepsilon)\int
g\,d\mu\,\frac{\int f\,d\mu}{\int g\,d\mu}.$
Taking the limit as $\varepsilon\to 0$, we obtain the result. ∎
Now we are ready to prove the main result of the paper, but first we give the
definition of the average (or second-order) density, which was introduced by
Bedford and Fisher [BF].
###### Definition 5.4.
Let $\alpha>0$ and $\eta$ a positive finite Borel measure on
${{\mathbb{R}}}^{d}$. The average $\alpha$-dimensional density of $\eta$ at
$x$ is
$A^{\alpha}(\eta,x)=\lim_{k\to\infty}\frac{1}{k}\int_{0}^{k}\frac{\eta(B_{e^{-t}}(x))}{(2e^{-t})^{\alpha}}\,dt,$
if the limit exists. Note that we can replace $e^{-t}$ by $\lambda^{-t}$ for
$\lambda>1$, without changing the value of $A^{\alpha}(\eta,x)$. For $d=1$,
the right average density is defined as above, replacing
$\eta(B_{e^{-t}}(x))/(2e^{-t})^{\alpha}$ by $\eta([x,x+e^{-t}))/e^{-\alpha
t}$.
It is known that for a graph-directed self-similar set $K$ satisfying the Open
Set Condition, the $\alpha$-dimensional average density of the Hausdorff
measure ${\mathcal{H}}^{\alpha}$ (where $\alpha$ is the dimension of the set)
restricted to $K$, exists and is constant ${\mathcal{H}}^{\alpha}$-a.e. This
is proved in [BF] for srtandard IFS, and the extension to the graph-directed
sets is straightforward (as we essentially show below).
###### Theorem 5.5.
Let $\mathbb{X}=(\Omega_{\mathcal{G}},\mu,{{\mathbb{R}}}^{d})$ be the tiling
dynamical system corresponding to a tile substitution $\mathcal{G}$. Suppose
that the tiling system satisfies the assumptions of Section 2.3. Assume that
$\mu$ is an infinite ($\sigma$-finite) invariant measure, positive and finite
on $\Omega_{\mathcal{B}}$, where $\Omega_{\mathcal{B}}$ is the set of tilings
which have a type ${\mathcal{B}}$ tile containing the origin.
Then there exist positive parameters $\alpha$ and $c$ such that for
$\mu$-almost every tiling $\mathcal{T}\in\Omega$ and for every function $f\in
L^{1}(\Omega,\mu)$, we have
(5.1)
$\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{0}^{t}\frac{\int_{B_{R}}f({\mathcal{T}}-u)\,du}{c(2R)^{\alpha}}\,\frac{dR}{R}=\int_{\Omega_{\mathcal{G}}}fd\mu.$
Here $\alpha=\log(\rho(B))/\log(\lambda)$ is the Hausdorff dimension of the
graph-directed self-similar sets from Section 3.1 and
$c=\gamma\cdot\lim\limits_{k\to\infty}\frac{1}{k}\int_{0}^{k}\frac{\mathcal{H}^{\alpha}(B_{\lambda^{-t}}(u)\cap
K_{Q})}{(2\lambda^{-t})^{\alpha}}\,dt$
for $\mathcal{H}^{\alpha}$-a.e. $u\in K_{Q}$ and for every $Q\in\mathcal{B}$,
where ${\mathcal{H}}^{\alpha}$ is the $\alpha$-dimensional Hausdorff measure
on $K_{Q}$ . The parameter $c$ is the average $\alpha$-dimensional density of
${\mathcal{H}}^{\alpha}$ restricted to $K_{Q}$, up to the normalizing constant
$\gamma$:
$\gamma^{-1}=\sum_{Q\in{\mathcal{B}}}\xi_{Q}{\mathcal{H}}^{\alpha}(Q),\ \
\mbox{where}\ \
\sum_{Q\in{\mathcal{B}}}\xi_{Q}{\mathcal{L}}^{d}(Q)=\mu(\Omega_{\mathcal{B}}),$
and $(\xi_{Q})_{Q\in{\mathcal{B}}}$ is a right Perron-Frobenius eigenvector of
the matrix $B$.
###### Proof.
(1) First note that, without loss of generality, we can normalize $\mu$ in
such a way that (4.7) holds, so that $\gamma=1$. We then define $\nu$ as in
Theorem 4.10 and consider the ergodic probability-preserving transformation
$(\Omega_{0},\nu,\mathcal{G}^{-1})$. We follow, in part, the argument of [BF,
Theorem 3.1] (see also [Fa, Theorem 6.6]). Recall that $B_{R}$ denotes the
closed ball of radius $R$ centered at the origin. Define a function
$\psi:\Omega_{0}\rightarrow\mathbb{R}$ by
$\psi(\mathcal{T})=\int\limits_{0}^{1}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap{B}_{\lambda^{t}})}{(2\lambda^{t})^{\alpha}}\,dt.$
Since the Hausdorff measure $\mathcal{H}^{\alpha}$ is finite on sets
$\mathcal{K}_{T}$, $T\in\mathcal{B}$, and the ball ${B}_{\lambda}$ contains
only a finite number of tiles, the function $\psi$ is bounded. It is
straightforward to check that the function $\psi$ is measurable.
(2) Recall that $\varphi=\lambda\cdot O$, where $O$ is an orthogonal matrix,
hence
${\mathcal{H}}^{\alpha}(\varphi^{-1}E)=\lambda^{-\alpha}{\mathcal{H}}^{\alpha}(E)$
for any Borel set $E$. Note also that
$\varphi(B_{\lambda^{t}})=B_{\lambda^{t+1}}$. Applying Proposition 3.6, we
obtain that
$\displaystyle\psi(\mathcal{G}^{-1}(\mathcal{T}))$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{1}\frac{\mathcal{H}^{\alpha}(\varphi^{-1}\mathcal{(}C(\mathcal{T}))\cap{B}_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt$
$\displaystyle=$
$\displaystyle\int\limits_{0}^{1}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap{B}_{\lambda^{t+1}})}{2^{\alpha}\lambda^{(t+1)\alpha}}\,dt$
$\displaystyle=$
$\displaystyle\int\limits_{1}^{2}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap{B}_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt.$
It follows that
$\sum_{i=0}^{k-1}\psi(\mathcal{G}^{-i}\mathcal{T})=\int\limits_{0}^{k}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap{B}_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt.$
Thus, applying the Birkhoff Ergodic Theorem to the system
$(\Omega_{0},\nu,\mathcal{G}^{-1})$ and the function $\psi$, we get that
(5.2)
$\lim\limits_{k\to\infty}\frac{1}{k}\int\limits_{0}^{k}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap{B}_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt=\int_{\Omega_{0}}\psi(\mathcal{S})\,d\nu(\mathcal{S})$
for $\nu$-a.e. tiling $\mathcal{T}\in\Omega_{0}$. Substituting $\lambda^{t}=R$
into (5.2), we obtain that
(5.3)
$\int_{\Omega_{0}}\psi(\mathcal{S})\,d\nu(\mathcal{S})=\lim\limits_{z\to\infty}\frac{1}{\log(z)}\int\limits_{1}^{z}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap{B}_{R})}{2^{\alpha}R^{\alpha}}\,\frac{dR}{R}$
for $\nu$-a.e. $\mathcal{T}\in\Omega_{0}$. (Passing from $z=\lambda^{k}$ for
$k\in{\mathbb{N}}$ to an arbitrary $z>0$, $z\to\infty$, in the limit above is
justified, since $\psi$ is a bounded function.)
(3) We have
$\Omega_{\mathcal{G}}=\bigcup_{Q\in{\mathcal{A}}}(\Gamma_{Q}-{\rm supp}(Q)).$
The sets in the right-hand side have intersections of zero $\mu$ measure in
view of (4.1), since ${\mathcal{L}}^{d}(\partial({\rm supp}(Q)))=0$. Consider
the function
$g:=\sum_{Q\in{\mathcal{B}}}\frac{{\mathcal{H}}^{\alpha}(Q)}{{\mathcal{L}}^{d}(Q)}\cdot\chi_{{}_{\Gamma_{Q}-{\rm
supp}(Q)}}\in L^{\infty}(\Omega_{\mathcal{G}},\mu).$
That is, $g({\mathcal{T}})$ is nonzero if and only if the origin lies in a
${\mathcal{T}}$-tile $Q-x$ of type ${\mathcal{B}}$, and then the value of the
function is $\frac{{\mathcal{H}}^{\alpha}(Q)}{{\mathcal{L}}^{d}(Q)}$ (this is
well-defined on a set of full $\mu$ measure). Then (4.1) implies
$\int_{\Omega_{\mathcal{G}}}g({\mathcal{T}})\,d\mu({\mathcal{T}})=\sum_{Q\in{\mathcal{B}}}\mu^{{\rm
tr}}(\Gamma_{Q}){\mathcal{H}}^{\alpha}(Q)=1.$
In view of Lemma 5.3, it suffices to establish the second order ergodic
theorem just for the function $g$.
Given a tiling $\mathcal{T}\in\Omega_{\mathcal{G}}$, denote
$V_{R}({\mathcal{T}})=\int_{{B}_{R}}g({\mathcal{T}}-u)\,du.$
Observe that for every tile $T=Q-x\in{\mathcal{T}}$, with $Q\in{\mathcal{B}}$,
such that ${\rm supp}(T)\subset B_{R}$, integrating $g({\mathcal{T}}-u)$ over
${\rm supp}(T)$ contributes ${\mathcal{H}}^{\alpha}(K_{T})$ to
$V_{R}({\mathcal{T}})$. Exactly the same contribution from $T$ comes to
${\mathcal{H}}^{\alpha}({\mathcal{C}}({\mathcal{T}})\cap B_{R})$. Therefore,
the difference between
${\mathcal{H}}^{\alpha}({\mathcal{C}}({\mathcal{T}})\cap B_{R})$ and
$V_{R}({\mathcal{T}})$ is bounded (in modulus) by the sum of
${\mathcal{H}}^{\alpha}(K_{T})$ over those $T\in{\mathcal{T}}$ of type
${\mathcal{B}}$ whose supports intersect $\partial B_{R}$. Thus, denoting by
$d_{M}$ the maximal diameter of a prototile, we obtain
$\left|\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap
B_{R})-V_{R}(\mathcal{T})\right|\leq{\mathcal{L}}^{d}(B_{R}\setminus
B_{R-d_{M}})\cdot\max_{Q\in{\mathcal{B}}}\frac{{\mathcal{H}}^{\alpha}(Q)}{{\mathcal{L}}^{d}(Q)}=O(R^{d-1}),$
with the implied constant in $O(\cdot)$ depending only on the tiling
${\mathcal{T}}$.
Since $\alpha>d-1$ (one of our standing assumptions), we obtain that
$\lim\limits_{z\to\infty}\frac{1}{\log(z)}\int\limits_{1}^{z}\frac{|\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap
B_{R})-V_{R}(\mathcal{T})|}{2^{\alpha}R^{\alpha+1}}\,dR=0.$
Using Equation (5.3), we conclude that
(5.4)
$\lim\limits_{z\to\infty}\frac{1}{\log(z)}\int\limits_{1}^{z}\frac{\int_{B_{R}}g(\mathcal{T}-u)du}{2^{\alpha}R^{\alpha+1}}\,dR=\int_{\Omega_{0}}\psi(\mathcal{S})\,d\nu(\mathcal{S})$
for $\nu$-a.e. tiling $\mathcal{T}\in\Omega_{0}$. Denote by $Y$ the set of all
tilings in $\Omega_{\mathcal{G}}$ for which Equation (5.4) holds. Observe that
if $\mathcal{T}\in Y$, then $\mathcal{T}-v\in Y$ for any $v\in\mathbb{R}^{d}$
(here we use $\alpha>d-1$ again), i.e. $Y$ is translation-invariant.
Translation invariance of $Y$ implies that
$Y\cap(\Gamma_{Q}-K_{Q})=(Y\cap\Gamma_{Q})-K_{Q}$ for each prototile
$Q\in\mathcal{B}$. Since $\nu(Y\cap\Omega_{0})>0$, there is a prototile
$Q\in\mathcal{B}$ with $\nu(Y\cap(\Gamma_{Q}-K_{Q}))>0$. Then Theorem 4.10(ii)
implies that
$0<\nu(Y\cap(\Gamma_{Q}-K_{Q}))=\nu((Y\cap\Gamma_{Q})-K_{Q})=\mu^{{\rm
tr}}(Y\cap\Gamma_{Q})\cdot{\mathcal{H}}^{\alpha}(K_{Q}),$
hence $\mu^{{\rm tr}}(Y\cap\Gamma_{Q})>0$. Again using translation-invariance
of $Y$ and (4.1) we obtain
$\mu(Y)\geq\mu((Y\cap\Gamma_{Q})-{\rm supp}(Q))=\mu^{{\rm
tr}}(Y\cap\Gamma_{Q})\cdot{\mathcal{L}}^{d}({\rm supp}(Q))>0,$
and ergodicity of the tiling dynamical system
$\Omega_{\mathcal{G}},\mu,{{\mathbb{R}}}^{d})$ implies that (5.4) holds for
$\mu$-a.e. $\mathcal{T}\in\Omega_{\mathcal{G}}$. Setting
$c=\int_{\Omega_{0}}\psi(\mathcal{S})\,d\nu(\mathcal{S}),$
we get the result.
(4) It remains to show that the parameter $c$ can be interpreted as the
average density of the Hausdorff measure on the graph-directed set. Using the
same arguments as in (2) above, we obtain that
$\displaystyle\psi(\mathcal{G}^{k}(\mathcal{T}))$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{1}\frac{\mathcal{H}^{\alpha}(\varphi^{k}(\mathcal{C}(\mathcal{T}))\cap
B_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{1}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap
B_{\lambda^{t-k}})}{2^{\alpha}\lambda^{(t-k)\alpha}}\,dt$ $\displaystyle=$
$\displaystyle\int\limits_{-k}^{-k+1}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap
B_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt.$
Applying the Birkhoff Ergodic Theorem to the system
$(\Omega_{0},\nu,\mathcal{G})$ and the function $\psi$, we see that for
$\nu$-a.e. $\mathcal{T}\in\Omega_{0}$,
(5.5)
$\lim\limits_{k\to\infty}\frac{1}{k}\int_{-k}^{0}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap
B_{\lambda^{t}})}{2^{\alpha}\lambda^{t\alpha}}\,dt=\lim\limits_{k\to\infty}\frac{1}{k}\int_{0}^{k}\frac{\mathcal{H}^{\alpha}(\mathcal{C}(\mathcal{T})\cap
B_{\lambda^{-t}})}{2^{\alpha}\lambda^{-t\alpha}}\,dt=c.$
We have
$\Omega_{0}=\bigcup_{Q\in{\mathcal{B}}}(\Gamma_{Q}-K_{Q}),$
and the sets in the right-hand side have intersections of zero $\nu$ measure,
in view of Theorem 4.10(ii) and (3.4). Then the set of tilings
${\mathcal{T}}\in\Omega_{0}$, such that (5.5) holds and ${\mathcal{T}}$
belongs to $\Gamma_{Q}-{\mathcal{K}}_{Q}$ for a unique $Q\in{\mathcal{B}}$,
has full $\nu$ measure. Denote this set of tilings by $Z$. Observe that the
behavior of the limit in (5.5) depends only on the small neighborhood of the
origin (because the limit doesn’t change if we replace $\int_{0}^{k}$ by
$\int_{i}^{k}$ for any fixed $i\in{\mathbb{N}}$), hence ${\mathcal{T}}-u\in Z$
for $T\in\Gamma_{Q}$ and $u\in K_{Q}$ implies ${\mathcal{T}}^{\prime}-u\in Z$
for any ${\mathcal{T}}^{\prime}\in\Gamma_{Q}$. Thus, for every
$Q\in{\mathcal{B}}$,
$Z\cap(\Gamma_{Q}-K_{Q})=\Gamma_{Q}-K_{Q}^{\prime}$
for some $K_{Q}^{\prime}\subset K_{Q}$. We have
$\displaystyle\mu^{{\rm tr}}(\Gamma_{Q})\cdot{\mathcal{H}}^{\alpha}(K_{Q})$
$\displaystyle=$ $\displaystyle\nu(\Gamma_{Q}-K_{Q})$ $\displaystyle=$
$\displaystyle\nu(Z\cap(\Gamma_{Q}-K_{Q}))$ $\displaystyle=$
$\displaystyle\nu(\Gamma_{Q}-K_{Q}^{\prime})$ $\displaystyle=$
$\displaystyle\mu^{{\rm
tr}}(\Gamma_{Q})\cdot{\mathcal{H}}^{\alpha}(K_{Q}^{\prime}).$
It follows that ${\mathcal{H}}^{\alpha}$-a.e. $u\in K_{Q}$ is such that
${\mathcal{T}}-u\in Z$ for $T\in\Gamma_{Q}$, which means, rewriting (5.5),
that
$c=\lim\limits_{k\to\infty}\frac{1}{k}\int_{0}^{k}\frac{\mathcal{H}^{\alpha}(K_{Q}\cap
B_{\lambda^{-t}}(u))}{2^{\alpha}\lambda^{-t\alpha}}\,dt\ \ \mbox{for\
${\mathcal{H}}^{\alpha}$-a.e.\ $u\in K_{Q}$,\ for all\ $Q\in\mathcal{B}$},$
as desired. The proof is complete. ∎
###### Remark 5.6.
1\. Since there exists $f\in L^{1}(\Omega_{\mathcal{G}},\mu)$ such that
$0<\int_{\Omega_{\mathcal{G}}}f\,d\mu<\infty$, it is immediate that the
paramaters $\alpha$ and $c$ in Theorem 5.5 are invariants of measure-theoretic
isomorphism.
2\. We considered averaging over Euclidean balls in Theorem 5.5. This was
needed in the equality $\varphi(B_{R})=B_{\lambda R}$. If we restrict
ourselves to the case when $\varphi$ is a pure dilation, i.e.
$\varphi(x)=\lambda x$ for $\lambda>1$, then we can use averaging over balls
in any norm.
## 6\. Substitution Dynamical Systems
In this section, we derive the second order ergodic theorem for a class of
one-dimensional substitution systems. We begin with a brief review of the
background. This will be reminiscent of our discussion of the structure of
tiling substitutions, however, there are certain fundamental differences. One
of the principal differences between symbolic substitution systems and their
tiling counterparts is that symbolic substitutions may have finite ergodic
invariant measures supported off the minimal components.
Now $\mathcal{A}$ is a finite alphabet (usually
${\mathcal{A}}=\\{1,\ldots,N\\}$) and $\mathcal{A}^{+}$ is the set of all
finite non-empty words over $\mathcal{A}$. A map
$\sigma:\mathcal{A}\rightarrow\mathcal{A}^{+}$ is called a substitution; it is
extended to $\mathcal{A}^{+}$ by concatenation. Given two words
$v,w\in\mathcal{A}^{+}$, we will write $v\prec w$ if $v$ is a subword of $w$.
Denote by $L(\sigma)$ the set of all words $w\in\mathcal{A}^{+}$ such that
$w\prec\sigma^{n}(a)$ for some $a\in\mathcal{A}$ and $n\geq 1$. The family
$L(\sigma)$ is called the language of the substitution. For a word $w$, its
length will be denoted by $|w|$.
###### Definition 6.1.
The substitution dynamical system determined by a substitution $\sigma$ is a
pair $(X_{\sigma},S)$, where
$X_{\sigma}=\\{x\in\mathcal{A}^{\mathbb{Z}}:\ x[-n,n]\in L(\sigma)\mbox{ for
all }n\geq 1\\}$
and $S:\mathcal{A}^{\mathbb{Z}}\rightarrow\mathcal{A}^{\mathbb{Z}}$ is the
left shift. The set $X_{\sigma}$ is $S$-invariant and closed in
$\mathcal{A}^{\mathbb{Z}}$ with respect to the product topology.
Given $a,b\in{\mathcal{A}}$, denote by $m_{a,b}$ the number of occurrences of
$a$ in the word $\sigma(b)$. The matrix
$M_{\sigma}=(m_{a,b})_{a,b\in\mathcal{A}}$ is the substitution matrix of
$\sigma$. Reordering the letters in the alphabet $\mathcal{A}$, the matrix
$M_{\sigma}$ can be transformed to have an upper block-triangular form as in
(2.5). The diagonal matrices $F_{i}$ determine the structure of invariant
subsets and the spectral properties of $F_{i}$ determine the structure of
invariant measures. We will not discuss these results here and refer the
reader to [BKMS] for details.
Our approach to the second order ergodic theorem of substitution systems is
based on Theorem 5.5, which we apply to the self-similar substitution system
on the line ${{\mathbb{R}}}$, arising from the symbolic substitution system
$(X_{\sigma},S)$ as a suspension flow. For this to exist, however, it is
necessary and sufficient that the substitution matrix $M_{\sigma}$ should have
a strictly positive left eigenvector, whose components will serve as the
lengths of the prototiles. This is a significant restriction: it is known that
a strictly positive left eigenvector for the matrix $M_{\sigma}$ in the form
(2.5) exists if and only if all the matrices $F_{1},\ldots,F_{s}$,
corresponding to the minimal components, have the same spectral radius, which
is strictly greater than the spectral radii of all the remaining diagonal
blocks $F_{s+1},\ldots,F_{m}$, see [G, Th.III.6, p.92].
Standing assumption. We will assume for simplicity that
(6.1) $M_{\sigma}=\left(\begin{array}[]{cc}A&C\\\ 0&B\end{array}\right),$
where $A$ and $B$ are primitive, with $\rho(A)>\rho(B)>1$, and $C$ is non-
zero. We expect that our method works in the more general case, when
$M_{\sigma}$ has a strictly positive left eigenvector, but we have not
verified the details.
Under our standing assumption, plus a technical condition stated below, the
system $(X_{\sigma},S)$ has a unique, up to scaling, invariant measure that is
positive and finite on at least one open set, and this measure is infinite
$\sigma$-finite. This follows from Corollary 5.6 in [BKMS] in the case when
the substitution system is non-periodic. The non-periodicity was needed to
ensure the recognizability property [BKM, Theorem 5.17], however, it is
possible to extend the proof of Theorem 5.17 from [BKM] to the needed
generality.
First we need a technical lemma. Given two sequences $\\{x_{n}\\}$ and
$\\{y_{n}\\}$ of reals, the notation $x_{n}\approx y_{n}$ means that
$x_{n}/y_{n}\to 1$ as $n\to\infty$.
###### Lemma 6.2.
Let $\mathcal{A}=\\{1,2,\ldots,N\\}$ be a finite alphabet and
$\sigma:\mathcal{A}\rightarrow\mathcal{A}^{+}$ a substitution with the
substitution matrix of the form (6.1). Assume that the matrices $A$ and $B$
are primitive and $\rho(A)>\rho(B)>1$. Then
(6.2) $|\sigma^{k}(i)|\approx\xi_{i}\rho(A)^{k},\ \ i=1,\ldots,N,$
where $\overline{\xi}=(\xi_{i})_{i=1}^{N}$ is a left Perron-Frobenius
eigenvector for $M=M_{\sigma}$, i.e.
$[\xi_{1}\ldots\xi_{N}]M=\rho(A)[\xi_{1}\ldots\xi_{N}].$
###### Proof.
We have $|\sigma^{k}(i)|=\langle M^{k}{\bf e}_{i},\overline{1}\rangle$ where
${\bf e}_{i}$ is the $i$-th unit vector and $\overline{1}=[1\ldots 1]^{t}$.
Asymptotics of the entries of powers of a non-negative (not necessarily
irreducible) matrix are known. It follows e.g. from Theorem (9.4) in [S] that
there exist $\xi_{i}>0$ such that $|\sigma^{k}(i)|\approx\xi_{i}\rho(A)^{k}$,
$i\leq N$. It remains to show that $\overline{\xi}=\\{\xi_{i}\\}$ is a left
eigenvector.
Notice that
$M^{k+1}{\bf e}_{j}=\sum_{i=1}^{N}M(i,j)M^{k}{\bf e}_{i}.$
Hence
$\langle M^{k+1}{\bf e}_{j},\overline{1}\rangle=\sum_{i=1}^{N}\langle
M^{k}{\bf e}_{i},\overline{1}\rangle M(i,j),$
which implies
$\xi_{j}\rho(A)^{k+1}\approx\sum_{i=1}^{N}\xi_{i}\rho(A)^{k}M(i,j)\mbox{ as
}k\to\infty.$
This implies that $\overline{\xi}$ is a left eigenvector for $M$, as desired.
∎
###### Definition 6.3.
Set $\lambda=\rho(A)$. Let $\overline{\xi}$ be the left eigenvector for the
matrix $M_{\sigma}$ in the lemma above, satisfying (6.2). For each letter
$a\in\mathcal{A}$, denote by $I_{a}$ the interval of length $\xi_{a}$ centered
at the origin. We will consider these intervals as tiles in $\mathbb{R}$,
labeled by their letters. Set $\varphi(x)=\lambda x$. Define the tile
substitution $\mathcal{G}$ on the tiles $\\{I_{a}\\}_{a\in{\mathcal{A}}}$ as
follows. Consider the inflated tile $\varphi(I_{v})=\lambda I_{v}$. Since
$\textrm{length}(\lambda
I_{v})=\lambda\xi_{v}=\sum_{w\in\mathcal{A}}M(w,v)\xi_{w},$
we can subdivide the interval $\lambda I_{v}$ into the intervals $\\{I_{w}\\}$
according to the sequence of all the letters of $\sigma(v)$. Define
$\mathcal{G}(I_{v})$ as the collection of the corresponding translates of the
intervals $\\{I_{w}\\}_{w\in\sigma(v)}$. We will call $\mathcal{G}$ the tile
substitution associated with $\sigma$ and denote by $\Omega_{\mathcal{G}}$ the
corresponding tiling space.
Denote by ${\mathcal{B}}$ the set of letters corresponding to the matrix $B$.
###### Lemma 6.4.
(i) The tiling substitution ${{\mathbb{R}}}$-action
$(\Omega_{\mathcal{G}},{{\mathbb{R}}})$ is isomorphic (canonically
topologically conjugate) to the suspension flow over the symbolic substitution
${\mathbb{Z}}$-action $(X_{\sigma},S)$, with the “roof function” equal to the
constant $\xi_{j}>0$ on the cylinder sets $[j]$ for $j\in{\mathcal{A}}$.
(ii) Assume that the substitution ${\mathcal{G}}$ satisfies the conditions of
Section 2.3. Then there is a unique infinite ($\sigma$-finite) invariant
measure $\nu$ for the system $(X_{\sigma},S)$, normalized so that
$\sum_{b\in{\mathcal{B}}}\xi_{b}\nu([b])=1.$
This measure may be identified with the transverse measure $\mu^{{\rm tr}}$ of
the invariant measure $\mu$ for the system
$(\Omega_{\mathcal{G}},{{\mathbb{R}}})$, normalized so that
$\mu(\Omega_{\mathcal{B}})=1$, where $\Omega_{\mathcal{B}}$ is the set of
tilings from $\Omega_{\mathcal{G}}$ having a tile of type ${\mathcal{B}}$
containing the origin.
###### Proof.
This follows from definitions and the results of [CS]. We just observe that
the transversal of $\Omega_{\mathcal{G}}$ may be naturally identified with
$X_{\sigma}$, and transversal measures correspond to invariant measures for
$(X_{\sigma},S)$. ∎
For the technical assumptions from Section 2.3 to hold, it is enough that
(6.3) $\forall\,b\in{\mathcal{B}},\ \sigma(b)\ \mbox{starts and ends with a
letter from}\ {\mathcal{B}},$
and
(6.4) $\forall\,b\in{\mathcal{B}},\ \exists\,k\in{\mathbb{N}}\ \mbox{such
that}\ \sigma^{k}(b)\ \mbox{has at least one ``interior'' letter from}\
{\mathcal{B}}.$
###### Definition 6.5.
We will call a substitution $\sigma$ admissible if it has the form (6.1), with
$\rho(A)>\rho(B)>1$, and both (6.3) and (6.4) are satisfied.
###### Remark 6.6.
Actually, condition (6.3) may be omitted: it implies the “non-periodic border
condition”, see Definition 2.11, which was needed for recognizability of non-
periodic tilings. In fact, in the setting of one-dimensional self-similar
tiling substutions, the proof of recognizability from [CS] works without it.
Let $(X,T,\nu)$ be an infinite ergodic measure-preserving transformation. The
system is called conservative if it has no wandering sets of positive measure,
i.e. there is no set $W\subset X$ with $\nu(W)>0$ and $W\cap
T^{-n}W=\emptyset$ for every $n\geq 1$. We need conservativity of our systems,
since we will consider one-sided averages for the substitution
${\mathbb{Z}}$-action.
###### Lemma 6.7.
The substitution dynamical system with an infinite invariant measure
$(X_{\sigma},\nu,S)$ , corresponding to an admissible substitution $\sigma$,
is conservative.
###### Proof.
We will use Maharam’s recurrence theorem (see [A, 1.1.7]), which says that if
there exists a subset $Y$ of finite measure, such that
$X_{\sigma}=\bigcup_{n=0}^{\infty}S^{-n}Y$ mod $\nu$, then $S$ is
conservative. Let $Y$ be the set of sequences $(y_{n})_{n\in{\mathbb{Z}}}\in
X_{\sigma}$ such that $y_{1}\in{\mathcal{B}}$. Then $\nu(Y)<\infty$ by Lemma
6.4. We have $y\not\in\bigcup_{n=0}^{\infty}S^{-n}Y$ if and only if there
exists $k\in{\mathbb{Z}}$ such that
$y_{n}\in{\mathcal{A}}\setminus{\mathcal{B}}$ for all $n>k$. Since $\nu$ is
supported on the set of sequences which contain at least one
${\mathcal{B}}$-symbol, it suffices to show that
$\nu(Y_{0})=0,\ \ \mbox{where}\ \ Y_{0}=\\{x\in X_{\sigma}:\
x_{n}\in{\mathcal{A}}\setminus{\mathcal{B}}\ \mbox{for all}\ n>0\ \mbox{and}\
x_{0}\in{\mathcal{B}}\\}.$
For every $y\in Y_{0}$ and $n>0$ there exist $b\in\mathcal{B}$ and
$i=0,\ldots,|\sigma^{n}(b)|-1$ such that $y\in S^{i}[\sigma^{n}(b)]$, where
$[\sigma^{n}(b)]=\\{x\in X_{\sigma}:x[0,|\sigma^{n}(b)|-1]=\sigma^{n}(b)\\}$.
Since every word $\sigma^{n}(b)$, $b\in\mathcal{B}$, ends with a letter from
$\mathcal{B}$, we immediately get that $i=|\sigma^{n}(b)|-1$. It follows that
$Y_{0}\subset\bigcup_{b\in{\mathcal{B}}}\bigcap_{n\geq
1}S^{|\sigma^{n}(b)|-1}[\sigma^{n}(b)]$.
Since the measure $\nu$ is non-atomic, we have that
$\nu(S^{|\sigma^{n}(b)|-1}[\sigma^{n}(b)])=\nu([\sigma^{n}(b)])\to 0\mbox{ as
}n\to\infty.$
This yields the result. ∎
The following simple “folklore” lemma gives the so-called “accordion”
representation of words from $X_{\sigma}$.
###### Lemma 6.8.
Let $x\in X_{\sigma}$ and $n\geq 1$. Then
(6.5)
$x[1,n]=u_{0}\sigma(u_{1})\sigma^{2}(u_{2})\ldots\sigma^{m}(u_{m})\sigma^{m}(v_{m})\sigma^{m-1}(v_{m-1})\ldots\sigma(v_{1})v_{0},$
where $m\geq 1$ and $u_{i},v_{j},\ i,j=1,\ldots,m$, are subwords (possibly
empty) of $\sigma(a)$, $a\in\mathcal{A}$. Moreover, at least one of
$u_{m},v_{m}$ is nonempty.
###### Proof.
Set $w=x[1,n]$. By the definition of $X_{\sigma}$, we can choose
$a\in{\mathcal{A}}$ and the minimal $k\in{\mathcal{N}}$ such that
$w\prec\sigma^{k}(a)$. Writing $\sigma^{k-1}(a)=a_{1}\ldots a_{m}$ we obtain
that $w\prec\sigma(a_{1})\ldots\sigma(a_{m})$, hence
$w=u_{0}\sigma(w^{(1)})v_{0},$
where $w^{(1)}$ is a subword $\sigma^{k-1}(a)$ (possibly empty), $u_{0}$ is a
suffix of some $\sigma(a_{i})$ (possibly empty), and $v_{0}$ is a prefix of
some $\sigma(a_{j})$ (possibly empty). Repeating this process with $w^{(1)}$,
etc., by induction, we obtain the desired representation (6.5). ∎
For a word $w\in{\mathcal{A}}^{+}$ we define its “population vector” by
$\overline{\ell}(w)=(\ell_{i}(w))_{i=1}^{N}$ where $\ell_{i}(w)$ is the number
of symbols $i$ in the word $w$. For $w\in{\mathcal{A}}^{+}$ denote
(6.6) $|w|_{\mathcal{T}}:=\langle\overline{\ell}(w),\overline{\xi}\rangle$
and call this quantity the tiling length of the word $w$. Note that
$\overline{\ell}(\sigma(w))=M_{\sigma}\overline{\ell}(w)$ by definition of the
substitution matrix.
###### Lemma 6.9.
Let $\sigma$ be an admissible substitution. Then for any $x\in X_{\sigma}$ we
have
$\lim_{n\to\infty}\frac{|x[1,n]|_{\mathcal{T}}}{n}=1.\ $
###### Proof.
Given $x\in X_{\sigma}$ and $n\geq 1$, consider the accordion representation
(6.5). Note that for all $i$,
$|u_{i}|,|v_{i}|\leq\max_{a}|\sigma(a)|=:L_{\max}.$
Recall that for each letter $a\in{\mathcal{A}}$ we have
$|\sigma^{k}(a)|\approx\xi_{a}\lambda^{k}=\lambda^{k}|a|_{\mathcal{T}}$ by
Lemma 6.2. Hence
(6.7) $|\sigma^{k}(u)|\approx\lambda^{k}|u|_{\mathcal{T}},\ \ k\to\infty,$
uniformly for all $u$ with $|u|\leq L_{\max}$. Note also
$\langle\overline{\ell}(\sigma^{k}(j)),\overline{\xi}\rangle=\langle
M_{\sigma}^{k}{\bf e}_{j},\overline{\xi}\rangle=\langle{\bf
e}_{j},(M_{\sigma}^{t})^{k}\overline{\xi}\rangle=\langle{\bf
e}_{j},\lambda^{k}\overline{\xi}\rangle=\lambda^{k}\xi_{j},$
which yields $|\sigma^{k}(u)|_{\mathcal{T}}=\lambda^{k}|u|_{\mathcal{T}}$.
Using the accordion representation of $x[1,n]$, we obtain
$\frac{|x[1,n]|_{\mathcal{T}}}{n}=\frac{\sum_{i=0}^{m}\lambda^{i}|u_{i}|_{\mathcal{T}}+\sum_{i=0}^{m}\lambda^{i}|v_{i}|_{\mathcal{T}}}{\sum_{i=0}^{m}|\sigma^{i}(u_{i})|+\sum_{i=0}^{m}|\sigma^{i}(v_{i})|}\,.$
Now the desired statement follows from (6.7) and the fact that $m\to\infty$ as
$n\to\infty$ and at least one of $u_{m},v_{m}$ is nonempty. ∎
The next lemma gives an upper bound for the number of $\mathcal{B}$-tiles in
the interval $[0,t]$. Recall that $\Omega_{\mathcal{G}}$ is the tiling space
of $\mathcal{G}$, the tile substitution associated with $\sigma$. Given
$\mathcal{T}\in\Omega_{\mathcal{G}}$, denote by
$N_{\mathcal{T}}({\mathcal{B}},t)$ the total number of ${\mathcal{B}}$-tiles
of ${\mathcal{T}}$, contained (completely) in the interval $[0,t]$.
###### Lemma 6.10.
There exists a constant $K>0$ such that for every
$\mathcal{T}\in\Omega_{\mathcal{G}}$ and $t>0$, we have
$N_{\mathcal{T}}({\mathcal{B}},t)\leq Kt^{\alpha}$, where
$\alpha=\log(\rho(B))/\log(\rho(A))$.
###### Proof.
Since the inequality will persist (with a slightly larger constant $K$) if we
shift ${\mathcal{T}}$ by a fixed vector, we can assume that ${\mathcal{T}}$
belongs to the transversal of $\Omega_{\mathcal{G}}$. For every integer $s>0$,
find a tiling $\mathcal{T}_{s}$ such that
$\mathcal{G}^{s}(\mathcal{T}_{s})=\mathcal{T}$, see Proposition 2.8. Choose an
integer $k>0$ such that $\rho(A)^{k}\xi_{i}>2$ for every $i\in\mathcal{A}$.
Let $T_{s}$ be the tile of ${\mathcal{T}}_{s}$ containing the origin; it is
centered at the origin for all $s$: since ${\mathcal{T}}$ is in the
transversal, all $T_{s}$ are actual prototiles, see Definition 6.3. Then the
interval $[0,\rho(A)^{s}]$ is covered by the patch $\mathcal{G}^{s+k}(T_{s})$
by our choice of $k$. Thus, $N_{\mathcal{T}}({\mathcal{B}},\rho(A)^{s})$ does
not exceed the number of occurrences of $\mathcal{B}$-tiles in the patch
$\mathcal{G}^{s+k}(T_{s})$. By the Perron-Frobenius theorem applied to the
primitive matrix $B$, the number of $\mathcal{B}$-tiles in
$\mathcal{G}^{s+k}(T_{s})$ asymptotically grows not faster than $K\rho(B)^{s}$
for some constant $K$ independent of $s$. The constant $K$ can be adjusted so
that $N_{\mathcal{T}}({\mathcal{B}},\rho(A)^{s})\leq K\rho(B)^{s}$ for every
tiling $\mathcal{T}$ and every positive real number $s$. Setting
$t=\rho(A)^{s}$, noting that $\rho(A)^{\alpha}=\rho(B)$ and adjusting the
constant again, we obtain the desired inequality for all $t>0$. ∎
Now we are ready to prove the main result on substitutions. It will be
convenient to write elements of $X_{\sigma}$ as $(x(n))_{n\in{\mathbb{Z}}}$.
###### Theorem 6.11.
Let $\sigma:\mathcal{A}\rightarrow\mathcal{A}^{+}$ be an admissible
substitution, see Definition 6.5. Let $\nu$ be the infinite invariant measure
on $X_{\sigma}$ from Lemma 6.4.
Then for every function $f\in L^{1}(X_{\sigma},\nu)$ and $\nu$-a.e. $x\in
X_{\sigma}$, we have that
$\int_{X_{\sigma}}f(y)d\nu(y)=\lim\limits_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{\sum_{i=0}^{k-1}f(S^{i}x)}{ck^{\alpha}}\,\frac{1}{k}\,,$
where $\alpha=\log(\rho(B))/\log(\rho(A))$ and $c>0$ is the right average
$\alpha$-dimensional density of ${\mathcal{H}}^{\alpha}$ restricted to any of
the graph-directed sets associated with $\mathcal{G}$.
###### Proof.
(1) Consider the one-dimensional tile substitution system
$(\Omega_{\mathcal{G}},\mathbb{R})$ associated with $\mathcal{G}$, see
Definition 6.3 and Lemma 6.4. Denote by $\mu$ the (unique) translation-
invariant measure on $\Omega_{\mathcal{G}}$ corresponding to $\nu$. Recall
that $X_{\sigma}$ can be identified with the transversal of
$\Omega_{\mathcal{G}}$. More precisely, for $x\in X_{\sigma}$, let
${\mathcal{T}}(x)\in\Omega_{\mathcal{G}}$ be the tiling, which has the
prototile $I_{x(0)}$ as its tile, and the other tiles are $I_{x(n)}+y_{n}$, so
that the left endpoint of $I_{x(n+1)}+y_{n+1}$ is the right endpoint of
$I_{x(n)}+y_{n}$ for all $n\in{\mathbb{Z}}$.
(2) In view of Theorem 4 in [Fi1], which is a one-sided version of Lemma 5.3,
it is enough to establish the result for a single function $f\in
L^{1}(X_{\sigma},\nu)$ with $\int fd\nu\neq 0$. Theorem 4 from [Fi1] was
established under the assumption that the system is conservative, so we use
Lemma 6.7 here. Consider the function $f$ on $X_{\sigma}$ given by: $f(x)=1$
if $x(1)\in\mathcal{B}$ and $f(x)=0$ otherwise.
Let $F$ be the function on $\Omega_{\mathcal{G}}$ such that
$F(\mathcal{T})=1/\xi_{i}$ if the tile of $\mathcal{T}$ containing the origin
is a translate of $I_{i}$ for some $i\in\mathcal{B}$, and $F(\mathcal{T})=0$
otherwise. This is well-defined $\mu$-a.e. Repeating the arguments of Theorem
5.5, we obtain that for $\mu$-a.e. tiling
$\mathcal{T}\in\Omega_{\mathcal{G}}$,
(6.8)
$\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{1}^{t}\frac{\int_{0}^{R}F(\mathcal{T}-u)\,du}{R^{\alpha+1}}\,dR=c\int_{\Omega_{\mathcal{G}}}F(\mathcal{S})\,d\mu(\mathcal{S}):=\theta>0,$
where $\alpha=\log(\rho(B))/\log(\rho(A))$ and $c>0$ is the right average
$\alpha$-dimensional density of ${\mathcal{H}}^{\alpha}$ restricted to any of
the graph-directed sets associated with $\mathcal{G}$
Recall that $N_{\mathcal{T}}({\mathcal{B}},R)$ is the number of
${\mathcal{B}}$-tiles contained in $[0,R]$. Thus,
$N_{\mathcal{T}}({\mathcal{B}},R)\leq\int_{0}^{R}F(\mathcal{T}-u)\,du\leq
N_{\mathcal{T}}({\mathcal{B}},R)+1.$
Since $\mu$ is (locally) a product of the transverse measure $\nu$ and the
Lebesgue measure on $\mathbb{R}$, it follows from Equation (6.8) that for
$\nu$-a.e. $x\in X_{\sigma}$,
(6.9)
$\lim\limits_{t\to\infty}\frac{1}{\log(t)}\int_{0}^{t}\frac{N_{{\mathcal{T}}(x)}({\mathcal{B}},R)}{R^{\alpha+1}}\,dR=\theta.$
(3) We want to show that
$\theta=\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{\sum_{i=0}^{k-1}f(S^{i}x)}{k^{\alpha+1}}$
for all $x\in X_{\sigma}$ satisfying Equation (6.9). Denote by
$\ell_{\mathcal{B}}(w)$ the number of ${\mathcal{B}}$-letters in a word
$w\in{\mathcal{A}}^{+}$, and observe that
$\sum_{i=0}^{k-1}f(S^{i}x)=\ell_{\mathcal{B}}(x[1,k]).$
Note that (6.9) is equivalent to
$\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{N_{{\mathcal{T}}(x)}({\mathcal{B}},k)}{k^{\alpha+1}}=\theta,$
so we just need to estimate
$|N_{{\mathcal{T}}(x)}({\mathcal{B}},k)-\ell_{\mathcal{B}}(x[1,k])|$. We claim
that
(6.10) $\ell_{\mathcal{B}}(x[1,k])=N_{\mathcal{B}}({\mathcal{T}}(x),R_{k}),\ \
\ \mbox{where}\ \ R_{k}=|x[1,k]|_{\mathcal{T}}+(1/2)\xi_{x(0)}.$
Indeed, the left-hand side represents the number of ${\mathcal{B}}$-letters in
$x[1,k]$ and the right-hand side equals the number of ${\mathcal{B}}$-tiles,
(completely) contained in the interval $[0,R_{k}]$. According to the
definition of ${\mathcal{T}}(x)$ at the beginning of the proof, it has the
prototile $I_{x(0)}$ centered at the origin, so that half of its length
$(1/2)\xi_{x(0)}$ is in $[0,R_{k}]$. After that the sequence of tiles which
fits in $[0,R_{k}]$ exactly corresponds to $x[1,k]$, by the definition of the
tile length. Thus, both sides of (6.10) count the same quantity.
Now, by Lemma 6.9 we have $R_{k}\approx k$, hence $R_{k}=k+o(k)$, as
$k\to\infty$, using the standard $o(\cdot)$ notation. In view of (6.10) and
Lemma 6.10, we have
$\displaystyle|N_{{\mathcal{T}}(x)}({\mathcal{B}},k)-\ell_{\mathcal{B}}(x[1,k])|$
$\displaystyle=$
$\displaystyle|N_{{\mathcal{T}}(x)}({\mathcal{B}},k)-N_{{\mathcal{T}}(x)}({\mathcal{B}},R_{k})|$
$\displaystyle\leq$ $\displaystyle K|R_{k}-k|^{\alpha}+1=o(k^{\alpha}).$
Indeed,
$N_{{\mathcal{T}}(x)}({\mathcal{B}},R_{k})-N_{{\mathcal{T}}(x)}({\mathcal{B}},k)$
equals the number of ${\mathcal{B}}$-tiles of ${\mathcal{T}}(x)$ in the
interval $[k,R_{k}]$ (assume that $k\leq R_{k}$ for definiteness) plus one, if
a ${\mathcal{B}}$-tile contains $k\in{{\mathbb{R}}}$ in its interior, and the
number of ${\mathcal{B}}$-tiles of ${\mathcal{T}}(x)$ in the interval
$[k,R_{k}]$ equals $N_{\mathcal{B}}({\mathcal{T}}(x)-k,R_{k}-k)$, to which we
can apply Lemma 6.10. Thus,
$\displaystyle\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{\sum_{i=0}^{k-1}f(S^{i}x)}{k^{\alpha+1}}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{\ell_{\mathcal{B}}(x[1,k])}{k^{\alpha+1}}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{N_{{\mathcal{T}}(x)}({\mathcal{B}},k)}{k^{\alpha+1}}$
$\displaystyle+$
$\displaystyle\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{o(k^{\alpha})}{k^{\alpha+1}}=\theta+0,$
as desired. Noticing that
$\theta=c\int_{\Omega_{\mathcal{G}}}F\,d\mu=c\sum_{b\in\mathcal{B}}\nu([b])=c\int_{X_{\sigma}}f\,d\nu$
by Lemma 6.4, we get the result. ∎
###### Remark 6.12.
(1) We note that the parameters $\alpha$ and $c$ appearing in the second order
ergodic theorem are invariants of the measure-theoretical isomorphism between
infinite measure preserving systems. This is immediate, since there exists
$f\in L^{1}(X_{\sigma},\nu)$ such that $\int_{X_{\sigma}}fd\nu$ is positive
and finite.
As an example, consider two symbolic substitution systems on the alphabet
$\mathcal{A}=\\{0,1\\}$ given by $\sigma_{1}(0)=0^{3}$ (three zeros),
$\sigma_{1}(1)=101$; and $\sigma_{2}(0)=0^{9}$, $\sigma_{2}(1)=1^{4}01^{4}$.
Then, $\alpha_{1}=\log(2)/\log(3)$, whereas $\alpha_{2}=\log(8)/\log(9)$.
Since $\alpha_{1}\neq\alpha_{2}$, these systems cannot be measure-
theoretically isomorphic with respect to the invariant infinite measures, and
hence, cannot be topologically conjugate.
The parameter $c$ can also be used to distinguish substitution systems,
although the computation is more involved. For example, consider for
$k=0,\ldots,3$ the substitutions $\sigma_{k}(0)=0^{9}$ and
$\sigma_{k}(1)=10^{k}10^{6-k}1$. For all of them we have $\alpha=1/2$, but the
average densities of the corresponding graph-directed sets are likely to be
different, which would imply that the substitution dynamical systems
associated with $\sigma_{k}$ are pairwise non-isomorphic.
(2) In general, symmetric and one-sided average densities need not be equal,
except in symmetric cases, such as the middle-thirds Cantor set.
As a corollary of Theorem 6.11, we establish that almost every sequence in
$X_{\sigma}$ admits an “$\alpha$-dimensional frequency”.
###### Corollary 6.13.
Let $(X_{\sigma},\nu,S)$ be a substitution system satisfying the assumptions
of Theorem 6.11. Then for every letter $b\in\mathcal{B}$, the limit
$\lim\limits_{n\to\infty}\frac{1}{\log(n)}\sum_{1\leq k\leq
n,\;x_{k}=b}\frac{1}{k^{\alpha}}$
exists and equals to $\alpha\cdot c\cdot\nu([b])$ for $\nu$-a.e. $x=(x_{k})\in
X_{\sigma}$.
###### Proof.
We will use the same notation as in the proof of Theorem 6.11. Fix a letter
$b\in\mathcal{B}$. Consider the function $f:X_{\sigma}\rightarrow\mathbb{R}$
such that $f(x)=1$ if $x_{0}=b$ and $f(x)=0$ otherwise. Given a sequence $x\in
X_{\sigma}$, denote by $\ell_{b}(x,k)=\ell_{b}(x[1,k])$ the number of
occurrences of the symbol $b$ in the word $x[1,k]$. Theorem 6.11 implies that
(6.11)
$\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{\infty}\frac{\ell_{b}(x,k)}{k^{\alpha+1}}=c\nu([b])$
for $\nu$-a.e. $x\in X_{\sigma}$. Fix a sequence $x\in X_{\sigma}$ satisfying
Equation (6.11). Using summation by parts, we get
$\sum_{k=1}^{n}\frac{f(S^{k}x)}{k^{\alpha}}=\sum_{k=1}^{n-1}\ell_{b}(x,k)\left(\frac{1}{k^{\alpha}}-\frac{1}{(k+1)^{\alpha}}\right)+\frac{\ell_{b}(x,n)}{n^{\alpha}}.$
Lemmas 6.9 and 6.10 imply that $\ell_{b}(x,n)/\alpha^{d}$ is uniformly bounded
in $n$. Notice that
$\left(\frac{1}{k^{\alpha}}-\frac{1}{(k+1)^{\alpha}}\right)\approx\frac{\alpha}{k^{\alpha+1}}.$
Thus, (6.11) yields
$\displaystyle\lim\limits_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\frac{f(S^{k}x)}{k^{\alpha}}$
$\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^{n}\ell_{b}(x,k)\frac{d}{k^{\alpha+1}}$
$\displaystyle=$ $\displaystyle\alpha\int fd\nu=\alpha\cdot c\cdot\nu([b])$
for $\nu$-a.e. $x\in X_{\sigma}$. ∎
This may be compared with a result of [Be], which implies that all “morphic”
sequences $x$ have the logarithmic frequency of letters. This means that for
$a\in\mathcal{A}$ the following limit exists:
$\lim\limits_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1,\;x_{k}=a}^{n}\frac{1}{k}\,.$
For our substitutions it is immediate that the logarithmic frequency of
$b\in{\mathcal{B}}$ is zero for all $x\in X_{\sigma}$, since already the
ordinary frequency $\lim_{n\to\infty}\frac{1}{n}\\#\\{k\leq n:\ x_{k}=b\\}$
equals zero.
## 7\. Examples and open questions
In this section we consider a few examples of tiling and substitution systems
and determine the parameter $\alpha$ appearing in the second order ergodic
theorem.
###### Example 7.1.
(“Cantor” substitution). Let $\mathcal{A}=\\{0,1\\}$ and $\sigma(0)=000$,
$\sigma(1)=101$. Consider the substitution system $(X_{\sigma},S)$ associated
to $\sigma$. Observe that the graph-directed set of $\sigma$ is the middle-
thirds Cantor set. The dynamical system $(X_{\sigma},S)$ admits a unique
ergodic measure $\mu$ on $X_{\sigma}$ with the property $\mu([1])=1$. Then
Theorem 6.11 holds for the system $(X_{\sigma},S)$ with parameters
$\alpha=\log(2)/\log(3)$ and $c>0$, where $c$ is the right average density of
the Hausdorff measure on the middle-thirds Cantor set. Actually, in this case
the right and left average densities are equal to the symmetric average
density; its numerical value has been computed in [PZ]. The second order
ergodic theorem for the system $(X_{\sigma},S)$ was originally established by
A. Fisher [Fi1].
###### Example 7.2.
Returning to Example 2.4, we see that the associated graph-directed set is the
classical Sierpiński carpet. So Theorem 5.5 with parameter
$\alpha=\log(8)/\log(3)$ applies to this tiling system.
###### Example 7.3.
Here we consider a tiling dynamical systems with prototiles having fractal
boundaries. Our example is a modification of the system described in [So1,
Section 7.2], belonging to the family of tilings constructed in [K2, Section
6].
Let $r\approx.34115+1.1616i$ be a root of the equation $x^{3}+x+1=0$. Let
$T_{a}$, $T_{b}$, and $T_{c}$ be sets (prototiles) as described in Lemma 7.7
of [So1]. We use the same notation as in [So1].
We note that $T_{a}$, $T_{b}$, and $T_{c}$ are compact subsets of
$\mathbb{C}$. Set $\theta(z)=rz$. Then $\theta(T_{a})=T_{b}$; $\theta(T_{b})$
is the union of a translation of $T_{b}$ and of $T_{c}$; and $\theta(T_{c})$
is a translation of $T_{a}$. This subdivision rule uniquely determines a tile
substitution $\Theta$.
Set $\mathcal{G}=\Theta^{2}$. Thus, $\mathcal{G}(T_{a})$ is the union of a
translation of $T_{b}$ and of $T_{c}$; $\mathcal{G}(T_{b})$ is the union of a
translation of $T_{b}$, of $T_{c}$, and of $T_{a}$; and $\mathcal{G}(T_{c})$
is a copy of $T_{b}$. Set $\varphi(z)=r^{2}z$. Then after the “realification”
of $\mathbb{C}$, the map $\varphi$ can be represented as
$\varphi((x,y)^{T})=\lambda\cdot O\cdot(x,y)^{T}$, where $\lambda=|r^{2}|$ and
$O=\left(\begin{array}[]{cc}\beta&-\gamma\\\ \gamma&\beta\\\
\end{array}\right)$
with $\beta+\gamma i=r^{2}/|r^{2}|$. Then $\lambda$ is the expansion constant
of $\varphi$.
Assuming that the tiles $T_{a}$, $T_{b}$, and $T_{c}$ are colored white,
denote by $S_{a}$, $S_{b}$, and $S_{c}$ their respective copies colored black.
Extend the tile substitution $\mathcal{G}$ on $\\{S_{a},S_{b},S_{c}\\}$ as
follows: $S_{a}$ is mapped into a union of $S_{b}$ and $S_{c}$ in the same way
as $T_{a}$; $S_{c}$ is mapped into a copy of $S_{b}$ as $T_{c}$; and the tile
$S_{b}$ is mapped into a union of $S_{a}$, $S_{b}$, and $S_{c}$ exactly as
$T_{b}$, but with the tile $S_{b}$ being replaced by the tile $T_{b}$ (of the
same shape but of a different color).
Denote by $\mathcal{A}$ the set of prototiles
$\\{T_{a},T_{b},T_{c},S_{a},S_{b},S_{c}\\}$. Consider the tiling dynamical
system $(\Omega_{\mathcal{G}},\mathbb{R}^{2})$ associated to
$\mathcal{G}:\mathcal{A}\rightarrow\mathcal{A}^{+}$. This system has a unique
minimal component determined by white tiles, see [CS, Lemma 2.10]. The minimal
component is non-periodic, see [So1, Section 7.2]. Hence, the substitution
$\mathcal{G}$ satisfies all the conditions of Theorem 2.16 yielding that this
system admits a unique “natural” infinite invariant measure $\mu$ up to
scaling. The substitution matrix $M_{\mathcal{G}}$ is given by
$\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\ 1&1&1&0&1&0\\\ 1&1&0&0&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&1&0&1\\\ 0&0&0&1&1&0\\\ \end{array}\right)$
It follows that the matrix $B$ is equal to $\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 1&1&0\\\ \end{array}\right).$ Since $\rho(B)\approx 1.618$ and the
expansion constant of $\mathcal{G}$ is $\lambda\approx 1.466$, we get that
$\alpha=\log(\rho(B))/\log(\lambda)>1$. Hence, the second order theorem
(Theorem 5.5) with parameter $\alpha\approx 1.258$ applies to the system
$(\Omega_{\mathcal{G}},\mathbb{R}^{2})$.
###### Example 7.4.
This example is a non-minimal extension of the well-known Rauzy tiling [R]. We
start with the Rauzy tiling itself. Let $r\approx-0.7771845+1.11514i$ be the
complex root of the equation $1-r-r^{2}-r^{3}=0$. The tiles may be described
using digit expansions in the base of $r$. There are three prototiles $T_{a}$,
$T_{b}$, and $T_{c}$, which may be represented as follows:
$T:=\Bigl{\\{}\sum_{n=0}^{\infty}a_{n}r^{-n}:\ a_{n}\in\\{0,1\\},\
a_{n}a_{n+1}a_{n+2}\neq 111\ \mbox{for all}\ n\Bigr{\\}}.$
Then
$T_{a}:=r^{-1}T,\ T_{b}:=1+r^{-2}T,\ T_{c}:=1+r^{-1}+r^{-3}T.$
Clearly, $rT_{a}=T_{a}\cup T_{b}\cup T_{c}$, $rT_{b}=r+T_{a}$, and
$rT_{c}=r+T_{b}$. This determines the substitution rule. (Strictly speaking,
these prototiles do not satisfy our definition, since $T_{b}$ and $T_{c}$ do
not contain the origin in the interior of their support, but this can be
easily rectified, translating the tiles. However, the given form of the tiles
is more convenient.) All the tiles of the Rauzy tiling can also be described
using base $r$ expansions: for any finite sum $z=\sum_{n=-N}^{-1}a_{n}r^{-n}$
with the property that $a_{n}\in\\{0,1\\},\ a_{n}a_{n+1}a_{n+2}\neq 111$ for
all $n$, we have $z+T_{a}\in{\mathcal{T}}$ in all cases,
$z+T_{b}\in{\mathcal{T}}$ iff $a_{-2}a_{-1}\neq 11$, and
$z+T_{c}\in{\mathcal{T}}$ iff $a_{-1}\neq 1$.
Now consider the “extended” tiling system, with the prototiles
$T_{a},T_{b},T_{c}$ and $S_{a}$, $S_{b}$, which have the same support as
$T_{a},T_{b}$ respectively, but have a different color (label). The
substitution acts as before on $T_{a},T_{b},T_{c}$, and
$rS_{a}=S_{a}\cup S_{b}\cup T_{c},\ \ rS_{b}=1+S_{a}.$
The matrix of the substitution is
$M_{\mathcal{G}}=\left(\begin{array}[]{cc}A&C\\\ 0&B\end{array}\right)$, where
$A=\left(\begin{array}[]{ccc}1&1&0\\\ 1&0&1\\\ 1&0&0\end{array}\right),\ \ \
B=\left(\begin{array}[]{cc}1&1\\\ 1&0\end{array}\right),\ \ \
C=\left(\begin{array}[]{cc}0&0\\\ 0&0\\\ 1&0\end{array}\right).$
The expansion $\lambda=|r|\approx 1.3562$ is the same as for the Rauzy tiling,
and $\rho(B)\approx 1.618$ is the golden ratio. All the assumptions from
Section SectionAssumptions are easily verified. We get
$\alpha=\log(\rho(B))/\log(\lambda)\approx 1.57935>1$, so Theorem 5.5 applies.
Figure 1 shows the “cantorization” of the tiling, so it gives an idea of both
“large-scale” structure of the tiling and the “small-scale” structure of the
graph-directed sets.
It is interesting to note that the “cantorization” of the tiling has a simple
description using base $r$ expansions: instead of all expansions using the
digits $(a_{n})$ with $111$ forbidden, one should consider all expansion with
the sequence of digits from the “golden mean” shift, that is, $11$ is
forbidden.
### 7.1. Open questions
1\. We had to impose some technical conditions on the substitution to prove
the second order ergodic theorem. For example, we do not know if it holds for
the following substitution on $\\{0,1,2\\}$:
$0\mapsto 00000,\ \ 1\mapsto 1111,\ \ 2\mapsto 20212.$
The matrix of the substitution is
$M_{\sigma}=\left(\begin{array}[]{ccc}5&0&1\\\ 0&4&1\\\
0&0&3\end{array}\right)$. Thus, there is an infinite ($\sigma$-finite)
invariant measure positive and finite on cylinder sets containing $2$,
however, there is no positive left eigenvector, so our methods do not work.
2\. We proved that (in appropriate contexts) converge the logarithmic averages
of the expressions
$R^{-\alpha}\int_{B_{R}}f({\mathcal{T}}-u)\,du\ \ \mbox{and}\ \ \
k^{-\alpha}\sum_{i=0}^{k-1}f(S^{i}x).$
But one can also view them as random variables (with ${\mathcal{T}}$ or $x$
taken randomly from the substitution space, according to the invariant measure
normalized on the appropriate cylinder set), and inquire whether they converge
in distribution as $R$ (resp. $k$) tend to infinity along a subsequence? For
instance, it appears that for the “integer Cantor” substitution from Example
1, we get that $2^{-n}\sum_{i=0}^{3^{n}-1}f(S^{i}x)$ tends to the uniform
distribution on $[0,1]$ as $n\to\infty$, for $f$ the characteristic function
of $[1]$ (and then for all $f\in L^{1}(X_{\sigma},\mu)$ with $\int
f\,d\mu=1$).
Acknowledgment. We are grateful to Karl Petersen, who asked whether the
second-order ergodic theorem holds for the Sierpiński gasket tiling system and
who told us about A. Fisher’s paper [Fi1]. We would also like to thank the
referee for his/her valuable comments.
## References
* [A] J. Aaronson, _An introduction to infinite ergodic theory_. Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997.
* [ADF] J. Aaronson, M. Denker, and A. Fisher, _Second order ergodic theorems for ergodic transformations of infinite measure spaces_. Proc. Amer. Math. Soc. 114 (1992), no. 1, 115–127.
* [AP] J. Anderson and I. Putnam, _Topological invariants for substitution tilings and their associated $C^{*}$-algebras_. Ergodic Theory Dynam. Systems 18 (1998), no. 3, 509–537.
* [Beck] M. Becker, A ratio ergodic theorem for groups of measure-preserving transformations. Illinois J. Math. 27 (1983), 562–570.
* [BF] T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. (3) 64 (1992), 95–124.
* [Be] J. Bell, Logarithmic frequency in morphic sequences. J. Theor. Nombres Bordeaux 20 (2008), no. 2, 227–241.
* [BKM] S. Bezuglyi, J. Kwiatkowski, K. Medynets, _Aperiodic substitution systems and their Bratteli diagrams_. Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37–72.
* [BKMS] S. Bezuglyi, J. Kwiatkowski, K. Medynets, B. Solomyak, _Invariant measures on stationary Bratteli diagrams_. Ergodic Theory Dynam. Systems 30 (2010), no. 4, 973–1007.
* [CS] M.I. Cortez and B. Solomyak, _Invariant measures for non-primitive tiling substitutions_. J. Anal. Math. 115 (2011), 293–342.
* [D] L. Danzer, Inflation species of planar tilings which are not of locally finite complexity. Proc. Steklov Inst. Math. 239 (2002), 108–116.
* [E] G. Edgar, Integral, Probability, and Fractal Measures. Springer, 1997.
* [Fa] K. Falconer, _Techniques in fractal geometry_. John Wiley & Sons, Ltd., Chichester, 1997.
* [Fi1] A. Fisher, _Integer Cantor sets and an order-two ergodic theorem_. Ergodic Theory Dynam. Systems 13 (1993), no. 1, 45–64.
* [FR] N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness. Trans. Amer. Math. Soc. 360 (2008), 1163–1177.
* [G] F. R. Gantmacher, Applications of the theory of martrices. Interscience Publishers, Inc., New York, 1959.
* [HY] M. Hama and H. Yuasa, Invariant measures for subshifts arising from substitutions of some primitive components. Hokkaido Math. J. 40 (2011), no. 2, 279–312.
* [H] M. Hochman, _A ratio ergodic theorem for multiparameter non-singular actions_. J. Eur. Math. Soc. 12 (2010), no. 2, 365–383.
* [HuW] W. Hurewicz, and H. Wallman, _Dimension Theory_. Princeton Mathematical Series, v. 4. Princeton University Press, Princeton, N. J., 1941.
* [K1] R. Kenyon, _Self-replicating tilings. Symbolic dynamics and its applications_ (New Haven, CT, 1991), 239-263, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.
* [K2] R. Kenyon, _The construction of self-similar tilings_. Geom. Funct. Anal. 6 (1996), no. 3, 471–488.
* [LS] F. Ledrappier and O. Sarig, _Fluctuations of ergodic sums for horocycle flows on $\mathbb{Z}^{d}$-covers of finite volume surfaces_. Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 247–325.
* [MW] R.D. Mauldin, S.C. Williams, _Hausdorff dimension in graph directed constructions_. Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829.
* [PZ] N. Patzschke and M. Zähle, M. Fractional differentiation in the self-affine case. III. The density of the Cantor set. Proc. Amer. Math. Soc. 117 (1993), no. 1, 137–144.
* [P] B. Praggastis, Numeration systems and Markov partitions from self similar tilings. Trans. Amer. Math. Soc. 351 (1999), no. 8, 3315–3349.
* [RW] C. Radin, M. Wolff, _Space tilings and local isomorphism_. Geom. Dedicata 42 (1992), no. 3, 355–360.
* [R] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110(2) (1982), 147–178.
* [Ro] E. A. Robinson, Jr., Symbolic dynamics and tilings of ${{\mathbb{R}}}^{d}$. Symbolic dynamics and its applications, Proc. Sympos. Appl. Math., Vol. 60, Amer. Math. Soc., Providence, RI, 2004, pp. 81–119.
* [S] H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey. Proceedings of the symposium on operator theory (Athens, 1985). Linear Algebra Appl. 84 (1986), 161–189.
* [So1] B. Solomyak, _Dynamics of self-similar tilings_. Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695–738.
* [So2] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), 265–279.
* [Y] H. Yuasa, Invariant measures for the subshifts arising from non-primitive substitutions. J. D’Analyse Math. 102 (2007), 143–180.
|
arxiv-papers
| 2012-09-18T19:51:10 |
2024-09-04T02:49:35.280823
|
{
"license": "Public Domain",
"authors": "Konstantin Medynets and Boris Solomyak",
"submitter": "Konstantin Medynets",
"url": "https://arxiv.org/abs/1209.4073"
}
|
1209.4157
|
# AutoAmp : An Open-Source Analog Amplifier Design Tool - For Classroom and
Lab Purposes
Om Prasad Patri K Sanmukh Rao
###### Abstract
This correspondence presents an open-source tool _AutoAmp_ developed at the
Indian Institute of Technology, Guwahati. It is available at
http://sourceforge.net/projects/autoamp-iitg/ This tool helps the user to
design different types of electronic amplifiers, using solid state devices,
for a given specification. It can handle several types of designs namely
common-emitter BJT amplifier (single and two-stage), operational amplifiers
(inverting and non-inverting) and power amplifier. Not only does it design the
amplifier, it also simulates the designed amplifier using SPICE simulator and
displays the performance curves. This tool is deemed to prove invaluable in
undergraduate teaching and labs. Especially in electronics-design related
laboratories, the student need not design the amplifiers which are mostly the
heart of many electronic designs.
$\\{$o.patri, kuppannagari$\\}$ⓐiitg.ernet.in
Supervised by : Dr. Amit K. Mishra, Assistant Professor, Department of ECE,
IIT Guwahati
Department of Computer Science and Engineering,
Indian Institute of Technology Guwahati
## I Introduction
Analog amplifiers are the building blocks of many electronic circuits.
Different types of analog electronic amplifiers are commonly used in radio and
television transmitters and receivers, high fidelity (hi-fi) stereo equipment,
micro computers and other electronic equipment. Transistor amplifiers are
among the most commonly used kinds of amplifiers. Most common active devices
used in transistor based amplifiers are bipolar junction transistors (BJTs)
and metal oxide semiconductor field-effect transistors (MOSFETs), with BJTs
being a preferred choice for lab level circuit-design. BJT based amplifiers
find use in audio amplifiers in a home stereo or public address system, RF
high power generation for semiconductor equipment, RF and microwave
applications such as radio transmitters. An operational amplifier(_op-amp_) is
an amplifier circuit with very high open loop gain and differential inputs
which employs external negative feedback for control of its transfer function
and gain. These attributes form the basis for op-amp applications in
integrated circuits and its extensive study and use in experimental circuits.
We have developed a simple open-source amplifier design tool, named AutoAmp,
[1] for the following types of amplifiers, given some design specifications:
1. 1.
Single Stage BJT CE Amplifier
2. 2.
Two-Stage BJT CE Amplifier
3. 3.
Operational Amplifier: Inverting/Non-Inverting
4. 4.
Op-Amp Difference Amplifier
5. 5.
Class-A Power Amplifier
For each type of design the software requires minimum design specifications
assuming the lab working environment. The assumptions can however be modified
in the source code as per requirements. Net voltage gain is the key design
parameter that the software uses across most amplifier types. However, few
designs require more specific information like the maximum available
resistance in the case of operational amplifier design; power transmitted to
load, $V_{CC}$ and load resistance in the case of power amplifier. Given the
choice of type followed by minimum design specifications, the software
generates a netlist file to be opened in LTSpice™ [2] for necessary circuit
analysis. AutoAmp is open-source and can be run on both Windows and Linux-
based systems. The necessary adaptations for the OS are mentioned in the
AutoAmp website [1]. Autoamp is available for free (along with the source
code) at [1].
There are not many free tools which automatically design basic amplifier
circuits given the design specifications. It is expected that industries may
be maintaining customized circuit design tools to solve their purpose. However
use of such commercial tools for academic purposes is likely to be
prohibitively expensive. Basic amplifier-circuit design along with its
analysis is required in any complex circuit in electronics. Thus availability
of such a tool will be a boon for teachers and students alike.
Among the existing tools for amplifier circuit design, tools for operational
amplifiers are available, including online tools, like [3], however, there is
no open-source design tool available for designing amplifier circuits with BJT
amplifiers and class-A power amplifiers. Most importantly, none of the
available tools have a provision for the analysis of the circuit generated. We
aim to provide an interface where the user can get the design of the circuit
in LTSpice™ after providing certain design specifications. Such a tool will be
very useful in classrooms and for other non-industrial purposes where such
circuit design is warranted.
The existing design tools are pretty complicated (especially for classroom
purposes), difficult to use, expensive, not open-source (user cannot change
the source to suit his own purpose) and lack a Spice or similar interface.
Moreover, design tools for BJTs, power amplifiers are very hard to find. The
commercial tools come with a whole package of electronic design automation
tools with lot of circuit-options, which makes them complicated. For learning
or teaching a course in analog electronics, only a few numbers of these
circuits are required. Further, addition or deletion of components and
changing the source according to individual requirements can not be done.
Our design tool tries to overcome most of these problems. It is a simple and
user-freindly tool. AutoAmp is easy to operate, takes minimum input and
generates an LTSpice™ netlist which can be used to design the circuit in
LTSpice™ directly. Being open-source, customized changes can be easily made to
the source code to give the desired results; components can be easily added or
removed by writing some extra functions in the source code.
Section II describes our design approach in detail including the software
design methodology and the circuit design approach. Section III shows our
demo/experimentation results and includes screenshots from the working of the
tool. The elaborate theoretical and mathematical analysis for each type of
amplifier can be found in the Appendix -A . Section IV sums up the proposal in
the conclusion and talks about possible future work related to the tool.
## II Design approach
This section describes our design approach of the tool in detail. A blackbox
representation of the tool is given by Fig. 1.
Figure 1: Blackbox Diagram for AutoAmp
### II-A Software Design Methodology
The tool is a command line software designed in C++ programming language. The
program has a class named autoAmp which consists of various functions for
computation of the amplifiers’ components and one function for printing in the
file. A struct data type is defined to store all the computed values and is
finally used to create the output file. The user is asked for the name of the
input file, to select an amplifier of her choice in a menu based environment
and finally to enter voltage gain and other parameters based on the type of
amplifier chosen. Based on user’s choice the respective functions are called
which compute the values of components and store then into the struct defined.
Now another function uses this struct to create the netlist of the respective
type of amplifier into the file specified by the user in the beginning.
### II-B Circuit Design Approach
#### II-B1 Single Stage BJT CE Amplifier
We have a designed a small-signal voltage amplifier operating in the audio
frequency range. We have used an n-p-n transistor, namely, 2N2222. Two port
h-parameters are used for circuit analysis. Maximum, minimum, and typical
values as required, of the h-parameters are obtained from the transistor
datasheet [4]. One method for obtaining the hybrid parameters of the BJT
amplifier is given by Al-Zobi et al [5]. These values along with the other
known values are used by the software to get optimized values of the circuit
components, i.e., resistances and capacitors. The detailed theoretical and
mathematical analysis for this part can be found in the appendix -A.
#### II-B2 Two Stage BJT CE Amplifier
This design consists of two CE amplifier stages in cascade. Two amplifying
stages thus give us a higher overall gain. The design methodology remains the
same as in single stage CE amplifier but is applicable over two stages in this
case. The detailed theoretical and mathematical analysis for this part can be
found in the appendix -B.
#### II-B3 Operating Amplifiers
This is the simplest ampifier designing strategy in which we are using the
inverting and non-inverting configurations of the ideal Universal OpAmp2 (as
given in LTSpice™). The detailed analysis is given by Sedra and Smith [6].
Assuming an ideal op amp with infinite open-loop gain, $R_{in}$ = $R_{1}$. Now
to avoid the loss of signal strength, voltage amplifiers are required to have
high input resistance. In the case of the inverting op-amp confguration we are
studying, to make $R_{in}$ high we should select a high value for $R$.
However, if the required gain $\frac{R_{2}}{R_{1}}$ is also high then $R$
could become impratically large (e.g. greater than a few megaohms). Hence in
our design we use a different feedback mechanism by which the circuit is able
to realize a large voltage gain without using large resistances in the
feedback path. The details of the design can be found in the Appendix.
Standard circuit design can be used for the non-inverting input configuration
as the input resistance is infinity as desired. The detailed theoretical and
mathematical analysis for this part can be found in the appendix -C.
#### II-B4 Op-amp Difference Amplifier
In this design, we implement a difference amplifier again using Universal
OpAmp2 that responds to the difference between the two signals applied at its
input and ideally rejects signals that are common to the two inputs. The
circuit design uses four resistances. The detailed theoretical and
mathematical analysis for this part (along with the circuit diagram) can be
found in the appendix -D.
#### II-B5 Class-A Power Amplifier
Here we have designed a power amplifier by simply implementing a CE BJT
amplifier with a high power output stage for which a transformer is used. The
detailed theoretical and mathematical analysis for this part can be found in
the appendix -E.
## III Demonstration
Here are some screenshots which are obtained after the netlist file created by
AutoAmp is input to LTSpice™. We simulated the netlist files generated by
AutoAmp in LTSpice. The input-output voltage graphs for sinusoidal input
voltages are presented in the figures 2 \- 6. Figure 2 shows a voltage gain of
$20$ for an input of $20mV$ peak-to-peak in single-stage BJT amplifier. Figure
3 shows a total voltage gain of $100$ for an input of $0.5mV$ peak-to-peak in
a two-stage BJT amplifier. Figure 4 and Figure 5 show a voltage gain of $10$
for an input of $20mV$ peak-to-peak in inverting and non-inverting operational
amplifier. Figure 6 shows the graph of voltage gain for difference operational
amplifier.
Figure 2: $V_{in}$ v/s $V_{out}$ in BJT Single stage Amplifier Figure 3:
$V_{in}$ v/s $V_{out}$ in BJT two-stage Amplifier Figure 4: $V_{in}$ v/s
$V_{out}$ in Inverting Op-amp Figure 5: $V_{in}$ v/s $V_{out}$ in Non-
Inverting Op-amp Figure 6: $V_{in}$ v/s $V_{out}$ in Op-amp based Difference
Amplifier
## IV Conclusion
This paper presents a simple open-source design tool using C++ which can help
in designing and analyzing an amplifier given some design specifications. The
software is able to demonstrate five design spec-combinations. The complete
package is deemed to be of utility to circuit designers and instructors,
especially in a technical college environment.
## Acknowledgements
The authors are grateful to Dr. Amit Kumar Mishra, Assistant Professor,
Department of Electronics and Communication Engineering, IIT Guwahati for
supervising this work. Also, they thank their colleague Mr. Nitin Dua,
Department of Computer Science and Engineering, IIT Guwahati for his help with
the theoretical analysis of the various amplifier circuits.
## References
* [1] “Autoamp, a simple, open-source amplifier design tool.” [Online]. Available: http://sourceforge.net/projects/autoamp-iitg/
* [2] “Ltspice ™.” [Online]. Available: http://www.linear.com/designtools/software/ltspice.jsp
* [3] “Webench, national semiconductor.” [Online]. Available: http://www.webench.national.com
* [4] P. A. Harden, _The Handyman’s Guide to Understanding Transistor Datasheets and Specifications_.
* [5] Q. Al-Zobi and A. Al-Smadi, “A compact method for obtaining the hybrid parameters of the bjt amplifier,” in _Proceedings of the 2004 11th IEEE International Conference on Electronics, Circuits and Systems, 2004. ICECS 2004_ , 2004, pp. 399–402.
* [6] A. S. Sedra and K. C. Smith, _Microelectronic Circuits_. Oxford University Press.
* [7] _Student Resources, Sedra and Smith_.
### -A Design of single stage CE BJT Amplifier
We show the design of the common-emitter single stage BJT audio frequency
amplifier. $A_{V}$ and $V_{0}$ is input by the user. We use voltage divider
circuit as it provides Q point independent of temperature and beta. We use
H-parameters of the CE BJT to analyse the circuit. The H parameters are
$h_{fe}$, $h_{oe}$, $h_{ie}$ and $h_{re}$, where $h_{fe}$ is the current gain,
$h_{ie}$ is the input impedance and $h_{oe}$ is the output admittance.
Figure 7: Single stage BJT CE amplifier
_Selection of $R_{C}$_
$R_{L^{\prime}}=R_{C}\parallel R_{L}$ (1)
If $R_{L}$ not given, $R_{L^{\prime}}$ = $R_{C}$
$R_{L}$ = load resistance connected between $V_{0}$ and ground.
$A_{V}=\frac{h_{fe}\ast R_{L^{\prime}}}{h_{ie}+(h\ast R_{C})}$ (2)
Where
$h=(h_{ie}\ast h_{oe})-(h_{fe}\ast h_{re})\\\ $ (3)
We got $h_{fe}$, $h_{re}$, $h_{ie}$, $h_{oe}$ from data sheet [4].
$h_{fe}=\beta,~{}h_{fe,max}=\beta_{max},~{}h_{fe,min}=\beta_{min}$ (4)
From the above equations we calculated $R_{L^{\prime}}$ and $R_{C}$. Select
higher std value for $R_{C}$ to increase voltage gain if min voltage gain is
specified or nothing is specified. If max voltage gain is specified use lower
std value. If some specific voltage gain is specified, use nearest std val.
_Selection of operating point ( $V_{CEQ}$, $I_{CQ}$)_
If $V_{CC}$ is given,
$V_{CEQ}=\frac{V_{CC}}{2}$ (5)
If VCC is not given,
$V_{CEQ}=1.5~{}\ast~{}(V_{0,peak}+V_{CE,sat})$ (6)
Then,
$I_{C,peak}=\frac{V_{0,peak}}{R_{L^{\prime}}}$ (7)
$I_{CQ}=I_{C,peak}+I_{C,min}$ (8)
Assume $I_{C,min}$ = $0$ or $0.005mA$
_Selection of $R_{e}$_
If $V_{CC}$ is not given, assume
$V_{re}=1$
If $V_{CC}$ is given,
$V_{re}=10\%~{}of~{}V_{CC}$
For either case,
$R_{e}=\frac{V_{re}}{I_{CQ}}$ (9)
Select lower std value of $R_{e}$ so that voltage drop across $R_{e}$ is less
which increases the voltage swing of o/p.
_Selection of $V_{CC}$_
If $V_{CC}$ is not given,
$V_{CC}=V_{CEQ}+I_{CQ}\ast(R_{C}+R_{e})$ (10)
Assume higher std val (typically 9,12,15,18).
_Selection of $R_{1}$ and $R_{2}$_
If stability factor is not given, assume $s$ = 8
$s=\frac{1+h_{fe,max}}{1+\frac{h_{fe,max}~{}\ast~{}R_{e}}{R_{b}+R_{e}}}$ (11)
$R_{b}$ is found and it is not standardized.
$V_{r2}=V_{be}+V_{re}\qquad V_{r1}=V_{CC}-V_{r2}$ (12)
Assume $V_{be}$ = 0.7 V (for Si)
$\frac{R_{1}}{R_{2}}=\frac{V_{r1}}{V_{r2}}$ (13)
We get $R_{1}$ in terms of $R_{2}$. Substitute in $R_{b}$.
$R_{b}=R_{1}\parallel R_{2}=\frac{R_{1}\ast R_{2}}{R_{1}+R_{2}}$ (14)
$R_{2}$ is found. Select lower standard value to make circuit independent of
$\beta$. Substitute in (13) to find $R_{1}$. We should select higher standard
value so that circuit draws minimum current from supply.
_Selection of coupling capacitors_
Select higher standard value for all capacitors.
Selection of $C_{E}$
$X_{CE}=\frac{R_{e}}{10}$ (15) $C_{E}=\frac{1}{2~{}\ast\pi~{}\ast f_{L}~{}\ast
X_{CE}}$ (16)
where $f_{L}$ = lower cutoff frequency. Assume $f_{L}$ = 20 Hz.
Selection of $C_{B}$
If $R_{S}$ (Source resistance) is not specified, assume $R_{S}=0$
$X_{CB}=R_{S}+R_{b}\parallel h_{ie}$ (17) $C_{B}=\frac{1}{2~{}\ast\pi~{}\ast
f_{L}~{}\ast X_{CB}}$ (18)
Selection of $C_{C}$
$R_{b}=R_{1}\parallel R_{2}$ (19) $X_{CC}=R_{C}+R_{L}$ (20)
If $R_{L}$ (load resistance) is not specified, then we assume amplifier is
connected to a similar next stage.
Hence,
$R_{L}=R_{b}\parallel h_{ie}$ (21)
$C_{C}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}\ast~{}X_{CC}}$ (22)
### -B Design of two-stage BJT CE Amplifier
Value of $A_{V}$ is input by the user.
Overall voltage gain,
$A_{V}=A_{v1}\ast A_{v2}$ (23) $\frac{A_{v1}}{A_{v2}}=\frac{R_{C1}}{R_{C2}}$
(Assumed)
$A_{v1}$ and $A_{v2}$ can be found from the above equations.
Figure 8: Two-stage BJT CE amplifier
#### -B1 Part 1: Design of second stage
_Calculation of $R_{L}$:_
$R_{L}=\frac{V_{0,peak}}{I_{0,peak}}$ (24)
Not necessary if $R_{L}$ is not given or $V_{0,peak},~{}I_{0,peak}$ is not
given.
_Selection of $R_{C2}$:_
$A_{v2}=\frac{h_{fe}\ast R_{L2^{\prime}}}{h_{ie}}$ (25)
$R_{L^{\prime}}=R_{C2}\parallel R_{L}$ (26)
or, if $R_{L}$ is not given, then,
$R_{L^{\prime}}=R_{C2}$ (27)
Hence, we calculate $R_{L}^{\prime}$ and $R_{C2}$
_Selection of $V_{CEQ}$:_
Case 1: $V_{CC}$ is given
$V_{CEQ,2}=1.5~{}\ast~{}(V_{0,peak}+V_{CE,saturation})$ (28)
If $V_{0}$ is not given, then
$V_{CEQ}=\frac{V_{CC}}{2}$ (29)
$V_{re2}=10~{}\text{to}~{}20~{}\%~{}\text{of}V_{CC}$ (30)
$V_{rc2}=V_{CC}-V_{CEQ,2}-V_{re2}$ (31) $I_{CQ,2}=\frac{V_{rc2}}{R_{C2}}$ (32)
$R_{e2}=\frac{V_{re2}}{I_{CQ,2}}$ (33)
Select lower standard value so that drop across $R_{e}$ is less which
increases gain of the output.
Case 2: $V_{C}C$ is not given
$V_{CEQ,2}=1.5~{}\ast~{}(V_{0,peak}+V_{CE,saturation})$ (34)
$I_{C2,peak}=\frac{V_{0,peak}}{R_{L2}}$ (35)
Assume $V_{re2}=2$ V
$V_{CC}=V_{CEQ}+I_{CQ}~{}\ast~{}(R_{C2}+R_{e2})$ (36)
Select higher std value.
_Selection of $R_{3}$, $R_{4}$_
If stability factor is not given, assume $s$ = 8
$s=\frac{h_{fe}+1}{1+\frac{h_{fe}~{}\ast~{}R_{e2}}{R_{b2}+R_{e2}}}$ (37)
$R_{b}$ is found.
$V_{r4}=V_{be}+V_{re}$ (38) $V_{r3}=V_{CC}-V_{r2}$ (39)
Assume $V_{be}$ = 0.7V (for Si)
$\frac{R_{3}}{R_{4}}=\frac{V_{r3}}{V_{r4}}$ (40)
Get $R_{3}$ in terms of $R_{4}$ and substitute in $R_{b2}$
$R_{b2}=R_{3}\parallel R_{4}=\frac{R_{3}~{}\ast~{}R_{4}}{R_{3}+R_{4}}$ (41)
Find $R_{4}$. Select lower standard value to make circuit independent of beta.
Substitute in (40) to find $R_{3}$. Select higher standard value so that
circuit draws minimum current from supply.
#### -B2 Part 2: Design of first stage
$A_{v1}=\frac{A_{V}}{A_{v2}}$ (42)
_Selection of $R_{C1}$_
$A_{v1}=\frac{h_{fe}~{}\ast~{}R_{L1^{\prime}}}{h_{ie}}$ (43)
$R_{L^{\prime}}=R_{C1}\parallel R_{b2}\parallel h_{ie}$ (44)
$R_{L^{\prime}}$ and $R_{C1}$ is calculated.
Let
$V_{CEQ,1}=V_{CEQ,2},~{}V_{rc1}=V_{rc2},~{}V_{re1}=V_{re2},~{}I_{CQ,1}=\frac{V_{rc1}}{R_{C1}}~{}and~{}R_{e1}=\frac{V_{re1}}{I_{CQ,1}}$
_Selection of $R_{1}$, $R_{2}$_
$s=\frac{1+h_{fe,max}}{1+\frac{h_{fe,max}~{}\ast~{}R_{e}}{R_{b}+R_{e}}}$ (45)
$R_{b}$ is found.
$V_{r2}=V_{be}+V_{re}$ (46) $V_{r1}=V_{CC}-V_{r2}$ (47)
Assume $V_{be}$ = 0.7V
$\frac{R_{1}}{R_{2}}=\frac{V_{r1}}{V_{r2}}$ (48)
We get $R_{1}$ in terms of $R_{2}$ and substitute in $R_{b}$.
$R_{b}=R_{1}\parallel R_{2}=\frac{R_{1}~{}\ast~{}R_{2}}{R_{1}+R_{2}}$ (49)
Find $R_{2}$. We select lower standard value to make circuit independent of
$\beta$. Substitute in (48) to find $R_{1}$. Select higher standard value so
that circuit draws minimum current from supply.
_Selection of coupling capacitors:_ Select higher standard value for all
capacitors.
Selection of $C_{E1}$:
$X_{CE1}=\frac{R_{e1}}{10}$ (50)
$C_{E1}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}~{}\ast~{}X_{CE1}}$ (51)
$f_{L}$ = lower cutoff frequency. We assume $f_{L}$ = 20 Hz.
Selection of $C_{E2}$:
$X_{CE2}=\frac{R_{e2}}{10}$ (52)
$C_{E2}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}~{}\ast~{}X_{CE2}}$ (53)
Selection of $C_{B1}$:
$R_{b}=R_{1}\parallel R_{2}$ (54)
If $R_{S}$ (Source resistance) is not specified, assume $R_{S}$ = 0
$X_{CB1}=R_{b}\parallel h_{ie}$ (55)
$C_{B1}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}~{}\ast~{}X_{CB}}$ (56)
Selection of $C_{B2}$:
$R_{b2}=R_{3}\parallel R_{4}$ (57) $X_{CB2}=R_{C1}+R_{b}\parallel h_{ie}$ (58)
$C_{B2}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}~{}\ast~{}X_{CB}}$ (59)
Selection of $C_{0}$:
$R_{b2}=R_{3}\parallel R_{4}$ (60) $X_{CC}=R_{C}+R_{L}$ (61)
If $R_{L}$ (load resistance) is not specified, then assume amplifier is
connected to a similar next stage. Hence,
$R_{L}=R_{b1}\parallel h_{ie}$ (62)
$C_{C}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}~{}\ast~{}X_{CC}}$ (63)
### -C Design of Operational Amplifiers based Amplifiers
Depending upon whether the value of $A_{V}$ input by the user is positive or
negative, the circuit is interpreted to be of the configuration non-inverting
or inverting, respectively.
#### -C1 Non-Inverting Amplifier
Figure 9: Non-Inverting Operational Amplifier [7]
The user inputs the gain $A_{V}$ and the resistances $R_{1}$ and $R_{2}$ are
found by the following formula :
$A_{V}=1+\frac{R_{2}}{R_{1}}$ (64)
#### -C2 Inverting Amplifier
Figure 10: Inverting Operational Amplifier [7]
At the inverting terminal of the op-amp, the voltage is
$V_{1}=-\frac{v_{0}}{A}=-\frac{v_{0}}{\infty}=0$ (65)
Here we have assumed that the circuit is producing a finite output voltage
$v_{0}$.
$i_{1}=\frac{v_{I}-v_{1}}{R_{1}}=\frac{v_{I}-0}{R_{1}}=\frac{v_{I}}{R_{1}}$
(66)
Since zero current flows into the inverting input terminal, all of $i_{1}$
will flow through $R_{2}$, and thus :
$i_{2}=i_{1}=\frac{v_{in}}{R_{1}}$ (67)
Voltage at $x$ :
$V_{x}=v_{1}-i_{2}~{}\ast~{}R_{2}=0-\frac{v_{I}}{R_{1}}~{}\ast~{}R_{2}=-\frac{R_{2}}{R_{1}}~{}\ast~{}v_{I}$
(68)
This in turn enables us to find $i_{3}$ :
$i_{3}=\frac{0-V_{x}}{R_{3}}=\frac{R_{2}}{R_{1}~{}\ast~{}R_{3}}~{}\ast~{}v_{I}$
(69)
Next, a node equation at $x$ yields $i_{4}$ :
$i_{4}=i_{2}+i_{3}=\frac{v_{I}}{R_{1}}+\frac{R_{2}}{R_{1}~{}\ast~{}R_{3}}~{}\ast~{}v_{I}$
(70)
Finally, we can determine $v_{0}$ from
$v_{0}=V_{x}-i_{4}~{}\ast~{}R_{4}=(-\frac{R_{2}}{R_{1}})~{}\ast~{}v_{I}-(\frac{v_{I}}{R_{1}}+\frac{R_{2}}{R_{1}~{}\ast~{}R_{3}}~{}\ast~{}v_{I})~{}\ast~{}R_{4}$
(71)
Thus, the voltage gain is given by :
$\frac{v_{0}}{v_{I}}=\frac{R_{2}}{R_{1}}+\frac{R_{4}}{R_{1}}~{}\ast~{}(1+\frac{R_{2}}{R_{3}})$
(72)
which can be written in the form
$\frac{v_{0}}{v_{I}}=-\frac{R_{2}}{R_{1}}~{}\ast~{}(1+\frac{R_{4}}{R_{2}}+\frac{R_{4}}{R_{3}})$
(73)
### -D Design of Op-Amp based Difference Amplifier
User inputs $A_{d}$.
In ideal case,
$A_{CM}=0\qquad\text{and}\qquad A_{d}=\frac{R_{2}}{R_{1}}$ (74)
### -E Design of Power Amplifier Class A
Figure 11: Power Amplifier Class A
_Selection of operating point :_
$V_{re}=\frac{V_{C}C}{10}$ (75) $V_{CEQ}=V_{CC}-V_{re}$ (76)
$V_{CE,peak}=V_{CEQ}-V_{CE,sat}$ (77)
$I_{C,peak}=\frac{2~{}\ast~{}P_{L^{\prime}}}{V_{CE,peak}}$ (78)
$I_{CQ}=I_{C,peak}+I_{C,min}$ (79)
Assume $I_{C,min}$ = 0.
Hence, we calculate $I_{CQ}$
_Selection of $R_{e}$ and $C_{e}$ :_
$R_{e}=\frac{V_{re}}{I_{CQ}}$ (80) $P_{re}=\frac{V_{re}^{2}}{R_{e}}$ (81)
$C_{e}=\frac{1}{2~{}\ast~{}\pi~{}\ast~{}f_{L}~{}\ast~{}R_{L}}$ (82)
We assume $f_{L}$ = 50 Hz. Since $C_{e}$ is very high, we leave $R_{e}$
unbypassed.
_Selection of $R_{1}$ and $R_{2}$ :_
Assume $s$ = 10
$s=\frac{1+h_{fe,max}}{1+\frac{h_{fe,max}~{}\ast~{}R_{e}}{R_{b}+R_{e}}}$ (83)
$R_{b}$ is found.
$V_{r2}=V_{be}+I_{CQ}\ast R_{e}$ (84) $V_{r1}=V_{CC}-V_{r2}$ (85)
Assume $V_{be}$ = 0.7V
$\frac{R_{1}}{R_{2}}=\frac{V_{r1}}{V_{r2}}$ (86)
We get $R_{1}$ in terms of $R_{2}$ and then substitute in $R_{b}$
$R_{b}=R_{1}\parallel R_{2}=\frac{R_{1}~{}\ast~{}R_{2}}{R_{1}+R_{2}}$ (87)
$R_{2}$ is found. Select lower standard value to make circuit independent of
$\beta$. Then it is substituted in (86) to find $R_{1}$. Select higher
standard value so that circuit draws minimum current from supply.
_Selection of output transformer :_
$R_{L^{\prime}}=\frac{V_{CE,peak}}{I_{C,peak}}$ (88)
$R_{L^{\prime}}$ is calculated.
$R_{L^{\prime}}=\frac{N_{1}^{2}}{N_{2}^{2}}~{}\ast~{}R_{L}$ (89)
$\frac{N_{1}}{N_{2}}$ is calculated.
|
arxiv-papers
| 2012-09-19T06:14:28 |
2024-09-04T02:49:35.292976
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Om Prasad Patri and K. Sanmukh Rao",
"submitter": "Om Patri",
"url": "https://arxiv.org/abs/1209.4157"
}
|
1209.4226
|
# Thulium and ytterbium-doped titanium oxide thin films deposited by
ultrasonic spray pyrolysis
S. Forissier H. Roussel P. Chaudouet Laboratoire des Matériaux et du Génie
Physique, CNRS, Grenoble Institute of Technology, MINATEC, 3 parvis Louis
Néel, 38016 Grenoble, France A. Pereira Laboratoire de Physico-Chimie des
Matériaux Luminescents UMR 5620 CNRS / UCBL Domaine Scientifique de la Doua,
Université Claude Bernard Lyon 1 Bâtiment Alfred Kastler 10 rue Ada Byron
69622 Villeurbanne cedex, France J-L. Deschanvres jean-
[email protected] Laboratoire des Matériaux et du Génie
Physique, CNRS, Grenoble Institute of Technology, MINATEC, 3 parvis Louis
Néel, 38016 Grenoble, France B. Moine Laboratoire de Physico-Chimie des
Matériaux Luminescents UMR 5620 CNRS / UCBL Domaine Scientifique de la Doua,
Université Claude Bernard Lyon 1 Bâtiment Alfred Kastler 10 rue Ada Byron
69622 Villeurbanne cedex, France
###### Abstract
Thin films of thulium and ytterbium-doped titanium oxide were grown by metal-
organic spray pyrolysis deposition from titanium(IV)oxide
bis(acetylacetonate), thulium(III)
tris(2,2,6,6-tetramethyl-3,5-heptanedionate) and ytterbium(III)
tris(acetylacetonate). Deposition temperatures have been investigated from
300°C to 600°C. Films have been studied regarding their crystallity and doping
quality. Structural and composition characterisations of TiO2:Tm,Yb were
performed by electron microprobe, X-ray diffraction and Fourier transform
infrared spectroscopy. The deposition rate can reach 0.8 $\upmu$m/h. The
anatase phase of TiO2 was obtained after synthesis at 400°C or higher. Organic
contamination at low deposition temperature is eliminated by annealing
treatments.
CVD, thulium, ytterbium, titanium oxide, thin film
## Introduction
Over the last decades, titanium dioxide has been attracting great interest due
to their relevance for a variety of applications. For example, TiO2 has been
investigated for use in photocatalytic processDiebold2003 ; Carp2004 ;
Foster2011 , in protective coatingsPoulios1999 and gas sensing
applicationsSavage2001 . TiO2 is also interesting for photovoltaic
applicationsRichards2004 . TiO2 thin films have been the most commonly used
antireflective (AR) coating in the photovoltaic industry. Owing to its
excellent optical properties, mechanical properties and good chemical
resistance, TiO2 still remains attractive for such application.
The physical and chemical properties of oxide matrix can be tuned by doping
with rare-earth (RE) elements. For example, the incorporation of RE ions into
oxide matrix received great attention for applications in photovoltaic
devicesZhang2010 ; Ende2009 ; Guille2012 . Indeed, one solution to reduce the
energy losses in the UV region due to thermalization of the charge carriers is
to adapt the solar spectrum to better match the bandgap of solar cells. This
approach, which involves energy transfer between rare-earth ions, is well
known for lighting issuesZhang2010 ; Beauzamy2008 and has been recently
proposed for photovoltaic applicationsRichards2006a . Depending on both the
host matrix and the RE ions used, down-shifting or quantum-cutting can be
observed. In this context, RE-doped titanium dioxide can be a potential low-
cost material for the down-conversion. The use of titanium oxide being well-
established in the photovoltaics industry, added functionality with rare-earth
could be easily integrated. However, depending on the size of the RE ions such
doping can be challenging. It is thus expected that the choice of the RE ions,
the method used to synthesize the RE-doped TiO2 and the experimental
parameters prevailing during the synthesis will have a strong influence on the
final material.
Most studies concerning rare-earth doped titanium oxide have been performed on
thin films, which have been fabricated by sol-gelFrindell2003 ; Hu2007 ;
Saif2007 ; Jia2006 ; Amlouk2008 and sputtering processesProciow2007 . In most
of these case, the studies were mainly concerned with samariumFrindell2003 ;
Hu2007 ; Saif2007 , europiumFrindell2003 ; Saif2007 ; Prociow2007 ,
terbiumFrindell2003 ; Saif2007 ; Jia2006 , erbiumFrindell2003 ,
neodymiumFrindell2003 , praseodymiumAmlouk2008 and ytterbiumFrindell2003 .
All of these can be used as photocatalysts or optical layers for light
applications. Up to now, few reports have appeared in the literature
concerning spray-pyrolysis synthesis of RE-doped titanium dioxide thin
filmsKanarjov2008 . This technique is well known as a powerful method for the
synthesis of all kind of materials (e.g. fluorides, oxides, metals) at
atmospheric pressure. Moreover, it allows excellent control of the elemental
composition of the resultant films through the addition of dopant elements in
the desired ratio to the solution medium. Thin films with large area can thus
be obtained with a simple technique at relatively low cost and easily scalable
(e.g. roll to roll processes). Previous studies have demonstrated the
possibility of titanium oxide synthesis by this technologyCastaneda2003 ;
Conde-Gallardo2005 ; Duminica2007 ; Senthilnathan2010 . For example, Castaneda
et al. and Conde-Gallardo et al. have shown by X-ray diffraction that
crystallization in anatase phase occurs over 300-400°C, Castaneda2003 ; Conde-
Gallardo2005 whereas Duminica et al. have reported a fraction of rutile phase
at high temperature Duminica2007 .
In this study, we investigated the structural and spectroscopic properties of
thulium and ytterbium-co-doped titanium dioxide films prepared by spray
pyrolysis. The influence of different deposition parameters on the doping
level and the crystalline structure of the films were investigated. Energy
transfer mechanisms in the co-doped TiO2 films are also discussed.
## I Experimental
Thulium and ytterbium-doped titanium dioxide thin films were deposited by mean
of ultrasonic spray pyrolysis methodMooney1982 . The precursor solution, was
composed of titanium(IV) oxide bis(acetylacetonate), TiO(acac)2, thulium(III)
tris(2,2,6,6-tetramethyl-3,5-heptanedionate) and ytterbium(III)(acac)
dissolved in high purity butanol (99%) at a total concentration of 0.03 mol/l.
All these precursors were purchased from STREM Chemicals Inc (Bischheim,
France) and butanol was purchased from Alfa Aesar GmbH (Schiltigheim, France).
Precursors were selected for non-toxicity, good stability at room temperature,
easy handling, high volatility and low cost Ryabova1968 . The films were
deposited on (100) silicon substrate. A similar ultrasonic spray pyrolysis
apparatus used in this study is presented in Deschanvres1990 ; Deschanvres1993
. However this study’s apparatus has a vertical sampler holder instead of a
horizontal one.
The aerosol was produced by means of a flat piezoelectric transducer excited
at 800 kHz which generated an ultrasonic beam in a solution containing the
reactant of the material to be deposited. The source solution is delivered to
the piezoelectric transducer through a constant level burette to ensure a
constant rate of vapour formation during deposition. This ultrasonic spraying
system guarantees a narrow dispersion of the droplet size, and based on the
experimental formula by LangLang1962 the size of the droplets is 3.6
$\upmu$m.
Droplets are carried with two air fluxes, dried and purified, to the heated
sample holder, the lower air flux (12.7 l/m) is the main carrier gas while the
upper air flux (10.1 l/m) lengthens the time that the vapour stay in the
vicinity of the sample holder. The overall air flow is parallel to the
substrate’s surface. For deposition, the substrate was fixed by clips on the
sample holder heated by an electrical resistance with a built-in thermocouple.
Due to this narrow distribution, in an appropriate temperature range the
pyrolysis of the aerosol correspond to CVD process as described by Spitz and
ViguiéViguie1975 .
The X-ray diffraction profiles were obtained with a Bruker D8 Advance using Cu
$K\alpha_{1}$ radiation at 0.15406 nm with an applied voltage of 40 kV and 40
mA anode current in $\theta/2\theta$ configuration. Absorption FT-IR
spectroscopy was used to study the structural evolution of the films versus
the deposition conditions. Spectra were obtained between 250 and 4000 cm-1
with 4 cm-1 resolution with a Bio-Rad Infrared Fourier Transform spectrometer
FTS165 and after performing Si substrate subtraction. SEM pictures were taken
with a FEI Quanta FEG 250 microscope using an Everhart-Thornley detector in
the secondary electron mode. The AFM pictures were taken with a Veeco
Dimension 3100 with a Quadrex equipment. The composition of the doped films
were measured by electron probe microanalysis (EPMA Cameca SX50) and computed
by help of special software dedicated to the thin film analysis, called
Stratagem and edited by the SAMx society Pouchou1984 . Luminescence spectra
were taken using a F900 spectrofluorimeter Edinburgh with a high spectral
resolution. A Xenon Arc lamp (450 W) is used for the excitation, the detector
is a photomultiplier Hamamatsu R2658P cooled by Peltier effect.
## II Results and discussion
In a first step we compared the deposition temperatures’ effect on the
samples’ composition and morphology; then their structural properties are
analysed. The last step will expose some of the spectroscopic properties of
the thin films.
### II.1 Morphology and composition
Figure 1: Deposition rate versus temperature.
First we studied the deposition rate of the thin film measuring their
thickness by SEM cross-section. We observe a increase of this rate until 500°C
with a maximum value of 760 nm/h (Figure1). Above 500°C the decrease of the
deposition rate correspond to the depletion regime where the precursor vapour
reacts before reaching the substrate and produced non-adherent thin films.
Then we synthesized different samples at different temperature from the same
solution with a rare-earth cationic concentration of 3% of Tm and 3% Yb in the
solution ($\frac{RE}{\Sigma RE+Ti}*100$); rare-earth concentration in the film
measured by electron microprobe, measurement of this batch are shown on Figure
2a. As we noticed that the rare-earth concentration in the films is always
lower than 3%, the rare-earth precursor reactivity is shown to be less than
that of the titanium oxide precursor. The Yb precursor reactivity is higher
that the one of the Tm precursor and the highest doping efficiency from the
solution to the film is obtained at 400°C.
Subsequent experiments were mainly conducted at this temperature and even in
this condition the rare-earth precursor reactivity is lower than the titanium
oxide precursor reactivity. Due to the reactor’s geometry, the thickness of
the sample are not homogeneous on the whole surface. Nevertheless the doping
level uniformity is quite good on the whole surface as shown on Figure 2b
(drawing).
(a)
(b)
Figure 2: 2a) Rare-earth cationic concentration ($\frac{RE}{\Sigma RE+Ti}$) in
the film versus deposition temperature from an original solution which
contained a concentration of 3% of Tm and 3% of Yb. 2b) Rare-earth cationic
concentration and thickness (measured by EPMA and analysed by Strata) of the
films deposited at 400°C at different points on the substrate surface
(drawing). Initial solution : 5% Tm and 6% Yb.
The as-deposited and even annealed sample exhibit a smooth surface as shown by
AFM picture (Figure 3a) with a measured RMS roughness of less than 4 nm for a
film thickness of 420 nm (measured by EPMA). The SEM cross-section picture
shows a good density in the thin films and a thickness of the same order.
(a) AFM surface view.
(b) SEM cross-section view.
Figure 3: Morphological pictures of a annealed sample at 800°C.
FT-IR measurements (Figure 4) showed that as-deposited thin films still have
some organic residues with typical hydroxyl band (3000–3500 cm-1) along with
C-O and C-C bonds signatures (1700–1300 cm-1). Those organic residues of the
precursors are eliminated by post-annealing treatment (800°C for 1h).
Figure 4: Infrared spectra of as-deposited and annealed sample. Organic
residues are disappearing with the annealing.
### II.2 Structural properties
(a) As-deposited samples at different synthesis temperatures.
(b) As-deposited and annealed samples for 1h at 500°C or 800°C.
Figure 5: X-ray diffraction spectra.
XRD spectroscopy showed that samples are amorphous when deposited below 400°C
(Figure 5a). From 400°C to 600°C, we observe the anatase phase of titanium
oxide even for total rare-earth concentration up to 8%. On the basis of the
literature, we assume that the RE ions enter the anatase structure as
substitutional defects with respect to Ti (Ghigna2007 ). We tried different
annealing conditions, first 500°C for 1h then 800°C for 1h in air; each
annealing improves the crystalline quality of the films as shown on Figure 5b.
With increasing annealing temperature the anatase peaks exhibit higher
intensity and smaller width. After annealing at 800°C RE-doped samples are
still crystallised in the anatase phase contrary to undoped samples which are
crystallized in the rutile phase. The presence of rare-earth dopants prevents
the phase transition as reported in Graf2007 .
### II.3 Luminescence properties
(a) Emission spectra, both ions are luminescing.
(b) Excitation spectra, the thin film is absorbing light through the matrix.
Figure 6: Luminescence spectra of a Tm and Yb-doped TiO2, respectively 0,37%
and 0,85% measured by EPMA.
Several RE couples have been studied for down-conversion, most of them are
doped with ytterbium as emitter and with another RE as absorber such as
praseodymiumChen2008 , terbiumTing2003 ; Das2011 or europiumStrek2000 . The
thulium and ytterbium couple luminescence in oxide matrix is also known in the
literature, Dominiak-Dzik et al have shown energy transfer between the RE
ionsDominiak-Dzik2008 .
The emission scans (Figure 6a) were recorded exciting at 330 nm. At this
wavelength the excitation is near the ${}^{1}D_{2}$ level of thulium and
absorbed by the TiO2 matrix. On mono-doped samples (not figured) we see the
respective ion transition : the Tm ${}^{3}F_{4}-^{3}H_{6}$ luminescence around
800 nm and the Yb ${}^{2}F_{5/2}-^{2}F_{7/2}$ luminescence at 980 nm. On Tm,Yb
co-doped samples (Figure 6a) both transitions are recorded. The exciting
energy appears to be transferred to both RE ions through the TiO2 matrix
leading to a down-conversion mechanism with thulium and ytterbium. The Tm
luminescence is well inside the absorption range and the Yb luminescence is
right before the band gap of the silicon solar cells.
We recorded excitation scans for both above-mentioned Tm and Yb transitions
(Figure 6b). Their luminescence is triggered through near-UV absorption
between 300 and 350 nm. This corroborates the idea of the energy transfer
between the matrix and the rare-earth ions.
## Conclusion
In summary, we succeeded in depositing smooth and uniform film of doped
titanium dioxide with thulium and ytterbium (up to 8%) by the ultrasonic spray
pyrolysis method and to grow them partially crystallized mainly in the anatase
phase. The crystallization is improved after proper annealing.
## Acknowledgement
The authors would like to thank Mr. Olivier Raccurt from the CEA-Liten
Grenoble for providing access to the spectrophotometer.
Funding for this project was provided by a grant from the French Research
National Agency (ANR) through Habitat intelligent et solaire photovoltaïque
program (project MULTIPHOT n° ANR-09-HABISOL-009), the CARNOT institute
Energie du futur and la Région Rhône-Alpes. Sébastien Forissier held a
doctoral fellowship from la Région Rhône-Alpes.
## References
* [1] Ulrike Diebold. The surface science of titanium dioxide. Surf. Sci. Rep., 48(5–8):53 – 229, 2003.
* [2] O. Carp, C.L. Huisman, and A. Reller. Photoinduced reactivity of titanium dioxide. Prog. Solid State Chem., 32(1–2):33 – 177, 2004.
* [3] Howard Foster, Iram Ditta, Sajnu Varghese, and Alex Steele. Photocatalytic disinfection using titanium dioxide: spectrum and mechanism of antimicrobial activity. Appl. Microbiol. Biotechnol., 90:1847–1868, 2011. 10.1007/s00253-011-3213-7.
* [4] I. Poulios, P. Spathis, A. Grigoriadou, K. Delidou, and P. Tsoumparis. Protection of marbles against corrosion and microbial corrosion with TiO2 coatings. Journal of Environmental Science and Health, Part A, 34(7):1455–1471, 1999.
* [5] Nancy O. Savage, Sheikh A. Akbar, and Prabir K. Dutta. Titanium dioxide based high temperature carbon monoxide selective sensor. Sensors and Actuators B: Chemical, 72(3):239 – 248, 2001.
* [6] BS Richards. Comparison of TiO2 and other dielectric coatings for buried-contact solar cells: a review. Progress in photovoltaics, 12(4):253–281, JUN 2004.
* [7] Q.Y. Zhang and X.Y. Huang. Recent progress in quantum cutting phosphors. Progress in Materials Science, 55(5):353 – 427, 2010.
* [8] Bryan M. van der Ende, Linda Aarts, and Andries Meijerink. Lanthanide ions as spectral converters for solar cells. Phys. Chem. Chem. Phys., 11(47):11081–11095, 2009.
* [9] A. Guille, A. Pereira, G. Breton, A. Bensalah-Ledoux, and B. Moine. Energy transfer in CaYAlO4: Ce3+ Pr3+ for sensitization of quantum cutting with the Pr3+–Yb3+ couple. J. Appl. Phys., 2012. AIP ID: 074204JAP (posted 22 September 2011, in press).
* [10] Léna Beauzamy, Bernard Moine, Richard S. Meltzer, Yi Zhou, Patrick Gredin, Anis Jouini, and Kyoung Jin Kim. Quantum cutting effect in $ky_{3}f_{10}:tm^{3+}$. Phys. Rev. B, 78:184302, Nov 2008.
* [11] B.S. Richards. Enhancing the performance of silicon solar cells via the application of passive luminescence conversion layers. Solar Energy Materials and Solar Cells, 90(15):2329 – 2337, 2006\.
* [12] KL Frindell, MH Bartl, MR Robinson, GC Bazan, A Popitsch, and GD Stucky. Visible and near-ir luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls. J. Solid State Chem., 172(1):81–88, APR 2003.
* [13] Lanying Hu, Hongwei Song, Guohui Pan, Bin Yan, Ruifei Qin, Qilin Dai, Libo Fan, Suwen Li, and Xue Bai. Photoluminescence properties of samarium-doped TiO2 semiconductor nanocrystalline powders. J. Lumin., 127(2):371 – 376, 2007.
* [14] Mona Saif and M. S. A. Abdel-Mottaleb. Titanium dioxide nanomaterial doped with trivalent lanthanide ions of Tb, Eu and Sm: Preparation, characterization and potential applications. Inorg. Chim. Acta, 360(9):2863–2874, JUN 10 2007.
* [15] CW Jia, EQ Xie, AH Peng, R Jiang, F Ye, HF Lin, and T Xu. Photoluminescence and energy transfer of terbium doped titania film. Thin Solid Films, 496(2):555–559, FEB 21 2006.
* [16] A. Amlouk, L. El Mir, S. Kraiem, M. Saadoun, S. Alaya, and A. C. Pierre. Luminescence of TiO2 : Pr nanoparticles incorporated in silica aerogel. Materials science and engineering b-solid state materials for advanced technology, 146(1-3):74–79, JAN 15 2008.
* [17] E. L. Prociow, J. Domaradzki, A. Podhorodecki, A. Borkowska, D. Kaczmarek, and J. Misiewicz. Photoluminescence of eu-doped TiO2 thin films prepared by low pressure hot target magnetron sputtering. Thin Solid Films, 515(16):6344–6346, JUN 4 2007.
* [18] P. Kanarjov, V. Reedo, I. Oja Acik, L. Matisen, A. Vorobjov, V. Kiisk, M. Krunks, and I. Sildos. Luminescent materials based on thin metal oxide films doped with rare earth ions. Phys. Solid State, 50(9):1727–1730, SEP 2008.
* [19] L. Castañeda, J. C. Alonso, A. Ortiz, E. Andrade, J. M. Saniger, and J. G. Bañuelos. Spray pyrolysis deposition and characterization of titanium oxide thin films. Mater. Chem. Phys., 77(3):938 – 944, 2003.
* [20] A. Conde-Gallardo, M. Guerrero, N. Castillo, A.B. Soto, R. Fragoso, and J.G. Cabañas-Moreno. TiO2 anatase thin films deposited by spray pyrolysis of an aerosol of titanium diisopropoxide. Thin Solid Films, 473(1):68 – 73, 2005.
* [21] F.-D. Duminica, F. Maury, and R. Hausbrand. Growth of TiO2 thin films by ap-mocvd on stainless steel substrates for photocatalytic applications. Surface & coatings technology, 201(22-23):9304–9308, SEP 25 2007\.
* [22] V. Senthilnathan and S. Ganesan. Novel spray pyrolysis for dye-sensitized solar cell. Journal of Renewable and Sustainable Energy, 2(6):063102, 2010.
* [23] J B Mooney and S B Radding. Spray pyrolysis processing. Annual Review of Materials Science, 12(1):81–101, 1982.
* [24] L.A. Ryabova and Y. S. Savitskaya. The preparation of thin films of some oxides by the pyrolysis method. Thin Solid Films, 2:141, 1968.
* [25] J.L. Deschanvres, M. Langlet, and J.C. Joubert. New spray pyrolysis deposition and magnetic properties of ferrite thin films for microwave applications. Journal of Magnetism and Magnetic Materials, 83(1–3):437 – 438, 1990.
* [26] J.-L. Deschanvres, B. Bochu, and J.-C. Joubert. Deposition conditions for the growth of textured ZnO thin films by aerosol CVD process. J. Phys. IV France, 03:C3–485–C3–491, 1993.
* [27] Robert J. Lang. Ultrasonic atomization of liquids. The Journal of the Acoustical Society of America, 34(1):6–8, 1962\.
* [28] J.C. Viguié and J Spitz. Chemical vapor deposition at low temperatures. J. Electrochem. Soc., 122(4):585, 1975.
* [29] J. L. Pouchou and F. Pichoir. La Recherche Aérospatiale, 5:349–367, 1984.
* [30] Paolo Ghigna, Adolfo Speghini, and Marco Bettinelli. “unusual ln(3+) substitutional defects”: The local chemical environment of pr3+ and nd3+ in nanocrystalline TiO2 by ln-k edge exafs. J. Solid State Chem., 180(11):3296–3301, NOV 2007.
* [31] Corinna Graf, Renate Ohser-Wiedemann, and Guenter Kreisel. Preparation and characterization of doped metal supported TiO2-layers. Journal of photochemistry and photobiology A-chemistry, 188(2-3):226–234, MAY 20 2007.
* [32] Daqin Chen, Yuansheng Wang, Yunlong Yu, Ping Huang, and Fangyi Weng. Near-infrared quantum cutting in transparent nanostructured glass ceramics. Opt. Lett., 33(16):1884–1886, Aug 2008.
* [33] Chu-Chi Ting and San-Yuan Chen. Physical characteristics and infrared fluorescence properties of sol–gel derived Er3+–Yb3+ codoped TiO2. J. Appl. Phys., 94(3):2102, august 2003.
* [34] Sandip Das and Krishna C. Mandal. Optical downconversion in rare earth (Tb3+ and Yb3+) doped CdS nanocrystals. Mater. Lett., 66(1):46 – 49, 2012.
* [35] W Strek, P Deren, and A Bednarkiewicz. Cooperative processes in KYb(WO4)2 crystal doped with Eu3+ and Tb3+ ions. J. Lumin., 87-9:999–1001, MAY 2000.
* [36] G Dominiak-Dzik, W Ryba-Romanowski, R Lisiecki, I Foldvári, and E Beregi. YAl3(BO3)4:Yb, Tm a nonlinear crystal : Up- and down-conversion phenomena and excited state relaxations. Optical materials, 31(6):989 – 994, 2009.
|
arxiv-papers
| 2012-09-19T12:47:18 |
2024-09-04T02:49:35.301241
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Forissier, H. Roussel, P. Chaudouet, A. Pereira, J.-L. Deschanvres,\n B. Moine",
"submitter": "S\\'ebastien Forissier",
"url": "https://arxiv.org/abs/1209.4226"
}
|
1209.4248
|
# A simple function for calculating the interaction between a molecule and a
graphene sheet
Xiongce Zhao [email protected] Joint Institute for Computational Sciences
and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA Current address: NIDDK, National Institutes
of Health, Bethesda, MD 20892, USA
###### Abstract
We present a novel potential model for calculating the interaction between a
molecule and a single graphene sheet. The dispersion/repulsion, induction,
dipole-quadrupole, quadrupole-quadrupole interactions between a fluid molecule
and a graphene sheet are described by integrated functions that are only
dependent on the separation between the molecule and the graphene along its
planar normal. The derived potential functions are in excellent agreement with
the computationally demanding atom-explicit summation method. Typical errors
of the integrated potential are less than 2% in the energy minimum compared
with the exact atom-explicit summation. To examine the practical effectiveness
of the newly developed functions, Monte Carlo simulations were performed to
model the adsorption of two representative gases in graphene sheets using both
the integrated and atom-explicit potentials. The integrated potential results
in same adsorption isotherms and density profiles for the adsorbed phase while
it only requires negligible computing time compared with that using the atom-
explicit method. The newly developed potential functions provide a simple and
accurate approach to calculating the physical interaction between molecules
and graphene sheets.
## I Introduction
Graphene, or individual layers of graphite in which each carbon atom is bonded
to three other carbon atoms, has been the subject of considerable research
interest recently geim07 . Being a representative two dimensional crystal,
graphene has many peculiar properties such as low dimensionality, surface
homogeneity, structure stability, conductivity, and charge transport ability.
It is deemed as one of the most promising materials for future applications in
various fields such as electronics, novel materials, sensors, biodevices etc.
Lee2008 ; Ponomarenko2008 ; wang2008 ; li2008a ; li2008b ; ruoff08 ; geim07 ;
schedin07 ; fasolino07 ; dikin07 ; abanin06 ; novoselov05 ; zhang05 .
In the investigations of fundamental properties of graphene, it is often of
great interest to understand the interaction of molecules with graphene on a
molecule level. For example, graphene sheets have been used as high-
sensitivity sensors schedin07 or background membranes Meyer2008 in studying
the behavior of individual molecules. In such applications, interaction
between the target molecule and the graphene is one of the key properties that
needs to be understood in order to design effective devices. Therefore, a
simple and reliable approach to estimating the graphene-fluid interaction is
essential. Furthermore, computer simulation, which is becoming a useful
partner of experiments in studying fluid-solid interactions, requires
realistic and computationally affordable method with sufficient accuracy to
calculate the molecule-solid interactions, such as that between molecules and
graphene sheets. However, to our knowledge, a simple and accurate function
describing the interaction between a molecule and a single graphene sheet is
still not available. In this paper, we attempt to develop a novel potential to
solve this problem.
There are two general approaches to calculating the interactions between a
molecule and a crystal solid like graphene. One can use an atom-explicit
potential to describe the interaction between each atom in the molecule and
each atom in crystal, calculating the total potential energy by summing up all
pairs of atoms. Dispersion, repulsion, electrostatic and induction
interactions can all be computed in this way. But such an approach is
computationally costly, especially for large systems involving significant
number of solid atoms. Alternatively, one can develop integrated potential
functions that accounts for all the solid atoms in an effective way, making
use of the periodicity and homogeneity in the distribution of atoms in the
crystal. This has been a popular and useful approach for modeling solid-fluid
interactions for many systems, including graphite and metals steele73 .
Integrated potentials are easy to program and very efficient computationally.
In this paper the second approach was employed to derive a new set of
effective potential force fields for molecules on a graphene sheet. The types
of solid-fluid interactions of interest include the dispersion and repulsive
(Lennard-Jones, or LJ), the induction due to the dipole in the molecule and
the polarizability of carbon atoms in graphene, and the interaction between
multipoles in the molecule and the permanent quadrupole in graphene. The
potential functions derived in this work are extensions of a previous work by
us zhao05 for modeling interaction between gases and a semi infinite
graphitic surface. The accuracy of the derived potentials are evaluated by
comparing the energies calculated from the integrated expressions and those
from atomistic summation approach. The effectiveness of the potentials are
demonstrated by using the derived formulas in simulating the adsorption of two
representative gases into slits composed of single graphene sheets.
In Section II, the functions are derived. Section III and IV are examinations
of the new potentials via comparison with atomistic summation methods. Section
V is concluding remarks.
## II Potential Development
We assume that a molecule is located right above the center of a single
graphene sheet. The origin of the system was set as the mass center of the
molecule. If we only consider the pairwise interactions, the total potential
energy between the molecule and the graphene can be expressed by a summation
$U=\sum_{i}^{\infty}u(r_{i}),$ (1)
where $r_{i}=(x_{i},y_{i},z)$ denotes the position of a carbon atom in
graphene relative to the molecule, $u(r_{i})$ is the pairwise potential
between the molecule and carbon atom $i$ in graphene, which is only dependent
on the separation between them, $r_{i}=(x_{i}^{2}+y_{i}^{2}+z^{2})^{1/2}$. For
a graphene sheet of infinite size along its lateral directions ($x$ and $y$),
$U$ only depends on the separation between the molecule and graphene along its
surface normal $z$, and Eq.(1) becomes
$U(z)=\sum_{i}^{\infty}u(x_{i},y_{i},z).$ (2)
For example, the Lennard-Jones interaction between the molecule and a carbon
atom is given as
$u(r)=4\varepsilon_{\mathrm{sf}}\left[\left(\frac{\sigma_{\mathrm{sf}}}{r}\right)^{12}-\left(\frac{\sigma_{\mathrm{sf}}}{r}\right)^{6}\right],$
(3)
where $\varepsilon_{\mathrm{sf}}$ and $\sigma_{\mathrm{sf}}$ are the LJ
potential parameters for the cross interaction between the molecule and a
graphene carbon atom. Substituting Eq.(3) into Eq.(2) gives
$U(z)=\sum_{i}^{\infty}4\varepsilon_{\mathrm{sf}}\left[\frac{\sigma_{\mathrm{sf}}^{12}}{(x_{i}^{2}+y_{i}^{2}+z^{2})^{6}}-\frac{\sigma_{\mathrm{sf}}^{6}}{(x_{i}^{2}+y_{i}^{2}+z^{2})^{3}}\right],$
(4)
If we assume that the graphene sheet is homogeneous and continuous along its
$x$ and $y$ directions, $U(z)$ can be approximated by an integral over $x$ and
$y$
$U(z)\\!=\\!4\varepsilon_{\mathrm{sf}}d_{\mathrm{s}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\left[\frac{\sigma_{\mathrm{sf}}^{12}}{(x^{2}\\!+\\!y^{2}\\!+\\!z^{2})^{6}}\\!-\\!\frac{\sigma_{\mathrm{sf}}^{6}}{(x^{2}\\!+\\!y^{2}\\!+\\!z^{2})^{3}}\right]\\!dxdy,$
(5)
where $d_{\mathrm{s}}$ is the density of carbon atoms in a unit graphene area
$dxdy$. The integration can be simplified by letting $x^{2}+y^{2}=S$, which
gives $dxdy=\pi dS$. Then Eq.(5) reduces to
$U(z)=4\pi\varepsilon_{\mathrm{sf}}d_{\mathrm{s}}\\!\\!\int_{0}^{\infty}\\!\\!\left[\frac{\sigma_{\mathrm{sf}}^{12}}{(S+z^{2})^{6}}-\frac{\sigma_{\mathrm{sf}}^{6}}{(S+z^{2})^{3}}\right]dS.$
(6)
Eq.(6) leads to an integrated potential for the LJ interaction between a
molecule and an infinitely large graphene sheet
$U_{\mathrm{LJ}}(z)=2\pi\varepsilon_{\mathrm{sf}}\sigma_{\mathrm{sf}}^{2}d_{\mathrm{s}}\left[\frac{2}{5}\left(\frac{\sigma_{\mathrm{sf}}}{z}\right)^{10}-\left(\frac{\sigma_{\mathrm{sf}}}{z}\right)^{4}\right].$
(7)
The derived expression Eq.(7) for the LJ interaction between a molecule and a
graphene is analogous to a potential by Steele steele73 . The interaction is
only a function of the distance of the molecule from the graphene along its
planar normal, $z$, thus the numerical calculation is substantially
simplified. It is noteworthy to point out that unlike the potential for
graphite steele73 the new function for graphene, Eq.(7), does not contain any
empirical term that requires fitted parameters. This is due to the fact that
no approximation was involved in the derivation to account for the semi-
infinite layers of graphene for a graphite surface. Only assumption here is
that the atoms in graphene sheet are continuous, otherwise the derivation is
strictly exact.
For LJ particles, Eq. (7) should be a sufficient approximation for calculating
the molecule-graphene potential. However, for molecules that has partial
charges or permanent multipole moments, additional polar interactions have to
be included. For this purpose, similar procedures are applied to derive
integrated forms for the induction and multipolar interactions between the
molecule and a graphene sheet. We note that the above derivation procedure is
applicable as long as the pairwise potentials are dependent only of $r$ by an
inverse power $r^{-n}$ with $n>1$. This condition is satisfied if we use point
or angle-averaged dipoles and point quadrupoles for the molecule-graphene
interaction. For example, the angle-averaged dipole-induced dipole interaction
between a polar molecule and a carbon atom in the graphene is given by
maitland81
$u_{\mu}(r)=-\frac{\mu_{\mathrm{f}}^{2}\alpha_{\mathrm{s}}}{(4\pi\varepsilon_{0})^{2}r^{6}},$
(8)
where $\mu_{\mathrm{f}}$ is the permanent dipole moment of the molecule,
$\alpha_{\mathrm{s}}$ is the isotropic polarizability of a carbon atom in
graphene, and $\varepsilon_{0}$ is the vacuum permittivity.
The angle-averaged dipole-quadrupole interaction is given by maitland81
$u_{\mu\Theta}(r)=-\frac{\mu_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{kT(4\pi\varepsilon_{0})^{2}r^{8}},$
(9)
where $\Theta_{\mathrm{s}}$ is the permanent quadrupole moment on each carbon
atom in graphene, $k$ is Boltzmann’s constant and $T$ is the absolute
temperature.
Finally, the angle-averaged quadrupole-quadrupole interaction for the molecule
and carbon atom is given by maitland81
$u_{\Theta\Theta}(r)=-\frac{14\Theta_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{5kT(4\pi\varepsilon_{0})^{2}r^{10}},$
(10)
where $\Theta_{\mathrm{f}}$ is the permanent quadrupole moment of the
molecule.
Substituting Eqs. (8)$-$(10) into Eq.(2) and integrating, we obtain the
following expressions. The integrated induction potential is
$U_{\mu}(z)=-\frac{\pi
d_{\mathrm{s}}\mu_{\mathrm{f}}^{2}\alpha_{\mathrm{s}}}{2(4\pi\varepsilon_{0})^{2}}\frac{1}{z^{4}}.$
(11)
The integrated dipole-quadrupole potential is
$U_{\mu\Theta}(z)=-\frac{\pi
d_{\mathrm{s}}\mu_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{3kT(4\pi\varepsilon_{0})^{2}}\frac{1}{z^{6}}.$
(12)
The integrated quadrupole-quadrupole potential is
$U_{\Theta\Theta}(z)=-\frac{7\pi
d_{\mathrm{s}}\Theta_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{10kT(4\pi\varepsilon_{0})^{2}}\frac{1}{z^{8}}.$
(13)
Likewise, the potential functions in Eq.(11)$-$(13) are only dependent on the
separation between the multipole and graphene along its planar normal $z$. By
using Eq.(7) and Eqs.(11)$-$(13), one can calculate the interaction potential
energy between a polar molecule and a graphene sheet without using the costly
pairwise atomistic summation.
## III Comparison with atomistic potentials
To examine the accuracy of the derived functions, we calculated the potential
energies of several representative nonpolar and polar molecules interacting
with a single graphene sheet using the integrated functions Eqs.(7) and
(11)$-$(13), to compare with those from atomistic summation of potentials Eqs.
(3) and (8)$-$(10). The graphene sheet in the atomistic summation was modeled
as a square about 10 nm in a side, containing 3680 carbon atoms. Trial
calculations indicate that such a system size contains more than enough carbon
atoms to approximate a graphene sheet that is infinite along its lateral
directions. A single molecule was placed over the center of the graphene sheet
and its interaction potential energies were computed by summing up
interactions from each carbon atom.
Table 1: The Lennard-Jones and multipole parameters for the example molecules studied. | $\varepsilon_{\mathrm{f}}$ (kcal/mol) | $\sigma_{\mathrm{f}}$ (Å) | $\mu_{\mathrm{f}}$ (Debye) | $\Theta_{\mathrm{f}}$ (10-20C Å2)
---|---|---|---|---
CH4 | 0.2378 | 3.527 | $-$ | $-$
H2O | 0.1554 | 3.152 | 2.351 | $-$
Cl2 | 0.7101 | 4.115 | $-$ | 10.79
The LJ and multipole potential parameters for the example molecules chosen are
shown in Table 1. The nonpolar molecule studied is methane, modeled by a
single LJ site jiang93 . The representative polar molecules selected are water
and chlorine. Water is modeled by the SPC/E potential Berendsen87 , with a
strong dipole moment of 2.351 Debye. Please note that the LJ parameters of
water in Table 1 are for the O atom. The chlorine molecule is modeled by a
single LJ site plus a quadrupole moment of 10.79$\times 10^{-20}$ C Å2 rowley
. The purpose of the calculation was to study the accuracy of the integrated
potential compared with atomistic summation. Therefore, the potential models
for the molecules were selected arbitrarily. The values of potential
parameters for carbon atoms in graphene are $\sigma_{\mathrm{s}}$=3.40 Å,
$\varepsilon_{\mathrm{s}}$=0.05569 kcal/mol, $d_{\mathrm{s}}$=0.382 Å-2 zhao05
, $\alpha_{\mathrm{c}}$=1.76 Å3 bates77 ,
$\Theta_{\mathrm{s}}$=$-3.03\times$10-20 C Å2 whitehouse93 . The cross
interaction parameters $\varepsilon_{\mathrm{sf}}$ and $\sigma_{\mathrm{sf}}$
were calculated using the Lorentz-Bertholet rules.
Figure 1: Interaction of methane with a single graphene sheet calculated
using the atom-explicit pairwise summation of Eq.(3) (line) and the integrated
function Eq.(7) (symbols).
Figure 2: Interaction of a water molecule with a single graphene sheet
calculated using the atomistic summation (lines) and integrated functions
(symbols). Atomistic summations: LJ interaction Eq.(3) (solid line), LJ plus
induction interactions Eq.(3)$+$Eq.(8) (long dashed line), and LJ plus
induction plus dipole-quadrupole interactions Eq.(3)$+$Eq.(8)$+$Eq.(9) (dashed
line). Integrated functions: LJ interaction Eq.(7) (circle), LJ plus induction
interactions Eq.(7)$+$Eq.(11) (square), and LJ plus induction plus dipole-
quadrupole interactions Eq.(7)$+$Eq.(11)$+$Eq.(12) (diamond).
Figure 3: Interaction a chlorine molecule with a single graphene sheet
calculated using the atomistic summation (lines) and integrated functions
(symbols). Atomistic summations: LJ interaction Eq.(3) (solid line), LJ plus
quadrupole-quadrupole interactions Eq.(3)$+$Eq.10 (long dashed line).
Integrated functions: LJ interaction Eq.(7) (circle), LJ plus quadrupole-
quadrupole interactions Eq.(7)$+$Eq.(13) (square).
In Figs. 1 to 3 we compare the potential energies calculated from the
integrated potentials and the atom-explicit potentials at 300 K, for CH4, H2O,
and Cl2. It is seen that the integrated expressions are in excellent agreement
with the atomistic summation results. If we take the atomistic summation
results as the standard, the typical errors in the integrated potential are
less than 2% at the energy minimum. The integrated functions always give a
slightly more attractive potential compared to the atom-explicit methods,
possibly due to the fact that we treat the graphene carbon atoms as continuum
in the integrated potential.
The integrated potentials are most accurate when the graphene sheet is
infinitely large. However, we also carried out series of calculations using
graphene sheets of various sizes, to test the applicability of the derived
functions to the finite-size graphene. We found that the integrated potentials
are in excellent agreement with the atomistic potentials as long as the
square-shaped graphene sheet contains more than $\sim$500 carbon atoms (or a
graphene sheet of $\sim$38$\times$38 Å2). This suggests that the derived
potentials can also be readily used in estimating the interaction of molecules
with finite graphene sheets that are nanometer-scale in size.
For the different types of interactions between a molecule and graphene, it is
interesting to see that, for example from Fig. 2, the induction and dipole-
quadrupole interactions for strongly polar molecules such as water are non-
negligible compared with the LJ energy. On the other hand, based on the Cl2
example, the quadrupole-quadrupole interaction is usually not significant,
even for molecules with a strong quadrupole moment like chlorine.
## IV Applications in adsorption simulation
One of important applications of the potential functions developed in this
work is molecular simulation of adsorption of molecules in pores composed of
single graphene sheets. In turn, such simulations serve as a verification of
the accuracy and effectiveness of the integrated functions in practical
applications. For this purpose, we chose to simulate the adsorption of two
typical nonpolar and polar gases, methane and water, confined in single
graphene sheets. The simulations were performed using the grand canonical
Monte Carlo (GCMC) method, the detailed description of which can be found in
literature allen .
The potentials parameters describing the interactions of water and methane
with graphene are given in Table 1. The methane is modeled as a simple one-
site LJ particle, with a cutoff of 9 Å. The LJ interaction between water
molecules is also modeled by a cutoff of 9 Å, without long-range correction
applied, as suggested by the original literature Berendsen87 . The
electrostatic interaction between water molecules are modeled by the partial
charges distributed on the three charge sites on the water model, without long
range correction. The interaction between the molecules and graphenes is
calculated by either the atomistic summation approach or integrated functions
for comparison.
The GCMC cell for adsorption simulations is a rectangular box. Two graphene
sheets are placed in the cell with their planes parallel to the $x-y$ plane of
the box. The distance between the two graphene layers along the $z$ direction
is 15 Å. Again, this separation is chosen arbitrarily. Periodic boundary
conditions are applied in all three directions. Therefore, the height of
simulation box in the $z$ direction is 30 Å. The typical box sizes in the $x$
and $y$ directions are set as 32 and 34 Å respectively, leading to a cell
volume of about 33 nm3. The types of move attempted during a GCMC simulation
were selected randomly with probability of 0.40, 0.40, 0.10, and 0.10 for
displacements, rotations, creations, and deletions of a fluid molecule
respectively. For simulation of the spherical methane, the rotation moves were
merged to displacement moves. Each simulation included equilibration of
2$\times 10^{6}$ MC moves and production of 2$\times 10^{6}$ MC moves.
The simulation of methane adsorption in graphene sheets were performed at 77 K
jiang93 . Previous theoretical studies predict that water only wets graphitic
carbon surfaces when temperature is above $\sim$510 K gatica04 ; zhao07 .
Therefore, we chose to perform simulations of water/graphene at 550 K. In GCMC
simulations the reduced chemical potential was varied to obtain isotherms and
density profiles of adsorbed methane or water, using either atomistic or
integrated potential for comparison.
Figure 4: Adsorption isotherm of methane in singlet graphene sheets at 77 K.
Circles were calculated using the atomistic summation potential and squares
were calculated using the integrated potential. Adsorbed amount ($\rho^{*}$)
and chemical potential ($\mu^{*}$) are in reduced unit.
Figure 5: Density profiles of methane adsorption in graphene sheets at 77 K
and reduced chemical potential of $-6.5$. (a) was calculated using the
atomistic summation potential and (b) was calculated using the integrated
potential. Two curves are almost identical so (a) was shifted by 0.01 in
$P(z)$ for clarity. The dashed lines at $z=\pm 7.5$ Å represent the two
graphene sheets in the system.
Figure 6: Adsorption isotherm of water in graphene sheets at 550 K. Circles
were calculated using the atomistic summation potential and squares were
calculated using the integrated potential. Adsorbed amount ($\rho^{*}$) and
chemical potential ($\mu^{*}$) are in reduced unit.
Figure 7: Density profiles of water adsorption in graphene sheets at 550 K
and reduced chemical potential of $-26$. (a) was calculated using the
atomistic summation potential and (b) was calculated using the integrated
potential. Two curves are almost identical so (a) was shifted by 0.005 in
$P(z)$ for clarity. The dashed lines at $z=\pm 7.5$ Å represent the two
graphene sheets in the system.
Calculated adsorption isotherms for methane in graphene sheets at 77 K are
presented in Fig. 4. The circles are simulation results computed from the
atomistic summation of Eq.(3), while the squares are from simulations using
the integrated potential Eq.(7). It can be seen that, within statistical
uncertainty, the isotherms from these two different potentials are in
excellent agreement. The first plateau in the isotherm corresponds to the
first layers of methane adsorbed on graphene sheets, and the second plateau is
from the methane adsorbed onto first layers as chemical potential increases.
This is confirmed by density profile distribution of the adsorbed phase shown
in Fig. 5. The transition from nonadsorption to the first layer occurs at
about $\mu^{*}=-8.6$, and the transition from the first to the second layer
occurs at about $\mu^{*}=-7.3$.
It is known that the isotherm shapes, especially the layering transition of
adsorbed phase calculated from simulations, are very sensitive to the solid-
fluid interactions zhao02 . A small deviation in the solid-fluid potential can
result in significant shift in the layer transition location in isotherms. The
excellent agreement in the adsorption layering transition for the two
potentials indicates that the integrated functions are robust alternatives to
the atomistic models.
One example of density profiles of methane adsorbed in graphene layers is
given in Fig. 5. Sharp peaks are observed at $z=\pm 4$, $\pm 11$ Å, which
correspond to the first adsorbed methane layers on the graphene sheets.
Additional broad peaks are observed at $z=0$, $\pm 15$ Å, representing the
adsorbed second layers between the first layers. Again, it is seen that the
density distribution of adsorbed methane in graphenes calculated using the
atomistic and integrated potentials are in excellent agreement. Note that the
density profiles calculated from the two potentials are almost identical to
each other, so that we have to present them by shifting one curve along the
vertical axis by a constant value to have a clear view.
The results for water adsorption in singlet graphenes at 550 K are shown in
Figs. 6 and 7. It is seen from the water adsorption isotherm that a continuous
wetting transition occurs at about $\mu^{*}=-33.4$. The density profile peaks
corresponding to the first adsorbed layers on the graphene sheets locate at
$z=\pm 4.4$ and $\pm 10.6$ Å. Slightly different from that of methane, the
adsorbed phase of water forms two second layers between the first layers, at
$z=\pm 1.5$ and $\pm 13.5$ Å, due to the relatively smaller size of water
molecule compared to methane. The isotherms and density profiles of water in
graphene sheets calculated from the integrated and atomistic potentials are
also in excellent agreement. This indicates that the integrated potentials for
multipolar interactions are also excellent approximations to the atomistic
approach.
We monitored the computational time required for the simulations using two
different approaches. It is found that typically the computational time for
the integrated potential is $<1\%$ of that required for the atomistic
summations.
## V Conclusions
A set of effective potentials were derived for calculating the Lennard-Jones,
dipole-induced dipole, dipole-quadrupole, and quadrupole-quadrupole
interactions between fluid molecules and a graphene sheet. The integrated
potential functions depend only on the separation between a molecule and the
graphene surface. They are mathematically simple and easy to use in either
estimating the interaction between a single molecule and a graphene sheet or
in large-scale molecular simulations. The potential energies calculated from
the integrated potentials are in excellent agreement with the results
calculated from direct atomistic summations, while the integrated ones are
computationally negligible compared with their atomistic counterparts.
Adsorption simulations of two representative gases in singlet graphene sheets
were performed to further test the derived potential functions. The adsorption
isotherms and density profiles calculated from the derived potentials and
atomistic models are in excellent agreement, indicating that the derived
potential can predict both the equilibrium and the structural properties of
adsorbed phase with excellent accuracy. The potentials developed in this work
provide a simple, accurate, and robust method for calculating the physical
interaction of molecules and single graphene sheets.
Finally, we note that the self-consistency in the polarization of carbon atoms
by a polar fluid molecule was neglected in the derivation. However, we believe
that the effect is relatively small compared with the dispersion, induction,
and dipole-quadrupole interactions. Also, by using an integrated potential,
one assumes the graphene surface is smooth and the impact of the surface
corrugation on the molecule-solid interaction is ignored.
###### Acknowledgements.
The author thanks Peter T. Cummings for many insightful discussions. This
research was conducted at the Center for Nanophase Materials Sciences, which
is sponsored at Oak Ridge National Laboratory by the Division of Scientific
User Facilities, U.S. Department of Energy. This research used resources of
the National Energy Research Scientific Computing Center, which is supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231.
## References
* (1) Geim, A. K.; Novoselov, K. S. _Nature Materials_ 2007, _6_ , 183\.
* (2) Lee, C.; Wei, X. D.; Kysar, J. W.; Hone, J. _Science_ 2008, _321_ , 385–388.
* (3) Ponomarenko, L. A.; Schedin, F.; Katsnelson, M. I.; Yang, R.; Hill, E. W.; Novoselov, K. S.; Geim, A. K. _Science_ 2008, _320_ , 356–358.
* (4) Wang, F.; Zhang, Y. B.; Tian, C. S.; Girit, C.; Zettl, A.; Crommie, M.; Shen, Y. R. _Science_ 2008, _320_ , 206–209.
* (5) Li, X. L.; Wang, X. R.; Zhang, L.; Lee, S. W.; Dai, H. J. _Science_ 2008, _319_ , 1229–1232.
* (6) Li, D.; Müller, M. B.; Gilje, S.; Kaner, R. B.; Wallace, G. G. _Nature Nanotechnology_ 2008, _3_ , 101–105.
* (7) Ruoff, R. _Nature Nanotechnology_ 2008, _3_ , 10–11.
* (8) Schedin, F.; Geim, A. K.; Morozov, S. V.; Hill, E. W.; Blake, P.; Katsnelson, M. I.; Novoselov, K. S. _Nature Materials_ 2007, _6_ , 652–655.
* (9) Fasolino, A.; Los, J. H.; Katsnelson, M. I. _Nature Materials_ 2007, _6_ , 858–861.
* (10) Dikin, D. A.; Stankovich, S.; Zimney, E. J.; Piner, R. D.; Dommett, G. H. B.; Evmenenko, G.; Nguyen, S. T.; Ruoff, R. S. _Nature_ 2007, _448_ , 457–460.
* (11) Abanin, D. A.; Lee, P. A.; Levitov, L. S. _Phys. Rev. Lett._ 2006, _96_ , 176803.
* (12) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. _Nature_ 2005, _438_ , 197–200.
* (13) Zhang, Y.; Zhang, Y. B.; Tan, Y. W.; Stormer, H. L.; Kim, P. _Nature_ 2005, _438_ , 201–204.
* (14) Meyer, J. C.; Girit, C. O.; Crommie, M. F.; Zettl, A. _Nature_ 2008, _454_ , 319–322.
* (15) Steele, W. A. _Surf. Sci._ 1973, _36_ , 317–352.
* (16) Zhao, X. C.; Johnson, J. K. _Molecular Simulation_ 2005, _31_ , 1–10.
* (17) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. _Intermolecular Forces_ ; Claredon Press: Oxford, 1981.
* (18) Jiang, S. Y.; Gubbins, K. E.; Zollweg, J. A. _Mol. Phys._ 1993, _80_ , 103–116.
* (19) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. _J. Phys. Chem._ 1987, _91_ , 6269.
* (20) Rowley, R. L. _Statistical Mechanics for Thermophysical Property Calculations_ ; Prentice Hall: Englewood Cliffs, NJ, 1994.
* (21) Miller, T. M.; Bederson, B. In _Atomic and molecular polarizabilities: A Review of recent advances_ ; Bates, D. R., Bederson, B., Eds.; Academic Press: London, 1977; Vol. 13, pp 1–55.
* (22) Whitehouse, D. B.; Buckingham, A. D. _J. Chem. Soc., Faraday Trans._ 1993, _89_ , 1909–1913.
* (23) Allen, M. P.; Tildesley, D. J. _Computer Simulation of Liquids_ ; Clarendon Press: Oxford, 1987.
* (24) Gatica, S. M.; Johnson, J. K.; Zhao, X. C.; Cole, M. W. _J. Phys. Chem. B_ 2004, _108_ , 11704.
* (25) Zhao, X. C. _Phys. Rev. B_ 2007, _76_ , 041402.
* (26) Zhao, X. C.; Kwon, S. J.; Vidic, R.; Borguet, E.; Johnson, J. K. _J. Chem. Phys._ 2002, _117_ , 7719–7731.
|
arxiv-papers
| 2012-09-19T14:13:28 |
2024-09-04T02:49:35.307151
|
{
"license": "Public Domain",
"authors": "Xiongce Zhao",
"submitter": "Xiongce Zhao",
"url": "https://arxiv.org/abs/1209.4248"
}
|
1209.4284
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-263 LHCb-PAPER-2012-024 19 September 2012
Differential branching fraction
and angular analysis of
the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay
The LHCb collaboration†††Authors are listed on the following pages.
The angular distribution and differential branching fraction of the decay
${B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}}$ are studied with a dataset
corresponding to 1.0$\mbox{\,fb}^{-1}$ of integrated luminosity, collected by
the LHCb experiment. The angular distribution is measured in bins of dimuon
invariant mass squared and found to be consistent with Standard Model
expectations. Integrating the differential branching fraction over the full
dimuon invariant mass range yields a total branching fraction of ${\cal
B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})=(4.36\pm 0.15\pm 0.18)\times
10^{-7}$. These measurements are the most precise to date of the
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay.
To be submitted to Journal of High Energy Physics
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, E.E. Bowen46,37, C. Bozzi16, T. Brambach9, J. van den Brand39, J.
Bressieux36, D. Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49,
A. Büchler-Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P.
Campana18,35, A. Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15,
L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M.
Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P.
Collins35, A. Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G.
Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47,
C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, C. Gaspar35, R.
Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V.
Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A.
Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado
Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T.
Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51,
P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P.
Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52,
M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N.
Hussain52, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P.
Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F.
Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K.
Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M.
Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D.
Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45, C.
Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M.
Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac
Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O. Maev27,35,
J. Magnin1, M. Maino20, S. Malde52, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, A. Massafferri1, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E.
Maurice6, A. Mazurov16,30,35, J. McCarthy42, G. McGregor51, R. McNulty12, M.
Meissner11, M. Merk38, J. Merkel9, D.A. Milanes13, M.-N. Minard4, J. Molina
Rodriguez54, S. Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38,
F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47,
N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N.
Nikitin29, T. Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53,
A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S.
Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro36, W. Qian3, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N. Rauschmayr35, G.
Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A.
Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35, V.
Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, P. Schaack50, M.
Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B.
Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I.
Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35,
P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L. Shekhtman31, O.
Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K. Sobczak5, F.J.P.
Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, H.
Voss10, C. Voß55, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R.
Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J.
Wicht35, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M.
Williams50,p, F.F. Wilson46, J. Wishahi9, M. Witek23,35, W. Witzeling35, S.A.
Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z.
Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a,
F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay111Charge conjugation is
implied throughout this paper unless explicitly stated otherwise. is a
$b\rightarrow s$ flavour changing neutral current process that is mediated in
the Standard Model (SM) by electroweak box and penguin diagrams. In many well
motivated extensions to the SM, new particles can enter in competing loop
diagrams, modifying the branching fraction of the decay or the angular
distribution of the dimuon system. The differential decay rate of the $B^{+}$
($B^{-}$) decay, as a function of $\cos\theta_{\ell}$, the cosine of the angle
between the $\mu^{-}$ ($\mu^{+}$) and the $K^{+}$ ($K^{-}$) in the rest frame
of the dimuon system, can be written as
$\frac{1}{\Gamma}\frac{\mathrm{d}\Gamma[B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}]}{\mathrm{d}\\!\cos{\theta_{l}}}=\frac{3}{4}(1-F_{\rm
H})(1-\cos^{2}\theta_{l})+\frac{1}{2}F_{\rm H}+A_{\rm FB}\cos{\theta_{l}}~{},$
(1)
which depends on two parameters, the forward-backward asymmetry of the dimuon
system, $A_{\rm FB}$, and the parameter $F_{\rm H}$ [1, 2]. If muons were
massless, $F_{\rm H}$ would be proportional to the contributions from
(pseudo-)scalar and tensor operators to the partial width, $\Gamma$. The
partial width, $A_{\rm FB}$ and $F_{\rm H}$ are functions of the dimuon
invariant mass squared ($q^{2}=m_{\mu^{+}\mu^{-}}^{2}$).
In contrast to the case of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ [3,
*LHCb-CONF-2012-008] decay, $A_{\rm FB}$ is zero for $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ in the SM. Any non-zero value for $A_{\rm FB}$ would
point to a contribution from new particles that would extend the set of SM
operators. In models with (pseudo-)scalar or tensor-like couplings $|A_{\rm
FB}|$ can be enhanced by up to 15% [5, 2]. Similarly, $F_{\rm H}$ is close to
zero in the SM (see Fig. 3), but can be enhanced in new physics models, with
(pseudo-)scalar or tensor-like couplings, up to $F_{\rm H}{~{}\raise
1.49994pt\hbox{$<$}\kern-8.50006pt\lower 3.50006pt\hbox{$\sim$}~{}}0.5$.
Recent predictions for these parameters in the SM are described in Refs. [2,
6, 7]. Any physics model has to satisfy the constraint $|A_{\rm FB}|\leq
F_{\rm H}/2$ for Eq. (1) to stay positive in all regions of phase space. The
contributions of scalar and pseudoscalar operators to $A_{\rm FB}$ and $F_{\rm
H}$ are constrained by recent limits on the branching fraction of
$B^{0}_{s}\\!\rightarrow\mu^{+}\mu^{-}$ [8, *Chatrchyan:2012rg]. The
differential branching fraction of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
can be used to constrain the contributions from (axial-)vector couplings in
the SM operator basis [10, 7, 11].
The relative decay rate of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ to
$B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ has previously been studied by the
LHCb collaboration in the context of a measurement of the isospin asymmetry
[12]. This paper presents a measurement of the differential branching fraction
($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$), $F_{\rm H}$ and $A_{\rm FB}$ of the
decay $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ in seven bins of $q^{2}$ and a
measurement of the total branching fraction. The analysis is based on
1.0$\mbox{\,fb}^{-1}$ of integrated luminosity collected in
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions by the LHCb
experiment in 2011.
## 2 Experimental setup
The LHCb detector [13] is a single-arm forward spectrometer, covering the
pseudorapidity range $2<\eta<5$, that is designed to study $b$ and $c$ hadron
decays. A dipole magnet with a bending power of 4 Tm and a large area tracking
detector provide a momentum resolution ranging from 0.4% for tracks with a
momentum of 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% for a momentum of
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. A silicon micro-strip detector,
located around the $pp$ interation region, provides excellent separation of
$B$ meson decay vertices from the primary $pp$ interaction and an impact
parameter resolution of 20$\,\upmu\rm m$ for tracks with high transverse
momentum ($p_{\rm T}$). Two ring-imaging Cherenkov (RICH) detectors provide
kaon-pion separation in the momentum range
$2-100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Muons are identified based on
hits created in a system of multiwire proportional chambers interleaved with
iron filters. The LHCb trigger comprises a hardware trigger and a two-stage
software trigger that performs a full event reconstruction.
Samples of simulated events are used to estimate the contribution from
specific sources of exclusive backgrounds and the efficiency to trigger,
reconstruct and select the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ signal.
The simulated $pp$ interactions are generated using Pythia 6.4 [14] with a
specific LHCb configuration [15]. Decays of hadronic particles are then
described by EvtGen [16] in which final state radiation is generated using
Photos [17]. Finally, the Geant4 toolkit [18, *Agostinelli:2002hh] is used to
simulate the detector response to the particles produced by Pythia/EvtGen, as
described in Ref. [20]. The simulated samples are corrected for differences
between data and simulation in the $B^{+}$ momentum spectrum, the detector
impact parameter resolution, particle identification and tracking system
performance.
## 3 Selection of signal candidates
The $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates are selected from
events that have been triggered by a single high transverse-momentum muon,
with $\mbox{$p_{\rm T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, in the
hardware trigger. In the first stage of the software trigger, candidates are
selected if there is a reconstructed track in the event with high impact
parameter with respect to the primary $pp$ interaction and high $p_{\rm T}$
[21]. In the second stage of the software trigger, candidates are triggered on
the kinematic properties of the partially or fully reconstructed $B^{+}$
candidate [22].
Signal candidates are then selected for further analysis based on the
following requirements: the $B^{+}$ decay vertex is separated from the primary
$pp$ interaction; the $B^{+}$ candidate impact parameter is small, and the
kaon and muon impact parameters large, with respect to the primary $pp$
interaction; the $B^{+}$ candidate momentum vector points along the $B^{+}$
line of flight to one of the primary $pp$ interactions in the event.
A tighter multivariate selection, using a Boosted Decision Tree (BDT) [23]
with the AdaBoost algorithm[24], is then applied to select a clean sample of
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates. The BDT uses kinematic
variables including the reconstructed $B^{+}$ decay time, the angle between
the $B^{+}$ line of flight and the $B^{+}$ momentum vector, the quality of the
vertex fit of the reconstructed $B^{+}$ candidate, impact parameter (with
respect to the primary $pp$ interaction) and $p_{\rm T}$ of the $B^{+}$ and
muons and the track quality of the kaon. The variables that are used in the
BDT provide good separating power between signal and background, while
minimising acceptance effects in $q^{2}$ and $\cos\theta_{\ell}$ that could
bias the differential branching fraction, $A_{\rm FB}(q^{2})$ or $F_{\rm
H}(q^{2})$. The multivariate selection is trained on data, using
$B^{+}\\!\rightarrow K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$)
candidates as a proxy for the signal and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates from the upper mass sideband
($5350<m_{K^{+}\mu^{+}\mu^{-}}<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
for the background. The training and testing of the BDT is carried out using a
data sample corresponding to 0.1$\mbox{\,fb}^{-1}$ of integrated luminosity,
that is not used in the subsequent analysis. The BDT selection is $85-90\%$
(depending on $q^{2}$) efficient on simulated candidates that have passed the
earlier selection.
Finally, a neural network, using information from the RICH [25], calorimeters
and muon system is used to reject backgrounds where a pion is incorrectly
identified as the kaon from the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
decay. The network is trained on simulated event samples to give the posterior
probability for charged hadrons to be correctly identified. The particle
identification performance of the network is calibrated using pions and kaons
from the decay chain $D^{*+}\\!\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+})\pi^{+}$ in the data. Based on simulation, the efficiency of the
neural network particle identification requirement is estimated to be
${~{}\raise 1.49994pt\hbox{$>$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}95\%$ on the signal.
The contribution from combinatorial backgrounds, where the reconstructed
$K^{+}$, $\mu^{+}$ and $\mu^{-}$ do not come from the same $b$-hadron decay,
is reduced to a small level by the multivariate selection. Remaining
backgrounds come from exclusive $b$-hadron decays. The decays
$B^{+}\\!\rightarrow K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$B^{+}\\!\rightarrow K^{+}\psi{(2S)}$ are rejected by removing the regions of
dimuon invariant mass around the charmonium resonances
($2946<m_{\mu^{+}\mu^{-}}<3176{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$3586<m_{\mu^{+}\mu^{-}}<3776{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$).
Candidates with
$m_{K^{+}\mu^{+}\mu^{-}}<5170{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ were
also removed to reject backgrounds from partially reconstructed $B$ decays,
such as $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$. The potential background
from $B^{+}\\!\rightarrow K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$),
where the kaon is identified as a muon and a muon as the kaon, is reduced by
requiring that the kaon candidate fails the muon identification criteria if
the $K^{+}\mu^{-}$ mass is consistent with that of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$. Candidates
with a $K^{+}\mu^{-}$ mass consistent with coming from a misidentified $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow K^{+}\pi^{-}$ decay
are rejected to remove contributions from $B^{+}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$. After the application of
all of the selection criteria, the dominant sources of exclusive background
are $B^{+}\\!\rightarrow K^{+}\pi^{-}\pi^{+}$ [26] and
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ [27, *LHCb-CONF-2012-006]. These
are determined from simulation to be at the level of $(1.5\pm 0.7)\%$ and
$(1.2\pm 0.2)\%$ of the signal, respectively.
## 4 Differential and total branching fraction
The $K^{+}\mu^{+}\mu^{-}$ invariant mass distribution of the selected
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates is shown in Fig. 1. The
number of signal candidates is estimated by performing an extended unbinned
maximum likelihood fit to the $K^{+}\mu^{+}\mu^{-}$ invariant mass
distribution of the selected candidates. The signal line-shape is extracted
from a fit to a $B^{+}\\!\rightarrow
K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$)
control sample (which is two orders of magnitude larger than the signal
sample), and is parameterised by the sum of two Crystal Ball functions [29].
The combinatorial background is parameterised by a slowly falling exponential
distribution. Contributions from $B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$ and
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decays are included in the fit. The
line shapes of these peaking backgrounds are taken from simulated events. In
total, $1232\pm 40$ $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ signal
candidates are observed in the
$0.05<q^{2}<22.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ range. The yields in
each of the $q^{2}$ bins used in the subsequent analysis are shown in Table 1.
Figure 1: Invariant mass of selected $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
candidates with $0.05<q^{2}<22.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.
Candidates with a dimuon invariant mass consistent with that of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$ are excluded.
The peaking background contribution from the decays $B^{+}\\!\rightarrow
K^{+}\pi^{+}\pi^{-}$ and $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
indicated in the figure. Figure 2: Differential branching fraction of
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ as a function of the dimuon
invariant mass squared, $q^{2}$. The SM theory prediction (see text) is given
as the continuous cyan (light) band and the rate-average of this prediction
across the $q^{2}$ bin is indicated by the purple (dark) region. No SM
prediction is included for the regions close to the narrow $c\overline{}c$
resonances.
The differential branching fraction in each of the $q^{2}$ bins is estimated
by normalising the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ event yield,
$N_{\text{sig}}$, in the $q^{2}$ bin to the total event yield of the
$B^{+}\\!\rightarrow K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
sample, $N_{K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and
correcting for the relative efficiency between the two decays in the $q^{2}$
bin, $\varepsilon_{K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\varepsilon_{K^{+}\mu^{+}\mu^{-}}$,
$\frac{\mathrm{d}{\cal
B}}{\mathrm{d}q^{2}}=\frac{1}{q^{2}_{\text{max}}-q^{2}_{\text{min}}}\frac{N_{\text{sig}}}{N_{K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\frac{\varepsilon_{K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\varepsilon_{K^{+}\mu^{+}\mu^{-}}}\times{\cal B}(B^{+}\\!\rightarrow
K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})\times{\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})~{}.$ (2)
The branching fractions of $B^{+}\\!\rightarrow
K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ are
${\cal B}(B^{+}\\!\rightarrow K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})=(1.014\pm 0.034)\times 10^{-3}$ and ${\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})=(5.93\pm 0.06)\times 10^{-2}$ [30]. The
resulting differential branching fraction is shown in Fig. 2.
The bands shown in Fig. 2 indicate the theoretical prediction for the
differential branching fraction and are calculated using input from Refs. [7]
and [31]. In the low $q^{2}$ region, the calculations are based on QCD
factorisation and soft collinear effective theory (SCET) [32], which profit
from having a heavy $B^{+}$ meson and an energetic kaon. In the soft-recoil,
high $q^{2}$ region, an operator product expansion (OPE) is used to estimate
the long-distance contributions from quark loops [33, 34]. No theory
prediction is included in the region close to the narrow $c\overline{}c$
resonances (the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$) where the assumptions from QCD factorisation/SCET and the OPE
break down. The form-factor calculations are taken from Ref. [35]. A
dimensional estimate is made on the uncertainty on the decay amplitudes from
QCD factorisation/SCET [36].
The total branching fraction is measured to be
${{\cal B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})=(4.36\pm 0.15\pm
0.18)\times 10^{-7}~{},}$
by summing over the partial branching fractions and accounting for the $q^{2}$
regions that are not used in the differential branching fraction analysis.
These regions account for $\sim 14.3\%$ of the total branching fraction (no
uncertainty is assigned to this number). This estimate ignores long distance
effects and uses a model for $\mathrm{d}\Gamma/\mathrm{d}q^{2}$ described in
Ref. [1] to extrapolate across the $c\overline{}c$ resonance region. The
values of the Wilson coefficients and the form-factors used in this model have
been updated according to Refs. [37] and [35].
## 5 Angular analysis
In each bin of $q^{2}$, $A_{\rm FB}$ and $F_{\rm H}$ are estimated by
performing a simultaneous unbinned maximum likelihood fit to the
$K^{+}\mu^{+}\mu^{-}$ invariant mass and $\cos\theta_{\ell}$ distribution of
the $B^{+}$ candidates. The candidates are weighted to account for the effects
of the detector reconstruction, trigger and the event selection. The weights
are derived from a SM simulation of the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ decay in bins of width
$0.5\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ in $q^{2}$ and 0.1 in
$\cos\theta_{\ell}$. This binning is investigated as a potential source of
systematic uncertainty. The largest weights (and largest acceptance effects)
apply to events with extreme values of $\cos\theta_{\ell}$
($|\cos\theta_{\ell}|\sim 1$) at low $q^{2}$. This distortion arises mainly
from the requirement for a muon to have $p{~{}\raise
1.49994pt\hbox{$>$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to reach the
LHCb muon system. This effect is well modelled in the simulation.
Equation (1) is used to describe the signal angular distribution. The
background angular and mass shapes are treated as independent in the fit. The
angular distribution of the background is parameterised by a second-order
Chebychev polynomial, which is observed to describe well the background away
from the signal mass window
($5230<m_{K^{+}\mu^{+}\mu^{-}}<5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$).
The resulting values of $A_{\rm FB}$ and $F_{\rm H}$ in the bins of $q^{2}$
are indicated in Fig. 3 and in Table 1. The measured values of $A_{\rm FB}$
are consistent with the SM expectation of zero asymmetry. The 68% confidence
intervals on $A_{\rm FB}$ and $F_{\rm H}$ are estimated using pseudo-
experiments and the Feldman-Cousins technique [38]. This avoids potential
biases in the estimate of the parameter uncertainties that come from using
event weights in the likelihood fit or from the boundary condition ($|A_{\rm
FB}|\leq F_{\rm H}/2$). When estimating the uncertainty on $A_{\rm FB}$
($F_{\rm H}$), $F_{\rm H}$ ($A_{\rm FB}$) is treated as a nuisance parameter
(along with the background parameters in the fit). The maximum-likelihood
estimate of the nuisance parameters is used when generating the pseudo-
experiments. The resulting confidence intervals ignore correlations between
$A_{\rm FB}$ and $F_{\rm H}$ and are not simultaneously valid at the 68%
confidence level.
Figure 3: Dimuon forward-backward asymmetry, $A_{\rm FB}$, and the parameter
$F_{\rm H}$ for $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ as a function of the
dimuon invariant mass squared, $q^{2}$. The SM theory prediction (see text)
for $F_{\rm H}$ is given as the continuous cyan (light) band and the rate-
average of this prediction across the $q^{2}$ bin is indicated by the purple
(dark) region. No SM prediction is included for the regions close to the
narrow $c\overline{}c$ resonances. Table 1: Signal yield ($N_{\text{sig}}$),
differential branching fraction ($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$), the
parameter $F_{\rm H}$ and dimuon forward-backward asymmetry ($A_{\rm FB}$) for
the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay in the $q^{2}$ bins used
in the analysis. Results are also given in the
$1<q^{2}<6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ range where theoretical
uncertainties are best under control.
$q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $N_{\text{sig}}$ | $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ $(10^{-8}\mathrm{\,Ge\kern-1.00006ptV}^{-2}c^{4})$ | $F_{\rm H}$ | $A_{\rm FB}$
---|---|---|---|---
$0.05-2.00$ | $159\pm 14$ | $2.85\pm 0.27\pm 0.14$ | $0.00\,^{\,+0.12}_{\,-0.00}\,{}^{\,+0.06}_{\,-0.00}$ | $\phantom{-}0.00\,^{\,+0.06}_{\,-0.05}\,{}^{\,+0.03}_{\,-0.01}$
$2.00-4.30$ | $164\pm 14$ | $2.49\pm 0.23\pm 0.10$ | $0.14\,^{\,+0.16}_{\,-0.10}\,{}^{\,+0.04}_{\,-0.02}$ | $\phantom{-}0.07\,^{\,+0.08}_{\,-0.05}\,{}^{\,+0.02}_{\,-0.01}$
$4.30-8.68$ | $327\pm 20$ | $2.29\pm 0.16\pm 0.09$ | $0.04\,^{\,+0.10}_{\,-0.04}\,{}^{\,+0.06}_{\,-0.04}$ | $-0.02\,^{\,+0.03}_{\,-0.05}\,{}^{\,+0.03}_{\,-0.03}$
$10.09-12.86$ | $211\pm 17$ | $2.04\pm 0.18\pm 0.08$ | $0.11\,^{\,+0.20}_{\,-0.08}\,{}^{\,+0.02}_{\,-0.01}$ | $-0.03\,^{\,+0.07}_{\,-0.07}\,{}^{\,+0.01}_{\,-0.01}$
$14.18-16.00$ | $148\pm 13$ | $2.07\pm 0.20\pm 0.08$ | $0.08\,^{\,+0.28}_{\,-0.08}\,{}^{\,+0.02}_{\,-0.01}$ | $-0.01\,^{\,+0.12}_{\,-0.06}\,{}^{\,+0.01}_{\,-0.01}$
$16.00-18.00$ | $141\pm 13$ | $1.77\pm 0.18\pm 0.09$ | $0.18\,^{\,+0.22}_{\,-0.14}\,{}^{\,+0.01}_{\,-0.04}$ | $-0.09\,^{\,+0.07}_{\,-0.09}\,{}^{\,+0.02}_{\,-0.01}$
$18.00-22.00$ | $114\pm 13$ | $0.78\pm 0.10\pm 0.04$ | $0.14\,^{\,+0.31}_{\,-0.14}\,{}^{\,+0.01}_{\,-0.02}$ | $\phantom{-}0.02\,^{\,+0.11}_{\,-0.11}\,{}^{\,+0.01}_{\,-0.01}$
$1.00-6.00$ | $357\pm 21$ | $2.41\pm 0.17\pm 0.14$ | $0.05\,^{\,+0.08}_{\,-0.05}\,{}^{\,+0.04}_{\,-0.02}$ | $\phantom{-}0.02\,^{\,+0.05}_{\,-0.03}\,{}^{\,+0.02}_{\,-0.01}$
## 6 Systematic uncertainties
For the differential branching fraction measurement, the largest source of
systematic uncertainty comes from an uncertainty of $\sim 4\%$ on the
$B^{+}\\!\rightarrow K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
branching fractions [30]. The systematic uncertainties are largely correlated
between the $q^{2}$ bins. The uncertainties coming from the corrections used
to calibrate the performance of the simulation to match that of the data are
at the level of $1-2$%. The uncertainties on these corrections are limited by
the size of the $D^{*+}\\!\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$
and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
control samples that are used to estimate the particle identification and
tracking performance in the data. The signal and background mass models are
also explored as a source of possible systematic uncertainty. In the fit to
the $K^{+}\mu^{+}\mu^{-}$ invariant mass it is assumed that the signal line-
shape is the same as that of the $B^{+}\\!\rightarrow
K^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay. In the simulation,
small differences are seen in the $B^{+}$ mass resolution due to the different
daughter kinematics between low and high $q^{2}$. A 4% variation of the mass
resolution is considered as a source of uncertainty and the effect on the
result found to be negligible.
For the extraction of $A_{\rm FB}$ and $F_{\rm H}$, the largest sources of
uncertainty are associated with the event weights that are used to correct for
the detector acceptance. The event weights are estimated from the simulation
in $0.5\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ wide $q^{2}$ bins (driven by
the size of the simulated event sample). At low $q^{2}$, the acceptance
variation can be large (at extreme values of $\cos\theta_{\ell}$) over the
$q^{2}$ bin size. The order of the uncertainty associated with this binning is
estimated by varying the event weights by half the difference between
neighbouring $q^{2}$ bins and forms the dominant source of systematic
uncertainty. The size of these effects on $A_{\rm FB}$ and $F_{\rm H}$ are at
the level of $0.01-0.03$ and $0.01-0.05$ respectively, and are small compared
to the statistical uncertainties. Variations of the background mass model are
found to have a negligible impact on $A_{\rm FB}$ and $F_{\rm H}$.
The background angular model is cross-checked by fitting a template to the
angular distribution in the upper mass sideband and fixing this shape in the
fit for $A_{\rm FB}$ and $F_{\rm H}$ in the signal mass window. This yields
consistent results in every $q^{2}$ bin. Therefore, no systematic uncertainty
is assigned to the background angular model. Two further cross checks have
been performed. Firstly, $A_{\rm FB}$ has been determined by counting the
number of forward- and backward-going events, after subtracting the
background. Secondly, $F_{\rm H}$ has been measured by fitting the
$\left|\cos\theta_{\ell}\right|$ distribution, which is independent of $A_{\rm
FB}$. Consistent results are found in both cases.
## 7 Conclusions
The measured values of $A_{\rm FB}$ and $F_{\rm H}$ are consistent with the SM
expectations of no forward-backward asymmetry and $F_{\rm H}\sim 0$. The
differential branching fraction of the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ decay is, however, consistently below the SM prediction
at low $q^{2}$. The results are in good agreement with, but statistically more
precise than, previous measurements of $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$
and $A_{\rm FB}$ from BaBar [39, *:2012vw], Belle [41] and CDF [42].
Integrating the differential branching fraction, over the full $q^{2}$ range,
yields a total branching fraction of ${(4.36\pm 0.15\pm 0.18)\times 10^{-7}}$,
which is more precise than the current world average of $(4.8\pm 0.4)\times
10^{-7}$ [30].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] A. Ali, P. Ball, L. T. Handoko, and G. Hiller, Comparative study of the decays $B\rightarrow(K,K^{*})\ell^{+}\ell^{-}$ in the standard model and supersymmetric theories, Phys. Rev. D61 (2000) 074024, arXiv:hep-ph/9910221
* [2] C. Bobeth, G. Hiller, and G. Piranishvili, Angular distributions of $\bar{B}\rightarrow K\bar{\ell}\ell$ decays, JHEP 12 (2007) 040, arXiv:0709.4174
* [3] LHCb Collaboration, R. Aaij et al., Differential branching fraction and angular analysis of the decay $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$, Phys. Rev. Lett. 108 (2012) 181806, arXiv:1112.3515
* [4] LHCb collaboration, Differential branching fraction and angular analysis of the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay, LHCb-CONF-2012-008
* [5] A. K. Alok, A. Dighe, and S. U. Sankar, Large forward-backward asymmetry in $B\rightarrow K\mu^{+}\mu^{-}$ from new physics tensor operators, Phys. Rev. D78 (2008) 114025, arXiv:0810.3779
* [6] A. Khodjamirian, T. Mannel, A. A. Pivovarov, and Y.-M. Wang, Charm-loop effect in $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ and $B\rightarrow K^{*}\gamma$, JHEP 09 (2010) 089, arXiv:1006.4945
* [7] C. Bobeth, G. Hiller, D. van Dyk, and C. Wacker, The decay $\bar{B}\rightarrow\bar{K}\ell^{+}\ell^{-}$ at low hadronic recoil and model-independent $\Delta B=1$ constraints, JHEP 01 (2012) 107, arXiv:1111.2558
* [8] LHCb collaboration, R. Aaij et al., Strong constraints on the rare decays $B_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$, Phys. Rev. Lett. 108 (2012) , arXiv:1203.4493
* [9] CMS collaboration, S. Chatrchyan et al., Search for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ decays, JHEP 04 (2012) 033, arXiv:1203.3976
* [10] W. Altmannshofer, P. Paradisi, and D. M. Straub, Model-independent constraints on new physics in $b\rightarrow s$ transitions, JHEP 04 (2012) 008, arXiv:1111.1257
* [11] W. Altmannshofer and D. M. Straub, Cornering new physics in $b\rightarrow s$ transitions, JHEP 08 (2012) 121, arXiv:1206.0273
* [12] LHCb collaboration, R. Aaij et al., Measurement of the isospin asymmetry in $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays, JHEP 07 (2012) 133, arXiv:1205.3422
* [13] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [14] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 Physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [15] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [16] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [17] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [18] GEANT4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [19] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [20] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. of Phys: Conf. Ser. 331 (2011) 032023
* [21] V. V. Gligorov, A single track HLT1 trigger, LHCb-PUB-2011-003
* [22] V. V. Gligorov, C. Thomas, and M. Williams, The HLT inclusive $B$ triggers, LHCb-PUB-2011-016
* [23] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
* [24] Y. Freund and R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Jour. Comp. and Syst. Sc. 55 (1997) 119
* [25] R. Forty, RICH pattern recognition for LHCb, Nucl. Instrum. Meth. A433 (1999) 257
* [26] BaBar Collaboration, B. Aubert et al., Evidence for direct CP violation from Dalitz-plot analysis of $B^{\pm}\rightarrow K^{\pm}\pi^{\mp}\pi^{\pm}$, Phys. Rev. D78 (2008) 012004, arXiv:0803.4451
* [27] LHCb collaboration, First observation of the decay $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$, LHCb-PAPER-2012-020
* [28] LHCb collaboration, First observation of $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$, LHCb-CONF-2012-006
* [29] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [30] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001
* [31] C. Bobeth, G. Hiller, and D. van Dyk, More benefits of semileptonic rare $B$ decays at low recoil: CP violation, JHEP 07 (2011) 067, arXiv:1105.0376
* [32] M. Beneke, T. Feldmann, and D. Seidel, Systematic approach to exclusive $B\rightarrow V\ell^{+}\ell^{-},\,V\gamma$ decays, Nucl. Phys. B612 (2001) 25, arXiv:hep-ph/0106067
* [33] B. Grinstein and D. Pirjol, Exclusive rare $B\rightarrow K^{*}\ell^{+}\ell^{-}$ decays at low recoil: controlling the long-distance effects, Phys. Rev. D70 (2004) 114005, arXiv:hep-ph/0404250
* [34] M. Beylich, G. Buchalla, and T. Feldmann, Theory of $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ decays at high $q^{2}$: OPE and quark-hadron duality, Eur. Phys. J. C71 (2011) 1635, arXiv:1101.5118
* [35] P. Ball and R. Zwicky, New results on $B\rightarrow\pi,K,\eta$ decay form factors from light-cone sum rules, Phys. Rev. D71 (2005) 014015, arXiv:hep-ph/0406232
* [36] U. Egede et al., New observables in the decay mode $\bar{B}_{d}\rightarrow\bar{K}^{*0}\ell^{+}\ell^{-}$, JHEP 11 (2008) 032, arXiv:0807.2589
* [37] A. Ali, E. Lunghi, C. Greub, and G. Hiller, Improved model independent analysis of semileptonic and radiative rare $B$ decays, Phys. Rev. D66 (2002) 034002, arXiv:hep-ph/0112300
* [38] G. J. Feldman and R. D. Cousins, Unified approach to the classical statistical analysis of small signals, Phys. Rev. D57 (1998) 3873, arXiv:physics/9711021
* [39] BaBar collaboration, B. Aubert et al., Measurements of branching fractions, rate asymmetries, and angular distributions in the rare decays $B\rightarrow K\ell^{+}\ell^{-}$ and $B\rightarrow K^{*}\ell^{+}\ell^{-}$, Phys. Rev. D73 (2006) 092001, arXiv:hep-ex/0604007
* [40] BaBar collaboration, J. P. Lees et al., Measurement of branching fractions and rate asymmetries in the rare decays $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$, arXiv:1204.3933
* [41] Belle collaboration, J.-T. Wei et al., Measurement of the differential branching fraction and forward-backword asymmetry for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$, Phys. Rev. Lett. 103 (2009) 171801, arXiv:0904.0770
* [42] CDF collaboration, T. Aaltonen et al., Measurements of the angular distributions in the decays $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ at CDF, Phys. Rev. Lett. 108 (2012) 081807, arXiv:1108.0695
|
arxiv-papers
| 2012-09-19T15:31:31 |
2024-09-04T02:49:35.315978
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben.-Haim, M. Benayoun, G. Bencivenni,\n S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van\n Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, E.\n E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler.-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma.-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.\n A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P. N. Y.\n David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, J.\n Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro.-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J.-C.\n Garnier, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, S. Hall, T. Hampson, S. Hansmann.-Menzemer, N. Harnew, S. T. Harnew,\n J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M.\n Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, R. S.\n Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean.-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov,\n R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E.\n Lopez Asamar, N. Lopez.-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S.\n Malde, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J.\n Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie.-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, C. Nguyen.-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska.-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel.-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez.-Calero Yzquierdo, P. Perret, M. Perrin.-Terrin, G.\n Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie\n Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi,\n A. Richards, K. Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, E.\n Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A.\n Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido\n Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K. Sobczak, F. J.\n P. Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp.-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, D.\n Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen,\n B. Viaud, I. Videau, D. Vieira, X. Vilasis.-Cardona, J. Visniakov, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, C.\n Vo{\\ss}, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K.\n Watson, A. D. Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Thomas Blake",
"url": "https://arxiv.org/abs/1209.4284"
}
|
1209.4327
|
arxiv-papers
| 2012-09-19T18:38:54 |
2024-09-04T02:49:35.323272
|
{
"license": "Public Domain",
"authors": "Jacob Farinholt",
"submitter": "Jacob Farinholt",
"url": "https://arxiv.org/abs/1209.4327"
}
|
|
1209.4384
|
# FPCP 2012 Summary Talk on Experiments
Jeffrey A. Appel Fermilab, Batavia, IL 60510, USA
###### Abstract
In over forty presentations on experiments at the 2012 conference on Flavor
Physics and CP Violation (FPCP 2012), there was an abundance of beautiful and
significant results. This summary of these experiment presentations begins
with a reminder of the context in which the measurements have been made and
the motivations for making the specific measurements reported at the
symposium. Given the number and breadth of physics topics covered at the
meeting, this review covers only a limited set of highlights, sort of a
traveler’s set of souvenir postcards of favorite slides. The selected slides
are grouped into eight overlapping categories as an aid to flipping through
the postcards and being reminded of the high points of the conference.
Finally, there are some summarizing comments about how the experiment results
presented here compare to expectations and what we may hope for the future.
## I Introduction
Before turning to the experiment presentations at FPCP 2012, it may be useful
to review the broader context of the FPCP conference and of the measurements.
We often say that particle physics is the study of matter, energy, space, and
time. What do we really want to know?
For matter, we want to know:
* •
Why is the universe so dominantly matter; why is there so little antimatter
around?
* •
Why is matter made of quarks and leptons, antiquarks and antileptons?
* •
Why are these constituents spin $\frac{1}{2}$ particles?
* •
Why do quarks and leptons come in three generations? (And, are there only
three?)
* •
Why do the quark generations have such different masses?
* •
Why are neutrino masses so small; and why different by generation? Are
neutrinos Dirac or Majorana particles?
* •
Why do quarks come in three colors?
For energy:
* •
Why is so little of the energy density in the universe composed of the mass
that I just listed?
* •
What is the dark matter we claim is the rest of the matter?
* •
Why are the force carriers spin 1; not spin 0 for example? Why are gravitons
spin 2?
* •
Why are the strong and electroweak forces flavor independent?
* •
Why aren’t all interactions flavor independent?
And, for space and time:
* •
Why are there three obvious spatial dimensions? Are there more?
* •
Why is the expansion of the universe accelerating? Or, do we not understand
gravity/space?
As an aside, note the preponderance of the number three in my list of
questions: three dimensions, three generations, and three colors. Are all
these trinities related? Are any of them related?
Sometimes we have ideas about possible keys to unlocking the answers to some
of these questions:
* •
Some kind of substructure to explain the pattern of quarks and leptons and
generations we see.
* •
The Higgs mechanism as what nature has chosen for ElectroWeak Symmetry
breaking? Do the masses of the $W$ and $Z$ come from the same mechanism as
that for quark masses.
* •
A seesaw mechanism involving very massive right-handed partners as the source
of the very light neutrino masses.
* •
The neutrino sector as the source of the matter/antimatter asymmetry of the
visible universe. We know now that the asymmetry is not from the CKM matrix in
the quark sector (inadequate by 8-10 orders of magnitude).
Not all these ideas are directly testable. Those that are, we are working hard
to test. We are doing it by going to higher energies at the LHC and by making
more and more precise measurements in the flavor sector. The latter is the
focus of our FPCP conferences, of course.
I am reminded of the story of the drunk and the lamp post. Maybe this is only
well known in the West, so indulge me if I tell it here. A drunk has lost his
keys and is spending a long time looking for them under a lamp post. When
asked why he is still looking there, he says that that is where there is
enough light to see them! We need to be careful not to make the same mistake.
Certainly, look where we think we may find the answers because we have good
models to test. However, in parallel, probe as deeply as we can where we don’t
have such light to guide us. We have vast new data sets, and we need to check
for the unexpected, too. As a trivial example, in studying decays of particles
with heavy flavor to $h^{\pm}\ell^{+}\ell^{-}$, look for
$h^{-}\ell^{+}\ell^{+}$, etc. as well. No matter what our theorist friends
tell us about where the answers lie, we have already seen the preferred space
of minimal SUSY disappear. And, the space for the Higgs to hide is also
closing down. Sorry, I am supposed to say that we are closing in on the Higgs
at about 125 $GeV/c^{2}$! (I note in this post conference write-up the
announcements from the LHC and Tevatron after the FPCP 2012 conference
indicating the possible discovery of a Standard-Model Higgs boson ATLAS-Higgs
; CMS-Higgs !)
## II Results shown at FPCP 2012
This brings us to the beautiful results shown at FPCP 2012, many results new
since the last meeting, some shown here publicly for the first time. Since I
cannot include all my favorite slides in the write-up of the talk, I can only
refer the reader to the slides for this summary available from the Proceedings
link on the conference web site FPCP as an accompaniment to reading this
article. As you look over the slides I have selected from the meeting, you
will see a personal selection. Afterall, choosing post cards when you are on
travel is a personal matter. So, I show some of my favorite postcards from
FPCP 2012, those I found especially pretty or revealing.
I won’t repeat all the excellent explanations of the results. I could not do
as well as we have heard from the presenters themselves. These explanations
are available in slides from the other talks, also available at the conference
web site, or in the individual write-ups from the presenters, also available
on links from the conference web site.
I would certainly choose a postcard each to remind me of the conference site,
the Chinese opera we saw, and the Bao Gong Memorial Park. However, there are
also lots of postcards to select with pretty plots and physics content.
My souvenir postcards are organized in eight categories:
* •
Standard-Model confirmations
* •
Significant reductions in uncertainty
* •
New-Physics space ruled out
* •
Tension with the Standard Model
* •
New signals and structures
* •
Hints of new physics
* •
New techniques and looking beyond the lamp post
* •
Postcards of the future
Remember, these are just souvenir post cards, visually impressive views of the
various physics topics presented, not part of a guide book. See the individual
talks for details at the level of a guide book. It should also be obvious that
most measurements that I reference could be listed for more than one category.
### II.1 Standard-Model confirmations
New measurements of the speed of neutrinos have confirmed that the earlier
measurement of neutrino speed by the OPERA experiment (now revised also by
OPERA itself) was wrong. Neutrinos are not superluminal. They do not travel
faster than the speed of light. In his talk, Andrew Cohen noted cohen that we
should have realized this from known physics, in particular the lack of
radiation by high-energy neutrinos from very far away, for example. The OPERA
result did motivate serious thinking about the issue, and generated
calculations of new, very sensitive limits on the violation of Lorentz
invariance by neutrinos.
Andrzej Bozek showed Belle’s result on the rate of $D_{S}\rightarrow\tau\nu$,
consistent with lepton universality bozek . There is also continuing progress
in measurements of $CP$ violation in the quark sector. Giovanni Marchiori
showed BaBar’s first three-sigma evidence of $CP$ violation in the decay of
the $B$ to three $K_{S}^{0}$’s from an analysis of the time-dependence of the
decays marchiori ; and we saw the first three-sigma evidence of $CP$ violation
in $B_{S}$ decays from LHCb as shown by Irina Nasteva nasteva . Also, note the
evidence shown by Yuehong Xie that the heavier $B_{S}$ lives longer than its
lighter sister (also from LHCb) xie .
Finally, the window for a Standard-Model Higgs continues to be better defined
with the more precise measurements of the $W$ mass by CDF and DZero at the
Tevatron charlton . The continued reduction in the “oval of uncertainty” in
the Higgs mass from ever more precise top-quark and $W$ mass measurements at
the Tevatron (consistent with a light Higgs as predicted from electroweak
measurements and possibly observed as announced soon after the FPCP meeting
ATLAS-Higgs ; CMS-Higgs ).
### II.2 Significant reductions in uncertainty
Perhaps the most startling reduction in a measurement uncertainty has come
with the surprisingly-quick measurement of $\theta_{13}$ of the neutrino-
mixing matrix parameterization as shown in the talks of Werner Rodejohann,
Jianglai Liu, and Phillip Litchfield rodejohann ; liu ; litchfield . The value
measured, first and best so far by the Daya Bay experiment, is near the
previous upper limit on this parameter. The optimists were right in this case.
There is also the improvement in the uncertainties in the parameters of the
so-called unitarity triangles of the CKM quark-mixing matrix as shown in
individual presentations and summarized in the talk by Sebastien Descotes-
Genon genon .
Also impressive are the measurements of the properties of the $h_{C}$ by
BESIII shown by Guangshun Huang huang and improvements in the now-lower value
of $y_{CP}$ in $D^{0}$ decays, including the new measurements from BaBar and
Belle, reported by Chunhul Chen c-chen .
Finally, I note separately the improvements in $B_{S}$ mixing parameters,
including the value of $\phi_{S}$ reported by Sebastien, Fabrizio Ruffini, and
Yuehong Xie genon ; ruffini ; xie , progress in reducing the semileptonic
decay uncertainties, e.g., in bottom decays as reported by Vera Luth luth ,
and in charm form factors from BESIII as reported by Jonas Rademacker
rademacker .
### II.3 New physics space ruled out
The parameter space for physics beyond the Standard Model is multidimensional.
We have become used to presentations of limits when models are reduced to two
relevant parameters, whether we are talking about dark matter in terms of
cross sections and particle mass or SUSY models at the selected internal-
parameter level. Some of the slides presented show measurement limits for
physical parameters with a range of model-possibility predictions of the
parameters as generated by Monte Carlo techniques to give a sense of how
effective the measurements are in restricting the range of model parameters.
Other limits come directly from two-dimensional plots of possible model
parameters with sections of the space ruled out by the measurements. Some of
the plots are quite colorful and artistic in appearance, as well as providing
physics insight.
Some of my favorite plots come from the presentation on dark-matter search
limits by Xinchou Lou lou both for generic dark matter and for a possible
dark Higgs. Improved limits on new physics come from lepton-flavor violation
searches as in the slide on $\mu\rightarrow e\gamma$ shown by Francesco Renga
renga and from the ratio of leptonic two-body $K$ decays,
$R_{K}=\Gamma(K\rightarrow e\nu)$ over $\Gamma(K\rightarrow\mu\nu)$ shown by
Evgueni Goudzovski goudzovski . Two of the most colorful plots are those
showed by Vincenzo Chiochia where constraints on new physics come from top-
quark-production asymmetries (forward-backward and charge asymmetries)
measured at the Tevatron and LHC chiochia and by Nicola Serra showing limits
from the rare processes $B\rightarrow\mu\mu$ and $B_{S}\rightarrow\mu\mu$
serra taken from the presentation by David Straub at the EW Rencontres de
Moriond this year. Finally, I would include in my postcard collection, the
distributions of observed events in the search for $\tau\rightarrow\mu\mu\mu$
at LHCb shown by Paul Seyfert seyfert .
### II.4 Tension with the Standard Model
Various fits to measurements of CKM unitarity triangles have been shown to
highlight possible discrepancies in the single-phase paradigm of the Standard
Model. Tensions with the Standard-Model overall fits have been observed,
mention being made by Sebastien Descotes-Genon genon and Koji Hara hara of
issues between the value of $\sin(2\beta)$ and the rate of $B$ decay to
$\tau\nu$. Also mentioned by Sebastien were the semileptonic asymmetry in
$B$-decay and $B_{s}$-decay parameters and the same-sign dimuon charge
asymmetry measured by CDF and DZero at the Tevatron. Amarjit Soni showed a
nice plot from a fit without inputs from semileptonic decays to address
concerns over sensitivity to $V_{cb}$ soni .
Vera Luth noted that the “tension” between inclusive and exclusive analyses of
semileptonic B decays remains luth , while stated uncertainties on the
branching fractions and on $|V_{ub}|$ and $|V_{cb}|$ are being reduced. The
search for discrepancies between measurements and Standard-Model predictions
of $B$-decay rates has revealed a significant excess of events in
$B\rightarrow D\tau\nu$ and in $B\rightarrow D^{*}\tau\nu$ (3.4 $\sigma$ when
the two BaBar excesses are combined). This feature cannot be easily explained.
Finally, I would point to the nice plots in the presentation by Rick Van
Kooten vankooten showing some tension in $B_{S}$ decay parameters.
### II.5 New signals and structures
Clearly, the observation of electron neutrinos coming from the oscillation of
muon neutrinos, giving the large observed value of $\theta_{13}$, is a new
signal. I show two more plots on this, which include the individual
measurements from Daya Bay, Reno, Double Chooz, T2K, MINOS, KamLAND, and
solar-neutrino measurements liu ; litchfield .
We also saw unexpected structures in baryonic $B$ decays from BaBar as shown
by Irina Nasteva nasteva and, at 6.3 $\sigma$, the first observation of
$B_{S}\rightarrow\phi\mu\mu$ from CDF shown by Rick Van Kooten vankooten .
The evidence for new $Z$-onium states is becoming more and more compelling,
given the nice plots of $Z_{1,2}\rightarrow h_{b}(1,2)\pi$ from Belle shown by
Jin Li li . Another visually compelling set of plots demonstrated the
suppression of the production the heavier upsilon mesons relative to the
ground-state upsilon in heavy-ion collisions at CMS as shown by Zebo Tang tang
.
Another signal that is becoming clear with the recent increase of data at LHCb
is the zero-crossing point in the forward-backward asymmetry in the decay of
$B\rightarrow K^{*}\mu\mu$ shown by Nicola Serra serra . I have also selected
three slides from the presentation of BESIII results shown by Guangsung Huang
huang : the newly observed isospin-breaking decay of $\eta(1405)\rightarrow
f_{0}(980)\pi^{0}$, an anomalous lineshape of the $f_{0}$ in the decay
$J/\psi\rightarrow\gamma f_{0}\pi^{0}$, and the first evidence of
$\psi(2S)\rightarrow\gamma\gamma J/\psi$.
Perhaps the most unusual new signal was the first direct observation of time-
reversal-symmetry violation in any system. The direct observation comes from
using entangled $\Upsilon(4S)$ decays to tagged $B$ mesons. The results shown
by Pablo Villanueva Perez perez come from BaBar.
As already noted, from a time-dependent Dalitz-plot analysis, we saw the first
evidence of CPV in $B_{S}\rightarrow K_{S}K_{S}K_{S}$ in the talk by Giovanni
Marchiori.
### II.6 Hints of new physics
Among the hints of new physics is the forward-back asymmetry in $t\bar{t}$
production at CDF and DZero, increasing with the mass of the $t\bar{t}$ system
at CDF as shown by Marc Besancon besancon . There are also hints of the Higgs
at ATLAS and CMS shown by David Charlton charlton , preceding the already-
mentioned announcements after the FPCP conference. I continue to think of this
as new physics, though most of you may already consider this a part of the
Standard Model.
The surprisingly-large $D^{0}$ mixing observed has led to suggestions that
there might be $CP$-symmetry violation in the charm sector. Combining LHCb and
CDF results on the difference of $CP$ asymmetries in $D^{0}$ decays to
$K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ led Vincenzo Vagnoni to say that the data
“is consistent with no $CP$ violation at 0.006% CL” vagoni . Even stated in
this way, of course, at this point, we can only say that the data is
inconsistent with no asymmetry at the given confidence level. The issue of
$CP$ violation is still being debated among theorists! Can the asymmetry be
due to Standard-Model long-distance effects?
### II.7 New techniques and looking beyond the lamp post
New techniques include both the application of new analysis methods and
improvements made to previous analyses. Manuel Tobias Schiller showed LHCb
multibody-decay results using combined Gronau-London-Wyler (GLW) and Atwood-
Dunietz-Soni (ADS) methods and also taking advantage of the strong variation
of hadronic parameters over the Daliz-plot phase space schiller . David
Charlton showed how analysis improvements allowed CDF to obtain more precise
results in their search for the Higgs boson than what one would expect from
the simple increase in data as the integrated luminosity grew over time
charlton . Similar improvements have been made at DZero, and we may expect
similar things from the LHC experiments too, as they accumulate more data and
experience.
As mentioned earlier, searching for the unexpected is looking beyond the lamp
post. Liang Sun showed the event distributions for searches for an unexpected
$B^{+}\rightarrow D^{-}\ell^{+}\ell^{+}$ by Belle sun .
### II.8 Postcards of the future
I was going to reserve the last slide for postcard images of the new
facilities shown in the first session of the final day. However, I decided to
focus on the physics we expect most. The fact is, I am less certain what to
expect than I have been for many years. So, I have left space in my postcard
collection for next year’s souvenirs!
## III Summary
As we flipped through the souvenir picture post cards of experiment results
selected from all those presented at FPCP 2012, we have seen a wonderfully-
rich abundance of new results. Yet, the selection presented here is
necessarily incomplete. At best, it gives a sense of the impressive range of
activity in flavor-physics and $CP$-violation experiments, and of the very
high quality of the data and analyses being generated.
There were over 40 experiment talks! In these talks, there were presentations
of results which included Standard-Model confirmations, significant reductions
in uncertainty, New Physics spaces ruled out, new signals and structures,
hints of New Physics, and still some tension with the Standard Model. And,
there were also places where results were presented which explored the
possibility of unexpected signals, looking beyond the lamp post.
There were, perhaps, fewer outstanding experimental issues relative to what
was presented at FPCP 2011, less tension with the Standard Model this year.
Nevertheless, there is growing disquiet over not seeing directions for the
answers to the questions about matter, energy, space, and time that were
listed in my introduction. There is no certainty about the direction of the
needed New Physics. [The signals around 125 $GeV/c^{2}$ shown by ATLAS and CMS
after FPCP 2012 also seems consistent with the Standard-Model Higgs boson,
though this is by no means yet proven.] We may hope for one or more bright new
ideas that could provide additional lamp posts to light our way. But this hope
is not something that we can count on soon. Thus, it is hard to predict what
the post cards will look like next year at FPCP 2013 in Buzios, Brazil.
Nevertheless, given the huge data sets collected and anticipated, there is
excellent reason for hope! As has happened in the past, data may be the key to
future progress in our understanding.
###### Acknowledgements.
I close this last talk of the Flavor Physics and CP Violation 2012 meeting
with a big and sincere thank you to all the presenters. Obviously, I have
taken freely from their presentations for this summary. I also want to thank
especially the organizers of this very enjoyable, interesting, and informative
meeting. My work is supported, in part, by the US Department of Energy through
Fermilab, which is operated by Fermi Research Alliance, LLC under Contract No.
DE-AC02-07CH11359 with the Department of Energy.
## References
* (1) http://www.atlas.ch/news/2012/latest-results- from-higgs-search.html
* (2) http://cms.web.cern.ch/news/observation-new- particle-mass-125-gev
* (3) http://hepg-work.ustc.edu.cn/fpcp2012/
* (4) A. Cohen,“Theoretical aspects of neutrino speed anomaly”, FPCP2012, Hefei, China, May, 2012.
* (5) A. Bozek, “Leptonic $D$ decays”, FPCP2012, Hefei, China, May, 2012.
* (6) G. Marchiori, “$CP$ violation and rare $B$ decays at the $B$ factories”, FPCP2012, Hefei, China, May, 2012.
* (7) I. Nasteva, “Hadronic $B$ decays”, FPCP2012, Hefei, China, May, 2012.
* (8) Y. Xie, “$CP$ violation in $J/\psi\phi$”, FPCP2012, Hefei, China, May, 2012.
* (9) D. Charlton, “Higgs searches at LHC and Tevatron”, FPCP2012, Hefei, China, May, 2012.
* (10) W. Rodejohann, “Status of the PMNS matrix”, FPCP2012, Hefei, China, May, 2012.
* (11) J. Liu, “Status report from Daya Bay”, FPCP2012, Hefei, China, May, 2012.
* (12) P. Litchfield, “$Theta_{1}3$”, FPCP2012, Hefei, China, May, 2012.
* (13) S. Descotes-Genon, “CKM matrix and constraints on new physics from quark flavour observables”, FPCP2012, Hefei, China, May, 2012.
* (14) G. Huang, “Recent results from BESIII”, FPCP2012, Hefei, China, May, 2012.
* (15) C. Chen, “$D$-$D$bar mixing, direct and indirect $CP$ violation”, FPCP2012, Hefei, China, May, 2012.
* (16) F. Ruffini, “Hadronic $B_{s}$ decays”, FPCP2012, Hefei, China, May, 2012.
* (17) V. Luth, “Semileptonic $B$ decays”, FPCP2012, Hefei, China, May, 2012.
* (18) J. Rademacker, “Semileptonic $D$ decays”, FPCP2012, Hefei, China, May, 2012.
* (19) X. Lou, “Searches for low mass new physics”, FPCP2012, Hefei, China, May, 2012.
* (20) F. Renga, “Searches for $LFV$ and $LNV$ in charged lepton decays”, FPCP2012, Hefei, China, May, 2012.
* (21) E. Goudzovski, “Recent kaon physics results and prospects”, FPCP2012, Hefei, China, May, 2012.
* (22) V. Chiochia, “Top quark properties and top decays”, FPCP2012, Hefei, China, May, 2012.
* (23) N. Serra, “$B_{s}\rightarrow\mu\mu,K^{*}\mu\mu$”, FPCP2012, Hefei, China, May, 2012.
* (24) P. Seyfert, “Searches for $LFV$ and $LNV$ in hadron decays”, FPCP2012, Hefei, China, May, 2012.
* (25) K. Hara, “$B$ leptonic decays”, FPCP2012, Hefei, China, May, 2012.
* (26) A. Soni, “New ideas and directions in flavor physics/$CP$ violation”, FPCP2012, Hefei, China, May, 2012.
* (27) R. Van Kooten, “Semileptonic $B_{s}$ decays”, FPCP2012, Hefei, China, May, 2012\.
* (28) J. Li, “New and conventional bottomonium states”, FPCP2012, Hefei, China, May, 2012.
* (29) Z. Tang, “Heavy flavor production in heavy ion collisions”, FPCP2012, Hefei, China, May, 2012.
* (30) P. Villanueva Perez, “First direct measurement of time-reversal violation $B$ decays”, FPCP2012, Hefei, China, May, 2012.
* (31) M. Besancon, “Top quark production at the Tevatron”, FPCP2012, Hefei, China, May, 2012.
* (32) V. Vagnoni, “$CP$ violation in $D$ decays”, FPCP2012, Hefei, China, May, 2012\.
* (33) M. Tobias Schiller, “Measurements of CKM angle gamma from tree-dominated decays”, FPCP2012, Hefei, China, May, 2012.
* (34) L. Sun, “$b\rightarrow s\gamma$”, FPCP2012, Hefei, China, May, 2012.
|
arxiv-papers
| 2012-09-19T22:48:18 |
2024-09-04T02:49:35.328213
|
{
"license": "Public Domain",
"authors": "Jeffrey A. Appel",
"submitter": "Jeffrey A. Appel",
"url": "https://arxiv.org/abs/1209.4384"
}
|
1209.4529
|
# A Monte Carlo simulation study on the wetting behavior of water on graphite
surface
Xiongce Zhao [email protected] Joint Institute for Computational Sciences
and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA Current address: NIDDK, National Institutes
of Health, Bethesda, MD 20892, USA
###### Abstract
This paper is an expanded edition of the rapid communication published several
years ago by the author (_Phys. Rev. B, v76, 041402(R), 2007_) on the
simulation of wetting transition of water on graphite, aiming to provide more
details on the methodology, parameters, and results of the study which might
be of interest to certain readers. We calculate adsorption isotherms of water
on graphite using grand canonical Monte Carlo simulations combined with
multiple histogram reweighting, based on the empirical potentials of SPC/E for
water, the 10-4-3 van der Waals model, and a recently developed induction and
multipolar potential for water and graphite. Our results show that wetting
transition of water on graphite occurs at 475-480 K, and the prewetting
critical temperature lies in the range of 505-510 K. The calculated wetting
transition temperature agrees quantitatively with a previously predicted value
using a simple model. The observation of the coexistence of stable and
metastable states at temperatures between the wetting transition temperature
and prewetting critical temperature indicates that the transition is first
order.
###### pacs:
68.35.Rh, 64.70.Fx, 82.20.Wt
## I INTRODUCTION
When a fluid adsorbs on a solid surface at temperatures below its liquid-vapor
critical temperature ($T_{\rm c}$), the adsorbed film either spreads across
the surface (wetting) or beads up as a droplet (nonwetting) as the pressure
approaches the saturated vapor pressure Psvp of the fluid. Wetting transition
describes the transition between those two kinds of behavior. Physically the
wetting transition corresponds to the phenomena when the contact angle of the
liquid drop on the surface changes from a nonzero value to zero. Analysis of
wetting transition was first presented 30 years ago by Cahn Cahn (1977) and
Ebner and Saam. Ebner and Saam (1977) They showed that if a fluid does not wet
a particular surface at low temperature, then the system ought to exhibit
wetting transition at some temperature Tw below $T_{\rm c}$. In terms of
adsorption isotherms, the wetting phenomenon should manifest itself as
following three different patterns. (1) At temperatures below Tw, adsorption
beginning with a thin film increases slightly as the pressure increases
towards the saturation pressure Psvp. At Psvp the bulk vapor condenses
completely, and the adsorption coverage becomes infinite. On a coverage versus
pressure diagram, the adsorption isotherm reaches Psvp with a discontinuous
jump (infinite slope). (2) In the temperature range between Tw and the
prewetting critical temperature Tpwc, the thin film grows as the pressure
increases until it jumps to a thick, liquid like film of finite thickness at
some pressure less than Psvp. This thin-to-finite film transition or wetting
transition is followed by continuous growth until condensation occurs at Psvp.
(3) At temperatures higher than Tpwc, the film grows continuously and the
prewetting transition disappears. A schematic diagram showing these three
types of adsorption patterns is given in FIG. 1.
Figure 1: Schematic diagram of adsorption isotherms near a wetting transition.
$T_{\mathrm{pwc}}>T_{3}>T_{2}>T_{\mathrm{w}}>T_{1}$.
Since the first theory on wetting transition was developed a variety of
experimental Mistura et al. (1994); Ross et al. (1995); Hallock (1995);
Demolder et al. (1995); Wyatt et al. (1995); Yao and Hensel (1996); Hess et
al. (1997); Ross et al. (1997); Kozhevnikov et al. (1997); Ross et al. (1998);
Hensel and Yao (1998); Ohmasa et al. (1998, 2001); Kozhevnikov et al. (1998)
and theoretical studies Ebner and Saam (1987); Finn and Monson (1989); Cheng
et al. (1991, 1993); Wagner and Ceperley (1994); Bojan et al. (1998);
Boninsegni and Cole (1998); Ancilotto and Toigo (1999); Bojan et al. (1999);
Curtarolo et al. (2000); Ancilotto et al. (2001); Shi et al. (2003); Gatica et
al. (2004) have been performed on the wetting transition of fluids on various
solid surfaces. Most of these studies were focused on simple fluids such as He
and H2 isotopes on alkali metal surfaces. One common feature of these systems
is that the fluid-surface interaction is only weakly attractive. This implies
that the bead-up of fluid on the surface at Psvp will be favorable over the
continuous growth of a film.
Finn and Monson Finn and Monson (1989) were among the first to calculate the
wetting temperature of fluid on solid surface using molecular simulations.
They predicted the wetting behavior of Ar on solid CO2 surface using isobaric-
isothermal Monte Carlo simulations. Shi et al. Shi et al. (2002) reevaluated
this system using grand canonical Monte Carlo (GCMC) simulations plus multiple
histogram reweighting techniques. Errington Errington (2004) studied the same
system by employing a new simulation method. Bojan et al. Bojan et al. (1999)
studied wetting behavior of Ne on surfaces with various interaction strengths.
Curtarolo et al. Curtarolo et al. (2000) used GCMC simulations to study the
wetting behavior of inert gases on alkali and Mg surfaces. Shi et al. Shi et
al. (2003) studied the wetting transition of hydrogen isotopes on Rb surface
by including the quantum effects of the fluids using path integral hybrid
Monte Carlo simulations.
The wetting transition of fluid-fluid systems were studied by both experiments
and theory, Bonn and Ross (2001) but the wetting behavior of fluids on solid
surfaces was only reported for atomistic molecules such as inert gases. No
wetting transition has ever been seen for any molecular fluids on solids other
than hydrogen and its isotopes, which are essentially spherical molecules. To
our knowledge, the wetting transition involving water has not been studied
until very recently Gatica et al. (2004) although water is an extensively
studied molecule. It is known that water does not wet many surfaces (such as
graphite) at room temperature. Theoretically, the wetting transition of water
on graphite is expected to occur at a temperature below its bulk critical
temperature. In a recent paper by Gatica et al., Gatica et al. (2004) wetting
temperatures of water on graphite had been predicted using a simple model
based on the water-solid interaction calculated from empirical potentials. The
recommended Tw from their calculations for water on graphite is 474 K.
In this study, we report evidence for the first-order wetting transition of
water on graphite from molecular simulations. We estimate the wetting
transition temperature and prewetting critical temperature of water on
graphite using grand canonical Monte Carlo simulations. The paper is organized
as the following: The next section describes the potential models and
simulation methodology. Section III presents the results and discussion.
Section IV summarizes our findings.
## II POTENTIALS AND METHODS
Water-water interaction is described by the SPC/E model. Berendsen et al.
(1987) This model is widely used in modeling systems involving water. The
model includes a Lennard-Jones site located on the oxygen atom and three
partial charge sites on each atom. The parameters for this model are given in
TABLE 1. The critical temperature of water calculated using the SPC/E model is
635 K. Hayward and Svishchev (2001) This value is the closest to the
experimental value (647 K) compared with the predictions by many other popular
nonpolarizable water potentials. Guillot (2002) Up to date there is no
potential available being able to reproduce in every detail the properties of
real water. Guillot (2002) It is known that the ability of an interaction
potential to describe the bulk critical behavior is a necessary (though not
sufficient) requirement in order for it to predict the wetting transition
behavior of the fluid on a surface. Bonn and Ross (2001) Therefore, we chose
the SPC/E model among tens of water potentials available.
The graphite surface is modeled as a smooth basal plane. The Lennard-Jones
interaction between a water molecule and the graphite surface is given by the
10-4-3 potential Steele (1973)
$\begin{split}V_{\mathrm{sfLJ}}(z)=2\pi\varepsilon_{\mathrm{sf}}\sigma_{\mathrm{sf}}^{2}\Delta\rho_{\mathrm{s}}&\bigg{[}\frac{2}{5}\left(\frac{\sigma_{\mathrm{sf}}}{z}\right)^{10}-\left(\frac{\sigma_{\mathrm{sf}}}{z}\right)^{4}\\\
&-\frac{\sigma_{\mathrm{sf}}^{4}}{3\Delta(0.61\Delta+z)^{3}}\bigg{]},\end{split}$
(1)
where z is the distance between the oxygen atom in a water molecule and the
graphite surface in the surface normal direction, $\sigma_{\mathrm{s}}$ is the
number density of carbon atoms in graphite, and $\Delta$ is the distance
between the graphene sheets in graphite. The graphite surface corrugation is
not included in this potential. The graphite-water interaction parameters
$\varepsilon_{\mathrm{sf}}$ and $\sigma_{\mathrm{sf}}$ are calculated from the
Lorentz-Berthelot rules,
$\varepsilon_{\mathrm{sf}}=(\varepsilon_{\mathrm{s}}\varepsilon_{\mathrm{f}})^{1/2},\quad\sigma_{\mathrm{sf}}=(\sigma_{\mathrm{s}}+\sigma_{\mathrm{f}})/2.$
The values of the parameters are: $\rho_{\mathrm{s}}$=114 nm-3, $\Delta$=0.335
nm, $\varepsilon_{\mathrm{s}}$=0.05569 kcal/mol, and
$\sigma_{\mathrm{s}}$=0.340 nm for graphite,
$\varepsilon_{\mathrm{f}}=\varepsilon_{\mathrm{O}}$=0.1554 kcal/mol, and
$\sigma_{\mathrm{f}}=\sigma_{\mathrm{O}}$=0.3165 nm for water.
A recently developed effective potential for the dipole-induced dipole,
dipole-quadrupole, and quadrupole-quadrupole interactions between polar fluids
and graphite Zhao and Johnson (2005) is used to calculate the water-graphite
polar interactions. For water/graphite system, only the induction and dipole-
quadrupole terms are important,
$\begin{split}V_{\mathrm{polar}}(z)=&-\frac{\pi\Delta\rho_{\mathrm{s}}\mu_{\mathrm{f}}^{2}}{(4\pi\varepsilon_{0})^{2}}\bigg{[}\frac{\alpha_{\mathrm{C}}}{2}\left(\frac{1}{z^{4}}+\frac{1}{3\Delta(\Delta+z)^{3}}\right)\\\
&\,\,\,\,\,\,\,+\frac{\Theta_{\mathrm{C}}^{2}}{3k_{\mathrm{B}}T}\left(\frac{1}{z^{6}}+\frac{1}{5\Delta(\Delta+z)^{5}}\right)\bigg{]},\end{split}$
(2)
where $\varepsilon_{0}$ is the vacuum permittivity, $k_{\mathrm{B}}$ is the
Boltzmann’s constant, $\mu_{\mathrm{f}}$ is the dipole moment of the water
molecule, $\alpha_{\mathrm{C}}$ is the isotropic polarizability of a carbon
atom in graphite, and $\Theta_{\mathrm{C}}$ is the permanent quadrupole moment
on each carbon atom in graphite. The values for these parameters are
$\mu_{\mathrm{f}}$=1.85 Debye, $\alpha_{\mathrm{C}}=1.76\times 10^{-3}$ nm3,
Miller and Bederson (1978) and $\Theta_{\mathrm{C}}=-3.03\times 10^{-40}$C m2.
Whitehouse and Buckingham (1993) We note that $\mu_{\mathrm{f}}$ calculated
for SPC/E model is 2.07 Debye, but 1.85 Debye is chosen here so that
comparison can be made between this study and Gatica et al.’s prediction.
Ewald summations were applied in simulations to account for the long-range
correction to electrostatic interactions. Since only two dimensional periodic
boundary conditions along x and y directions were applied in the adsorption
simulations, a pseudo-two-dimensional Ewald summation method Yeh and Berkowitz
(1999) was used. The total electrostatic energy is given by
$\begin{split}V_{el}=&\frac{1}{2V_{0}\varepsilon_{0}}\sum_{k{\neq}0}^{\infty}\frac{e^{-k^{2}/{4\alpha^{2}}}}{k^{2}}\bigg{|}\sum_{j}^{N}{q_{j}e^{-i\bf{k}{\cdot}{\bf{r}}_{j}}}{\bigg{|}^{2}}\\\
&+{\frac{1}{4\pi\varepsilon_{0}}}\bigg{[}\sum_{n<j}^{N}{\frac{q_{n}q_{j}}{r_{nj}}\mathrm{erfc}(\alpha
r_{nj})}\\\
&-\sum_{mole}{\bigg{(}{\frac{\alpha}{\sqrt{\pi}}}{\sum_{j=1}^{site}q_{j}^{2}}+\sum_{n,j}^{list}{\frac{q_{n}q_{j}}{r_{nj}}}\bigg{)}}\bigg{]}+{\frac{M_{z}^{2}}{2V_{0}\varepsilon_{0}}},\end{split}$
(3)
where V0 is the volume of the simulation cell and $\alpha$ is the Ewald
convergence parameter. In Eq. (3), the first term is the reciprocal sum for
all the Gaussian charges, k is the reciprocal lattice vector, qj is the
partial charge on interacting site j, rnj is the vector from rn to rj. The
second term is the standard real-space sum, erfc is the complementary error
function. The third term is the self-exclusion on each molecule, plus the 1-2
and 1-3 intramolecular exclusions, where site is the total number of partial
charge sites in the molecule, and list contains all the 1-2, 1-3 exclusions.
The last term is the 2D correction term, where Mz is the z component of the
total dipole moment of the simulation box.
Table 1: Potential parameters for SPC/E model. Berendsen et al. (1987) $r_{\mathrm{OH}}$ is the O-H bond length, $\theta_{\mathrm{HOH}}$ is the H-O-H bond angle. $\varepsilon_{\mathrm{O}}$ and $\sigma_{\mathrm{O}}$ are Lennard-Jones parameters for the O atom. $q_{\mathrm{O}}$ and $q_{\mathrm{H}}$ are the partial charges on O and H atoms, respectively. $r_{\mathrm{OH}}$[nm] | $\theta_{\mathrm{HOH}}$ | $\varepsilon_{\mathrm{O}}$[kcal/mol] | $\sigma_{\mathrm{O}}$[nm] | $q_{\mathrm{O}}$[$e$] | $q_{\mathrm{H}}$[$e$]
---|---|---|---|---|---
0.1 | 109.47∘ | 0.1554 | 0.3165 | $-0.8476$ | 0.4238
We have used grand canonical Monte Carlo simulations combined with multiple
histogram reweighting (MHR) method Ferrenberg and Swendsen (1988, 1989) to
compute the saturated coexistence chemical potentials Shi and Johnson (2001)
and adsorption isotherms Shi et al. (2002) of water on graphite at various
temperatures. More details on the MHR method can be found from the original
literature Ferrenberg and Swendsen (1988, 1989) and several other articles on
its applications. Shi and Johnson (2001); de Pablo et al. (1999) The GCMC cell
for adsorption simulations is a rectangular box with volume of
2000$\sigma_{\mathrm{f}}^{3}$. The height of adsorption box in the z
direction, $H$, is 15$\sigma_{\mathrm{f}}$, the sides in x and y directions
are 11.5$\sigma_{\mathrm{f}}$. The lower plane normal to the z axis is modeled
as the graphite surface.
Special care was taken to avoid capillary condensation effects on the wetting
transition properties of the system. The plane opposite to the graphite
surface is modeled as a hard repulsive wall. The hard wall is always dry to
the liquid phase and wet to the gas phase. Therefore it helps to suppress the
capillary condensation. Evans and Marini Bettolo Marconi (1985); van Swol and
Henderson (1986); Parry and Evans (1990, 1992); Finn and Monson (1989) The
capillary condensation can also be eliminated by using a simulation cell with
sufficient separations between the adsorbent surface and the opposite hard
wall. Finn and Monson (1989); Bojan et al. (1999); Curtarolo et al. (2000);
Shi et al. (2003) However, an overall small cell volume is preferred for the
MHR technique. Therefore, we have performed series of trial simulations using
various cell heights to search for an appropriate value of $H$. Trial
simulations were performed using cell heights ranging from 10$\sigma_{\rm f}$
to 40$\sigma_{\rm f}$ at interested temperatures and chemical potentials
(pressures). It was found that the adsorption properties such as isotherms
obtained at 15$\sigma_{\rm f}$ are consistent with those obtained at
20$\sigma_{\rm f}$, 30$\sigma_{\rm f}$, and 40$\sigma_{\rm f}$ within the
statistical fluctuations (see FIG. 2 for an example). This indicates that
15$\sigma_{\rm f}$ is adequate for effectively avoiding the influence of
capillary condensation for the systems of interest. Based on this we chose
15$\sigma_{\rm f}$ as the cell height in most of our GCMC simulations.
Additional simulations with $H$=20$\sigma_{\rm f}$ were performed as
verifications at each temperature.
The type of move to attempt during a GCMC simulation was selected randomly
with probability of 0.45, 0.45, 0.05, and 0.05 for displacements, rotations,
creations, and deletions of a water molecule, respectively. Each simulation
included equilibration of 80$\times 10^{6}$ MC moves and production of
20$\times 10^{6}$ MC moves. Histograms were collected every 20 MC moves during
the production. The cutoff of pair-wise Lennard-Jones interaction between
water molecules was 0.9 nm as suggested by the original literature, without
long-range correction applied. Berendsen et al. (1987)
In the light of Gatica et al.’s prediction, Gatica et al. (2004) we chose to
perform GCMC simulations at 460, 470, 480, 490, 500, 510 K with varying
reduced chemical potentials to obtain the histograms for bulk water and
water/graphite adsorption systems. The values of bulk saturation chemical
potentials ($\mu_{\mathrm{svp}}^{*}$) at each different temperature can be
determined using the MHR method by combining the histograms collected. The MHR
provide very precise values of $\mu$svp through the equal area criterion, Shi
and Johnson (2001) which is very important for studying wetting transitions.
Sufficient overlap between histograms of adjacent state points is necessary in
order to use the MHR technique. We employed the method proposed by Shi et al.
Shi et al. (2002) to check the overlap of any two adjacent state points.
According to this method, the grand canonical partition functions extrapolated
by histogram reweighting method for any two adjacent state points should
approximately satisfy
$\frac{\Xi(\mu_{i},V,T_{i})}{\Xi(\mu_{j},V,T_{j})}\bigg{|}_{\rm
HR}\times\frac{\Xi(\mu_{j},V,T_{j})}{\Xi(\mu_{i},V,T_{i})}\bigg{|}_{\rm
HR}=1\pm\delta,$ (4)
where the subscript HR indicates that the partition function in the numerator
has been extrapolated from the histogram reweighting of the state point in the
denominator. The recommended value for $\delta$ is 0.65 for checking the
overlap of two adjacent state points. Shi et al. (2002) Whenever the overlap
criterion defined in Eq. (4) is not satisfied by any of two adjacent state
points, additional simulations at state points that bridge them were
performed. From preliminary simulations we found that the wetting transition
of water on graphite occurs approximately between 470 and 480 K and the
wetting critical temperature is between 500 and 510 K. In order to narrow down
the values of $T_{\mathrm{w}}$ and $T_{\mathrm{pwc}}$, we performed additional
simulations at 475 K and 505 K.
## III RESULTS AND DISCUSSIONS
GCMC adsorption simulations were carried out for reduced chemical potentials
up to saturation under each selected temperature. Isotherms were calculated
using MHR based on the histograms collected during the simulations. Three
representative isotherms are plotted in FIG. 2. Some of the data points
calculated directly from GCMC simulations but not included in the MHR
calculation are also plotted in the figure to compare with isotherms from MHR.
It can be seen that the difference between the densities obtained from
direction GCMC simulations and those from MHR are small. This serves as a test
of the accuracy of the MHR isotherms.
Figure 2: Typical adsorption isotherms of water on graphite from GCMC
simulations and MHR, where $\rho^{*}$ is the reduced number density and
$\mu^{*}$ is the reduced chemical potential. The curves correspond to $T$=475,
490, 510 K, from left to right, which were computed from MHR using histograms
collected from simulations with cell height $H$=15$\sigma_{\rm f}$. The
symbols are data from individual GCMC simulations and were not included in the
MHR calculations. Circles: $H$=15$\sigma_{\rm f}$, squares: $H$=20$\sigma_{\rm
f}$, diamonds: $H$=30$\sigma_{\rm f}$, triangles: $H$=40$\sigma_{\rm f}$.
To present the isotherms obtained at different temperatures in a concise and
clear fashion, we define a parameter as
$\chi^{*}=\exp\left(\frac{\mu^{*}-\mu_{\mathrm{svp}}^{*}}{T^{*}}\right),$
where $\mu^{*}$ is the reduced chemical potential, $\mu_{\mathrm{svp}}^{*}$ is
the saturation chemical potential at the reduced temperature $T^{*}$. Plotting
the adsorption coverage versus $\chi^{*}$ gives a diagram similar to FIG. 1,
which helps one to identify the wetting transition points without ambiguity.
The parameter $\chi^{*}$ is the ratio of the activity to the activity at
saturation, with $\chi^{*}=1$ corresponding to $\mu=\mu_{\mathrm{svp}}^{*}$,
or $P=P_{\mathrm{svp}}$. Additionally, $\chi^{*}=P/P_{\mathrm{svp}}$ if ideal
behavior is assumed in the bulk vapor phase. However, water vapor cannot be
treated as an ideal gas under the interested simulation temperatures in this
study. For example, the experimental compressibility factor of the saturated
water vapor is about 0.865 Smith et al. (2001) at 500 K.
As the chemical potential was increased toward saturation, three different
types of behavior in the growth of the water adsorption film on graphite were
observed, corresponding to three ranges of temperature. Adsorption isotherms
for water/graphite at several representative temperatures are shown in FIG. 3.
Figure 3: Adsorption isotherms of water on graphite from GCMC simulations and
MHR. The curves correspond to $T$=510, 505, 500, 490, 480, 475 K, from left to
right.
At temperatures below 475 K, the adsorption coverage is minuscule until the
saturation chemical potential is reached, which indicates partial wetting or
nonwetting. The isotherm jumps to the saturated liquid density at
$\chi_{\mathrm{svp}}^{*}$, which can be seen from FIGs. 2, 3, and from the
density profile growth patterns shown in FIG. 4. The sharp increase of density
between $\chi^{*}$=0.991 and $\chi^{*}$=1.008 corresponds to a first-order
transition from nonwetting to liquid condensation (FIG. 4). At
$\chi^{*}$=1.008, much of the density profile becomes comparable with the
saturated liquid density profile (the saturated liquid density at 475 K is
$\rho^{*}\approx$0.89 from bulk GCMC simulations) except for the first peak
corresponds to the density of liquid film in contact with the wall.
In contrast, the simulation results at temperatures between 480 K and 505 K
manifest quite different behavior. Taking the isotherm at 490 K as an example,
there is sudden jump in adsorption from minimum to a finite coverage of about
$\rho^{*}$=0.25 at $\chi^{*}$=0.965 to $\chi^{*}$=0.972 (FIG. 5). But
apparently the increased coverage does not correspond to a liquid condensation
(the saturated liquid density at 490 K is $\rho^{*}\approx$0.85). As the
chemical potential is increased further, the film thickens, which indicates a
wetting behavior (FIGs. 2, 3, 5). By comparing the results shown in FIG. 3 and
FIG. 1 we readily see that wetting transition of water on graphite occurs
somewhere between 475 K and 480 K, i. e., $T_{\mathrm{w}}$=475-480 K. At
temperatures in the range of 480 K and 505 K, the prewetting jump in density
occurs further to the saturation chemical potential with the smaller density
jump as temperature increases (FIG. 3).
Figure 4: The local density profiles for water adsorption on graphite as a
function of reduced distance from the surface, $z^{*}=z/\sigma_{\mathrm{f}}$,
at 475 K. The values of $\chi^{*}$ at which the calculations were performed
are indicated by the labels in the graph. Figure 5: The local density profiles
for water on graphite as a function of reduced distance from the surface at
490 K. The values of $\chi^{*}$ at which the calculations were performed are
indicated by the labels in the graph.
At $T$=510 K, the adsorption isotherm becomes continuous as the chemical
potential increases (FIG. 3), which indicates $T_{\mathrm{pwc}}\leq$510 K.
This is more clearly presented by the growth of density profiles shown in FIG.
6. The adsorption film builds from a thin to a thick one continuously with the
increase of the chemical potential. The density of the first peak in the
profiles increases gradually to that of an adsorbed liquid. At saturation
chemical potential, the density profile evolves to the one corresponding to
the liquid density except for the first peak adjacent to the wall. Comparison
of the isotherms obtained at 505 K and 510 K in FIG. 3 with FIG. 1 indicates
that the prewetting critical temperature of water on graphite lies somewhere
between these two values of temperature, i.e. $T_{\mathrm{pwc}}$=505-510 K.
The nature of the prewetting jump of water on graphite at temperatures 480-
505 K can be further shown by the results obtained from simulations at 490 K,
$\chi^{*}=$0.992, with varying simulation cell dimension in the surface normal
direction. Shown in FIG. 7 are the density profiles obtained from simulations
with cell heights of $H^{*}=h/\sigma_{\mathrm{f}}$=10, 20, 30, 40,
respectively. In order to compare the results with consistency, in those four
simulations the area of the graphite wall is kept at
10$\sigma_{\mathrm{f}}$$\times$10$\sigma_{\mathrm{f}}$, but the height, or the
volume, of the cell varies. It can be seen from FIG. 7 that the rapid rise of
the film thickness to a finite value is independent of the height of the
simulation cell. The film thickness keeps at about 7.5$\sigma_{\mathrm{f}}$
under various $H^{*}$. This is a clear indication that the transition is
prewetting rather than capillary condensation. We also notice that the density
profile at $H^{*}$=10 has more fluctuations compared with those at $H^{*}$=20,
30, and 40. That could be due to the finite size effects in $H^{*}$. This
confirms the importance of using a simulation cell with sufficient height in
order to obtain reliable wetting transition information. Theoretically the
density profiles obtained with different $H^{*}$ should coincide with each
other. But it is hard to achieve this in simulations because of statistical
fluctuations and metastability nature of the problem.
Figure 6: The local density profiles for water on graphite as a function of
reduced distance from the surface at 510 K. Profile curves are for $\chi^{*}=$
0.731, 0.878, 0.892, 0.919, 0.940, 0.962, 0.977, 0.995, 1.003, from bottom to
top. Figure 7: Film density for water on graphite at T=490 K and
$\chi^{*}=$0.992. Curves correspond to varying heights of the simulation cell:
$H^{*}$=10, 20, 40, 30, from left to right.
One of the clearest demonstrations of the first-order nature of many wetting
transitions is the observation of one stable and one metastable states shown
in the systems of interest in the temperature range of Tw to Tpwc Bonn and
Ross (2001). The experimental work by Bonn et al. Bonn et al. (1992) showed
that two different stable values of the film thickness could be found in the
binary liquid mixture of methanol/cyclohexane at a temperature between 295 K
and 308 K. The thin film of 1.0 nm is the metastable state, and the thick film
of 40.0 nm is the stable state. Shi et al. Shi et al. (2003) observed the
similar switching behavior between thin and thick films in the simulation
study of hydrogen isotopes on alkali metal surfaces.
We also observed the coexistence of stable and metastable states for water on
graphite, which is shown in FIG. 8. The probability density function of the
water density distribution in the box is collected during the simulation with
T=490 K, $\chi^{*}=$0.968. One major peak is located at $\rho^{*}\approx$0.012
corresponding to a thin film, at which the simulation samples most frequently.
Another small peak exists at $\rho^{*}\approx$0.25, which corresponds to a
thick film. The bimodal feature of the probability density function of the
distribution indicates the wetting transition at this temperature is first
order. The switching between the thin and thick films is further confirmed by
the fluctuation of the number density of adsorbate in the box. In the inset of
FIG. 8 we show the evolution of water density in the simulation cell as a
function of simulation steps. At about $11\times 10^{6}$ configurations the
system abruptly jumps to a higher density of $\rho^{*}\approx$0.25 from
$\rho^{*}\approx$0.012, corresponding to a switching from thin film to thick
film. This thick film subsequently evaporates to the thin film at about
$14\times 10^{6}$ configurations. Comparing the sizes of the two peaks and
evolution of the number density fluctuation we readily conclude that at this
state point, the thin film is stable and the thick film was metastable.
In this work we did not attempt to determine the exact values of the wetting
temperature or the prewetting critical temperature, but only provide estimated
ranges for them, which are 475-480 K and 505-510 K, respectively. One reason
is the lack of reliable theoretical method for determination of exact
$T_{\mathrm{w}}$ and $T_{\mathrm{pwc}}$. One of the popular techniques being
used previously is a power law extrapolation method Ancilotto and Toigo
(1999). Theoretical predictions indicate that
$\Delta\mu^{*}=(\mu_{\mathrm{svp}}^{*}-\mu_{\mathrm{w}}^{*})\propto(T^{*}-T_{\mathrm{w}}^{*})^{3/2}$
Ancilotto and Toigo (1999). Hence, a plot of $\Delta\mu^{*}$ versus $T^{*}$
can be used to identify $T_{\mathrm{w}}^{*}$ by extrapolating the curve to
$\Delta\mu^{*}=0$ Mistura et al. (1994), with the saturation and wetting
transition chemical potentials ($\mu_{\mathrm{svp}}^{*}$ and
$\mu_{\mathrm{w}}^{*}$) at various temperatures up to $T_{\mathrm{c}}$ of the
fluid being calculated from MHR. However, Shi et al. Shi et al. (2002) found
that this power law extrapolation method could be quite inaccurate in
predicting the wetting temperature of certain system.
Another important concern is the realism of the water potential employed in
this study. It is known that the transition temperature calculated from
simulations is sensitively dependent on the solid-fluid interactions Curtarolo
et al. (2000); Shi et al. (2003). For example, Shi et al. found that a
$\sim$10% increase in the surface-fluid attraction decreases the wetting
transition temperature of Ar on a CO2 surface by 3 K Shi et al. (2003). In
this work, the choice of the water potential will affect the graphite-water
interaction implicitly, and thus the calculated $T_{\rm w}$. The SPC/E water
potential gives by far the best predictions for the critical properties of the
bulk water among all the available nonpolarizable water models, while it still
cannot simulate exactly the coexistence properties of real water. The accuracy
of the estimated $T_{\rm w}$ and $T_{\rm pwc}$ may be improved by using more
accurate polarizable models such as the Gaussian charge polarizable model
Paricaud et al. (2005).
In addition, by modeling the graphite as a smooth surface, we neglected the
possible impact from surface corrugations and dynamics of the surface
structure during the adsorption. A previous simulation work indicates that
impact of surface corrugation of the adsorbent on the wetting transition
behavior of Ne is minimal Bojan et al. (1999). But it is unclear if the same
conclusion is applicable to the graphite-water system.
Figure 8: The switching between the thin and thick film of water adsorption on
graphite at T=490 K, $\chi^{*}=$0.968. The inset shows the density evolution
during the simulation.
Cheng et al. Cheng et al. (1993) proposed a simple model (CCST hereafter)
which interprets the wetting transitions in terms of a balance between the
surface tension cost of producing a thicker film and the energy gain
associated with the film’s interaction with the surface, V(z). The theory
results in an implicit equation for the wetting temperature
$I=-\int_{z_{\mathrm{min}}}^{\infty}V(z)dz=\left(\frac{2\gamma}{\rho_{l}-\rho_{v}}\right)_{T_{\mathrm{w}}},$
(5)
where $\rho_{l}$ and $\rho_{v}$ are the number densities of the adsorbate
liquid and vapor at coexistence, $\gamma$ is the surface tension of the
liquid, and $z_{\mathrm{min}}$ minimizes the fluid-surface interaction
potential V(z). The wetting transition temperature can be calculated by
solving the equation since the right hand side of Eq. (5) is dependent
implicitly on temperature. The impact of solid-fluid interaction on $T_{w}$ is
reflected by $I$, and the fluid-fluid interaction is incorporated in the model
by $\gamma$ and $\rho_{l}-\rho_{v}$.
The wetting transition of water on graphite calculated in this work agree
quantitatively with the previous prediction Gatica et al. (2004) using the
CCST model, although different water potentials are employed in the two
studies. It has been pointed out by Shi et al. Shi et al. (2003) that the
wetting behavior predicted theoretically depends on both the well depth and
well shape of the solid-fluid interacting potential. Here the potential width
is defined as the full width at half minimum of the attractive part of the
potential. In Gatica et al.’s work, TIP4P was used instead of SPC/E. But we
note that the well depth (D) and well width (w) of the water/graphite
potential for SPC/E and TIP4P are almost identical, both with $D=$9.35 kJ/mol
and w=0.135 nm, if evaluated at T=475 K and using the water dipole moment of
1.85 D. Therefore we expect that the wetting transition temperature calculated
from the CCST model using these two potentials be comparable. If the
simulation results in this work are taken to be standard, the CCST model
predicted $T_{\mathrm{w}}$ of 474 K is very accurate indeed. It has been shown
that the CCST model usually works well in predicting the wetting behavior
involving spherical fluids such as inert gases, but it has not been tested
extensively with nonspherical molecules. The fact that this simple model works
well in predicting the wetting of water on graphite, although water has a very
different kind of potential than inert gases, indicates that the CCST model
contains the essential physics of wetting.
## IV CONCLUSIONS
In summary, we report the first simulation study of the wetting transition of
water on graphite surface. The wetting transition temperature calculated from
GCMC simulations is 475-480 K, and the prewetting critical temperature is
505-510 K. The wetting transition is first order. The simulation results in
this work agrees well with the prediction by the CCST model, although the CCST
model is designed on the basis of the simple fluids such as inert gases.
Finally, we point out that the wetting temperature and prewetting critical
temperature calculated in this work depends on the accuracy of the water
potential employed. Improvement in the predictions may be made if more
accurate water potential is available. Future investigations can be performed
by including the corrugation of graphite surface, the finite size effect of
the system, and by using the more robust simulation techniques such as the one
proposed by Errington Errington (2004). Experimental search for the predicted
wetting behavior is also warranted.
###### Acknowledgements.
The author thanks Peter T. Cummings and Milton W. Cole for many helpful
discussions throughout this work. This research was conducted at the Center
for Nanophase Materials Sciences, which is sponsored at Oak Ridge National
Laboratory by the Division of Scientific User Facilities, U.S. Department of
Energy.
## References
* Cahn (1977) J. W. Cahn, J. Chem. Phys. 66, 3667 (1977).
* Ebner and Saam (1977) C. Ebner and W. F. Saam, Phys. Rev. Lett. 38, 1486 (1977).
* Mistura et al. (1994) G. Mistura, H. C. Lee, and M. H. W. Chan, J. Low Temp. Phys. 96, 221 (1994).
* Ross et al. (1995) D. Ross, P. Taborek, and J. E. Rutledge, Phys. Rev. Lett. 74, 4483 (1995).
* Hallock (1995) R. B. Hallock, J. Low Temp. Phys. 101, 31 (1995).
* Demolder et al. (1995) B. Demolder, N. Bigelow, P. Nacher, and J. Dupont-Roc, J. Low Temp. Phys. 98, 91 (1995).
* Wyatt et al. (1995) A. F. G. Wyatt, J. Klier, and P. Stefanyi, Phys. Rev. Lett. 74, 1151 (1995).
* Yao and Hensel (1996) M. Yao and F. Hensel, J. Phys.: Condens. Matter 8, 9547 (1996).
* Hess et al. (1997) G. B. Hess, M. J. Sabatini, and M. H. W. Chan, Phys. Rev. Lett. 78, 1739 (1997).
* Ross et al. (1997) D. Ross, J. A. Phillips, J. E. Rutledge, and P. Taborek, J. Low. Temp. Phys. 106, 81 (1997).
* Kozhevnikov et al. (1997) V. F. Kozhevnikov, D. I. Arnold, S. P. Naurzakov, and M. E. Fisher, Phys. Rev. Lett. 78, 1735 (1997).
* Ross et al. (1998) D. Ross, P. Taborek, and J. E. Rutledge, Phys. Rev. B 58, R4274 (1998).
* Hensel and Yao (1998) F. Hensel and M. Yao, Ber-Bunsen-Ges. Phys. Chem. 102, 1798 (1998).
* Ohmasa et al. (1998) Y. Ohmasa, Y. Kajihara, and M. Yao, J. Phys.: Condens. Matter 10, 11589 (1998).
* Ohmasa et al. (2001) Y. Ohmasa, Y. Kajihara, and M. Yao, Phys. Rev. E 63, 051601 (2001).
* Kozhevnikov et al. (1998) V. F. Kozhevnikov, D. I. Arnold, S. P. Naurzakov, and M. E. Fisher, Fluid Phase Equilib. 150, 625 (1998).
* Ebner and Saam (1987) C. Ebner and W. F. Saam, Phys. Rev. B 35, 1822 (1987).
* Finn and Monson (1989) J. E. Finn and P. A. Monson, Phys. Rev. A 39, 6402 (1989).
* Cheng et al. (1991) E. Cheng, M. W. Cole, W. F. Saam, and J. Treiner, Phys. Rev. Lett. 67, 1007 (1991).
* Cheng et al. (1993) E. Cheng, M. W. Cole, W. F. Saam, and J. Treiner, Phys. Rev. B 48, 18214 (1993).
* Wagner and Ceperley (1994) M. Wagner and D. M. Ceperley, J. Low. Temp. Phys. 94, 185 (1994).
* Bojan et al. (1998) M. J. Bojan, M. W. Cole, J. K. Johnson, W. A. Steele, and Q. Wang, J. Low Temp. Phys. 110, 653 (1998).
* Boninsegni and Cole (1998) M. Boninsegni and M. W. Cole, J. Low Temp. Phys. 110, 685 (1998).
* Ancilotto and Toigo (1999) F. Ancilotto and F. Toigo, Phys. Rev. B 60, 9019 (1999).
* Bojan et al. (1999) M. J. Bojan, G. Stan, S. Curtarolo, W. A. Steele, and M. W. Cole, Phys. Rev. E 59, 864 (1999).
* Curtarolo et al. (2000) S. Curtarolo, G. Stan, M. J. Bojan, M. W. Cole, and W. A. Steele, Phys. Rev. E 61, 1670 (2000).
* Ancilotto et al. (2001) F. Ancilotto, S. Curtarolo, F. Toigo, and M. W. Cole, Phys. Rev. Lett. 87, 206103 (2001).
* Shi et al. (2003) W. Shi, J. K. Johnson, and M. W. Cole, Phys. Rev. B. 68, 125401 (2003).
* Gatica et al. (2004) S. M. Gatica, J. K. Johnson, X. C. Zhao, and M. W. Cole, J. Phys. Chem. B 108, 11704 (2004).
* Shi et al. (2002) W. Shi, X. C. Zhao, and J. K. Johnson, Mol. Phys. 100, 2139 (2002).
* Errington (2004) J. R. Errington, Langmuir 20, 3798 (2004).
* Bonn and Ross (2001) D. Bonn and D. Ross, Rep. Prog. Phys. 64, 1085 (2001).
* Berendsen et al. (1987) H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, J. Phys. Chem. 91, 6269 (1987).
* Hayward and Svishchev (2001) T. M. Hayward and I. M. Svishchev, Fluid Phase Equilib. 182, 65 (2001).
* Guillot (2002) B. Guillot, J. Mole. Liquids 101, 219 (2002).
* Steele (1973) W. A. Steele, Surf. Sci. 36, 317 (1973).
* Zhao and Johnson (2005) X. C. Zhao and J. K. Johnson, Mol. Sim. 31, 1 (2005).
* Miller and Bederson (1978) T. M. Miller and B. Bederson, in _Advances in Atomic and Molecular Physics_ , edited by D. R. Bates and B. Bederson (Academic Press, London, 1978), vol. 13, p. 1.
* Whitehouse and Buckingham (1993) D. B. Whitehouse and A. D. Buckingham, J. Chem. Soc. Faraday Trans. 89, 1909 (1993).
* Yeh and Berkowitz (1999) I. C. Yeh and M. L. Berkowitz, J. Chem. Phys. 111, 3155 (1999).
* Ferrenberg and Swendsen (1988) A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988).
* Ferrenberg and Swendsen (1989) A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989).
* Shi and Johnson (2001) W. Shi and J. K. Johnson, Fluid Phase Equilib. 187, 171 (2001).
* de Pablo et al. (1999) J. J. de Pablo, Q. L. Yan, and F. A. Escobedo, Annu. Rev. Phys. Chem. 50, 377 (1999).
* Evans and Marini Bettolo Marconi (1985) R. Evans and U. Marini Bettolo Marconi, Phys. Rev. A. 32, 3817 (1985).
* van Swol and Henderson (1986) F. van Swol and J. R. Henderson, J. Chem. Soc. Faraday Trans. II 82, 1685 (1986).
* Parry and Evans (1990) A. O. Parry and R. Evans, Phys. Rev. Lett. 64, 439 (1990).
* Parry and Evans (1992) A. O. Parry and R. Evans, Physica A 181, 250 (1992).
* Smith et al. (2001) J. M. Smith, H. C. van Ness, and M. M. Abbott, _Introduction to Chemical Engineering Thermodynamics_ (McGraw-Hill, New York, 2001), 6th ed.
* Bonn et al. (1992) D. Bonn, H. Kellay, and G. H. Wegdam, Phys. Rev. Lett. 69, 1975 (1992).
* Paricaud et al. (2005) P. Paricaud, M. Predota, A. A. Chialvo, and P. T. Cummings, J. Chem. Phys. 122, 4511 (2005).
|
arxiv-papers
| 2012-09-20T13:41:21 |
2024-09-04T02:49:35.338075
|
{
"license": "Public Domain",
"authors": "Xiongce Zhao",
"submitter": "Xiongce Zhao",
"url": "https://arxiv.org/abs/1209.4529"
}
|
1209.4544
|
# Minimal dilaton model
Tomohiro Abe tomohiro˙[email protected] Institute of Modern Physics and
Center for High Energy Physics, Tsinghua University, Beijing 100084, China
Ryuichiro Kitano [email protected] Department of Physics, Tohoku
University, Sendai 980-8578, Japan Yasufumi Konishi
[email protected] Department of Physics, Saitama
University, Saitama 355-8570, Japan Kin-ya Oda
[email protected] Department of Physics, Kyoto University,
Kyoto 606-8502, Japan Joe Sato [email protected] Department of Physics,
Saitama University, Saitama 355-8570, Japan Shohei Sugiyama
[email protected] Institute for Cosmic Ray Research (ICRR), University
of Tokyo, Kashiwa, Chiba 277-8582, Japan
###### Abstract
We construct a minimal calculable model of a light dilaton based on the
scenario where only top and Higgs sectors are involved in a quasiconformal
dynamics. The model consistently accommodates the electroweak precision tests
even when the Higgs boson is very heavy, thereby allowing one to consider the
possibility that the particle at around 125 GeV, discovered at the LHC
experiments, is identified as the light dilaton rather than the Higgs boson.
We find that the current LHC data allow distinct parameter regions where the
observed particle is either mostly the Higgs boson or the dilaton.
KUNS-2417, STUPP-12-211, TU-919
## I Introduction
It has been reported recently that both the ATLAS and the CMS experiments
observed resonances in the $\gamma\gamma$, $ZZ$, and $WW$ channels at around
125 GeV obs:2012gk ; obs:2012gu . The resonance can naturally be interpreted
as a signal of the Higgs boson in the Standard Model (SM). The mass range
suggests that the Higgs sector of the SM is fairly weakly coupled, perfectly
consistent with the precision measurements at the LEP experiments.
The observation of the light Higgs boson excludes various models of
electroweak symmetry breaking via strong dynamics. For example, typical
technicolor models TC and their proposed effective descriptions such as the
Higgsless model Csaki:2003dt are in trouble with the weakly coupled
descriptions of the Higgs sector.
There is, however, a logical possibility that the excess is not due to the
Higgs boson but rather a Nambu-Goldstone boson associated with an approximate
scale invariance in the dynamical sector. In Ref. WTC_TD , it has been
proposed that the dilaton in the walking technicolor WTC_TD_original may
explain the signals, see also Ref. light_dilaton for an effective theory
description of dilaton systems. (There is an attempt to identify a dilaton as
the Higgs Higgs as PNGB .) In Ref. radion in RS as 125GeV , the radion in the
Randall-Sundrum model has been discussed as a possible candidate, see also
Ref. radion further development .
It has been reported that the excess of events at around 126.5 (125) GeV at
ATLAS (CMS) in the diphoton final state is enhanced from the expectation of
the SM by a factor $1.8\pm 0.5$ ($1.6\pm 0.4$) obs:2012gk ; obs:2012gu .
Though this tendency is not to the level of significance, it is noteworthy
that the above stated models can naturally account for the trend.
If the excesses at around 125 GeV are due to such a non-Higgs particle, one
needs to consider a consistency with the constraints on the Peskin-Takeuchi
$S$ and $T$ parameters PeskinTakeuchi . This non-Higgs scenario requires the
real Higgs boson (if it exists) to be heavier than $\sim 600$ GeV in order to
be consistent with the LHC data, and such a heavy Higgs boson does not give a
good fit to the precision measurements. It is required to have some other
contributions to the $S$ and $T$ parameters. Since the models based on
(unknown) technicolor theories or an extra dimension have no predictability or
only weak predictability on the $S$ and $T$ parameters, it is difficult to
judge if such models are really viable.
In this paper, we construct an effective (minimal) model of such a framework
and see if there is a viable parameter region. The model consists of a massive
vectorlike top partner fermion. The top partner mass $M$ represents a mass gap
in the dynamical sector, to which the dilaton naturally couples in order to
recover the scale invariance: $M\to Me^{-\phi/f}$. The coupling in turn
provides interactions between the dilaton $\phi$ and the photons/gluons
through loop diagrams of the top partners, explaining the excesses at the
ATLAS and CMS. The top partner can also contribute to the Peskin-Takeuchi $S$,
$T$ parameters, which turns out to be a good direction to come back to the
allowed region, canceling the heavy SM Higgs boson contribution to the $T$
parameter.
We discuss whether such a dilaton explanation of the excesses is viable and
how such a scenario can be discriminated by measuring various cross sections
and decay branching ratios at the LHC experiments. The minimal model we study
below catches essential features of dilaton/radion models, and the model
parameters we discuss can easily be translated into those in other models.
In Sec. II, we present the minimal dilaton model as an effective
renormalizable theory equipped with a linearized dilaton field $S$ and the
vectorlike top partner $T$ 111 There would be no room for confusion from the
notational abuse that the linearized dilaton field $S$ and the top partner $T$
should not be misunderstood as the Peskin-Takeuchi parameters which appear
with superscripts only in Eqs. (36)–(40).. In Sec. III, we show signal
strengths for all the Higgs decay final states and give constraints on model
parameters, namely a Higgs-dilaton mixing angle $\theta_{H}$ and a dilaton
decay constant $f$, which is nothing but a vacuum expectation value of $S$. In
Sec. IV, constraints from the electroweak precision measurements are examined
on the top partner sector, namely on the left-handed mixing $\theta_{L}$
between the top and its partner $T$ and on the heavier $t^{\prime}$ quark
mass. The last section is devoted to summary and discussions.
## II The minimal dilaton model
The dilaton field, defined as the Nambu-Goldstone particle associated with an
approximate scale invariance of the theory, can be produced through the gluon
fusion process at the LHC and can decay into two photons through the following
effective operators:
$\displaystyle\phi G^{a}_{\mu\nu}G^{a\mu\nu},\ \ \ \phi F_{\mu\nu}F^{\mu\nu},$
(1)
where $\phi$, $G^{a}_{\mu\nu}$, and $F_{\mu\nu}$ are the dilaton, the gluon
field strength, and the photon field strength, respectively. These effective
operators are generated if there is a colored and charged field which obtains
mass through the spontaneous breaking of the approximate scale invariance. For
example, in an approximately scale invariant technicolor theory, one can
expect such operators to appear at low energy if there are colored and charged
techniquarks. Also, there can be a dual hadronic description of such a theory
where the approximate scale invariance is nonlinearly realized. An example is
a model with a warped extra dimension where $\phi$ represents the radius of
the extra dimension. In either example, predictions to the effective couplings
and also constraints from the electroweak precision measurements are pretty
model dependent, and moreover, there are often technical difficulties in the
estimations due to large nonperturbative effects or incalculable corrections
from the cutoff scale physics.
We, therefore, consider an effective minimal model of the dilaton, based on a
weakly coupled renormalizable theory. The model allows us to perform explicit
computations of the Peskin-Takeuchi $S$, $T$ parameters and also
production/decay processes of the dilaton at the LHC. This exercise not only
provides us with a sense of how such a model is constrained, but is also
practically useful since the obtained allowed range of parameters can easily
be translated into other calculable models.
The operators in Eq. (1) are obtained by integrating out a field which is
colored and charged. We choose the field to have the same quantum numbers as
the right-handed top quark. This is somewhat a natural choice. When we
consider the origin of the large top-quark mass, one may need to assume that
the top quark is (semi) strongly coupled to a dynamical sector, such as in the
topcolor Hill:1991at or the top seesaw models TopSeeSaw . It is then
reasonable to assume an existence of a resonance with the same quantum number
as the top quarks. The resonance can decay into a bottom quark and a $W$
boson, and thus does not have a problem with a exotic stable state. As a
minimal choice, we consider a vectorlike top partner with the same gauge
quantum number as the right-handed $SU(2)_{L}$ singlet top quark rather than
the left-handed doublet that also includes bottom quark partner.
We write down the following Lagrangian for the dilaton system:
$\displaystyle{\cal L}={\cal L}_{\rm SM}-{e^{-2\phi/f}\over
2}\partial_{\mu}\phi\partial^{\mu}\phi-\overline{T}\left(\not{D}+Me^{-\phi/f}\right)T-\left[y^{\prime}\overline{T_{R}}(q_{3L}\cdot
H)+\text{h.c.}\right]-V(\phi,H),$ (2)
where $T$ is the heavy vectorlike top partner representing the resonance,
$q_{3L}$ is the left-handed (top and bottom) quark doublet, and $H$ is the SM
Higgs doublet field. The Lagrangian ${\cal L}_{\rm SM}$ is the Standard Model,
and the term with a coupling constant $y^{\prime}$ provides a mixing between
the top quark and $T$. The Lagrangian has a nonlinearly realized scale
invariance except for the scalar potential term $V(\phi,H)$ which contains
terms with small explicit breaking of the scale invariance. The potential
terms provide mass terms for $\phi$ as well as a mixing between $\phi$ and the
Higgs boson. We choose the origin of the field $\phi$ so that
$\langle\phi\rangle=0$. A mass term of $\overline{u_{3R}}T_{L}$, with $u_{3R}$
being the right-handed top quark, can be eliminated by an appropriate field
redefinition; see Appendix A. It may be interesting to consider this model as
the low-energy effective theory of the top condensation model TopCondensation
, where the coupling $y^{\prime}$ and the quartic coupling constant of the
Higgs field blows up simultaneously at a high-energy scale. We do not impose
such a constraint in this paper in order to leave the discussion general.
### II.1 Linearized model
By a field redefinition,
$\displaystyle S=fe^{-\phi/f},$ (3)
the Lagrangian given in Eq. (2) is equivalent to
$\displaystyle{\cal L}={\cal L}_{\rm SM}-{1\over
2}\partial_{\mu}S\partial^{\mu}S-\overline{T}\left(\not{D}+{M\over
f}S\right)T-\left[y^{\prime}\overline{T}(q_{3L}\cdot
H)+\text{h.c.}\right]-\tilde{V}(S,H).$ (4)
The scale invariance is now linearly realized. The potential $\tilde{V}(S,H)$
should be arranged so that $\langle S\rangle=f$ and $\langle
H^{0}\rangle=v/\sqrt{2}$. The explicit form of $\tilde{V}$ is shown in
Appendix B for completeness, though we do not need to specify it for the
following analysis as we will be discussing with physical quantities such as
the masses and mixings. We propose this Lagrangian as a minimal effective
description of an approximately scale invariant theory of (dynamical)
electroweak symmetry breaking involving the only top and Higgs sectors.
If the mixing between the dilaton and the Higgs boson is small and if the
dilaton is the one which explains the observed resonance at the LHC, the Higgs
boson must be heavier than about $600$ GeV in order to be consistent with the
Higgs boson searches at the LHC. With such a heavy Higgs boson, constraints
from the Peskin-Takeuchi $S$, $T$ parameters require a new contribution to
come back to the ellipse in the $S$-$T$ plane. As we will see shortly, such a
contribution is already there in this model since the loop diagrams of the
$t^{\prime}$ field can push $S$ and $T$ parameters towards the right
direction. This Lagrangian, therefore, provides a compact realistic model of
the light dilaton, which can be used for LHC studies.
### II.2 Model parameters
This model has new parameters in addition to the Standard Model ones. We here
list them and their definitions:
* •
$f$
This is the decay constant of the dilaton. The size of $f$ controls the
strength of the coupling of the dilaton $\phi$ to photons and gluons and to
all the fields involved in the quasiconformal dynamics. For later use, we
define a dimensionless quantity,
$\displaystyle\eta={v\over f}N_{T},$ (5)
where $v=\sqrt{2}\langle H^{0}\rangle=246$ GeV, and $N_{T}$ is the number of
the $T$ fields. This parameter appears when we discuss the production and
decay of $\phi$. If necessary, one can obtain a large value of $\eta$ with a
large number of $N_{T}$ without requiring too small $f$. For the minimal model
$N_{T}=1$ which we will assume hereafter.
* •
$m_{s}$, $m_{h}$, and $\theta_{H}$
These are masses ($m_{s}<m_{h}$) and mixing of the scalar fields. We take
lighter mass eigenstate to explain the LHC excesses, i.e.,
$\displaystyle m_{s}\simeq 125\,{\rm GeV}.$ (6)
The Higgs-dilaton mixing angle is defined as
$\displaystyle S$ $\displaystyle=f+s\cos\theta_{H}-h\sin\theta_{H},$
$\displaystyle H^{0}$
$\displaystyle={v+s\sin\theta_{H}+h\cos\theta_{H}\over\sqrt{2}},$ (7)
where $s$ and $h$ are the lighter and the heavier mass eigenstates,
respectively. We impose
$\displaystyle m_{h}>600~{}{\rm GeV},$ (8)
to be consistent with the data from LHC. When the mixing angle is so large
that the lighter one is almost the Standard Model Higgs boson, it is not
necessary to impose the above constraint for a large $f$. Since we are
particularly interested in a small mixing region, we always impose the above
constraint in the following analysis.
* •
$m_{t^{\prime}}$ and $\theta_{L}$
These are the $T$ mass and the left-handed mixing of the top sector. The mass
matrix for the top quark and its partner,
$\displaystyle\mathcal{M}_{t}$
$\displaystyle=\begin{bmatrix}y_{t}v/\sqrt{2}&y^{\prime}v/\sqrt{2}\\\
0&M\end{bmatrix},$ (9)
are diagonalized as
$\displaystyle\begin{bmatrix}\cos\theta_{L}&-\sin\theta_{L}\\\
\sin\theta_{L}&\cos\theta_{L}\end{bmatrix}\mathcal{M}_{t}\mathcal{M}_{t}^{\dagger}\begin{bmatrix}\cos\theta_{L}&\sin\theta_{L}\\\
-\sin\theta_{L}&\cos\theta_{L}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}m_{t}^{2}&0\\\
0&m_{t^{\prime}}^{2}\end{bmatrix},$ (10)
where $m_{t}$ is the top quark mass. One can trade the new parameters
$y^{\prime}$ and $M$ by the observables $m_{t^{\prime}}$ and $\theta_{L}$,
while $y_{t}$ should be adjusted to reproduce the top quark mass. See Appendix
A for more detailed discussion. In Sec. IV, we verify constraints on the
parameters $\theta_{L}$ and $m_{t^{\prime}}$ from the electroweak precision
data.
### II.3 Coupling to the Standard Model fields
The loop diagrams of $T$ generate the effective couplings in Eq. (1). In the
limit of $2m_{t^{\prime}}\gg m_{s}$, the effective couplings for the
production and decays of $s$ are approximated by momentum independent pieces
which are insensitive to $m_{t^{\prime}}$ or $\theta_{L}$. This behavior can
most easily be understood by identifying the effective coupling as the
dilaton/Higgs dependence of the running gauge coupling constants:
$\displaystyle{\cal L}_{\rm eff}=-{1\over
4g_{s}^{2}}G^{a}_{\mu\nu}G^{a\mu\nu},$ (11)
where we take the gluon as an example. The running coupling constant at a low-
energy scale is given by
$\displaystyle{1\over g_{s}^{2}(\mu)}={1\over g_{s}^{2}(\Lambda)}-{2(b_{\rm
SM}+\Delta b)\over(4\pi)^{2}}\log{m_{t^{\prime}}\over\Lambda}-{2b_{\rm
SM}\over(4\pi)^{2}}\log{m_{t}\over m_{t^{\prime}}}-{2(b_{\rm SM}-\Delta
b)\over(4\pi)^{2}}\log{\mu\over m_{t}},$ (12)
at one-loop level, where $b_{\rm SM}=-7$ and $\Delta b=2/3$. This can be
rearranged to
$\displaystyle{1\over g_{s}^{2}(\mu)}$ $\displaystyle=$ $\displaystyle{1\over
g_{s}^{2}(\Lambda)}-{2(b_{\rm SM}-\Delta
b)\over(4\pi)^{2}}\log{\mu\over\Lambda}-{2\Delta
b\over(4\pi)^{2}}\log{(y_{t}\langle H^{0}\rangle)(M\langle
S\rangle/f)\over\Lambda^{2}},$ (13)
where we have used the fact that
$\displaystyle m_{t}m_{t^{\prime}}=y_{t}\langle H^{0}\rangle\,{M\langle
S\rangle\over f},$ (14)
which is derived from the mass matrix in Eq. (9). By recovering the field
fluctuations $\langle H^{0}\rangle,\langle S\rangle$ $\to H^{0},S$ and
considering the mixing factors in Eq. (7), we obtain the effective coupling of
$s$ to gluons from Eq. (11) as
$\displaystyle{\cal L}_{\rm eff}^{(sgg)}={1\over
4}\,{g_{s}^{2}\over(4\pi)^{2}}{2\over v}\,{2\over
3}\left(\eta\cos\theta_{H}+\sin\theta_{H}\right)sG^{a}_{\mu\nu}G^{a\mu\nu},$
(15)
where we have canonically normalized the kinetic term of the gluon. The terms
proportional to $\sin\theta_{H}$ and $\eta\cos\theta_{H}$ are contributions
from the SM top and its partner, respectively. We can explicitly see that the
coupling is independent of $m_{t^{\prime}}$ or $\theta_{L}$. For the $s$ to
photon coupling, we need to include a loop of the $W$ bosons. The result is
$\displaystyle{\cal L}_{\rm eff}^{(s\gamma\gamma)}={1\over
4}\,{e^{2}\over(4\pi)^{2}}{2\over v}\left({4\over
3}N_{c}Q_{t}^{2}\eta\cos\theta_{H}+A_{\text{SM}}\sin\theta_{H}\right)sF_{\mu\nu}F^{\mu\nu},$
(16)
where $N_{c}=3$ and $Q_{t}=2/3$ are the color factor and the top quark charge,
respectively, and the explicit form of the loop factor
$A_{\text{SM}}\simeq-6.5$ can be found e.g. in (2.45) in Ref. Djouadi:2005gi .
The first term in the parentheses is the contribution from the top partner
loop whereas the second is from the SM top and $W$ ones.
The particle $s$ can also couple to the $W$ and $Z$ bosons and the fermions
through the $\theta_{H}$ mixing. The couplings are simply given by those of
the SM Higgs boson times a factor of $\sin\theta_{H}$.
Note here that the model is not the same as the Higgs-dilaton model studied in
Refs. Giardino:2012ww ; Giardino:2012dp ; Carmi:2012in , where the coupling
between the dilaton and the Standard Model fields are assumed to have the
form:
$\displaystyle{\cal L}_{\rm int}={\phi\over f}T^{\mu}_{\ \mu}.$ (17)
Here, $T^{\mu}_{\ \mu}$ is the trace of the energy-momentum tensor of the
Standard Model. Through this term, the dilaton directly couples to the
violation of the scale invariance in the Standard Model, i.e., to the $W$, $Z$
bosons and fermions with strength proportional to their masses. Also, the
couplings to the photons and gluons are proportional to the beta functions.
The effective interaction term in Eq. (17) is generated at low energy if the
whole Standard Model sector is a part of the scale invariant theory in the UV;
all the gauge bosons and fermions are composite particles.
In contrast, we take a more conservative picture that the Standard Model
except for the top/Higgs sector is a spectator of the dynamics, and thus the
dilaton couples to the $W$, $Z$ bosons and fermions only through the mixing
with the Higgs fields. The couplings to the gluons and photons are generated
only through the loops of $t$ and $t^{\prime}$. Due to these different origin
of the couplings between two models, the production and decay properties are
quite different. Indeed, we will see that our model can give better fit to the
LHC data compared to the SM Higgs boson, while Refs. Giardino:2012ww ;
Giardino:2012dp ; Carmi:2012in have reported that the dilaton scenario based
on Eq. (17) is rather disfavored.
In terms of the parameters in Refs. Giardino:2012ww ; Giardino:2012dp ;
Carmi:2012in , our model corresponds to
$\displaystyle c_{V}=c_{F}$ $\displaystyle=\sin\theta_{H},$ $\displaystyle
c_{t}$
$\displaystyle=\cos^{2}\theta_{L}\sin\theta_{H}+\eta\sin^{2}\theta_{L}\cos\theta_{H},$
(18) $\displaystyle c_{g}$ $\displaystyle=\eta\cos\theta_{H}+\sin\theta_{H},$
$\displaystyle c_{\gamma}$ $\displaystyle=\eta
A_{t^{\prime}}\cos\theta_{H}+A_{\text{SM}}\sin\theta_{H},$ $\displaystyle
c_{\text{inv}}$ $\displaystyle=0,$ (19)
where $F$ stands for all the SM fermions except the top quark. (The parameter
$c_{X}$ for a production/decay process $X$ is written as $\kappa_{X}$ in a
recent analysis by ATLAS ATLAS coupling . Our notation in the forthcoming Eqs.
(21)–(27) reads $R_{X}=c_{X}^{2}=\kappa_{X}^{2}$.) It is worth noting that a
negative value for $c_{F}$ can be easily obtained in our model, which tends to
be more favored than the SM value $c_{F}=1$ in order to suppress the $c_{g}$
coupling while keeping $c_{\gamma}$ large Giardino:2012ww ; Giardino:2012dp ;
Carmi:2012in .
## III Dilaton at the LHC
As we discussed in Sec. II, there are two mass eigenstates in the scalar
sector, $s$ and $h$, and their couplings are determined by two parameters,
$\eta$ and $\theta_{H}$ in Eqs. (5) and (7). In the small $\theta_{H}$ region,
$s$ is dilatonlike as we can see from Eq. (7). In the following, we assume the
lighter mass eigenstate $s$ to be around 125 GeV and study the production and
decays of $s$ at the LHC.
### III.1 Production
As one can see from the above discussion, the $s$ particle has suppressed
couplings to $W$, $Z$ and fermions, and either enhanced or suppressed
couplings to $\gamma$ and the gluon compared to the Higgs boson in the
Standard Model. The production cross section of $s$ compared to that of the SM
Higgs boson (at the same mass as $s$) through a process $X$,
$\displaystyle R_{X}$
$\displaystyle:={\sigma_{X}\over\sigma_{X}^{\text{SM}}},$ (20)
is given by
$\displaystyle R_{\text{GF}}$
$\displaystyle=\left(\eta\cos\theta_{H}+\sin\theta_{H}\right)^{2},$ (21)
$\displaystyle R_{\text{VBF}}=R_{\text{VH}}$
$\displaystyle=\sin^{2}\theta_{H},$ (22) $\displaystyle R_{\text{ttH}}$
$\displaystyle=\left(\cos^{2}\theta_{L}\sin\theta_{H}+\eta\sin^{2}\theta_{L}\cos\theta_{H}\right)^{2},$
(23)
for the gluon fusion (GF), the vector boson fusion (VBF), the Higgs-strahlung
(VH), and the associated production with a $t\bar{t}$ pair (ttH),
respectively; see e.g. Ref. Djouadi:2005gi for a review. Here and hereafter
$V$ ($VV$) denotes either $W$ or $Z$ ($WW$ or $ZZ$). Note that the SM cross
section in the denominator in Eq. (20) is evaluated at $m_{h}\simeq 125$ GeV
for comparison to the experimental data, while its value in our model is
$m_{h}\gtrsim 600$ GeV. As said above, $\eta$ appearing in Eq. (21) is not a
ratio of the Yukawa couplings but that of the vacuum expectation values which
can also be checked by a direct loop computation in the linearized model given
in Eq. (4). We plot $R_{\text{GF}}$ in the left panel of Fig. 1. The ttH ratio
(23) reduces to $R_{\text{ttH}}\to\sin^{2}\theta_{H}$ in the small top mixing
limit $\theta_{L}\ll 1$, which we will assume hereafter. The validity of this
approximation will be confirmed in Sec. IV. We note that the ttH process gives
negligible contribution to our diphoton analysis in Eq. (32), with
$\varepsilon^{i}_{\text{ttH}}$ being at most 4%, and that we include it just
for completeness.
Figure 1: Ratio to the SM for the dominant GF production cross section given
in Eq. (21) (left) and to the total decay width given in Eq. (29) (right).
Contours 0, 0.5, 1, 1.5, and 2 are drawn, with 0 and 1 being dotted and thick
lines, respectively. A denser region gives larger value, with density changing
for each increase of the ratio by 0.1 from 0 to 2. Both sides
$\theta_{H}=\pm\pi/2$ correspond to the SM.
### III.2 Decay
For each decay process $s\to X$, we define the decay width ratio to that of
the SM Higgs at 125 GeV,
$\displaystyle R(s\to X)$ $\displaystyle={\Gamma_{s\to X}\over\Gamma_{h\to
X}^{\text{SM}}}.$ (24)
The minimal dilaton model predicts
$\displaystyle R(s\to\text{others})$ $\displaystyle=\sin^{2}\theta_{H},$ (25)
$\displaystyle R(s\to gg)$
$\displaystyle=\left(\eta\cos\theta_{H}+\sin\theta_{H}\right)^{2},$ (26)
$\displaystyle R(s\to\gamma\gamma)$
$\displaystyle=\left(\eta{A_{t^{\prime}}\over
A_{\text{SM}}}\cos\theta_{H}+\sin\theta_{H}\right)^{2},$ (27)
where the subscript “others” denotes the tree-level processes
$bb,VV,\tau\tau,cc$, etc. and
$\displaystyle A_{t^{\prime}}$
$\displaystyle:=N_{c}Q_{t^{\prime}}^{2}\,A_{1\over 2}\\!\left(m_{s}^{2}\over
4m_{t^{\prime}}^{2}\right)\simeq{16\over 9},$ (28)
with the loop function $A_{1\over 2}$ given in Eq. (2.46) in Ref.
Djouadi:2005gi . The ratio for the total decay width is
$\displaystyle R(s\to\text{all})$
$\displaystyle={\Gamma_{s\to\text{others}}+\Gamma_{s\to
gg}+\Gamma_{s\to\gamma\gamma}\over\Gamma_{h\to\text{all}}^{\text{SM}}}$
$\displaystyle=\operatorname{BR}_{\text{others}}^{\text{SM}}\,\sin^{2}\theta_{H}+\operatorname{BR}_{gg}^{\text{SM}}\,\left(\eta\cos\theta_{H}+\sin\theta_{H}\right)^{2}+\operatorname{BR}_{\gamma\gamma}^{\text{SM}}\,\left(\eta{A_{t^{\prime}}\over
A_{\text{SM}}}\cos\theta_{H}+\sin\theta_{H}\right)^{2},$ (29)
where at around 125 GeV, branching ratios in the SM are given e.g. in Ref.
Giardino:2012ww
$\displaystyle\operatorname{BR}_{\text{others}}^{\text{SM}}$
$\displaystyle=91.3\%,$ $\displaystyle\operatorname{BR}_{gg}^{\text{SM}}$
$\displaystyle=8.5\%,$
$\displaystyle\operatorname{BR}_{\gamma\gamma}^{\text{SM}}$
$\displaystyle=0.2\%.$ (30)
We plot $R(s\to\text{all})$ in the right panel of Fig. 1.
### III.3 Dilaton vs SM Higgs signal strengths
Both ATLAS and CMS experiments discovered a new particle at around 125 GeV in
the diphoton CMS_diphoton ; ATLAS_diphoton , $ZZ\to 4l$ CMS_ZZ4l ; ATLAS_ZZ4l
, and $WW\to l\nu l\nu$ CMS_WWlnln ; ATLAS_WWlnln channels. The obtained data
for each channel are translated into the signal strength, which is an expected
production cross section for a particle that decays the same as in the SM
Higgs at the same mass. We constrain the model parameters $\theta_{H}$ and
$\eta^{-1}=f/v$ from these three channels.
The minimal dilaton model predicts different production cross sections between
GF and VBF/VH/ttH processes. In $H\to\gamma\gamma$ search, composition of
these production channels differs category by category and are summarized in
Table 2 in Ref. obs:2012gu for CMS and in Table 6 in Ref. ATLAS_diphoton for
ATLAS. We define $\varepsilon^{i}_{X}$ as the proportion of the production
process $X$ within a category $i$. Note that $\sum_{X}\varepsilon^{i}_{X}=1$
by definition for each category $i$, where a summation over $X$ is always
understood as for all the relevant production channels: GF, VBF, VH, and ttH.
GF is the dominant production process and satisfies
$\varepsilon^{i}_{\text{GF}}\lesssim 90\%$ in production processes other than
dijet category. In the dijet category, the dominant production process is VBF,
and $\varepsilon_{\text{VBF}}\lesssim 70\%$.
When acceptance of a production channel $X$ for a category $i$ is $a^{i}_{X}$,
the estimated value of a signal fraction under the given set of cuts $i$
becomes
$\displaystyle\varepsilon^{i}_{X}$
$\displaystyle={a^{i}_{X}\sigma^{\text{SM}}_{X}\over\sum_{Y}a^{i}_{Y}\sigma^{\text{SM}}_{Y}},$
(31)
where $\sigma^{\text{SM}}_{X}$ is the Higgs production cross section in the SM
through the channel $X$. Given $\\{\varepsilon^{i}_{X}\\}$, we can compute the
signal strength under the imposed cuts for each category $i$
$\displaystyle\hat{\mu}_{i}(h\to\gamma\gamma)$
$\displaystyle={\sum_{X}a^{i}_{X}\,\sigma_{X}\over\sum_{Y}a^{i}_{Y}\,\sigma_{Y}^{\text{SM}}}\,{\operatorname{BR}(s\to\gamma\gamma)\over\operatorname{BR}(h\to\gamma\gamma)_{\text{SM}}}=\sum_{X}\varepsilon^{i}_{X}R_{X}\,{R(s\to\gamma\gamma)\over
R(s\to\text{all})}.$ (32)
We have assumed that the acceptance $a^{i}_{X}$ under the category $i$ does
not change from that of the SM for each production channel $X$.
Figure 2: Diphoton $s\to\gamma\gamma$ signal strength $\hat{\mu}$ when the
production is purely from the GF (VBF/VH/ttH) process in the left (right)
panel. Drawn the same as in Fig. 1.
Figure 3: Signal strength $\hat{\mu}$ for processes other than diphoton and
digluon, namely for final states $s\to ZZ$, $WW$, $\tau\tau$, $bb$, etc. Drawn
the same as in Fig. 2.
Figure 4: Signal strength $\hat{\mu}$ for the digluon $s\to gg$ process. Drawn
the same as in Fig. 2. Dashed and dot-dashed contours are added for
$\hat{\mu}=10$ and 30, respectively, in the left panel.
As an illustration, we plot each contribution from the initial state $X$ in
the signal strength $\hat{\mu}(s\to F)$:
$\displaystyle\hat{\mu}_{X}(s\to F)$ $\displaystyle=R_{X}{R(s\to F)\over
R(s\to\text{all})},$ (33)
where explicit form of all the rates in right-hand side has already been
presented in Sec. III.2. Figures 2 and 3 are, respectively, for the diphoton
final state and for the others than diphoton and digluon ones. Though it is
hardly observable at the LHC, we also plot the signal strength for digluon
final states in Fig. 4 for completeness. For real experimental data under a
given set of cuts $i$, signal strength becomes a mixture of those from GF and
VBF,VH,ttH processes shown in the left and right panels, respectively, with
coefficients $\varepsilon^{i}_{X}$ ($X=\text{GF, VBF, VH, and ttH}$) being
multiplied as in Eq. (32). Figures 2–4 are the prediction of our model.
From Fig. 2, we see that the diphoton signal strength can be larger than unity
when the dilaton decay constant $f$ is not much larger than the SM Higgs
vacuum expectation value (VEV), namely when $\eta^{-1}=f/v\lesssim 1$. In a
pure VBF/VH/ttH production channel, only the negative $\theta_{H}$ region can
give an enhancement of the diphoton signal strength. We can see from the right
panels in Figs. 2–4 that in purely dilatonic region $\theta_{H}\simeq 0$, the
VBF/VH/ttH production is suppressed for all the decay modes. In particular,
the other decay modes $WW,ZZ,bb,\tau\tau$ etc. are always suppressed in the
VBF/VH/ttH channel. The signal strengths for decay modes other than
$\gamma\gamma$ are generally enhanced with GF production for a positive
$\theta_{H}$, as can be seen in left panels of Figs. 3 and 4. Especially the
digluon signal strength can be enhanced as large as 30.
### III.4 Constraints on dilaton/Higgs sector
Figure 5: Favored regions within 90, 95 and 99% confidence intervals, enclosed
by solid, dashed, and dotted lines, respectively. Density (area) of favored
region decreases (increases) in according order. Results are shown for ATLAS
(left), CMS (center), and combined (right). See text for details.
As all the signal strengths are obtained, we perform a chi-square test with
the Gaussian approximation for all the errors
$\displaystyle\chi^{2}$
$\displaystyle=\sum_{i}\left(\hat{\mu}_{i}-\mu_{i}\over\sigma_{i}\right)^{2},$
(34)
where summation over $i$ is for all the diphoton categories as well as the
$WW$ and $ZZ$ channels. For the $ZZ\to 4l$ and $WW\to l\nu l\nu$ decay
channels, we assume that all the signals are coming from GF and, hence, we
approximate
$\displaystyle\hat{\mu}(s\to VV)$
$\displaystyle=\left|c_{g}\right|^{2}\,{R(s\to\text{others})\over
R(s\to\text{all})}$ (35)
for $VV=WW$ and $ZZ$. For ATLAS, the central value $\mu_{i}$ and deviation
$\sigma_{i}$ are read off from Fig. 14 in Ref. ATLAS_diphoton for diphoton
channels and from Fig. 10 in Ref. obs:2012gk for $WW$ and $ZZ$ channels. For
CMS, Fig. 6(b) in Ref. CMS_diphoton and Table 7 in Ref. obs:2012gu are used
for diphoton and $VV$ channels, respectively. The resultant number of degrees
of freedom is 22 and 13 for ATLAS and CMS, respectively.
The results are shown in Fig. 5. We see that both experiments have allowed
dilatonlike region $\left|\theta_{H}\right|<\pi/4$ within 90% confidence
interval though the ATLAS disfavors the purely dilatonic case
$\theta_{H}\simeq 0$ outside the 95% confidence interval. This is one of our
main results.
As an illustration, we have also presented in right panel of Fig. 5 a
“theorist combination” plot with the data from Fig. 3 in Ref. Giardino:2012dp
. We have assumed that $WW$, $ZZ$, and $\gamma\gamma$ ($bbV$, $WWV$ and
$\tau\tau$) are all coming from GF (VBF/VH/ttH) processes whereas
$\gamma\gamma jj$ has 70% from VBF/VH/ttH and 30% from GF. In this naive
treatment, we see that the SM is already outside the 90% confidence interval
whereas the minimal dilaton model has the allowed regions with a dilatonlike
scalar.
## IV Constraints on top sector from electroweak data
When the Higgs-dilaton mixing $\theta_{H}$ is small, the relevant parameters
for the Peskin-Takeuchi $S$, $T$ parameters are the top partner mass
$m_{t^{\prime}}$ and the left-handed top mixing $\theta_{L}$. As we will see
later, physics of the dilaton at the LHC is independent of those two
parameters $m_{t^{\prime}}$, $\theta_{L}$ in the top sector. Therefore one can
discuss the electroweak constraints and the LHC physics independently. In this
section, we present allowed region of the parameters $m_{t^{\prime}}$ and
$\theta_{L}$ from the electroweak precision measurements.
As is well known, the electroweak precision tests prefer a light Higgs boson
in the Standard Model. The upper bound is 185 GeV at the 95% confidence level
(CL) ALEPH:2010aa . On the other hand, the assumption that the 125 GeV
excesses at the LHC as the dilaton requires the SM Higgs boson (if exists) to
be heavier than about 600 GeV. It is then necessary that the $t^{\prime}$
loops provide a correction with an appropriate size and sign to push back to
the allowed region in the $S$-$T$ plane.
We have obtained contributions to the Peskin-Takeuchi $S$, $T$ parameters from
the $t$ and $t^{\prime}$ loops as
$\displaystyle S^{\text{top}}=$
$\displaystyle\sin^{2}\theta_{L}\frac{N_{c}}{6\pi}\left[\left(\frac{1}{3}-\cos^{2}\theta_{L}\right)\ln
x+\left(\frac{(1+x)^{2}}{(1-x)^{2}}+\frac{2x^{2}(3-x)}{(1-x)^{3}}\ln
x-\frac{8}{3}\right)\cos^{2}\theta_{L}\right],$ (36) $\displaystyle
T^{\text{top}}=$
$\displaystyle\sin^{2}\theta_{L}\frac{N_{c}}{16\pi}\frac{1}{s_{W}^{2}c_{W}^{2}}\frac{m_{t}^{2}}{m_{Z}^{2}}\left[\frac{\sin^{2}\theta_{L}}{x}-(1+\cos^{2}\theta_{L})-\frac{2}{1-x}\cos^{2}\theta_{L}\ln
x\right],$ (37)
where
$\displaystyle x:=\frac{m_{t}^{2}}{m_{t^{\prime}}^{2}}<1$ (38)
and $\theta_{L}$ is the mixing angle between $t$ and $t^{\prime}$ defined in
Appendix A. If $\theta_{L}=0$, then $t^{\prime}$ decouples, and
$S^{\text{top}}$ and $T^{\text{top}}$ become 0. This is because we have
already subtracted the SM contributions from the definition of $S$ and $T$
parameters, as usual. For a fixed $\theta_{L}$ and a large $m_{t^{\prime}}$,
$T^{\text{top}}$ is enhanced as $\propto m_{t^{\prime}}^{2}$, whereas
$S^{\text{top}}$ only has a logarithmic dependence. Therefore, we generically
obtain a large positive contribution to $T^{\text{top}}$ and
$|S^{\text{top}}|\ll T^{\text{top}}$. Interestingly, this is indeed the
required direction to come back to the allowed region when the Higgs boson is
heavy.
We also need to calculate the contributions from scalar sectors because now we
have two scalars, $s$ and $h$, and their couplings to the gauge boson are
different from the SM Higgs couplings. We find
$\displaystyle S^{\text{scalar}}=$
$\displaystyle-\frac{\cos^{2}\theta_{H}}{12\pi}\ln\frac{m_{\text{$h$ref}}^{2}}{m_{h}^{2}}-\frac{\sin^{2}\theta_{H}}{12\pi}\ln\frac{m_{\text{$h$ref}}^{2}}{m_{s}^{2}}+\cos^{2}\theta_{H}f_{S}(m_{h})+\sin^{2}\theta_{H}f_{S}(m_{s})-f_{S}(m_{\text{$h$ref}}),$
(39) $\displaystyle T^{\text{scalar}}=$
$\displaystyle\frac{3\cos^{2}\theta_{H}}{16\pi
c_{W}^{2}}\ln\frac{m_{\text{$h$ref}}^{2}}{m_{h}^{2}}+\frac{3\sin^{2}\theta_{H}}{16\pi
c_{W}^{2}}\ln\frac{m_{\text{$h$ref}}^{2}}{m_{s}^{2}}+\cos^{2}\theta_{H}f_{T}(m_{h})+\sin^{2}\theta_{H}f_{T}(m_{s})-f_{T}(m_{\text{$h$ref}}),$
(40)
where $m_{\text{$h$ref}}$ is the reference Higgs boson mass and $f_{S}(m_{h})$
and $f_{T}(m_{h})$ are small nonlogarithmic contributions whose explicit
expression are
$\displaystyle f_{S}(m_{h})$
$\displaystyle=-\frac{1}{12\pi}\frac{(9m_{h}^{2}+m_{Z}^{2})m_{Z}^{4}}{(m_{h}^{2}-m_{Z}^{2})^{3}}\ln\frac{m_{Z}^{2}}{m_{h}^{2}}-\frac{(2m_{h}^{2}+3m_{Z}^{2})m_{Z}^{2}}{6\pi(m_{h}^{2}-m_{Z}^{2})^{2}},$
(41) $\displaystyle f_{T}(m_{h})$ $\displaystyle=-\frac{3m_{Z}^{2}}{16\pi
s_{W}^{2}c_{W}^{2}(m_{h}^{2}-m_{Z}^{2})}\ln\frac{m_{h}^{2}}{m_{Z}^{2}}+\frac{3m_{W}^{2}}{16\pi
s_{W}^{2}(m_{h}^{2}-m_{W}^{2})}\ln\frac{m_{h}^{2}}{m_{W}^{2}}.$ (42)
Note that the $S$ and $T$ parameters given in Eqs. (39) and (40) are
independent of the sign of the Higgs-dilaton mixing angle $\theta_{H}$ since
they are functions of $\sin^{2}\theta_{H}$.
The contributions from $S^{\text{scalar}}$ and $T^{\text{scalar}}$ tend to be
smaller than the contributions from the top sector. The region in which they
give non-negligible contributions is around $\sin^{2}\theta_{H}=1$. In this
region, we can not ignore the sizable contribution from the term which is
proportional to $\ln(m_{\text{$h$ref}}^{2}/m_{s}^{2})$. However, in this
region, the $s$ couplings to the SM particles become almost the same as the SM
Higgs boson couplings, and $h$ behaves like SM singlet particle. Then this
region is nothing but the SM limit, which is not the interest in this paper.
Therefore we can conclude that the dominant contributions to the $S$ and $T$
parameters arise from the top sector, and $\theta_{H}$ dependence of the $S$
and $T$ parameters is mild.
Figure 6: Favored region plot from the Peskin-Takeuchi $S$, $T$. White regions
are excluded at 95% CL.
The numerical values of parameters we use are
$\displaystyle s_{W}^{2}$ $\displaystyle=0.23,$ $\displaystyle v$
$\displaystyle=246\,{\rm GeV},$ $\displaystyle m_{s}$ $\displaystyle=125\,{\rm
GeV},$ $\displaystyle m_{h}$ $\displaystyle=600,\,1000\,{\rm GeV},$
$\displaystyle\left|\theta_{H}\right|$ $\displaystyle=0,\,{\pi\over
6},\,{\pi\over 3}.$ (43)
In Fig. 6, we show favored region in the $m_{t^{\prime}}$ \- $\sin\theta_{L}$
plane. White regions are excluded at 95% CL by $S$ and $T$ parameters.
There are other experimental constraints as well as $S$ and $T$ parameters.
The mass bound on $t^{\prime}$ from the direct search at the LHC is t' mass
$\displaystyle m_{t^{\prime}}>560~{}{\rm GeV}~{}~{}~{}~{}(95{\rm\%~{}CL}).$
(44)
We can find a constraint on $\theta_{L}$ from the bound on $V_{tb}$ because
the mixing angle $\theta_{L}$ changes the top quark couplings, such as
$g_{Wtb}$. The bound on $V_{tb}$ without assuming the unitarity triangle is
Abazov:2012vd
$\displaystyle 0.81<|V_{tb}|\leq 1~{}~{}~{}~{}(95{\rm\%~{}CL}).$ (45)
If we assume the top quark never mixed with light quarks, then the above
constraint gives
$\displaystyle 0.81<|\cos\theta_{L}|\leq 1,$ (46)
namely,
$\displaystyle|\sin\theta_{L}|<0.59.$ (47)
We find that the constraints given in Eqs. (44) and (47) are easily satisfied
in the allowed region in Fig. 6.
We also study the constraint from $Zb_{L}b_{L}$ coupling. In the SM case, the
flavor-dependent corrections to this coupling are proportional to the squared
top-Yukawa coupling. In this model, this correction is modified due to the
mixing between $t$ and $t^{\prime}$. We parametrize $Zb_{L}\overline{b}_{L}$
coupling as follows
$\displaystyle\frac{e}{s_{W}c_{W}}\left(-\frac{1}{2}+\delta
g_{L}+\frac{1}{3}s_{W}^{2}\right).$ (48)
We focus on only the flavor-dependent correction hereafter because we use the
constraint on $R_{b}$ pdg:2012 to derive the constraint on $Zb_{L}b_{L}$. We
find that the flavor-dependent part of $\delta g_{L}$ is given by
$\displaystyle\delta g_{L}$
$\displaystyle=\frac{m_{t}^{2}}{(4\pi)^{2}v^{2}}+\delta g_{L}^{\text{new}},$
(49)
where the first term is the SM contribution, and the second terms is the
additional contributions due to the $t^{\prime}$ and the mixing angle
$\theta_{L}$. We find
$\displaystyle\delta g_{L}^{\text{new}}$
$\displaystyle=-\frac{m_{t}^{2}}{16\pi^{2}v^{2}}\frac{\sin^{2}\theta_{L}}{(1-\sin^{2}\theta_{L}(1-x))^{3}}$
$\displaystyle\qquad\times\Biggl{(}-1-\sin^{4}\theta_{L}(1-x)^{3}+2x+\sin^{2}\theta_{L}(2-5x+2x^{2})+\frac{(1-\sin^{2}\theta_{L})x(1+x)\ln
x}{2(1-x)}\Biggr{)}$
$\displaystyle\simeq+\frac{m_{t}^{2}}{16\pi^{2}v^{2}}\frac{\sin^{2}\theta_{L}}{(1-\sin^{2}\theta_{L})}\qquad(x\ll
1),$ (50)
where $x$ is defined in Eq. (38). The constraint on $\delta
g_{L}^{\text{new}}$ Abe:2009ni , which is derived from the constraint on
$R_{b}$, is
$\displaystyle\delta g_{L}^{\text{new}}$ $\displaystyle=\left(-5.8\pm
8.6\right)\times 10^{-4}.$ (51)
Comparing Eqs. (50) and (51), we find the region $|\sin\theta_{L}|>0.52$ is
excluded at 95% CL. This region is already excluded from the constraint on $S$
and $T$ parameters in Fig. 6. Hence we conclude that the $Zb_{L}b_{L}$
constraint is not important in this model.
Before closing this section, we comment on the validity of our perturbative
calculation. In the top sector, there are three parameters in the Lagrangian,
$m$, $y_{t}$, and $y^{\prime}$. These three can be expressed by observables
$m_{t}$, $m_{t^{\prime}}$, and $\theta_{L}$ as shown in Appendix A. In the
limit $m_{t^{\prime}}\gg m_{t}$,
$\displaystyle y_{t}$ $\displaystyle\simeq{\sqrt{2}\over
v}{m_{t}\over\cos^{2}\theta_{L}},$ $\displaystyle y^{\prime}$
$\displaystyle\simeq{\sqrt{2}\over v}m_{t^{\prime}}\sin\theta_{L},$
$\displaystyle M$ $\displaystyle\simeq m_{t^{\prime}}\cos\theta_{L}.$ (52)
We have seen that the small $\theta_{L}$ region is allowed by the $S$-$T$
bound. Taking $\theta_{L}\ll 1$ limit, we get
$\displaystyle y_{t}$ $\displaystyle\simeq{\sqrt{2}m_{t}\over v}\simeq 1,$
$\displaystyle y^{\prime}$ $\displaystyle\simeq{m_{t^{\prime}}\over
m_{t}}\sin\theta_{L},$ $\displaystyle M$ $\displaystyle\simeq m_{t^{\prime}}.$
(53)
We see that only $y^{\prime}$ provides a nontrivial constraint, especially in
large $m_{t^{\prime}}$ region. In Fig. 6, we see that the allowed region of
$\sin\theta_{L}$ is about less than 0.2 in large $m_{t^{\prime}}$ limit. Then,
using Eq. (53), we see that
$\displaystyle y^{\prime}$ $\displaystyle\lesssim
0.2\,\frac{m_{t^{\prime}}}{m_{t}}.$ (54)
From this relation, we find that if we impose that $y^{\prime}$ should be
lower than 4 to keep perturbativity, $m_{t^{\prime}}$ should be lighter than
3400 GeV. This upper bound on $m_{t^{\prime}}$ does not spoil our discussion
in this paper. Thus our perturbative calculation with $y^{\prime}$ is valid
unless we take $m_{t^{\prime}}$ to be extremely large.222 A tighter upper
bound would be imposed, roughly $M\lesssim 1$ TeV depending on $\eta$, if one
requires perturbativity for the coupling ${M\over f}S\overline{T}T$, though it
is irrelevant for the computations in this paper.
## V Conclusions
We have considered a possibility that the Higgs-like excesses observed at the
LHC experiments are actually the signals of the dilaton associated with
spontaneously broken scale invariance. We have constructed a minimal model of
the dilaton which can be produced through the gluon fusion process at the LHC,
and can decay into two photons. The effective coupling is obtained through the
loop diagrams of a new vectorlike state $T$ that has the same gauge charges as
the right-handed top quark. The $T$ field contributes to the Peskin-Takeuchi
$S$, $T$ parameters in the electroweak precision tests. This contribution
allows one to push the Higgs boson mass above the experimental constraint, 600
GeV, providing a consistent framework for the light dilaton plus a heavy Higgs
scenario.
We find that the current experimental data allow distinct parameter regions
where the excesses are either Higgs-like or dilatonlike. Once the excesses are
confirmed with more statistics, it is possible to distinguish two scenarios.
Note added.—After completion of this manuscript, there appeared works treating
a similar subject note added .
### Acknowledgements
The authors thank the Yukawa Institute for Theoretical Physics at Kyoto
University, where this work was initiated during the YITP workshop on “Beyond
the Standard Model Physics (YITP-W-11-27),” March 19–23, 2012\. The authors
also acknowledge the participants of the workshop for very active discussions.
We thank Masaya Ishino, Koji Nakamura, Yoshiko Ohno, Masaharu Tanabashi, and
Junichi Tanaka for useful comments. This work is supported in part by the
Grants-in-Aid for Scientific Research No. 23740165 (R. K.), No. 23104009, No.
20244028, No. 23740192 (K. O.), and No. 24340044 (J. S.) of JSPS.
## Appendix A Mixing of top and its partner
As said in the text, we have chosen the basis on which the mass mixing between
top and its partner $\overline{T_{L}}u_{3R}$ is rotated away
$\displaystyle\begin{bmatrix}\overline{q_{3L}}&\overline{T_{L}}\end{bmatrix}\begin{bmatrix}m&m^{\prime}\\\
0&M\end{bmatrix}\begin{bmatrix}u_{3R}\\\ T_{R}\end{bmatrix},$ (55)
where $m=y_{t}v/\sqrt{2}$ and $m^{\prime}=y^{\prime}v/\sqrt{2}$. Switching to
mass eigenstates
$\displaystyle\begin{bmatrix}q_{3L}\\\ T_{L}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos\theta_{L}&\sin\theta_{L}\\\
-\sin\theta_{L}&\cos\theta_{L}\end{bmatrix}\begin{bmatrix}t_{L}\\\
t^{\prime}_{L}\end{bmatrix},$ $\displaystyle\begin{bmatrix}u_{3R}\\\
T_{R}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos\vartheta_{R}&\sin\vartheta_{R}\\\
-\sin\vartheta_{R}&\cos\vartheta_{R}\end{bmatrix}\begin{bmatrix}t_{R}\\\
t^{\prime}_{R}\end{bmatrix},$ (56)
we may diagonalize as
$\displaystyle\begin{bmatrix}\overline{q_{3L}}&\overline{T_{L}}\end{bmatrix}\begin{bmatrix}m&m^{\prime}\\\
0&M\end{bmatrix}\begin{bmatrix}u_{3R}\\\ T_{R}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\overline{t_{L}}&\overline{t^{\prime}_{L}}\end{bmatrix}\begin{bmatrix}m_{t}&0\\\
0&m_{t^{\prime}}\end{bmatrix}\begin{bmatrix}t_{R}\\\
t^{\prime}_{R}\end{bmatrix},$ (57)
where
$\displaystyle\tan\theta_{L}$
$\displaystyle={\sqrt{\left(M^{2}-m^{2}+m^{\prime 2}\right)^{2}+4m^{\prime
2}m^{2}}-M^{2}+m^{2}+m^{\prime 2}\over 2m^{\prime}M}={m^{\prime}\over
M}+O(M^{-3}),$ $\displaystyle\tan\vartheta_{R}$
$\displaystyle={\sqrt{\left(M^{2}-m^{2}+m^{\prime 2}\right)^{2}+4m^{\prime
2}m^{2}}-M^{2}+m^{2}-m^{\prime 2}\over 2m^{\prime}m}={m^{\prime}m\over
M^{2}}+O(M^{-4}),$ (58)
and the mass eigenvalues are
$\displaystyle\left\\{\begin{array}[]{c}m_{t}^{2}\\\
m_{t^{\prime}}^{2}\end{array}\right\\}$ $\displaystyle={M^{2}+m^{2}+m^{\prime
2}\mp\sqrt{\left(M^{2}+m^{2}+m^{\prime 2}\right)^{2}-4m^{2}M^{2}}\over 2}.$
(61)
For large $M$,
$\displaystyle m_{t}$ $\displaystyle=\left(1-{m^{\prime 2}\over
2M^{2}}\right)m+O(M^{-4}),$ $\displaystyle m_{t^{\prime}}$
$\displaystyle=M+{m^{\prime 2}\over 2M}+O(M^{-3}).$ (62)
Conversely, parameters in the Lagrangian can be written in terms of the
observables:
$\displaystyle M$
$\displaystyle=\sqrt{m_{t}^{2}\sin^{2}\theta_{L}+m_{t^{\prime}}^{2}\cos^{2}\theta_{L}},$
(63) $\displaystyle y_{t}$
$\displaystyle=\frac{\sqrt{2}}{v}\frac{m_{t}m_{t^{\prime}}}{\sqrt{m_{t}^{2}\sin^{2}\theta_{L}+m_{t^{\prime}}^{2}\cos^{2}\theta_{L}}},$
(64) $\displaystyle y^{\prime}$
$\displaystyle=\frac{\sqrt{2}}{v}\frac{(m_{t^{\prime}}^{2}-m_{t}^{2})\sin\theta_{L}\cos\theta_{L}}{\sqrt{m_{t}^{2}\sin^{2}\theta_{L}+m_{t^{\prime}}^{2}\cos^{2}\theta_{L}}}.$
(65)
Instead of our choice (55), we may choose another basis where the Yukawa
mixing $\overline{q_{3L}}T_{R}$ is erased
$\displaystyle\begin{bmatrix}\overline{q_{3L}}&\overline{T_{L}}\end{bmatrix}\begin{bmatrix}m\cos\vartheta+m^{\prime}\sin\vartheta&0\\\
M\sin\vartheta&M\cos\vartheta\end{bmatrix}\begin{bmatrix}\widetilde{u}_{3R}\\\
\widetilde{T}_{R}\end{bmatrix},$ (66)
where
$\displaystyle\begin{bmatrix}\widetilde{u}_{3R}\\\
\widetilde{T}_{R}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos\vartheta&\sin\vartheta\\\
-\sin\vartheta&\cos\vartheta\end{bmatrix}\begin{bmatrix}u_{3R}\\\
T_{R}\end{bmatrix},$ $\displaystyle\tan\vartheta$
$\displaystyle={m^{\prime}\over m}={y^{\prime}\over y_{t}}.$ (67)
Note that $\vartheta$ is not necessarily a small mixing angle and that the
right-handed tops in this basis are related to the mass eigenstates by
$\displaystyle\begin{bmatrix}t_{R}\\\ t^{\prime}_{R}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos(\vartheta+\vartheta_{R})&-\sin(\vartheta+\vartheta_{R})\\\
\sin(\vartheta+\vartheta_{R})&\cos(\vartheta+\vartheta_{R})\end{bmatrix}\begin{bmatrix}\widetilde{u}_{3R}\\\
\widetilde{T}_{R}\end{bmatrix}.$ (68)
Although $\theta_{L}$ is more directly related to physical observables, one
may trade it with $\vartheta+\vartheta_{R}$, which is the angle denoted by
$\theta_{u}^{R}$ and is constrained in Ref. Cacciapaglia:2011fx . We find the
following relation between $\theta_{L}$ and $\vartheta+\vartheta_{R}\
(=\theta_{u}^{R})$:
$\displaystyle\sin\theta_{L}=$
$\displaystyle\frac{m_{t}}{\sqrt{m_{t}^{2}\sin^{2}\theta_{u}^{R}+m_{t^{\prime}}^{2}\cos^{2}\theta_{u}^{R}}}\sin\theta_{u}^{R}.$
(69)
Using this relation, we find our result is compatible with the result given in
Ref. Cacciapaglia:2011fx .
## Appendix B Linear realization potential
Though the precise form of the potential $\tilde{V}$ in Eq. (4) is irrelevant
for the experimental consequences which are governed by the Higgs-dilaton
mixing angle $\theta_{H}$ and the dilaton decay constant in units of the
electroweak scale $\eta^{-1}=f/v$, let us write down a renormalizable
linearized version of our potential just for completeness
$\displaystyle\tilde{V}(S,H)$ $\displaystyle={m_{S}^{2}\over
2}S^{2}+{\lambda_{S}\over 4!}S^{4}+{\kappa\over
2}S^{2}\left|H\right|^{2}+m_{H}^{2}\left|H\right|^{2}+{\lambda_{H}\over
2^{2}}\left|H\right|^{4}.$ (70)
The VEVs for the SM Higgs $\left\langle H\right\rangle=v/\sqrt{2}$ and for the
singlet $\left\langle S\right\rangle=f$ are obtained as
$\displaystyle\begin{bmatrix}f^{2}\\\ v^{2}\end{bmatrix}$
$\displaystyle={1\over{\lambda_{S}\lambda_{H}\over
6}-\kappa^{2}}\begin{bmatrix}\lambda_{H}&-2\kappa\\\
-2\kappa&{2\lambda_{S}\over 3}\end{bmatrix}\begin{bmatrix}-m_{S}^{2}\\\
-m_{H}^{2}\end{bmatrix}.$ (71)
The mass eigenvalues are
$\displaystyle\left\\{\begin{matrix}m_{s}^{2}\\\
m_{h}^{2}\end{matrix}\right\\}$ $\displaystyle={\left({\lambda_{S}\over
3}f^{2}+{\lambda_{H}\over 2}v^{2}\right)\mp\sqrt{\left({\lambda_{S}\over
3}f^{2}+{\lambda_{H}\over
2}v^{2}\right)^{2}-4\left({\lambda_{S}\lambda_{H}\over
6}-\kappa^{2}\right)f^{2}v^{2}}\over 2}$ (72)
with the Higgs-dilaton mixing (7) being
$\displaystyle\tan 2\theta_{H}$ $\displaystyle={2\kappa
fv\over{\lambda_{S}\over 3}f^{2}-{\lambda_{H}\over 2}v^{2}}.$ (73)
Since we want
$m_{s}^{2}/m_{h}^{2}\lesssim\left(125\,\text{GeV}\right)^{2}/\left(600\,\text{GeV}\right)^{2}\simeq
4\%\ll 1$, we can write
$\displaystyle m_{s}^{2}$
$\displaystyle=m_{\sigma}^{2}+O\\!\left(m_{\sigma}^{4}\over M_{h}^{2}\right),$
$\displaystyle m_{h}^{2}$
$\displaystyle=M_{h}^{2}-m_{\sigma}^{2}+O\\!\left(m_{\sigma}^{4}\over
M_{h}^{2}\right),$ (74)
with
$\displaystyle m_{\sigma}^{2}$
$\displaystyle:=\left({\lambda_{S}\lambda_{H}\over
6}-\kappa^{2}\right){f^{2}v^{2}\over M_{h}^{2}},$ $\displaystyle M_{h}^{2}$
$\displaystyle:={\lambda_{S}\over 3}f^{2}+{\lambda_{H}\over 2}v^{2},$ (75)
where ${\lambda_{S}\lambda_{H}\over 6}-\kappa^{2}>0$ is required in order not
to have a tachyon. Finally we note that a tree-level vacuum stability
condition for a large VEVs reads ${2\lambda_{S}\over
3}-4\kappa+\lambda_{H}>0$.
## References
* (1) G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B [arXiv:1207.7214 [hep-ex]].
* (2) S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B [arXiv:1207.7235 [hep-ex]].
* (3) S. Weinberg, “Implications of Dynamical Symmetry Breaking,” Phys. Rev. D 13, 974 (1976);
L. Susskind, “Dynamics of Spontaneous Symmetry Breaking in the Weinberg-Salam
Theory,” Phys. Rev. D 20, 2619 (1979).
* (4) C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, “Gauge theories on an interval: Unitarity without a Higgs,” Phys. Rev. D 69, 055006 (2004) [hep-ph/0305237].
* (5) S. Matsuzaki and K. Yamawaki, “Techni-dilaton signatures at LHC,” Prog. Theor. Phys. 127 (2012) 209 [arXiv:1109.5448 [hep-ph]];
S. Matsuzaki and K. Yamawaki, “Techni-dilaton at 125 GeV,” Phys. Rev. D 85
(2012) 095020 [arXiv:1201.4722 [hep-ph]];
S. Matsuzaki and K. Yamawaki, “Discovering 125 GeV techni-dilaton at LHC,”
arXiv:1206.6703 [hep-ph];
S. Matsuzaki and K. Yamawaki, “Is 125 GeV techni-dilaton found at LHC?,”
arXiv:1207.5911 [hep-ph];
D. Elander and M. Piai, “The decay constant of the holographic techni-dilaton
and the 125 GeV boson,” arXiv:1208.0546 [hep-ph];
S. Matsuzaki and K. Yamawaki, “Holographic techni-dilaton at 125 GeV,”
arXiv:1209.2017 [hep-ph].
* (6) K. Yamawaki, M. Bando and K. -i. Matumoto, “Scale Invariant Technicolor Model and a Technidilaton,” Phys. Rev. Lett. 56, 1335 (1986);
M. Bando, K. -i. Matumoto and K. Yamawaki, “Technidilaton,” Phys. Lett. B 178,
308 (1986).
* (7) W. D. Goldberger, B. Grinstein and W. Skiba, “Distinguishing the Higgs boson from the dilaton at the Large Hadron Collider,” Phys. Rev. Lett. 100 (2008) 111802 [arXiv:0708.1463 [hep-ph]];
J. Fan, W. D. Goldberger, A. Ross and W. Skiba, “Standard Model couplings and
collider signatures of a light scalar,” Phys. Rev. D 79 (2009) 035017
[arXiv:0803.2040 [hep-ph]].
* (8) R. Foot, A. Kobakhidze and R. R. Volkas, “Electroweak Higgs as a pseudo-Goldstone boson of broken scale invariance,” Phys. Lett. B 655 (2007) 156 [arXiv:0704.1165 [hep-ph]];
R. Foot, A. Kobakhidze and K. L. McDonald, “Dilaton as the Higgs boson,” Eur.
Phys. J. C 68 (2010) 421 [arXiv:0812.1604 [hep-ph]].
* (9) K. Cheung and T. -C. Yuan, “Could the excess seen at 124-126 GeV be due to the Randall-Sundrum Radion?,” Phys. Rev. Lett. 108, 141602 (2012) [arXiv:1112.4146 [hep-ph]].
* (10) B. Grzadkowski, J. F. Gunion and M. Toharia, “Higgs-Radion interpretation of the LHC data?,” Phys. Lett. B 712 (2012) 70 [arXiv:1202.5017 [hep-ph]];
Y. Tang, “Comment on ‘Could the Excess Seen at 124-126 GeV Be due to the
Randall-Sundrum Radion?’,” arXiv:1204.6145 [hep-ph];
H. Kubota and M. Nojiri, “Radion-higgs mixing state at the LHC with the KK
contributions to the production and decay,” arXiv:1207.0621 [hep-ph].
* (11) M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections,” Phys. Rev. D 46 (1992) 381;
M. E. Peskin and T. Takeuchi, “A New constraint on a strongly interacting
Higgs sector,” Phys. Rev. Lett. 65 (1990) 964.
* (12) C. T. Hill, “Topcolor: Top quark condensation in a gauge extension of the standard model,” Phys. Lett. B 266, 419 (1991).
* (13) B. A. Dobrescu and C. T. Hill, “Electroweak symmetry breaking via top condensation seesaw,” Phys. Rev. Lett. 81, 2634 (1998) [hep-ph/9712319];
R. S. Chivukula, B. A. Dobrescu, H. Georgi and C. T. Hill, “Top quark seesaw
theory of electroweak symmetry breaking,” Phys. Rev. D 59, 075003 (1999) [hep-
ph/9809470];
H. -J. He, C. T. Hill and T. M. P. Tait, “Top quark seesaw, vacuum structure
and electroweak precision constraints,” Phys. Rev. D 65, 055006 (2002) [hep-
ph/0108041].
* (14) V. A. Miransky, M. Tanabashi and K. Yamawaki, “Dynamical Electroweak Symmetry Breaking with Large Anomalous Dimension and t Quark Condensate,” Phys. Lett. B 221, 177 (1989);
V. A. Miransky, M. Tanabashi and K. Yamawaki, “Is the t Quark Responsible for
the Mass of W and Z Bosons?,” Mod. Phys. Lett. A 4, 1043 (1989);
W. A. Bardeen, C. T. Hill and M. Lindner, “Minimal Dynamical Symmetry Breaking
of the Standard Model,” Phys. Rev. D 41, 1647 (1990);
Y. Nambu, “Bootstrap Symmetry Breaking In Electroweak Unification,” EFI-89-08.
* (15) A. Djouadi, “The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model,” Phys. Rept. 457 (2008) 1 [hep-ph/0503172].
* (16) P. P. Giardino, K. Kannike, M. Raidal and A. Strumia, “Reconstructing Higgs boson properties from the LHC and Tevatron data,” JHEP 1206 (2012) 117 [arXiv:1203.4254 [hep-ph]].
* (17) P. P. Giardino, K. Kannike, M. Raidal and A. Strumia, “Is the resonance at 125 GeV the Higgs boson?,” arXiv:1207.1347 [hep-ph].
* (18) D. Carmi, A. Falkowski, E. Kuflik, T. Volansky and J. Zupan, “Higgs After the Discovery: A Status Report,” arXiv:1207.1718 [hep-ph].
* (19) The ATLAS Collaboration, “Coupling properties of the new Higgs-like boson observed with the ATLAS detector at the LHC,” ATLAS-CONF-2012-127.
* (20) The ATLAS Collaboration, “Observation of an excess of events in the search for the Standard Model Higgs boson in the $\gamma\gamma$ channel with the ATLAS detector,” ATLAS-CONF-2012-091.
* (21) The CMS Collaboration, “Evidence for a new state decaying into two photons in the search for the standard model Higgs boson in pp collisions,” CMS PAS HIG-12-015.
* (22) The ATLAS Collaboration, “Observation of an excess of events in the search for the Standard Model Higgs boson in the $H\to ZZ\to 4\ell$ channel with the ATLAS detector,” ATLAS-CONF-2012-092.
* (23) The CMS Collaboration, “Evidence for a new state in the search for the standard model Higgs boson in the $H\to ZZ\to 4\ell$ channel in pp collisions at $\sqrt{s}=7$ and 8 TeV,” CMS PAS HIG-12-016.
* (24) The ATLAS Collaboration, “Observation of an excess of events in the search for the Standard Model Higgs boson in the $H\to WW\to\ell\nu\ell\nu$ channel with the ATLAS detector,” ATLAS-CONF-2012-098.
* (25) The CMS Collaboration, “Search for the standard model Higgs boson decaying to $W^{+}W^{-}$ in the fully leptonic final state in pp collisions at $\sqrt{s}=8$ TeV,” CMS PAS HIG-12-017.
* (26) [The ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the Tevatron Electroweak Working Group, and the SLD electroweak and heavy flavour groups], “Precision Electroweak Measurements and Constraints on the Standard Model,” arXiv:1012.2367 [hep-ex].
* (27) S. Chatrchyan et al. [CMS Collaboration], “Search for a Vector-like Quark with Charge 2/3 in t + Z Events from pp Collisions at sqrt(s) = 7 TeV,” Phys. Rev. Lett. 107, 271802 (2011) [arXiv:1109.4985 [hep-ex]];
S. Chatrchyan et al. [CMS Collaboration], “Search for heavy, top-like quark
pair production in the dilepton final state in $pp$ collisions at $\sqrt{s}=7$
TeV,” arXiv:1203.5410 [hep-ex];
The CMS Collaboration, “Search for the pair production of a fourth-generation
up-type $t^{\prime}$ quark in events with a lepton and at least four jets,”
CMS PAS EXO-11-099.
* (28) V. M. Abazov et al. [D0 Collaboration], “An Improved determination of the width of the top quark,” Phys. Rev. D 85, 091104 (2012) [arXiv:1201.4156 [hep-ex]].
* (29) J. Beringer et al. [Particle Data Group Collaboration], “Review of Particle Physics (RPP),” Phys. Rev. D 86 (2012) 010001.
* (30) T. Abe, R. S. Chivukula, N. D. Christensen, K. Hsieh, S. Matsuzaki, E. H. Simmons and M. Tanabashi, “Z to b bbar and Chiral Currents in Higgsless Models,” Phys. Rev. D 79, 075016 (2009) [arXiv:0902.3910 [hep-ph]].
* (31) Z. Chacko and R. K. Mishra, “Effective Theory of a Light Dilaton,” arXiv:1209.3022 [hep-ph];
Z. Chacko, R. Franceschini and R. K. Mishra, “Resonance at 125 GeV: Higgs or
Dilaton/Radion?,” arXiv:1209.3259 [hep-ph];
B. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, “A Higgslike
Dilaton,” arXiv:1209.3299 [hep-ph].
* (32) G. Cacciapaglia, A. Deandrea, L. Panizzi, N. Gaur, D. Harada and Y. Okada, “Heavy Vector-like Top Partners at the LHC and flavour constraints,” JHEP 1203 (2012) 070 [arXiv:1108.6329 [hep-ph]].
|
arxiv-papers
| 2012-09-20T14:23:10 |
2024-09-04T02:49:35.345426
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tomohiro Abe, Ryuichiro Kitano, Yasufumi Konishi, Kin-ya Oda, Joe\n Sato, and Shohei Sugiyama",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1209.4544"
}
|
1209.4553
|
††thanks: Present address: Joint Quantum Institute, University of Maryland and
NIST, College Park, Maryland, 20742
# Universal three-body recombination via resonant $d$-wave interactions
Jia Wang Department of Physics and JILA, University of Colorado, Boulder, CO
80309, USA Department of Physics, University of Connecticut, Storrs, CT
06269, USA J. P. D’Incao Department of Physics and JILA, University of
Colorado, Boulder, CO 80309, USA Yujun Wang Department of Physics and JILA,
University of Colorado, Boulder, CO 80309, USA Chris H. Greene Department of
Physics and JILA, University of Colorado, Boulder, CO 80309, USA Department
of Physics, Purdue University, West Lafayette, IN 47907, USA
###### Abstract
For a system of three identical bosons interacting via short-range forces,
when two of the atoms are about to form a two-body $s$-wave dimer, the Efimov
effect takes place leading to the formation of an infinite number of three-
body (Efimov) states. The lowest Efimov state crosses the three-body break-up
threshold when the $s$-wave two-body scattering length is $a\approx-9.73r_{\rm
vdW}$, $r_{\rm vdW}$ being the van der Waals length. This article focuses on a
generalized version of this Efimov scenario, where two of the atoms are about
to form a two-body $d$-wave dimer, resulting in strong $d$-wave interactions.
Bo Gao has argued, in [Phys. Rev. A. 62, 050702(R) (2000)], that for broad
resonances the $d$-wave dimer is always formed when $a\approx 0.956r_{\rm
vdW}$. Our results demonstrate that a single universal three-body state
associated with the $d$-wave dimer is also formed near the three-body break-up
threshold at $a\approx 1.09r_{\rm vdW}$, or alternatively $a_{2}=0.902r_{\rm
vdW}$, where $a_{2}$ is the two-body $d$-wave scattering length. Such a
universal three-body state is signaled experimentally by an enhancement of the
three-body recombination rate. The three-body effective potential curves that
are crucial for understanding the recombination dynamics are also calculated
and analyzed. An improved method to calculate the couplings, effective
potential curves, and recombination rate coefficients is presented.
## I Introduction
In recent years, the investigation of Efimov physics Efimov (1971) has
attracted much interest and has sparked a tremendous advance in our
understanding of its fundamental aspects from both experimental Kraemer et al.
(2006); Berninger et al. (2011); Pollack et al. (2009); Gross et al. (2009,
2010); Ottenstein et al. (2008); Lompe et al. (2010); Huckans et al. (2009);
Williams et al. (2009); Wild et al. (2012); Barontini et al. (2009) and
theoretical Petrov (2004); Wang et al. (2011a); Braaten and Hammer (2006);
Hammer and Platter (2007); von Stecher et al. (2009); Wang et al. (2011b)
viewpoints. Efimov physics describes a universal phenomenon in three-body
systems: when the atoms are about to form a two-body $s$-wave dimer, i.e., the
$s$-wave scattering length $a$ goes to infinity, and an infinite number of
three-body Efimov states exists. The ratio between the energies of nearby
Efimov states is given by a universal scaling factor
$E_{n+1}/E_{n}=e^{-2\pi/s_{0}}$ , where $s_{0}$ depends only on the mass
ratios, number of resonant interactions and quantum statistics of the atoms,
e.g., $e^{\pi/s_{0}}\approx 22.7$ for three identical bosons.
Therefore, a single parameter, called the three-body parameter, is needed to
determine the whole Efimov spectrum and other scattering observables. One
possible definition of the three-body parameter is $a_{-}^{*}$, the value of
$a$ at which the first resonance in the three-body recombination rate $K_{3}$
appears, i.e., at which the lowest Efimov state energy crosses the three-body
break-up threshold ($E=0$). One fundamental assumption underlying all Efimov
physics is that the three-body parameter encapsulates the short-range details
of two- and three-body interactions, and thus should not be universal.
Surprisingly, however, ultracold experiments with alkali atoms near Fano-
Feshbach resonances Kraemer et al. (2006); Berninger et al. (2011); Pollack et
al. (2009); Gross et al. (2009, 2010); Ottenstein et al. (2008); Lompe et al.
(2010); Huckans et al. (2009); Williams et al. (2009); Wild et al. (2012);
Barontini et al. (2009) have observed a universal value, namely
$a_{-}^{*}\approx-9.1r_{\rm vdW}$ in homonuclear atomic systems, where $r_{\rm
vdW}$ is the van der Waals length. The van der Waals length $r_{\rm
vdW}=\left({2\mu_{2b}C_{6}}\right)^{1/4}/2$ is a characteristic length scale
for the van der Waals interaction $-C_{6}/r^{6}$ between two neutral atoms
with two-body reduced mass $\mu_{2b}$. A relevant energy scale, called the van
der Waals energy, can be defined as $E_{\rm
vdW}=\hbar^{2}/\left({2\mu_{2b}r_{\rm vdW}^{2}}\right)$.
This newly found universality in the three-body parameter was subsequently
studied in several different theoretical models Schmidt et al. (2012); Chin
(2011); Sørensen et al. (2012); Wang et al. (2012a); Naidon et al. (2012). In
particular, Refs. Wang et al. (2012a); Naidon et al. (2012) conclude that the
universality of the three-body parameter comes from a universal barrier in the
three-body potential curves arising from interatomic distances around 2
$r_{\rm vdW}$. This barrier originates from the suppression of the probability
to find two atoms at distances shorter than $r_{\rm vdW}$ due to the increase
of the classical local velocity and it is a universal property of van der
Waals interactions between neutral atoms. Our previous studies Wang et al.
(2012a, b) have indicated that the Lennard-Jones potential is an excellent
model potential to study the universality of three-body physics in ultracold
atomic gases, and for this reason it is also adopted in this analysis.
The present study, however, employs the Lennard-Jones potential model to study
three-body systems with atom pairs close to a $d$-wave resonance.
Interestingly, Gao Gao (2000) predicted that $d$-wave dimers form at a
universal value of $a$, $a\approx 0.956r_{\rm vdW}$, for potentials with a van
der Waals tail. Evidently, since that prediction was obtained for a single
channel potential model its validity is expected to hold only for broad Fano-
Feshbach resonances, and possibly is limited only to single-channel two-body
interactions. Nevertheless, one natural question is: will there also be a
universality of three-body physics near that value of scattering length, i.e.,
when $d$-wave interactions are expected to be strong? Here, the three-body
recombination rate is studied to answer this question, and it is seen to
exhibit two universal enhancements. Analysis of the three-body effective
potential curves yields an intuitive understanding of these enhancements. One
of these enhancements is caused by the simple fact that when the two-body
$d$-wave state becomes barely bound an additional decay channel is formed,
leading to an enhancement in recombination to $d$-wave dimers. The second
enhancement, however, has a different nature. It corresponds to the formation
of a universal three-body state that crosses the collision threshold for three
free particles. Our analysis indicates that other three-body states associated
with excited 2-body angular momenta are also possible, however, their
signatures in recombination will be suppressed in the low energy limit due to
the Wigner threshold laws for recombination Esry et al. (2001). It is
important to mention that since in experiments using ultracold atoms $s$\- and
$d$-wave states are coupled by the magnetic anisotropic dipole interaction the
$s$-wave scattering actually diverges when the $d$-wave dimer becomes bound.
Although the connection between these three-body states associated with
$d$-wave resonant interactions and the multichannel physics in ultracold atoms
still remains to be understood more deeply, the present study can offer an
alternative parameterization in terms of the properties of the $d$-wave
scattering length. Moreover, since the three-body effective potential curves
are an important tool for understanding universality in few-body systems at
ultracold temperatures Wang et al. (2012a); Naidon et al. (2012); Wang et al.
(2012b), an improved method to calculate the coupling matrix elements is also
developed in this paper.
The remainder of the paper is organized as follows. Section II tests the Gao
prediction numerically for a Lennard-Jones potential, and demonstrates very
good agreement. The three-body recombination rate is then calculated using the
Lennard-Jones two-body potential with 2 $s$-wave bound states in the vicinity
of the $d$-wave resonance. The improved numerical method is discussed in the
subsection II.A. Analysis of the recombination rate using effective potential
curves is made in subsection II.B. Section III checks the universality using
the Lennard-Jones potential with 3 $s$-wave dimer bound states.
## II Universal two-body $d$-wave dimer
One of the key experimental tools used to study ultracold atomic gases is the
Fano-Feshbach resonance, which can be utilized to magnetically tune the
scattering length. Although the multichannel character of the hyperfine
interactions leads to a great deal of complication, the single-channel van der
Waals interactions has been shown to offer a good model for studying a broad
Fano-Feshbach resonance in ultracold atomic gases Chin et al. (2011). In 2000,
Bo Gao predicted that for single-channel interactions with a van der Waals
tail, $-C_{6}/r^{6}$, there is always a $d$-wave dimer (and dimers with higher
angular momentum $l=4j+2$, where $j=1,2,3...$) that becomes bound at a
universal value of the two-body $s$-wave scattering length
$a=a^{*}=4\pi/[\Gamma(1/4)]^{2}r_{\rm{vdW}}\approx 0.956r_{\rm{vdW}}$ Gao
(2000). [Note that $a^{*}$ has the same value as the so-called mean scattering
length $\bar{a}$ as defined by Gribakin and Flambaum Gribakin and Flambaum
(1993); Chin et al. (2011), which in turn should not be confused with the
usual scattering length $a$.] Another special property of the two-body system
at $a=a^{*}$ is that the effective range $r_{\rm{eff}}$ has the value
$2\left[{\Gamma\left({1/4}\right)}\right]^{2}/\left({3\pi}\right)r_{\rm{vdW}}\approx
2.789r_{\rm{vdW}}$, which equals the effective range when $s$-wave scattering
length is infinity. In Gao’s work Gao (2000), the universality of $a^{*}$ is
based on an $l$-independent quantum-defect theory, where the short-range
physics can be described by a single short-range parameter, $K^{c}$, which is
itself approximately $l$-independent under proper conditions that are
satisfied by most diatomic systems. In combination with solutions for the
long-range interactions, $K^{c}$ uniquely determines both the scattering and
the bound state properties of diatomic systems.
Figure 1: (Color online) The values of the two-body $s$-wave scattering length
$a^{*}_{l}$ at the point where a $d$-wave ($l=2$) dimer (black curve with
square symbols) and where an $i$-wave ($l=6$) dimer just becomes bound (red
curve with circular symbols) are shown as functions of the number of two-body
$s$-wave bound states.
In this section, the universal value of the two-body $s$-wave scattering
length $a_{l}^{*}$ [where a $d$-wave ($l=2$) or $i$-wave ($l=6$) dimer becomes
bound] is studied numerically as a function of the number of two-body $s$-wave
bound states and using the Lennard-Jones potential model for the two-body
interaction:
$v(r)=-\frac{C_{6}}{r^{6}}\left(1-\frac{\lambda^{6}}{r^{6}}\right).$ (1)
Note that in the present work the parameter $\lambda$ is used to adjust the
values of $a$ as well as the number of bound states. (Some additional details
about the Lennard-Jones potential are discussed in Appendix A.) In Fig. 1, the
black square symbols (red circular symbols) indicates the numerical values of
the two-body $s$-wave scattering length $a^{*}_{2}$ ($a^{*}_{6}$) at the point
where a $d$-wave ($i$-wave) dimer becomes bound. The more $s$-wave bound
states supported by the system, the deeper is the potential, implying
different short-range physics. The universal prediction from Gao’s work,
$a^{*}\approx 0.956r_{\rm vdW}$ Gao (2000) (horizontal dashed line in Fig. 1),
however, suggests that the actual form of the potential at short-distances is
not important. In fact, our numerical results obtained for $a^{*}_{l}$ agree
well with the Gao prediction of two-body binding energies [within $1\%$
($6\%$) in the case of 10 $s$-wave bound states for a $d$-wave ($i$-wave)] and
this agreement tends to improve as one increases the number of $s$-wave
states. In the vicinity of the point where the dimer is just barely bound,
i.e., when $a\leq a^{*}_{l}$, the binding energy can be expressed as a linear
function of the scattering length , i.e.,
$E_{l}/\left[{\hbar^{2}/(2\mu_{2b}{r_{\rm{vdW}}^{2}})}\right]\approx
d_{l}\left({a^{*}_{l}-a}\right)/r_{\rm{vdW}},$ (2)
where $d_{2}\approx 5.8$ and $d_{4}\approx 43$, for $d$-wave and $i$-wave
states, respectively, are approximately universal. It is also interesting to
note that near $a=a^{*}_{2}$ one expects strong $d$-wave interactions. In
order to parametrize $d$-wave interactions in the ultracold regime, scattering
properties must also be explored for two atoms colliding in the $l=2$ channel.
In this case, due to the $1/r^{6}$ dependence of the Lennard-Jones potential,
the $d$-wave elastic scattering phase-shift $\delta_{2}\left(k\right)$ is
known to have the following low energy expansion:
$\tan\delta_{2}\left(k\right)=-\lambda_{2}k^{4}-a_{2}^{5}k^{5},$ (3)
instead of the usual $k^{2l+1}$ dependence found for $l<2$. Here,
$\lambda_{2}=\left[{\pi\Gamma\left(5\right)\Gamma\left({1/2}\right)}\right]/\left[{4\Gamma^{2}\left(3\right)\Gamma\left({11/2}\right)}\right]r_{\rm
vdW}^{4}\approx 0.160r_{\rm vdW}^{4}$ is a constant that only depends on the
van der Walls length, and $a_{2}$ diverges when a $d$-wave state is just about
to be bound Willner and Gianturco (2006). Therefore, $a_{2}$ is denoted the
“$d$-wave scattering length” in this study (which should not be confused with
$a_{2}^{*}$), and is also used in order to characterize our findings in terms
of the $d$-wave interactions. This parameterization, therefore, allows a
better comparison to experiments in ultracold quantum gases. It is important,
however, to emphasize that the complicated multichannel structure of alkali
atoms used in ultracold quantum gases can strongly affect the relations
explored above between the three-body states found for $d$-wave interactions
and the single channel parameters that characterize the present study. This is
particularly true near narrow Feshbach resonances Wang et al. (2011a).
## III Three-body recombination
This section focuses on the three-body recombination rate near the region
where the $d$-wave dimer is about to be bound. In particular, the example of a
Lennard-Jones potential with two $s$-wave bound states is elaborated. In this
numerical example, the $d$-wave dimer becomes bound at about
$a=0.995r_{\rm{vdW}}$ and the $i$-wave dimer becomes bound at about
$1.206r_{\rm{vdW}}$. Later, the three-body recombination rate with more two-
body $s$-wave bound states will be computed to test the universality of those
results.
### III.1 Hyperspherical approach
The hyperspherical approach is utilized here to study the three-body systems.
The version of that approach adopted here has been discussed in detail in
other recent studies Wang et al. (2011c, 2012a). To briefly summarize, the
three-body system is described in hyperspherical coordinates: a hyperradius
$R$ that describes the overall size of the three-body system, two hyperangles
$\theta$ and $\phi$ that describe the shape of the three-body system, and the
three usual Euler angles $\alpha,\beta,\gamma$ that describe rigid-body
rotations. In these coordinates, the three-body Schrödinger equation is
written as,
$\left[{-\frac{{\hbar^{2}}}{{2\mu_{3b}}}\left({\frac{{\partial^{2}}}{{\partial
R^{2}}}-\frac{{\Lambda^{2}+15/4}}{{R^{2}}}}\right)+V\left({R,\theta,\varphi}\right)-E}\right]\psi_{E}=0,$
(4)
where $\Lambda^{2}\equiv\Lambda^{2}(\Omega)$
[$\Omega\equiv\\{\theta,\varphi,\alpha,\beta,\gamma\\}$] is the “grand
angular-momentum operator” and $\mu_{3b}=m/\sqrt{3}$ is the three-body reduced
mass of three identical atoms with mass $m$ Kendrick et al. (1999); Suno et
al. (2002).
Figure 2: (Color online) Adiabatic three-body potential curves for $a\approx
0.977r_{\rm vdW}$. The $2+1$ channels, which corresponding to a dimer plus a
free atom at very large distance, are labeled by a combination of a letter and
a number. The letter denotes the angular momentum quantum number $l$ of the
dimer, and the number labels the channels for the same dimer angular momentum
from low-to-high dimer binding energies.
To solve this Schrödinger equation, the first step in our method is to
diagonalize the adiabatic Hamiltonian $H_{\rm{ad}}\left({R,\Omega}\right)$ and
find the adiabatic potentials $U_{\nu}(R)$ and channel functions
$\Phi_{\nu}\left({R;\Omega}\right)$:
$H_{\rm{ad}}\left({R,\Omega}\right)\Phi_{\nu}\left({R;\Omega}\right)=U_{\nu}\left(R\right)\Phi_{\nu}\left({R;\Omega}\right),$
(5)
whose solutions depend parametrically on $R$. The adiabatic Hamiltonian,
containing all angular dependence and interactions, is defined as
$H_{\rm{ad}}\left({R,\Omega}\right)=\left[{\frac{{\hbar^{2}\Lambda^{2}}}{{2\mu_{3b}R^{2}}}+\frac{{15\hbar^{2}}}{{8\mu_{3b}R^{2}}}+V\left({R,\theta,\varphi}\right)}\right].$
(6)
For illustration, Fig. 2 shows the three-body adiabatic hyperspherical
potential curves for the case of $a\approx 0.977r_{\rm{vdW}}$. [Note that in
this figure the labels for the final recombination channels are also indicated
and are useful for further analysis.]
The adiabatic potentials $U_{\nu}(R)$ and channel functions
$\Phi_{\nu}(R;\Omega)$ obtained by solving Eq. (5) for fixed values of $R$
contain all the correlations relevant to this problem. For each $R$, the set
of $\Phi_{\nu}\left({\Omega;R}\right)$ is orthogonal,
$\int{d\Omega\Phi_{\mu}\left({R;\Omega}\right)^{*}}\Phi_{\nu}\left({R;\Omega}\right)=\delta_{\mu\nu},$
(7)
and complete
$\sum\limits_{\tau}{\Phi_{\tau}\left({R;\Omega}\right)\Phi_{\tau}\left({R;\Omega^{\prime}}\right)^{*}=\delta\left({\Omega-\Omega^{\prime}}\right)}.$
(8)
In practice, calculation of all the channel functions is time consuming and
impractical. However, numerical studies have shown that only a small number of
channels are needed as a truncated basis-set to expand the total wave
function, e.g.,
$\psi_{E}\left({R,\Omega}\right)=\sum\limits_{\nu=1}^{N_{c}}{F_{\nu}^{E}\left(R\right)\Phi_{\nu}\left({\Omega;R}\right)},$
(9)
where $N_{c}$ is the number of channels adopted. Insertion of Eq. (9) into Eq.
(4) leads to a set of coupled one-dimensional equations:
$\left[{-\frac{\hbar^{2}}{{2\mu_{3b}}}\frac{d^{2}}{{dR^{2}}}+U_{\nu}\left(R\right)}-E\right]F_{\nu\nu^{\prime}}^{E}\left(R\right)-\frac{\hbar^{2}}{{2\mu_{3b}}}\sum\limits_{\mu}{\left[{2P_{\nu\mu}\left(R\right)\frac{d}{{dR}}+Q_{\nu\mu}\left(R\right)}\right]}F_{\mu\nu^{\prime}}^{E}\left(R\right)=0,$
(10)
where $\underline{P}$ and $\underline{Q}$ are the coupling matrices defined
below, and $\nu^{\prime}$ denotes the $\nu^{\prime}-$th independent solution.
(Hereafter, unless otherwise specified, we use an underline to denote the
matrix form, e.g., $\underline{P}$ denotes a matrix with matrix element
$P_{\nu\mu}$.) In the equation above, the non-adiabatic coupling matrices are
defined as
$P_{\nu\mu}\left(R\right)=\int{d\Omega}\Phi_{\nu}\left({R;\Omega}\right)^{*}\frac{\partial}{{\partial
R}}\Phi_{\mu}\left({R;\Omega}\right),$ (11)
$Q_{\nu\mu}\left(R\right)=\int{d\Omega}\Phi_{\nu}\left({R;\Omega}\right)^{*}\frac{{\partial^{2}}}{{\partial
R^{2}}}\Phi_{\mu}\left({R;\Omega}\right),$ (12)
and are responsible for inelastic scattering processes as well as the finite
width of three-body resonant states. In practice, only the
$P_{\nu\mu}^{2}\left(R\right)=-\int{d\Omega}\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R;\Omega}\right)^{*}\frac{\partial}{{\partial
R}}\Phi_{\mu}\left({R;\Omega}\right),$ (13)
component of $Q_{\nu\mu}$ is needed to solve the coupled equations Burke
(1999). The relation between $\underline{Q}$ and $\underline{P}^{2}$ is given
by $\frac{d}{{dR}}\underline{P}=-\underline{P^{2}}+\underline{Q}$. From the
definition of the $\underline{P}$ and $\underline{Q}$ matrices, it is easy to
see that the coupling matrices have the following properties:
$P_{\nu\mu}=-P_{\mu\nu}$ and $P_{\nu\mu}^{2}=P_{\mu\nu}^{2}$, which leads to
$P_{\nu\nu}=0$, and $Q_{\nu\nu}=P_{\nu\nu}^{2}$. In the hyperspherical
adiabatic representation, the effective potentials are usually defined as
$\widetilde{U}_{\nu}\left(R\right)\equiv
W_{\nu\nu}\left(R\right)=U_{\nu}\left(R\right)-\frac{{\hbar^{2}}}{{2\mu_{3b}}}P_{\nu\nu}^{2}\left(R\right),$
(14)
and differs for the hyperspherical potential $U_{\nu}(R)$ by the addition of
the diagonal correction $P_{\nu\nu}^{2}(R)$. The effective potentials are
usually more physical than the raw Born-Oppenheimer potentials because they
include hyperadial kinetic energy contributions through the $P_{\nu\nu}^{2}$
term. For example, the effective potential gives physical asymptotic behaviors
of the system at large $R$ with the correct coefficient of $R^{-2}$. In some
cases, one can even neglect the nonadiabatic couplings between different
adiabatic channels and approximately solve the Schrödinger equation as a
single channel problem:
$\left[{-\frac{{\hbar^{2}}}{{2\mu_{3b}}}\frac{{d^{2}}}{{dR^{2}}}+\tilde{U}_{\nu}\left(R\right)-E}\right]F_{\nu\nu}^{E}\left(R\right)=0,$
(15)
called the adiabatic approximation Kokoouline and Greene (2003). Although a
semi-quantitative picture can emerge within this approximation, the fully
coupled system of equation has to be solved to obtain a more quantitative
description of the properties of the three-body system including inelasticity
in particular.
Quantitative solutions of Eq. (10) hinge on the ability to calculate non-
adiabatic couplings accurately. A traditional method for calculating the
coupling matrices is to apply a simple finite difference scheme for the
derivative of $\Phi_{\mu}\left({R;\Omega}\right)$, i.e.,
$\frac{\partial}{{\partial
R}}\Phi_{\mu}\left({R;\Omega}\right)\approx\frac{{\Phi_{\mu}\left({R+\Delta
R;\Omega}\right)-\Phi_{\mu}\left({R-\Delta R;\Omega}\right)}}{{2\Delta R}}.$
(16)
This scheme, however, is only accurate up to the first order in $\Delta R$.
Here, we use instead an improved method to calculate the
$\partial\Phi_{\mu}\left({R;\Omega}\right)/\partial R$. This method only
relies on $H_{\rm{ad}}\left({R,\Omega}\right)$ and $\partial
H_{\rm{ad}}\left({R,\Omega}\right)/\partial R$ at the desired value of $R$ and
is shown to be in principle numerically exact (see Appendix B for a detailed
discussion).
### III.2 Three-body recombination rate
The numerical method described in the previous subsection yields more accurate
coupling matrix elements and hence improved effective potential curves. These
effective potentials are not only crucial for intuitively understanding the
system, but also for quantitative calculations of three-body observables like
the recombination rate. Together with the $R$-matrix propagation method
discussed in Ref. Wang et al. (2011c), the three-body recombination rate for
$J^{\pi}=0^{+}$ symmetry, where $J$ is the three-body total angular momentum
and $\pi$ the parity, are calculated and shown in Fig. 3. The solid black
curve with square symbols shows the total recombination rate as a function of
the scattering length $a$, near the $d$-wave resonance, i.e., for values of
$a$ near $a^{*}_{2}$. The total rate exhibits two clear enhancements labeled
by “A” ($a_{\rm A}\approx 1.09r_{\rm{vdW}}$) and “B” ($a_{\rm B}\approx
0.98r_{\rm{vdW}}$). The enhancements “A” and “B”, indicated in Fig. 3 by the
vertical dashed lines, occur right after and right before the $d$-wave dimer
becomes bound (see vertical solid line in Fig. 3), respectively. Analysis of
the partial recombination rates helps to elucidate the mechanism responsible
for these enhancements. The enhancement around the peak “A” appears in all the
partial three-body recombination rates. After the $d$-wave becomes bound and
forms a new decay channel, the partial $K_{3}$ rates near the enhancement “B”
have a different characteristic. The total rate is now dominated only by the
partial rate into this new $d$-wave channel [see the inset of Fig. 3]. These
features suggest that peak “A” might be a resonance due to a three-body state
that crosses the three-body threshold, while peak “B” seems to relate to some
other threshold behavior related to the formation of the new $d$-wave decay
channel. The effective potentials near these enhancements, however, can yield
insights into their origin.
Figure 3: (Color online) Total and partial $J^{\pi}=0^{+}$ three-body
recombination rates as functions of the two-body scattering length $a$, shown
on a linear scale, near a $d$-wave resonance associated with the $d$-wave
dimer formation. The inset shows the same graph with a logarithmic scale for
the y-axis. The units of $K_{3}$ are converted to $\rm{cm^{6}/s}$ by using the
van der Waals length $r_{\rm vdW}=101.0$ bohr and mass $132.905429$ amu of
133Cs. The solid vertical line indicates the value of the $s$-wave scattering
length at the point where the $d$-wave dimer becomes bound. The two dashed
vertical lines indicate the two values of $a$ at which recombination rate is
enhanced, denoted A and B respectively.
First, consider the physical origin of the enhancement A. Figure. 4 (a) shows
the adiabatic hyperspherical potential curves at peak A. Only the channels
relevant to the resonances are shown in this figure. The red dashed-line shows
the effective hyperspherical potential. We have diabatized the potential near
an avoided crossing since it plays no role, and display it as the black solid
curve. The black solid curve shows an outer barrier, and a three-body short-
ranged state could be supported for this potential. To check whether there is
a three-body state near the threshold, a calculation of the WKB phase at zero
scattering energy is carried out:
$\phi_{\rm
WKB}=\int_{R_{a}}^{R_{b}}{\frac{\hbar}{{\sqrt{-2\mu_{3b}W_{\nu\nu}\left(R\right)}}}}dR,$
(17)
where $R_{a}$ and $R_{b}$ are the classical turning points shown in Fig. 4
(a). The calculated WKB phase $\phi_{\rm WKB}$ is about $0.51\pi$, which is
strong evidence for the existence of a three-body state. [Below, we calculate
the actual energy of this state and describe some of its properties.] It is
this three-body state associated with the $d$-wave dimer state crossing the
three-body break-up threshold that causes the enhancement A in the three-body
recombination rate. The physical picture is that the three-body state at
threshold can trap the system within the potential well for a long time, hence
making it more likely that the system will decay into deeper channels.
Therefore, this decay process only depends on the initial channel, which is
consistent with the fact noted above that near the enhancement A all partial
rates display the resonance effect.
Next consider the enhancement mechanism responsible for peak B. Figure 4 (b)
shows the effective potentials $W_{\nu\nu}(R)$ as functions of $R$ for three
different scattering lengths around the enhancement peak B as solid curves,
dashed curves, and dash-dotted curves. The corresponding $K_{3}$ rates are
indicated by the three points $(i)$, $(ii)$, and $(iii)$ in the inset of Fig.
4. Here, the dominant pathway at low energy is from the lowest entrance
channel to the newly formed $d$-wave channel. For each scattering length, the
thin curves in Fig. 4 are the entrance channels, and as one can readily see,
they do not change substantially in all three cases. This means that the
incident channel potential curve depends on the atom-atom scattering length
only weakly. The thick curves in Fig. 4 are the most important recombination
channel associated with the new $d$-wave channel. It is clear that from point
$(i)$ to $(iii)$, the potential barrier decreases appreciably and can explain
the increase of recombination as it gradually allows the three-body wave
function to approach shorter distances, where non-adiabatic couplings are
expected to be stronger.
From Ref. Wang et al. (2011c), the scattering matrix element describing the
recombination from the lowest incident channel to the newly formed $d$-wave
channel can be approximated by,
$S_{fi}\approx 2\pi
i\int_{0}^{\infty}{dRF_{f}\left(R\right)W_{fi}\left(R\right)F_{i}\left(R\right)}$
(18)
where $F_{i}$ ($F_{f}$) are the regular solutions of Eq. (15) — single channel
approximation — for the initial (final) channel $i$ ($f$). As mentioned above,
the incident channel is dominated by the centrifugal barrier which, in turn,
is independent of the scattering length. Hence the energy-normalized $F_{i}$
are well approximated by Bessel functions:
$F_{i}\left(R\right)\approx\sqrt{\mu_{3b}R}J_{2}\left({k_{i}R}\right),$ (19)
which is, of course, also independent of scattering length. The coupling
matrix element $W_{fi}\left(R\right)$ is also found, numerically, to depend on
scattering length only weakly for the three cases we studied here. Therefore,
we assume that the product of $W_{fi}\left(R\right)F_{i}\left(R\right)$ does
not depend appreciably on the scattering length. In addition, this product is
shown numerically to be significant only at small hyperradius (around 1.3
$r_{\rm vdW}$ to 2.8 $r_{\rm vdW}$). Therefore, the scattering length
dependence of $S_{fi}$ is fully determined by $F_{f}\left(R\right)$ at short
distances. There are two competing mechanisms controlling the amplitude of
$F_{f}\left(R\right)$: one is the tunneling through the outer barrier, which
makes the amplitude become increasingly larger, as the barrier becomes
increasingly lower from $(i)$ to $(iii)$; the other is the overall amplitude
for an energy normalized wave function, which makes the amplitude becomes
increasingly smaller, since the threshold energy associated to the energy of
the $d$-wave dimer become lower and lower, and the out-going wave vector
become larger and larger. These two competing mechanisms help to form peak
structure B. The WKB phase calculated for these three cases is smaller than
$\pi/2$, which is another evidence that this enhancement is not due to a
three-body state resonance. This is also consistent with the fact that such an
enhancement is not shown in the partial recombination rates for channels other
than the newly formed $d$-wave.
Figure 4: (a) (Color online) $W_{\nu\nu}(R)$ as a function of hyperradius $R$
at the $K_{3}$ resonance peak A. (b) (Color online) $W(R)_{\nu\nu}$ as a
function of hyperradius $R$ around the $K_{3}$ enhancement peak B. The inset
shows the three points where the potential curves correspond to.
## IV Three-body state associated with the $d$-wave dimer
While the Efimov states can be viewed as three-body states associated with a
$s$-wave dimer near the three-body breakup threshold, the new three-body state
discussed in the previous section is a three-body state that can be associated
with a $d$-wave dimer. Because of the existence of deeper atom-dimer
thresholds, this three-body state is actually a quasi-bound state. Figure 5
shows the three-body state energy as a function of the scattering length $a$
(red dots), with the error bars indicating the width of the quasi-bound state.
The red line represents a fitting formula to this energy:
$\frac{E_{3b}}{\hbar^{2}/\left({2\mu_{2b}r_{\rm{vdW}}^{2}}\right)}=d_{3b}\frac{\left({a^{*}_{3b}-a}\right)}{r_{\rm
vdW}}+e_{3b}\frac{\left({a^{*}_{3b}-a}\right)^{2}}{r_{\rm vdW}^{2}},$ (20)
where $a^{*}_{3b}\approx 1.09r_{\rm{vdW}}$, $d_{3b}\approx 8.21$ and
$e_{3b}\approx 10.01$ are fitting parameters. For comparison, the figure also
shows the $d$-wave energy as the black curve with solid square symbols. The
three-body energy crosses the three-body breakup threshold at $a\approx
1.09r_{\rm{vdW}}$, which corresponds to peak A in $K_{3}$. As one can see, the
three-body state is formed before the $d$-wave state becomes bound and its
energy is always below the $d$-wave dimer energy (the black curve with square
symbols in Fig. 5). The difference between the dimer and trimer energy also
increases when the scattering length becomes smaller, and the dimer becomes
deeper. However, the width of the three-body quasi-bound state does not seem
to change appreciably.
Figure 5: (Color online) Energy of the three-body bound state associated with
a $d$-wave dimer as a function of scattering length $a$, both in van der Waals
units.
The Gao analysis in Ref. Gao (2000), suggests that there is also an $i$-wave
dimer that becomes bound at a nearby scattering length. However, the $i$-wave
dimer does not affect these two peaks. The $i$-wave channel has a very sharp
avoided crossing with all the other channels. As a consequence, the partial
$K_{3}$ rate into the $i$-wave dimer channel is negligible compared with the
other partial rates. One explanation may be that forming a $i$-wave ($l=6$)
dimer results in the exchange of a large amount of angular momentum between
the dimer and the free atom. Therefore, although higher partial waves
$l=10,14,18...$ might also be formed at nearby scattering lengths, they are
not expected to show any strong features in the three-body recombination rates
at ultracold energies.
Figure 6: (Color online) The enhancements for the total three-body
recombination rates at about $a=0.995r_{\rm vdW}$ for Lennard-Jones potential
with 2 and 3 $s$-wave bound states. $K_{3}$ is convert to $\rm{cm^{6}/s}$ by
using van der Waals length $r_{\rm vdW}=101.0$ bohr and mass $132.905429$ amu
of 133Cs Figure 7: (Color online) The total three-body recombination rates as
a function of $r_{\rm vdW}^{5}/a_{2}^{5}$. The units of $K_{3}$ are the same
as Fig. 6.
Finally, to test the universality of our results, the three-body recombination
enhancements for the model potentials of Lennard-Jones type, having either 2
or 3 $s$-wave bound states, are shown in Fig. 6. In this figure, the red
(black) solid vertical line shows where the $d$-wave dimer crosses the
threshold for LJ with 2 (3) $s$-waves. The red (black) dashed vertical line
indicates where the three-body state associated with the $d$-wave level
crosses the threshold for LJ with 2 (3) $s$-wave states. The values of the
$s$-wave scattering length for such enhancements from these two different
models differs by only 0.01 $r_{\rm vdW}$. This small difference is consistent
with the small shift of the scattering length value where the $d$-wave dimer
crosses the three-body threshold (see Fig. 1). As discussed before, the
$d$-wave scattering length is a better parameter to describe $d$-wave
interactions. We have therefore analyzed the three-body recombination rate as
a function of $1/a_{2}^{5}$, which is shown in Fig. 7. The agreement between
calculations using different potential models in fact looks better in this
parameterization, where the three-body state crosses the threshold at
$a_{2}\approx 0.902r_{\rm vdW}$ for both LJ with 2 and 3 $s$-waves bound
states.
This result suggests (although it does not prove rigorously) that the three-
body state associated with the $d$-wave dimer is also universal. We also note
that at the threshold for formation of the three-body state, the effective
potential relevant for that state does also display an inner repulsive barrier
[see Fig. 4 (a)] preventing atoms from approaching to hyperradii smaller than
$1.5r_{\rm vdW}$. This indicates that the properties of such a three-body
state should also be insensitive to the details of nonadditive short-range
three-body interactions.
## V Summary
In summary, we have calculated the three-body recombination rates for Lennard-
Jones potentials with 2 and 3 $s$-wave bound states for small positive values
of the scattering length. Two universal enhancement peaks are found at about
$a=0.995$ $r_{\rm vdW}$. In particular, one of the enhancement peaks
corresponds to a new universal three-body state that is associated with a
$d$-wave dimer and is expected to exist in general for potentials having a van
der Waals tail.
## Acknowledgements
The authors want to thank Eric A. Cornell, John Bohn, Cheng Chin, Robin Côté
and Doerte Blume for helpful discussions. This work was supported in part by
NSF and the AFOSR-MURI.
## Appendix A Lennard-Jones potential
The two-body model potentials applied here are the Lennard-Jones potential,
which can be expressed as a function of interatomic distances $r$:
$V\left(r\right)=-\frac{{C_{6}}}{{r^{6}}}\left({1-\frac{{\lambda^{6}}}{{r^{6}}}}\right).$
(21)
This form of potential has a van der Waals tail $-C_{6}/r^{6}$ at long range
and a repulsive core near the origin controlled by the parameter $\lambda$. In
the present study, the values of $\lambda$ are adjusted to give the desired
$s$-wave scattering length $a$ and number of bound states. Figure 8 shows two
Lennard-Jones potentials with a zero-energy $d$-wave state and their
corresponding spectrum of bound state with even $l$. (The zero energy $d$-wave
state is not shown in Fig. 8). The black thicker curve corresponds to
$\lambda\approx 0.503r_{\rm vdW}$ and has two $s$-wave bound states with
$a\approx 0.995r_{\rm vdW}$. The red thinner curve corresponds to
$\lambda\approx 0.414r_{\rm vdW}$ and has three $s$-wave bound states with
$a\approx 0.983r_{\rm vdW}$. It is shown that although the potentials at
short-range are very different for the two different parameters, the
scattering length and the spectrum near threshold are similar.
Figure 8: (Color online) Lennard-Jones potential with $\lambda\approx
0.503r_{\rm vdW}$ (black thicker curve) and $\lambda\approx 0.414r_{\rm vdW}$
(red thinner curve). The horizontal lines show the bound state spectrum with
even angular momentum quantum number $l$. The line style solid, dashed, dash-
dotted, dash-dot-dotted, and short-dashed indicates $l=0,2,4,6,8$
correspondingly.
## Appendix B Improved method for calculating nonadiabatic coupling matrices
Equation (16) is only accurate up to the first order in $\Delta R$. In
addition, the value chosen for $\Delta R$ in a realistic numerical calculation
can sometimes be tricky. When $\Phi_{\mu}\left({R;\Omega}\right)$ changes very
rapidly, e.g., near a sharp avoided crossing, we need to choose a very small
step size $\Delta R$. In contrast, when $\Phi_{\mu}\left({R;\Omega}\right)$
changes very slowly, e.g., at very large distances $R$, we need to choose a
relatively larger step size $\Delta R$, or else $\Phi_{\mu}\left({R+\Delta
R;\Omega}\right)-\Phi_{\mu}\left({R-\Delta R;\Omega}\right)$ would be too
small, and the accuracy would be limited by the machine precision.
One way to improve the accuracy is to apply the Hellmann-Feynman theorem. This
theorem can give us analytical formulas for the coupling matrices if we know
the derivative of the adiabatic Hamiltonian $\frac{\partial}{{\partial
R}}H_{\rm{ad}}$ from the following derivation. First, taking the derivative of
both sides of Eq. (5) leads to
$\displaystyle\left[{H_{\rm{ad}}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}\right]\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R;\Omega}\right)=$
$\displaystyle-\left[{\frac{\partial}{{\partial
R}}H_{\rm{ad}}\left({R,\Omega}\right)-\frac{\partial}{{\partial
R}}U_{\nu}\left(R\right)}\right]\Phi_{\nu}\left({R;\Omega}\right).$ (22)
Next, multiplying $\Phi_{\mu}\left({R;\Omega}\right)^{*}$ on both sides of Eq.
(B) and integrating over $\Omega$ gives
$\displaystyle P_{\mu\nu}$ $\displaystyle=$
$\displaystyle\int{d\Omega\Phi_{\mu}\left({R;\Omega}\right)^{*}}\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R;\Omega}\right)$ (23) $\displaystyle=$
$\displaystyle-\frac{\int{d\Omega\Phi_{\mu}\left({R;\Omega}\right)^{*}\left[{\frac{\partial}{{\partial
R}}H_{\rm{ad}}\left({R,\Omega}\right)}\right]\Phi_{\nu}\left({R;\Omega}\right)}}{\left[{U_{\mu}\left(R\right)-U_{\nu}\left(R\right)}\right]},$
where $\mu\neq\nu$, and
$\frac{\partial}{{\partial
R}}U_{\nu}\left(R\right)=\int{d\Omega\Phi_{\nu}\left({R;\Omega}\right)^{*}}\left[{\frac{\partial}{{\partial
R}}H_{\rm{ad}}\left({R,\Omega}\right)}\right]\Phi_{\nu}\left({R;\Omega}\right)$
(24)
after some manipulation of algebra. The Hellmann–Feynman theorem is in
principle exact. The matrix elements for $P^{2}$ can be obtained by
$P_{\mu\nu}^{2}=\sum\limits_{\tau=1}^{N_{c}}{P_{\mu\tau}P_{\tau\nu}},$ (25)
where $N_{c}$ is the number of channels. However, numerical studies show that
the convergence of $P_{\mu\nu}^{2}$ with respect to number of channels is very
slow, making this method impractical.
We now introduce an improved method to calculate $\frac{\partial}{{\partial
R}}\Phi_{\mu}\left({R;\Omega}\right)$. It is in principle numerically exact.
The first hint of the derivation of this method is that Eq. (B) seems
plausible to be directly solved for $\frac{\partial}{{\partial
R}}\Phi_{\mu}\left({R;\Omega}\right)$ by
$\displaystyle\frac{\partial}{{\partial R}}\Phi_{\nu}\left({R;\Omega}\right)=$
$\displaystyle-\left[{H_{\rm{ad}}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}\right]^{-1}\times$
$\displaystyle\left[{\frac{\partial}{{\partial
R}}H_{\rm{ad}}\left({R,\Omega}\right)-\frac{\partial}{{\partial
R}}U_{\nu}\left(R\right)}\right]\Phi_{\nu}\left({R;\Omega}\right).$
However, this solution is forbidden since
${H_{\rm{ad}}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}$ is singular:
$\left|{H_{\rm{ad}}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}\right|=0$,
meaning that ${H_{\rm{ad}}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}$ is
not invertible. The singularity can also be understood from the fact that the
equation
$\displaystyle\left[{H_{ad}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}\right]\chi_{\nu}\left({R;\Omega}\right)=$
(27) $\displaystyle-\left[{\frac{\partial}{{\partial
R}}H_{ad}\left({R,\Omega}\right)-\frac{\partial}{{\partial
R}}U_{\nu}\left(R\right)}\right]\Phi_{\nu}\left({R;\Omega}\right)$
does not have a unique solution, $\chi_{\nu}\left({R;\Omega}\right)$. In fact,
any functions with the form of
$\chi_{\nu}\left({R;\Omega}\right)=\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R;\Omega}\right)+c\Phi_{\nu}\left({R;\Omega}\right),$ (28)
(where $c$ is an arbitrary number) can be a solution of Eq. (27). The
singularity of matrix
${H_{\rm{ad}}\left({R,\Omega}\right)-U_{\nu}\left(R\right)}$ can be removed by
considering the additional condition that
$\int{d\Omega\Phi_{\nu}\left({R,\Omega}\right)^{*}}\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R,\Omega}\right)=0,$ (29)
which can be derived from the normalization condition of
$\Phi_{\nu}\left({R,\Omega}\right)$, as shown in Ref. Nelson (1976).
Nevertheless, our numerical studies show that even without removing the
singularity, the use of numerical solution packages such as “Linear Algebra
PACKage” (LAPACK) Anderson et al. (1992) or PARDISO Naumann and Schenk (2012)
to solve Eq. (27) directly still gives an accurate solution
$\chi_{\nu}\left({R;\Omega}\right)$ in the form of Eq. (28) with an unknown
$c$. And once we have the numerical solution
$\chi_{\nu}\left({R;\Omega}\right)$, $c$ can be calculated by
$c=\int{d\Omega\Phi_{\nu}\left({R;\Omega}\right)^{*}}\chi_{\nu}\left({R;\Omega}\right).$
(30)
Finally, the derivative of $\Phi_{\nu}\left({R;\Omega}\right)$ can be written
as
$\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R;\Omega}\right)=\chi_{\nu}\left({R;\Omega}\right)-c\Phi_{\nu}\left({R;\Omega}\right),$
(31)
which can be inserted into Eq. (11) and Eq. (13) for the coupling matrices.
The $P$ matrices obtained in this way are found to be numerically the same as
the one calculated from the Hellmann–Feynman theorem up to machine precision,
proving that our $\frac{\partial}{{\partial
R}}\Phi_{\nu}\left({R;\Omega}\right)$ are numerically accurate. Therefore,
this method can give numerically very accurate coupling matrices. The major
limitation of this approach is that it becomes computationally more expensive
to obtain nonadiabatic couplings as the number of requested channels
increases. Nevertheless, for our typical calculations the number of relevant
three-body channels is not so large and this method has been shown to be quite
efficient.
## References
* Efimov (1971) V. Efimov, Sov. J. Nucl. Phys. 12, 589 (1971).
* Kraemer et al. (2006) T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, et al., Nature 440, 315 (2006).
* Berninger et al. (2011) M. Berninger, A. Zenesini, B. Huang, W. Harm, H. C. Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M. Hutson, Phys. Rev. Lett. 107, 120401 (2011).
* Pollack et al. (2009) S. E. Pollack, D. Dries, and R. G. Hulet, Science 326, 1683 (2009).
* Gross et al. (2009) N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich, Phys. Rev. Lett. 103, 163202 (2009).
* Gross et al. (2010) N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich, Phys. Rev. Lett. 105, 103203 (2010).
* Ottenstein et al. (2008) T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S. Jochim, Phys. Rev. Lett. 101, 203202 (2008).
* Lompe et al. (2010) T. Lompe, T. B. Ottenstein, F. Serwane, K. Viering, A. N. Wenz, G. Zürn, and S. Jochim, Phys. Rev. Lett. 105, 103201 (2010).
* Huckans et al. (2009) J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, and K. M. O’Hara, Phys. Rev. Lett. 102, 165302 (2009).
* Williams et al. (2009) J. R. Williams, E. L. Hazlett, J. H. Huckans, R. W. Stites, Y. Zhang, and K. M. O’Hara, Phys. Rev. Lett. 103, 130404 (2009).
* Wild et al. (2012) R. J. Wild, P. Makotyn, J. M. Pino, E. A. Cornell, and D. S. Jin, Phys. Rev. Lett. 108, 145305 (2012).
* Barontini et al. (2009) G. Barontini, C. Weber, F. Rabatti, J. Catani, G. Thalhammer, M. Inguscio, and F. Minardi, Phys. Rev. Lett. 103, 043201 (2009).
* Petrov (2004) D. S. Petrov, Phys. Rev. Lett. 93, 143201 (2004).
* Wang et al. (2011a) Y. Wang, J. P. D’Incao, and B. D. Esry, Phys. Rev. A 83, 042710 (2011a).
* Braaten and Hammer (2006) E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259 (2006).
* Hammer and Platter (2007) H. W. Hammer and L. Platter, Eur. Phys. J. A 32, 113 (2007).
* von Stecher et al. (2009) J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature Physics 5, 417 (2009).
* Wang et al. (2011b) Y. Wang, J. P. D’Incao, , and C. H. Greene, Phys. Rev. Lett. 106, 233201 (2011b).
* Schmidt et al. (2012) R. Schmidt, S. P. Rath, and W. Zwerger, arxiv: p. 1201.4310 (2012).
* Chin (2011) C. Chin, arxiv: p. 1111.1484 (2011).
* Sørensen et al. (2012) P. K. Sørensen, D. V. Fedorov, A. S. Jensen, and N. T. Ainner, arxiv: p. 1206.2274 (2012).
* Wang et al. (2012a) J. Wang, J. P. D’Incao, B. D. Esry, and C. H. Greene, Phys. Rev. Lett. 108, 263001 (2012a).
* Naidon et al. (2012) P. Naidon, S. Endo, and M. Ueda, arxiv: p. 1208.3912 (2012).
* Wang et al. (2012b) Y. Wang, J. Wang, J. P. D’Incao, and C. H. Greene, arxiv: p. 1207.6439 (2012b).
* Gao (2000) B. Gao, Phys. Rev. A 62, 050702(R) (2000).
* Esry et al. (2001) B. D. Esry, C. H. Greene, and H. Suno, Phys. Rev. A 65, 010705 (2001).
* Chin et al. (2011) C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev. Mod. Phys. 107, 120401 (2011).
* Gribakin and Flambaum (1993) G. F. Gribakin and V. V. Flambaum, Phys. Rev. A 48, 546 (1993).
* Willner and Gianturco (2006) K. Willner and F. A. Gianturco, Phys. Rev. A 74, 052715 (2006).
* Wang et al. (2011c) J. Wang, J. P. D’Incao, and C. H. Greene, Phys. Rev. A 84, 052721 (2011c).
* Kendrick et al. (1999) B. K. Kendrick, R. T. Pack, R. B. Walker, and E. F. Hayes, J. Chem. Phys. 110, 6673 (1999).
* Suno et al. (2002) H. Suno, B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. A 65, 042725 (2002).
* Burke (1999) J. P. Burke, Ph.D. thesis, University of Colorado at Boulder (1999).
* Kokoouline and Greene (2003) V. Kokoouline and C. H. Greene, Phys. Rev. A 68, 012703 (2003).
* Nelson (1976) R. B. Nelson, AIAA Journal 14, 1201 (1976).
* Anderson et al. (1992) E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. D. Cros, A. Greenbaum, S. Hammerling, A. Mckenney, S. Ostouchov, et al., _LAPACK Users Guide_ (SIAM, 1992).
* Naumann and Schenk (2012) U. Naumann and O. Schenk, _Combinatorial Scientific Computing_ (Champman and Hall, 2012).
|
arxiv-papers
| 2012-09-20T14:52:41 |
2024-09-04T02:49:35.353331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jia Wang, J. P. D'Incao, Yujun Wang and Chris H. Greene",
"submitter": "Jia Wang",
"url": "https://arxiv.org/abs/1209.4553"
}
|
1209.4582
|
# Inter-critical NLS:
critical $\dot{H}^{s}$-bounds imply scattering
Jason Murphy Department of Mathematics, University of California at Los
Angeles, Los Angeles, CA 90095-1555, USA [email protected]
###### Abstract.
We consider a class of power-type nonlinear Schrödinger equations for which
the power of the nonlinearity lies between the mass- and energy-critical
exponents. Following the concentration-compactness approach, we prove that if
a solution $u$ is bounded in the critical Sobolev space throughout its
lifespan, that is, $u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$, then $u$ is
global and scatters.
###### Contents
1. 1 Introduction
1. 1.1 Outline of the proof of Theorem 1.3
2. 2 Notation and useful lemmas
1. 2.1 Some notation
2. 2.2 Basic harmonic analysis
3. 2.3 Strichartz estimates
4. 2.4 Concentration-compactness
3. 3 Local well-posedness
4. 4 Reduction to almost periodic solutions
5. 5 Long-time Strichartz estimates
6. 6 The rapid frequency-cascade scenario
7. 7 The frequency-localized interaction Morawetz inequality
8. 8 The quasi-soliton scenario
9. A Some basic estimates
## 1\. Introduction
We consider the initial-value problem for defocusing nonlinear Schrödinger
equations of the form
$\left\\{\begin{array}[]{ll}(i\partial_{t}+\Delta)u=|u|^{p}u\\\
u(0,x)=u_{0}(x),\end{array}\right.$ (1.1)
where $p>0$ is chosen to lie between the mass- and energy-critical exponents,
that is, $\tfrac{4}{d}<p<\tfrac{4}{d-2}$. Here
$u:\mathbb{R}_{t}\times\mathbb{R}_{x}^{d}\to\mathbb{C}$ is a complex-valued
function of time and space.
The class of solutions to (1.1) is left invariant by the scaling
$u(t,x)\mapsto\lambda^{\frac{2}{p}}u(\lambda^{2}t,\lambda x)$
for $\lambda>0$. This scaling defines a notion of _criticality_. In
particular, one can check that the only homogeneous $L_{x}^{2}$-based Sobolev
space that is left invariant by this scaling is
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, where the _critical regularity_ $s_{c}$
is given by $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}.$ If we take
$u_{0}\in\dot{H}_{x}^{s}(\mathbb{R}^{d})$ in (1.1), then for $s=s_{c}$, we
call the problem _critical_. For $s>s_{c}$, we call the problem _subcritical_
, while for $s<s_{c}$ the problem is _supercritical_.
We consider the critical problem for (1.1) in the _inter-critical_ regime,
that is, $0<s_{c}<1$. For $(d,s_{c})$ satisfying an appropriate set of
constraints, we prove that any maximal-lifespan solution that is uniformly
bounded (throughout its lifespan) in $\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$
must be global and scatter.
We begin by making the notion of a solution more precise.
###### Definition 1.1 (Solution).
A function $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ on a non-empty time interval
$I\ni 0$ is a _solution_ to (1.1) if it belongs to
$C_{t}\dot{H}_{x}^{s_{c}}(K\times\mathbb{R}^{d})\cap
L_{t,x}^{p(d+2)/2}(K\times\mathbb{R}^{d})$ for every compact $K\subset I$ and
obeys the Duhamel formula
$u(t)=e^{it\Delta}u_{0}-i\int_{0}^{t}e^{i(t-s)\Delta}(|u|^{p}u)(s)\,ds$
for all $t\in I$. We call $I$ the _lifespan_ of $u$; we say $u$ is a _maximal-
lifespan solution_ if it cannot be extended to any strictly larger interval.
If $I=\mathbb{R}$, we say $u$ is _global_.
We define the _scattering size_ of a solution $u$ to (1.1) on a time interval
$I$ by
$S_{I}(u):=\int_{I}\int_{\mathbb{R}^{d}}|u(t,x)|^{\frac{p(d+2)}{2}}\,dx\,dt.$
(1.2)
Standard arguments show that if a solution $u$ to (1.1) is global, with
$S_{\mathbb{R}}(u)<\infty$, then it _scatters_ ; that is, there exist unique
$u_{\pm}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ such that
$\lim_{t\to\pm\infty}\big{\|}u(t)-e^{it\Delta}u_{\pm}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}=0.$
The goal of this paper is to address some cases of the following
###### Conjecture 1.2.
Let $d\geq 1$ and $p>0$ such that $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}\geq 0$.
Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan solution to
(1.1) such that $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$. Then $u$ is global
and scatters, with
$S_{\mathbb{R}}(u)\leq
C(\big{\|}u\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(\mathbb{R}\times\mathbb{R}^{d})})$
for some function $C:[0,\infty)\to[0,\infty).$
For two special cases, it is unnecessary to take $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$ as an additional hypothesis, as this bound
follows from conservation laws. In particular, in the mass-critical case,
$s_{c}=0$ (i.e. $p=\tfrac{4}{d}$), the fact that $u\in
L_{t}^{\infty}L_{x}^{2}$ follows from the conservation of _mass_ , defined by
$M[u(t)]:=\int_{\mathbb{R}^{d}}|u(t,x)|^{2}\,dx,$
while in the energy-critical case, $s_{c}=1$ (i.e. $p=\tfrac{4}{d-2}$, $d\geq
3$), the fact that $u\in L_{t}^{\infty}\dot{H}_{x}^{1}$ follows from the
conservation of _energy_ , defined by
$E[u(t)]:=\int_{\mathbb{R}^{d}}\tfrac{1}{2}|\nabla
u(t,x)|^{2}+\tfrac{1}{p+2}|u(t,x)|^{p+2}\,dx.$
Due to the presence of conserved quantities at critical regularity, the mass-
and energy-critical equations have been the most widely studied; in fact,
Conjecture 1.2 has been settled in these cases. For $s_{c}\notin\\{0,1\\}$,
the hypothesis that $u$ stays bounded in $\dot{H}_{x}^{s_{c}}$ in Conjecture
1.2 plays the role of the ‘missing conservation law’ at critical regularity;
the aim of this paper is to show that with this extra assumption, the
techniques developed to treat the mass- and energy-critical NLS can be applied
in the inter-critical regime, $0<s_{c}<1.$
The defocusing energy-critical case was handled first by Bourgain [3],
Grillakis [23], and Tao [49] for radial data, and subsequently by
Colliander–Keel–Staffilani–Takaoka–Tao [15], Ryckman–Vişan [40], and Vişan
[55, 56] for arbitrary data. (See also [25, 32] for results in the focusing
case.) The primary obstacle to establishing these results was the lack of any
a priori estimates with critical scaling (besides the conservation of energy);
that is to say, none of the known monotonicity formulae for NLS (i.e. Morawetz
inequalities) scale like the energy (in contrast to the energy-critical
nonlinear wave equation, for example). It was ultimately the ‘induction on
energy’ technique of Bourgain that showed how one can move beyond this
difficulty: by finding a bubble of concentration inside a solution, one
introduces a characteristic length scale into a scale-invariant problem.
Having ‘broken’ the scaling symmetry in this way, the available Morawetz
inequalities come back into play, despite their non-critical scaling. All
subsequent work for NLS at critical regularity has built upon this fundamental
idea.
In the energy-critical case, the critical regularity associated to the
available Morawetz estimates is lower than the critical regularity of the
problem. Bourgain was able to make use of the Lin–Strauss Morawetz inequality
(appearing first in [38]), which scales like $\dot{H}_{x}^{\frac{1}{2}}$ and
is well-suited to the radial case; to remove the radial assumption,
Colliander–Keel–Staffilani–Takaoka–Tao introduced the interaction Morawetz
inequality (see [14]), which has the scaling of $\dot{H}_{x}^{\frac{1}{4}}$
(but still requires control over at least half of a derivative, cf. (1.3)
below). These considerations lead us to believe that the techniques developed
to handle the energy-critical problem should be applicable to resolve
Conjecture 1.2 in the case $s_{c}\geq\tfrac{1}{2}$.
In particular, we will be making use of the concentration-compactness (or
‘minimal counterexample’) approach to induction on energy. Minimal
counterexamples were introduced over the course of several papers in the
context of the mass-critical problem (see, for example, [1, 2, 7, 28, 29,
39]); however, the first application of minimal counterexamples to establish a
global well-posedness result was carried out by Kenig–Merle [25], who
developed the technique in the focusing, energy-critical setting.
Minimal counterexamples were also used to establish Conjecture 1.2 in the
mass-critical setting, first for spherically-symmetric data in dimensions
$d\geq 2$ (see [30, 37, 52]), and subsequently for arbitrary initial data in
all dimensions by Dodson [17, 18, 19]. (For results in the focusing case, see
[20, 30, 37].) Notice that in the mass-critical case, the critical regularity
of the problem is lower than that of the available Morawetz estimates; thus
one needs to prove additional regularity (instead of decay) to access these
estimates. We pause here to point out [36, 57], as well, which revisit the
defocusing energy-critical problem from the perspective of minimal
counterexamples.
The first case of Conjecture 1.2 at non-conserved critical regularity to be
addressed was the case $d=3$ and $s_{c}=\tfrac{1}{2}$, in which case the
nonlinearity is cubic. Kenig–Merle [26] were able to handle this case by using
their concentration-compactness technique (as in [25]), together with the
Lin–Strauss Morawetz inequality (which is scaling-critical in this case). As
we will see, this case also falls into the range of cases that we consider,
although we will opt to use the interaction Morawetz inequality instead.
Some cases of Conjecture 1.2 in the energy-supercritical regime (i.e.
$s_{c}>1$) have also been handled by Killip–Vişan [33], also through the use
of minimal counterexamples. In particular, they deal with the case of a cubic
nonlinearity for $d\geq 5$, along with some other cases for which $s_{c}>1$
and $d\geq 5$. Their restriction to high dimensions stems ultimately from
their use of the so-called ‘double Duhamel trick’; for more details, see [33]
and the references cited therein.
Before we discuss our contribution, we note that the analogous conjecture has
also been studied for the nonlinear wave equation. For progress in the energy-
supercritical case, one can refer to works of Kenig–Merle [27], Killip–Vişan
[34, 35], and Bulut [4, 5, 6]. For some results in the energy-subcritical case
with radial data, see [42, 43].
Finally, we are in a position to describe the cases of Conjecture 1.2 that we
will address in this paper. As mentioned above, we will work in the inter-
critical regime, $0<s_{c}<1$ (that is, $\tfrac{4}{d}<p<\tfrac{4}{d-2}$). Our
primary restriction is technical; namely, we only consider cases for which
$p\geq 1$. This restriction serves to simplify the analysis, which still
becomes a bit complicated. For example, when we need to estimate a fractional
number of derivatives of the function $G(z)=|z|^{p}$, things are quite a bit
simpler when $G$ is locally Lipschitz, rather than merely Hölder continuous.
Next, in order to make use of the interaction Morawetz inequality, we restrict
to the cases $d\geq 3$ and $s_{c}\geq\tfrac{1}{2}$ (cf. (1.3) below). For this
restriction to be compatible with $p\geq 1$, we must then restrict to $d\leq
5$. The use of the interaction Morawetz inequality ultimately leads to a
further (more severe) restriction: when $d=3$, we must exclude the cases
$\tfrac{3}{4}<s_{c}<1$. Let us briefly describe the reason for this additional
restriction.
The standard interaction Morawetz inequality may be written as follows: for
$u$ solving (1.1), we have
$\displaystyle-\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}$
$\displaystyle|u(t,x)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u(t,y)|^{2}\,dx\,dy\,dt$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}^{2}\big{\|}|\nabla|^{1/2}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}^{2}.$
(1.3)
As we are in the case $s_{c}\geq\tfrac{1}{2}$, we see that to guarantee that
the right-hand side is finite, we must truncate the solution $u$ to high
frequencies (that is, we must work with $u_{\geq N}$ for some $N>0$). In
Section 7, we do exactly this. However, $u_{\geq N}$ is no longer a solution
to (1.1); thus, the truncation results in error terms for (1.3) that must be
handled to arrive at a useful estimate. When $d=3$ and $s_{c}>\tfrac{3}{4}$,
we find that there is one error term that we cannot handle unless we also
impose a spatial truncation on the weight we use to derive (1.3); see, for
example, [15, 36], which address the case $d=3$, $s_{c}=1$. This spatial
truncation, however, results in even _more_ error terms; it turns out that
some of these additional error terms then require control over the
$\dot{H}_{x}^{1}$-regularity of the solution. Thus, in the energy-critical
case, one can push the argument through, while in our case we cannot succeed
without significant additional input. For further comments, see Remark 7.4.
Our final restriction is to exclude the case $(d,s_{c})=(5,\tfrac{1}{2})$, for
which $p=1$. In this case, the proof of Lemma 4.2, which is essential to the
reduction to almost periodic solutions in Section 4, breaks down. For further
discussion, see Remark 4.3.
To summarize, our main results will apply to (1.1) with $(d,s_{c})$ satisfying
the following:
$\left\\{\begin{array}[]{llllll}\tfrac{1}{2}\leq
s_{c}\leq\tfrac{3}{4}&\text{if }d=3,\vspace{1mm}\\\ \tfrac{1}{2}\leq
s_{c}<1&\text{if }d=4,\vspace{1mm}\\\ \tfrac{1}{2}<s_{c}<1&\text{if
}d=5.\end{array}\right.$ (1.4)
It is worth mentioning, however, that many of our results will apply to a less
restrictive set of $(d,s_{c})$. In particular, many of our results will apply
to
$d\in\\{3,4,5\\}\quad\text{and}\quad\tfrac{1}{2}\leq
s_{c}<1,\quad\text{excluding}\quad(d,s_{c})=(5,\tfrac{1}{2}).$ (1.5)
As we proceed, we will keep track of which restrictions are necessary for
which results.
Our main result is the following:
###### Theorem 1.3.
Suppose $(d,s_{c})$ satisfies (1.4). Let
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan solution to (1.1)
such that $u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d}).$
Then $u$ is global and scatters, with
$S_{\mathbb{R}}(u)\leq
C(\big{\|}u\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(\mathbb{R}\times\mathbb{R}^{d})})$
for some function $C:[0,\infty)\to[0,\infty).$
To establish Theorem 1.3, we will model our approach after several sources,
including [31, 32, 33, 36, 57]. In particular, Section 3 follows the
presentation in [31, 33]; Section 4 draws heavily from [32]; and the
presentation of the remaining sections is inspired largely by [36, 57]. We
note also that we rely on [31] for several standard results regarding almost
periodic solutions in the outline below.
The first step towards a global-in-time theory for (1.1) is to develop a good
local-in-time theory for this equation. In particular, building off arguments
of Cazenave–Weissler [8], one can prove the following
###### Theorem 1.4 (Local well-posedness).
Let $(d,s_{c})$ satisfy (1.5). Then, given
$u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, there exists a unique maximal-
lifespan solution $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1). Moreover,
this solution satisfies the following:
$\bullet$ (Local existence) $I$ is an open neighborhood of $0$.
$\bullet$ (Blowup criterion) If $\sup I$ is finite, then $u$ blows up forward
in time, in the sense that $S_{[0,\sup I)}(u)=\infty$. If $\inf I$ is finite,
then $u$ blows up backwards in time, in the sense that $S_{(\inf
I,0]}(u)=\infty.$
$\bullet$ (Scattering) If $\sup I=+\infty$ and $u$ does not blow up forward in
time, then $u$ scatters forward in time; that is, there exists a unique
$u_{+}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ such that
$\lim_{t\to\infty}\big{\|}u(t)-e^{it\Delta}u_{+}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}=0.$
(1.6)
Conversely, for any $u_{+}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, there is a
unique solution $u$ to (1.1) so that (1.6) holds. The analogous statements
hold backward in time.
$\bullet$ (Small-data global existence) There exists $\eta_{0}=\eta_{0}(d,p)$
such that if
$\big{\|}u_{0}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{2}<\eta_{0},$
then $u$ is global and scatters, with
$S_{\mathbb{R}}(u)\lesssim\big{\|}u_{0}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{p(d+2)/2}.$
###### Remark 1.5.
We note here that the notion of blowup described above corresponds exactly to
the impossibility of extending the solution to a larger time interval in the
class described in Definition 1.1.
In Section 3, we will establish this theorem as a corollary of a local well-
posedness result of Cazenave–Weissler [8] (Theorem 3.1) and a stability result
(Theorem 3.4). This stability result plays an essential role in the argument
we present, specifically in the proof of Theorem 1.12.
### 1.1. Outline of the proof of Theorem 1.3
The proof is by contradiction. We first recall that Theorem 1.3 holds if we
restrict to sufficiently small initial data (cf. Theorem 1.4); thus, the
failure of Theorem 1.3 would imply the existence of a ‘threshold’ size, below
which Theorem 1.3 holds, but above which we can find (almost) counterexamples.
Using a limiting argument, we then find blowup solutions _at_ this threshold,
so-called ‘minimal blowup solutions’. By carefully analyzing such solutions,
we can show that they must have so many nice properties that in fact, they
cannot exist at all.
The main property of these special counterexamples is that of almost
periodicity modulo symmetries:
###### Definition 1.6 (Almost periodic solutions).
Let $s_{c}>0$. A solution $u$ to (1.1) with lifespan $I$ is said to be _almost
periodic (modulo symmetries)_ if $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$ and there exist
functions $N:I\to\mathbb{R}^{+},$ $x:I\to\mathbb{R}^{d}$, and
$C:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that for all $t\in I$ and all
$\eta>0$,
$\int_{|x-x(t)|\geq\frac{C(\eta)}{N(t)}}\big{|}|\nabla|^{s_{c}}u(t,x)\big{|}^{2}\,dx+\int_{|\xi|\geq
C(\eta)N(t)}|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta.$
We call $N$ the _frequency scale function_ , $x$ the _spatial center function_
, and $C$ the _compactness modulus function_.
###### Remark 1.7.
By the Arzelà–Ascoli theorem, a family of functions is precompact in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ if and only if it is norm-bounded and
there exists a compactness modulus function $C$ such that
$\int_{|x|\geq
C(\eta)}\big{|}|\nabla|^{s_{c}}f(x)\big{|}^{2}\,dx+\int_{|\xi|\geq
C(\eta)}|\xi|^{2s_{c}}|\widehat{f}(\xi)|^{2}\,d\xi\leq\eta$
uniformly for all functions $f$ in the family. Thus, an equivalent formulation
of Definition 1.6 is the following: $u$ is almost periodic (modulo symmetries)
if and only if
$\\{u(t):t\in
I\\}\subset\\{\lambda^{\frac{2}{p}}f(\lambda(x+x_{0})):\lambda\in(0,\infty),\
x_{0}\in\mathbb{R}^{d},\ \text{and}\ f\in K\\}$
for some compact $K\subset\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d}).$
Furthermore, Sobolev embedding gives that every compact set in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ is also compact in
$L_{x}^{\frac{dp}{2}}(\mathbb{R}^{d})$; thus, for any almost periodic solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$, we also have
$\int_{|x-x(t)|\geq\frac{C(\eta)}{N(t)}}|u(t,x)|^{\frac{dp}{2}}\,dx\leq\eta$
for all $t\in I$ and $\eta>0$.
###### Remark 1.8.
Another consequence of almost periodicity modulo symmetries is the existence
of a function $c:\mathbb{R}^{+}\to\mathbb{R}^{+}$ so that for all $t\in I$ and
all $\eta>0$,
$\int_{|x-x(t)|\leq\frac{c(\eta)}{N(t)}}\big{|}|\nabla|^{s_{c}}u(t,x)\big{|}^{2}\,dx+\int_{|\xi|\leq
c(\eta)N(t)}|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta.$
One can show (see [31, Lemma 5.18], for example) that the modulation
parameters of almost periodic solutions obey the following local constancy
property:
###### Lemma 1.9 (Local constancy).
Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan almost
periodic solution to (1.1). Then there exists $\delta=\delta(u)>0$ such that
if $t_{0}\in I$, then
$[t_{0}-\delta N(t_{0})^{-2},t_{0}+\delta N(t_{0})^{-2}]\subset I,$
with
$N(t)\sim_{u}N(t_{0})\indent\text{for}\quad|t-t_{0}|\leq\delta N(t_{0})^{-2}.$
Given a maximal-lifespan almost periodic solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1), Lemma 1.9 allows us to
subdivide $I$ into _characteristic subintervals_ $J_{k}$ on which $N(t)$ is
constant and equal to some $N_{k}$, with $|J_{k}|\sim_{u}N_{k}^{-2}$. To do
this, we need to modify the compactness modulus function by a time-independent
multiplicative factor.
The local constancy property also has the following consequence (see [31,
Corollary 5.19]):
###### Corollary 1.10 ($N(t)$ at blowup).
Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan almost
periodic solution to (1.1). If $T$ is any finite endpoint of $I$, then
$N(t)\gtrsim_{u}|T-t|^{-1/2}$; in particular, $\lim_{t\to T}N(t)=\infty.$ If
$I$ is infinite or semi-infinite, then for any $t_{0}\in I$, we have
$N(t)\gtrsim\langle t-t_{0}\rangle^{-1/2}.$
Finally, we need the following result, which relates the frequency scale
function of an almost periodic solution to its Strichartz norms.
###### Lemma 1.11 (Spacetime bounds).
Let $(d,s_{c})$ satisfy (1.5), and suppose $u$ is an almost periodic solution
to (1.1) on a time interval $I$. Then
$\int_{I}N(t)^{2}\,dt\lesssim_{u}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}^{2}\lesssim_{u}1+\int_{I}N(t)^{2}\,dt.$
One can prove this result by adapting the proof of [31, Lemma 5.21]; the key
is to notice that $\int_{I}N(t)^{2}\,dt$ is approximately the number of
characteristic subintervals inside $I$. The restriction on $(d,s_{c})$ is not
actually necessary, but it covers our cases of interest.
With these preliminaries established, we can now describe the first major step
in the proof of Theorem 1.3.
###### Theorem 1.12 (Reduction to almost periodic solutions).
Suppose that Theorem 1.3 fails for $(d,s_{c})$ satisfying (1.5). Then there
exists a maximal-lifespan solution $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to
(1.1) such that $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$, $u$ is almost
periodic modulo symmetries, and $u$ blows up both forward and backward in
time. Moreover, $u$ has minimal $L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$-norm among
all blowup solutions; i.e.,
$\sup_{t\in
I}\big{\|}u(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}\leq\sup_{t\in
J}\big{\|}v(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}$
for all maximal-lifespan solutions $v:J\times\mathbb{R}^{d}\to\mathbb{C}$ that
blow up in at least one time direction.
We sketch a proof of Theorem 1.12 in Section 4. By now, the reduction to
almost periodic solutions is a fairly standard technique in the study of
dispersive equations at critical regularity. Keraani [29] first proved the
existence of minimal blowup solutions (in the mass-critical setting), while
Kenig–Merle [25] were the first to use them as a tool to prove global well-
posedness (in the energy-critical setting). For many more examples of these
techniques, one can refer to [26, 27, 30, 32, 31, 33, 34, 35, 37, 53, 52], for
example.
Still, while the underlying ideas are well-established, we will see that to
carry out the reduction in the cases we consider will require some new ideas
and careful analysis; indeed, the resolution of this problem is the chief
novelty of this paper. One of the principal difficulties arises in the proof
of Lemma 4.2, in which we establish a decoupling of nonlinear profiles in
order to show that a sequence of approximate solutions to (1.1) converges in
some sense to a true solution.
For the mass- and energy-critical cases, one can use pointwise estimates and
well-known arguments of [28] to establish this decoupling; in our setting, the
nonlocal nature of $|\nabla|^{s_{c}}$ prevents the direct use of any pointwise
estimates. The authors of [33], who dealt with some cases in the energy-
supercritical regime, overcame this difficulty by establishing analogous
pointwise estimates for a square function of Strichartz that shares estimates
with $|\nabla|^{s_{c}}$ (see [46]). With such estimates in hand, the usual
arguments can then be pushed through. Their approach does not work in our
setting, however, as it strongly relies on the fact that $s_{c}>1$. In [26],
the authors treat a cubic nonlinearity in dimension $d=3$ (in which case
$s_{c}=\tfrac{1}{2}$); by exploiting the polynomial nature of the nonlinearity
and employing a paraproduct estimate, they too overcome the nonlocal nature of
$|\nabla|^{s_{c}}$ and put themselves in a position where the standard
arguments are applicable. In our setting, the combination of fractional
derivatives and non-polynomial nonlinearities presents a new technical
challenge. Ultimately, the resolution of this problem comes from a careful
reworking of the proof of the fractional chain rule, in which the
Littlewood–Paley square function allows us to work at the level of individual
frequencies. By making use of some tools from harmonic analysis (including
maximal and vector maximal inequalities), we are eventually able to adapt the
standard arguments to establish the decoupling in our setting. For further
discussion, see Section 4.
After establishing Theorem 1.12, we make some further refinements to the class
of solutions we consider. First, we can use a rescaling argument to restrict
our attention to almost periodic solutions that do not escape to arbitrarily
low frequencies on at least half of their maximal lifespan, say $[0,T_{max})$.
We will not include the details here; one can instead refer to Section 4 in
any of [30, 32, 52]. Next, following [17], we will divide these solutions into
two classes that depend on the control given by the interaction Morawetz
inequality; these classes will correspond to the ‘rapid frequency-cascade’ and
‘quasi-soliton’ scenarios. Finally, as described above, we use Lemma 1.9 to
subdivide $[0,T_{max})$ into characteristic subintervals $J_{k}$ and set
$N(t)$ to be constant and equal to $N_{k}$ on each $J_{k}$, with
$|J_{k}|\sim_{u}N_{k}^{-2}$. In this way, we arrive at
###### Theorem 1.13 (Two special scenarios for blowup).
Suppose that Theorem 1.3 fails for $(d,s_{c})$ satisfying (1.5). Then there
exists an almost periodic solution
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ that blows up forward in
time, with
$N(t)\equiv N_{k}\geq 1$
for each $t\in J_{k}$, where $[0,T_{max})=\cup_{k}J_{k}$. Furthermore,
$\text{either}\quad\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty\quad\text{or}\quad\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$
Thus, to establish Theorem 1.3, it remains to preclude the existence of the
almost periodic solutions described in Theorem 1.13. The main technical tool
we will use to achieve this will be a long-time Strichartz inequality,
Proposition 5.1. Such inequalities were originally developed by Dodson [17]
for almost periodic solutions in the mass-critical setting; for variants in
the energy-critical setting, see [36, 57]. In this paper, we establish a long-
time Strichartz estimate for the first time in the inter-critical regime. The
proof of Proposition 5.1 is by induction; the recurrence relation is derived
with the aid of Strichartz estimates, together with a paraproduct estimate
(Lemma 2.6) and a bilinear Strichartz inequality (Lemma 2.10).
In Section 6, we preclude the rapid frequency-cascade scenario, that is,
almost periodic solutions as in Theorem 1.13 for which
$\textstyle\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty.$ This proof requires
two ingredients. The first ingredient is the long-time Strichartz inequality
(Proposition 5.1), while the second is the following
###### Proposition 1.14 (No-waste Duhamel formula).
Let $u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}$. Then for all $t\in[0,T_{max}),$ we have
$u(t)=i\lim_{T\to T_{max}}\int_{t}^{T}e^{i(t-s)\Delta}(|u|^{p}u)(s)\,ds,$
where the limits are taken in the weak $\dot{H}_{x}^{s_{c}}$ topology.
To prove Proposition 1.14, one can adapt the proof of [31, Proposition 5.23];
we omit the details. Using Proposition 1.14 and Strichartz estimates, we can
upgrade the information given by Proposition 5.1 to see that a rapid
frequency-cascade solution must have finite mass. In fact, we can show that
the solution has zero mass, which contradicts that the solution blows up.
In Section 8, we preclude the quasi-soliton scenario, that is, almost periodic
solutions as in Theorem 1.13 for which
$\textstyle\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$ The main ingredient
is a frequency-localized interaction Morawetz inequality (Proposition 7.1),
which we prove in Section 7. To establish this estimate, we begin with the
usual interaction Morawetz inequality, truncate to high frequencies, and use
Proposition 5.1 to control the resulting error terms. (As described above, one
of these error terms eventually forces us to exclude the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ from Theorem 1.3; see Remark 7.4.)
To rule out the quasi-soliton scenario and thereby complete the proof of
Theorem 1.3, we notice that the frequency-localized interaction Morawetz
inequality provides uniform control over $\int_{I}N(t)^{3-4s_{c}}\,dt$ for all
compact time intervals $I\subset[0,T_{max})$; thus we can derive a
contradiction by taking $I$ to be sufficiently large inside of $[0,T_{max})$.
### Acknowledgements
I owe many thanks to my advisors, Rowan Killip and Monica Vişan, for all of
their guidance and support. I am very grateful to them, not only for bringing
this problem to my attention, but also for engaging in many helpful
discussions, and for a careful reading of the manuscript. This work was
supported in part by NSF grant DMS-1001531 (P.I. Rowan Killip).
## 2\. Notation and useful lemmas
### 2.1. Some notation
We write $X\lesssim Y$ or $Y\gtrsim X$ whenever $X\leq CY$ for some $C>0$. If
$X\lesssim Y\lesssim X$, we write $X\sim Y$. If the implicit constant $C$
depends on the dimension $d$ or the power $p$, this dependence will be
suppressed; dependence on additional parameters will be indicated with
subscripts. For example, $X\lesssim_{u}Y$ indicates that $X\leq CY$ for some
$C=C(u).$
For a spacetime slab $I\times\mathbb{R}^{d}$, we write
$L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})$ to denote the Banach space of
functions $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ equipped with the norm
$\big{\|}u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:=\left(\int_{I}\big{\|}u(t)\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}^{q}\,dt\right)^{1/q},$
with the usual conventions when $q$ or $r$ is infinity. If $q=r$, we
abbreviate $L_{t}^{q}L_{x}^{q}=L_{t,x}^{q}.$ At times we will also abbreviate
$\big{\|}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$ to
$\big{\|}f\big{\|}_{L_{x}^{r}}$ or $\big{\|}f\big{\|}_{r}.$
We define the Fourier transform on $\mathbb{R}^{d}$ by
$\widehat{f}(\xi):=(2\pi)^{-d/2}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)\,dx.$
For $s\in\mathbb{R}$, we can then define the fractional differentiation
operator $|\nabla|^{s}$ via
$\widehat{|\nabla|^{s}f}(\xi):=|\xi|^{s}\widehat{f}(\xi),$
which in turn defines the homogeneous Sobolev norm
$\big{\|}f\big{\|}_{\dot{H}_{x}^{s}(\mathbb{R}^{d})}:=\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}.$
### 2.2. Basic harmonic analysis
Let $\varphi$ be a radial bump function supported in the ball
$\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\tfrac{11}{10}\\}$ and equal to 1 on the
ball $\\{\xi\in\mathbb{R}^{d}:|\xi|\leq 1\\}.$ For $N\in 2^{\mathbb{Z}}$, we
define the Littlewood–Paley projection operators via
$\displaystyle\widehat{P_{\leq N}f}(\xi):=\widehat{f_{\leq
N}}(\xi):=\varphi(\xi/N)\widehat{f}(\xi),$
$\displaystyle\widehat{P_{>N}f}(\xi):=\widehat{f_{>N}}(\xi):=(1-\varphi(\xi/N))\widehat{f}(\xi),$
$\displaystyle\widehat{P_{N}f}(\xi):=\widehat{f_{N}}(\xi):=(\varphi(\xi/N)-\varphi(2\xi/N))\widehat{f}(\xi).$
We define $P_{<N}$ and $P_{\geq N}$ similarly. We also define
$P_{M<\cdot\leq N}:=P_{\leq N}-P_{\leq M}=\sum_{M<N^{\prime}\leq
N}P_{N^{\prime}}$
for $M<N.$ All such summations are understood to be over $N\in
2^{\mathbb{Z}}.$ Being Fourier multiplier operators, these Littlewood–Paley
projection operators commute with $e^{it\Delta}$, as well as differential
operators (for example, $i\partial_{t}+\Delta$). We will need the following
standard estimates for these operators:
###### Lemma 2.1 (Bernstein estimates).
For $1\leq r\leq q\leq\infty$ and $s\geq 0$,
$\displaystyle\big{\|}|\nabla|^{\pm
s}P_{N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$ $\displaystyle\sim N^{\pm
s}\big{\|}P_{N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}|\nabla|^{s}P_{\leq
N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$ $\displaystyle\lesssim
N^{s}\big{\|}P_{\leq N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}P_{\geq N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\lesssim N^{-s}\big{\|}|\nabla|^{s}P_{\geq
N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$ $\displaystyle\big{\|}P_{\leq
N}f\big{\|}_{L_{x}^{q}(\mathbb{R}^{d})}$ $\displaystyle\lesssim
N^{\frac{d}{r}-\frac{d}{q}}\big{\|}P_{\leq
N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}.$
###### Lemma 2.2 (Littlewood–Paley square function estimates).
For $1<r<\infty$,
$\displaystyle\big{\|}\big{(}\sum|P_{N}f(x)|^{2}\big{)}^{1/2}\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\sim\big{\|}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}\big{(}\sum
N^{2s}|f_{N}(x)|^{2}\big{)}^{1/2}\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\sim\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}\quad\text{for
}s\in\mathbb{R},$ $\displaystyle\big{\|}\big{(}\sum
N^{2s}|f_{>N}(x)|^{2}\big{)}^{1/2}\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\sim\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}\quad\text{for
}s>0.$
We will also need the following general inequalities, which appear originally
in [10]. For a textbook treatment, one can refer to [48].
###### Lemma 2.3 (Fractional product rule, [10]).
Let $s\in(0,1]$ and $1<r,r_{1},r_{2},q_{1},q_{2}<\infty$ such that
$\tfrac{1}{r}=\tfrac{1}{r_{i}}+\tfrac{1}{q_{i}}$ for $i=1,2$. Then
$\big{\|}|\nabla|^{s}(fg)\big{\|}_{r}\lesssim\big{\|}f\big{\|}_{r_{1}}\big{\|}|\nabla|^{s}g\big{\|}_{q_{1}}+\big{\|}|\nabla|^{s}f\big{\|}_{r_{2}}\big{\|}g\big{\|}_{q_{2}}.$
###### Lemma 2.4 (Fractional chain rule, [10]).
Suppose $G\in C^{1}(\mathbb{C}),$ $s\in(0,1]$, and $1<r,r_{1},r_{2}<\infty$
are such that $\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{1}{r_{2}}.$ Then
$\big{\|}|\nabla|^{s}G(u)\big{\|}_{r}\lesssim\big{\|}G^{\prime}(u)\big{\|}_{r_{1}}\big{\|}|\nabla|^{s}u\big{\|}_{r_{2}}.$
We will also make use of the following refinement of the fractional chain
rule, which appears in [35]:
###### Lemma 2.5 (Derivatives of differences, [35]).
Fix $p>1$ and $0<s<1$. Then for $1<r,r_{1},r_{2}<\infty$ such that
$\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{p-1}{r_{2}},$ we have
$\big{\|}|\nabla|^{s}[|u+v|^{p}-|u|^{p}]\big{\|}_{r}\lesssim\big{\|}|\nabla|^{s}u\big{\|}_{r_{1}}\big{\|}v\big{\|}_{r_{2}}^{p-1}+\big{\|}|\nabla|^{s}v\big{\|}_{r_{1}}\big{\|}u+v\big{\|}_{r_{2}}^{p-1}.$
Finally, we prove a paraproduct estimate, very much in the spirit of Lemma 2.3
in [57].
###### Lemma 2.6 (Paraproduct estimate).
Fix $d\in\\{3,4,5\\}.$
(a) For $p>0$ such that $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}\in[\tfrac{1}{2},1)$,
we have
$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(fg)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}.$
(b) We also have
$\big{\|}|\nabla|^{-\frac{2}{5}}(fg)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}|\nabla|^{-\frac{2}{5}}f\big{\|}_{L_{x}^{\frac{10d}{5d-11}}}\big{\|}|\nabla|^{\frac{2}{5}}g\big{\|}_{L_{x}^{\frac{2d}{5}}}.$
###### Proof.
For (a), we will prove the equivalent estimate
$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\left(|\nabla|^{\frac{1}{2}s_{c}}f\,|\nabla|^{-\frac{1}{2}s_{c}}g\right)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}$
by decomposing the left-hand side into low-high and high-low frequency
interactions. More precisely, we introduce the projections $\pi_{l,h}$ and
$\pi_{h,l}$, defined for any pair of functions $\phi,\psi$ by
$\pi_{l,h}(\phi,\psi):=\sum_{N\lesssim
M}\phi_{N}\psi_{M}\quad\text{and}\quad\pi_{h,l}(\phi,\psi):=\sum_{N\gg
M}\phi_{N}\psi_{M}.$
We first consider the low-high interactions: by Sobolev embedding, we have
$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{4dp}{p(3d+4)-4}}}.$
(2.1)
We note here that when $d=3$, the assumption $s_{c}<1$ guarantees that
$\tfrac{4dp}{p(3d+4)-4}>1.$
If we now consider the multiplier of the operator given by
$T(f,g):=\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g),$
that is,
$\sum_{N\lesssim
M}|\xi_{1}|^{\frac{1}{2}s_{c}}\widehat{f_{N}}(\xi_{1})|\xi_{2}|^{-\frac{1}{2}s_{c}}\widehat{g_{M}}(\xi_{2}),$
then we see that this multiplier is a symbol of order zero with
$\xi=(\xi_{1},\xi_{2})$. Thus, continuing from (2.1), we can cite a theorem of
Coifman–Meyer (see [11, 13], for example) to conclude
$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}.$
We now consider the high-low interactions: if we consider the multiplier of
the operator given by
$S(f,h):=|\nabla|^{-\frac{1}{2}s_{c}}\pi_{h,l}(|\nabla|^{\frac{1}{2}s_{c}}f,h),$
that is,
$\sum_{N\gg
M}|\xi_{1}+\xi_{2}|^{-\frac{1}{2}s_{c}}|\xi_{1}|^{\frac{1}{2}s_{c}}\widehat{f_{N}}(\xi_{1})\widehat{h_{M}}(\xi_{2}),$
then we see that this multiplier is also a symbol of order zero. Thus, using
the result cited above, along with Sobolev embedding, we can estimate
$\displaystyle\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\pi_{h,l}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}g\big{\|}_{L_{x}^{\frac{d}{2}}}$
$\displaystyle\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}.$
Combining the low-high and high-low interactions, we recover (a). Mutis
mutandis, the exact same proof gives (b). ∎
### 2.3. Strichartz estimates
Let $e^{it\Delta}$ be the free Schrödinger propagator,
$[e^{it\Delta}f](x)=\tfrac{1}{(4\pi
it)^{d/2}}\int_{\mathbb{R}^{d}}e^{i|x-y|^{2}/4t}f(y)\,dy$
for $t\neq 0$. This explicit formula immediately implies the dispersive
estimate
$\big{\|}e^{it\Delta}f\big{\|}_{L_{x}^{\infty}(\mathbb{R}^{d})}\lesssim|t|^{-\frac{d}{2}}\big{\|}f\big{\|}_{L_{x}^{1}(\mathbb{R}^{d})}$
for $t\neq 0$. Interpolating with
$\big{\|}e^{it\Delta}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}=\big{\|}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}$
(cf. Plancherel), one arrives at
$\big{\|}e^{it\Delta}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}\lesssim|t|^{-(\frac{d}{2}-\frac{d}{r})}\big{\|}f\big{\|}_{L_{x}^{r^{\prime}}(\mathbb{R}^{d})}$
(2.2)
for $t\neq 0$ and $2\leq r\leq\infty$, with
$\tfrac{1}{r}+\tfrac{1}{r^{\prime}}=1.$ This estimate can be used to prove the
standard Strichartz estimates, which we state below. First, we need the
following
###### Definition 2.7 (Admissible pairs).
For $d\geq 3$, we call a pair of exponents $(q,r)$ _Schrödinger admissible_ if
$\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}\indent\text{and}\indent 2\leq
q,r\leq\infty.$
For a spacetime slab $I\times\mathbb{R}^{d}$, we define
$\big{\|}u\big{\|}_{S^{0}(I)}:=\sup\left\\{\big{\|}u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:(q,r)\text{
admissible}\right\\}.$
We define $S^{0}(I)$ to be the closure of the test functions under this norm,
and denote the dual of $S^{0}(I)$ by $N^{0}(I)$. We note
$\big{\|}u\big{\|}_{N^{0}(I)}\lesssim\big{\|}u\big{\|}_{L_{t}^{q^{\prime}}L_{x}^{r^{\prime}}(I\times\mathbb{R}^{d})}\quad\text{for
any admissible pair }(q,r).$
We now state the Strichartz estimates in the form we will need them.
###### Lemma 2.8 (Strichartz).
Let $s\geq 0$, let $I$ be a compact time interval, and let
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to the forced Schrödinger
equation
$(i\partial_{t}+\Delta)u=F.$
Then
$\big{\|}|\nabla|^{s}u\big{\|}_{S^{0}(I)}\lesssim\big{\|}|\nabla|^{s}u(t_{0})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s}F\big{\|}_{N^{0}(I)}$
for any $t_{0}\in I$.
###### Proof.
As mentioned, the key ingredient is (2.2). For the endpoint
$(q,r)=(2,\tfrac{2d}{d-2})$ in $d\geq 3$, see [24]. For the non-endpoint
cases, see [22, 47], for example. ∎
The free propagator also obeys some local smoothing estimates (see [16, 44,
54] for the original results). We will make use of the following, which
appears as Proposition 4.14 in [31]:
###### Lemma 2.9 (Local smoothing).
For any $f\in L_{x}^{2}(\mathbb{R}^{d})$ and any $\varepsilon>0$,
$\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\big{|}[|\nabla|^{\frac{1}{2}}e^{it\Delta}f](x)\big{|}^{2}\langle
x\rangle^{-1-\varepsilon}\,dx\,dt\lesssim_{\varepsilon}\big{\|}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}.$
Next, we record the following bilinear Strichartz estimates. The version we
need can be deduced from (the proof of) Corollary 4.19 in [31].
###### Lemma 2.10 (Bilinear Strichartz).
Let $0<s_{c}<\tfrac{d-1}{2}$. For any spacetime slab $I\times\mathbb{R}^{d}$
and any frequencies $M>0$ and $N>0$, we have
$\big{\|}u_{\leq M}v_{\geq
N}\big{\|}_{L_{t,x}^{2}(I\times\mathbb{R}^{d})}\lesssim
M^{(\frac{d-1}{2}-s_{c})}N^{-(\frac{1}{2}+s_{c})}\big{\|}|\nabla|^{s_{c}}u_{\leq
M}\big{\|}_{S_{0}^{*}(I)}\big{\|}|\nabla|^{s_{c}}v_{\geq
N}\big{\|}_{S_{0}^{*}(I)},$
where
$\big{\|}u\big{\|}_{S_{0}^{*}(I)}:=\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}+\big{\|}(i\partial_{t}+\Delta)u\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}.$
###### Remark 2.11.
We will use Lemma 2.10 in the proof of Proposition 5.1. In that context, we
will have $u=v$ an almost periodic solution to (1.1) and $I=J_{k}$, a
characteristic subinterval. In this case, interpolating between $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$ and Lemma 1.11 gives
$\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(J_{k})}\lesssim_{u}1,$
so that we can use the fractional chain rule and Sobolev embedding to estimate
$\displaystyle\big{\|}|\nabla|^{s_{c}}(|u|^{p}u)\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}(J_{k}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(J_{k}\times\mathbb{R}^{d})}^{p}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}(J_{k}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(J_{k})}^{p+1}$
$\displaystyle\lesssim_{u}1.$
Thus, in this setting, an application of Lemma 2.10 gives
$\big{\|}u_{\leq M}u_{\geq
N}\big{\|}_{L_{t,x}^{2}(J_{k}\times\mathbb{R}^{d})}\lesssim_{u}M^{(\frac{d-1}{2}-s_{c})}N^{-(\frac{1}{2}+s_{c})}.$
### 2.4. Concentration-compactness
We record here the linear profile decomposition that we will use to prove the
reduction in Theorem 1.12. We begin with the following
###### Definition 2.12 (Symmetry group).
For any position $x_{0}\in\mathbb{R}^{d}$ and scaling parameter $\lambda>0$,
we define a unitary transformation
$g_{x_{0},\lambda}:\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})\to\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$
by
$[g_{x_{0},\lambda}f](x):=\lambda^{-\frac{2}{p}}f\left(\lambda^{-1}(x-x_{0})\right)$
(recall $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}$). We let $G$ denote the collection
of such transformations. For a function
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$, we define
$T_{g_{x_{0},\lambda}}u:\lambda^{2}I\times\mathbb{R}^{d}\to\mathbb{C}$ by the
formula
$[T_{g_{x_{0},\lambda}}u](t,x):=\lambda^{-\frac{2}{p}}u\left(\lambda^{-2}t,\lambda^{-1}(x-x_{0})\right),$
where $\lambda^{2}I:=\\{\lambda^{2}t:t\in I\\}.$ Note that if $u$ is a
solution to (1.1), then $T_{g}u$ is a solution to (1.1) with initial data
$gu_{0}$.
###### Remark 2.13.
It is easily seen that $G$ is a group under composition. The map $u\mapsto
T_{g}u$ maps solutions to (1.1) to solutions with the same scattering size,
that is, $S(T_{g}u)=S(u)$. Furthermore, $u$ is a maximal-lifespan solution if
and only if $T_{g}u$ is a maximal-lifespan solution.
We now state the linear profile decomposition in the form that we need. For
$s_{c}=0$, this result was originally proven in [1, 7, 39], while for
$s_{c}=1$ it was established in [28]. In the generality we need, a proof can
be found in [41].
###### Lemma 2.14 (Linear profile decomposition, [41]).
Fix $0<s_{c}<1$ and let $\\{u_{n}\\}_{n\geq 1}$ be bounded sequence in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d}).$ Then, after passing to a subsequence if
necessary, there exist functions $\\{\phi^{j}\\}_{j\geq
1}\subset\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d}),$ group elements $g_{n}^{j}\in
G$, and times $t_{n}^{j}\in\mathbb{R}$ such that for all $J\geq 1$, we have
the decomposition
$u_{n}=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}$
with the following properties:
$\bullet$ For all $n$ and all $J\geq 1$, we have
$w_{n}^{J}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, with
$\lim_{J\to\infty}\limsup_{n\to\infty}\
\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(\mathbb{R}\times\mathbb{R}^{d})}=0.$
(2.3)
$\bullet$ For any $j\neq k$, we have the following asymptotic orthogonality of
parameters:
$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}+\frac{\lambda_{n}^{k}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{k}|^{2}}{\lambda_{n}^{j}\lambda_{n}^{k}}+\frac{|t_{n}^{j}(\lambda_{n}^{j})^{2}-t_{n}^{k}(\lambda_{n}^{k})^{2}|}{\lambda_{n}^{j}\lambda_{n}^{k}}\to\infty\quad\text{as}\
n\to\infty.$ (2.4)
$\bullet$ For any $J\geq 1$, we have the decoupling properties:
$\lim_{n\to\infty}\bigg{[}\big{\|}|\nabla|^{s_{c}}u_{n}\big{\|}_{2}^{2}-\sum_{j=1}^{J}\big{\|}|\nabla|^{s_{c}}\phi^{j}\big{\|}_{2}^{2}-\big{\|}|\nabla|^{s_{c}}w_{n}^{J}\big{\|}_{2}^{2}\bigg{]}=0,$
(2.5)
and for any $1\leq j\leq J$,
$e^{-it_{n}^{j}\Delta}[(g_{n}^{j})^{-1}w_{n}^{J}]\rightharpoonup
0\quad\text{weakly in }\dot{H}_{x}^{s_{c}}\ \text{as }n\to\infty.$ (2.6)
###### Remark 2.15.
In this linear profile decomposition, we may always choose the scaling
parameters $\lambda_{n}^{k}$ so that they belong to $2^{\mathbb{Z}}$.
## 3\. Local well-posedness
In this section, we develop a local theory for (1.1). We begin by recording a
standard local well-posedness result proven by Cazenave–Weissler [8]; see also
[9, 31, 50]. We will also need to establish a stability result (appearing as
Theorem 3.4), which will be essential in the reduction to almost periodic
solutions in Section 4. For stability results in the mass- and energy-critical
settings, see [15, 40, 51, 53].
For the following local well-posedness result, one must assume that the
initial data belongs to the inhomogeneous Sobolev space
$H_{x}^{s_{c}}(\mathbb{R}^{d}).$ This assumption serves to simplify the proof
(allowing for a contraction mapping argument in mass-critical spaces); we can
remove it a posteriori by using the stability result we prove below.
###### Theorem 3.1 (Standard local well-posedness [8]).
Let $d\geq 1$, $0<s_{c}<1$, and $u_{0}\in H_{x}^{s_{c}}(\mathbb{R}^{d}).$ Then
there exists $\eta_{0}>0$ so that if $0<\eta\leq\eta_{0}$ and $I$ is an
interval containing zero such that
$\big{\|}|\nabla|^{s_{c}}e^{it\Delta}u_{0}\big{\|}_{L_{t}^{p+2}L_{x}^{\frac{2d(p+2)}{2(d-2)+dp}}(I\times\mathbb{R}^{d})}\leq\eta,$
(3.1)
then there exists a unique solution $u$ to (1.1) that obeys the following
bounds:
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{p+2}L_{x}^{\frac{2d(p+2)}{2(d-2)+dp}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq 2\eta,$
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{0}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\eta^{p+1},$
$\displaystyle\big{\|}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\big{\|}u_{0}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}.$
###### Remark 3.2.
By Strichartz, we have
$\big{\|}|\nabla|^{s_{c}}e^{it\Delta}u_{0}\big{\|}_{L_{t}^{p+2}L_{x}^{\frac{2d(p+2)}{2(d-2)+dp}}(I\times\mathbb{R}^{d})}\lesssim\big{\|}|\nabla|^{s_{c}}u_{0}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})},$
so that (3.1) holds with $I=\mathbb{R}$ for sufficiently small initial data.
One can also guarantee that (3.1) holds by taking $|I|$ sufficiently small
(cf. monotone convergence).
We now turn to the question of stability for (1.1). We will prove a stability
result for $(d,s_{c})$ satisfying (1.5), in which case we always have $p\geq
1$. As we will see, this assumption allows for a very simple stability theory.
On the other hand, when $p<1$, developing a stability theory can become quite
delicate. For a discussion in the energy-critical case, see [31, Section 3.4]
and the references cited therein. See also [33] for a stability theory in the
energy-supercritical regime, as well as [21] for a stability theory in the
inter-critical regime in high dimensions.
Following the arguments in [31], we begin with the following
###### Lemma 3.3 (Short-time perturbations).
Fix $(d,s_{c})$ satisfying (1.5). Let $I$ be a compact interval and
$\tilde{u}:I\times\mathbb{R}^{d}\to\mathbb{C}$ a solution to
$(i\partial_{t}+\Delta)\tilde{u}=|\tilde{u}|^{p}\tilde{u}+e$
for some function $e$. Assume that
$\big{\|}\tilde{u}\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})}\leq
E.$ (3.2)
Let $t_{0}\in I$ and $u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$. Then there
exist $\varepsilon_{0},$ $\delta>0$ (depending on $E$) such that for all
$0<\varepsilon<\varepsilon_{0}$, if
$\displaystyle\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq\delta,$ (3.3)
$\displaystyle\big{\|}u_{0}-\tilde{u}(t_{0})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}$
$\displaystyle\leq\varepsilon,$ (3.4)
$\displaystyle\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\leq\varepsilon,$ (3.5)
then there exists $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ solving
$(i\partial_{t}+\Delta)u=|u|^{p}u$ with $u(t_{0})=u_{0}$ satisfying
$\displaystyle\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\varepsilon,$ (3.6)
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim E,$ (3.7)
$\displaystyle\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(I)}$
$\displaystyle\lesssim\varepsilon.$ (3.8)
###### Proof.
We prove the lemma under the additional hypothesis $u_{0}\in
L_{x}^{2}(\mathbb{R}^{d})$; this allows us (by Theorem 3.1) to find a solution
$u$, so that we are left to prove all of the estimates as a priori estimates.
Once the lemma is proven, we can use approximation by
$H_{x}^{s_{c}}(\mathbb{R}^{d})$ functions (along with the lemma itself) to see
that the lemma holds for $u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$.
Define $w=u-\tilde{u}$, so that
$(i\partial_{t}+\Delta)w=|u|^{p}u-|\tilde{u}|^{p}\tilde{u}-e$. Without loss of
generality, assume $t_{0}=\inf I$, and define
$A(t)=\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}([t_{0},t))}.$
We first note that by Duhamel, Strichartz, (3.4), and (3.5), we get
$\displaystyle\big{\|}$
$\displaystyle|\nabla|^{s_{c}}w\big{\|}_{S^{0}([t_{0},t))}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}w(t_{0})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}([t_{0},t))}+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\lesssim\varepsilon+A(t).$ (3.9)
Using this fact, together with Lemma 2.5, (3.3), and Sobolev embedding, we can
estimate (with all spacetime norms over $[t_{0},t)\times\mathbb{R}^{d}$)
$\displaystyle A(t)$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(|\tilde{u}+w|^{p}(\tilde{u}+w)-|\tilde{u}|^{p}\tilde{u})\big{\|}_{L_{t}^{\frac{p(d+2)}{2(p+1)}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+6dp-8}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}}\big{\|}w\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{p}$
$\displaystyle\ \ \ \ \ \ \
+\big{\|}|\nabla|^{s_{c}}w\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}}\big{\|}\tilde{u}+w\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{p}$
$\displaystyle\lesssim\delta[\varepsilon+A(t)]^{p}+[\varepsilon+A(t)][\delta^{p}+(\varepsilon+A(t))^{p}].$
Thus, recalling $p\geq 1$ and choosing $\delta,\varepsilon$ sufficiently
small, we conclude $A(t)\lesssim\varepsilon$ for all $t\in I$, which gives
(3.8). Combining (3.8) with (3.9), we also get (3.6). Finally, we can prove
(3.7) as follows: by Strichartz, (3.6), (3.2), (3.5), (3.3), the fractional
chain rule, and Sobolev embedding,
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I)}+\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\varepsilon+\big{\|}|\nabla|^{s_{c}}\tilde{u}(t_{0})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s_{c}}(|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(I)}+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\lesssim\varepsilon+E+\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}(I\times\mathbb{R}^{d})}\big{\|}\widetilde{u}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(I\times\mathbb{R}^{d})}^{p}$
$\displaystyle\lesssim E+\varepsilon+\delta^{p+1}$ $\displaystyle\lesssim E$
for $\varepsilon$ and $\delta$ sufficiently small depending on $E$. ∎
With Lemma 3.3 established, we now turn to
###### Theorem 3.4 (Stability).
Fix $(d,s_{c})$ satisfying (1.5). Let $I$ be a compact time interval and
$\tilde{u}:I\times\mathbb{R}^{d}\to\mathbb{C}$ a solution to
$(i\partial_{t}+\Delta)\tilde{u}=|\tilde{u}|^{p}\tilde{u}+e$
for some function $e$. Assume that
$\displaystyle\big{\|}\tilde{u}\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq E,$ (3.10) $\displaystyle S_{I}(\tilde{u})$
$\displaystyle\leq L.$ (3.11)
Let $t_{0}\in I$ and $u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$. Then there
exists $\varepsilon_{1}=\varepsilon_{1}(E,L)$ such that if
$\displaystyle\big{\|}u_{0}-\tilde{u}(t_{0})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}$
$\displaystyle\leq\varepsilon,$ (3.12)
$\displaystyle\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\leq\varepsilon$ (3.13)
for some $0<\varepsilon<\varepsilon_{1}$, then there exists a solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to $(i\partial_{t}+\Delta)u=|u|^{p}u$
with $u(t_{0})=u_{0}$ satisfying
$\displaystyle\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I)}$
$\displaystyle\leq C(E,L)\varepsilon,$ (3.14)
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$ $\displaystyle\leq
C(E,L).$ (3.15)
###### Proof.
Once again, we may assume $t_{0}=\inf I$. To begin, we let $\eta>0$ be a small
parameter to be determined shortly. By (3.11), we may subdivide $I$ into
(finitely many, depending on $\eta$ and $L$) intervals $J_{k}=[t_{k},t_{k+1})$
so that
$\big{\|}\tilde{u}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(J_{k}\times\mathbb{R}^{d})}\sim\eta$
for each $k$. Then by Strichartz, (3.10), (3.13), and the fractional chain
rule, we have
$\displaystyle\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(J_{k})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}\tilde{u}(t_{k})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s_{c}}(|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(J_{k})}+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(J_{k})}$
$\displaystyle\lesssim
E+\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}\big{\|}\tilde{u}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(J_{k}\times\mathbb{R}^{d})}^{p}+\varepsilon$
$\displaystyle\lesssim
E+\varepsilon+\eta^{p}\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}.$
Thus for $\varepsilon\leq E$ and $\eta$ sufficiently small, we find
$\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(J_{k})}\lesssim E.$
Adding these bounds, we find
$\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}\leq C(E,L).$ (3.16)
Now, we take $\delta>0$ as in Lemma 3.3 and subdivide $I$ into finitely many,
say $J_{0}=J_{0}(C(E,L),\delta)$ intervals $I_{j}=[t_{j},t_{j+1})$ so that
$\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}(I_{j}\times\mathbb{R}^{d})}\leq\delta$
for each $j$. We now wish to proceed inductively. We may apply Lemma 3.3 on
each $I_{j}$, provided we can guarantee
$\big{\|}u(t_{j})-\tilde{u}(t_{j})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}\leq\varepsilon$
(3.17)
for some $0<\varepsilon<\varepsilon_{0}$ and each $j$ (where $\varepsilon_{0}$
is as in Lemma 3.3). In the event that (3.17) holds for some $j$, applying
Lemma 3.3 on $I_{j}=[t_{j},t_{j+1})$ gives
$\displaystyle\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I_{j})}$
$\displaystyle\leq C(j)\varepsilon,$ (3.18)
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I_{j})}$
$\displaystyle\leq C(j)E,$ (3.19)
$\displaystyle\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(I_{j})}$
$\displaystyle\leq C(j)\varepsilon.$ (3.20)
Now, we first note that (3.17) holds for $j=0$, provided we take
$\varepsilon_{1}<\varepsilon_{0}$. Next, assuming that (3.17) holds for $0\leq
k\leq j-1$, we can use Strichartz, (3.12), (3.13), and the inductive
hypothesis (3.20) to estimate
$\displaystyle\big{\|}u$
$\displaystyle(t_{j})-\tilde{u}(t_{j})\big{\|}_{\dot{H}^{s_{c}}_{x}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}u(t_{0})-\tilde{u}(t_{0})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}\\!+\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}([t_{0},t_{j}))}\\!+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}([t_{0},t_{j}))}$
$\displaystyle\lesssim\varepsilon+\sum_{k=0}^{j-1}C(k)\varepsilon+\varepsilon$
$\displaystyle<\varepsilon_{0},$
provided $\varepsilon_{1}=\varepsilon_{1}(\varepsilon_{0},J_{0})$ is taken
sufficiently small. Thus, by induction, we get (3.18) and (3.19) on each
$I_{j}$. Adding these bounds over the $I_{j}$ yields (3.14) and (3.15). ∎
###### Remark 3.5.
As mentioned above, with this stability result in hand, we can see that
Theorem 3.1 holds without the assumption $u_{0}\in L_{x}^{2}(\mathbb{R}^{d})$.
Using this updated version of Theorem 3.1 (along with the original proof of
Theorem 3.1), one can then derive Theorem 1.4. We omit the standard arguments;
one can refer instead to [8, 9].
## 4\. Reduction to almost periodic solutions
The goal of this section is to sketch a proof of Theorem 1.12. We will follow
the argument presented in [32, Section 3]. By now, the general procedure is
fairly standard; see, for example, [25, 26, 27, 31, 53] for other instances in
different contexts. Thus, we will merely outline the main steps of the
argument, providing full details only when significant new difficulties arise
in our setting.
We suppose that Theorem 1.3 fails for some $(d,s_{c})$ satisfying (1.5). We
then define the function $L:[0,\infty)\to[0,\infty]$ by
$\displaystyle
L(E):=\sup\\{S_{I}(u):u:I\\!\times\\!\mathbb{R}^{d}\to\mathbb{C}\text{ solving
}\eqref{nls}\text{ with }\sup_{t\in
I}\big{\|}u(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{2}\leq E\\},$
where $S_{I}(u)$ is defined as in (1.2). We note that $L$ is a non-decreasing
function, and that Theorem 1.4 implies
$L(E)\lesssim E^{\frac{p(d+2)}{4}}\indent\text{for}\indent E<\eta_{0},$ (4.1)
where $\eta_{0}$ is the small-data threshold. Thus, there exists a unique
‘critical’ threshold $E_{c}\in(0,\infty]$ such that $L(E)<\infty$ for
$E<E_{c}$ and $L(E)=\infty$ for $E>E_{c}$. The failure of Theorem 1.3 implies
that $0<E_{c}<\infty.$
The key ingredient to proving Theorem 1.12 is a Palais–Smale condition modulo
the symmetries of the equation; indeed, once the following proposition is
proven, deriving Theorem 1.12 is standard (see [32]).
###### Proposition 4.1 (Palais–Smale condition modulo symmetries).
Let $(d,s_{c})$ satisfy (1.5). Let
$u_{n}:I_{n}\times\mathbb{R}^{d}\to\mathbb{C}$ be a sequence of solutions to
(1.1) such that
$\limsup_{n\to\infty}\sup_{t\in
I_{n}}\big{\|}u_{n}(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{2}=E_{c},$
and suppose $t_{n}\in I_{n}$ are such that
$\lim_{n\to\infty}S_{[t_{n},\sup I_{n})}(u_{n})=\lim_{n\to\infty}S_{(\inf
I_{n},t_{n}]}(u_{n})=\infty.$ (4.2)
Then the sequence $u_{n}(t_{n})$ has a subsequence that converges in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ modulo symmetries; that is, there exist
$g_{n}\in G$ such that $g_{n}[u_{n}(t_{n})]$ converges along a subsequence in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, where $G$ is as in Definition 2.12.
We now sketch the proof of this proposition, following the argument as it
appears in [32]. As in that setting, the main ingredients will be a linear
profile decomposition (Lemma 2.14 in our case) and a stability result (Theorem
3.4 in our case). However, as we will see, combining fractional derivatives
with non-polynomial nonlinearities will present some significant new
difficulties in our setting when it comes time to apply the stability result.
We begin by translating so that each $t_{n}=0$, and apply the linear profile
decomposition Lemma 2.14 to write
$u_{n}(0)=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}$ (4.3)
along some subsequence. Refining the subsequence for each $j$ and
diagonalizing, we may assume that for each $j$, we have $t_{n}^{j}\to
t^{j}\in[-\infty,\infty].$ If $t^{j}\in(-\infty,\infty)$, then we replace
$\phi^{j}$ by $e^{it^{j}\Delta}\phi^{j}$, so that we may take $t^{j}=0$.
Moreover, we can absorb the error $e^{it_{n}^{j}\Delta}\phi^{j}-\phi^{j}$ into
the error term $w_{n}^{J}$, and so we may take $t_{n}^{j}\equiv 0$. Thus,
without loss of generality, either $t_{n}^{j}\equiv 0$ or
$t_{n}^{j}\to\pm\infty$.
Next, appealing to Theorem 1.4, for each $j$ we define
$v^{j}:I^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ to be the maximal-lifespan
solution to (1.1) such that
$\left\\{\begin{array}[]{ll}v^{j}(0)=\phi^{j}&\text{if }t_{n}^{j}\equiv 0,\\\
\\\ v^{j}\text{ scatters to }\phi^{j}\text{ as }t\to\pm\infty&\text{if
}t_{n}^{j}\to\pm\infty.\end{array}\right.$
We now define the nonlinear profiles
$v_{n}^{j}:I_{n}^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ by
$v_{n}^{j}(t)=g_{n}^{j}v^{j}\left((\lambda_{n}^{j})^{-2}t+t_{n}^{j}\right),$
where $I_{n}^{j}=\\{t:(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\in I^{j}\\}.$ Now, the
$\dot{H}_{x}^{s_{c}}$ decoupling of the profiles $\phi^{j}$, (2.5),
immediately tells us that the $v_{n}^{j}$ are global and scatter for $j$
sufficiently large, say $j\geq J_{0}$; indeed, for large enough $j$, we are in
the small-data regime. We want to show that there exists some $1\leq
j_{0}<J_{0}$ such that
$\limsup_{n\to\infty}S_{[0,\sup I_{n}^{j_{0}})}(v_{n}^{j_{0}})=\infty.$ (4.4)
Once we obtain at least one such ‘bad’ nonlinear profile, we can show that in
fact, there can only be one profile. To see this, one needs to adapt the
argument in [32, Lemma 3.3] to see that the $\dot{H}_{x}^{s_{c}}$ decoupling
of the profiles persists in time (this does not follow immediately, as the
$\dot{H}_{x}^{s_{c}}$-norm is not a conserved quantity for (1.1)). Then, the
‘critical’ nature of $E_{c}$ can be used to rule out the possibility of
multiple profiles.
Comparing with (4.3), one sees that once we show that there is only one
profile $\phi^{j_{0}}$, the proof of Proposition 4.1 is nearly complete; one
needs only to rule out the cases $t_{n}^{j_{0}}\to\pm\infty$. This can be
easily done by applying the stability theory; we omit the details and instead
refer the reader to [32].
We turn now to proving that there is at least one bad profile. We suppose
towards a contradiction that there are no bad nonlinear profiles. In this
case, we can show
$\sum_{j\geq 1}S_{[0,\infty)}(v_{n}^{j})\lesssim_{E_{c}}1$ (4.5)
for $n$ sufficiently large (to control the tail of the sum, for example, we
recall that for $j\geq J_{0}$, we are in the small-data regime; thus we can
use (2.5) and (4.1) to bound the tail by $E_{c}$ for $n$ sufficiently large).
We would like to use (4.5) and the stability result (Theorem 3.4) to deduce a
bound on the scattering size of the $u_{n}$, thus contradicting (4.2).
To this end, we define the approximations
$u_{n}^{J}(t)=\sum_{j=1}^{J}v_{n}^{j}(t)+e^{it\Delta}w_{n}^{J}.$
By the construction of the $v_{n}^{j}$, it is easy to see that
$\limsup_{n\to\infty}\big{\|}u_{n}(0)-u_{n}^{J}(0)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}=0.$
(4.6)
We also claim that we have
$\lim_{J\to\infty}\limsup_{n\to\infty}S_{[0,\infty)}(u_{n}^{J})\lesssim_{E_{c}}1.$
(4.7)
To see why (4.7) holds, first note that by (2.3) and (4.5), it suffices to
show
$\lim_{J\to\infty}\limsup_{n\to\infty}\bigg{|}S_{[0,\infty)}\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}-\sum_{j=1}^{J}S_{[0,\infty)}(v_{n}^{j})\bigg{|}=0.$
(4.8)
To establish (4.8), we can first use the pointwise inequality
$\bigg{|}\
\bigg{|}\sum_{j=1}^{J}v_{n}^{j}\bigg{|}^{\frac{p(d+2)}{2}}-\sum_{j=1}^{J}|v_{n}^{j}|^{\frac{p(d+2)}{2}}\bigg{|}\lesssim_{J}\sum_{j\neq
k}|v_{n}^{j}|^{\frac{p(d+2)}{2}-1}|v_{n}^{k}|$
along with Hölder’s inequality to see
$\displaystyle\text{LHS}\eqref{ok version3}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}([0,\infty)\times\mathbb{R}^{d})}^{\frac{p(d+2)}{2}-2}\big{\|}v_{n}^{j}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{4}}([0,\infty)\times\mathbb{R}^{d})}.$
(4.9)
Then, following an argument of Keraani (cf. [28, Lemma 2.7]), given $j\neq k$,
we can approximate $v^{j}$ and $v^{k}$ by compactly supported functions in
$\mathbb{R}\times\mathbb{R}^{d}$ and use the asymptotic orthogonality of
parameters (2.4) to show
$\limsup_{n\to\infty}\big{\|}v_{n}^{j}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{4}}([0,\infty)\times\mathbb{R}^{d})}=0.$
(4.10)
Thus, continuing from (4.9), we get that (4.8) (and therefore (4.7)) holds.
With (4.6) and (4.7) in place, we see that if we can show that the $u_{n}^{J}$
asymptotically solve (1.1), that is,
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{[}(i\partial_{t}+\Delta)u_{n}^{J}-|u_{n}^{J}|^{p}u_{n}^{J}\big{]}\big{\|}_{N^{0}([0,\infty))}=0,$
then we will be able to apply Theorem 3.4 to deduce bounds on the scattering
size of the $u_{n}$. Writing $F(z)=|z|^{p}z$, the proof of Proposition 4.1
therefore reduces to showing the following
###### Lemma 4.2 (Decoupling of nonlinear profiles).
$\lim_{J\to\infty}\limsup_{n\to\infty}\bigg{\|}|\nabla|^{s_{c}}\bigg{(}F\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}-\sum_{j=1}^{J}F(v_{n}^{j})\bigg{)}\bigg{\|}_{N^{0}([0,\infty))}=0,$
(4.11)
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\left(F(u_{n}^{J}-e^{it\Delta}w_{n}^{J})-F(u_{n}^{J})\right)\big{\|}_{N^{0}([0,\infty))}=0.$
(4.12)
While many of the ideas needed to establish this lemma may be found in [28],
we will see that new difficulties appear in our setting. Consider, for
example, (4.11). In the mass-critical setting, i.e. $s_{c}=0$, one has the
pointwise estimate
$\bigg{|}\
F\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}-\sum_{j=1}^{J}F(v_{n}^{j})\bigg{|}\lesssim_{J}\sum_{j\neq
k}|v_{n}^{j}|\,|v_{n}^{k}|^{p}.$ (4.13)
To see that the contribution of the terms on the the right-hand side of (4.13)
is acceptable, one can follow the argument of Keraani just described above;
that is, one can use the asymptotic orthogonality of parameters to derive an
estimate like (4.10), which in turn gives (4.11).
In the energy-critical setting, i.e. $s_{c}=1$, one can instead use the
pointwise estimate
$\bigg{|}\nabla\bigg{(}F\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}-\sum_{j=1}^{J}F(v_{n}^{j})\bigg{)}\bigg{|}\lesssim_{J}\sum_{j\neq
k}|\nabla v_{n}^{j}|\,|v_{n}^{k}|^{p}.$
A similar argument can then be used to prove (4.11); the key in both cases is
to exhibit terms that all contain some $v_{n}^{j}$ paired against some
$v_{n}^{k}$ for $j\neq k$.
In the energy-supercritical case, the authors of [33] were able to establish
analogous pointwise estimates (in terms of the Hardy–Littlewood maximal
function) for a square function of Strichartz that shares estimates with
fractional differentiation operators (see [46]). With the appropriate
pointwise estimates in place, the usual arguments can then be applied; in this
way, a potentially complicated analysis is handled quite efficiently. The
approach of [33], however, does not work in our setting, as it relies
fundamentally on the fact that $s_{c}>1$.
See also [26], which deals with the case $d=3$ and $s_{c}=\tfrac{1}{2}$ (in
which case $p=2$). In that setting, one also has to face the nonlocal nature
of $|\nabla|^{\frac{1}{2}}$; however, by using the polynomial nature of the
nonlinearity, along with the well-developed theory of paraproducts (see [12,
48]), the authors are able to place themselves back into a situation where the
usual arguments apply. In this way, they are able to overcome the difficulty
of fractional derivatives while still providing a very clean analysis.
In our case, we must deal simultaneously with a non-polynomial nonlinearity
and a fractional number of derivatives; as we will see, this necessitates a
fairly delicate and technical analysis. The main difficulty of our task stems
from the fact the nonlocal operator $|\nabla|^{s_{c}}$ does not respect
pointwise estimates in the spirit of (4.13). We will deal with this problem by
opening up the proof of the fractional chain rule (Lemma 2.4) as given in [48,
$\S 2.4$]; in particular, we will employ the Littlewood–Paley square function
(specifically, Lemma 2.2), which allows us to work at the level of individual
frequencies. By making use of maximal function and vector maximal function
estimates, we can then find a way to adapt the standard arguments.
###### Proof of (4.11).
By induction, it will suffice to treat the case of two summands; to simplify
notation, we write $f=v_{n}^{j}$ and $g=v_{n}^{k}$ for some $j\neq k$, and we
are left to show
$\displaystyle\big{\|}|\nabla|^{s_{c}}\big{(}|f+g|^{p}(f+g)-|f|^{p}f-|g|^{p}g\big{)}\big{\|}_{N^{0}([0,\infty))}\to
0$ (4.14)
as $n\to\infty$.
As alluded to above, the key will be to perform a decomposition in such a way
that all of the resulting terms we need to estimate have $f$ paired against
$g$ inside of a single integrand; for such terms, we will be able to use the
asymptotic orthogonality of parameters (2.4) to our advantage.
We first rewrite
$\displaystyle|f+g$ $\displaystyle|^{p}(f+g)-|f|^{p}f-|g|^{p}g$
$\displaystyle=\big{(}|f+g|^{p}-|f|^{p}\big{)}f+\big{(}|f+g|^{p}-|g|^{p}\big{)}g.$
By symmetry, it will suffice to treat the first term. We turn therefore to
estimating
$\big{\|}|\nabla|^{s_{c}}\big{[}(|f+g|^{p}-|f|^{p})f\big{]}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
By Lemma 2.2, it will suffice to consider
$\bigg{\|}\left(\sum\big{|}N^{s_{c}}P_{N}\big{[}(|f+g|^{p}-|f|^{p})f\big{]}\big{|}^{2}\right)^{1/2}\bigg{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
(4.15)
Thus, we restrict our attention to a single frequency $N\in 2^{\mathbb{Z}}$.
We let $\delta_{y}f(x):=f(x-y)-f(x)$, and let $\check{\psi}$ denote the
convolution kernel of the Littlewood–Paley projection $P_{1}$. As $\psi(0)=0$,
we have
$\textstyle\int\check{\psi}(y)\,dy=0,$
so that exploiting cancellation, we can write
$\displaystyle
P_{N}\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}$
$\displaystyle\quad=\textstyle\int
N^{d}\check{\psi}(Ny)\delta_{y}\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}\,dy.$
(4.16)
We now rewrite
$\displaystyle\delta_{y}$
$\displaystyle\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}$
$\displaystyle=\delta_{y}f(x)\big{[}|f(x-y)+g(x-y)|^{p}-|f(x-y)|^{p}\big{]}$
(4.17) $\displaystyle\
+\\!f(x)\big{[}|f(x)+g(x-y)|^{p}-|f(x)+g(x)|^{p}\big{]}$ (4.18)
$\displaystyle\
+\\!f(x)\big{[}|f(x-y)\\!+\\!g(x-y)|^{p}\\!-\\!|f(x-y)|^{p}\\!+\\!|f(x)|^{p}\\!-\\!|f(x)\\!+\\!g(x-y)|^{p}\big{]}.$
(4.19)
We estimate each term individually. First, we have
$\displaystyle|\eqref{expansion1}|\lesssim|\delta_{y}f(x)|\,|g(x-y)|\left\\{|f(x-y)|^{p-1}+|g(x-y)|^{p-1}\right\\}.$
Next, we see
$\displaystyle|\eqref{expansion2}|\lesssim|f(x)|\,|\delta_{y}g(x)|\left\\{|f(x)|^{p-1}+|g(x)|^{p-1}+|g(x-y)|^{p-1}\right\\}.$
We now turn to (4.19). First, if $1<p\leq 2$, a simple argument using the
fundamental theorem of calculus implies
$\displaystyle|\eqref{expansion3}|\lesssim|f(x)|\,|\delta_{y}f(x)|\,|g(x-y)|^{p-1}$
(see Lemma A.1 for details). For $p>2$, one instead finds
$\displaystyle|\eqref{expansion3}|\lesssim|f(x)|\,|\delta_{y}f(x)|\,|g(x-y)|\left\\{|f(x)|^{p-2}+|f(x-y)|^{p-2}+|g(x-y)|^{p-2}\right\\}.$
###### Remark 4.3.
Let us pause here to note that if $p=1$, the approach above breaks down.
Notice that each term in the bounds for (4.17), (4.18), and (4.19) has two
essential properties: (i) it features $f$ paired against powers of $g$, and
(ii) the derivative (in the form of $\delta_{y}$) lands on either $f$ or $g$.
When $p=1$, the same approach does not yield a decomposition that is
satisfactory in this sense; it is for this reason that we have excluded the
case $(d,s_{c})=(5,\tfrac{1}{2})$ from this paper.
To ease the exposition, we will restrict our attention here and below to the
more difficult case $1<p\leq 2$; once we have dealt with this case, it should
be clear how to proceed when $p>2$.
Collecting terms, we continue from (4.16) to see
$\displaystyle\big{|}$ $\displaystyle
P_{N}\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}\big{|}$
$\displaystyle\lesssim\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}f(x)|\,|g(x-y)|\left\\{|f(x-y)|^{p-1}+|g(x-y)|^{p-1}\right\\}\\!\,dy$
(4.20) $\displaystyle\ +\int
N^{d}|\check{\psi}(Ny)|\,|f(x)|\,|\delta_{y}g(x)|\left\\{|f(x)|^{p-1}\\!+\\!|g(x)|^{p-1}\\!+\\!|g(x-y)|^{p-1}\right\\}\,\\!dy$
(4.21) $\displaystyle\ +\int
N^{d}|\check{\psi}(Ny)|\,|f(x)|\,|\delta_{y}f(x)||g(x-y)|^{p-1}\,dy.$ (4.22)
One can see that we are already faced with several terms to estimate;
moreover, to estimate any single term will require further decomposition.
However, in the end, the same set of tools will suffice to handle every term
that appears. Thus, let us deal with only (4.20) in detail; once we have seen
how to handle this term, it should be clear that the same techniques apply to
handle (4.21) and (4.22).
Turning to (4.20), we first write
$\displaystyle\eqref{11}$ $\displaystyle=\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}f(x)|\,|g(x-y)|\,|f(x-y)|^{p-1}\,dy$
(4.23) $\displaystyle\ \ +\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}f(x)|\,|g(x-y)|^{p}\,dy.$ (4.24)
For both of these terms, we will need to make use of some auxiliary
inequalities in the spirit of [48, $\S$2.3], which we record in Lemma A.2.
We turn to (4.23). If we first write
$\displaystyle|\delta_{y}$ $\displaystyle
f(x)|\lesssim|f_{>N}(x)|+|f_{>N}(x-y)|+\sum_{K\leq N}|\delta_{y}f_{K}(x)|,$
(4.25)
then putting Lemma A.2 to use, we arrive at
$\displaystyle\eqref{11f}\lesssim$ $\displaystyle\
|f_{>N}(x)|\,M(g\,|f|^{p-1})(x)$ (4.26)
$\displaystyle+M(f_{>N}\,g\,|f|^{p-1})(x)$ (4.27) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(f_{K})(x)M(g\,|f|^{p-1})(x)$ (4.28) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(M(f_{K})\,g\,|f|^{p-1})(x).$ (4.29)
Similarly, we can decompose
$\displaystyle\eqref{11g}\lesssim$ $\displaystyle\ |f_{>N}(x)|\,M(|g|^{p})(x)$
(4.30) $\displaystyle+M(f_{>N}|g|^{p})(x)$ (4.31) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(f_{K})(x)M(|g|^{p})(x)$ (4.32) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(M(f_{K})|g|^{p})(x).$ (4.33)
Let us now consider the contribution of (4.26) to the left-hand side of
(4.14). Comparing with (4.15), we see it will suffice to estimate
$\displaystyle\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}f_{>N}M(g\,|f|^{p-1})\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
Using Hölder’s inequality and maximal function estimates, we can control this
term by
$\displaystyle\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}f_{>N}\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}|g|\,|f|^{p-1}\big{\|}_{L_{t,x}^{\frac{d+2}{2}}}.$
We now recall that $f=v_{n}^{j}$ and $g=v_{n}^{k}$ for some $j\neq k$. Then,
the first term is controlled by
$\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{S^{0}}$ (cf. Lemma 2.2), which in
turn is bounded (recall that by assumption, all of the $v_{n}^{j}$ have
scattering size $\lesssim E_{c}$). The second term can be handled in the
standard way; that is, this term vanishes in the limit due to the asymptotic
orthogonality of parameters (2.4) (cf. [28, Lemma 2.7]). Thus, we see that
(4.26) is under control. A similar approach (this time using the vector
maximal inequality) handles (4.27).
To estimate the contribution of (4.28) to the left-hand side of (4.14), we
need to estimate
$\displaystyle\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}\sum_{K\leq
N}\tfrac{K}{N}M(f_{K})M(g\,|f|^{p-1})\big{|}^{2}\big{)}^{1/2}\big{\|}_{N^{0}([0,\infty))}.$
(4.34)
For this term, we need to make use of the following basic inequality: for a
nonnegative sequence $\\{a_{K}\\}_{K\in 2^{\mathbb{Z}}}$ and $0<s<1$, one has
$\sum_{N\in 2^{\mathbb{Z}}}N^{2s}\big{|}\sum_{K\leq
N}\tfrac{K}{N}a_{K}\big{|}^{2}\lesssim\sum_{K\in
2^{\mathbb{Z}}}K^{2s}|a_{K}|^{2}$ (4.35)
(cf. [48, Lemma 4.2]). Using this inequality, along with Hölder, we can
estimate
$\displaystyle\eqref{113 again}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{K}\big{|}K^{s_{c}}M(f_{K})\big{|}^{2}\big{)}^{1/2}M(g\,|f|^{p-1})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{K}|K^{s_{c}}M(f_{K})|^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}|g|\,|f|^{p-1}\big{\|}_{L_{t,x}^{\frac{d+2}{2}}}\to
0$
as $n\to\infty$, exactly as before. Thus, (4.28) is under control; the same
approach handles (4.29) (after an application of the vector maximal
inequality).
Let us now turn to (4.30). As before, we sum over $N\in 2^{\mathbb{Z}}$ and
find that we need to estimate
$\bigg{\|}\left(\sum\big{|}N^{s_{c}}f_{>N}\big{|}^{2}\right)^{1/2}M(|g|^{p})\bigg{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
(4.36)
Recalling that $f=v_{n}^{j}$ and $g=v_{n}^{k}$ for some $j\neq k$, we see that
we are once again in a position to use the argument from [28]. To begin, we
may assume without loss of generality that both
$\Phi_{1}:=\left(\sum|N^{s_{c}}P_{>N}v^{j}|^{2}\right)^{1/2}\quad\text{and}\quad\Phi_{2}:=M(|v^{k}|^{p})$
belong to $C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$; indeed,
$C_{c}^{\infty}$-functions are dense in both $L_{t,x}^{\frac{2(d+2)}{d}}$ and
$L_{t,x}^{\frac{d+2}{2}}.$ We now wish to use the asymptotic orthogonality of
parameters, that is,
$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}+\frac{\lambda_{n}^{k}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{k}|^{2}}{\lambda_{n}^{j}\lambda_{n}^{k}}+\frac{|t_{n}^{j}(\lambda_{n}^{j})^{2}-t_{n}^{k}(\lambda_{n}^{k})^{2}|}{\lambda_{n}^{j}\lambda_{n}^{k}}\to\infty\quad\text{as}\
n\to\infty,$ (4.37)
to show (4.36)$\to 0$.
Consider first the case $\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\to c>0$
(along a subsequence, say). If we unravel the definition of the nonlinear
profiles and change variables to move the symmetries onto $\Phi_{2}$, we
arrive at
$\displaystyle\eqref{115again}^{\frac{2(d+2)}{d+4}}$
$\displaystyle=\left(\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\right)^{\frac{4(d+2)}{d+4}}\iint\bigg{|}\Phi_{1}(s,y)\Phi_{2}(\big{(}t_{n}^{k}+\big{(}\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\big{)}^{2}(s-t_{n}^{j}),\big{(}\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\big{)}y+\tfrac{x_{n}^{j}-x_{n}^{k}}{\lambda_{n}^{k}}\big{)}\bigg{|}^{\frac{2(d+2)}{d+4}}\,dy\,ds.$
Then, recalling (4.37), we see that as $n\to\infty$, either the spatial or
temporal argument of $\Phi_{2}$ must escape the support of $\Phi_{1}$. Thus,
in this case, we get (4.36)$\to 0$.
If instead we have $\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\to 0$, then
continuing from above, we can estimate
$\displaystyle\eqref{115again}$
$\displaystyle\lesssim\big{(}\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\big{)}^{2}\big{\|}\Phi_{1}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}\big{\|}\Phi_{2}\big{\|}_{L_{t,x}^{\infty}}.$
As $\Phi_{1},\Phi_{2}\in C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$, we
see that (4.36)$\to 0$ in this case, as well.
Finally, we can treat the case
$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\to\infty$ just like the previous
case; the only difference is that we change variables to move the symmetries
onto $\Phi_{1}$, instead of $\Phi_{2}$. Thus, we have that (4.36)$\to 0$ in
this third and final case.
We have now shown that (4.30) is under control. The same ideas can be used to
handle (4.31), (4.32), and (4.33).
As mentioned above, this same set of ideas suffices to deal with all the
remaining terms stemming from (4.11). ∎
###### Proof of (4.12).
For this term, we will need to make use (2.3). As we will see, the terms in
which $e^{it\Delta}w_{n}^{J}$ appears without derivatives will be relatively
easy to handle, as (2.3) will apply directly. On the other hand, the terms
that only contain $|\nabla|^{s_{c}}e^{it\Delta}w_{n}^{J}$ will require a more
careful analysis; in particular, we will need to carry out a local smoothing
argument before we can make effective use of (2.3).
Defining $g:=\sum_{j=1}^{J}v_{n}^{j}$ and $h:=e^{it\Delta}w_{n}^{J}$, we are
left to show
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\left(|g+h|^{p}(g+h)-|g|^{p}g\right)\big{\|}_{N^{0}([0,\infty))}=0.$
(4.38)
We write
$\displaystyle|g+h|^{p}(g+h)-|g|^{p}g=$ $\displaystyle\ |g+h|^{p}h$ (4.39)
$\displaystyle+(|g+h|^{p}-|g|^{p})g$ (4.40)
and first restrict our attention to (4.39). We proceed as before, working at a
single frequency and exploiting cancellation to write
$\displaystyle|P_{N}(|g+h|^{p}h)(x)|$ $\displaystyle=\big{|}\textstyle\int
N^{d}\check{\psi}(Ny)\delta_{y}\big{[}|g(x)+h(x)|^{p}h(x)\big{]}\,dy\big{|}$
$\displaystyle\leq\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|\delta_{y}h(x)|\,dy$ (4.41)
$\displaystyle\quad+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}\big{[}|g(x)+h(x)|^{p}\big{]}\,|h(x)|\,dy.$
(4.42)
We will deal only with (4.41), which is the more difficult term. Indeed, in
all of the terms that stem from (4.42), we will have a copy of
$e^{it\Delta}w_{n}^{J}$ appearing without derivatives, so that (2.3) will
suffice. (For completeness, we will later show how to handle such a term; cf.
(4.54) below.)
Proceeding as in (4.25), we write
$\displaystyle\eqref{221}$ $\displaystyle\lesssim\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|h_{>N}(x)|\,dy$ (4.43)
$\displaystyle+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|h_{>N}(x-y)|\,dy$ (4.44)
$\displaystyle+\sum_{K\leq N}\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|\delta_{y}h_{K}(x)|\,dy.$
(4.45)
Let us now deal only with (4.45); in doing so, we will see all of the ideas
necessary to handle (4.43) and (4.44), as well. We first write
$\displaystyle\eqref{2213}$ $\displaystyle\lesssim\sum_{K\leq N}\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)|^{p}|\delta_{y}h_{K}(x)|\,dy$ (4.46)
$\displaystyle+\sum_{K\leq N}\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|h(x-y)|^{p}|\delta_{y}h_{K}(x)|\,dy.$ (4.47)
We only consider (4.46), as the contribution of (4.47) is easier to estimate
(again, due to the presence of $e^{it\Delta}w_{n}^{J}$ without derivatives).
Employing the inequalities of Lemma A.2, we find
$\displaystyle\eqref{22131}\lesssim\sum_{K\leq
N}\tfrac{K}{N}M(|g|^{p})(x)M(h_{K})(x)+\sum_{K\leq
N}\tfrac{K}{N}M(|g|^{p}M(h_{K}))(x).$
Let us now concern ourselves only with the first term above, as the second is
similar. As before, to estimate the contribution of this term to (4.38) (and
thereby complete our treatment of (4.39)), we need to sum over $N\in
2^{\mathbb{Z}}$. Using (4.35) and recalling the definitions of $g$ and $h$, we
write
$\displaystyle\big{\|}\big{(}$
$\displaystyle\sum_{N}\big{|}N^{s_{c}}\sum_{K\leq
N}\tfrac{K}{N}M(|g|^{p})M(h_{K})\big{|}^{2}\big{)}^{1/2}\big{\|}_{N^{0}}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(h_{N})\big{|}^{2}\big{)}^{1/2}M(|g|^{p})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})|^{2}\big{)}^{1/2}M\big{(}\big{|}\sum_{j=1}^{J}v_{n}^{j}\big{|}^{p}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
Thus, to complete our treatment of (4.39), we are left to show
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})|^{2}\big{)}^{1/2}M\big{(}\big{|}\sum_{j=1}^{J}v_{n}^{j}\big{|}^{p}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{d+2}{2}}=0.$
(4.48)
To begin, we let $\eta>0$; then using (4.5), we see that there exists some
$J_{1}=J_{1}(\eta)$ so that
$\sum_{j\geq
J_{1}}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}<\eta.$
Using Hölder’s inequality, maximal function and vector maximal function
estimates, and Lemma 2.2, we can argue as we did to obtain (4.8) to see
$\displaystyle\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}$
$\displaystyle\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})|^{2}\big{)}^{1/2}M\big{(}\big{|}\sum_{j\geq
J_{1}}v_{n}^{j}\big{|}^{p}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{d+2}{2}}$
$\displaystyle\lesssim\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{d+2}{2}}\sum_{j\geq
J_{1}}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}$
$\displaystyle\lesssim\eta.$
As $\eta>0$ was arbitrary, we see that to establish (4.48), it will suffice to
show
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\big{)}^{1/2}M(|v_{n}^{j}|^{p})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}=0$
(4.49)
for $1\leq j<J_{1}$.
Restricting our attention to a single $j$ and recalling the definition of
$v_{n}^{j}$, we change variables and find we need to estimate
$\big{\|}\big{(}\sum_{N}\big{|}(\lambda_{n}^{j})^{\frac{2}{p}}N^{s_{c}}MP_{N}\big{[}e^{i[(\lambda_{n}^{j})^{2}(t-t_{n}^{j})]\Delta}w_{n}^{J}(\lambda_{n}^{j}x+x_{n}^{j})\big{]}\big{|}^{2}\big{)}^{1/2}M(|v^{j}|^{p})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
We will now carry out some reductions, inspired by the proof of [28,
Proposition 3.4]: as $M(|v^{j}|^{p})$ shares bounds with $|v^{j}|^{p}$, and
$v^{j}$ obeys good bounds (it has scattering size $\lesssim E_{c}$), we may
replace $M(|v^{j}|^{p})$ with some function $\Phi$ in
$C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$. If we then use Hölder’s
inequality, we find it suffices to estimate the first term in
$L_{t,x}^{2}(K)$, where $K$ is the (compact) support of this function $\Phi$.
The next step will be to use a local smoothing estimate on this (fixed) set
$K$. Now, the norms that will appear in these estimates will have critical
scaling; that is, they will be invariant under the change of variables that
eliminates the parameters $\lambda_{n}^{j}$, $x_{n}^{j},$ and $t_{n}^{j}$.
Thus, without loss of generality, we will ignore them from the start.
To establish (4.49) and complete our treatment of (4.39), we are therefore
left to show
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{2}(K)}=0$
(4.50)
for a fixed compact set $K\subset\mathbb{R}\times\mathbb{R}^{d}$.
To establish (4.50), we will need to rely on the fact that we are working on a
compact set, so that we can carry out a local smoothing argument. Indeed, the
term appearing above is morally like $|\nabla|^{s_{c}}e^{it\Delta}w_{n}^{J}$,
over which we do not have sufficient control (cf. (2.6)). However, we do have
good control over $e^{it\Delta}w_{n}^{J}$, in the form of (2.3). Thus, to
succeed, we need to find a way to estimate the term above using fewer than
$s_{c}$ derivatives; this is exactly the role of local smoothing.
For the proof of (4.50), we will use a standard local smoothing result for the
free propagator (Lemma 2.9), along with a few results from [45, Chapter V]. In
particular, we need the following: if we choose $\varepsilon>0$ so that
$-d<-1-\varepsilon$, then $|x|^{-1-\varepsilon}$ is an $A_{2}$ weight, so that
$M$ is bounded on $L^{2}(|x|^{-1-\varepsilon}\,dx)$.
###### Proof of (4.50).
We can write $K\subset[-T,T]\times\\{|x|\leq R\\}$ for some $T,R>0$. We fix
some $N_{0}\in 2^{\mathbb{Z}}$ and break into low and high frequencies:
$\displaystyle\iint_{K}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt\lesssim$
$\displaystyle\sum_{N\leq
N_{0}}\iint_{K}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt$
$\displaystyle+\sum_{N>N_{0}}\iint_{K}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt.$
For the low frequencies, we use Hölder and maximal function estimates to write
$\displaystyle\sum_{N\leq N_{0}}\iint_{K}\big{|}$ $\displaystyle
M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt$
$\displaystyle\lesssim\sum_{N\leq
N_{0}}T^{\frac{p(d+2)-4}{p(d+2)}}R^{\frac{d(p(d+2)-4)}{p(d+2)}}\big{\|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{2}$
$\displaystyle\lesssim_{K}\sum_{N\leq
N_{0}}N^{2s_{c}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{2}$
$\displaystyle\lesssim_{K}N_{0}^{2s_{c}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{2}.$
For the high frequencies, we choose $\varepsilon>0$ so that
$-d<-1-\varepsilon$. Then, using Lemma 2.9, Bernstein, and the fact that
$|x|^{-1-\varepsilon}\in A_{2}$, we can estimate
$\displaystyle\sum_{N>N_{0}}\iint_{K}\big{|}$ $\displaystyle
M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt$
$\displaystyle\lesssim
R^{1+\varepsilon}\sum_{N>N_{0}}\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})|^{2}\langle
x\rangle^{-1-\varepsilon}\,dx\,dt$
$\displaystyle\lesssim_{K}\sum_{N>N_{0}}N^{2s_{c}}\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\big{|}P_{N}e^{it\Delta}w_{n}^{J}\big{|}^{2}\langle
x\rangle^{-1-\varepsilon}\,dx\,dt$
$\displaystyle\lesssim_{K}\sum_{N>N_{0}}N^{2s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}}P_{N}w_{n}^{J}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\lesssim_{K}\sum_{N>N_{0}}N^{-1}\big{\|}|\nabla|^{s_{c}}w_{n}^{J}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\lesssim_{K}N_{0}^{-1}\big{\|}|\nabla|^{s_{c}}w_{n}^{J}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}.$
Optimizing in the choice of $N_{0}$ now yields
$\big{\|}\big{(}\sum_{N}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{2}(K)}\lesssim_{K}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{1}{2s_{c}+1}}\big{\|}w_{n}^{J}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{\frac{2s_{c}}{2s_{c}+1}},$
which, by (2.3), gives (4.50). ∎
We have now dealt with (4.39), and so we finally turn to (4.40). As usual, we
first restrict our attention to a single frequency $N$. We have dealt with a
term of this form before (cf. (4.16)); proceeding in exactly the same way, we
arrive at
$\displaystyle\big{|}$ $\displaystyle
P_{N}\big{(}[|g(x)+h(x)|^{p}-|g(x)|^{p}]g(x)\big{)}\big{|}$
$\displaystyle\lesssim\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}g(x)|\,|h(x-y)|\big{\\{}|g(x-y)|^{p-1}+|h(x-y)|^{p-1}\big{\\}}\,dy$
(4.51) $\displaystyle+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x)|\,|\delta_{y}g(x)|\,|h(x-y)|^{p-1}\,dy$ (4.52)
$\displaystyle+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x)|\,|\delta_{y}h(x)|\,\big{\\{}|g(x)|^{p-1}+|h(x)|^{p-1}+|h(x-y)|^{p-1}\big{\\}}\,dy,$
(4.53)
at least in the case $p\leq 2$ (as above, we will only consider this case).
Note that all of the terms above are similar to terms we have handled before.
Thus, we proceed in the same way, decomposing terms exactly as before.
Whenever a term includes a copy of $e^{it\Delta}w_{n}^{J}$ without
derivatives, things will be relatively straightforward, as one can rely on
(2.3) (see (4.54) below for details); for the one term stemming from (4.53) in
which $e^{it\Delta}w_{n}^{J}$ only appears with derivatives, we have to go
through the same local smoothing argument given above (cf. the proof of
(4.50)).
Thus, to conclude the proof of (4.12), we will see how to estimate the
contribution of the term
$\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}g(x)|\,|h(x-y)|\,|g(x-y)|^{p-1}\,dy.$
(4.54)
Estimating $|\delta_{y}g(x)|$ as before, we find we need to bound the terms
$\displaystyle M(h|g$ $\displaystyle|^{p-1})g_{>N}+M(h|g|^{p-1}g_{>N})$
$\displaystyle+\sum_{K\leq N}\tfrac{K}{N}M(h|g|^{p-1})M(g_{K})+\sum_{K\leq
N}M(h|g|^{p-1}M(g_{K})).$
Let us now see how to handle the contribution of the first term only, as the
other three are similar. We begin by summing over $N\in 2^{\mathbb{Z}}$ and
recalling the definitions of $g$ and $h$; then, using Hölder, maximal function
estimates, and Lemma 2.2, we can argue as we did to obtain (4.8) to see
$\displaystyle\lim_{J\to\infty}$
$\displaystyle\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}g_{>N}\big{|}^{2}\big{)}^{1/2}M(h|g|^{p-1})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{p(d+2)}{2(p-1)}}$
$\displaystyle\lesssim\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}^{\frac{p(d+2)}{2(p-1)}}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2(p-1)}}\sum_{j=1}^{J}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}.$
(4.55)
We turn to estimating the first term above. We first write
$\displaystyle\lim_{J\to\infty}$
$\displaystyle\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}$
$\displaystyle\lesssim\lim_{J\to\infty}\limsup_{n\to\infty}\bigg{(}\sum_{j=1}^{J}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}+\sum_{j\neq
k}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}|\nabla|^{s_{c}}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{d+2}{d}}}\bigg{)}.$
(4.56)
Arguing as we did to obtain (4.10), we immediately get that
$\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j\neq
k}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}|\nabla|^{s_{c}}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{d+2}{d}}}=0.$
(4.57)
Next, we let $\eta>0$; then, using (2.5), we can find $J(\eta)>0$ so that
$\sum_{j>J(\eta)}\big{\|}|\nabla|^{s_{c}}\phi^{j}\big{\|}_{L_{x}^{2}}^{2}<\eta.$
Taking $\eta$ sufficiently small and applying a standard bootstrap argument,
we find
$\sum_{j>J(\eta)}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}\lesssim\sum_{j>J(\eta)}\big{\|}|\nabla|^{s_{c}}\phi^{j}\big{\|}_{L_{x}^{2}}^{2}\lesssim\eta.$
(4.58)
On the other hand, the fact that each $v_{n}^{j}$ has scattering size
$\lesssim E_{c}$ implies
$\sum_{j=1}^{J(\eta)}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}\lesssim_{E_{c}}1.$
(4.59)
Combining (4.57), (4.58), and (4.59), we can continue from (4.56) to see
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}\lesssim_{E_{c}}1.$
Thus, continuing from (4.55) and using (4.5) and (2.3), we find
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}^{\frac{p(d+2)}{2(p-1)}}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2(p-1)}}\sum_{j=1}^{J}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}=0,$
as needed. This completes the proof of (4.12). ∎
Having established (4.11) and (4.12), we are now done with the proof of Lemma
4.2, as well as the sketch of the proof of Proposition 4.1.
## 5\. Long-time Strichartz estimates
In this section, we prove a long-time Strichartz estimate. Such estimates were
first developed by Dodson [17] in the study of the mass-critical NLS, but have
since appeared in the energy-critical setting (see [36, 57]). In this paper,
we establish a long-time Strichartz estimate for the first time in the inter-
critical setting, modeling our approach after [17, 36, 57]. The long-time
Strichartz estimate will be an important technical tool in Section 6, in which
we rule out rapid frequency-cascade solutions, as well as in Section 7, in
which we establish a frequency-localized interaction Morawetz inequality.
We will prove long-time Strichartz estimates for $(d,s_{c})$ satisfying (1.5).
This guarantees $p>1$, which simplifies the proof. Actually, as we will point
out below, the same ideas can be used to handle $(d,s_{c})=(5,\tfrac{1}{2})$,
in which case $p=1$.
###### Proposition 5.1 (Long-time Strichartz estimates).
Take $(d,s_{c})$ satisfying (1.5). Let
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}\subset[0,T_{max})$. Then on any compact time interval
$I\subset[0,T_{max})$, which is a union of contiguous characteristic
subintervals $J_{k}$, and for any $N>0$, we have
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}\lesssim_{u}1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}},$
(5.1)
where $K:=\int_{I}N(t)^{3-4s_{c}}\,dt.$ Moreover, for any $\varepsilon>0$,
there exists $N_{0}=N_{0}(\varepsilon)$ such that for all $N\leq N_{0}$,
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}\lesssim_{u}\varepsilon(1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}).$
(5.2)
We also note that the implicit constants in (5.1) and (5.2) are independent of
$I$.
###### Proof.
Fix a compact interval $I\subset[0,T_{max})$, which is a contiguous union of
characteristic subintervals $J_{k}$; throughout the proof, all spacetime norms
will be taken over $I\times\mathbb{R}^{d}$ unless stated otherwise. Let
$\eta_{0}>0$ and $\eta>0$ be small parameters to be chosen later, and note
that by Remark 1.8, we may find $c=c(\eta)$ so that
$\big{\|}|\nabla|^{s_{c}}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\leq\eta.$ (5.3)
For $N>0$, we define
$A(N):=\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}\quad\text{and}\quad
A_{k}(N):=\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(J_{k}\times\mathbb{R}^{d})}$
for an individual characteristic subinterval $J_{k}$. We first note that by
Lemma 1.11, (5.1) holds whenever $N\geq\sup_{J_{k}\subset I}N_{k}$. Indeed, in
this case, we have
$\displaystyle A(N)$
$\displaystyle\lesssim_{u}1+\left(\int_{I}N(t)^{2}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}1+\left(\int_{I}N(t)^{3-4s_{c}}N^{4s_{c}-1}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
We will establish (5.1) for arbitrary $N>0$ by induction, beginning by
establishing a recurrence relation for $A(N)$:
###### Lemma 5.2 (Recurrence relation for $A(N)$).
$\displaystyle A(N)$ $\displaystyle\lesssim_{u}\ \inf_{t\in
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}$
$\displaystyle\quad+\eta^{\nu}A\big{(}\tfrac{N}{\eta_{0}}\big{)}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M)$
(5.4)
uniformly in $N$, for some positive constants $B(\eta,\eta_{0})$ and $\nu$.
###### Proof of Lemma 5.2.
We first apply Strichartz to see
$A(N)\lesssim\inf_{t\in I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}}+\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(|u|^{p}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}.$ (5.5)
Our next step is to decompose the nonlinearity $|u|^{p}u$ and estimate the
resulting pieces; the particular decomposition we choose depends on the
ambient dimension.
Case 1. When $d=3$, we have $2\leq p<4$, and we decompose as follows:
$\displaystyle|u|^{p}u=$ $\displaystyle\ (|u|^{p}+|u|^{p-2}\bar{u}u_{\leq
N/\eta_{0}})u_{>N/\eta_{0}}$ $\displaystyle+|u|^{p-2}\bar{u}(P_{>cN(t)}u_{\leq
N/\eta_{0}})u_{\leq N/\eta_{0}}$ (5.6) $\displaystyle+|u|^{p-2}\bar{u}(P_{\leq
cN(t)}u_{\leq N/\eta_{0}})u_{\leq N/\eta_{0}}.$
To estimate the contribution of the first term on the right-hand side of (5.6)
to (5.5), we let
$G:=|u|^{p}+|u|^{p-2}\bar{u}u_{\leq N/\eta_{0}}$
and use Bernstein, Lemma 2.6, and Hölder to estimate
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(Gu_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(Gu_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\displaystyle\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.7)
To estimate the contribution of the first term above, we first use the
fractional chain rule and Sobolev embedding to see
$\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{3p+4}}}\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}\lesssim_{u}1,$
while by the fractional product rule, the fractional chain rule, and Sobolev
embedding we get
$\displaystyle\big{\|}|\nabla|^{\frac{1}{2}s_{c}}$
$\displaystyle(|u|^{p-2}\bar{u}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}$
$\displaystyle\lesssim\
\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-12}}}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{3p+4}}}$
$\displaystyle\lesssim\
\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-2}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{3p+4}}}+\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}$
$\displaystyle\lesssim\
\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}$
$\displaystyle\lesssim_{u}\ 1.$
Thus, continuing from (5.7), we see
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle\big{(}(|u|^{p}+|u|^{p-2}\bar{u}u_{\leq
N/\eta_{0}})u_{>N/\eta_{0}}\big{)}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim_{u}\\!\\!\sum_{M>N/\eta_{0}}\\!\\!\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}\\!A(M).$
(5.8)
Next, we turn to estimating the contribution of the second term in (5.6) to
(5.5). We begin by restricting our attention to an individual
$J_{k}\times\mathbb{R}^{d}$. Note that we only need to consider the case
$cN_{k}\leq N/\eta_{0}$; in this case, we can use Bernstein, Hölder, Sobolev
embedding, Lemma 1.11, and the fact that $s_{c}\geq\tfrac{1}{2}$ to estimate
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u|^{p-2}\bar{u}(P_{>cN_{k}}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}|u|^{p-2}\bar{u}(P_{>cN_{k}}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$
$\displaystyle\lesssim_{u}N^{s_{c}}(cN_{k})^{-s_{c}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}^{2}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}$
(5.9)
for some positive constant $B(\eta,\eta_{0})$. Summing the estimates (5.9)
over the characteristic subintervals $J_{k}\subset I$ then gives
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u|^{p-2}\bar{u}(P_{>cN(t)}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.10)
Before proceeding to the next term in (5.6), we note that in obtaining
estimate (5.9), we could have held onto the term
$\big{\|}|\nabla|^{s_{c}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}},$
which (by interpolation) we can estimate by
$\displaystyle\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}.$
In this case, summing the estimates yields
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u|^{p-2}\bar{u}(P_{>cN(t)}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim_{u}\sup_{J_{k}\subset I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{2}}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.11)
This variant of (5.10) will be important when we eventually need to exhibit
smallness in (5.2).
To estimate the contribution of the final term in (5.6) to (5.5), we begin
with an application of the fractional product rule and Hölder to see
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u|^{p-2}\bar{u}(P_{\leq cN(t)}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim$
$\displaystyle\
\big{\|}|\nabla|^{s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{6p}{7p-8}}}\big{\|}P_{\leq
cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$ (5.12)
$\displaystyle+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{s_{c}}P_{\leq
cN(t)}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$ (5.13)
$\displaystyle+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}P_{\leq
cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}.$ (5.14)
We first note that by the fractional chain rule and Sobolev embedding, we get
$\big{\|}|\nabla|^{s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{6p}{7p-8}}}\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-2}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}1.$
Using Sobolev embedding, interpolation, and (5.3), we also see
$\displaystyle\big{\|}P_{\leq cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}P_{\leq cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}P_{\leq cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}P_{\leq
cN(t)}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}^{\frac{1}{2}}$
$\displaystyle\lesssim\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}^{\frac{1}{2}}.$
Estimating similarly gives
$\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}\lesssim_{u}A\big{(}\tfrac{N}{\eta_{0}}\big{)}^{\frac{1}{2}}.$
Plugging these last three estimates into (5.12), (5.13), and (5.14) and
employing a few more instances of Sobolev embedding and (5.3) finally gives
$\big{\|}|\nabla|^{s_{c}}P_{\leq N}(|u|^{p-2}\bar{u}(P_{\leq cN(t)}u_{\leq
N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}\lesssim_{u}\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
(5.15)
Collecting the estimates (5.8), (5.10), and (5.15), we see that in the case
$d=3$, the estimate (5.5) becomes
$\displaystyle A(N)\lesssim_{u}$ $\displaystyle\inf_{t\in
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}$
$\displaystyle\quad+\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.16)
Comparing (5.16) to (5.2), we see that Lemma 5.2 holds for $d=3$.
Case 2. In this case, we have $d\in\\{4,5\\}$ and $\tfrac{4}{d-1}\leq
p<\tfrac{4}{d-2}$, with both inequalities strict for $d=5$. In particular, we
have $1<p<2$.
Again, we wish to decompose the nonlinearity and continue from (5.5). This
time, we decompose as follows:
$\displaystyle|u|^{p}u$ $\displaystyle=|u|^{p}u_{>N/\eta_{0}}$
$\displaystyle\quad+|u_{>cN(t)}|^{p}P_{\leq cN(t)}u_{\leq N/\eta_{0}}$ (5.17)
$\displaystyle\quad+|u_{>cN(t)}|^{p}P_{>cN(t)}u_{\leq N/\eta_{0}}$
$\displaystyle\quad+(|u|^{p}-|u_{>cN(t)}|^{p})u_{\leq N/\eta_{0}}.$
We estimate the contribution of the first term on the right-hand side of
(5.17) to (5.5) similarly to the case $d=3$; in particular, by Bernstein,
Hölder, and Lemma 2.6, we have
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u|^{p}u_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(|u|^{p}u_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.18)
As we can use the fractional chain rule and Sobolev embedding to estimate
$\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{dp+4}}}\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}\lesssim_{u}1,$
we can continue from (5.18) to get
$\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(|u|^{p}u_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim_{u}\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.19)
Next, we turn to estimating the second term in (5.17) to (5.5). Restricting
our attention to an individual characteristic subinterval $J_{k}$, we first
apply Bernstein, Hölder, and the fractional product rule to see
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u_{>cN_{k}}|^{p}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$ $\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}\big{\|}|\nabla|^{\frac{1}{4}}(|u_{>cN_{k}}|^{p}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}\big{\|}|\nabla|^{\frac{1}{4}}|u_{>cN_{k}}|^{p}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2dp}{p(d+3)-4}}}\big{\|}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2dp}{4-p}}}$ (5.20)
$\displaystyle\ \ \
+\\!N^{s_{c}-\frac{1}{4}}\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{4dp^{2}}{p(2d+5)-8}}}^{p}\big{\|}|\nabla|^{\frac{1}{4}}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{4dp}{8-p}}}.$ (5.21)
Using Hölder, the fractional chain rule, Sobolev embedding, Bernstein,
interpolation, (5.3), and Young’s inequality, we can estimate
$\displaystyle\eqref{young 1}$ $\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{4}}u_{>cN_{k}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}|\nabla|^{s_{c}}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}$
$\displaystyle\lesssim_{u}N^{s_{c}-\frac{1}{4}}(cN_{k})^{\frac{1}{4}-s_{c}}\big{\|}|\nabla|^{s_{c}}u_{>cN_{k}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}B(\eta)\left(\tfrac{N}{N_{k}}\right)^{s_{c}-\frac{1}{4}}\eta^{\frac{1}{2}}A_{k}\big{(}\tfrac{N}{\eta_{0}}\big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}B(\eta)\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}+\eta
A_{k}\big{(}\tfrac{N}{\eta_{0}}\big{)},$
for some positive constant $B(\eta)$. Using Lemma 1.11 as well, we can
estimate similarly
$\displaystyle\eqref{young 2}$ $\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}(cN_{k})^{\frac{1}{4}-s_{c}}\big{\|}|\nabla|^{(s_{c}-\frac{1}{4})/p}u_{>cN_{k}}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{4dp^{2}}{p(2d+5)-8}}}^{p}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}$ $\displaystyle\lesssim
B(\eta)\left(\tfrac{N}{N_{k}}\right)^{s_{c}-\frac{1}{4}}\big{\|}|\nabla|^{s_{c}}u_{>cN_{k}}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{2dp}{dp-1}}}^{p}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}B(\eta)\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}+\eta
A_{k}\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
Collecting the estimates for (5.20) and (5.21) and summing over the intervals
$J_{k}\subset I$, we arrive at
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u_{>cN(t)}|^{p}P_{\leq cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim_{u}B(\eta)N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta
A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$ (5.22)
Before proceeding, we note that for both (5.20) and (5.21), we could have
instead estimated
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
cN_{k}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{4}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{4}}$
$\displaystyle\lesssim\eta^{\frac{1}{4}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{4}}.$
If we had done this, upon summing we could have ended up with the alternate
estimate
$\displaystyle\big{\|}|\nabla$ $\displaystyle|^{s_{c}}P_{\leq
N}(|u_{>cN(t)}|^{p}P_{\leq cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\sup_{J_{k}\subset I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{2}}B(\eta)N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
(5.23)
This variant of (5.22) will be important when we need to exhibit smallness in
(5.2).
To estimate the contribution of the third term in (5.17) to (5.5), we first
define the following:
$\left\\{\begin{array}[]{ll}\theta:=\tfrac{dp-4-p}{4-p}\in[0,1),&\sigma:=\tfrac{p^{2}(d^{2}+2d-2)-4p(4d+1)+48}{4p(dp-8)}\in(0,s_{c}),\\\
\\\
r_{1}:=\tfrac{4dp(dp-8)}{p^{2}(d^{2}-2d-2)+p(28-8d)-16},&r_{2}:=\tfrac{4dp(dp-8)}{p^{2}(d^{2}+2d-2)-4p(2d+1)-16}.\end{array}\right.$
With this choice of parameters, we have
$\left\\{\begin{array}[]{ll}s_{c}+\theta(\tfrac{d-1}{2}-s_{c})=2s_{c}-\tfrac{1}{2},\\\
\\\
-\theta(s_{c}+\tfrac{1}{2})-2\sigma(1-\theta)=-(2s_{c}-\tfrac{1}{2})\end{array}\right.$
and (by Sobolev embedding)
$\begin{array}[]{ll}\dot{H}^{s_{c},\frac{2d}{d-2}}\hookrightarrow\dot{H}^{\sigma,r_{1}},&\dot{H}^{s_{c},2}\hookrightarrow\dot{H}^{\sigma,r_{2}}.\end{array}$
Then restricting our attention to an individual $J_{k}$, we can use Bernstein,
Hölder, the bilinear Strichartz estimate (Lemma 2.10), and Sobolev embedding
to estimate
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u_{>cN_{k}}|^{p}P_{>cN_{k}}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}u_{>cN_{k}}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t,x}^{2}}^{\theta}$
$\displaystyle\quad\times\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{r_{1}}}^{1-\theta}\big{\|}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{r_{2}}}^{1-\theta}$
$\displaystyle\lesssim_{u}N^{s_{c}}\left(\tfrac{N}{\eta_{0}}\right)^{\theta(\frac{d-1}{2}-s_{c})}(cN_{k})^{-\theta(s_{c}+\frac{1}{2})}$
$\displaystyle\quad\times\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{r_{1}}}^{1-\theta}\big{\|}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{r_{2}}}^{1-\theta}$
$\displaystyle\lesssim_{u}B(\eta_{0})N^{2s_{c}-\frac{1}{2}}(cN_{k})^{-\theta(s_{c}+\frac{1}{2})-2\sigma(1-\theta)}$
$\displaystyle\quad\times\big{\|}|\nabla|^{\sigma}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{r_{1}}}^{1-\theta}\big{\|}|\nabla|^{\sigma}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{r_{2}}}^{1-\theta}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{1-\theta}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{1-\theta}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}$
(5.24)
for some positive constant $B(\eta,\eta_{0})$. If we sum the estimates (5.24)
over the intervals $J_{k}\subset I$, we arrive at
$\big{\|}|\nabla|^{s_{c}}P_{\leq N}(|u_{>cN(t)}|^{p}P_{>cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim_{u}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.25)
Before moving on to the fourth (and final) term in (5.17), we note that if we
had held on to the term $\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{1-\theta}$ when deriving
(5.24), then upon summing we would get
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u_{>cN(t)}|^{p}P_{>cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim_{u}\sup_{J_{k}\subset I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{1-\theta}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.26)
This variant of (5.25) will be important when we eventually need to exhibit
smallness in (5.2).
We now turn to the final term in (5.17), beginning with an application of the
fractional product rule and Hölder:
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}\left((|u|^{p}-|u_{>cN(t)}|^{p})u_{\leq
N/\eta_{0}}\right)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(|u|^{p}-|u_{>cN(t)}|^{p})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2dp}{p(d+4)-4}}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{dp}{2-p}}}$ (5.27)
$\displaystyle\quad+\big{\|}|u|^{p}-|u_{>cN(t)}|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{d}{2}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}.$ (5.28)
By Lemma 2.5, Sobolev embedding, and (5.3), we first estimate
$\displaystyle\eqref{differences 1}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}A\big{(}\tfrac{N}{\eta_{0}}\big{)}$
$\displaystyle\quad+\big{\|}|\nabla|^{s_{c}}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}A\big{(}\tfrac{N}{\eta_{0}}\big{)}$
$\displaystyle\lesssim_{u}(\eta^{p-1}+\eta)A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
On the other hand, by Sobolev embedding, Hölder, and (5.3), we get
$\displaystyle\eqref{differences 2}$
$\displaystyle\lesssim\big{(}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}+\big{\|}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{)}\big{\|}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}\lesssim_{u}\eta
A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
Thus we can estimate the contribution of the final term in (5.17) by
$\big{\|}|\nabla|^{s_{c}}P_{\leq N}\left((|u|^{p}-|u_{>cN(t)}|^{p})u_{\leq
N/\eta_{0}}\right)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim_{u}\eta^{p-1}A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
(5.29)
Collecting the estimates (5.19), (5.22), (5.25), and (5.29), we see that in
Case 2, the estimate (5.5) becomes
$\displaystyle A(N)$ $\displaystyle\lesssim_{u}\inf_{t\in
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}$
$\displaystyle\quad+\eta^{\min\\{\frac{1}{2},p-1\\}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.30)
Comparing (5.30) to (5.2), we see that Lemma 5.2 holds for $d\in\\{4,5\\}$. ∎
###### Remark 5.3.
We have omitted the case $(d,s_{c})=(5,\tfrac{1}{2})$, in which $p=1$; this
scenario is not handled under Case 2 due to the use of Lemma 2.5. However, by
using the alternate decomposition
$|u|u=|u|u_{>N/\eta_{0}}+(|u|-|u_{\leq cN(t)}|)u_{\leq N/\eta_{0}}+|u_{\leq
cN(t)}|u_{\leq N/\eta_{0}},$
one can use the same ideas as above to establish the recurrence relation in
this case.
With the recurrence relation (5.2) in hand, we can now use induction to
complete the proof of Proposition 5.1. First, recall that (5.1) holds for
$N\geq\sup_{J_{k}\subset I}N_{k}.$; i.e. we have
$A(N)\leq C(u)\left[1+N^{2s_{c}-1/2}K^{1/2}\right]$ (5.31)
for $N\geq\sup_{J_{k}\subset I}N_{k}.$ Of course, this inequality remains true
if we replace $C(u)$ by any larger constant.
We now suppose (5.31) holds at frequency $N$ and use the recurrence relation
(5.2) to show it holds at frequency $N/2$. Let us first rewrite (5.2) as
$\displaystyle A(N)$
$\displaystyle\leq\widetilde{C}(u)\big{[}1+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta^{\nu}A(\tfrac{N}{\eta_{0}})+\\!\\!\\!\\!\\!\\!\sum_{M>N/\eta_{0}}\\!\\!\\!\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M)\big{]}.$
(5.32)
To simplify notation, we will let $\alpha:=2s_{c}-\frac{1}{2}$ and write
$B(\eta,\eta_{0})=B$; then, if we take $\eta_{0}<\tfrac{1}{2}$ and use the
inductive hypothesis, (5.32) becomes
$\displaystyle A(\tfrac{N}{2})$
$\displaystyle\leq\widetilde{C}(u)\big{[}1+B(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}+\eta^{\nu}C(u)(1+\eta_{0}^{-\alpha}(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}})$
$\displaystyle\quad+C(u)\\!\\!\\!\\!\\!\\!\sum_{M>N/2\eta_{0}}\\!\\!\\!\left(\tfrac{N}{2M}\right)^{\frac{3}{2}s_{c}}(1+M^{\alpha}K^{\frac{1}{2}})\big{]}$
$\displaystyle\leq\widetilde{C}(u)\big{[}1+B(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}+\eta^{\nu}C(u)(1+\eta_{0}^{-\alpha}(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}})$
$\displaystyle\quad+C(u)\eta_{0}^{\frac{3}{2}s_{c}}+C(u)\eta_{0}^{\frac{1}{2}(1-s_{c})}(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\big{]}$
$\displaystyle=\widetilde{C}(u)\left[1+B(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\right]+C(u)\big{[}(\eta^{\nu}+\eta_{0}^{\frac{3}{2}s_{c}})\widetilde{C}(u)$
$\displaystyle\quad+\big{(}\eta_{0}^{-\alpha}\eta^{\nu}+\eta_{0}^{\frac{1}{2}(1-s_{c})}\big{)}\widetilde{C}(u)(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\big{]}.$
(5.33)
Notice that we had convergence of the sum above precisely because $s_{c}<1$.
If we choose $\eta_{0}$ possibly even smaller depending on $\widetilde{C}(u)$,
and $\eta$ sufficiently small depending on $\widetilde{C}(u)$ and $\eta_{0}$,
we can guarantee
$\displaystyle\eqref{induction
term}\leq\widetilde{C}(u)\left[1+B(\eta,\eta_{0})(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\right]+\tfrac{1}{2}C(u)\left[1+(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\right].$
If we now choose $C(u)$ possibly larger so that $C(u)\geq
2(1+B(\eta,\eta_{0}))\widetilde{C}(u)$, then this inequality implies that
(5.31) holds at $N/2$, as was needed to show. This completes the proof of
(5.1).
It remains to establish (5.2). To begin, fix $\varepsilon>0$. To exhibit the
smallness in (5.2), we need to revisit the proof of the recurrence relation
for $A(N)$, paying closer attention to the terms that gave rise to the
expression $N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$ More precisely, if we use
(5.11) instead of (5.10); (5.23) instead of (5.22); and (5.26) instead of
(5.25); then continuing from (5.5), the recurrence relation for $A(N)$ takes
the form
$\displaystyle A(N)\lesssim_{u}$ $\displaystyle\
f(N)+f(N)N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta^{\nu}A(\tfrac{N}{\eta_{0}})+\sum_{M>N/\eta_{0}}(\tfrac{N}{M})^{\frac{3}{2}s_{c}}A(M),$
(5.34)
where $f(N)$ has the form
$\displaystyle f(N)=\ $ $\displaystyle\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}$
$\displaystyle+B(\eta,\eta_{0})\sum_{i=1}^{4}\sup_{J_{k}\subset
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\theta_{i}}$
(5.35)
for some $\theta_{i}\in(0,1]$. Here the particular values of the $\theta_{i}$
are not important; we will only need the fact that each $\theta_{i}>0$.
Combining the updated recurrence relation (5.34) with the newly proven
estimate (5.1) and once again simplifying notation via
$\alpha=2s_{c}-\frac{1}{2}$, we see
$\displaystyle A(N)$
$\displaystyle\lesssim_{u}f(N)+f(N)N^{\alpha}K^{\frac{1}{2}}+\eta^{\nu}(1+\eta_{0}^{-\alpha}N^{\alpha}K^{\frac{1}{2}})+\eta_{0}^{\frac{3}{2}s_{c}}(1+\eta_{0}^{-\alpha}N^{\alpha}K^{\frac{1}{2}})$
$\displaystyle\lesssim_{u}f(N)+\eta^{\nu}+\eta_{0}^{\frac{3}{2}s_{c}}+\left[f(N)+\eta^{\nu}\eta_{0}^{-\alpha}+\eta_{0}^{\frac{1}{2}(1-s_{c})}\right]N^{\alpha}K^{\frac{1}{2}}.$
(5.36)
To complete the argument, we will need the fact that for fixed
$\eta,\eta_{0}>0$, we have
$\lim_{N\to 0}f(N)=0,$ (5.37)
which is a consequence of almost periodicity and the fact that
$\inf_{t\in[0,T_{max})}N(t)\geq 1.$
Then, continuing from (5.36), we choose $\eta_{0}$ small enough that
$\eta_{0}^{\frac{3}{2}s_{c}}+\eta_{0}^{\frac{1}{2}(1-s_{c})}<\varepsilon$, and
choose $\eta$ sufficiently small depending on $\eta_{0}$ so that
$\eta^{\nu}+\eta_{0}^{-\alpha}\eta^{\nu}<\varepsilon$. Finally, using (5.37),
we choose $N_{0}=N_{0}(\varepsilon)$ so that $f(N)<\varepsilon$ for $N\leq
N_{0}$. With this choice of parameters, (5.36) becomes
$A(N)\lesssim_{u}\varepsilon(1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}})$
for $N\leq N_{0},$ which completes the proof of (5.2). ∎
## 6\. The rapid frequency-cascade scenario
In this section, we preclude the existence of almost periodic solutions as in
Theorem 1.13 for which $\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty.$ We show
that their existence is inconsistent with the conservation of mass. The main
tool we will use is the long-time Strichartz estimate established in the
previous section; as such, we will prove the following result for $(d,s_{c})$
satisfying (1.5).
###### Theorem 6.1 (No rapid frequency-cascades).
Let $(d,s_{c})$ satisfy (1.5). Then there are no almost periodic solutions
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) with $N(t)\equiv
N_{k}\geq 1$ on each characteristic subinterval $J_{k}\subset[0,T_{max})$ such
that
$\big{\|}u\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}([0,T_{max})\times\mathbb{R}^{d})}=\infty$
(6.1)
and
$\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty.$ (6.2)
###### Proof.
We argue by contradiction. Suppose $u$ were such a solution; then by Corollary
1.10, we have
$\lim_{t\to T_{max}}N(t)=\infty,$
whether $T_{max}$ is finite or infinite (recall $s_{c}>\tfrac{1}{4}$). Thus by
Remark 1.8, we see
$\lim_{t\to T_{max}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}=0\indent\text{for any }N>0.$ (6.3)
Now, we let $I_{n}$ be a nested sequence of compact subintervals of
$[0,T_{max})$, each of which is a contiguous union of characteristics
intervals $J_{k}$. On each $I_{n}$, we will now apply Proposition 5.1;
specifically, for fixed $\eta,\eta_{0}>0$, we use the recurrence relation
(5.2), the estimate (5.1), and the hypothesis (6.2) to see
$\displaystyle A_{n}(N)$ $\displaystyle:=\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I_{n}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}\left(\int_{I_{n}}N(t)^{3-4s_{c}}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\quad+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A_{n}(M)$
$\displaystyle\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}\bigg{(}\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt\bigg{)}^{\frac{1}{2}}$
$\displaystyle\quad+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A_{n}(M)$
$\displaystyle\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A_{n}(M).$
Arguing as we did to obtain (5.1), we conclude
$A_{n}(N)\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+N^{2s_{c}-\frac{1}{2}}.$
Letting $n\to\infty$ and using (6.3) then gives
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}([0,T_{max})\times\mathbb{R}^{d})}\lesssim_{u}N^{2s_{c}-\frac{1}{2}}\indent\text{for
all }N>0.$ (6.4)
We now claim that (6.4) implies
###### Lemma 6.2.
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim_{u}N^{2s_{c}-\frac{1}{2}}\indent\text{for
all }N>0.$ (6.5)
###### Proof of Lemma 6.2.
Fix $N>0$; we first use Proposition 1.14 and Strichartz to estimate
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(|u|^{p}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}([0,T_{max})\times\mathbb{R}^{d})}.$
(6.6)
To proceed, we decompose the nonlinearity and estimate the individual pieces;
as before, the particular decomposition we use depends on the ambient
dimension. In the estimates that follow, spacetime norms will be taken over
$[0,T_{max})\times\mathbb{R}^{d}$.
Case 1. When $d=3$, we decompose
$|u|^{p}u=|u|^{p-2}\bar{u}u_{\leq
N}^{2}+(|u|^{p-2}\bar{u}u_{>N}+2|u|^{p-2}\bar{u}u_{\leq N})u_{>N}.$
We can use Hölder, the fractional product rule, fractional chain rule, Sobolev
embedding, interpolation, and (6.4) to estimate the contribution of the first
piece as follows:
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u|^{p-2}\bar{u}u_{\leq N}^{2})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{6p}{7p-8}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}^{2}$
$\displaystyle\quad+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{s_{c}}(u_{\leq
N}^{2})\big{\|}_{L_{t}^{2}L_{x}^{\frac{6p}{4+p}}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-2}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{4}L_{x}^{3}}^{2}$
$\displaystyle\quad+\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{6}}+N^{2s_{c}-\frac{1}{2}}$
$\displaystyle\lesssim_{u}N^{2s_{c}-\frac{1}{2}}.$
To estimate the contribution of the second piece, we denote
$G=|u|^{p-2}\bar{u}u_{>N}+2|u|^{p-2}\bar{u}u_{\leq N}$
and use Bernstein, Hölder, Lemma 2.6, and (6.4) to see
$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{s_{c}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}N^{2s_{c}-\frac{1}{2}}.$
(6.7)
A few applications of the fractional product rule, fractional chain rule, and
Sobolev embedding give
$\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}\lesssim_{u}1,$
so that continuing from (6.7), we get
$\big{\|}|\nabla|^{s_{c}}P_{\leq
N}\big{(}(|u|^{p-2}\bar{u}u_{>N}+2|u|^{p-2}\bar{u}u_{\leq
N})u_{>N}\big{)}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}\lesssim_{u}N^{2s_{c}-1/2}.$
Thus we see that the claim holds in this first case.
Case 2. When $d\in\\{4,5\\}$, we decompose
$|u|^{p}u=|u|^{p}u_{\leq N}+|u|^{p}u_{>N}.$
We employ Hölder, the fractional product rule, the fractional chain rule,
Sobolev embedding, and (6.4) to estimate the contribution of the first piece
as follows:
$\displaystyle\big{\|}|\nabla$ $\displaystyle|^{s_{c}}P_{\leq
N}(|u|^{p}u_{\leq N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2dp}{p(d+4)-4}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{dp}{2-p}}}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}+N^{2s_{c}-\frac{1}{2}}$
$\displaystyle\lesssim_{u}N^{2s_{c}-\frac{1}{2}}.$
For the second piece, we use Hölder, Bernstein, Lemma 2.6, the fractional
chain rule, and Sobolev embedding to see
$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u|^{p}u_{>N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(|u|^{p}u_{>N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{dp+4}}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{s_{c}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}N^{2s_{c}-\frac{1}{2}}$
$\displaystyle\lesssim_{u}N^{2s_{c}-\frac{1}{2}}.$
Thus we see that the claim holds in this second case, completing the proof of
Lemma 6.2. ∎
We now wish to use (6.5) to prove
###### Lemma 6.3.
$u\in
L_{t}^{\infty}\dot{H}_{x}^{-\varepsilon}([0,T_{max})\times\mathbb{R}^{d})\indent\text{for
some }\varepsilon>0.$
###### Proof of Lemma 6.3.
For $s_{c}>\tfrac{1}{2},$ this is easy; indeed, choosing $\varepsilon>0$ such
that $s_{c}-\tfrac{1}{2}-\varepsilon>0$, we can use Bernstein and (6.5) to see
$\displaystyle\big{\|}|\nabla|^{-\varepsilon}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}N^{-s_{c}-\varepsilon}\big{\|}|\nabla|^{s_{c}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}N^{-s_{c}-\varepsilon}\big{\|}|\nabla|^{s_{c}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq
1}N^{-s_{c}-\varepsilon}N^{2s_{c}-\frac{1}{2}}+1$
$\displaystyle\lesssim_{u}1.$
When $s_{c}=\tfrac{1}{2}$ (that is, $p=\tfrac{4}{d-1}$), we need to work a bit
harder. To begin, we note that by Bernstein and (6.5), we have
$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}N^{-\frac{1}{10}}\big{\|}|\nabla|^{\frac{1}{2}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}N^{-\frac{1}{10}}\big{\|}|\nabla|^{\frac{1}{2}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq 1}N^{\frac{2}{5}}+1$
$\displaystyle\lesssim_{u}1.$ (6.8)
We wish to show that in fact, we have the more quantitative statement
$\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim_{u}N^{\frac{1}{2}}\indent\text{for
all }N>0.$ (6.9)
Once we have established (6.9), we can complete the proof of Lemma 6.3 as
follows: choosing $0<\varepsilon<\tfrac{1}{10}$, we use Bernstein, (6.8), and
(6.9) to estimate
$\displaystyle\big{\|}|\nabla|^{-\varepsilon}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}N^{-\frac{2}{5}-\varepsilon}\big{\|}|\nabla|^{\frac{2}{5}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}N^{-\frac{2}{5}-\varepsilon}\big{\|}|\nabla|^{\frac{2}{5}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq 1}N^{\frac{1}{10}-\varepsilon}+1$
$\displaystyle\lesssim_{u}1.$
Thus, to complete the proof of Lemma 6.3, it remains to establish (6.9). We
begin by fixing $N>0$. The proof of (6.9) will be a second iteration of the
arguments that gave (6.5), this time using (6.8) as additional input.
We first use (6.8) (and the uniqueness of weak limits) to see that the no-
waste Duhamel formula (Proposition 1.14) also holds in the weak
$\dot{H}_{x}^{\frac{2}{5}}$ topology; thus, using Strichartz as well, we can
estimate
$\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim\big{\|}|\nabla|^{\frac{2}{5}}P_{\leq
N}(|u|^{\frac{4}{d-1}}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}([0,T_{max})\times\mathbb{R}^{d})}.$
Once again, we decompose the nonlinearity, and again the decomposition depends
on the ambient dimension. The estimates that follow will be very similar in
spirit to the estimates that gave (6.5); all estimates will be taken over
$[0,T_{max}).$
Case 1. When $d=3$, we decompose
$|u|^{2}u=\bar{u}u_{\leq N}^{2}+(\bar{u}u_{>N}+2\bar{u}u_{\leq N})u_{>N}.$
We estimate the first piece as follows: by Hölder, the fractional product
rule, the fractional chain rule, Sobolev embedding, interpolation, (6.4) and
(6.8),
$\displaystyle\big{\|}$ $\displaystyle|\nabla|^{\frac{2}{5}}P_{\leq
N}(\bar{u}u_{\leq N}^{2})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}{u}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{4}L_{x}^{6}}^{2}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{30/11}}\big{\|}|\nabla|^{\frac{2}{5}}(u_{\leq
N})^{2}\big{\|}_{L_{t}^{2}L_{x}^{15/7}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{4}L_{x}^{3}}^{2}+\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{3}}\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{15/2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\quad+\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$ $\displaystyle\lesssim_{u}N^{\frac{1}{2}}.$
For the second piece, we first let $G:=\bar{u}u_{>N}+2\bar{u}u_{\leq N},$ and
use Bernstein, Hölder, Lemma 2.6, Sobolev embedding, and (6.4) to see
$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}P_{\leq
N}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{-\frac{2}{5}}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}\big{\|}|\nabla|^{-\frac{2}{5}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{15/2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{15/2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{1}{2}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}N^{\frac{1}{2}}.$
(6.10)
A few applications of the fractional product rule, Sobolev embedding, and
(6.8) give
$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{3}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}1,$
so that (6.10) becomes
$\big{\|}|\nabla|^{\frac{2}{5}}P_{\leq
N}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}\lesssim_{u}N^{\frac{1}{2}}.$
We see that (6.9) holds in this first case.
Case 2. When $d\in\\{4,5\\}$, we decompose
$|u|^{\frac{4}{d-1}}u=|u|^{\frac{4}{d-1}}u_{\leq
N}+|u|^{\frac{4}{d-1}}u_{>N}.$
We first note that interpolating between $u\in
L_{t}^{\infty}\dot{H}_{x}^{\frac{2}{5}}$ and $u\in
L_{t}^{\infty}\dot{H}_{x}^{\frac{1}{2}},$ we have
$\displaystyle u\in L_{t}^{\infty}\dot{H}_{x}^{\frac{21-d}{40}}.$ (6.11)
Thus, to estimate the contribution of the first piece, we can use Hölder, the
fractional product rule, the fractional chain rule, Sobolev embedding, (6.4),
(6.5), and (6.11) to see
$\displaystyle\big{\|}$
$\displaystyle|\nabla|^{\frac{2}{5}}(|u|^{\frac{4}{d-1}}u_{\leq
N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}|u|^{\frac{4}{d-1}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{5}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-3}}}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{40d}{21(d-1)}}}^{\frac{4}{d-1}}\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{10d}{5d-11}}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{d-1}}}^{\frac{5-d}{d-1}}\\!\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}\\!+\\!\big{\|}|\nabla|^{\frac{21-d}{40}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{4}{d-1}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{5-d}{d-1}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}N^{\frac{1}{2}}+N^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}.$
For the second piece, we use Bernstein, Hölder, Lemma 2.6, the fractional
chain rule, Sobolev embedding, (6.4), (6.5), and (6.8) to see
$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}$ $\displaystyle P_{\leq
N}(|u|^{\frac{4}{d-1}}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{-\frac{2}{5}}(|u|^{\frac{4}{d-1}}u_{\geq
N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$ $\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}|u|^{\frac{4}{d-1}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{5}}}\big{\|}|\nabla|^{-\frac{2}{5}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{10d}{5d-11}}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{d-1}}}^{\frac{5-d}{d-1}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{10d}{5d-11}}}$
$\displaystyle\lesssim_{u}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{1}{2}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}.$
Thus (6.9) holds in this second case. This completes the proof of Lemma 6.3.∎
With Lemma 6.3 at hand, we are ready to complete the proof of Theorem 6.1. Fix
$t\in[0,T_{max})$ and $\eta>0$. By Remark 1.8, we may find $c(\eta)>0$ so that
$\int_{|\xi|\leq
c(\eta)N(t)}|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta.$
Interpolating with $u\in L_{t}^{\infty}\dot{H}_{x}^{-\varepsilon},$ we get
$\int_{|\xi|\leq c(\eta)N(t)}|\widehat{u}(t,\xi)|^{2}\
d\xi\lesssim_{u}\eta^{\frac{\varepsilon}{s_{c}+\varepsilon}}.$
On the other hand, we have
$\int_{|\xi|\geq
c(\eta)N(t)}|\widehat{u}(\xi,t)|^{2}\,d\xi\leq(c(\eta)N(t))^{-2s_{c}}\int|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\lesssim_{u}(c(\eta)N(t))^{-2s_{c}}.$
Adding these last estimates and using Plancherel, we conclude that for all
$t\in[0,T_{max}),$ we have
$0\leq{M}(u(t)):=\int|u(t,x)|^{2}\,dx\lesssim_{u}\eta^{\frac{\varepsilon}{s_{c}+\varepsilon}}+(c(\eta)N(t))^{-2s_{c}}.$
Thus, recalling $\lim_{t\to T_{max}}N(t)=\infty$, we can conclude that for all
$\eta>0$, we may find $t_{0}\in[0,T_{max})$ so that for all
$t\in(t_{0},T_{max}),$ we have ${M}(u(t))\leq\eta.$ But by conservation of
mass, ${M}(u(t))\equiv{M}(u_{0}),$ and so we find that ${M}(u_{0})\leq\eta$
for all $\eta>0$. Of course, this gives $u\equiv 0$, which contradicts that
$u$ blows up (cf. (6.1)). ∎
###### Remark 6.4.
We have omitted the case $(d,s_{c})=(5,\tfrac{1}{2})$ from Theorem 6.1 only
because we omitted this case from the long-time Strichartz estimate,
Proposition 5.1. Of course, as remarked in the proof of Proposition 5.1, the
long-time Strichartz estimates continue to hold when
$(d,s_{c})=(5,\tfrac{1}{2})$; thus we see that Theorem 6.1 holds in this case
as well.
## 7\. The frequency-localized interaction Morawetz inequality
In this section, we prove spacetime bounds for the high-frequency portions of
almost periodic solutions to (1.1); these bounds can be used to preclude the
existence of quasi-solitons (Section 8). As we will see, establishing these
bounds will lead to the most non-trivial restrictions on the set of
$(d,s_{c})$ to which our main theorem (Theorem 1.3) applies; see the proof
below for a more detailed discussion. The main result of this section is the
following
###### Proposition 7.1 (Frequency-localized interaction Morawetz inquality).
Let $(d,s_{c})$ satisfy (1.4). Suppose
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ is an almost periodic
solution to (1.1) such that $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}\subset[0,T_{max})$, and let $I\subset[0,T_{max})$ be a
compact time interval, which is a union of contiguous subintervals $J_{k}$.
Then for any $\eta>0$, there exists $N_{0}=N_{0}(\eta)$ such that for any
$N\leq N_{0}$, we have
$-\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|u_{\geq
N}(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u_{\geq
N}(t,x)|^{2}\,dx\,dy\,dt\lesssim_{u}\eta(N^{1-4s_{c}}+K),$ (7.1)
where $K:=\int_{I}N(t)^{3-4s_{c}}\,dt.$ Furthermore, $N_{0}$ and the implicit
constants above do not depend on $I$.
Before we begin the proof of Proposition 7.1, we recall a general form of the
interaction Morawetz inequality, introduced originally in [14] (for more
discussion, see also [31] and the references cited therein). We will
essentially follow the presentation in [55, Section 5].
For a fixed function $a:\mathbb{R}^{d}\to\mathbb{R}$ and $\varphi$ solving
$(i\partial_{t}+\Delta)\varphi=\mathcal{N}$, we define the interaction
Morawetz action by
$M(t)=2\,\text{Im}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}a_{k}(x-y)(\varphi_{k}\bar{\varphi})(t,x)\,dx\,dy,$
where subscripts denote spatial derivatives and repeated indices are summed.
If we define the mass bracket
$\\{f,g\\}_{m}:=\text{Im}(f\bar{g})$
and the momentum bracket
$\\{f,g\\}_{\mathcal{P}}:=\text{Re}(f\nabla\bar{g}-g\nabla\bar{f}),$
then one can show
$\displaystyle\partial_{t}M(t)=$
$\displaystyle-\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}a_{jjkk}(x-y)|\varphi(t,x)|^{2}\,dx\,dy$
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}4a_{jk}(x-y)\text{Re}(\bar{\varphi}_{j}\varphi_{k})(t,x)\,dx\,dy$
(7.2) $\displaystyle-\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\indent
2\,\text{Im}(\bar{\varphi}\varphi_{k})(t,y)a_{jk}(x-y)2\,\text{Im}(\bar{\varphi}\varphi_{j})(t,x)\,dx\,dy$
(7.3)
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}2\\{\mathcal{N},\varphi\\}_{m}(t,y)a_{j}(x-y)\
2\,\text{Im}(\bar{\varphi}\varphi_{j})(t,x)\,dx\,dy$
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\
2\,\nabla a(x-y)\cdot\\{N,\varphi\\}_{\mathcal{P}}(t,x)\,dx\,dy.$
To prove Proposition 7.1, we will use $a(x)=|x|$. Note that in this case, we
have
$\left\\{\begin{array}[]{ll}a_{j}(x)=\tfrac{x_{j}}{|x|},\\\ \\\
a_{jk}(x)=\tfrac{\delta_{jk}}{|x|}-\tfrac{x_{j}x_{k}}{|x|^{3}},\\\ \\\ \Delta
a(x)=\tfrac{d-1}{|x|},\\\ \\\ \Delta\Delta
a(x)=-(d-1)\Delta(\tfrac{1}{|x|}).\end{array}\right.$
For this choice of $a$, one can also show $\eqref{mor2}+\eqref{mor3}\geq 0$
(for details, see for example [55, Lemma 5.4]). Thus, integrating
$\partial_{t}M$ over $I$, we arrive at the following
###### Lemma 7.2 (Interaction Morawetz inequality).
$\displaystyle-\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|\varphi(t,x)|^{2}\,dx\,dy\,dt$
$\displaystyle+\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\tfrac{x-y}{|x-y|}\cdot\left\\{\mathcal{N},\varphi\right\\}_{\mathcal{P}}(t,x)\,dx\,dy\,dt$
$\displaystyle\ \ \ \lesssim\sup_{t\in
I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(t,x)\bar{\varphi}(t,x)\,dx\,dy$
$\displaystyle\ \ \ \
+\bigg{|}\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\left\\{\mathcal{N},\varphi\right\\}_{m}(t,y)\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(t,x)\bar{\varphi}(t,x)\,dx\,dy\,dt\
\bigg{|}.$
To prove Proposition 7.1, we will apply this estimate with $\varphi=u_{\geq
N}$, with $N$ chosen small enough to capture ‘most’ of the solution. To make
this idea more precise, we first need to record the following corollary of
Proposition 5.1.
###### Corollary 7.3 (Low and high frequencies control).
Let $(d,s_{c})$ satisfy (1.5), and let
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}\subset[0,T_{max}).$ Then on any compact time interval
$I\subset[0,T_{max}),$ which is a union of continuous subintervals $J_{k}$,
and for any frequency $N>0$, we have
$\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}\lesssim_{u}N^{-s_{c}}(1+N^{4s_{c}-1}K)^{\frac{1}{q}}$
(7.4)
for all $\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$ with $q>4-\tfrac{2p}{dp-4},$
where $K:=\int_{I}N(t)^{3-4s_{c}}\,dt.$
Moreover, for any $\eta>0$, there exists $N_{0}=N_{0}(\eta)$ such that for all
$N\leq N_{0}$, we have
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}\lesssim_{u}\eta(1+N^{4s_{c}-1}K)^{\frac{1}{q}}$
(7.5)
for all $\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$ with $q\geq 2$.
Furthermore, $N_{0}$ and the implicit constants in (7.4) and (7.5) do not
depend on $I$.
###### Proof of Corollary 7.3.
We first show (7.4). For fixed $\alpha>s_{c}-\tfrac{1}{2}$, we can use
Bernstein and (5.1) to see
$\displaystyle\big{\|}|\nabla|^{-\alpha}u_{\geq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\sum_{M\geq N}M^{-\alpha-
s_{c}}\big{\|}|\nabla|^{s_{c}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{u}\sum_{M\geq N}M^{-\alpha-
s_{c}}(1+M^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}})$
$\displaystyle\lesssim_{u}N^{-\alpha-s_{c}}(1+N^{4s_{c}-1}K)^{\frac{1}{2}}.$
(7.6)
Now, take $(q,r)$ with $2<q\leq\infty$ and
$\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$, and define
$\alpha=\tfrac{(q-2)(dp-4)}{4p}.$ Notice that $\alpha>s_{c}-\tfrac{1}{2}$
exactly when $q>4-\tfrac{2p}{dp-4}.$ Thus, in this case, we get by
interpolation and (7.6) that
$\displaystyle\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{-\alpha}u_{\geq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}^{\frac{2}{q}}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}^{1-\frac{2}{q}}$
$\displaystyle\lesssim_{u}\left[N^{-\frac{qs_{c}}{2}}(1+N^{4s_{c}-1}K)^{\frac{1}{2}}\right]^{\frac{2}{q}},$
which gives (7.4). As for (7.5), we first note that since $\inf_{t\in
I}N(t)\geq 1$, for any $\eta>0$ we may find $N_{0}(\eta)$ so that
$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}\leq\eta$
for all $N\leq N_{0}$ (cf. Remark 1.8). The estimate (7.5) then follows by
interpolating with (5.1). ∎
We are now ready for the
###### Proof of Proposition 7.1.
Take $I\subset[0,T_{max})$, a compact time interval, which is a contiguous
union of subintervals $J_{k}$, and let $K:=\int_{I}N(t)^{3-4s_{c}}\,dt$.
Throughout the proof, all spacetime norms will be taken over
$I\times\mathbb{R}^{d}$.
Fix $\eta>0$, and choose $N_{0}=N_{0}(\eta)$ small enough that (7.5) holds;
recall that (7.4) holds without any restriction on $N$. Next, we claim that
for $N_{0}$ possibly even smaller, we can guarantee that for $N\leq N_{0}$, we
have
$\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}\eta^{10}N^{-s_{c}}$ (7.7)
and
$\big{\|}|\nabla|^{1-s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}\eta^{10}N^{1-2s_{c}}.$ (7.8)
Indeed, by Remark 1.8 and the fact that $\inf_{t\in I}N(t)\geq 1$, we may find
$c(\eta)>0$ so that
$\big{\|}|\nabla|^{s_{c}}u_{\leq
c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\leq\eta^{10};$
combining this inequality with Bernstein, we get
$\displaystyle N^{s_{c}}\big{\|}u_{\geq N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim N^{s_{c}}\big{\|}u_{N\leq\cdot\leq
c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+N^{s_{c}}\big{\|}u_{>c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\tfrac{N^{s_{c}}}{c(\eta)^{s_{c}}}\big{\|}|\nabla|^{s_{c}}u_{>c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\eta^{10}+N^{s_{c}}.$
Thus, taking $N$ sufficiently small, we recover (7.7). A similar argument
yields (7.8).
Next, we record the following inequality that will be useful below:
$\sup_{y\in\mathbb{R}^{d}}\bigg{|}\int_{\mathbb{R}^{d}}\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(x)\bar{\varphi}(x)\
dx\bigg{|}\lesssim\big{\|}|\nabla|^{s}\varphi\big{\|}_{2}\big{\|}|\nabla|^{1-s}\varphi\big{\|}_{2}$
(7.9)
for $0\leq s\leq 1$. Indeed, for fixed $y\in\mathbb{R}^{d}$, we can first
write
$\displaystyle\bigg{|}\int_{\mathbb{R}^{d}}\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(x)\bar{\varphi}(x)\
dx\bigg{|}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s}\tfrac{x-y}{|x-y|}\varphi\big{\|}_{2}\big{\|}|\nabla|^{-s}\nabla\varphi\big{\|}_{2}$
$\displaystyle\sim\big{\|}|\nabla|^{s}\tfrac{x-y}{|x-y|}\varphi\big{\|}_{2}\big{\|}|\nabla|^{1-s}\varphi\big{\|}_{2}.$
Thus, to prove (7.9), we need to see that the operator
$|\nabla|^{s}\tfrac{x-y}{|x-y|}|\nabla|^{-s}$ is bounded on $L_{x}^{2}$
(uniformly in $y$). When $s=0$, this is clear. When $s=1$, this follows from
the chain rule, Hardy’s inequality, and the boundedness of Riesz transforms.
The general case then follows from complex interpolation.
We now wish to apply the interaction Morawetz inequality (Lemma 7.2) with
$\varphi=u_{\geq N}$ and $\mathcal{N}=P_{\geq N}(|u|^{p}u)$, with $N\leq
N_{0}$. Together with (7.7), (7.8), (7.9), Bernstein, and the fact that $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$, an application of
Lemma 7.2 gives
$\displaystyle-$ $\displaystyle\iiint|u_{\geq
N}(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u_{\geq N}(t,x)|^{2}\,dx\,dy\,dt$
$\displaystyle+\quad\\!\\!\\!\\!\\!\\!\iiint|u_{\geq
N}(t,y)|^{2}\tfrac{x-y}{|x-y|}\cdot\\{P_{\geq N}(|u|^{p}u),u_{\geq
N}\\}_{\mathcal{P}}(t,x)\,dx\,dy\,dt$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}|\nabla|^{1/2}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}$
$\displaystyle\quad+\big{\|}\\{P_{\geq N}(|u|^{p}u),u_{\geq
N}\\}_{m}\big{\|}_{L_{t,x}^{1}}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{1-s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\eta^{20}N^{1-4s_{c}}+\eta^{10}N^{1-2s_{c}}\big{\|}\\{P_{\geq
N}(|u|^{p}u),u_{\geq N}\\}_{m}\big{\|}_{L_{t,x}^{1}}.$ (7.10)
Thus, to prove Proposition 7.1, we need to get sufficient control over the
mass and momentum bracket terms appearing above.
To begin, we consider the contribution of the momentum bracket term. We can
write
$\displaystyle\\{P_{\geq N}$ $\displaystyle(|u|^{p}u),u_{\geq
N}\\}_{\mathcal{P}}$ $\displaystyle=\\{|u|^{p}u,u\\}_{\mathcal{P}}-\\{|u_{\leq
N}|^{p}u_{\leq N},u_{\leq N}\\}_{\mathcal{P}}$
$\displaystyle\quad\quad-\\{|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N},u_{\leq
N}\\}_{\mathcal{P}}-\\{P_{\leq N}(|u|^{p}u),u_{\geq N}\\}_{\mathcal{P}}$
$\displaystyle=-\tfrac{p}{p+2}\nabla(|u|^{p+2}-|u_{\leq
N}|^{p+2})-\\{|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N},u_{\leq N}\\}_{\mathcal{P}}$
$\displaystyle\quad\quad-\\{P_{\leq N}(|u|^{p}u),u_{\geq N}\\}_{\mathcal{P}}$
$\displaystyle=:I+II+III.$
After an integration by parts, we see that term $I$ contributes to the left-
hand side of (7.10) a multiple of
$\displaystyle\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq
N}(t,x)|^{p+2}}{|x-y|}\,dx\,dy\,dt$ $\displaystyle-\iiint\frac{|u_{\geq
N}(t,y)|^{2}(|u|^{p+2}-|u_{\geq N}|^{p+2}-|u_{\leq
N}|^{p+2})(t,x)}{|x-y|}\,dx\,dy\,dt.$
For term $II$, we use
$\\{f,g\\}_{\mathcal{P}}=\nabla\text{\O}(fg)+\text{\O}(f\nabla g);$ when the
derivative hits the product, we integrate by parts, while for the second term
we simply bring absolute values inside the integral. In this way, we find that
term $II$ contributes to the right-hand side of (7.10) a multiple of
$\displaystyle\iiint\frac{|u_{\geq N}(t,y)|^{2}|(|u|^{p}u-|u_{\leq
N}|^{p}u_{\leq N})u_{\leq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$
$\displaystyle+\iiint|u_{\geq N}(t,y)|^{2}|(|u|^{p}u-|u_{\leq N}|^{p}u_{\leq
N})(t,x)|\ |\nabla u_{\leq N}(t,x)|\,dx\,dy\,dt.$
Finally, for term $III$, we integrate by parts when the derivative falls on
$u_{\geq N}$; in this way, we see that term $III$ contributes to the right-
hand side of (7.10) a multiple of
$\displaystyle\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|P_{\leq
N}(|u|^{p}u)(t,x)|}{|x-y|}\,dx\,dy\,dt$ $\displaystyle+\iiint|u_{\geq
N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|\nabla P_{\leq
N}(|u|^{p}u)(t,x)|\,dx\,dy\,dt.$
We next consider the mass bracket term in (7.10). Exploiting the fact that
$\\{|u_{\geq N}|^{p}u_{\geq N},u_{\geq N}\\}_{m}=0,$
we can write
$\displaystyle\\{P_{\geq N}(|u|^{p}u),u_{\geq N}\\}_{m}$
$\displaystyle=\\{P_{\geq N}(|u|^{p}u)-|u_{\geq N}|^{p}u_{\geq N},u_{\geq
N}\\}_{m}$ $\displaystyle=\\{P_{\geq N}(|u|^{p}u-|u_{\geq N}|^{p}u_{\geq
N}-|u_{\leq N}|^{p}u_{\leq N}),u_{\geq N}\\}_{m}$ $\displaystyle\ +\\{P_{\geq
N}(|u_{\leq N}|^{p}u_{\leq N}),u_{\geq N}\\}_{m}-\\{P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N}),u_{\geq N}\\}_{m}.$
We will now collect the contributions of the mass and momentum bracket terms
and insert them back into (7.10). We will also make use of the pointwise
inequalities
$\displaystyle\big{|}|f+g|^{p}(f+g)-|f|^{p}f\big{|}\lesssim|g|^{p+1}+|g|\
|f|^{p},$
$\displaystyle\big{|}|f+g|^{p+2}-|f|^{p+2}-|g|^{p+2}\big{|}\lesssim|f|\
|g|^{p+1}+|f|^{p+1}|g|.$
In this way, (7.10) becomes
$\displaystyle-\iiint|u_{\geq
N}(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u_{\geq N}(t,x)|^{2}\,dx\,dy\,dt$
(7.11) $\displaystyle\quad+\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq
N}(t,x)|^{p+2}}{|x-y|}\,dx\,dy\,dt$
$\displaystyle\lesssim_{u}\eta^{20}N^{1-4s_{c}}$ (7.12)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}|u_{\leq N}|^{p}u_{\geq
N}^{2}\big{\|}_{L_{t,x}^{1}}$ (7.13)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}|u_{\geq N}|^{p+1}u_{\leq
N}\big{\|}_{L_{t,x}^{1}}$ (7.14)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}P_{\geq N}(|u_{\leq N}|^{p}u_{\leq
N})u_{\geq N}\big{\|}_{L_{t,x}^{1}}$ (7.15)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}P_{\leq N}(|u_{\geq N}|^{p}u_{\geq
N})u_{\geq N}\big{\|}_{L_{t,x}^{1}}$ (7.16)
$\displaystyle+\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|u_{\leq
N}(t,x)|^{p+1}}{|x-y|}\,dx\,dy\,dt$ (7.17) $\displaystyle+\iiint\frac{|u_{\geq
N}(t,y)|^{2}|u_{\geq N}(t,x)|^{p+1}|u_{\leq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$
(7.18) $\displaystyle+\iiint\frac{|u_{\geq N}(t,y)|^{2}|P_{\leq
N}(|u|^{p}u)(t,x)|\,|u_{\geq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$ (7.19)
$\displaystyle+\iiint|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|u_{\leq
N}(t,x)|^{p}|\nabla u_{\leq N}(t,x)|\,dx\,dy\,dt$ (7.20)
$\displaystyle+\iiint|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|^{p+1}|\nabla
u_{\leq N}(t,x)|\,dx\,dy\,dt$ (7.21) $\displaystyle+\iiint|u_{\geq
N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|\nabla P_{\leq
N}(|u|^{p}u)(t,x)|\,dx\,dy\,dt.$ (7.22)
To complete the proof of Proposition 7.1, we need to show that the error terms
(7.12) through (7.22) are acceptable, in the sense that they can be controlled
by $\eta(N^{1-4s_{c}}+K).$ Clearly, (7.12) is acceptable.
Next, we consider (7.13). Using Hölder, Sobolev embedding, (7.4), and (7.5),
we get
$\displaystyle\big{\|}|u_{\leq N}|^{p}u_{\geq N}^{2}\big{\|}_{L_{t,x}^{1}}$
$\displaystyle\lesssim\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{dp}}^{p}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}^{2}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{\frac{2dp}{dp-2}}}^{p}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}^{2}$
$\displaystyle\lesssim_{u}\eta^{p}N^{-2s_{c}}(1+N^{4s_{c}-1}K),$
which renders (7.13) acceptable.
We now turn to (7.14). For this term, we can again use Hölder, Sobolev
embedding, (7.4), and (7.5) to see
$\displaystyle\big{\|}|u_{\geq N}|^{p+1}u_{\leq N}\big{\|}_{L_{t,x}^{1}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$ $\displaystyle\lesssim_{u}\eta
N^{-2s_{c}}(1+N^{4s_{c}-1}K).$
Thus this term is acceptable as well. Before proceeding, however, we note that
it is this term that has forced us to exclude the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ from this paper; we postpone
further discussion until Remark 7.4 below.
We next turn to (7.15); using Hölder, Bernstein, the fractional chain rule,
Sobolev embedding, (7.4), and (7.5), we see
$\displaystyle\big{\|}P_{\geq N}$ $\displaystyle(|u_{\leq N}|^{p}u_{\leq
N})u_{\geq N}\big{\|}_{L_{t,x}^{1}}$ $\displaystyle\lesssim
N^{-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}|\nabla|^{s_{c}}(|u_{\leq
N}|^{p}u_{\leq N})\big{\|}_{L_{t}^{\frac{4}{3}}L_{x}^{\frac{2d}{d+1}}}$
$\displaystyle\lesssim N^{-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{2dp}{3}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$ $\displaystyle\lesssim
N^{-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{2dp}{dp-1}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\eta^{p+1}N^{-2s_{c}}(1+N^{4s_{c}-1}K),$
so that (7.15) is also acceptable.
For the final term originating from the mass bracket, (7.16), we use Hölder,
Bernstein, Sobolev embedding, (7.4), and (7.7) to see
$\displaystyle\big{\|}P_{\leq N}(|u_{\geq N}|^{p}u_{\geq N})u_{\geq
N}\big{\|}_{L_{t,x}^{1}}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d+4}}}$
$\displaystyle\lesssim N^{s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|u_{\geq N}|^{p}u_{\geq
N}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3dp}{3dp+2p-6}}}$
$\displaystyle\lesssim N^{s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{3}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}N^{-2s_{c}}(1+N^{4s_{c}-1}K),$
which shows that (7.16) is acceptable.
We now turn to the terms originating from the momentum bracket. First,
consider (7.17). By Hölder, Hardy–Littlewood–Sobolev, Sobolev embedding,
Bernstein, (7.4), (7.5), and (7.7), we can estimate
$\displaystyle\eqref{error 6}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{3d}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$
$\displaystyle\quad\times\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{3dp}{6-p}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-8}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim\big{\|}|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3d}{3d-2}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}N^{1-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}$
$\displaystyle\lesssim_{u}\eta^{12}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
so that (7.17) is acceptable.
For (7.18), we consider two cases. If $|u_{\leq N}|\leq 10^{-100}|u_{\geq
N}|,$ then we can absorb this term into the left-hand side of the inequality,
provided we can show
$\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq
N}(t,x)|^{p+2}}{|x-y|}\,dx\,dy\,dt<\infty.$ (7.23)
On the other hand, if $|u_{\geq N}|\leq 10^{100}|u_{\leq N}|$, then we are
back in the situation of (7.17), which we have already handled. Thus, to
render (7.18) acceptable, it remains to prove (7.23). To this end, we define
$\theta=\tfrac{4dp-16-3p}{2(dp-4)}\in(0,p+2),$
and use Hölder, Hardy–Littlewood–Sobolev, Sobolev embedding, Lemma 1.11, and
interpolation to estimate
$\displaystyle\text{LHS}\eqref{finiteness of bootstrap terms}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{4}L_{x}^{4d}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\theta}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\frac{2p(2dp-5)}{3(dp-4)}}L_{x}^{\frac{dp(2dp-5)}{dp+2}}}^{p+2-\theta}$
$\displaystyle\lesssim\big{\|}|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{4}L_{x}^{\frac{4d}{4d-3}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\theta}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\frac{2p(2dp-5)}{3(dp-4)}}L_{x}^{\frac{2dp(2dp-5)}{2(dp)^{2}-11dp+24}}}^{p+2-\theta}$
$\displaystyle\lesssim_{u}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{4d}{2d-3}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{1+\theta}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{1/4}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}N^{-s_{c}(1+\theta)}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}N^{1/4-s_{c}(2+\theta)}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}N^{1-4s_{c}}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{1}{4}+\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}N^{1-4s_{c}}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right),$
which gives (7.23), and thereby shows that (7.18) is acceptable.
Next, we turn to (7.19). Denoting
$G=|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N}-|u_{\geq N}|^{p}u_{\geq N},$
we begin by writing
$\displaystyle\eqref{error 8}$ $\displaystyle\quad=\iiint\frac{|u_{\geq
N}(t,y)|^{2}|P_{\leq N}(|u_{\leq N}|^{p}u_{\leq N})(t,x)|\,|u_{\geq
N}(t,x)|}{|x-y|}\,dx\,dy\,dt$ (7.24)
$\displaystyle\quad\quad+\iiint\frac{|u_{\geq N}(t,y)|^{2}|P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})(t,x)|\,|u_{\geq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$ (7.25)
$\displaystyle\quad\quad+\iiint\frac{|u_{\geq N}(t,\\!y)|^{2}|P_{\leq
N}G(t,x)|\,|u_{\geq N}(t,x)|}{|x-y|}\,dx\,dy\,dt.$ (7.26)
For (7.24), we can write
$\displaystyle\eqref{8a}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{3d}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{3dp}{6-p}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-8}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{12}N^{1-4s_{c}}(1+N^{4s_{c}-1}K)$
by the same arguments that dealt with (7.17).
For (7.25), we can use Hölder, Hardy–Littlewood–Sobolev, Bernstein, Sobolev
embedding, (7.4), (7.5), and (7.7) to estimate
$\displaystyle\eqref{8b}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}\tfrac{1}{|x|}\ast(P_{\leq
N}(|u_{\geq N}|^{p}u_{\geq N})u_{\geq
N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3d}{2}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})u_{\geq
N}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3d}{3d-1}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}P_{\leq N}(|u_{\geq N}|^{p}u_{\geq
N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d-2}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}N^{1+s_{c}}\big{\|}P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3dp}{3dp+2p-6}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}N^{1+s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{10}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
which renders (7.25) acceptable.
For (7.26), we first note
$|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N}-|u_{\geq N}|^{p}u_{\geq
N}=\text{\O}(u_{\geq N}u_{\leq N}|u|^{p-1}),$
so that using Hölder, Hardy–Littlewood–Sobolev, Bernstein, Sobolev embedding,
(7.4), (7.5), and (7.7), we get
$\displaystyle\eqref{8c}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{3d}}\big{\|}P_{\leq N}(\text{\O}(u_{\geq
N}u_{\leq
N}|u|^{p-1}))\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d+2}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$ $\displaystyle\lesssim
N\big{\|}|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3d}{3d-2}}}\big{\|}\text{\O}(u_{\geq
N}u_{\leq
N}|u|^{p-1})\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d+8}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$ $\displaystyle\lesssim
N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{21}N^{1-4s_{c}}(1+N^{4s_{c}-1}K).$
Thus (7.26), and so (7.19), is acceptable.
We now turn to (7.20). By Hölder, Sobolev embedding, Bernstein, (7.5), and
(7.7), we estimate
$\displaystyle\eqref{error 9}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{3}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{dp}}^{p}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$ $\displaystyle\lesssim
N^{1-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{3}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{\frac{2dp}{dp-2}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\eta^{p+31}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
so that (7.20) is acceptable.
For (7.21), we use Hölder, Sobolev embedding, Bernstein, (7.4), (7.5), and
(7.7) to get
$\displaystyle\eqref{error 10}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}$
$\displaystyle\lesssim_{u}N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$
$\displaystyle\lesssim_{u}\eta^{21}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
which renders (7.21) acceptable.
Finally, we consider (7.22). We begin by writing
$\displaystyle\eqref{error 11}\lesssim$ $\displaystyle\,\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq N}\nabla P_{\leq
N}(|u_{\leq N}|^{p}u_{\leq N})\big{\|}_{L_{t,x}^{1}}$ (7.27) $\displaystyle+$
$\displaystyle\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq N}\nabla P_{\leq
N}(|u_{\geq N}|^{p}u_{\geq N})\big{\|}_{L_{t,x}^{1}}$ (7.28) $\displaystyle+$
$\displaystyle\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq N}\nabla P_{\leq
N}(|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N}-|u_{\geq N}|^{p}u_{\geq
N})\big{\|}_{L_{t,x}^{1}}.$ (7.29)
To begin, we use Hölder, the chain rule, and the arguments that gave (7.20) to
see
$\displaystyle\eqref{11a}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{3}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{dp}}^{p}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\eta^{p+31}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
so that (7.27) is acceptable.
For (7.28), we argue essentially as we did for (7.16). That is, we use Hölder,
Bernstein, Sobolev embedding, (7.4), and (7.7) to estimate
$\displaystyle\eqref{11b}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}\nabla P_{\leq
N}(|u_{\geq N}|^{p}u_{\geq
N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d+4}}}$
$\displaystyle\lesssim N^{1+s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|u_{\geq N}|^{p}u_{\geq
N}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3dp}{3dp+2p-6}}}$
$\displaystyle\lesssim N^{1+s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{3}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{20}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
which gives that (7.28) is acceptable.
For (7.29), we argue similarly to the case of (7.26). In particular, we use
Hölder, Bernstein, Sobolev embedding, (7.4), (7.5), and (7.7) to see
$\displaystyle\eqref{11c}$ $\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}\text{\O}(u_{\leq
N}u_{\geq N}|u|^{p-1})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d+4}}}$
$\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{21}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$
which gives that (7.29). Collecting the estimates for (7.27), (7.28), and
(7.29), we see that (7.22) is acceptable. This completes the proof of
Proposition 7.1. ∎
###### Remark 7.4.
Let us discuss why (7.14) has forced us to exclude the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ from this paper. As one can see in
the proof above, in the cases we consider, this term is fairly harmless.
However, once $s_{c}>\tfrac{3}{4}$ in dimension $d=3$ (which corresponds to
$p>\tfrac{8}{3}$), this term becomes a problem; put simply, we end up with too
many copies of $u_{\geq N}$ to deal with.
This problem has already been encountered in the energy-critical setting
($s_{c}=1$) in dimension $d=3$; in this case, one can overcome the hurdle by
applying a spatial truncation to the weight $a$. One can refer to [15] for the
original argument, wherein spatial truncation is applied at various levels and
subsequently averaged. The authors of [36] revisit the result of [15] in the
context of minimal counterexamples; at this point in the argument, they choose
to work with a more carefully designed spatial truncation, which removes the
need for any subsequent averaging argument.
This discussion begs the question: why doesn’t spatial truncation work in our
setting? To answer this, we need to understand how spatial truncations affect
the argument that leads to Proposition 7.1. What we find is that spatial
truncations ruin the convexity properties of $a$ that made some of the terms
in the proof of Lemma 7.2 positive; thus, to establish Proposition 7.1 with a
further spatial truncation, we have to control additional error terms. It
turns out that one of these additional error terms requires uniform control
over $\big{\|}u\big{\|}_{L_{x}^{p+2}}$, while another requires uniform control
over $\big{\|}\nabla u\big{\|}_{L_{x}^{2}}$ (see [36, Lemma 6.5 and Lemma
6.6]). In the energy-critical case, one can use the conservation of energy to
push the argument through, while in our cases, we cannot proceed without some
significant new input. We have therefore abandoned the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ in this paper.
For a further discussion of these issues, refer to [36], especially Remark 6.9
therein.
## 8\. The quasi-soliton scenario
In this section we preclude the existence of almost periodic solutions as in
Theorem 1.13 for which $\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$ We will
show that their existence is inconsistent with the frequency-localized
interaction Morawetz inequality (Proposition 7.1).
Before we begin, we note that
$-\Delta(\tfrac{1}{|x|})=\left\\{\begin{array}[]{ll}4\pi\delta&d=3\\\
\tfrac{d-3}{|x|^{3}}&d\geq 4.\end{array}\right.$
Thus, if $d=3$, the conclusion of Proposition 7.1 reads
$\int_{I}\int_{\mathbb{R}^{3}}|u_{\geq
N}(t,x)|^{4}\,dx\,dt\lesssim_{u}\eta(N^{1-4s_{c}}+K)$ (8.1)
for $N\leq N_{0}(\eta)$, while if $d\in\\{4,5\\}$, the conclusion of
Proposition 7.1 reads
$\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{|u_{\geq
N}(t,x)|^{2}|u_{\geq
N}(t,y)|^{2}}{|x-y|^{3}}\,dx\,dy\,dt\lesssim_{u}\eta(N^{1-4s_{c}}+K)$ (8.2)
for $N\leq N_{0}(\eta)$.
We now turn to
###### Theorem 8.1 (No quasi-solitons).
Let $(d,s_{c})$ satisfy (1.4). Then there are no almost periodic solutions
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) with $N(t)\equiv
N_{k}\geq 1$ on each characteristic subinterval $J_{k}\subset[0,T_{max})$ that
satisfy both
$\big{\|}u\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}([0,T_{max})\times\mathbb{R}^{d})}=\infty$
and
$\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$ (8.3)
###### Proof.
Suppose towards a contradiction that such a solution $u$ exists. We will need
the following
###### Lemma 8.2.
There exist $N_{0}>0$ and $C(u)>0$ so that
$\inf_{t\in[0,T_{max})}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{2}\,dx\gtrsim_{u}1\indent\text{for all }N\leq N_{0}.$ (8.4)
We provide the proof of Lemma 8.2 below; let us first take it for granted and
use it to complete the proof of Theorem 8.1. We let $I$ be a compact time
interval, which is a union of contiguous subintervals $J_{k}$, and let
$\eta>0$ be a small parameter. We take $C(u)$ and $N_{0}$ as in Lemma 8.2;
then, choosing $N_{0}$ possibly smaller, we can guarantee by Proposition 7.1
that (8.1) or (8.2) holds (depending on the dimension) for all $N\leq N_{0}$.
We now consider two cases:
Case 1. When $d=3$, we first note by Hölder’s inequality that
$\displaystyle\big{(}\textstyle\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}$
$\displaystyle|u_{\geq
N}(t,x)|^{2}\,dx\big{)}^{2}\lesssim_{u}N(t)^{-3}\textstyle\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{4}\,dx.$
Using this inequality, followed by (8.4), we find
$\displaystyle\int_{I}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}$
$\displaystyle|u_{\geq N}(t,x)|^{4}\,dx\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\left(\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{2}\,dx\right)^{2}N(t)^{3}\,dt$
$\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-4s_{c}}\,dt$
for all $N\leq N_{0}$. Thus, appealing to (8.1), we find
$\displaystyle\eta\left[N^{1-4s_{c}}+\int_{I}N(t)^{3-4s_{c}}\right]$
$\displaystyle\gtrsim_{u}\int_{I}\int_{\mathbb{R}^{3}}|u_{\geq
N}(t,x)|^{4}\,dx\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{4}\,dx\,dt$ $\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-4s_{c}}\,dt$
(8.5)
for all $N\leq N_{0}$.
Case 2. If $d\in\\{4,5\\}$, we once again use (8.4), but use (8.2) instead of
(8.1). We find
$\displaystyle\eta\bigg{[}N^{1-4s_{c}}$
$\displaystyle+\int_{I}N(t)^{3-4s_{c}}\,dt\bigg{]}$
$\displaystyle\gtrsim_{u}\int_{I}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|u_{\geq
N}(t,x)|^{2}|u_{\geq N}(t,y)|^{2}}{|x-y|^{3}}\,dx\,dy\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\iint_{|x-y|\leq\frac{2C(u)}{N(t)}}\big{[}\tfrac{N(t)}{2C(u)}\big{]}^{3}|u_{\geq
N}(t,x)|^{2}|u_{\geq N}(t,y)|^{2}\,dx\,dy\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\big{[}\tfrac{N(t)}{C(u)}\big{]}^{3}\left(\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{2}\,dx\right)^{2}\,dt$
$\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-4s_{c}}\,dt$ (8.6)
for all $N\leq N_{0}$. Thus, continuing from (8.5) or (8.6), we see that in
either case, for $\eta$ sufficiently small depending on $u$, we get
$\int_{I}N(t)^{3-4s_{c}}\,dt\lesssim_{u}N^{1-4s_{c}}\indent\text{for all
}I\subset[0,T_{max})\text{ and all }N\leq N_{0}.$
Recalling (8.3), we now reach a contradiction by taking $I$ sufficiently large
inside $[0,T_{max}).$ ∎
Finally, we turn to the
###### Proof of Lemma 8.2.
Let $\eta_{0}>0$ be a small parameter to be determined later. As
$\inf_{t\in[0,T_{max})}N(t)\geq 1$, for $N_{0}=N_{0}(\eta_{0})$ sufficiently
small, we can guarantee
$\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}([0,T_{max})\times\mathbb{R}^{d})}\leq\eta_{0}\indent\text{for
}N\leq N_{0}.$
Then, given _any_ $C(u)>0$ and $N\leq N_{0}$, we can use Hölder’s inequality
and Sobolev embedding to estimate
$\displaystyle\bigg{|}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{>N}(t,x)|^{2}-|u(t,x)|^{2}\,dx\bigg{|}$
$\displaystyle\lesssim_{u}N(t)^{-2s_{c}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}$
$\displaystyle\lesssim_{u}\eta_{0}N(t)^{-2s_{c}}$
for all $t\in[0,T_{max}).$ Thus, if we can show that for $C(u)$ sufficiently
large, we have
$\inf_{t\in[0,T_{max})}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{2}\,dx\gtrsim_{u}1,$
(8.7)
we will have (8.4) by choosing $\eta_{0}=\eta_{0}(u)$ sufficiently small.
We now turn to (8.7). We first choose $c=c(\eta_{0})$ sufficiently large that
$\big{\|}|\nabla|^{s_{c}}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}<\eta_{0}.$
(8.8)
We then notice that by Hölder, Bernstein, Sobolev embedding, and (8.8), we
have
$\displaystyle\bigg{|}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|$ $\displaystyle
u(t,x)|^{2}-|u_{\leq cN(t)}(t,x)|^{2}\,dx\bigg{|}$ $\displaystyle\lesssim
N(t)^{-s_{c}}\big{\|}u_{>cN(t)}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}\big{\|}u(t)\big{\|}_{L_{x}^{\frac{dp}{2}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{u}\eta_{0}N(t)^{-2s_{c}}$ (8.9)
for all $t\in[0,T_{max})$. Thus, if we can show that for $C(u)$ sufficiently
large, we have
$\inf_{t\in[0,T_{max})}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx\gtrsim_{u}1,$ (8.10)
then (8.7) will follow by taking $\eta_{0}=\eta_{0}(u)$ sufficiently small.
Let us therefore turn to establishing (8.10). We begin by choosing $C(u)$
sufficiently large that
$\inf_{t\in[0,T_{max})}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{dp}{2}}\,dx\gtrsim_{u}1$
(cf. Remark 1.7). Then, with $c=c(\eta_{0})$ as above, we see by Hölder’s
inequality, Sobolev embedding, and (8.8) that
$\displaystyle\bigg{|}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)$
$\displaystyle|^{\frac{dp}{2}}-|u_{\leq
cN(t)}(t,x)|^{\frac{dp}{2}}\,dx\bigg{|}$
$\displaystyle\lesssim\big{\|}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{\frac{dp}{2}-1}$
$\displaystyle\lesssim_{u}\eta_{0}$
for all $t\in[0,T_{max}).$ Thus for $\eta_{0}=\eta_{0}(u)$ sufficiently small,
we have
$\inf_{t\in[0,T_{max})}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{\frac{dp}{2}}\,dx\gtrsim_{u}1.$ (8.11)
Finally, by Hölder and Bernstein, we estimate
$\displaystyle\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|$ $\displaystyle u_{\leq
cN(t)}(t,x)|^{\frac{dp}{2}}\,dx$ $\displaystyle\lesssim\big{\|}u_{\leq
cN(t)}(t)\big{\|}_{L_{x}^{\infty}(\mathbb{R}^{d})}^{\frac{dp}{2}-2}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx$
$\displaystyle\lesssim_{u}N(t)^{2s_{c}}\big{\|}u(t)\big{\|}_{L_{x}^{\frac{dp}{2}}(\mathbb{R}^{d})}^{\frac{dp}{2}-2}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx.$
$\displaystyle\lesssim_{u}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx$ (8.12)
for all $t\in[0,T_{max})$. Combining (8.11) and (8.12) now yields (8.10),
which completes the proof of Lemma 8.2. ∎
## Appendix A Some basic estimates
We collect here some basic estimates that are useful in Section 4. We begin
with
###### Lemma A.1.
Let $1<p\leq 2$. Then
$\displaystyle\big{|}\,|a+c|^{p}-|a|^{p}-|b+c|^{p}+|b|^{p}\,\big{|}\lesssim|a-b|\,|c|^{p-1}$
(A.1)
for all $a,b,c\in\mathbb{C}$.
###### Proof.
Defining $G(z):=|z+c|^{p}-|z|^{p}$, the fundamental theorem of calculus gives
$\displaystyle\text{LHS}\eqref{stupid stupid
stupid}=\bigg{|}(a-b)\int_{0}^{1}G_{z}(b+\theta(a-b))\,d\theta+\overline{(a-b)}\int_{0}^{1}G_{\bar{z}}(b+\theta(a-b))\,d\theta\bigg{|}.$
Thus, to establish (A.1), it will suffice to establish
$\displaystyle|G_{z}(z)|+|G_{\bar{z}}(z)|\lesssim|c|^{p-1}\quad\text{uniformly
for }z\in\mathbb{C}.$
That is, we need to show
$\big{|}\,|z+c|^{p-2}(z+c)-|z|^{p-2}z\,\big{|}\lesssim|c|^{p-1}$
uniformly in $z$. If $c=0$, this inequality is obvious. Otherwise, setting
$z=c\,\zeta$ reduces the problem to showing
$\big{|}\,|z+1|^{p-2}(z+1)-|z|^{p-2}z\,\big{|}\lesssim 1$ (A.2)
uniformly in $z$. For $|z|\lesssim 1$, we immediately get (A.2) from the
triangle inequality. For $|z|\gg 1$, we can use the fundamental theorem of
calculus and the fact that $p\leq 2$ to see
$\big{|}\,|z+1|^{p-2}(z+1)-|z|^{p-2}z\,\big{|}\lesssim|z|^{p-2}\lesssim 1.$
Thus, we see that (A.2) holds, which completes the proof of Lemma A.1. ∎
Next, we record a few inequalities in the spirit of [48, $\S$2.3].
###### Lemma A.2.
Let $M$ denote the Hardy–Littlewood maximal function, and let $\check{\psi}$
denote the convolution kernel of the Littlewood–Paley projection $P_{1}$. For
a fixed function $f,$ $y\in\mathbb{R}^{d},$ and $N\in 2^{\mathbb{Z}}$, we have
$\displaystyle\textstyle\int_{\mathbb{R}^{d}}N^{d}|\check{\psi}(Ny)|\,|f(x-y)|\,dy$
$\displaystyle\lesssim M(f)(x),$ (A.3) $\displaystyle|\delta_{y}f_{N}(x)|$
$\displaystyle\lesssim N|y|\left\\{M(f_{N})(x)+M(f_{N})(x-y)\right\\},$ (A.4)
$\displaystyle\textstyle\int_{\mathbb{R}^{d}}N^{d}|y|\,|\check{\psi}(Ny)|\,\,dy$
$\displaystyle\lesssim\tfrac{1}{N}.$ (A.5)
###### Proof.
We begin with (A.3). Note first that
$\eta:=N^{d}|\check{\psi}(Ny)|$
is a spherically symmetric, decreasing function of radius; thus, we can write
$\eta(y)=\int_{0}^{\infty}\chi_{B(0,r)}(y)(-\eta^{\prime}(r))\,dr,$
where $\eta^{\prime}:=\tfrac{\partial\eta}{\partial r}$. We can then use the
definition of the Hardy–Littlewood maximal function and integrate by parts to
estimate
$\displaystyle\text{LHS}\eqref{aux 1}$
$\displaystyle\lesssim\int_{0}^{\infty}\left(\int_{|y|\leq
r}|f(x-y)|\,dy\right)(-\eta^{\prime}(r))\,dr$
$\displaystyle\lesssim\left(\int_{0}^{\infty}\eta(r)r^{d-1}\,dr\right)M(f)(x)$
$\displaystyle\lesssim_{\psi}M(f)(x).$
For (A.4), we begin by defining
$\psi_{0}(\xi)=\psi(2\xi)+\psi(\xi)+\psi(\xi/2)$, the ‘fattened’
Littlewood–Paley multiplier. Then we can write
$|\delta_{y}f_{N}(x)|=\big{|}\textstyle\int[N^{d}\check{\psi_{0}}(N(z-y))-N^{d}\check{\psi_{0}}(Nz)]f_{N}(x-z)\,dz\big{|}.$
(A.6)
If $N|y|\geq 1$, we can use the triangle inequality and argue as above to see
that
$|\delta_{y}f_{N}(x)|\leq M(f_{N})(x-y)+M(f_{N})(x),$
giving (A.4) in this case. If instead $N|y|\leq 1$, we can use the fact that
$\check{\psi_{0}}$ is Schwartz to estimate
$|\check{\psi_{0}}(N(z-y))-\check{\psi_{0}}(Nz)|\lesssim
N|y|(1+N|z|)^{-100d}.$
Then continuing from (A.6) and once again arguing as for (A.3), we find
$|\delta_{y}f_{N}(x)|\lesssim N|y|M(f_{N})(x),$
which gives (A.4) in this case.
Finally, we note that since $\check{\psi}$ is Schwartz, we have
$\int_{\mathbb{R}^{d}}N^{d}|Ny|\,|\check{\psi}(Ny)|\,dy\lesssim 1,$
which immediately gives (A.5). ∎
## References
* [1] P. Bégout and A. Vargas, _Mass concentration phenomena for the $L^{2}$-critical nonlinear Schrödinger equation_. Trans. Amer. Math. Soc. 359 (2007), 5257–5282. MR2327030
* [2] J. Bourgain, _Refinements of Strichartz’ inequality and applications to 2d-NLS with critical nonlinearity._ Int. Math. Res. Not. (1998), 253–283. MR1616917
* [3] J. Bourgain, _Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case._ J. Amer. Math. Soc. 12 (1999), 145–171. MR1626257
* [4] A. Bulut, _Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation_. J. Funct. Anal. 263 (2012), 1609–1660. MR2948225
* [5] A. Bulut, _The radial defocusing energy-supercritical cubic nonlinear wave equation in $\mathbb{R}^{1+5}$_. Preprint arXiv:1104.2002
* [6] A. Bulut, _The defocusing energy-supercritical cubic nonlinear wave equation in dimension five_. Preprint arXiv:1112.0629
* [7] R. Carles and S. Keraani, _On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The $L^{2}$-critical case._ Trans. Amer. Math. Soc. 359 (2007), 33-62. MR2247881
* [8] T. Cazenave and F. B. Weissler, _The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$._ Nonlinear Anal. 14 (1990), 807–836. MR1055532
* [9] T. Cazenave, _Semilinear Schrödinger equations_. Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003. MR2002047
* [10] M. Christ and M. Weinstein, _Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation._ J. Funct. Anal. 100 (1991), 87–109. MR1124294
* [11] R. R. Coifman and Y. Meyer, _On commutators of singular integrals and bilinear singular integrals._ Trans. Amer. Math. Soc. 212 (1975), 315–331. MR0380244
* [12] R. R. Coifman and Y. Meyer, _Au délà des opérateaurs pseudo-différentiels,_ Astérique 57 (1979). MR0518170
* [13] R. R. Coifman and Y. Meyer, _Ondelletes et opérateurs III, Operateurs multilineaires._ Actualites Mathematiques, Hermann, Paris (1991). MR1160989
* [14] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, _Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^{3}$._ Comm. Pure Appl. Math. 57 (2004), 987–1014. MR2053757
* [15] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, _Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$._ Ann. Math. 167 (2008), 767–865. MR2415387
* [16] P. Constantin, J.-C. Saut, _Local smoothing properties of dispersive equations._ J. Amer. Math. Soc. 1 (1988), 413–439. MR0928265
* [17] B. Dodson, _Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d\geq 3$_. J. Amer. Math. Soc. 25 (2012), 429–463. MR2869023
* [18] B. Dodson, _Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$_. Preprint arXiv:1006.1375
* [19] B. Dodson, _Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=1$_. Preprint arXiv:1010.0040
* [20] B. Dodson, _Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state_. Preprint arXiv:1104.1114
* [21] L. Dong, X. Zhang, _Stability of solutions for nonlinear Schrödinger equations in critical spaces._ Sci. China Math. 54 (2011), 973–986. MR2800921
* [22] J. Ginibre and G. Velo, _Smoothing properties and retarded estimates for some dispersive evolution equations_. Comm. Math. Phys. 144 (1992), 163–188. MR1151250
* [23] G. Grillakis, _On nonlinear Schrödinger equations_. Comm. PDE 25 (2000), 1827–1844. MR1778782
* [24] M. Keel and T. Tao, _Endpoint Strichartz estimates_. Amer. J. Math. 120 (1998), 955–980. MR1646048
* [25] C. E. Kenig and F. Merle, _Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case._ Invent. Math. 166 (2006), 645–675. MR2257393
* [26] C. E. Kenig and F. Merle, _Scattering for $\dot{H}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions._ Trans. Amer. Math. Soc. 362 (2010), 1937–1962. MR2574882
* [27] C. E. Kenig and F. Merle, _Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications._ Amer. J. Math. 133 (2011), 1029–1065. MR2823870
* [28] S. Keraani, _On the defect of compactness for the Strichartz estimates for the Schrödinger equations._ J. Diff. Eq. 175 (2001), 353–392. MR1855973
* [29] S. Keraani, _On the blow up phenomenon of the critical nonlinear Schrödinger equation._ J. Funct. Anal. 235 (2006), 171–192. MR2216444
* [30] R. Killip, T. Tao, and M. Vişan, _The cubic nonlinear Schrödinger equation in two dimensions with radial data._ J. Eur. Math. Soc. (JEMS) 11 (2009), 1203–1258. MR2557134
* [31] R. Killip and M. Vişan, _Nonlinear Schrödinger equations at critical regularity._ To appear in proceedings of the Clay summer school “Evolution Equations”, June 23–July 18, 2008, Eidgenössische Technische Hochschule, Zürich.
* [32] R. Killip and M. Vişan, _The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher._ Amer. J. Math. 132 (2010), 361–424. MR2654778
* [33] R. Killip and M. Vişan, _Energy-supercritical NLS: critical $\dot{H}^{s}$-bounds imply scattering._ Comm. Partial Differential Equations _35_ (2010), 945–987. MR2753625
* [34] R. Killip and M. Vişan, _The defocusing energy-supercritical nonlinear wave equation in three space dimensions_. Trans. Amer. Math. Soc. 363 (2011), 3893–3934. MR2775831
* [35] R. Killip and M. Vişan, _The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions_. Proc. Amer. Math. Soc. 139 (2011), 1805–1817. MR2763767
* [36] R. Killip and M. Vişan, _Global well-posedness and scattering for the defocusing quintic NLS in three dimensions._ To appear in Analysis and PDE. Preprint arXiv:math/1102.1192
* [37] R. Killip, M. Vişan, X. Zhang, _The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher._ Analysis and PDE 1 (2008), 229–266. MR2472890
* [38] J. Lin and W. Strauss, _Decay and scattering of solutions of a nonliner Schrödinger equation._ J. Funct. Anal. 30 (1978), no. 2, 245–263. MR0515228
* [39] F. Merle and L. Vega, _Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in 2$D$._ Int. Math. Res. Not. (1998), 399–425. MR1628235
* [40] E. Ryckman and M. Vişan, _Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$._ Amer. J. Math. 129 (2007), 1–60. MR2288737
* [41] S. Shao, _Maximizers for the Strichartz inequalities and Sobolev-Strichartz inequalities for the Schrödinger equation._ Electron. J. Differential Equations (2009), 1–13. MR2471112
* [42] R. Shen, _Global well-posedness and scattering of defocusing energy subcritical nonlinear wave equation in dimension 3 with radial data._ Preprint arXiv:1111.2345
* [43] R. Shen, _On the energy subcritical, non-linear wave equation with radial data for $p\in(3,5)$_. Preprint arXiv:1208.2108
* [44] P. Sjölin, _Regularity of solutions to the Schrödinger equation._ Duke Math. J. 55 (1987), 699-715. MR0904948
* [45] E. M. Stein, _Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals._ Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993. MR1232192
* [46] R. S. Strichartz, _Multipliers on fractional Sobolev spaces._ J. Math. Mech. 16 (1967), 1031–1060. MR0215084
* [47] R. S. Strichartz, _Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations_. Duke Math. J. 44 (1977), 705–774. MR0512086
* [48] M. E. Taylor, _Tools for PDE_. Mathematical Surveys and Monographs, 81. American Mathematical Society, Providence, RI, 2000. MR1766415
* [49] T. Tao, _Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data._ New York J. of Math. 11 (2005), 57–80. MR2154347
* [50] T. Tao, _Nonlinear dispersive equations. Local and global analysis_. CBMS Regional Conference Series in Mathematics, 106. American Mathematical Society, Providence, RI, 2006. MR2233925
* [51] T. Tao and M. Vişan, _Stability of energy-critical nonlinear Schrödinger equations in high dimensions._ Electron. J. Diff. Eqns. (2005), 1–28 MR2174550
* [52] T. Tao, M. Vişan, and X. Zhang, _Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions._ Duke Math. J. 140 (2007), 165–202. MR2355070
* [53] T. Tao, M. Vişan, and X. Zhang, _Minimal-mass blowup solutions of the mass-critical NLS._ Forum Math. 20 (2008), 881–919. MR2445122
* [54] L. Vega, _Schrödinger equations: pointwise convergence to the initial data._ Proc. Amer. Math. Soc. 102 (1988), 874–878. MR0934859
* [55] M. Vişan, _The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions._ Duke Math. J. 138 (2007), 281–374. MR2318286
* [56] M. Vişan, _The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher._ Ph.D. Thesis, UCLA, 2006.
* [57] M. Vişan, _Global well-posedness and scattering for the defocusing cubic NLS in four dimensions._ Int. Math. Res. Not. (2011), 1037–1067.
|
arxiv-papers
| 2012-09-20T16:51:48 |
2024-09-04T02:49:35.364069
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jason Murphy",
"submitter": "Jason Murphy",
"url": "https://arxiv.org/abs/1209.4582"
}
|
1209.4879
|
scrheadings pageheadfoot sectioning
# Coloring $d$-Embeddable $k$-Uniform Hypergraphs
Carl Georg Heise Zentrum Mathematik, Technische Universität München, Germany,
{cgh, taraz}@ma.tum.de Konstantinos Panagiotou Mathematisches Institut,
Ludwig-Maximilians-Universität München, Germany, [email protected] Oleg
Pikhurko Mathematics Institute, University of Warwick, Coventry, UK,
[email protected] Anusch Taraz Zentrum Mathematik, Technische
Universität München, Germany, {cgh, taraz}@ma.tum.de
11footnotetext: Partially supported by the ENB graduate program TopMath and
DFG grant GR 993/10-1. The author gratefully acknowledges the support of the
TUM Graduate School’s Thematic Graduate Center TopMath at the Technische
Universität München22footnotetext: Partially supported by the National Science
Foundation, Grant DMS-1100215, and the Alexander von Humboldt
Foundation.33footnotetext: Partially supported by DFG grant TA 309/2-2.
## Abstract
This paper extends the scenario of the Four Color Theorem in the following
way. Let $\mathcal{H}_{d,k}$ be the set of all $k$-uniform hypergraphs that
can be (linearly) embedded into $\mathbb{R}^{d}$. We investigate lower and
upper bounds on the maximum (weak and strong) chromatic number of hypergraphs
in $\mathcal{H}_{d,k}$. For example, we can prove that for $d\geq 3$ there are
hypergraphs in $\mathcal{H}_{2d-3,d}$ on $n$ vertices whose weak chromatic
number is $\Omega(\log n/\log\log n)$, whereas the weak chromatic number for
$n$-vertex hypergraphs in $\mathcal{H}_{d,d}$ is bounded by
$\mathcal{O}(n^{(d-2)/(d-1)})$ for $d\geq 3$.
## 1 Introduction
The Four Color Theorem [AH77, AHK77] asserts that every graph that is
embeddable in the plane has chromatic number at most four. This question has
been one of the driving forces in Discrete Mathematics and its theme has
inspired many variations. For example, the chromatic number of graphs that are
embedabble into a surface of fixed genus has been intensively studied by
Heawood [Hea90], Ringel and Youngs [RY68], and many others.
In this paper, we consider graphs and hypergraphs that are embeddable into
$\mathbb{R}^{d}$ for $d\geq 3$ in such a way that their edges do not intersect
(see Definition 1 below). For graphs, however, this is not a very interesting
question because for any $n\in\mathbb{N}$ the vertices of the complete graph
$K_{n}$ can be embedded into $\mathbb{R}^{3}$ using the embedding
$\varphi(v_{i})=\left(i,i^{2},i^{3}\right)\quad\forall i\in\\{1,\ldots,n\\}.$
(1)
It is a well known property of the moment curve $t\mapsto(t,t^{2},t^{3})$ that
any two edges between four distinct vertices do not intersect. E. g., this
follows trivially from Corollary 11 in the case of $k=2$ and $d=3$.
As a consequence, we now focus our attention on hypergraphs, which are in
general not embeddable into any specific dimension. Some properties of these
hypergraphs (or more generally simplicial complexes) have been investigated
(see e. g. [Men28, MTW11, vK33, Flo34]), but to our surprise, we have not been
able to find any bounds on their chromatic number.
Before we can state our main results, we quickly recall and introduce some
useful notation. We say that $H=(V,E)$ is a _$k$ -uniform hypergraph_ if the
vertex set $V$ is a finite set and the edge set $E$ consists of $k$-element
subsets of $V$, i. e. $E\subseteq\binom{V}{k}$. For any hypergraph $H$, we
denote by $V(H)$ the vertex set of $H$ and by $E(H)$ its edge set. We define
$K_{n}^{(k)}:=\left(\\{1,2,\ldots,n\\},\binom{\\{1,2,\ldots,n\\}}{k}\right)$
and call any hypergraph isomorphic to $K_{n}^{(k)}$ a _complete $k$-uniform
hypergraph of order $n$_.
Let $H$ be a $k$-uniform hypergraph. A function
$\kappa:V(H)\to\\{1,\ldots,c\\}$ is said to be a _strong $c$-coloring_ if for
all $e\in E(H)$ the property $|\kappa(e)|=k$ holds. The function $\kappa$ is
said to be a _weak $c$-coloring_ if $|\kappa(e)|>1$ for all $e\in E(H)$. The
_strong/weak chromatic number_ of $H$ is defined as the minimum
$c\in\mathbb{N}$ such that there exists a strong/weak coloring of $H$ with $c$
colors. The chromatic number of $H$ is denoted by $\chi^{\text{{s}}}(H)$ and
$\chi^{\text{{w}}}(H)$ respectively. Obviously, for graphs, weak and strong
colorings are equivalent.
We next define what we mean when we say that a hypergraph is embedabble into
$\mathbb{R}^{d}$. Here, $\operatorname{aff}$ denotes the affine hull of a set
of points and $\operatorname{conv}$ the convex hull.
###### Definition 1 ($d$-embeddings).
Let $H$ be a $k$-uniform hypergraph and $d\in\mathbb{N}$. A (linear)
_embedding_ of $H$ into $\mathbb{R}^{d}$ is a function
$\varphi:V(H)\to\mathbb{R}^{d}$, where $\varphi(A)$ for $A\subseteq V(H)$ is
to be interpreted pointwise, such that
* •
$\dim\operatorname{aff}\varphi(e)=k-1$ for all $e\in E(H)$ and
* •
$\operatorname{conv}\varphi\left(e_{1}\cap
e_{2}\right)=\operatorname{conv}\varphi(e_{1})\cap\operatorname{conv}\varphi(e_{2})$
for all $e_{1},e_{2}\in E(H)$
The first property is needed to exclude functions mapping the vertices of one
edge to affinely non-independent points. The second guarantees that the
embedded edges only intersect in the convex hull of their common vertices.
Note that the inclusion from left to right always holds. A $k$-uniform
hypergraph $H$ is said to be _$d$ -embeddable_ if there exists an embedding of
$H$ into $\mathbb{R}^{d}$. Also, we denote by $\mathcal{H}_{d,k}$ the set of
all $d$-embeddable $k$-uniform hypergraphs.
One can easily see that our definition of 2-embeddability coincides with the
classical concept of planarity [Fár48]. Note that in general there are several
other notions of embeddability. The most popular thereof are piecewise linear
embeddings and general topological embeddings. A short and comprehensive
introduction is given in Section 1 in [MTW11]. We have decided to focus on
linear embeddings, as they lead to a very accessible type of geometry and, at
least in theory, the decision problem of whether a given $k$-uniform
hypergraph is $d$-embeddable is decidable and in PSPACE [Ren92]. The
aforementioned three types of embeddings have shown to be equivalent only in
the less than 3-dimensional case (see e. g. [Bre83, BG00]). Since piecewise
linear and topological embeddings are more general than linear embeddings, all
lower bounds for chromatic numbers can easily be transferred. Furthermore, we
prove all our results on upper bounds for piecewise linear embeddings.
We can now give a summary of our main results in the following Tables 1.1,
1.2, and 1.3, which contain upper or lower bounds for the maximum weak
respectively strong chromatic number of a $d$-embeddable $k$-uniform
hypergraph on $n$ vertices. All results which only follow non-trivially from
prior knowledge are indexed with a theorem number from which they can be
derived.
$d\diagdown k$ | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---
1 | 2 | 1 | 1 | 1 | 1 | 1
2 | 4 | 4 | 1 | 1 | 1 | 1
3 | $n$ | $n_{\langle\ref{thm:stLB2}\rangle}$ | $\Omega\left(\sqrt{n}\right)_{\langle\ref{thm:stLB1}\rangle}$ | 1 | 1 | 1
4 | $n$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $\Omega\left(\sqrt{n}\right)_{\langle\ref{cor:stLB1}\rangle}$ | 1 | 1
5 | $n$ | $n$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $\Omega\left(\sqrt{n}\right)_{\langle\ref{cor:stLB1}\rangle}$ | 1
6 | $n$ | $n$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $\Omega\left(\sqrt{n}\right)_{\langle\ref{cor:stLB1}\rangle}$
7 | $n$ | $n$ | $n$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $n_{\langle\ref{cor:stLB2}\rangle}$ | $n_{\langle\ref{cor:stLB2}\rangle}$
Table 1.1: Currently known _values_ for the maximum strong chromatic number of a $d$-embeddable $k$-uniform hypergraph on $n$ vertices as $n\to\infty$. The number in chevrons indicates the theorem number where we prove this bound. $d\diagdown k$ | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---
1 | 2 | 1 | 1 | 1 | 1 | 1
2 | 4 | 2 | 1 | 1 | 1 | 1
3 | $n$ | $\Omega\left(\frac{\log n}{\log\log n}\right)_{\langle\ref{thm:wlo1}\rangle}$ | 1 | 1 | 1 | 1
4 | $n$ | $\Omega\left(\frac{\log n}{\log\log n}\right)_{\langle\ref{thm:wlo1}\rangle}$ | 1 | 1 | 1 | 1
5 | $n$ | $\lceil n/2\rceil$ | $\Omega\left(\frac{\log n}{\log\log n}\right)_{\langle\ref{thm:wlo2}\rangle}$ | 1 | 1 | 1
6 | $n$ | $\lceil n/2\rceil$ | $\Omega\left(\frac{\log n}{\log\log n}\right)_{\langle\ref{thm:wlo2}\rangle}$ | 1 | 1 | 1
7 | $n$ | $\lceil n/2\rceil$ | $\lceil n/3\rceil$ | $\Omega\left(\frac{\log n}{\log\log n}\right)_{\langle\ref{thm:wlo2}\rangle}$ | 1 | 1
8 | $n$ | $\lceil n/2\rceil$ | $\lceil n/3\rceil$ | $\Omega\left(\frac{\log n}{\log\log n}\right)_{\langle\ref{thm:wlo2}\rangle}$ | 1 | 1
Table 1.2: Currently known _lower bounds_ for the maximum weak chromatic number of a $d$-embeddable $k$-uniform hypergraph on $n$ vertices as $n\to\infty$. The number in chevrons indicates the theorem number where we prove this bound. $d\diagdown k$ | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---
1 | 2 | 1 | 1 | 1 | 1 | 1
2 | 4 | 2 | 1 | 1 | 1 | 1
3 | $n$ | $\mathcal{O}(n^{1/2})_{\langle\ref{thm:wup1}\rangle}$ | $\mathcal{O}(n^{1/2})_{\langle\ref{thm:wup1}\rangle}$ | 1 | 1 | 1
4 | $n$ | $\lceil n/2\rceil$ | $\mathcal{O}(n^{2/3})_{\langle\ref{thm:wup1}\rangle}$ | $\mathcal{O}(n^{2/3})_{\langle\ref{thm:wup1}\rangle}$ | 1 | 1
5 | $n$ | $\lceil n/2\rceil$ | $\mathcal{O}(n^{26/27})_{\langle\ref{thm:wup2}\rangle}$ | $\mathcal{O}(n^{3/4})_{\langle\ref{thm:wup1}\rangle}$ | $\mathcal{O}(n^{3/4})_{\langle\ref{thm:wup1}\rangle}$ | 1
6 | $n$ | $\lceil n/2\rceil$ | $\lceil n/3\rceil$ | $\mathcal{O}(n^{35/36})_{\langle\ref{thm:wup2}\rangle}$ | $\mathcal{O}(n^{4/5})_{\langle\ref{thm:wup1}\rangle}$ | $\mathcal{O}(n^{4/5})_{\langle\ref{thm:wup1}\rangle}$
7 | $n$ | $\lceil n/2\rceil$ | $\lceil n/3\rceil$ | $\mathcal{O}(n^{107/108})_{\langle\ref{thm:wup2}\rangle}$ | $\mathcal{O}(n^{44/45})_{\langle\ref{thm:wup2}\rangle}$ | $\mathcal{O}(n^{5/6})_{\langle\ref{thm:wup1}\rangle}$
8 | $n$ | $\lceil n/2\rceil$ | $\lceil n/3\rceil$ | $\lceil n/4\rceil$ | $\mathcal{O}(n^{134/135})_{\langle\ref{thm:wup2}\rangle}$ | $\mathcal{O}(n^{53/54})_{\langle\ref{thm:wup2}\rangle}$
Table 1.3: Currently known _upper bounds_ for the maximum weak chromatic
number of a $d$-embeddable $k$-uniform hypergraph on $n$ vertices as
$n\to\infty$. The number in chevrons indicates the theorem number where we
prove this bound.
To conclude the introduction here is a rough outline for the rest of the
paper. In Section 2 the general concept of embedding hypergraphs into
$d$-dimensional space is discussed and we also show the embeddability of
certain structures needed later on, hereby extensively using known properties
of the moment curve $t\mapsto(t,t^{2},t^{3},\ldots,t^{d})$. Section 3 then
applies this to the strong coloring problem of hypergraphs and, finally,
Section 4 presents our current level of knowledge for the apparently more
difficult problem of weakly coloring hypergraphs.
## 2 Embeddability
The first part of this section gives insight into the structure of
neighborhoods of single vertices in a hypergraph $H\in\mathcal{H}_{d,k}$. We
will later use them to prove upper bounds on the number of edges in our
hypergraphs. This will then yield upper bounds on the weak chromatic number.
However, we must first take a small technical detour into piecewise linear
embeddings. As our hypergraphs are finite and of fixed uniformity we give a
slightly simplified definition.
###### Definition 2 (Piecewise linear $d$-embeddings).
Let $H$ be a $k$-uniform hypergraph and $d\in\mathbb{N}$. By the Menger-
Nöbeling Theorem (see [Men28, p. 295] and [Nöb31]) $H\in\mathcal{H}_{D,k}$ for
$D\geq 2k-1$. So, let $\varphi:V(H)\to\mathbb{R}^{D}$ be such a linear
embedding and define $\varphi(H)=\bigcup_{e\in
E(H)}\operatorname{conv}\varphi(e)$.
A _piecewise linear embedding_ of $H$ into $\mathbb{R}^{d}$ is a homeomorphism
$\psi:\varphi(H)\to\mathbb{R}^{d}$ such that there exists a subdivision $K$ of
$\varphi(H)$ (seen as a geometric simplicial complex) such that $\psi$ is
affine on all elements of $K$. We say $H$ to be _piecewise linearly
$d$-embeddable_ if there exists an embedding of $H$ into $\mathbb{R}^{d}$ and
we denote by $\mathcal{H}_{d,k}^{\text{{PL}}}$ the set of all piecewise
linearly $d$-embeddable $k$-uniform hypergraphs.
###### Definition 3 (Neighborhoods).
For a $k$-uniform hypergraph $H$ and a vertex $v\in V(H)$ we say the
_neighborhood_ of $v$ is $N_{H}(v)=\\{w\in V(H):w\neq v\text{ and there is an
edge in $E(H)$ incident with $w$ and $v$}\\}$. We define the _neighborhood
hypergraph_ (or link) of $v\in V(H)$ to be the the induced $(k-1)$-uniform
hypergraph
$\operatorname{NH}_{H}(v)=\left(N_{H}(v),\\{e\backslash\\{v\\}:e\in E(H),v\in
e\\}\right).$
The _degree_ $\deg_{H}(v)=\deg(v)$ is the number of edges in $E(H)$ incident
with $v$.
###### Lemma 4.
For a hypergraph $H\in\mathcal{H}_{d,k}^{\text{{PL}}}$ on $n$ vertices, $d\geq
k\geq 2$, and for any vertex $v$ we have that
$\operatorname{NH}_{v}(H)\in\mathcal{H}_{d-1,k-1}^{\text{{PL}}}$.
###### Proof..
Let $d\geq k\geq 2$, $H\in\mathcal{H}_{d,k}^{\text{{PL}}}$, $v\in V(H)$, and
$V_{v}=N_{H}(v)$ nonempty. Then there exist $\varphi:V(H)\to\mathbb{R}^{2k-1}$
a linear embedding and $\psi:\varphi(H)\to\mathbb{R}^{d}$ a piecewise linear
embedding of $H$ for some subdivision $K$ of $\varphi(H)$. Without restriction
assume that $\varphi(v)=\mathbf{0}_{2k-1}$ and
$\psi(\mathbf{0}_{2k-1})=\mathbf{0}_{d}$.
Let $H_{v}=(V_{v}\cup\\{v\\},\\{e\in E(H):v\in e\\})$ be the sub-hypergraph of
$H$ of all edges containing $v$. Obviously, $\psi|\varphi(H_{v})$ (the
restriction of $\psi$ onto $\varphi(H_{v})$) is a piecewise linear embedding
of $H_{v}$ for some subdivision $K_{v}\subseteq K$. Let $K^{1}_{v}=\\{e\in
K_{v}:\mathbf{0}_{2k-1}\in e\\}$. Then there exists an $\varepsilon>0$ such
that
$\varepsilon\cdot\varphi(H_{v})\subseteq\bigcup_{e\in K^{1}_{v}}e,$
i. e. all points in $\varepsilon\cdot\varphi(H_{v})$ are so close to
$\mathbf{0}_{2k-1}$ that they lie completely in elements of $K_{v}$ that
contain the origin.
Then
$\varphi^{\prime}:V_{v}\cup\\{v\\}\to\mathbb{R}^{2k-1},w\mapsto\varepsilon\cdot\varphi(w)$
is a linear and thus $\psi|\varphi^{\prime}(H_{v})$ a piecewise linear
embedding of $H_{v}$ for the subdivision
$K^{2}_{v}=\\{e\cap\varphi^{\prime}(H_{v}):e\in K^{1}_{v}\\}$. We now claim
that there exists a $1$-dimensional subspace $S$ of $\mathbb{R}^{d}$ such that
$\psi(\varphi^{\prime}(H_{v}))\cap S=\\{\mathbf{0}_{d}\\}$. This is true as
each simplex $\psi(e)$, $e\in K^{2}_{v}$, has dimension less or equal $k-1$
and hence “forbids” only a null set of possible lines through the origin when
$d\geq k$. Without loss of generality, let
$S=\mathbb{R}\times\\{\mathbf{0}_{d-1}\\}$.
Let $V_{K^{2}_{v}}\supseteq\varphi^{\prime}(V_{v})$ be the set of all
subdivision points of $K^{2}_{v}$ _without_ $\mathbf{0}_{d}$. Let
$\pi_{d-1}:\mathbb{R}^{d}\to\mathbb{R}^{d-1}$ and
$\pi_{1}:\mathbb{R}^{d}\to\mathbb{R}$ be the natural projections onto the last
$d-1$ coordinates and the first coordinate respectively. Further, put
$\delta=\min\\{\|\pi_{d-1}(\psi(w))\|:w\in V_{K^{2}_{v}}\\}>0$ and
$\Delta=\max\\{|\pi_{1}(\psi(w))|:w\in V_{K^{2}_{v}}\\}$. We take a regular
$(d-1)$-simplex $T\subseteq\mathbb{R}^{d-1}$ centered at the origin with sides
of length $\delta$ and set
$C=\partial\operatorname{conv}((\\{-2\Delta\\}\times
T)\cup\\{(2\Delta,\mathbf{0}_{d-1})\\})$. Thus, $C$ is the boundary of a
stretched $d$-simplex. Due to our choice of $\delta$ and $\Delta$, all
$\psi(w)$ for $w\in V_{K^{2}_{v}}$ lie outside of $C$ and for all $e\in
K^{2}_{v}$ the intersection $\psi(e)\cap C$ completely lies in
$C^{\prime}:=C\backslash(\\{-2\Delta\\}\times T)$, is connected, and the union
of finitely many at most $(k-2)$-dimensional simplices .
Thus, there exists a subdivision $K^{3}_{v}$ of $K^{2}_{v}$ such that for all
$e\in K^{3}_{v}$ with dimension $k-1$ we have that $\psi(e)\cap C^{\prime}$ is
a $(k-2)$-dimensional simplex and still $\mathbf{0}_{d}\in e$ . We denote the
set of subdivision points _without_ $\mathbf{0}_{d}$ by
$V_{K^{3}_{v}}\supseteq V_{K^{2}_{v}}$. Now, one can find a retraction
$\varrho:\psi(\varphi^{\prime}(H_{v}))\to\psi(\varphi^{\prime}(H_{v}))$ that
maps each $\psi(w)$, $w\in V_{K^{3}_{v}}$, to the intersection point of the
line segment $[\mathbf{0}_{d},\psi(w)]$ with $C^{\prime}$, such that $\varrho$
is linear on all $\psi(e)$ for $e\in K^{3}_{v}$.
Finally, set $\hat{K}=\\{\operatorname{conv}(e\cap V_{K^{3}_{v}}):e\in
K^{3}_{v}\\}$ which is now a subdivision of
$\varphi^{\prime}(\operatorname{NH}_{H}(v))\subseteq\varphi^{\prime}(H_{v})$.
Then the image of $\varrho\circ(\psi|\hat{K})$ lies completely in
$C^{\prime}$. Thus,
$\hat{\psi}=\pi_{d-1}\circ\varrho\circ(\psi|\hat{K})$
is a piecewise linear embedding of $\operatorname{NH}_{H}(v)$ into
$\mathbb{R}^{d-1}$ for the subdivision $\hat{K}$ and
$\operatorname{NH}_{H}(v)\in\mathcal{H}_{d-1,k-1}^{\text{{PL}}}$.
$\square$
###### Lemma 5.
a) For a hypergraph $H\in\mathcal{H}_{k,k}^{\text{{PL}}}$ on $n$ vertices,
$k\geq 2$, we have that $|E(H)|\leq\frac{6n^{k-1}-12n^{k-2}}{k!}$.
b) For a hypergraph $H\in\mathcal{H}_{k+1,k+1}^{\text{{PL}}}$ on $n$ vertices,
$k\geq 2$, and for any vertex $v$ we have that
$\deg_{H}(v)\leq\frac{6n^{k-1}-12n^{k-2}}{k!}$.
###### Proof..
If $k=2$, then (a) is equivalent to the fact that for $G$ planar $|E(G)|\leq
3n-6$. Given that (a) is true for some $k\geq 2$, we show that (b) holds for
$k$ as well. Let $H\in\mathcal{H}_{k+1,k+1}^{\text{{PL}}}$, $v$ one of the $n$
vertices. By Lemma 4,
$\operatorname{NH}_{H}(v)\in\mathcal{H}_{k,k}^{\text{{PL}}}$. By (a),
$|E(\operatorname{NH}_{H}(v))|\leq\frac{6n^{k-1}-12n^{k-2}}{k!}$ which implies
$\deg_{H}(v)\leq\frac{6n^{k-1}-12n^{k-2}}{k!}$.
Given that (b) is true for some $k\geq 2$, we show that (a) holds for $k+1$.
Let $H\in\mathcal{H}_{k+1,k+1}^{\text{{PL}}}$. Since (b) is true for every
vertex $v_{i}$, we have
$|E(H)|=\frac{\sum_{i=1}^{n}\deg_{H}(v_{i})}{k+1}\leq\frac{n(6n^{k-1}-12n^{k-2})}{(k+1)k!}=\frac{6n^{k}-12n^{k-1}}{(k+1)!}.$
$\square$
###### Corollary 6.
For a hypergraph $H\in\mathcal{H}_{k,k}^{\text{{PL}}}$ on $n$ vertices, $k\geq
3$, and for any edge $e\in E(H)$ there exist at most
$k(6n^{k-2}-12n^{k-3})/(k-1)!-1$ other edges adjacent to it.
###### Proof..
This follows from Lemma 5, since every edge has exactly $k$ vertices and each
of them has at most degree $\frac{6n^{k-2}-12n^{k-3}}{(k-1)!}$. As $e$ itself
counts for the degree as well, one can subtract 1. $\square$
We need to bound the number of edges in a $d$-embeddable hypergraph to prove
upper bounds for the chromatic number. The following proposition will also
help to do this. Note that there exist much stronger conjectured bounds (see
[Kal02, Conjecture 27] and [Gun09, Conjecture 1.4.4]).
###### Proposition 7 (Gundert [Gun09, Proposition 3.3.5]).
Let $k\geq 2$. For a $k$-uniform hypergraph on $n$ vertices that is
topologically embedabble into $\mathbb{R}^{2k-2}$, we have that
$|E(H)|<n^{k-3^{1-k}}$.
###### Corollary 8.
Using Lemma 4 and Proposition 7, we inductively obtain that for
$H\in\mathcal{H}_{2k-l,k}^{\text{{PL}}}$ on $n$ vertices, $k\geq l\geq 2$, we
have that $|E(H)|<n^{k-3^{l-1-k}}$.
###### Corollary 9.
For a hypergraph $H\in\mathcal{H}_{2k-l,k}^{\text{{PL}}}$ on $n$ vertices,
$k\geq l\geq 3$, and for any edge $e\in E(H)$ there exist at most
$kn^{k-1-3^{l-1-k}}-1$ other edges adjacent to it.
###### Proof..
This fact follows analogously to Corollary 6 from Corollary 8. $\square$
In order to find lower bounds for the chromatic number of hypergraphs later
on, we need to be able to prove embeddability. The following theorem from
Shephard will turn out to be very useful when embedding vertices of a
hypergraph on the moment curve.
###### Theorem 10 (Shephard [She68]).
Let $W=\\{w_{1},\ldots,w_{m}\\}\subseteq\mathbb{R}^{d}$ be distinct points on
the moment curve in that order and $P=\operatorname{conv}W$. We say a
$q$-element subset $\\{w_{i_{1}},w_{i_{2}},\ldots,w_{i_{q}}\\}\subseteq W$
where $i_{1}<i_{2}<\cdots<i_{q}$ is called contiguous if $i_{q}-i_{1}=q-1$.
Then $U\subseteq W$ is the set of vertices of a $(k-1)$-face of $P$ iff
$|U|=k$ and for some $t\geq 0$
$U=Y_{S}\cup X_{1}\cup\cdots\cup X_{t}\cup Y_{E},$
where all $X_{i}$, $Y_{S}$, and $Y_{E}$ are _contiguous_ sets,
$Y_{S}=\emptyset$ or $w_{1}\in Y_{S}$, $Y_{E}=\emptyset$ or $w_{m}\in Y_{E}$,
and at most $d-k$ sets $X_{i}$ have odd cardinality.
Shephard’s Theorem thus says that the absolute position of points on the
moment curve is irrelevant and only their relative order is important.
Furthermore, note that all points in $W$ are vertices of $P$. The following
corollary helps in proving that two given edges of a hypergraph intersect
properly.
###### Corollary 11.
In the setting of Theorem 10 assume that $W=U_{1}\cup U_{2}$ where $U_{1}$ and
$U_{2}$ are embedded edges of a $k$-uniform hypergraph. Then these edges do
not intersect in a way forbidden by Definition 1, if at least one of them is a
face of $P=\operatorname{conv}W$. It thus suffices to show that for one
$j\in\\{1,2\\}$
$U_{j}=Y_{S}\cup X_{1}\cup\cdots\cup X_{t}\cup Y_{E}$
holds where at most $d-k$ of the contiguous sets $X_{i}$ have odd cardinality.
In the $k=d=3$ case Corollary 11 allows zero odd sets $X_{i}$. Thus, we can
easily classify all possible configurations for two edges.
###### Corollary 12.
Given a 3-uniform hypergraph $H$ and $\varphi:V(H)\to\mathbb{R}^{3}$. Then
$\varphi$ is an embedding of $H$ if $\varphi$ maps all vertices one-to-one on
the moment curve and, for each pair of edges $e$ and $f$ sharing at most one
vertex, the order of the points $\varphi(e\cup f)$ on the moment curve has one
of Configurations 1–10 shown in Figure 2.1. The relative order of edges with
two common vertices is irrelevant.
Figure 2.1: Possible configurations for two edges $e$ and $f$ in
$\mathbb{R}^{3}$. The vertices of $e$ are marked on the top, those of $f$
marked on the bottom. Equivalent cases, one being the reverse of the other,
are only displayed once.
However, there is one more possible configuration which is not covered by
Shephard’s Theorem as both edges are not faces of the polytope of their
vertices.
###### Lemma 13.
Given a 3-uniform hypergraph $H$ and $\varphi:V(H)\to\mathbb{R}^{3}$. Then
$\varphi$ is an embedding of $H$ if $\varphi$ maps all vertices one-to-one on
the moment curve and for each pair of edges $e$ and $f$ sharing at most one
vertex, the order of the points $\varphi(e\cup f)$ on the moment curve has one
of the configurations shown in Figure 2.1. The relative order of edges with
two common vertices is irrelevant.
###### Proof..
Having in mind Corollary 12, it is sufficient to prove the following: For
$x_{0,0}<x_{1,0}<x_{0,1}<x_{2,0}<x_{1,1}<x_{2,1}\in\mathbb{R}$,
$\psi:\mathbb{R}\to\mathbb{R}^{3},\psi(x)=(x,x^{2},x^{3})$ the moment curve,
and $D_{i}=\\{x_{0,i},x_{1,i},x_{2,i}\\}$ we have that
$\operatorname{conv}\psi(D_{0})\cap\operatorname{conv}\psi(D_{1})=\varnothing$.
Assume otherwise. Note that if two triangles intersect in $\mathbb{R}^{3}$ the
intersection points must contain at least one point of the border of at least
one of the triangles. Thus, without loss of generality,
$\operatorname{conv}\\{\psi(x_{j_{1},0}),\psi(x_{j_{2},0})\\}\cap\operatorname{conv}\psi(D_{1})\neq\varnothing$.
However, by Theorem 10 we know that
$\operatorname{conv}\\{\psi(x_{j_{1},0}),\psi(x_{j_{2},0})\\}$ is a face of
the polytope
$P=\operatorname{conv}(\\{\psi(x_{j_{1},0}),\psi(x_{j_{2},0})\\}\cup\psi(D_{1}))$
which is a contradiction. $\square$
## 3 Strong colorings
For $d,k,n\in\mathbb{N}$ we define
$\chi_{d,k}^{\text{{s}}}(n)=\max\\{\chi^{\text{{s}}}(H):H\in\mathcal{H}_{d,k},|V(H)|=n\\}$
to be the maximum strong chromatic number of a $d$-embeddable $k$-uniform
hypergraph on $n$ vertices.
Clearly, $\chi_{d,k}^{\text{{s}}}(n)$ is monotonically increasing in $n$ and
in $d$ and monotonically decreasing in $k$ if the other parameters remain
fixed. Furthermore, it is not difficult to establish some kind of strict
simultaneous monotonicity as follows:
###### Lemma 14.
For $d,k,n\in\mathbb{N}$, we have
$\chi_{d+1,k+1}^{\text{{s}}}(n+1)\geq\chi_{d,k}^{\text{{s}}}(n)+1.$
###### Proof..
Let $H\in\mathcal{H}_{d,k}$ be such that
$c=\chi^{\text{{s}}}(H)=\chi_{d,k}^{\text{{s}}}(n)$. Let $\varphi$ be an
embedding of $H$ into $\mathbb{R}^{d}$. Set
$V^{\prime}=V(H)\cup\\{v^{\prime}\\}$ where $v^{\prime}\notin V(H)$.
Furthermore, set $E^{\prime}=\left\\{e\cup\\{v^{\prime}\\}:e\in E(H)\right\\}$
and consider the embedding
$\varphi^{\prime}:V^{\prime}\to\mathbb{R}^{d+1},\quad
v\mapsto\left\\{\begin{matrix}(\varphi(v),0)&\text{if $v\in V(H)$}\\\
(\mathbf{0}_{d},1)&\text{if $v=v^{\prime}$}\end{matrix}\right..$
Then, $H^{\prime}=(V^{\prime},E^{\prime})$ is in $\mathcal{H}_{d+1,k+1}$ using
the embedding $\varphi^{\prime}$. Assume that $H^{\prime}$ has a strong
coloring with at most $c$ colors. By the construction of $E^{\prime}$, this
would yield a coloring with less than $c$ colors for $H$, which contradicts
the choice of $c$. $\square$
We now give lower bounds on $\chi_{d,k}^{\text{{s}}}(n)$, which we essentially
derive by retreating to graphs.
###### Definition 15 (The shadow of a hypergraph).
Let $H$ be a $k$-uniform hypergraph, $k\geq 2$. Then we call, following
[GGL95, §7.1],
$\mathcal{S}(H)=\left(V(H),\left\\{\\{v_{1},v_{2}\\}:\\{v_{1},v_{2}\\}\subseteq
e\text{ for some }e\in E(H)\right\\}\right)$
the _(second) shadow_ of $H$.
For a $k$-uniform hypergraph $H$, we have
$\chi^{\text{{s}}}(H)=\chi(\mathcal{S}(H)),$ (2)
where $\chi(G)$ is the classical chromatic number of a graph $G$.
###### Remark 16.
a) The Four Color Theorem and Equation (2) imply that
$\chi_{2,k}^{\text{{s}}}(n)\leq 4$ for $k\in\\{2,3\\}$. Obviously, there are
graphs and hypergraphs for which this bound is sharp.
b) For $d\geq 2k-1$, we have $\chi_{d,k}^{\text{{s}}}(n)=n$ as $K_{n}^{(k)}$
is $(2k-1)$-embeddable for all $k\in\mathbb{N}$ by the Menger-Nöbeling Theorem
(see [Men28, p. 295] and [Nöb31]) and
$\chi^{\text{{s}}}\left(K_{n}^{(k)}\right)=n$.
c) For $d\leq k-2$, we know $\chi_{d,k}^{\text{{s}}}(n)=1$ as
$H\in\mathcal{H}_{d,k}$ cannot have any edge.
###### Theorem 17.
For $n\geq 4$ we have
$\chi_{3,4}^{\text{{s}}}(n)\geq\left\lfloor\sqrt{n}\right\rfloor$.
###### Proof..
It is sufficient that for $m\geq 4$ we can find a hypergraph
$H\in\mathcal{H}_{3,4}$ on $m^{2}$ vertices with strong chromatic number
larger or equal $m$. Let $m\geq 4$ and let $G=K_{m}$ be the complete graph on
$m$ vertices. We can use the embedding $\varphi$ from Equation (1) to embed
$G$ into $\mathbb{R}^{3}$.
Note that $G$ has only finitely many edges. So for every embedded edge
$e=\\{u,v\\}$ of $G$ there exists a small open convex set
$C_{e}\subseteq\mathbb{R}^{3}$ such that
$(\operatorname{conv}\varphi(e))\backslash\varphi(e)\subseteq C_{e}$,
$\varphi(e)\subseteq\partial C_{e}$, and
$(C_{e}\cap\operatorname{conv}\varphi(f))\subseteq\\{\varphi(u),\varphi(v)\\}$
for all $f\neq e\in E(G)$.
Figure 3.1: Construction of $C_{e}$.
Now for each edge $e\in E(G)$ define two new vertices $v^{\prime}_{e}$ and
$v^{\prime\prime}_{e}$. Further, let $w^{\prime}_{e}$ and
$w^{\prime\prime}_{e}\in\mathbb{R}^{3}$ be two arbitrary points in $C_{e}$
such that $\\{w^{\prime}_{e},w^{\prime\prime}_{e},\varphi(u),\varphi(v)\\}$
are in general position and thus form the edges of a tetrahedron in $C_{e}$.
We want to define the hypergraph $H$ using the vertex set
$V=V(G)\cup\\{v^{\prime}_{e},v^{\prime\prime}_{e}:e\in E(G)\\}$ thus adding
the newly created vertices. We set
$E=\\{e\cup\\{v^{\prime}_{e},v^{\prime\prime}_{e}\\}:e\in E(G)\\}$ and
$\varphi^{\prime}:V\to\mathbb{R}^{3},\quad
v\mapsto\left\\{\begin{matrix}\varphi(v)&\text{if $v\in V(G)$}\\\
w^{\prime}_{e}&\text{if $v=v^{\prime}_{e}$ for some $e$}\\\
w^{\prime\prime}_{e}&\text{if $v=v^{\prime\prime}_{e}$ for some
$e$}\end{matrix}\right..$
Then $\varphi^{\prime}$ is an embedding of $H$ as each edge forms a
tetrahedron inside its corresponding set $C_{e}$ and intersects with other
tetrahedrons at most at its two endpoints. Hence, $H\in\mathcal{H}_{3,4}$.
Obviously, $|V|=m^{2}$ and
$\chi^{\text{{s}}}(H)=\chi(\mathcal{S}(H))\geq\chi(G)=\chi(K_{m})=m$.
$\square$
###### Corollary 18.
Together with Lemma 14 we obtain for large $n$ and $d\geq 3$ that
$\chi_{d,d+1}^{\text{{s}}}(n)\geq\left\lfloor\sqrt{n-d+3}\right\rfloor+d-3.$
Using monotonicity the same holds for all $d\geq 3,k\leq d+1$ and
$\chi_{d,k}^{\text{{s}}}(n)$.
###### Theorem 19.
For $n\geq 1$ we have $\chi_{3,3}^{\text{{s}}}(n)=n$.
###### Proof..
First consider $n\in\mathbb{N}$ odd, $n=2m+1$. Set
$V_{m}=\\{v_{0},\ldots,v_{m},w_{1},\ldots,w_{m}\\}$,
$E_{m}=\\{\\{v_{i},v_{j},w_{j}\\}:0\leq i<j\leq
m\\}\cup\\{\\{w_{i},v_{j},w_{j}\\}:1\leq i<j\leq m\\}$ and look at the
3-uniform hypergraph $F_{m}=(V_{m},E_{m})$. It has $2m+1$ vertices and one can
easily see that
$\chi^{\text{{s}}}(F_{m})=\chi(\mathcal{S}(F_{m}))=\chi(K_{2m+1})=n$.
It is now left to show that $F_{m}$ is 3-embeddable for all $m$. Define
$\varphi(x)=\left(x,x^{2},x^{3}\right)$ for $x\in\mathbb{R}$. Set
$\psi_{m}(v_{0})=\varphi(1)$, $\psi_{m}(v_{i})=\varphi(2i)$, and
$\psi_{m}(w_{i})=\varphi(2i+1)$ for all $i\in\\{1,\ldots,m\\}$.
$\psi_{m}$ is an embedding of $F_{m}$: We use induction on $m$ and for $m=0$
or $m=1$ this is trivial. Now assume that this is true for some $m$. Take any
two edges $e_{1}\neq e_{2}\in E_{m+1}$. Observe that
$\psi_{m+1}|V_{m}=\psi_{m}$. If $e_{1},e_{2}\in E_{m}$, then they intersect
according to Definition 1 because $\psi_{m}$ was an embedding of $F_{m}$. If
both edges are in $E_{m+1}\backslash E_{m}$ then they share the two vertices
$v_{m+1}$ and $w_{m+1}$. Thus, they also intersect according to Definition 1.
When $e_{1}\in E_{m+1}\backslash E_{m}$ and $e_{2}\in E_{m}$, we have to
distinguish the several different cases of Corollary 12. If $e_{1}$ and
$e_{2}$ share one vertex, we have one of Cases 6–8 and we are done. So, assume
$e_{1}$ and $e_{2}$ to be disjoint. Let $0\leq i,j,k\leq m$ be pairwise
distinct. If $e_{1}=\\{v_{i},v_{m+1},w_{m+1}\\}$ and
$e_{2}=\\{w_{i},v_{k},w_{k}\\}$, then $i<k$ and we have Case 3. If
$e_{1}=\\{w_{i},v_{m+1},w_{m+1}\\}$ and $e_{2}=\\{v_{i},v_{k},w_{k}\\}$, we
have Case 2. The last possibility is that $e_{1}=\\{[v_{i}\text{ or
}w_{i}],v_{m+1},w_{m+1}\\}$ and $e_{2}=\\{[v_{j}\text{ or
}w_{j}],v_{k},w_{k}\\}$. However, then we are in Case 1 if $i>k>j$, in Case 2
if $k>i>j$, and in Case 3 if $k>j>i$. Hence, $\psi_{m+1}$ is an embedding of
$F_{m+1}$.
Now, let $n$ be even, $n=2m$. Note that by monotonicity we already have
$\chi_{3,3}^{\text{{s}}}(2m)\geq\chi_{3,3}^{\text{{s}}}(2m-1)\geq 2m-1$. Take
$F_{m-1}$ and add one more vertex to obtain
$V_{m}^{\prime}=V_{m-1}\cup\\{v_{0}^{\prime}\\}$. Furthermore, set
$E_{m}^{\prime}=E_{m-1}\cup\\{\\{v_{0}^{\prime},v_{i},w_{i}\\}:1\leq i\leq
m\\}\cup\\{\\{v_{0}^{\prime},v_{0},w_{m-1}\\}\\}$ and define
$F_{m}^{\prime}=(V_{m}^{\prime},E_{m}^{\prime})$. This hypergraph has $2m$
vertices and
$\chi^{\text{{s}}}(F_{m}^{\prime})=\chi(\mathcal{S}(F_{m}^{\prime}))=\chi(K_{2m})=n$.
We claim that it is 3-embeddable for all $m$ via the embedding
$\psi_{m}^{\prime}(v)=\psi_{m-1}(v)$ if $v\neq v_{0}^{\prime}$ and
$\psi_{m}^{\prime}(v_{0}^{\prime})=\varphi(0)$.
As before, take any two edges $e_{1}\neq e_{2}\in E_{m}^{\prime}$. If both
edges are in $E_{m-1}$, both edges are of the form
$\\{v_{0}^{\prime},v_{i},w_{i}\\}$, or both edges share two vertices, then
they obviously intersect according to Definition 1. If
$e_{1}=\\{v_{0}^{\prime},v_{0},w_{m-1}\\}$, we have Case 3 if the edges are
disjoint and one of Cases 5, 8, or 9 if not. Let $0\leq i,j,k\leq m-1$ be
pairwise distinct and let $e_{1}=\\{v_{0}^{\prime},v_{i},w_{i}\\}$. If
$e_{2}=\\{v_{0},v_{j},w_{j}\\}$, we then have Case 3 if $j<i$ and Case 2
otherwise. If $e_{2}=\\{[v_{j}\text{ or }w_{j}],v_{k},w_{k}\\}$, we are in
Case 1 if $k>j>i$, Case 2 if $k>i>j$, and Case 3 if $i>k>j$. Finally, if
$e_{2}=\\{v_{i},v_{k},w_{k}\\}$, we have Case 7 and if
$e_{2}=\\{w_{i},v_{k},w_{k}\\}$, Case 6. Hence, $\psi_{m}^{\prime}$ is an
embedding of $F_{m}^{\prime}$. $\square$
###### Corollary 20.
Together with Lemma 14 we obtain that
$\chi_{d,d}^{\text{{s}}}(n)=n.$
By monotonicity, $\chi_{d,k}^{\text{{s}}}(n)=n$ holds for all $d\geq 3,k\leq
d$.
Thus, except for the cases where $k=d+1$, the maximum strong coloring problem
was solved. In particular, we have shown that an unbounded number of colors
can be necessary for any strong coloring of a $d$-embeddable hypergraph if
$d>2$.
## 4 Weak colorings
For $d,k,n\in\mathbb{N}$ we define
$\chi_{d,k}^{\text{{w}}}(n)=\max\\{\chi^{\text{{w}}}(H):H\in\mathcal{H}_{d,k},|V(H)|=n\\}$
to be the maximum weak chromatic number of a $d$-embeddable $k$-uniform
hypergraph on $n$ vertices.
In this section, we give lower and upper bounds on
$\chi_{d,k}^{\text{{w}}}(n)$. Obviously, $\chi_{d,k}^{\text{{w}}}(n)$ is
monotonically increasing in $n$ and in $d$ and monotonically decreasing in $k$
if the other parameters remain fixed. Note that an equivalent of Lemma 14 is
not true for weak colorings as the existence of one vertex incident with all
edges automatically implies the existence of a weak 2-coloring.
###### Remark 21.
a) For $k=2$, the results in Tables 1.2 and 1.3 are the same as in Table 1.1
as weak and strong chromatic numbers coincide.
b) For $d\geq 2k-1$, we have $\chi_{d,k}^{\text{{w}}}(n)=\Theta(n)$ as
$K_{n}^{(k)}$ is $(2k-1)$-embeddable for all $k\in\mathbb{N}$ by the Menger-
Nöbeling Theorem (see [Men28, p. 295] and [Nöb31]) and
$\chi^{\text{{w}}}\left(K_{n}^{(k)}\right)=\lceil n/(k-1)\rceil$.
c) For $d\leq k-2$, we again know $\chi_{d,k}^{\text{{w}}}(n)=1$ as
$H\in\mathcal{H}_{d,k}$ cannot have any edge.
###### Proposition 22.
For all $n\geq 3$ we have $\chi_{2,3}^{\text{{w}}}(n)\leq 2$. (This bound is
obviously sharp.)
###### Proof..
Let $H\in\mathcal{H}_{2,3}$ and $V=V(H)$. Then $G=\mathcal{S}(H)$ is a planar
graph, thus $\chi(G)\leq 4$. Let $\kappa:V\to\\{1,2,3,4\\}$ be a 4-coloring of
$G$. Define
$\kappa^{\prime}:V\to\\{1,2\\},v\mapsto(\kappa(v)\mod 2)+1.$
In any triangle $\\{u,v,w\\}$ of $H$ under the coloring $\kappa$ these
vertices have exactly three different colors. Therefore, under the coloring
$\kappa^{\prime}$ at least one vertex with color 1 and one vertex with color 2
exists. Thus $\kappa^{\prime}$ is a valid 2-coloring of $H$. $\square$
###### Theorem 23.
Let $d\geq 3$. Then one has
$\chi_{d,d}^{\text{{w}}}(n)\leq\left\lceil\left(\frac{6ed}{(d-1)!}\right)^{\frac{1}{d-1}}n^{\frac{d-2}{d-1}}\right\rceil=\mathcal{O}\left(\left(\frac{n}{d}\right)^{\frac{d-2}{d-1}}\right).$
This result also holds for piecewise linear embeddings.
###### Proof..
Let $H\in\mathcal{H}_{d,d}^{\text{{PL}}}\supseteq\mathcal{H}_{d,d}$. By
Corollary 6 we know that every edge is at most adjacent to
$\Delta=d(6n^{d-2}-12n^{d-3})/(d-1)!-1$ other edges.
We want to apply the Lovász Local Lemma [EL75, Spe77] to bound the weak
chromatic number of $H$. Let $c\in\mathbb{N}$. In any $c$-coloring of the
vertices of $H$ an edge is called bad if it is monochromatic and good if not.
In any uniformly random $c$-coloring the probability for one edge to be bad is
$p=\frac{1}{c^{d-1}}$. Moreover, note that the events whether one edge and a
set of edges not adjacent to the first edge are bad are independent. Thus each
event whether an edge is bad depends on at most $\Delta$ other such events.
The Lovász Local Lemma guarantees us that there is positive probability that
all edges are good if $e\cdot p\cdot(\Delta+1)\leq 1$. This implies that $H$
is weakly $c$-colorable. Note that
$e\cdot p\cdot(\Delta+1)\leq
1\Leftrightarrow\frac{ed(6n^{d-2}-12n^{d-3})}{(d-1)!}\leq c^{d-1}.$
Choosing an integer
$c\geq\left(\frac{6ed}{(d-1)!}\right)^{\frac{1}{d-1}}n^{\frac{d-2}{d-1}}\geq\left(\frac{ed(6n^{d-2}-12n^{d-3})}{(d-1)!}\right)^{\frac{1}{d-1}},$
the hypergraph $H$ is $c$-colorable and $\chi^{\text{{w}}}(H)\leq c$.
$\square$
###### Theorem 24.
Let $d\geq l\geq 3$. Then one has
$\chi_{2d-l,d}^{\text{{w}}}(n)\leq\left\lceil(ed)^{\frac{1}{d-1}}n^{1-\frac{3^{l-1-d}}{d-1}}\right\rceil=\mathcal{O}\left(n^{1-\frac{3^{l-1-d}}{d-1}}\right).$
This result also holds for piecewise linear embeddings.
###### Proof..
By Corollary 9 we know that every edge is at most adjacent to
$\Delta=dn^{d-1-3^{l-1-d}}-1$ other edges. The rest of the proof is now
analogous to the proof of Theorem 23. $\square$
By monotonicity, the upper bounds presented here also hold if the uniformity
of the hypergraph is larger than stated in Theorems 23 and 24. In the
remaining part of this section, we now consider lower bounds for the chromatic
number of hypergraphs.
###### Theorem 25.
As $n\to\infty$ one has
$\chi_{3,3}^{\text{{w}}}(n)=\Omega\left(\frac{\log n}{\log\log n}\right).$
###### Proof..
We first define a sequence of hypergraphs $H_{m}$ for $m\geq 2$ such that
$\chi^{\text{{w}}}(H_{m})\geq m$. Set $H_{2}=K_{3}^{(3)}$ which has 3
vertices. Define $H_{m}$ for $m>2$ iteratively, assuming
$\chi^{\text{{w}}}(H_{m-1})\geq m-1$. Take $m$ new vertices
$\\{v_{0},\ldots,v_{m-1}\\}$ and $m(m-1)/2$ disjoint copies of $H_{m-1}$,
labeled $H_{m-1}^{[0,1]},\ldots,H_{m-1}^{[m-2,m-1]}$.
The edges of $H_{m}$ shall be all former edges of all $H_{m-1}^{[i,j]}$
together with all edges of the form $\\{v_{i},v_{j},w\\}$ where $i<j$ and
$w\in H_{m-1}^{[i,j]}$. Assume $H_{m}$ is weakly $(m-1)$-colorable. Given such
a coloring, one color must occur twice in $\\{v_{0},\ldots,v_{m-1}\\}$. Say,
these are the vertices $v_{i_{1}}$ and $v_{i_{2}}$ where $i_{1}<i_{2}$. This
color cannot occur anymore in the coloring of $H_{m-1}^{[i_{1},i_{2}]}$. Thus,
$H_{m-1}^{[i_{1},i_{2}]}$ must be weakly $(m-2)$-colorable. This is a
contradiction and $H_{m}$ is at least (and obviously exactly) weakly
$m$-chromatic.
Figure 4.1: Construction of $H_{m}$.
We now claim that $H_{m}\in\mathcal{H}_{3,3}$ for all $m\geq 2$. Therefor we
give a function $f_{m}:V(H_{m})\to\\{1,\ldots,n_{m}\\}$ where $n_{m}$ is the
number of vertices of $H_{m}$. This function defines the order in which the
vertices of $H_{m}$ will be arranged on the moment curve
$t\mapsto(t,t^{2},t^{3})$. Lemma 13 on possible configurations then guarantees
that $H_{m}$ is embeddable via arbitrary points on the moment curve. Note that
the absolute position of vertices on the moment curve is not important, only
their relative order.
The hypergraph $H_{2}=K_{3}^{(3)}$ can be embedded into $\mathbb{R}^{3}$ via
any three points on the moment curve, so $f_{2}:V(H_{2})\to\\{0,1,2\\}$ can be
chosen arbitrarily. Assume that $f_{m-1}$ has already been defined and that
the vertices of $H_{m-1}$ arranged in that order on the moment curve form an
embedding. Look at the vertices of $H_{m}$ as given before. We define
$f_{m}(v_{j})=n_{m-1}\cdot j(j-1)/2+j$ for $0\leq j\leq m-1$ and for any $w\in
H_{m-1}^{[i,j]}$ with $i<j$ we set
$f_{m}(w)=n_{m-1}\cdot(j(j-1)/2+i)+j+f_{m-1}(w)$. This gives exactly the order
shown in Figure 4.1.
Now, arrange the vertices of $H_{m}$ on the moment curve in that order and
pick any two edges $e_{1}$ and $e_{2}$. By Lemma 13 we can assume that they do
not share two vertices. If both edges are from the same subhypergraph
$H_{m-1}^{[i,j]}$ then they can only intersect according to Definition 1 as
their relative order reflects that of $f_{m-1}$. If they originate from
distinct subhypergraphs $H_{m-1}^{[i_{1},j_{1}]}$ and
$H_{m-1}^{[i_{2},j_{2}]}$, they are of Case 1 in Figure 2.1. Next, assume that
one of them is of the form $\\{v_{i_{1}},v_{j_{1}},v\\}$ where $v\in
H_{m-1}^{[i_{1},j_{1}]}$ and that the other is from some
$H_{m-1}^{[i_{2},j_{2}]}$. Then, by definition, $i_{1}<j_{1}$ and
$i_{2}<j_{2}$ and all the possible cases of Lemma 13 are listed in Table 4.1.
Case number | Relative order of $i_{1},i_{2},j_{1},j_{2}$
---|---
1 | $j_{1}<j_{2}$, $j_{1}=j_{2}$ and $i_{1}<i_{2}$, $i_{1}>j_{2}$
3 | $j_{1}>j_{2}$ and $i_{1}\leq j_{2}$, $j_{1}=j_{2}$ and $i_{1}>i_{2}$
1, 2, 3, 6–8, or 11 | $j_{1}=j_{2}$ and $i_{1}=i_{2}$
Table 4.1: Possible cases if one edge is newly constructed and one is an old
edge.
Finally, take $e_{1}=\\{v_{i_{1}},v_{j_{1}},v\\}$ and
$e_{2}=\\{v_{i_{2}},v_{j_{2}},w\\}$ (again $i_{1}<j_{1}$ and $i_{2}<j_{2}$).
We then have one of the cases listed in Table 4.2. Thus, the order given by
$f_{m}$ provides an embedding of $H_{m}$.
Case number | Relative order of $i_{1},i_{2},j_{1},j_{2}$
---|---
1 | $j_{1}<i_{2}$, $i_{1}>j_{2}$
2 | $i_{2}<i_{1}<j_{2}<j_{1}$, $i_{1}<i_{2}<j_{1}<j_{2}$
3 | $i_{1}<i_{2}<j_{2}<j_{1}$, $i_{2}<i_{1}<j_{1}<j_{2}$
7 | $j_{1}=i_{2}$, $i_{1}=j_{2}$
8 | $i_{1}=i_{2}$ and $j_{1}\neq j_{2}$
10 | $i_{2}\neq i_{1}$ and $j_{1}=j_{2}$
two shared vertices | $i_{1}=i_{2}$ and $j_{1}=j_{2}$
Table 4.2: Possible cases if both edges are newly constructed.
To estimate $n_{m}$, we use the following recursion
$\displaystyle n_{2}$ $\displaystyle=3,$ $\displaystyle n_{m}$
$\displaystyle=m+n_{m-1}\cdot m(m-1)/2\quad\text{for }m>2.$
This can be bounded by $n_{m}\leq m^{2m}=:\hat{n}_{m}$. Then
$\frac{\log\hat{n}_{m}}{\log\log\hat{n}_{m}}=2m\cdot\frac{\log m}{\log(2m\log
m)}=\mathcal{O}(m)$
and we finally get that
$m=\Omega\left(\frac{\log\hat{n}_{m}}{\log\log\hat{n}_{m}}\right)\geq\Omega\left(\frac{\log
n_{m}}{\log\log n_{m}}\right)$. $\square$
Note that by monotonicity also
$\chi_{4,3}^{\text{{w}}}(n)=\Omega\left(\frac{\log n}{\log\log n}\right)$.
Figure 4.2 gives examples for an embedding of $H_{3}$ and $H_{4}$.
Figure 4.2: Examples for an embedding of $H_{3}$ and $H_{4}$.
###### Theorem 26.
Let $d\geq 3$. Then, as $n\to\infty$ one has
$\chi_{2d-3,d}^{\text{{w}}}(n)=\Omega\left(\frac{\log n}{\log\log n}\right).$
###### Proof..
Induction over $d$. The case $d=3$ was shown in Theorem 25. Let $d>3$. Suppose
we have constructed a family $(H_{m}^{d-1})_{m\in\mathbb{N}}$ of hypergraphs
in $\mathcal{H}_{2d-5,d-1}$ such that $\chi^{\text{{w}}}(H_{m}^{d-1})\geq m$.
Let $H_{2}^{d}=K_{d}^{(d)}$. The hypergraph $H_{2}^{d}$ has $d$ vertices, one
edge, and is weakly 2-colorable. Define $H_{m}^{d}$ for $m>2$ iteratively,
given that $\chi^{\text{{w}}}(H_{m-1}^{d})\geq m-1$. Therefor take one copy of
$H_{m-1}^{d}$ and one copy of $(d-1)$-uniform $H_{m}^{d-1}$.
The edges of $H_{m}^{d}$ shall be all edges of $H_{m-1}^{d}$ and all edges of
the form $(\\{v\\}\cup e)$ for $v\in V(H_{m-1}^{d})$ and $e\in
E(H_{m}^{d-1})$. Assume that there exists a weak $(m-1)$-coloring of
$H_{m}^{d}$. Then there has to be at least one monochromatic edge $e\in
E(H_{m}^{d-1})$. No vertex of $H_{m-1}^{d}$ can be colored with this color, so
its edges must be weakly $(m-2)$-colored. This is a contradiction and thus
$\chi^{\text{{w}}}(H_{m}^{d})\geq m$.
Figure 4.3: Construction of $H_{m}^{d}$.
We now claim that $H_{m}^{d}\in\mathcal{H}_{2d-3,d}$ for all $m\geq 2$. As in
the proof of Theorem 25, we give a function
$f_{m}^{d}:V(H_{m}^{d})\to\\{1,\ldots,n_{m}^{d}\\}$ where $n_{m}^{d}$ is the
number of vertices of $H_{m}^{d}$. This defines the order in which the
vertices of $H_{m}^{d}$ will be arranged on the moment curve
$t\mapsto(t,\ldots,t^{2d-3})$. We then use Corollary 11 to prove that
$H_{m}^{d}$ is embeddable via arbitrary points on the moment curve. As before,
the absolute position of vertices on the moment curve is not important. For a
fixed uniformity $d$ and dimension $2d-3$, Corollary 11 guarantees that if for
two given edges the vertices of at least one edge have at most $d-3$ odd
contiguous subsets, they intersect properly according to Definition 1.
If $d=3$ we can set $f_{m}^{3}=f_{m}$ for all $m\geq 2$, where $f_{m}$ is as
in the proof of Theorem 25. For $d>3$ we have by assumption that there exists
a corresponding family of functions
$\left(f_{m}^{d-1}:V(H_{m}^{d-1})\to\\{1,\ldots,n_{m}^{d-1}\\}\right)_{m}$
such that the vertices of $H_{m}^{d-1}$ arranged in that order on the moment
curve form an embedding. We then have to give a family of functions
$f_{m}^{d}$ for $d$.
$H_{2}^{d}$ can be embedded into $\mathbb{R}^{2d-3}$ via any $d$ points on the
moment curve, so $f_{2}^{d}:V(H_{2}^{d})\to\\{1,\ldots,d\\}$ can be chosen
arbitrarily. Assume that $f_{m-1}^{d}$ has already been defined and gives an
embedding of $H_{m-1}^{d}$. We define $f_{m}^{d}(v)=f_{m-1}^{d}(v)$ for $v\in
V(H_{m-1}^{d})$ and for any $w\in V(H_{m}^{d-1})$ we set
$f_{m}^{d}(w)=n_{m-1}^{d}+f_{m}^{d-1}(w)$. This is also shown in Figure 4.3.
Arrange the vertices of $H_{m}^{d}$ on the moment curve in that order and pick
any two edges $g_{1}$ and $g_{2}$. If both edges are from the subhypergraph
$H_{m-1}^{d}$ then they intersect in accordance to Definition 1 as their
relative order reflects that of $f_{m-1}^{d}$. If one edge is from
$H_{m-1}^{d}$ and the other of the form $(\\{v\\}\cup e)$ where $v\in
V(H_{m-1}^{d})$ and $e\in E(H_{m}^{d-1})$, both edges have at most one odd
contiguous subset (except from the first and last one), which is no problem
for $d>3$.
Finally, we look at the case $g_{1}=(\\{v_{1}\\}\cup e_{1})$ and
$g_{2}=(\\{v_{2}\\}\cup e_{2})$. Then their vertex sets have at most one more
odd contiguous subset than the edges $e_{1}$ and $e_{2}$ had in the ordering
of $f_{m}^{d-1}$. The last number, by assumption, was bounded from above by
$(d-1)-3$ for at least on $e_{i}$, $i\in\\{1,2\\}$. So at least one $g_{i}$
has at most $d-3$ odd contiguous subsets. Thus, the order given by $f_{m}^{d}$
provides an embedding of $H_{m}^{d}$.
Note that there is one small exception when $d=4$. Here, $e_{1}$ and $e_{2}$
could be in the relative position of Case 11 in Figure 2.1 and consequently
have more than $(d-1)-3=0$ odd contiguous subsets. However, this is no problem
as in all possible extensions to $g_{1}$ and $g_{2}$ at least one of the edges
continues to have only one odd contiguous subset (see Figure 4.4).
Figure 4.4: All possible 4-uniform extensions of Case 11 in Figure 2.1 as
occuring in the construction of $H_{m}^{4}$.
To bound the number of vertices of $H_{m}^{d}$ we use
$\displaystyle n_{2}^{d}$ $\displaystyle=d,$ $\displaystyle n_{m}^{d}$
$\displaystyle=n_{m-1}^{d}+n_{m}^{d-1}\quad k>2.$
Iteratively, we get that $n_{m}^{d}=d+\sum_{r=3}^{m}n_{r}^{d-1}\leq m\cdot
n_{m}^{d-1}\leq\cdots\leq m^{d-3}\cdot\hat{n}_{m}=m^{2m+d-3}$. Now for large
$m$ and fixed $d$,
$\frac{\log n_{m}^{d}}{\log\log
n_{m}^{d}}\leq(2m+d-3)\cdot\frac{\log(m)}{\log\left((2m+d-3)\log(m)\right)}\leq
3m\cdot\frac{\log(m)}{\log\left(m\log(m)\right)}=\mathcal{O}(m).$
Hence, $m=\Omega\left(\frac{\log n_{m}^{d}}{\log\log n_{m}^{d}}\right)$.
$\square$
Note that by monotonicity also
$\chi_{2d-2,d}^{\text{{w}}}(n)=\Omega\left(\frac{\log n}{\log\log n}\right)$.
## References
* [AH77] K. Appel and W. Haken, _Every planar map is four colorable. Part I: Discharging_ , Illinois Journal of Mathematics 21 (1977), no. 3, 429–490.
* [AHK77] K. Appel, W. Haken, and J. Koch, _Every planar map is four colorable. Part II: Reducibility_ , Illinois Journal of Mathematics 21 (1977), no. 3, 491–567.
* [BG00] J. Bokowski and A. Guedes de Oliveira, _On the Generation of Oriented Matroids_ , Discrete and Computational Geometry 24 (2000), 197–208.
* [Bre83] U. Brehm, _A nonpolyhedral triangulated Möbius strip_ , Proceedings of the American Mathematical Society 89 (1983), no. 3, 519–522.
* [EL75] P. Erdős and L. Lovász, _Problems and results on 3-chromatic hypergraphs and some related questions_ , Infinite and Finite Sets (to Paul Erdős on his 60th birthday), Vol. II, North-Holland (1975), 609–627.
* [Flo34] A. Flores, _Über n-dimensionale Komplexe, die im $\mathbb{R}_{2n+1}$ absolut selbstverschlungen sind_, Ergebnisse Eines Mathematischen Kolloquiums 6 (1934), 4–7.
* [Fár48] I. Fáry, _On straight-line representation of planar graphs_ , Acta Scientiarum Mathematicarum Szeged 11 (1948), 229–233.
* [GGL95] R. Graham, M. Grötschel, and L. Lovász (eds.), _Handbook of Combinatorics, Vol. 1_ , North-Holland, 1995.
* [Gun09] A. Gundert, _On The Complexity of Embeddable Simplicial Complexes_ , Diplomarbeit, Technische Universität Berlin, 2009.
* [Hea90] J. C. Heawood, _Map-colour theorem_ , The Quarterly Journal of Pure and Applied Mathematics 24 (1890), 332–338.
* [Kal02] G. Kalai, _Algebraic Shifting_ , Computational Commutative Algebra and Combinatorics, Advanced Studies in Pure Mathematics 33 (2002), 121–163, Mathematical Society of Japan.
* [Men28] K. Menger, _Dimensionstheorie_ , Teubner, Leipzig, 1928.
* [MTW11] J. Matoušek, M. Tancer, and U. Wagner, _Hardness of Embedding Simplicial Complexes in $\mathbb{R}^{d}$_, Journal of the European Mathematical Society 13 (2011), 259–295.
* [Nöb31] G. Nöbeling, _Über eine $n$-dimensionale Universalmenge im $\mathbb{R}_{2n+1}$_, Mathematische Annalen 104 (1931), no. 1, 71–80.
* [Ren92] J. Renegar, _On the computational complexity and geometry of the first-order theory of the reals. I, II, III_ , Journal of Symbolic Computation 13 (1992), no. 3, 255–299,301–327,329–352.
* [RY68] G. Ringel and J. W. T. Youngs, _Solution of the Heawood map-coloring problem_ , Proceedings of the National Academy of Sciences 60 (1968), no. 2, 438–445.
* [She68] G. C. Shephard, _A Theorem on Cyclic Polytopes_ , Israel Journal of Mathematics 6 (1968), no. 4, 368–372.
* [Spe77] J. Spencer, _Asymptotic lower bounds for Ramsey functions_ , Discrete Mathematics 20 (1977), 69–76.
* [vK33] E. R. van Kampen, _Komplexe in euklidischen Räumen_ , Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9 (1933), no. 1, 72–78, corrections ibidem pp. 152–153.
|
arxiv-papers
| 2012-09-21T18:19:47 |
2024-09-04T02:49:35.380865
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz",
"submitter": "Carl Georg Heise",
"url": "https://arxiv.org/abs/1209.4879"
}
|
1209.4893
|
[nc-nd]Kasturi Varadarajan and Xin Xiao Billy Editor, Bill Editors2Conference
title on which this volume is based on111 10.4230/LIPIcs.xxx.yyy.p
# On the Sensitivity of Shape Fitting Problems111This material is based upon
work supported by the National Science Foundation under Grant No. 0915543.
Kasturi Varadarajan Department of Computer Science
University of Iowa, Iowa City, IA 52242, USA
[email protected] Xin Xiao Department of Computer Science
University of Iowa, Iowa City, IA 52242, USA
[email protected]
###### Abstract.
In this article, we study shape fitting problems, $\epsilon$-coresets, and
total sensitivity. We focus on the $(j,k)$-projective clustering problems,
including $k$-median/$k$-means, $k$-line clustering, $j$-subspace
approximation, and the integer $(j,k)$-projective clustering problem. We
derive upper bounds of total sensitivities for these problems, and obtain
$\epsilon$-coresets using these upper bounds. Using a dimension-reduction type
argument, we are able to greatly simplify earlier results on total sensitivity
for the $k$-median/$k$-means clustering problems, and obtain positively-
weighted $\epsilon$-coresets for several variants of the $(j,k)$-projective
clustering problem. We also extend an earlier result on $\epsilon$-coresets
for the integer $(j,k)$-projective clustering problem in fixed dimension to
the case of high dimension.
###### Key words and phrases:
Coresets, shape fitting, k-means, subspace approximation
###### 1991 Mathematics Subject Classification:
F.2.2 Analysis of Algorithms and Problem Complexity
## 1\. Introduction
In this article, we study shape fitting problem, coresets, and in particular,
total sensitivity. A shape fitting problem is specified by a triple
$(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, where $\mathbb{R}^{d}$ is the
$d$-dimensional Euclidean space, ${\mathcal{F}}$ is a family of subsets of
$\mathbb{R}^{d}$, and ${\rm
dist}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{+}$ is a continuous
function that we will refer to as a distance function. We also assume that (a)
$\mbox{dist}(p,q)=0$ if and only if $p=q$, and (b)
$\mbox{dist}(p,q)=\mbox{dist}(q,p)$. We refer to each $F\in{\cal F}$ as a
shape, and we require each shape $F$ to be a non-empty, closed, subset of
$\mathbb{R}^{d}$. We define the distance of a point $p\in\mathbb{R}^{d}$ to a
shape $F\in{\cal F}$ to be $\mbox{dist}(p,F)=\min_{q\in F}\mbox{dist}(p,q)$.
An instance of a shape fitting problem is specified by a finite point set
$P\subset\mathbb{R}^{d}$. We slightly abuse notation and use
$\mbox{dist}(P,F)$ to denote $\sum_{p\in P}\mbox{dist}(p,F)$ when $P$ is a set
of points in $\mathbb{R}^{d}$. The goal is to find a shape which best fits
$P$, that is, a shape minimizing $\sum_{p\in P}\mbox{dist}(p,F)$ over all
shapes $F\in{\mathcal{F}}$. This is referred to as the $L_{1}$ fitting
problem, which is the main focus of this paper. In the $L_{\infty}$ fitting
problem, we seek to find a shape $F\in{\mathcal{F}}$ minimizing $\max_{p\in
P}\mbox{dist}(p,F)$.
In this paper, we focus on the $(j,k)$-projective clustering problem. Given
non-negative integers $j$ and $k$, the family of shapes is the set of
$k$-tuples of affine $j$-subspaces (that is, $j$-flats) in $\mathbb{R}^{d}$.
More precisely, each shape is the union of some $k$ $j$-flats. The underlying
distance function is usually the $z^{\rm th}$ power of the Euclidean distance,
for a positive real number $z$. When $j=0$, ${\mathcal{F}}$ is the set of all
$k$-point sets of $\mathbb{R}^{d}$, so the $(0,k)$-projective clustering
problem is the $k$-median clustering problem when the distance function is the
Euclidean distance, and it is the $k$-means clustering problem when the
distance function is the square of the Euclidean distance; when $j=1$, the
family of shapes is the set of $k$-tuples of lines in $\mathbb{R}^{d}$; when
$k=1$, $(j,1)$-projective clustering is the subspace approximation problem,
where the family of shapes is the set of $j$-flats. Other than these
projective clustering problems where $j$ or $k$ is set to specific values,
another variant of the $(j,k)$-projective clustering problem is the integer
$(j,k)$-projective clustering problem, where we assume that the input points
have integer coordinates (but there is no restriction on $j$ and $k$), and the
magnitude of these coordinates is at most $n^{c}$, where $n$ is the number of
input points and $c>0$ is some constant. That is, the points are in a
polynomially large integer grid.
An $\epsilon$-coreset for an instance $P$ of a shape fitting problem is a
weighted set $S$, such that for any shape $F\in{\mathcal{F}}$, the summation
of distances from points in $P$ approximates the weighted summation of the
distances from points in $S$ up to a multiplicative factor of
$(1\pm\epsilon)$. A more precise definition (Definition 2.1) follows later.
Coresets can be considered as a succinct representation of the point set; in
particular, in order to obtain a $(1+\epsilon)$-approximation solution fitting
$P$, it is sufficient to find a $(1+\epsilon)$-approximation solution for the
coreset $S$. One usually seeks a small coreset, whose size $|S|$ is
independent of the cardinality of $P$. Coresets of size $o(n)$ for the
$(j,k)$-projective clustering problem for general $j$ and $k$ are not known to
exist. However, the $k$-median/$k$-means clustering, $k$-line clustering,
$j$-subspace approximation, and integer $(j,k)$-projective clustering problems
admit small coresets.
Langberg and Schulman [10] introduced a general approach to coresets via the
notion of sensitivity of points in a point set, which provides a natural way
to set up a probability distribution $\Pr{\cdot}$ on $P$. Roughly speaking,
the sensitivity of a point with respect to a point set measures the importance
of the point, in terms of fitting shapes in the given family of shapes
${\mathcal{F}}$. Formally, the sensitivity of point $p$ in a point set $P$ is
defined by
$\sigma_{P}(p):=\sup_{F\in{\mathcal{F}}}\mbox{dist}(p,F)/\mbox{dist}(P,F)$.
(In the degenerate case where the denominator in the ratio is $0$, the
numerator is also $0$, and we take the ratio to be $0$; the reader should feel
free to ignore this technicality.) The total sensitivity of a point set $P$ is
defined by $\mathfrak{S}_{P}:=\sum_{p\in P}\sigma_{P}(p)$. The nice property
of quantifying the “importance” of a point in a point set is that for any
$F\in{\mathcal{F}}$, $\mbox{dist}(p,F)/\mbox{dist}(P,F)\leq\sigma_{P}(p)$.
Setting the probability of selecting $p$ to be
$\sigma_{P}(p)/\mathfrak{S}_{P}$, and the weight of $p$ to be
$\mathfrak{S}_{P}/\sigma_{P}(p)$, $\forall p\in P$, one can show that the
variance of the sampling scheme is $O((\mathfrak{S}_{P})^{2})$. When
$\mathfrak{S}_{P}$ is $o(n)$, (for example, a constant or logarithmic in terms
of $n=|P|$), one can obtain an $\epsilon$-coreset by sampling a small number
of points. Langberg and Schulman [10] show that the total sensitivity of any
(arbitrarily large) point set $P\subset\mathbb{R}^{d}$ for
$k$-median/$k$-means clustering problem is a constant, depending only on $k$,
independent of the cardinality of $P$ and the dimension of the Euclidean space
where $P$ and ${\mathcal{F}}$ are from. Using this, they derived a coreset for
these problems with size depending polynomially on $d$ and $k$ and independent
of $n$. Their work can be seen as evolving from earlier work on coresets for
the $k$-median/$k$-means and related problems via other low variance sampling
schemes [3, 4, 7, 5].
Feldman and Langberg [6] relate the notion of an $\epsilon$-coreset with the
well-studied notion of an $\epsilon$-approximation of range spaces. They use a
“functional representation” of points: consider a family of functions
${\mathcal{P}}=\\{f_{p}(\cdot)|p\in P\\}$, where each point $p$ is associated
with a function $f_{p}:X\to\mathbb{R}$. The target here is to pick a small
subset $S\subseteq P$ of points, and assign weights appropriately, so that
$\sum_{p\in S}w_{p}f_{p}(x)$ approximates $\sum_{p\in P}f_{p}(x)$ at every
$x\in X$. When $X$ is ${\mathcal{F}}$ and $f_{p}(F)=\mbox{dist}(p,F)$, this is
just the original $\epsilon$-coreset for $P$. However, $f_{p}(\cdot)$ can be
any other function defined over ${\mathcal{F}}$, for example, $f_{p}(\cdot)$
can be the “residue distance” of $p$, i.e.,
$f_{p}(F)=|\mbox{dist}(p,F)-\mbox{dist}(p^{\prime},F)|$, where $p^{\prime}$ is
the projection of $p$ on the optimum shape $F^{\ast}$ fitting $P$. The
definitions of sensitivities and total sensitivity easily carry over in this
setting: $\sigma_{{\mathcal{P}}}(f_{p})=\sup_{x\in
X}f_{p}(x)/\sum_{f_{q}\in{\mathcal{P}}}f_{q}(x)$ (which coincides with
$\sigma_{P}(p)$ when $f_{p}(\cdot)$ is $\mbox{dist}(p,\cdot)$), and
$\mathfrak{S}_{{\mathcal{P}}}=\sum_{f_{p}\in{\mathcal{P}}}\sigma_{{\mathcal{P}}}(f_{p})$
(which coincides with $\mathfrak{S}_{P}$ similarly). One of the results in [6]
is that an approximating subset $S\subseteq P$ can be computed with the size
$|S|$ upper bounded by the product of two quantities:
$(\mathfrak{S}_{{\mathcal{P}}})^{2}$, and another parameter, the “dimension”
(see Definition 2.3) of a certain range space induced by ${\mathcal{P}}$,
denoted ${\rm dim}\left({\mathcal{P}}\right)$. We remark that ${\rm
dim}\left({\mathcal{P}}\right)$ depends on $d$, which is the dimension of
Euclidean space where $P$ is from, and some other parameters related to $X$;
when $X$ is the family of shapes for the $(j,k)$-projective clustering
problem, ${\rm dim}\left({\mathcal{P}}\right)$ also depends on $j$ and $k$.
This connection allows them to use many results from the well-studied area of
$\epsilon$-approximation of range spaces (such as deterministic construction
of small $\epsilon$-approximation of range spaces), thus constructing smaller
coreset deterministically, and removes some routine analysis in the
traditional way of obtaining coresets via random sampling.
### 1.1. Our Results
In this article, we prove upper bounds of total sensitivities for the
$(j,k)$-projective clustering problems. In particular, we show a careful
analysis of computing total sensitivities for shape fitting problems in high
dimension. Total sensitivity $\mathfrak{S}_{P}$ for a point set
$P\subset\mathbb{R}^{d}$ may depend on $d$: consider the shape fitting problem
where the family of shapes is the set of hyperplanes, and $P$ is a point set
of size $d$ in general position. Then clearly $\sigma_{P}(p)=1$ (since there
always exists a hyperplane containing all $d-1$ points other than $p$), so
$\mathfrak{S}_{P}=d$.
One question that arises naturally is that whether the dependence of the total
sensitivity on the dimension $d$ is essential. To answer this question, we
show that if the distance function is Euclidean distance, or the $z^{\rm th}$
power of Euclidean distance for $z\in[1,\infty)$, then the total sensitivity
function of a shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$ in the high dimensional space $\mathbb{R}^{d}$ is roughly the same as
that of the low-dimensional variant
$(\mathbb{R}^{d^{\prime}},{\mathcal{F}}^{\prime},{\rm dist})$, where
$d^{\prime}$ is the “intrinsic” dimension of the shapes in ${\mathcal{F}}$,
and ${\mathcal{F}}^{\prime}$ consists of shapes contained in the low
dimensional space $\mathbb{R}^{d^{\prime}}$. A reification of this statement
is that the total sensitivity function of the $(j,k)$-projective clustering is
independent of $d$. For the $(j,k)$-projective clustering problems, the shapes
are intrinsically low dimensional: each $k$-tuple of $j$-flats is contained in
a subspace of dimension at most $k(j+1)$. As we will see, the total
sensitivity function for $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, where
${\mathcal{F}}$ is the family of $k$-tuples of $j$-flats in $\mathbb{R}^{d}$,
is of the same magnitude as the total sensitivity function of
$(\mathbb{R}^{f(j,k)},{\mathcal{F}}^{\prime},{\rm dist})$, where $f(j,k)$ is a
function of $j$ and $k$ (which is independent of $d$), and
${\mathcal{F}}^{\prime}$ is the family of $k$-tuples of $j$-flats in
$\mathbb{R}^{f(j,k)}$.
We sketch our approach to upper bound the total sensitivity of the
$(j,k)$-projective clustering. We first make the observation (Theorem 3.2
below) that the total sensitivity of a point set $P$ is upper bounded by a
constant multiple of the total sensitivity of $P^{\prime}={\rm
proj}\left(P,F^{\ast}\right)$, which is the projection of $P$ on the optimum
shape $F^{\ast}$ fitting $P$ in ${\mathcal{F}}$. The computation of total
sensitivity of $P^{\prime}$ is very simple in certain cases; for example, for
$k$-median clustering, $P^{\prime}$ is a multi-set which contains $k$ distinct
points, whose total sensitivity can be directly bounded by $k$. Therefore, we
are able to greatly simplify the proofs in [10]. Another more important use of
this observation is that it allows us to get a dimension-reduction type result
for the $(j,k)$-projective clustering problems: note that although the point
set and the shapes might be in a high dimension space $\mathbb{R}^{d}$, the
projected point set $P^{\prime}$ lies in a subspace of dimension $(j+1)k$
(since each $k$-tuple of $j$-flats is contained in a subspace of dimension at
most $(j+1)k$), which is small under the assumption that both $j$ and $k$ are
constant. Therefore, $\mathfrak{S}_{P}$, which usually depends on $d$ if one
directly computes it in a high dimensional space, depends only on $j$ and $k$,
since $\mathfrak{S}_{P}$ is $O(\mathfrak{S}_{P^{\prime}})$.
Our method for bounding the total sensitivity directly translates into a
template for computing $\epsilon$-coresets:
1. (1)
Compute $F^{\ast}$, the optimal shape fitting $P$. (It suffices to use an
approximately optimal shape.) Compute $P^{\prime}$, the projection of $P$ onto
$F^{\ast}$.
2. (2)
Compute a bound on the sensitivity of each point in $P^{\prime}$ with respect
to $P^{\prime}$. Since the ambient dimension is $O(jk)$, we may use a method
that yields bounds on $\mathfrak{S}_{P^{\prime}}$ with dependence on the
ambient dimension. Use Theorem 3.2 to translate this into a bound for
$\sigma_{P}(p)$ for each $p\in P$.
3. (3)
Sample points from $P$ with probabilities proportional to $\sigma_{P}(p)$ to
obtain a coreset, as described in [10, 6].
We now point out the difference between our usage of total sensitivity in the
construction of coresets and the method in [6]. The construction of coresets
in [6] may also be considered as based on total sensitivity, however in a very
different way:
1. (1)
First obtain a small weighted point set $S\subseteq P$, such that
$\mbox{dist}(P,F)-\mbox{dist}(P^{\prime},F)$ is approximately the same as
$\mbox{dist}(S,F)-\mbox{dist}(S^{\prime},F)$ ($S^{\prime}$ is ${\rm
proj}\left(S,F^{\ast}\right)$) for every $F\in{\mathcal{F}}$.
2. (2)
Then compute an $\epsilon$-coreset $Q^{\prime}\subseteq P^{\prime}$ for the
projected point set $P^{\prime}$, that is, $\mbox{dist}(Q^{\prime},F)$
approximates $\mbox{dist}(P^{\prime},F)$ for every $F\in{\mathcal{F}}$. (Since
$P^{\prime}$ is from a low-dimensional subspace, the ambient dimension is
small, and the computation can exploit this.)
Therefore, for each $F\in{\cal F}$,
$\mbox{dist}(P,F)=(\mbox{dist}(P,F)-\mbox{dist}(P^{\prime},F))+\mbox{dist}(P^{\prime},F)\approx(\mbox{dist}(S,F)-\mbox{dist}(S^{\prime},F))+\mbox{dist}(Q^{\prime},F)$.
Thus the weighted set $Q^{\prime}\cup S\cup S^{\prime}$ is a coreset for $P$,
but notice that the points in $S^{\prime}$ have negative weights. In contrast,
the weights of points in the coreset in our construction are positive. The
advantage of getting coresets with positive weights is that in order to get an
approximate solution to the shape fitting problem, we may run algorithms or
heuristics developed for the shape fitting problem on the coreset, such as
[1]. When points have negative weights, on the other hand, some of these
heuristics do not work or need to be modified appropriately.
Another useful feature of the coresets obtained via our results is that the
coreset is a subset of the original point set. When each point stands for a
data item, the coreset inherits a natural interpretation. See [11] for a
discussion of this issue in a broader context.
The sizes of the coresets in this paper are somewhat larger than the size of
coresets in [6]. Roughly speaking, the size of the coreset in [6] is
$f_{1}(d)+f_{2}(j,k)$, where $f_{1}(d)$ (respectively $f_{2}(j,k)$) is a
function depending only on $d$ (respectively $j$ and $k$) for the
$(j,k)$-projective clustering problem, while the coreset size in our paper is
$f_{1}(d)\cdot f_{2}(j,k)$.
Organization of this paper: In this article, we focus on the construction that
establishes small total sensitivity for various shape fitting problems, and
the size of the resulting coreset. For clarity, we omit the description of
algorithms for computing such bounds on sensitivity. Efficient algorithms
result from the construction using a methodology that is now well-understood.
Also because the weights for points in the coreset are nonnegative, the
coreset lend itself to streaming settings, where points arrive one by one as
$p_{1},p_{2},\cdots$ [9][6]. In Section 2, we present necessary definitions
used through this article, and summarize related results from [6] and [12]. In
Section 3, we prove the upper bound of total sensitivity of an instance of a
shape fitting problem in high dimension by its low dimensional projection. In
Sections 4, 5, 6, and 7, we apply the upper bound from Section 3 to
$k$-median/$k$-means, clustering, $k$-line clustering, $j$-subspace
approximation, and the integer $(j,k)$-projective clustering problem,
respectively, to obtain upper bounds for their total sensitivities, and the
size of the resulting $\epsilon$-coresets.
## 2\. Preliminaries
In this section, we formally define some of the concepts studied in this
article, and state crucial results from previous work. We begin by defining an
$\epsilon$-coreset.
###### Definition 2.1 ($\epsilon$-coreset of a shape fitting problem).
Given an instance $P\subset\mathbb{R}^{d}$ of a shape fitting problem
$(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, and $\epsilon\in[0,1]$, an
$\epsilon$-coreset of $P$ is a (weighted) set $S\subseteq P$, together with a
weight function $w:S\to\mathbb{R}^{+}$, such that for any shape $F$ in
${\mathcal{F}}$, it holds that
${\lvert\mbox{dist}(P,F)-\mbox{dist}(S,F)\rvert}\leq\epsilon\cdot\mbox{dist}(P,F)$,
where by definition, $\mbox{dist}(P,F)=\sum_{p\in P}\mbox{dist}(p,F),\mbox{
and }\mbox{dist}(S,F)=\sum_{p\in S}w(p)\mbox{dist}(p,F)$. The size of the
weighted coreset $S$ is defined to be $|S|$.
We note that in the literature, the requirement that the weights be non-
negative, as well as the requirement that the coreset $S$ be a subset of the
original instance $P$, are sometimes relaxed. We include these requirements in
the definition to emphasize that the coresets constructed here do satisfy
them. We now define the sensitivities of points in a shape fitting instance,
and the total sensitivity of the instance.
###### Definition 2.2 (Sensitivity of a shape fitting instance [10]).
Given an instance $P\subset\mathbb{R}^{d}$ of a shape fitting problem
$(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, the sensitivity of a point $p$ in
$P$ is $\sigma_{P}(p):=\inf\\{\beta\geq
0|\mbox{dist}(p,F)\leq\beta\mbox{dist}(P,F),\forall F\in{\mathcal{F}}\\}.$
Note that an equivalent definition is to let
$\sigma_{P}(p)=\sup_{F\in{\mathcal{F}}}\mbox{dist}(p,F)/\mbox{dist}(P,F)$,
with the understanding that when the denominator in the ratio is $0$, the
ratio itself is $0$.
The total sensitivity of the instance $P$, is defined by
$\mathfrak{S}_{P}:=\sum_{p\in P}\sigma_{P}(p)$. The total sensitivity function
of the shape fitting problem is $\mathfrak{S}_{n}:=\sup_{{\lvert
P\rvert}=n}\mathfrak{S}_{P}$.
We now need a somewhat technical definition in order to be able to state an
important earlier result from [6]. On a first reading, the reader is welcome
to skip the detailed definition.
###### Definition 2.3 (The dimension of a shape fitting instance [6]).
Let $P\subset\mathbb{R}^{d}$ be an instance of a shape fitting problem
$(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$. For a weight function
$w:P\to\mathbb{R}^{+}$, consider the set system $(P,{\mathcal{R}})$, where
${\mathcal{R}}$ is a family of subsets of $P$ defined as follows: each element
in ${\mathcal{R}}$ is a set of the form $R_{F,r}$ for some $F\in{\mathcal{F}}$
and $r\geq 0$, and $R_{F,r}=\\{p\in P\ |\ w_{p}\cdot\mbox{dist}(p,F)\leq
r\\}$. That is, $R_{F,r}$ is the set of those points in $P$ whose weighted
distance to the shape $F$ is at most $r$. The dimension of the instance $P$ of
the shape fitting problem, denoted by ${\rm dim}\left(P\right)$, is the
smallest integer $m$, such that for any weight function $w$ and $A\subseteq P$
of size ${\lvert A\rvert}=a\geq 2$, we have: ${\lvert\\{A\cap
R_{F,r}|F\in{\mathcal{F}},r\geq 0\\}\rvert}\leq a^{m}$.
For instance, in the $(j,k)$-projective clustering problem with the underlying
distance function ${\rm dist}$ being the $z^{\rm th}$ power of the Euclidean
distance, the dimension ${\rm dim}\left(P\right)$ of any instance $P$ is
$O(jdk)$, independent of $|P|$ [6]. This is shown by methods similar to the
ones used to bound the VC-dimension of geometric set systems. In fact, this
bound is the only fact that we will need about the dimension of a shape
fitting instance.
The following theorem recalls the connection established in [6] between
coresets and sensitivity via the above notion of dimension.
###### Theorem 2.4 (Connection between total sensitivity and
$\epsilon$-coreset [6]).
Given any $n$-point instance $P\subset\mathbb{R}^{d}$ of a shape fitting
problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, and any
$\epsilon\in(0,1]$, there exists an $\epsilon$-coreset for $P$ of size
$O\left(\left(\frac{\mathfrak{S}_{n}}{\epsilon}\right)^{2}{\rm
dim}\left(P\right)\right)$.
Finally, we will need known bounds on the total sensitivity of
$(j,k)$-projective clustering problem. These earlier bounds involve the
dimension $d$ corresponding to shape fitting problem
$(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$.
###### Theorem 2.5 (Total sensitivity of $(j,k)$-projective clustering
problem in fixed dimension [12]).
We have the following upper bounds of total sensitivities for the
$(j,k)$-projective clustering problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$, where ${\rm dist}$ is the $z$-th power of the Eucldiean distance for
$z\in(0,\infty)$.
* •
$j=1$ ($k$-line center): $\mathfrak{S}_{n}$ is $O(k^{f(k,d)}\log n)$, where
$f(d,k)$ is a function depending only on $d$ and $k$.
* •
integer $(j,k)$-projective clustering problem: For any $n$-point instance $P$,
with each coordinate being an integer of magnitude at most $n^{c}$ for any
constant $c>0$, $\mathfrak{S}_{P}$ is $O((\log n)^{f(d,j,k)})$, where
$f(d,j,k)$ is a function depending only on $d$, $j$, and $k$.
## 3\. Bounding the Total Sensitivity via Dimension Reduction
In this section, we show that the total sensitivity of a point set $P$ is of
the same order as that of ${\rm proj}\left(P,F^{\ast}\right)$, which is the
projection of $P$ onto an optimum shape $F^{\ast}$ from ${\mathcal{F}}$
fitting $P$. This result captures the fact that total sensitivity of a shape
fitting problem quantifies the complexity of shapes, in the sense that total
sensitivity depends on the dimension of smallest subspace containing each
shape, regardless of the dimension of the ambient space where $P$ is from.
###### Definition 3.1 (projection of points on a shape).
For a shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$,
define ${\rm proj}:\mathbb{R}^{d}\times{\mathcal{F}}\to\mathbb{R}^{d}$, where
${\rm proj}\left(p,F\right)$ is the projection of $p$ on a shape $F$, that is,
${\rm proj}\left(p,F\right)$ is a point in $F$ which is nearest to $p$, with
ties broken arbitrarily. That is, $\mbox{dist}(p,{\rm
proj}\left(p,F\right))=\min_{q\in F}\mbox{dist}(p,q)$. We abuse the notation
to denote the multi-set $\\{{\rm proj}\left(p,F\right)|p\in P\\}$ by ${\rm
proj}\left(P,F\right)$ for $P\subset\mathbb{R}^{d}$.
We first show that $\mathfrak{S}_{P}$ is $O(\mathfrak{S}_{{\rm
proj}\left(P,F^{\ast}\right)})$, where $F^{\ast}$ is an optimum shape fitting
$P$ from ${\mathcal{F}}$. In particular, this implies that when $F^{\ast}$ is
a low-dimensional object, the total sensitivity of $P\subset\mathbb{R}^{d}$
can be upper bounded by the total sensitivity of a point set contained in a
low dimension subspace.
###### Theorem 3.2 (Dimension reduction, computing the total sensitivity of a
point set in high dimensional space with the projected lower dimensional point
set).
Given an instance $P$ of a shape fitting problem
$(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, let $F^{\ast}$ denote a shape
that minimizes $\mbox{dist}(P,F)$ over all $F\in{\mathcal{F}}$. Let
$p^{\prime}$ denote ${\rm proj}\left(p,F^{\ast}\right)$ and let $P^{\prime}$
denote ${\rm proj}\left(P,F^{\ast}\right)$. Assume that the distance function
satisfies the relaxed triangle inequality:
$\mbox{dist}(p,q)\leq\alpha(\mbox{dist}(p,r)+\mbox{dist}(r,q))$ for any
$p,q,r\in\mathbb{R}^{d}$ for some constant $\alpha\geq 1$. Then
1. (1)
the following inequality holds: $\mathfrak{S}_{P}\leq
2\alpha^{2}\mathfrak{S}_{P^{\prime}}+\alpha$.
2. (2)
if $\mbox{dist}(P,F^{\ast})=0$, then
$\sigma_{P}(p)=\sigma_{P^{\prime}}(p^{\prime})$ for each $p\in P$. If
$\mbox{dist}(P,F^{\ast})>0$, then
$\sigma_{P}(p)\leq\left(\alpha\frac{\mbox{dist}(p,p^{\prime})}{\mbox{dist}(P,F^{\ast})}+2\alpha^{2}\sigma_{P^{\prime}}(p^{\prime})\right).$
###### Proof 3.3.
If $\mbox{dist}(P,F^{\ast})=0$, then $P=P^{\prime}$, and clearly both parts of
the theorem hold.
Let us consider the case where $\mbox{dist}(P,F^{\ast})>0$. By definition,
$\displaystyle\sigma_{P}(p)$ $\displaystyle=\inf\\{\beta\geq 0\ |\
\mbox{dist}(p,F)\leq\beta\mbox{dist}(P,F),\forall F\in{\mathcal{F}}\\},$
$\displaystyle\sigma_{P^{\prime}}(p^{\prime})$
$\displaystyle=\inf\\{\beta^{\prime}\geq 0\ |\
\mbox{dist}(p^{\prime},F)\leq\beta^{\prime}\mbox{dist}(P^{\prime},F),\forall
F\in{\mathcal{F}}\\}.$
Let $F$ be an arbitrary shape in ${\mathcal{F}}$. Then we have
$\begin{split}\mbox{dist}(p,F)&\leq\alpha\mbox{dist}(p,p^{\prime})+\alpha\mbox{dist}(p^{\prime},F)\\\
&\leq\alpha\mbox{dist}(p,p^{\prime})+\alpha\sigma_{P^{\prime}}(p^{\prime})\mbox{dist}(P^{\prime},F)\\\
&\leq\alpha\mbox{dist}(p,p^{\prime})+2\alpha^{2}\sigma_{P^{\prime}}(p^{\prime})\mbox{dist}(P,F)\\\
&=\alpha\frac{\mbox{dist}(p,p^{\prime})}{\mbox{dist}(P,F)}\cdot\mbox{dist}(P,F)+2\alpha^{2}\sigma_{P^{\prime}}(p^{\prime})\mbox{dist}(P,F)\\\
&\leq\alpha\frac{\mbox{dist}(p,p^{\prime})}{\mbox{dist}(P,F^{\ast})}\cdot\mbox{dist}(P,F)+2\alpha^{2}\sigma_{P^{\prime}}(p^{\prime})\mbox{dist}(P,F)\\\
&=\left(\alpha\frac{\mbox{dist}(p,p^{\prime})}{\mbox{dist}(P,F^{\ast})}+2\alpha^{2}\sigma_{P^{\prime}}(p^{\prime})\right)\mbox{dist}(P,F).\end{split}$
The first inequality follows from the relaxed triangle inequality, the second
inequality follows from the definition of sensitivity of $p^{\prime}$ in
$P^{\prime}$, and third inequality follows from the fact that
$\mbox{dist}(P^{\prime},F)=\sum_{p^{\prime}\in
P^{\prime}}\mbox{dist}(p^{\prime},F)\leq\sum_{p\in
P}\alpha\left(\mbox{dist}(p,F)+\mbox{dist}(p,p^{\prime})\right)=\alpha(\mbox{dist}(P,F)+\mbox{dist}(P,F^{\ast}))\leq
2\alpha\mbox{dist}(P,F),$ since $\mbox{dist}(P,F^{\ast})\leq\mbox{dist}(P,F)$.
Thus the second part of the theorem holds. Now,
$\displaystyle\mathfrak{S}_{P}$ $\displaystyle=$ $\displaystyle\sum_{p\in
P}\sigma_{P}(p)$ $\displaystyle\leq$ $\displaystyle\sum_{p\in
P}\left(\alpha\frac{\mbox{dist}(p,p^{\prime})}{\mbox{dist}(P,F^{\ast})}+2\alpha^{2}\sigma_{P^{\prime}}(p^{\prime})\right)$
$\displaystyle=$ $\displaystyle\alpha+2\alpha^{2}\mathfrak{S}_{P^{\prime}}.$
We make a remark regarding the value of $\alpha$ in Theorem 3.2 when the
distance function is $z^{\rm th}$ power of Euclidean distance. It is used in
Sections 4, 5, 6, and 7 when we derive upper bounds of total sensitivities for
various shape fitting problems.
###### Remark 3.4 (Value of $\alpha$ when
$\mbox{dist}(\cdot,\cdot)=(\|\,\cdot\,\|_{2})^{z}$).
Let $z\in(0,\infty)$. Suppose $\mbox{dist}(p,q)=(\|\,p-q\,\|_{2})^{z}$. When
$z\in(0,1)$, the weak triangle inequality holds with $\alpha=1$; when $z\geq
1$, the weak triangle inequality holds with $\alpha=2^{z-1}$. For a proof,
see, for example, [8].
Theorem 3.2 bounds the total sensitivity of an instance $P$ of a shape fitting
problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$ in terms of the total
sensitivity of $P^{\prime}$. Suppose that there is an $m_{2}\ll d$ so that
each shape $F\in{\mathcal{F}}$ is in some subspace of dimension $m_{2}$. In
the $(j,k)$-projective clustering problem, for example, $m_{2}=k(j+1)$. Then
note that $P^{\prime}$ is contained in a subspace of dimension $m_{2}$.
Furthermore, when ${\rm dist}$ is the $z^{\rm th}$ power of the Euclidean
distance, it turns out that for many shape fitting problems the sensitivity of
$P^{\prime}$ can be bounded as if the shape fitting problem was housed in
$\mathbb{R}^{2m_{2}}$ instead of $\mathbb{R}^{d}$. To see why this is the case
for the $(j,k)$-projective clustering problem, fix an arbitrary subspace $G$
of dimension $\min\\{d,2m_{2}\\}$ that contains $P^{\prime}$. Then for for any
$F\in{\mathcal{F}}$, there is an $F^{\prime}\in{\mathcal{F}}$ such that (a)
$F^{\prime}$ is contained in $G$, and (b)
$\mbox{dist}(p^{\prime},F^{\prime})=\mbox{dist}(p^{\prime},F)$ for all
$p^{\prime}\in P^{\prime}$.
The following theorem summarizes this phenomenon. For simplicity, it is stated
for the $(j,k)$-projective clustering problem, even though the phenomenon
itself is somewhat more general.
###### Theorem 3.5 (Sensitivity of a lower dimensional point set in a high
dimensional space).
Let $P^{\prime}$ be an $n$-point instance of the $(j,k)$-projective clustering
problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$, where ${\rm dist}$ is the
$z^{\rm th}$ power of the Euclidean distance, for some $z\in(0,\infty)$.
Assume that $P^{\prime}$ is contained in a subspace of dimension $m_{1}$.
(Note that for each shape $F\in{\mathcal{F}}$, there is a subspace of
dimension $m_{2}=k(j+1)$ containing it.) Let $G$ be any subspace of dimension
$m=\min\\{m_{1}+m_{2},d\\}$ containing $P^{\prime}$; fix an orthonormal basis
for $G$, and for each $p^{\prime}\in P^{\prime}$, let
$p^{\prime\prime}\in\mathbb{R}^{m}$ be the coordinates of $p^{\prime}$ in
terms of this basis. Let $P^{\prime\prime}=\\{p^{\prime\prime}\ \mid\
p^{\prime}\in P^{\prime}\\}$, and view $P^{\prime\prime}$ as an instance of
the $(j,k)$-projective clustering problem
$(\mathbb{R}^{m},{\mathcal{F}}^{\prime},{\rm dist})$, where
${\mathcal{F}}^{\prime}$ is the set of all $k$-tuples of $j$-subspaces in
$\mathbb{R}^{m}$, and ${\rm dist}$ is the $z^{\rm th}$ power of the Eucldiean
distance. Then,
$\sigma_{P^{\prime}}(p^{\prime})=\sigma_{P^{\prime\prime}}(p^{\prime\prime})$
for each $p^{\prime}\in P^{\prime}$, and
$\mathfrak{S}_{P^{\prime}}=\mathfrak{S}_{P^{\prime\prime}}$.
## 4\. $k$-median/$k$-means Clustering Problem
In this section, we derive upper bounds for the total sensitivity function for
the $k$-median/$k$-means problems, and its generalizations, where the distance
function is $z^{\rm th}$ power of Euclidean distance, using the approach in
Section 4. These bounds are similar to the ones derived by Langberg and
Schulman [10], but the proof is much simplified. For the rest of the article,
${\rm dist}$ is assumed to be the $z^{\rm th}$ power of the Euclidean
distance.
###### Theorem 4.1 (Total sensitivity of $(0,k)$-projective clustering).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$, where ${\mathcal{F}}$ is the set of all $k$-point subsets of
$\mathbb{R}^{d}$. We have the following upper bound on the total sensitivity:
$\displaystyle\mathfrak{S}_{n}$ $\displaystyle\leq 2^{2z-1}k+2^{z-1},$
$\displaystyle z\geq 1,$ $\displaystyle\mathfrak{S}_{n}$ $\displaystyle\leq
2k+1,$ $\displaystyle z\in(0,1).$
In particular, the total sensitivity of the $k$-median problem (which
corresponds to the case when $z=1$) is at most $2k+1$, and the total
sensitivity of the $k$-means problem (which corresponds to the case when
$z=2$) is $8k+2$.
###### Proof 4.2.
Let $P$ be an arbitrary $n$-point set. Apply Theorem 3.2, and note that ${\rm
proj}\left(P,C^{\ast}\right)$, where $C^{\ast}$ is an optimum set of $k$
centers, contains at most $k$ distinct points. Assume that
$C^{\ast}=\\{c_{1}^{\ast},c_{2}^{\ast},\cdots,c_{k}^{\ast}\\}$. Let $P_{i}$ be
the set of points in $P$ whose projection is $c_{i}^{\ast}$, that is,
$P_{i}=\\{p\in P|{\rm proj}\left(p,C^{\ast}\right)=c_{i}^{\ast}\\}$. It is
easy to see that the summation of sensitivities of the ${\lvert P_{i}\rvert}$
copies of $c_{i}^{\ast}$ is at most 1: for any $k$-point set $C$ in
$\mathbb{R}^{d}$, ${\lvert
P_{i}\rvert}\cdot\frac{\mbox{dist}(c_{i}^{\ast},C)}{\mbox{dist}(C^{\ast},C)}=\frac{{\lvert
P_{i}\rvert}\mbox{dist}(c_{i}^{\ast},C)}{\sum_{j=1}^{k}{\lvert
P_{j}\rvert}\mbox{dist}(c_{j}^{\ast},C)}\leq 1$.
Therefore, the total sensitivity of ${\rm proj}\left(P,C^{\ast}\right)$ is at
most $k$. Substituting $\alpha$ from the remark after Theorem 3.2, we get the
above result.
###### Theorem 4.3 ($\epsilon$-coreset for $(0,k)$-projective clustering).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$, where ${\mathcal{F}}$ is the set of all $k$-point subsets of
$\mathbb{R}^{d}$. For any $n$-point instance $P$, there is an
$\epsilon$-coreset of size $O(k^{3}d\epsilon^{-2})$.
###### Proof 4.4.
Observe that the ${\rm dim}\left(P\right)$ is $O(kd)$. Using Theorem 2.4, and
Theorem 4.1, we obtain the above result.
## 5\. $k$-line Clustering Problem
In this section, we derive upper bounds on the total sensitivity function for
the $k$-line clustering problem, that is, the $(1,k)$-projective clustering
problem.
###### Theorem 5.1 (Total sensitivity for $k$-line clustering problem).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$, where ${\mathcal{F}}$ is the set of $k$-tuple of lines. The total
sensitivity function, $\mathfrak{S}_{n}$, is $O(k^{f(k)}\log n)$, where $f(k)$
is a function the depends only on $k$.
###### Proof 5.2.
Let $P$ be an arbitrary $n$-point set. Let $K^{\ast}$ denote an optimum set of
$k$ lines fitting $P$. Using Theorems 3.2 and 3.5, it suffices to bound the
sensitivity of an $n$-point instance of a $k$-line clustering problem housed
in $\mathbb{R}^{4k}$. By Theorem 2.5, the total sensitivity of this latter
shape fitting problem is $O(k^{f(k)}\log n)$, where $f(k)$ is a function
depending only on $k$. Therefore, $\mathfrak{S}_{n}$ is $O(k^{f(k)}\log n)$.
(Alternatively, one could use a recent result in [8]. Let $P^{\prime}$ denote
the projection of $P$ into $K^{\ast}$. Since $K^{\ast}$ is a union of $k$
lines, we can upper bound the sensitivity of $P^{\prime}$ by $k$ times the
sensitivity of an $n$-point set that lies on a single line. The sensitivity of
an $n$-point set that lies on a single line can be upper bounded by the
sensitivity of an $n$-point set for the weighted $(0,k)$-projective clustering
problem, for which the sensitivity bound is $O(k^{f(k)}\log n)$ as shown in
[8].)
Notice that for $k$-line clustering problem, the bound on the total
sensitivity depends logarithmically on $n$. We give below a construction of a
point set that shows that this is necessary, even for $d=2$.
###### Theorem 5.3 (The upper bound of total sensitivity for $k$-line
clustering problem is tight).
For every $n\geq 2$, there exists an $n$-point instance of the $k$-line
clustering problem $(\mathbb{R}^{2},{\mathcal{F}},{\rm dist})$, where ${\rm
dist}$ is the Euclidean distance, such that the total sensitivity of $P$ is
$\Omega(\log n)$.
###### Proof 5.4.
We construct a point set $P$ of size $n$, together with $n$ shapes
$F_{i}\in{\mathcal{F}}$, $i=1,\cdots,n$, such that
$\sum_{i=1}^{n}\mbox{dist}(p_{i},F_{i})/\mbox{dist}(P,F_{i})$ is $\Omega(\log
n)$. Note that this implies that $\mathfrak{S}_{P}$ is at least $\Omega(\log
n)$. Let $P$ be the following point set in $\mathbb{R}^{2}$:
$p_{i}=(1/2^{i-1},0)$, for $i=1,\cdots,n$. Let $F_{i}$ be a pair of lines: one
vertical line and one horizontal line, where the vertical line is the
$y$-axis, and the horizontal line is $\\{(x,1/2^{i})|x\in\mathbb{R}\\}$.
Consider the point $p_{i}$, where $i=1,\cdots,n$. We show that
$\mbox{dist}(p_{i},F_{i})/\mbox{dist}(P,F_{i})$ is at least $1/(2+i)$, for
$i=1,\cdots,n$. For $j\leq i$, note that $\mbox{dist}(p_{j},F_{i})=1/2^{i}$:
since the distance from $p_{j}$ to the horizontal line in $F_{i}$ is $1/2^{i}$
and the distance to the vertical line is $1/2^{j-1}$,
$\mbox{dist}(p_{j},F_{i})=\min\\{1/2^{j-1},1/2^{i}\\}=1/2^{i}$. For $i+1\leq
j\leq n$, on the other hand, $\mbox{dist}(p_{j},F_{i})=1/2^{j-1}$. Therefore,
$\sum_{j=i+1}^{n}\mbox{dist}(p_{j},F_{i})=\sum_{j=i+1}^{n}1/2^{j-1}=(1/2^{i-1})\cdot(1-(1/2)^{n-i})$.
Thus, we have
$\sigma_{P}(p_{i})=\sup_{F\in{\mathcal{F}}}\frac{\mbox{dist}(p_{i},F)}{\mbox{dist}(P,F)}\geq\frac{\mbox{dist}(p_{i},F_{i})}{\mbox{dist}(P,F_{i})}=\frac{1/2^{i}}{(1/2^{i-1}-1/2^{n-1})+i\cdot(1/2^{i})}>\frac{1}{2+i}$
Therefore,
$\mathfrak{S}_{P}\geq\sum_{i=1}^{n}\sigma_{P}(p_{i})>\sum_{i=1}^{n}\frac{1}{2+i}$,
which is $\Omega(\log n)$.
###### Theorem 5.5 ($\epsilon$-coreset for $k$-line clustering problem).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$, where ${\mathcal{F}}$ is the set of all $k$-tuples of lines in
$\mathbb{R}^{d}$. For any $n$-point instance $P$, there is an
$\epsilon$-coreset with size $O(k^{f(k)}d(\log n)^{2}/\epsilon^{2})$.
###### Proof 5.6.
This result follows from Theorem 5.1, Theorem 2.4, and the fact that ${\rm
dim}\left(P\right)$ in this case is $O(kd)$.
## 6\. Subspace approximation
In this section, we derive upper bounds on the sensitivity of the subspace
approximation problem, that is, the $(j,1)$-projective clustering problem. For
the applications of Theorems 3.2 and 3.5 in the other sections, we use
existing bounds on the sensitivity that have a dependence on the dimension
$d$. For the subspace approximation problem, however, we derive here the
dimension-dependent bounds on sensitivity by generalizing an argument from
[10] for the case $j=d-1$ and $z=2$. This derivation is somewhat technical.
With these bounds in hand, the derivation of the dimension-independent bounds
is readily accomplished in a manner similar to the other sections.
distance. Although the size of the $\epsilon$-coreset obtained in this way is
exponential in $j$, which is larger than the size of the coreset in
[6][feldmanarvix] and Theorem LABEL: in this section, it is still a constant
(as $j$ is considered as a constant) and in particular, independent of the
cardinality of the input point set. It can be considered as an simple and
straight-forward way to see why small $\epsilon$-coresets exist for
$j$-subspace approximation problems.
### 6.1. Dimension-dependent bounds on Sensitivity
We first recall the notion of an _$(\alpha,\beta,z)$ -conditioned basis_ from
[5], and state one of its properties (Lemma 6.2). We will use standard matrix
terminilogy: $m_{ij}$ denotes the entry in the $i$-th row and $j$-th column of
$M$, and $M_{i\cdot}$ is the $i$-th row of $M$.
###### Definition 6.1.
Let $M$ be an $n\times m$ matrix of rank $\rho$. Let $z\in[1,\infty)$, and
$\alpha,\beta\geq 1$. An $n\times\rho$ matrix $A$ is an
$(\alpha,\beta,z)$-conditioned basis for $M$ if the column vectors of $A$ span
the column space of $M$, and additionally $A$ satisfies that: (1)
$\sum_{i,j}{\lvert a_{ij}\rvert}^{z}\leq\alpha^{z}$, (2) for all
$u\in\mathbb{R}^{\rho}$, $\|\,u\,\|_{z^{\prime}}\leq\beta\|\,Au\,\|_{z}$,
where $\|\,\cdot\,\|_{z^{\prime}}$ is the dual norm for $\|\,\cdot\,\|_{z}$
(i.e. $1/z+1/z^{\prime}=1$).
###### Lemma 6.2.
Let $M$ be an $n\times m$ matrix of rank $\rho$. Let $z\in[1,\infty)$. Let $A$
be an $(\alpha,\beta,z)$-conditioned basis for $M$. For every vector
$u\in\mathbb{R}^{m}$, the following inequality holds: ${\lvert
M_{i\cdot}u\rvert}^{z}\leq\left(\|\,A_{i\cdot}\,\|_{z}^{z}\cdot\beta^{z}\right)\|\,Mu\,\|_{z}^{z}$.
###### Proof 6.3.
We have $M=A\tau$ for some $\rho\times m$ matrix $\tau$. Then,
${\lvert M_{i\cdot}u\rvert}^{z}={\lvert A_{i\cdot}\tau
u\rvert}^{z}\leq\|\,A_{i\cdot}\,\|_{z}^{z}\cdot\|\,\tau
u\,\|_{z^{\prime}}^{z}\leq\|\,A_{i\cdot}\,\|_{z}^{z}\cdot\beta^{z}\|\,A\tau
u\,\|_{z}^{z}=\|\,A_{i\cdot}\,\|_{z}^{z}\cdot\beta^{z}\|\,Mu\,\|_{z}^{z}.$
The second step is Holder’s inequality, and the third uses the fact that $A$
is $(\alpha,\beta,z)$-conditioned.
Using Lemma 6.2, we derive an upper bound on the total sensitivity when each
shape is a hyperplane.
###### Lemma 6.4 (total sensitivity for fitting a hyperplane).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$
where ${\mathcal{F}}$ is the set of all $(d-1)$-flats, that is, hyperplanes.
The total sensitivity of any $n$-point set is $O(d^{1+z/2})$ for $1\leq z<2$,
$O(d)$ for $z=2$, and $O(d^{z})$ for $z>2$.
###### Proof 6.5.
We can parameterize a hyperplane with a vector in $\mathbb{R}^{d+1}$,
$u=\begin{bmatrix}u_{1}&\cdots&u_{d+1}\end{bmatrix}^{T}$: the hyperplane
determined by $u$ is
$h_{u}=\\{x\in\mathbb{R}^{d}|\sum_{i=1}^{d}u_{i}x_{i}+u_{d+1}=0\\}$, where
$x_{i}$ denotes the $i^{\rm th}$ entry of the vector $x$. Without loss of
generality, we may assume that $\sum_{i=1}^{d}u_{i}^{2}=1$. The Euclidean
distance to $h_{u}$ from a point $q\in\mathbb{R}^{d}$ is
$\mbox{dist}(q,h_{u})={\lvert\sum_{i=1}^{d}u_{i}q_{i}+u_{d+1}\rvert}/\sqrt{\sum_{i=1}^{d}u_{i}^{2}}={\lvert\sum_{i=1}^{d}u_{i}q_{i}+u_{d+1}\rvert}.$
(the second equality follows from the assumption that
$\sum_{i=1}^{d}u_{i}^{2}=1$.)
Let $P=\\{p_{1},p_{2},\ldots,p_{n}\\}\subseteq\mathbb{R}^{d}$ be any set of
$n$ points. Let $\tilde{p_{i}}$ denote the row vector
$\begin{bmatrix}p_{i}^{T}&1\end{bmatrix}$, and let $M$ be the $n\times(d+1)$
matrix whose $i^{\rm th}$ row is $\tilde{p_{i}}$. Then,
$\mbox{dist}(p_{i},h_{u})={\lvert M_{i\cdot}u\rvert}^{z}$, and
$\mbox{dist}(P,h_{u})=\sum_{i=1}^{n}{\lvert
M_{i\cdot}u\rvert}^{z}=\|\,Mu\,\|_{z}^{z}$. Then using Lemma 6.2, we have
$\sigma_{P}(p_{i})=\sup_{u}\frac{{\lvert
M_{i\cdot}u\rvert}^{z}}{\|\,Mu\,\|_{z}^{z}}\leq\|\,A_{i\cdot}\,\|_{z}^{z}\cdot\beta^{z}$,
where $A$ is an $(\alpha,\beta,z)$-conditioned basis for $M$. Thus,
$\mathfrak{S}_{P}=\sum_{i=1}^{n}\sigma_{P}(p_{i})\leq\beta^{z}\sum_{i=1}^{n}\|\,A_{i\cdot}\,\|_{z}^{z}=\beta^{z}\sum_{i,j}{\lvert
a_{ij}\rvert}^{z}=(\alpha\beta)^{z}.$
For $1\leq z<2$, $M$ has $((d+1)^{1/z+1/2},1,z)$-conditioned basis; for $z=2$,
$M$ has $((d+1)^{1/2},1,z)$-conditioned basis; for $z>2$, $M$ has
$((d+1)^{1/z+1/2},(d+1)^{1/z^{\prime}-1/2},z)$-conditioned basis [5]. Thus the
total sensitivity for the three cases are $(d+1)^{1+z/2}$, $d+1$, and
$(d+1)^{z}$, respectively.
It is now easy to derive dimension-dependent bounds on the sensitivity when
each shape is a $j$-subspace.
###### Corollary 6.6 (Total sensitivity for fitting a $j$-subspace).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$
where ${\mathcal{F}}$ is the set of all $j$-flats. The total sensitivity of
any $n$-point set is $O(d^{1+z/2})$ for $1\leq z<2$, $O(d)$ for $z=2$, and
$O(d^{z})$ for $z>2$.
###### Proof 6.7.
Denote by ${\mathcal{F}}^{\prime}$ the set of hyperplanes in $\mathbb{R}^{d}$.
Let $P\subseteq\mathbb{R}^{d}$ be an arbitrary $n$-point set. We first show
that $\sigma_{P,{\mathcal{F}}}(p)\leq\sigma_{P,{\mathcal{F}}^{\prime}}(p)$,
where the additional subscript is being used to indicate which shape fitting
problem we are talking about (hyperplanes or $j$-flats). Let $p$ be an
arbitrary point in $P$. Let $F_{p}\in{\mathcal{F}}$ denote the $j$-subspace
such that
$\sigma_{P,{\mathcal{F}}}(p)=\mbox{dist}(p,F_{p})/\mbox{dist}(P,F_{p})$. Let
${\rm proj}\left(p,F_{p}\right)$ denote the projection of $p$ on $F_{p}$.
Consider the hyperplane $F^{\prime}$ containing $F_{p}$ and orthogonal to the
vector $p-{\rm proj}\left(p,F_{p}\right)$. We have
$\mbox{dist}(p,F^{\prime})=\mbox{dist}(p,F_{p})$, whereas
$\mbox{dist}(q,F^{\prime})\leq\mbox{dist}(q,F_{p})$ for each $q\in P$.
Therefore,
$\sigma_{P,{\mathcal{F}}^{\prime}}(p)\geq\mbox{dist}(p,F^{\prime})/\mbox{dist}(P,F^{\prime})\geq\mbox{dist}(p,F_{p})/\mbox{dist}(P,F_{p})=\sigma_{P,{\mathcal{F}}}(p).$
It follows that
$\mathfrak{S}_{P,{\mathcal{F}}}\leq\mathfrak{S}_{P,{\mathcal{F}}^{\prime}}$.
The statement in the corollary now follows from Lemma 6.4.
### 6.2. Dimension-independent Bounds on the Sensitivity
We now derive dimension-independent upper bounds for the total sensitivity for
the $j$-subspace fitting problem.
###### Theorem 6.8 (Total sensitivity for $j$-subspace fitting problem).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$
where ${\mathcal{F}}$ is the set of all $j$-flats. The total sensitivity of
any $n$-point set is $O(j^{1+z/2})$ for $1\leq z<2$, $O(j)$ for $z=2$, and
$O(j^{z})$ for $z>2$.
###### Proof 6.9.
Use Theorem 3.2, note that the projected point set $P^{\prime}$ is contained
in a $j$-subspace. Further, each shape is a $j$-subspace. So, applying Theorem
3.5 and Corollary 6.6, the total sensitivity is $O(j^{2+z/2})$ or $z\in[1,2)$,
$O(j)$ for $z=2$ and $O(j^{z})$ for $z>2$.
Using Theorem 6.8 and the fact that ${\rm dim}\left(P\right)$ for the
$j$-subspace fitting problem is $O(jd)$, we obtain small $\epsilon$-coresets:
###### Theorem 6.10 ($\epsilon$-coreset for $j$-subspace fitting problem).
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm dist})$
where ${\mathcal{F}}$ is the set of all $j$-flats. For any $n$-point set,
there exists an $\epsilon$-coreset whose size is $O(j^{3+z}d\epsilon^{-2})$
for $z\in[1,2)$, $O(j^{3}d\epsilon^{-2})$ for $z=2$ and
$O(j^{2z+1}d\epsilon^{-2})$ for $z\geq 2$.
###### Proof 6.11.
The result follows from Theorem 6.8, and Theorem 2.4.
We note that for the case $j=d-1$ and $z=2$, a linear algebraic result from
[2] yields a coreset whose size is an improved $O(d\epsilon^{-2})$.
## 7\. The $(j,k)$ integer projective clustering
###### Theorem 7.1.
Consider the shape fitting problem $(\mathbb{R}^{d},{\mathcal{F}},{\rm
dist})$, where ${\mathcal{F}}$ is the set of $k$-tuples of $j$-flats. Let
$P\subset\mathbb{R}^{d}$ be any $n$-point instance with integer coordinates,
the magnitude of each coordinate being at most $n^{c}$, for some constant $c$.
The total sensitivity $\mathfrak{S}_{P}$ of $P$ is $O((\log n)^{f(k,j)})$,
where $f(k,j)$ is a function of only $k$ and $j$. There exists an
$\epsilon$-coreset for $P$ of size $O((\log n)^{2f(k,j)}kjd\epsilon^{-2})$.
###### Proof 7.2.
Observe that the projected point set $P^{\prime}={\rm
proj}\left(P,\\{J_{1}^{\ast},\cdots,J_{k}^{\ast}\\}\right)$, where
$\\{J_{1}^{\ast},\cdots,J_{k}^{\ast}\\}$ is an optimum $k$-tuple of $j$-flats
fitting $P$, is contained in a subspace of dimension $O(jk)$. Using Theorem
2.5, Theorem 3.5, and Theorem 3.2, the total sensitivity $\mathfrak{S}_{P}$ is
upper bouned by $O((\log n)^{f(k,j)})$, where $f(k,j)$ is a function of $k$
and $j$. (A technical complication is that the coordinates of $P^{\prime}$, in
the appropriate orthonormal basis, may not be integers. This can be addressed
by rounding them to integers, at the expense of increasing the constant $c$. A
similar procedure is adopted in [12], and we omit the details here.)
Using Theorem 2.4 and the fact that ${\rm dim}\left(P\right)$ is $O(djk)$, we
obtain the bound on the coreset.
## 8\. Acknowledgements.
We thank the anonymous reviewers and Dan Feldman for their insightful
feedback.
## References
* [1] Pankaj K. Agarwal and Nabil H. Mustafa. $k$-Means projective clustering. In Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, PODS ’04, pages 155–165, New York, NY, USA, 2004. ACM.
* [2] Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-Ramanujan sparsifiers. In STOC, pages 255–262, 2009.
* [3] Ke Chen. On coresets for $k$-median and $k$-means clustering in metric and euclidean spaces and their applications. SIAM J. Comput., 39(3):923–947, 2009.
* [4] Kenneth L. Clarkson. Subgradient and sampling algorithms for $\ell_{1}$ regression. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 257–266, 2005.
* [5] Anirban Dasgupta, Petros Drineas, Boulos Harb, Ravi Kumar, and Michael W. Mahoney. Sampling algorithms and coresets for $\ell_{p}$ regression. SIAM J. Comput., 38(5):2060–2078, 2009.
* [6] Dan Feldman and Michael Langberg. A unified framework for approximating and clustering data. In STOC, pages 569–578. For an updated version, see http://arxiv.org/abs/1106.1379v1, 2011.
* [7] Dan Feldman, Morteza Monemizadeh, and Christian Sohler. A PTAS for $k$-means clustering based on weak coresets. In SCG ’07: Proceedings of the twenty-third annual symposium on Computational geometry, pages 11–18, New York, NY, USA, 2007. ACM.
* [8] Dan Feldman and Leonard J. Schulman. Data reduction for weighted and outlier-resistant clustering. In SODA, pages 1343–1354, 2012.
* [9] Sariel Har-Peled and Soham Mazumdar. On coresets for $k$-means and $k$-median clustering. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, STOC ’04, pages 291–300, New York, NY, USA, 2004. ACM.
* [10] Michael Langberg and Leonard J. Schulman. Universal $\epsilon$-approximators for integrals. In SODA, pages 598–607, 2010.
* [11] Michael W. Mahoney and Petros Drineas. CUR matrix decompositions for improved data analysis. Proceedings of the National Academy of Sciences, 106(3):697–702, 2009.
* [12] Kasturi Varadarajan and Xin Xiao. A near-linear algorithm for projective clustering integer points. In SODA, pages 1329–1342, 2012.
|
arxiv-papers
| 2012-09-21T19:55:53 |
2024-09-04T02:49:35.391571
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kasturi Varadarajan and Xin Xiao",
"submitter": "Xin Xiao",
"url": "https://arxiv.org/abs/1209.4893"
}
|
1209.4915
|
# Cartography of high-dimensional flows: A visual guide to sections and slices
Predrag Cvitanović [email protected]. Daniel Borrero-Echeverry Keith M.
Carroll Bryce Robbins Evangelos Siminos Center for Nonlinear Science and
School of Physics, Georgia Inst. of Technology, Atlanta, GA 30332, USA
(July 21, 2012)
###### Abstract
Symmetry reduction by the method of slices quotients the continuous symmetries
of chaotic flows by replacing the original state space by a set of charts,
each covering a neighborhood of a dynamically important class of solutions,
qualitatively captured by a ‘template’. Together these charts provide an atlas
of the symmetry-reduced ‘slice’ of state space, charting the regions of the
manifold explored by the trajectories of interest. Within the slice, relative
equilibria reduce to equilibria and relative periodic orbits reduce to
periodic orbits. Visualizations of these solutions and their unstable
manifolds reveal their interrelations and the role they play in organizing
turbulence/chaos.
symmetry reduction, equivariant dynamics, relative equilibria, relative
periodic orbits, slices, moving frames
###### pacs:
02.20.-a, 05.45.-a, 05.45.Jn, 47.27.ed, 47.52.+j, 83.60.Wc
> Today, it is possible to take a stroll through the high-dimensional state
> space of hydrodynamic turbulence and observe that turbulent trajectories are
> guided by close passes to invariant solutions of the Navier-Stokes
> equations. Charting how close these passes are is a geometer’s task, but in
> order to place them on a map, one first has to deal with families of
> solutions equivalent under the symmetries of a given flow. Evolution in time
> decomposes the state space into a ‘spaghetti’ of time trajectories.
> Continuous spatial symmetries foliate it like the layers of an onion. In
> this visual tour of dynamics, we use a low-dimensional flow to illustrate
> how this tangle can be unraveled (symmetry reduction), and how to pick a
> single representative point for each trajectory (section it) and group orbit
> (slice it). Once the symmetry induced degeneracies are out of the way, one
> can identify and describe the prominent turbulent structures by a taxonomy
> of invariant building blocks (numerically exact solutions of the Navier-
> Stokes equations, finite sets of relative equilibria and infinite
> hierarchies of relative periodic orbits) and describe the dynamics in terms
> of near passes to their heteroclinic connections.
## I Introduction
Over the last decade, new insights into the dynamics of moderate Re turbulent
flows Faisst and Eckhardt (2003); Wedin and Kerswell (2004); Hof _et al._
(2004); Kerswell (2005) have been gained through visualizations of their
$\infty$-dimensional state spaces by means of dynamically invariant,
representation independent coordinate frames constructed from physically
prominent unstable coherent structures, Gibson, Halcrow, and Cvitanović (2008)
hereafter referred to as templates. Navigating and charting the geometry of
these extremely high-dimensional state spaces necessitates a reexamination of
two of the basic tools of the theory of dynamical systems: Poincaré sections
and symmetry reduction.
In quantum-mechanical calculations, one always starts out by making sure that
the Hamiltonian has been brought to its symmetry-reduced block-diagonal,
irreducible form; anything else would be sheer masochism. As the dynamical
theory of turbulence is still in its infancy, symmetry reduction is not yet a
common practice in processing turbulence data collected in experimental
measurements and numerical simulations. Symmetry reduction of nonlinear flows
is much trickier than the more familiar theory of irreducible representations
for linear problems such as quantum mechanics, so most of our sketches
illustrate the simplest case, the 1-parameter compact continuous group
$\textrm{SO}(2)$ symmetry.
We show here how to bring the numerical or experimental data to a symmetry-
reduced format _before_ any further analysis of it takes place. Our tool of
choice is the linear implementation of the method of slices. Rowley and
Marsden (2000); Beyn and Thümmler (2004); Siminos and Cvitanović (2011);
Froehlich and Cvitanović (2011) Here, we extend this local method to a global
reduction of a turbulent flow by defining local ‘charts’, their borders, and
the ridges that glue these linear tiles into an atlas that spans the ergodic
state space region of interest. While ‘charts’ and ‘atlases’ are standard
tools in geometry, the prescription for explicit construction of a symmetry-
reduced state space presented here is, to our knowledge, new. We explain the
key geometrical ideas in simple but illustrative settings, eschewing the fluid
dynamical and group theoretical technicalities.
(a)
[rgb]0,0,0$v$[rgb]0,0,0$x(\tau)$[rgb]0,0,0$t_{1}$[rgb]0,0,0$t_{2}$[rgb]0,0,0${\cal
M}_{x}$[rgb]0,0,0$x$ (b) [rgb]0,0,0$x(\tau)$[rgb]0,0,0$x(0)$
(c) [rgb]0,0,0$x(0)$[rgb]0,0,0${\cal M}_{x(0)}$[rgb]0,0,0${\cal
M}_{x(\tau)}$[rgb]0,0,0$x(\tau)$ (d) [rgb]0,0,0$x(\tau)$[rgb]0,0,0$x(0)$
Figure 1: (a) In the presence of an $N$-continuous parameter symmetry, each
state space point $x$ owns $(N\\!+\\!1)$ tangent vectors: one $v(x)$ along the
time flow $x(\tau)$, and the $N$ group tangents
$t_{1}(x),\,t_{2}(x),\,\cdots,t_{N}(x)$ along infinitesimal symmetry shifts,
tangent to the $N$-dimensional group orbit ${\cal M}_{x}$. (b) Each point has
a unique trajectory (blue) under time evolution. (c) Each point also belongs
to a group orbit (green) of symmetry-related points. For $\textrm{SO}(2)$,
this is topologically a circle. Any two points on a group orbit are physically
equivalent, but may lie far from each other in state space. (d) Together,
time-evolution and group actions trace out a wurst of physically equivalent
solutions.
Let us begin by defining a dynamical system comprised of a flow $f^{t}$ and
the state space ${\cal M}$ on which it acts. If a group $G$ of continuous
transformations acts on a continuous time flow, each state space point owns a
set of tangent vectors (Fig. 1 (a)). Integrated in time, the velocity vector
$v(x)$ traces out a trajectory ${f^{\tau}(x)}$ (Fig. 1 (b)). Applying the
continuous transformations traces out a group orbit ${\cal M}_{x}=\\{g\,x\mid
g\in{G}\\}\,$ (Fig. 1 (c)). Together, time evolution and group actions trace
out a complicated smooth manifold, hereafter affectionately referred to as a
wurst (see Figs. 1 (d), 4 (b) and 8), which we shall here teach you how to
slice.
A flow is said to have symmetry $G$ if the form of evolution equations
$\dot{x}=v(x)$ is left invariant, $v(x)=g^{-1}\,v(g\,x)\,,$ by the set of
transformations $g\in{G}$. If a flow has symmetry, the simplest solutions are
highly symmetric invariant equilibria and relative equilibria studied in
bifurcation-theory approaches to the onset of turbulence. Physicists love
symmetries, Kerswell (2012) but nature often prefers solutions of no symmetry:
while the flow equations may be invariant under $G$, turbulent solutions are
not. The highly symmetric solutions often lie far from the regions of state
space explored by turbulence Willis, Cvitanović, and Avila (2012) and thus are
of limited usefulness in understanding its dynamics. In contrast, the relative
periodic orbits studied here are embedded in the turbulence, and capture its
geometry and statistics.
We can make headway in unraveling the tangle of 1-dimensional time
trajectories with the notion of recurrence. To quantify how close the state of
the system at a given time is to a previously visited state, we need the
notion of distance between two points in state space. The simplest (but far
from the only, or the most natural) is the Euclidean norm
$\|{x-x^{\prime}}\|^{2}=\langle{x-x^{\prime}}\vphantom{x-x^{\prime}}|\vphantom{x-x^{\prime}}{x-x^{\prime}}\rangle=\sum_{i=1}^{d}(x-x^{\prime})_{i}^{2}\,.$
(1)
For experimental data, a better norm, for example, might be a distance between
digitized images. While in this paper we simply assume that a norm is given,
its importance cannot be overstated: the construction of invariant, PDE
discretization independent state space coordinates, Gibson, Halcrow, and
Cvitanović (2008) the symmetry reduction by minimization of the distance
between group orbits undertaken in what follows, and the utility of the charts
so constructed all depend on a well-chosen notion of distance in the high-
dimensional state spaces we are charting here.
Given a notion of distance, we can talk about a ‘neighborhood’, an open set of
nearby states with qualitatively similar dynamics. Our main task in what
follows will be to make this precise by defining a chart over a neighborhood
and its borders. Given distances and neighborhoods, the next key notion is
_measure_ , or how likely a typical trajectory is to visit a given
neighborhood. After some observations of a given turbulent flow, one can
identify a set of _templates_ , Rowley and Marsden (2000) points
${\hat{x}^{\prime}}{}^{(j)}$, $j=1,2,\cdots$ in the state space representative
of the most frequently revisited features of the flow.
Our goals here are two-fold: (i) In sect. II, we review the method of Poincaré
sections, with emphasis on two particular aspects that are applicable to high-
dimensional flows: the construction of multiple local linear charts and the
determination of their borders. (ii) In sect. III, we discuss the effect of
continuous symmetries on nonlinear flows, and in sect. IV we use the lessons
learned from our discussion of Poincaré sections to aid us in the reduction of
continuous symmetries, and, thus, enable us to commence a systematic charting
of the long-time dynamics of high-dimensional flows (sect. V).
## II Section
In the Poincaré section method, one records the coordinates $\hat{x}_{n}$ of
the trajectory $x(\tau)$ at the instants $\tau_{n}$ when it traverses a fixed
oriented hypersurface ${\cal P}$ of codimension 1. For the high-dimensional
flows that we have in mind, the practical choice is a hyperplane, the only
type of Poincaré section (from now on, just a _section_) that we shall
consider here. One can choose a section such that it contains a template of
interest. Properly oriented, such a section can capture important features of
the flow in the neighborhood of the section-fixing template.
But how far does this neighborhood extend? The answer is that the section
captures neighboring trajectories as long as it cuts them transversally; it
fails the moment the velocity field at a point $\hat{x}^{\ast}$ fails to
pierce the section. At these locations, the velocity either vanishes
(equilibrium) or is orthogonal to the section normal $\hat{n}$,
$\hat{n}\cdot v(\hat{x}^{\ast})=0\,,\qquad\hat{x}^{\ast}\in\cal{S}\,.$ (2)
For a smooth flow in $d$ dimensions such points form a smooth
$(d\\!-\\!2)$-dimensional _section border_ ${\cal S}\subset{\cal P}$, which
encloses the open neighborhood of the template characterized by qualitatively
similar flow. We shall refer to this region of the section as a _chart_ of the
template neighborhood (see Fig. 2). Beyond the border, the flow pierces the
section in the ‘wrong’ direction and the dynamics are qualitatively different.
As an example consider the Rössler system Rössler (1976),
$\begin{split}\dot{x}&=-y\,-\,z\\\ \dot{y}&=x+ay\\\
\dot{z}&=b+z(x-c)\,,\end{split}$ (3)
where $a=b=0.2$ and $c=5.7$. This flow has two prominent invariant states, the
‘inner’ and the ‘outer’ unstable equilibria ${\hat{x}^{\prime}}{}^{(-)}$ and
${\hat{x}^{\prime}}{}^{(+)}$ (see Fig. 2 (a)) , which we choose as templates
for our sections.
We orient the sections so the plane ${\cal P}_{-}$ contains
${\hat{x}^{\prime}}{}^{(-)}$ and its 1-dimensional stable eigenvector (Fig. 2
(b)), and the other section ${\cal P}_{+}$ contains
${\hat{x}^{\prime}}{}^{(+)}$ and its 1-dimensional unstable eigenvector (Fig.
2 (c)), thus capturing the local spiral-in, spiral-out dynamics. The remaining
freedom to rotate each section can be used to orient them in such a way that
the ridge (the intersection of the two sections) lies approximately between
the two templates (Fig. 2 (d)). Choosing sections is a dark art: in the
example at hand the dynamics of interest is captured by the two charts - if
that were not the case, one would have had to interpolate, by inserting a
third chart between them.
(a)
[rgb]0,0,0${\hat{x}^{\prime}}{}^{(-)}$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(+)}$(b)
[rgb]0,0,0${\hat{x}^{\prime}}{}^{(-)}$
(c) [rgb]0,0,0${\hat{x}^{\prime}}{}^{(+)}$(d)
[rgb]0,0,0${\hat{x}^{\prime}}{}^{(-)}$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(+)}$
Figure 2: 2-chart atlas for Rössler flow. (a) The inner equilibrium
${\hat{x}^{\prime}}{}^{(-)}$ is a (spiral-out) saddle-focus with a
2-dimensional unstable manifold and a 1-dimensional stable manifold. The outer
equilibrium ${\hat{x}^{\prime}}{}^{(+)}$ is a (spiral-in) saddle-focus, with a
2-dimensional stable manifold (basin boundary for initial conditions that
either fall into the chaotic attractor, or escape to infinity) and a
1-dimensional unstable manifold. (b) Chart ${\cal P}_{-}$ of the
${\hat{x}^{\prime}}{}^{(-)}$ neighborhood carved out of a Poincaré section
plane through the inner equilibrium ${\hat{x}^{\prime}}{}^{(-)}$ and its
stable eigenvector, with section border drawn as the solid red line. Note the
ridge (dashed blue line): the chart stops at the ridge. (c) Chart ${\cal
P}_{+}$ (here viewed from below) is bounded by section border (solid red line)
of a section through the outer equilibrium ${\hat{x}^{\prime}}{}^{(+)}$ and
its unstable eigenvector. The chart stops at the ridge (dashed blue line), and
it does not intersect the strange attractor. (d) A two-chart atlas of Rössler
flow, with charts ${\cal P}_{-}$ and ${\cal P}_{+}$ oriented and combined so
that the ridge (intersection of the two sections, indicated by the dashed blue
line in the three figures) lies approximately between the templates. Section
hyperplanes beyond this ridge do not belong to the atlas.
For Rössler flow, the border condition (2) yields a quadratic condition in 3
dimensions, so the section borders drawn in Fig. 2 (b) and Fig. 2 (c) are
conic sections. The two charts meet at a ridge, and together do a pretty good
job as the 2-chart atlas of the interesting Rössler dynamics. Due to the
extreme contraction rate of the attractor, its intersection with the section
in Fig. 2 (b) is for all practical purposes 1-dimensional, and the associated
return map yields all periodic orbits of the 3-dimensional flow. Cvitanović
_et al._ (2012)
In 3 dimensions everything —sections, ridges, section borders— can be drawn
and the chart fits on a 2-dimensional sheet of papyrus. But what about for
hydrodynamic flows where the dimensionality $d$ of the state space is very
large? The point of the cartographical enterprize undertaken here is that
while it is impossible to visualize the $(d\\!-\\!2)$-dimensional section
border of the $(d\\!-\\!1)$-dimensional slab that is now our chart, Gibson and
Cvitanović (2012) a point is a point and a line is a line in a projection from
any number of dimensions, so a trajectory crossing of either a section or a
section border can be easily determined and visualized in any dimension.
To summarize: Evolution in time decomposes the state space into a spaghetti of
1-dimensional trajectories $x(\tau)$, each fixed by picking a single point
$x(0)$ on it. A well chosen set of section charts of codimension 1 allows us
to ‘quotient’ the continuous time parameter $\tau$, and reveal the dynamically
important transverse structure of the flow’s stable/unstable manifolds. For
unstable trajectories one needs, in addition, a notion of recurrence to the
section. The set of points $\\{\hat{x}_{n}\\}=\\{x(\tau_{n})\\}$, separated by
short time flights in between sections, captures the transverse dynamics
without losing any information about the chaotic flow. We can thus chart
interesting regions of state space by picking a sufficient number of templates
and using them to construct charts of their neighborhoods, each bounded by
section borders and ridges.
We close this section with a remark on what sections _are not_ : A Poincaré
section is not a projection onto a lower-dimensional space (in sense that a
photograph is a 2-dimensional projection of a 3-dimensional space). Rather, it
is a local change of coordinates to a direction along the flow $v(\hat{x})$,
and the remaining coordinates transverse to it. No information about the flow
is lost; the full space trajectory $x(\tau)$ can always be reconstructed by
integration from its point $\hat{x}$ in the section.
## III Dancers and drifters
(a) [rgb]0,0,0$z$[rgb]0,0,0$\theta$(b) [rgb]0,0,0$z$[rgb]0,0,0$\theta$
(c) [rgb]0,0,0$z$[rgb]0,0,0$\theta$(d) [rgb]0,0,0$z$[rgb]0,0,0$\theta$
Figure 3: A symmetry relates physically equivalent states; a pipe flow
solution translated or rotated is also a solution. (a) An instantaneous state
of the fluid is indicated by a ‘swirl’ - here the reader has to imagine a
particular instantaneous velocity field across the entire pipe. The same state
may be rigidly (b) translated by downstream shift $\ell$ (fluid states are
$\textrm{SO}(2)_{z}$ equivariant in a stream-wise periodic pipe), (c)
translated by $\ell$ and rotated azimuthally by ${\bf\phi}$ (the two states
are $\textrm{SO}(2)_{\theta}\times\textrm{SO}(2)_{z}$ equivariant), and (d)
reflected and rotated azimuthally by ${\bf\phi}$ (the two states are
$\textrm{O}(2)_{\theta}$ equivariant). Some symmetry-related states may also
be connected by time evolution. A _relative equilibrium_ is a solution of the
equations of motion that retains its shape while rotating and traveling
downstream with constant phase velocity $c$. A _relative periodic orbit_
${\cal M}_{p}$ is a _time dependent_ , shape-changing state of the fluid that
after a period ${T_{p}}$ reemerges as (b), (c), or (d), the initial state
translated by $\ell_{p}$, rotated by ${\bf\phi}_{p}$ and possibly also
azimuthally reflected.
What is a symmetry? A visualization of the fluid dynamics of a pipe flow, Fig.
3, affords an intuitive illustration. Solutions of pipe flow remain physically
the same under azimuthal rotations and stream-wise translations (which become
$\textrm{SO}(2)$ rotations in numerical stream-wise periodic pipes), but
rotated and shifted solutions may correspond to distant points in state space.
Each $\textrm{SO}(2)$ group orbit is topologically a circle, but it traces out
a complicated state space curve composed of many Fourier modes that are
nonlinearly coupled and thus of comparable magnitude. Together, the two
$\textrm{SO}(2)$ symmetries of numerical pipe flow sweep out contorted and
hard to visualize $T^{2}$ tori (see ref. Willis, Cvitanović, and Avila
(2012)), so we shall illustrate the key ideas by a much simpler example, the
$\textrm{SO}(2)$-equivariant Gibbon and McGuinness Gibbon and McGuinness
(1982); Fowler, Gibbon, and McGuinness (1982) complex Lorenz equations of
geophysics and laser physics,
$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle-\sigma x_{1}+\sigma
y_{1}\,,\qquad\dot{x}_{2}\,=\,-\sigma x_{2}+\sigma y_{2}$
$\displaystyle\dot{y}_{1}$ $\displaystyle=$
$\displaystyle(\rho_{1}-z)x_{1}-\rho_{2}x_{2}-y_{1}-ey_{2}$
$\displaystyle\dot{y}_{2}$ $\displaystyle=$
$\displaystyle\rho_{2}x_{1}+(\rho_{1}-z)x_{2}+ey_{1}-y_{2}$
$\displaystyle\dot{z}\;$ $\displaystyle=$ $\displaystyle-
bz+x_{1}y_{1}+x_{2}y_{2}\,.$ (4)
Here all coordinates and parameters are real. In our calculations the
parameters are set to those used by Siminos’ Siminos (2009)
$\rho_{1}=28,\,\rho_{2}=0,\,b=8/3,\,\sigma=10,\,e=1/10$. The complex Lorenz
equations are an example of a simple dynamical system with a continuous (but
no discrete) symmetry, equivariant under $\textrm{SO}(2)$ rotations by
$\displaystyle g({\bf\phi})$ $\displaystyle=$
$\displaystyle\exp{({{\bf\phi}}\mathbf{T})}\,=\,\left(\begin{array}[]{ccccc}\cos{\bf\phi}&\sin{\bf\phi}&0&0&0\\\
-\sin{\bf\phi}&\cos{\bf\phi}&0&0&0\\\ 0&0&\cos{\bf\phi}&\sin{\bf\phi}&0\\\
0&0&-\sin{\bf\phi}&\cos{\bf\phi}&0\\\ 0&0&0&0&1\end{array}\right)$ (10)
The group is 1-dimensional and compact, its elements parameterized by
${\bf\phi}\mbox{ mod }2\pi$. For historical background, Poincaré return maps,
symbolic dynamics and in-depth investigation of the model, see refs. Siminos
(2009); Siminos and Cvitanović (2011).
(a) [rgb]0,0,0$z$[rgb]0,0,0$x_{1}$[rgb]0,0,0$x_{2}$(b)
[rgb]0,0,0$z$[rgb]0,0,0$x_{1}$[rgb]0,0,0$x_{2}$
(c) [rgb]0,0,0$z$[rgb]0,0,0$y_{2}$[rgb]0,0,0$x_{2}$(d)
[rgb]0,0,0$z$[rgb]0,0,0$y_{2}$[rgb]0,0,0$x_{2}$
Figure 4: Complex Lorenz flow, $d=5\to 3$ dimensional $\\{x_{1},x_{2},z\\}$
projections: (a) The strange attractor. (b) The initial relative equilibrium
$TW_{1}$ point is shown by the red dot, and its group orbit / trajectory by
the dashed red line. One period of the $\overline{01}$ relative periodic orbit
is shown by the solid blue line. The group orbit of its (arbitrary) starting
point is shown by the dashed blue line: after one period the trajectory has
returned to the group orbit but with a different phase. The wurst, i.e., the
group orbit of the $\overline{01}$ trajectory (dark blue) is shown by the cyan
surface. Following $\overline{01}$ for 15 more periods (faint dotted lines)
starts filling out this torus; in that time, the slowly drifting relative
equilibrium $TW_{1}$ has advanced to the next red dot (red line). Symmetry-
reduced complex Lorenz flow, $d=4\to 3$ dimensional $\\{x_{2},y_{2},z\\}$
projections: (c) Strange attractor from frame (a) reduced to a single slice
hyperplane, using $TW_{1}$ as the template. $\overline{01}$ is now a periodic
orbit, shown by the solid black line. The dynamics exhibits singular jumps
(shown in red) due to forbidden crossings of the chart border. In contrast to
the 1-dimensional section borders of Fig. 2, here the chart borders are
3-dimensional and hard to visualize. (d) The 2-chart atlas (see the sketch of
Fig. 11) of the same strange attractor encounters no chart borders and
exhibits no singularities. The trajectory changes colors from red to blue as
it crosses between the slice hyperplanes of ${\hat{x}^{\prime}}{}^{(1)}$ and
${\hat{x}^{\prime}}{}^{(2)}$. The ridge (shown in brown) acts as a Poincaré
section ${\cal P}$ with red or blue ridge points $\hat{x}^{*}$ marking the
direction of the crossing. The charts are 4-dimensional, the ridge
3-dimensional, so the colored blocks and planes are only cartoon drawings of
their projections onto the 2-dimensional figure.
The strange attractor of the complex Lorenz flow , in its present state, is a
complete mess (Fig. 4 (a)). Solutions tend to drift along continuous symmetry
directions, with the physically important shape-changing dynamics hidden from
view.
The ultimate drifter, the signature invariant solution that signals the
presence of a continuous symmetry is a relative equilibrium (traveling wave,
rotational wave, etc.), a trajectory whose velocity field lies within the
group tangent space, $v(x)=c\cdot t(x)\,,$ and whose time evolution is thus
confined to the group orbit (see Fig. 4 (b)); think of an unchanging body
carried by a stream.
A relative periodic orbit behaves more like a dancer. ${\cal M}_{p}$ is a
trajectory that recurs exactly
$x(\tau)=g_{p}\,x(\tau+{T_{p}})\,,\qquad x(\tau)\in{\cal M}_{p}\,,$ (11)
after a fixed relative period ${T_{p}}$, but shifted by a fixed group action
${g_{p}}$ that maps the endpoint $x({T_{p}})$ back into the initial point
cycle point $x(0)$; think of a dancer moving across the stage through a set of
motions and then striking her initial pose, Shapere and Wilczek (2006) or
study the pipe flow sketches in Fig. 3.
Because the $\textrm{SO}(2)$ transformations act on the complex Lorenz flow
only through the simplest $m=1$ Fourier mode, here all group orbits are
circles and appear elliptical in $d=5\to 3$ dimensions projections.
Nevertheless, even the wurst traced out by one of the simplest relative
periodic orbits $\overline{01}$ Siminos and Cvitanović (2011) (shown in Fig. 4
(b)) is not so easy to get one’s head around: you are looking at a
3-dimensional projection of a _torus_ embedded in 5 dimensions.
(a) [rgb]0,0,0
${\cal M}_{x(\tau)}$
[rgb]0,0,0
${\cal M}_{x(0)}$
[rgb]0,0,0
$x(0)$
[rgb]0,0,0
$x(\tau)$
[rgb]0,0,0
${\cal M}$
(b) [rgb]0,0,0
$x(\tau)$
[rgb]0,0,0
$\hat{x}(0)$
[rgb]0,0,0
$\hat{x}(\tau)$
[rgb]0,0,0
$g(\tau)$
[rgb]0,0,0
$x(0)$
[rgb]0,0,0
$g(0)$
[rgb]0,0,0
${\cal M}$
(c) [rgb]0,0,0
$\hat{\cal M}$
[rgb]0,0,0
$\hat{x}(0)$
[rgb]0,0,0
$\hat{x}(\tau)$
(d) [rgb]0,0,0${\cal
M}_{{\hat{x}^{\prime}}}$[rgb]0,0,0$t^{\prime}$[rgb]0,0,0$\hat{x}$[rgb]0,0,0${\hat{x}^{\prime}}$
Figure 5: (a) The $N$-dimensional group orbit ${\cal M}_{x(0)}$ of state
space point $x(0)$ and the group orbit ${\cal M}_{x(\tau)}$ reached by the
trajectory $x(\tau)$ a time $\tau$ later. (b) Two physically equivalent
trajectories $x(\tau)$ and $\hat{x}(\tau)$ are related, in general, by an
arbitrary, time dependent moving frame transformation $g(\tau)$, such that
$x(\tau)=g(\tau)\,\hat{x}(\tau)$. (c) A symmetry reduction scheme ${\cal
M}\to\hat{\cal M}$ is a rule that prescribes $g(\tau)$ and thus replaces a
group orbit ${\cal M}_{x}\subset{\cal M}$ through $x$ by a single point
$\hat{x}\in\hat{\cal M}$. (d) In this paper, $g(\tau)$ is fixed variationally
by the extremal condition (13) for the point $\hat{x}$ on the group orbit
${\cal M}_{x}$ that is nearest to the template ${\hat{x}^{\prime}}$.
To summarize: continuous symmetries in the dynamics foliate the state space
into an onion, where each layer is a group orbit (Fig. 5 (a)). How are we to
sort out this mess? All the points on a group orbit are physically equivalent,
so we are free to replace a given flow $x(\tau)$ by any other
$\hat{x}$($\tau$), such that $x(\tau)=g(\tau)\,\hat{x}(\tau)$ by a moving
frame Fels and Olver (1998, 1999); Olver (1999) transformation $g(\tau)$. As
long as no symmetry reduction procedure is prescribed, $g(\tau)$ is free: it
can be any, in general time dependent, group transformation. For example, to
film our dancer, we can mount the camera on a cart moving alongside her. So,
in the presence of continuous symmetries, there are two kinds of motion: those
of a dancer, continuously changing shapes, and those of a drifter, merely
shuffling along the shape invariant directions. We will presently banish the
drifters and just enjoy the dance.
## IV Chart
Suppose you are computing a set of numerically exact invariant solutions of
the Navier-Stokes equations. Do you want to compute the same solution over and
over again, once for every point on the group orbit? No, you would like to
compute it only once. The strategy for picking out that one representative
solution is called _symmetry reduction_. Its goal is to replace each group
orbit by a unique point in a lower-dimensional symmetry-reduced state space
$\hat{\cal M}\subset{\cal M}/G$, as sketched in Fig. 5 (c).
What is a smart way to go about it? Intuition gained from pipe flow (see Fig.
3) will again prove helpful. A turbulent flow exhibits a myriad of unstable
structures, each traveling down the pipe with its own phase velocity. The
method of slices Rowley and Marsden (2000); Beyn and Thümmler (2004); Siminos
and Cvitanović (2011); Froehlich and Cvitanović (2011) that we now describe
tells you how to pull each solution back into a fixed frame called a _slice_
and compare it to your repertoire of precomputed solutions, or the templates
$\\{{\hat{x}^{\prime}}{}^{(j)}\\}$, using the poor geometer’s version of a
geodesic, the principle of the _closest distance_ to each. What follows is
similar to the construction of sections of sect. II; due to the linear action
of the symmetry group, slicing is easier than sectioning, but wholly
unfamiliar. This is why we reviewed the Poincaré sections first. We now offer
a pictorial tour of this (save for one bold incursion Willis, Cvitanović, and
Avila (2012)) hitherto uncharted territory.
First, pick a template ${\hat{x}^{\prime}}$ and use the freedom to shift and
rotate it (Fig. 5 (b)) until it overlies, as well as possible, the state $x$,
by minimizing the distance
$\|{x-g({\bf\phi})\,{\hat{x}^{\prime}}}\|\,.$ (12)
Now, replace the entire group orbit of $x$ by the closest match to the
template pattern, given by $\hat{x}=g^{-1}x$. From here on, we will use the
hat on $\hat{x}$ to indicate the unique point on the group orbit of $x$ that
is closest to the template $\hat{x}^{\prime}$. The symmetry-reduced state
space $\hat{\cal M}$ is comprised of such closest matches, a point for each
full state space group orbit.
The minimal distance satisfies the extremum condition (Fig. 5 (d))
$\frac{\partial}{\partial{\bf\phi}}\|{x-g({\bf\phi})\,{\hat{x}^{\prime}}}\|^{2}=2\,\langle{\hat{x}-{\hat{x}^{\prime}}}\vphantom{t^{\prime}}|\vphantom{\hat{x}-{\hat{x}^{\prime}}}{t^{\prime}}\rangle=0\,,\quad
t^{\prime}=\mathbf{T}{\hat{x}^{\prime}}\,,$
where the $[d\\!\times\\!d]$ matrix $\mathbf{T}$ is the generator of
infinitesimal symmetry transformations.
$\|{g({\bf\phi}){\hat{x}^{\prime}}}\|=\|{{\hat{x}^{\prime}}}\|$ is a constant.
To streamline the exposition, we shall assume here that the symmetry group is
$\textrm{SO}(n)$. In that case $\mathbf{T}$ is antisymmetric, so the group
tangent vector $t^{\prime}$ evaluated at ${\hat{x}^{\prime}}$ is normal to
${\hat{x}^{\prime}}$ and the term
$\langle{{\hat{x}^{\prime}}}\vphantom{\mathbf{T}\,{\hat{x}^{\prime}}}|\vphantom{{\hat{x}^{\prime}}}{\mathbf{T}\,{\hat{x}^{\prime}}}\rangle$
vanishes. Therefore $\hat{x}$, the point on the group orbit of $x$ that lands
in the slice satisfies the _slice condition_
$\langle{\hat{x}}\vphantom{t^{\prime}}|\vphantom{\hat{x}}{t^{\prime}}\rangle=0\,.$
(13)
As $x(\tau)$ varies in time, the template ${\hat{x}^{\prime}}$ tracks the
motion using the slice condition (13) to minimize
$\|{x(\tau)-g(\phi(\tau)){\hat{x}^{\prime}}}\|$, and the full-space trajectory
$x(\tau)$ is rotated into the reduced state space trajectory $\hat{x}(\tau)$
by appropriate time varying _moving frame_ angles $\\{{\bf\phi}(\tau)\\}$, as
depicted in Fig. 6 (a). $\hat{\cal M}$ is thus a $(d\\!-\\!N)$-dimensional
hyperplane normal to the $N$ group tangents evaluated at the
$\hat{x}^{\prime}$ as sketched in Fig. 6 in a highly idealized manner: A group
orbit is an $N$-dimensional manifold and, even for $\textrm{SO}(2)$, is
usually only topologically a circle and can intersect a hyperplane any number
of times (see Figs. 7 and 8).
(a) [rgb]0,0,0
$\hat{\cal M}$
[rgb]0,0,0
$g\,{\hat{x}^{\prime}}$
[rgb]0,0,0
${\hat{x}^{\prime}}$
[rgb]0,0,0
$t^{\prime}$
[rgb]0,0,0
$x(\tau)$
[rgb]0,0,0
$\hat{x}(\tau)$
[rgb]0,0,0
$g\,x(\tau)$
(b) [rgb]0,0,0
$\hat{\cal M}$
[rgb]0,0,0
$x(0)$
[rgb]0,0,0
$\hat{x}(\tau)$
[rgb]0,0,0
$x(\tau)$
[rgb]0,0,0
$x({T_{p}})$
[rgb]0,0,0
$\hat{x}(0)$
Figure 6: The method of slices, a state space visualization: (a) A chart
$\hat{\cal M}\subset{\cal M}/G$ lies in the $(d\\!-\\!N)$-dimensional slice
hyperplane (13) normal to $t^{\prime}_{1}...t^{\prime}_{N}$, which span the
$N$-dimensional space tangent to the group orbit $g\,{\hat{x}^{\prime}}$
(dotted line) evaluated at the template point ${\hat{x}^{\prime}}$. The
hyperplane intersects all full state space group orbits (green dashes). The
full state space trajectory $x(\tau)$ (blue) and the reduced state space
trajectory $\hat{x}(\tau)$ (green) are equivalent up to a ‘moving frame’
rotation $x(\tau)=g(\tau)\,\hat{x}(\tau)$, where $g(\tau)$ is a shorthand for
$g({\bf\phi}(\tau))$. (b) In the full state space, a relative periodic orbit
$x(0)\to x(\tau)\to x({T_{p}})$ returns to the group orbit of $x(0)$ after a
time ${T_{p}}$, such that $x(0)=g_{p}x({T_{p}})$. A generic relative periodic
orbit quasi-periodically fills out what is topologically a torus (Fig. 4 (b)).
In the slice, the symmetry-reduced trajectory is periodic,
$\hat{x}(0)=\hat{x}({T_{p}})$.
One can write the equations for the flow in the reduced state space
$\dot{\hat{x}}=\hat{v}(\hat{x})$ (for details see, for example, ref.
Cvitanović _et al._ (2012)) as
$\displaystyle\hat{v}(\hat{x})$ $\displaystyle=$ $\displaystyle
v(\hat{x})\,-\,\dot{{\bf\phi}}(\hat{x})\,t(\hat{x})$ (14)
$\displaystyle\dot{{\bf\phi}}(\hat{x})$ $\displaystyle=$
$\displaystyle\langle{v(\hat{x})}\vphantom{t^{\prime}}|\vphantom{v(\hat{x})}{t^{\prime}}\rangle/\langle{t(\hat{x})}\vphantom{t^{\prime}}|\vphantom{t(\hat{x})}{t^{\prime}}\rangle\,,$
(15)
which confines the motion to the slice hyperplane. Thus, the dynamical system
$\\{{\cal M},f^{t}\\}$ with continuous symmetry $G$ is replaced by the reduced
state space dynamics $\\{\hat{\cal M},\hat{f}^{t}\\}$: The velocity in the
full state space $v$ is the sum of $\hat{v}$, the velocity component in the
slice hyperplane, and $\dot{{\bf\phi}}\,t$, the velocity component along the
group tangent space. The integral of the reconstruction equation for
$\dot{{\bf\phi}}$ keeps track of the group shift in the full state space.
(a) [rgb]0,0,0$\hat{\cal
M}$[rgb]0,0,0$g{\hat{x}^{\prime}}$[rgb]0,0,0${\hat{x}^{\prime}}$[rgb]0,0,0$t^{\prime}$
(b) [rgb]0,0,0$\hat{\cal
M}$[rgb]0,0,0${\hat{x}^{\prime}}$[rgb]0,0,0$t^{\prime}$[rgb]0,0,0$g{\hat{x}^{\prime}}$
Figure 7: The chart border is the $(d\\!-\\!N\\!-1)$-dimensional hyperplane
that contains all the points $\hat{x}^{\ast}$ whose group tangents
$t(\hat{x}^{\ast})$ lie in the slice hyperplane or vanish and are thus normal
to $t^{\prime}$. Beyond this boundary, the group orbits pierce the slice
hyperplane in the wrong direction, so _only_ the half-hyperplane that contains
the template belongs to the slice. The chart border is not easy to visualize;
For the lack of dimensions, here it is drawn as a ‘line’, the $z$ axis in this
3-dimensional sketch. (a) If the equivariant coordinates transform only under
the $m=1$ representation of $\textrm{SO}(2)$, every group orbit is a circle,
and crosses any slice hyperplane exactly twice. However, if there are
coordinates that transform as higher $m$, the group orbit can pierce the
hyperplane up to $2m$ times, and the chart border lies closer to the template:
For example, (b) a group orbit for a combination of $m=1$ and $m=2$
equivariant coordinates resembles the seam of a baseball, and can cross the
_slice hyperplane_ 4 times, out of which only the point closest to the
template is in the _slice_.
The template ${\hat{x}^{\prime}}$ should be a generic state space point in the
sense that its group orbit has the full $N$ dimensions of the group $G$. The
set of the group orbit points closest to the template $\hat{x}^{\prime}$ forms
a neighborhood of $\hat{x}^{\prime}$ in which each group orbit intersects the
hyperplane _only once_. A slice hyperplane qualitatively captures neighboring
group orbits until, for a point $\hat{x}^{\ast}$ not so close to the template,
the group tangent vector $t(\hat{x}^{\ast})$ lies in the slice hyperplane. The
group orbits for such points are grazed tangentially rather than sliced
transversally, much like what happens at the section border (2) for evolution
in time. This is also a linear condition and defines the chart border ${\cal
S}$, Siminos and Cvitanović (2011); Froehlich and Cvitanović (2011) a
$(d\\!-\\!N\\!-1)$-dimensional manifold, which contains all the points
$\hat{x}^{\ast}$ whose group tangents lie in the slice hyperplane, i.e.,
$\langle{\hat{x}^{\ast}}\vphantom{t^{\prime}}|\vphantom{\hat{x}^{\ast}}{t^{\prime}}\rangle\,=\,0\mbox{
and
}\langle{t(\hat{x}^{\ast})}\vphantom{t^{\prime}}|\vphantom{t(\hat{x}^{\ast})}{t^{\prime}}\rangle\,=\,0\,.$
(16)
${\cal S}$ also contains all points for which $t(\hat{x}^{\ast})=0$. While for
the Poincaré sections (2) the analogous points were equilibria (captured only
if the section cut through them), for slice hyperplanes points with vanishing
group actions belong to invariant subspaces, and, by its definition, every
chart border automatically includes _all_ invariant subspaces.
[rgb]0,0,0$\hat{p}$[rgb]0,0,0$\hat{p}$[rgb]0,0,0$\hat{x}(0)$[rgb]0,0,0$x(0)$[rgb]0,0,0$x(T_{p})$
Figure 8: Wurst, sliced. Every slice hyperplane cuts every group orbit at
least twice (see Fig. 6), once at the orbit’s closest passage to the template,
and another time at the most distant passage, also satisfying the slice
condition (13). An $\textrm{SO}(2)$ relative periodic orbit ${\cal M}_{p}$ is
topologically a torus, so the two cuts are the two periodic orbit images of
the same relative periodic orbit, the good close one $\hat{x}_{p}$ (blue), and
the bad distant one (red), on the other side of chart border, and thus not in
the slice.
For the complex Lorenz equations (4), the invariant subspace is the
1-dimensional $z$-axis, with trivial dynamics, $z=-bz$, but in general
invariant subspaces are high-dimensional and have their own dynamics.
Physicists, for example general relativists, often work in invariant
subspaces, as this is easier than solving the full problem. Cvitanović,
Davidchack, and Siminos (2010) Such approaches yield highly symmetric
solutions, Stephani _et al._ (2009); Gibson, Halcrow, and Cvitanović (2009)
whose dynamics may be quite different from those that guide turbulence in the
full state space (for a striking example, see ref. Willis, Cvitanović, and
Avila (2012)).
There is yet another, much kinder type of a border: a _ridge_. Our initial
chart $\hat{\cal M}{}^{(1)}$ is a ($d\\!-\\!N$)-dimensional hyperplane. If we
pick another template point ${\hat{x}^{\prime}}{}^{(2)}$, it comes along with
its own slice hyperplane $\hat{\cal M}{}^{(2)}$. Any pair of
$(d\\!-\\!N)$-dimensional local slice hyperplanes intersects in a ridge, a
$(d\\!-\\!N-\\!1)$-dimensional hyperplane ${\cal P}$ of points $\hat{x}^{*}$
shared by a pair of charts and thus satisfying the slice condition (13) for
both,
$\langle{\hat{x}^{*}}\vphantom{t^{\prime}{}^{(1)}}|\vphantom{\hat{x}^{*}}{t^{\prime}{}^{(1)}}\rangle=0\mbox{
and
}\langle{\hat{x}^{*}}\vphantom{t^{\prime}{}^{(2)}}|\vphantom{\hat{x}^{*}}{t^{\prime}{}^{(2)}}\rangle=0\,.$
(17)
The ridge forms a Poincaré section ${\cal P}{}^{(ij)}$ that serves as a toll
bridge, crossed by any direct transit from a chart $\hat{\cal M}{}^{(j)}$ to
the adjacent chart $\hat{\cal M}{}^{(i)}$. In Fig. 10 (a) a ridge is
visualized as a ‘line’, and in Fig. 11 as a ‘plane’ of intersection of two
volumes. We shall refer to the neighborhood of a template
${\hat{x}^{\prime}}{}^{(j)}$ bounded by its chart border and the ridges to
other such linear neighborhoods as a _chart_ $\hat{\cal M}{}^{(j)}\subset{\cal
M}/G$, and to (16) and (17) as the border conditions.
## V Charting the slice
Let us summarize the voyage so far: we are charting a curved manifold, and it
would be nice to use tools of differential geometry, but this seems not
possible in the high-dimensional state space of hydrodynamics turbulence. The
only feasible way to chart this space is to (1) quotient all continuous
symmetries, and (2) tile the reduced state space with flat
$(d\\!-\\!N)$-dimensional tiles, or charts. We do this step by step, starting
with a set of templates and using them to construct charts of each
neighborhood, and then building up an atlas of the _slice_ , chart by chart,
which captures all of the reduced dynamics of interest (but not all possible
dynamics). Here are the steps along the way:
Template
Pick a template ${\hat{x}^{\prime}}$ such that $G$ acts on it regularly with a
group orbit of dimension $N$.
Slice hyperplane
The $(d\\!-\\!N)$-dimensional hyperplane satisfying
$\langle{\hat{x}}\vphantom{t^{\prime}_{a}}|\vphantom{\hat{x}}{t^{\prime}_{a}}\rangle=0\,,$
normal to group transformation directions at the template
${\hat{x}^{\prime}}$.
Moving frame
For any $x$, the slice condition
$\langle{\hat{x}}\vphantom{t^{\prime}}|\vphantom{\hat{x}}{t^{\prime}}\rangle=0$
on $x=g({\bf\phi})\hat{x}$ determines the moving frame, i.e., the group action
$g({\bf\phi})$ that brings $x$ into the slice hyperplane.
Chart border
The set of points $\hat{x}^{\ast}$ on a slice hyperplane whose group orbits
graze the hyperplane tangentially, such that
$\langle{\hat{x}^{\ast}}\vphantom{t^{\prime}}|\vphantom{\hat{x}^{\ast}}{t^{\prime}}\rangle\,=\,\langle{t(\hat{x}^{\ast})}\vphantom{t^{\prime}}|\vphantom{t(\hat{x}^{\ast})}{t^{\prime}}\rangle\,=\,0\,.$
Flow invariant subspace
If a subset or all of the group tangents of a chart border point
$\hat{x}^{\ast}$ vanish, $t_{a}(\hat{x}^{\ast})=0$, its time trajectory
remains within a flow-invariant subspace for all times.
Ridge
A hyperplane of points $\hat{x}^{*}\in{\cal P}{}^{(21)}$ formed by the
intersection of a pair of slice hyperplanes $\hat{\cal M}{}^{(1)}$ and
$\hat{\cal M}{}^{(2)}$.
Chart
The neighborhood of a template ${\hat{x}^{\prime}}{}^{(j)}$, bounded by the
chart border and the ridges to other linear neighborhoods, comprises a _chart_
$\hat{\cal M}{}^{(j)}\subset{\cal M}/G$. The borders ensure that there is no
more than one oriented group orbit traversal per chart; a group orbit either
pierces one chart, or no charts at all.
Atlas
A set of $(d\\!-\\!N)$-dimensional contiguous charts $\hat{\cal
M}{}^{(1)},\hat{\cal M}{}^{(2)},\cdots$
Slice
Let $G$ act on a $d$-dimensional manifold ${\cal M}$, with group orbits of
dimension $N$ or less. A _slice_ is a $(d\\!-\\!N)$-dimensional submanifold
$\hat{\cal M}$ such that all group orbits that intersect $\hat{\cal M}$ do so
transversally and only once.
(a)
[rgb]0,0,0${\hat{x}^{\prime}}{}^{(1)}$[rgb]0,0,0$t^{\prime}{}^{(1)}$[rgb]0,0,0$x^{\prime}{}^{(2)}$
(b)
[rgb]0,0,0$t^{\prime}{}^{(1)}$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(1)}$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(2)}$[rgb]0,0,0$t^{\prime}{}^{(2)}$[rgb]0,0,0$x^{\prime}{}^{(2)}$[rgb]0,0,0$\hat{\cal
M}{}^{(1)}$
Figure 9: A 2-chart atlas. Sketch (a) depicts two templates
${\hat{x}^{\prime}}{}^{(1)}$, $x^{\prime}{}^{(2)}$, each with its group orbit.
Start with the template ${\hat{x}^{\prime}}{}^{(1)}$. All group orbits
traverse its $(d\\!-\\!1)$-dimensional slice hyperplane, including the group
orbit of the second template $x^{\prime}{}^{(2)}$. (b) Replace the second
template by its closest group orbit point ${\hat{x}^{\prime}}{}^{(2)}$, i.e.,
the point in chart $\hat{\cal M}{}^{(1)}$. This is allowed as long as
${\hat{x}^{\prime}}{}^{(2)}$ is closer than the $\hat{\cal M}{}^{(1)}$ chart
border (red region), otherwise an interpolating, closer template needs to be
introduced.
(a)
[rgb]0,0,0${\hat{x}^{\prime}}{}^{(1)}$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(2)}$[rgb]0,0,0$\hat{x}(0)$[rgb]0,0,0$\hat{x}(\tau)$
(b) [rgb]0,0,0$\hat{\cal M}{}^{(1)}$[rgb]0,0,0$\hat{\cal
M}{}^{(2)}$[rgb]0,0,0$\hat{x}(0)$[rgb]0,0,0$\hat{x}(\tau)$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(2)}$[rgb]0,0,0${\hat{x}^{\prime}}{}^{(1)}$
Figure 10: A 2-chart atlas. (a) Now that the group orbits have been reduced
to points, erase them and consider the two slice hyperplanes through the two
templates. As these two templates are the closest points viewed from either
group orbit, they lie in both slice hyperplanes. However, the two tangent
vectors $t^{\prime}{}^{(1)}$ and $t^{\prime}{}^{(2)}$ have different
orientations, so they define two distinct charts $\hat{\cal M}{}^{(1)}$ and
$\hat{\cal M}{}^{(2)}$ which intersect in the _ridge_ , a hyperplane of
dimension $(d\\!-\\!2)$ (here drawn as a ‘line’, and in Fig. 11 as
intersection of two ‘volumes’) shared by the template pair that satisfies both
slice conditions (17). The chart for the neighborhood of each template (a page
of the atlas in part (b)) extends only as far as this ridge. If the templates
are sufficiently close, the chart border of each slice hyperplane (red region)
is beyond this ridge, and not encountered by the symmetry-reduced trajectory
$\hat{x}(\tau)$. The reduced trajectory is continuous in the slice comprised
of such charts - it switches the chart whenever it crosses a ridge. (b) The
slice (unique point for each group orbit) is now covered by an atlas
consisting of $(d\\!-\\!1)$-dimensional charts $\hat{\cal M}{}^{(1)},\hat{\cal
M}{}^{(2)},\cdots$.
In the literature, Mostow (1957); Palais (1961); Guillemin and Sternberg
(1990) ‘slice’ refers to any co-dimension $N$ manifold that slices
transversally a group orbit. Here, we define an atlas over a slice
constructively but more narrowly, as a contiguous set of flat charts, with
every group orbit accounted for by the atlas sliced only once, and belonging
to a single chart. A slice is not global, it slices only the group orbits in
an open neighborhood of the state space region of interest.
The physical task, for a given dynamical flow, is to pick a set of
qualitatively distinct templates (for a turbulent pipe flow there might be one
typical of 2-roll states, one for 4-roll states, and so on), which together
provide a good atlas for the region of ${\cal M}/G$ explored by chaotic
trajectories.
The rest is geometry of hyperplanes and has nothing to do with dynamics. Group
orbits ${\cal M}_{x^{(j)}}$ through $x^{(j)}$, group tangents
$t({\hat{x}^{\prime}}{}^{(j)})$, and the associated charts $\hat{\cal
M}{}^{(j)}$ are purely group-theoretic concepts. The slice, chart border and
ridge conditions (13), (16) and (17) are all linear conditions which depend on
the ray defined by the template $\hat{x}^{\prime}$, not its magnitude.
Checking whether the chart border is on the far side of the ridge between two
slice hyperplanes is a linear computation; for a symmetry-reduced trajectory
moving in $\hat{\cal M}{}^{(1)}$ chart one only has to keep checking the sign
of
$\langle{\hat{x}(\tau)}\vphantom{t^{\prime}{}^{(2)}}|\vphantom{\hat{x}(\tau)}{t^{\prime}{}^{(2)}}\rangle\,.$
(18)
Once the sign changes, the ridge has been crossed, and from then on the
trajectory should be reduced to the $\hat{\cal M}{}^{(2)}$ chart. For three or
more charts you will have to align the ridge of the current chart with a
previously-used chart. You’ll cross that ridge when you come to it (a hint:
the manifold is curved, so there will be a finite jump in phase).
How the charts are put together is best told as a graphic tale, in the 5
frames of Figs. 9, 10 and 11, and then illustrated by contrasting the mess of
the complex Lorenz equations strange attractor Fig. 4 (a) to the elegance of
its 2-chart atlas, Fig. 4 (d).
It is worthwhile to note that the only object that enters the slice
hyperplane, border and ridge conditions is the ray defined by the unit vector
$\hat{t}{}^{{}^{\prime}}=t^{\prime}/\|{t^{\prime}}\|$. This gives much freedom
in picking templates. In particular, the two rays
$\displaystyle\hat{t}{}^{{}^{\prime}(1)}$ $\displaystyle=$
$\displaystyle(0.263,-0.692,0.624,-0.251,0)$
$\displaystyle\hat{t}{}^{{}^{\prime}(2)}$ $\displaystyle=$
$\displaystyle(0.153,-0.610,0.747,-0.213,0)$ (19)
used to construct the complex Lorenz equations 2-chart atlas of Fig. 4 (d)
were found by numerical experimentation.
With the atlas in hand, the dynamics is fully charted: as explained in refs.
Cvitanović _et al._ (2012); Siminos and Cvitanović (2011), Poincaré return
maps then yield all admissible relative periodic orbits.
Three concluding remarks on what slices _are not_ :
(1) Symmetry reduction is not a dimensional-reduction scheme, a projection
onto fewer coordinates, or flow modeling by fewer degrees of freedom: It is a
local change of coordinates with one (or $N$) coordinate(s) pointing along the
continuous symmetry directions. No information is lost by symmetry
‘reduction’, one can go freely between solutions in the full and reduced state
spaces by integrating the associated reconstruction equations (15).
(2) If the flow is also invariant under discrete symmetries, these should be
reduced by methods described, for example, in ChaosBook.org.
(3) An atlas is _not needed_ for Newton determination of a single invariant
solution, or a study of its bifurcations. Golubitsky, Stewart, and Schaeffer
(1988) Any local section and slice plus time and shift constraints does the
job. Viswanath (2007); Duguet, Pringle, and Kerswell (2008); Mellibovsky and
Eckhardt (2011) It is possible to compute 60,000 relative periodic orbits this
way. Cvitanović, Davidchack, and Siminos (2010) Once we have more than one
invariant solution, the question is: how is this totality of solutions
interrelated? For that, a good atlas is a necessity.
[rgb]0,0,0$\hat{x}(0)$[rgb]0,0,0$\hat{x}(\tau)$[rgb]0,0,0
$\hat{\cal M}{}^{(2)}$
[rgb]0,0,0
$\hat{\cal M}{}^{(1)}$
[rgb]0,0,0$\hat{x}_{2}$[rgb]0,0,0$\hat{x}_{1}$ Figure 11: Here the two charts
of Fig. 10 (a) are drawn as two $(d\\!-\\!1)$-dimensional slabs. The ridge,
their $(d\\!-\\!2)$-dimensional intersection, can then be drawn as the shaded
plane. This hyperplane cuts across the symmetry-reduced trajectory
$\hat{x}(\tau)$ and thus serves as a Poincaré section ${\cal P}{}^{(21)}$ that
captures all transits from the neighborhood of template
${\hat{x}^{\prime}}{}^{(1)}$ to the neighborhood of template
${\hat{x}^{\prime}}{}^{(2)}$. Poincaré section transits are oriented, so
$\hat{x}_{1}$ and $\hat{x}_{2}$ are in the section, but the third point is
not.
## VI Bridges to nowhere
Everybody encounters a symmetry sooner or later, so the literature on symmetry
reduction is vast (for a historical overview, see remarks in ChaosBook.org and
ref. Siminos and Cvitanović (2011)). Before asking, “Why the method of slices
and not […]?” a brief tour of the more familiar symmetry reduction schemes is
called for. They all have one thing in common: they will not work for high-
dimensional nonlinear systems.
To start with, mastery of quantum-mechanics or bifurcation theory Ruelle
(1973); Golubitsky, Stewart, and Schaeffer (1988) symmetry reduction to linear
irreducible representations is only partially illuminating; linear theory
works quite well for linear unitary operators or close to a bifurcation, but,
as we tried to show in this pictorial tour, the way symmetries act on
nonlinear systems is much subtler. For flows with strongly nonlinearly coupled
modes, both time trajectories and group orbits are complicated, so choices of
sections and slices require insight into the geometry of the particular flow,
there exists no general theory of linear transformations into symmetry
irreducible coordinates that would do the job.
There are purely group-theoretical approaches, with no dynamics to inform
them, inspired by the observation that while coordinates $x_{i}$ are
equivariant, the squared length $r^{2}=\sum x_{i}^{2}$ is _invariant_ under
$\textrm{O}(n)$ transformations.
For $\textrm{SO}(2)$, an obvious idea is to go to polar coordinates. The
simplest nonlinear examples Armbruster, Guckenheimer, and Holmes (1988)
already run into $r_{j}\to 0$ type of singularities, and it is not altogether
clear how one would rewrite the Navier-Stokes equations in such a format, or
integrate them numerically. A more sophisticated approach is to rewrite the
dynamics in terms of invariant polynomial bases, described lucidly in ref.
Gilmore and Letellier (2007), with the equivariant state space coordinates
$(x_{1},x_{2},x_{3},...,x_{d})$ replaced by an invariant polynomial basis
$(u_{1},u_{2},u_{3},...,u_{m})$. As the dimension of the problem increases,
the number of these polynomials grows quickly, as does the number of syzygies,
the nonlinear relations amongst them. There is no guiding principle for
picking a set of such polynomials, and no practical way to implement the
scheme Gatermann (2000) for high-dimensional flows: how and why would one
replace the large number of equivariant state space coordinates of
hydrodynamic turbulence with a vast number of invariant polynomials?
Others approaches are informed by dynamics, foremost among them being the
method of co-moving frames. Visualizing a single ‘relative’ trajectory in its
co-moving frame, i.e., moving with that solution’s mean phase velocity, is
useful if one is concerned with that individual solution and the tiny relative
periodic orbits (modulated-amplitude waves) that bifurcate off it. Duguet,
Pringle, and Kerswell (2008); Mellibovsky and Eckhardt (2011) A co-moving
frame is useless, however, if we are concerned with studying collections of
these trajectories, as each solution travels with its own mean phase velocity
$c_{p}={\bf\phi}_{p}/{T_{p}}$, and there is no single co-moving frame that can
simultaneously reduce _all_ traveling solutions. The slice that we construct
here is not ‘co-moving’, but _emphatically_ stationary.
There exists a beautiful theory of symplectic symmetry reduction for the
mechanics of three-dimensional rigid bodies, Marsden and Weinstein (1974);
Abraham and Marsden (1978) or using Lie symmetry reduction to derive Eulerian
velocity fields from Lagrangian trajectories. Morrison and Greene (1980) These
approaches do not appear to be applicable to problems considered here, and
anyway, the goal is different. Rather than to reduce a particular set of
equations, we seek to formulate a computationally straightforward and general
method of reducing any continuous symmetry, for _any_ high-dimensional
chaotic/turbulent flow. One should also note that ‘symmetry reduction’ in
general relativity Stephani _et al._ (2009) and Lie theory often implies
restricting one’s solution space to a subspace of higher symmetry; here we
always work in the full state space.
(a)
[rgb]0,0,0$v$[rgb]0,0,0$v_{\bot}$[rgb]0,0,0$t_{1}$[rgb]0,0,0$t_{2}$[rgb]0,0,0${\cal
M}_{x}$(b) [rgb]0,0,0
$x(\tau)$
[rgb]0,0,0
$x(0)$
[rgb]0,0,0
$\hat{x}(\tau)$
[rgb]0,0,0
$g(\tau)$
[rgb]0,0,0
$v_{\bot}$
Figure 12: (a) By equivariance $v(x)$ can be replaced by $v_{\bot}(x)$, the
velocity normal to the group tangent directions at state space point $x$. (b)
The method of connections replaces $v(\hat{x})$ at every instant
$\hat{x}=\hat{x}(\tau)$ by $v_{\bot}(\hat{x})$, so in $\hat{x}(\tau)$’s
covariant frame there is no motion along the group tangent directions.
There is, however, one intriguing, compelling and physically informed
contender. In mechanics and field theory it is natural to separate the flow
locally into group dynamics and a transverse, ‘horizontal’ flow, Smale (1970);
Abraham and Marsden (1978) by the ‘method of connections’, Rowley _et al._
(2003) illustrated in Fig. 12. The method of connections, however, does not
reduce the dynamics to a lower-dimensional reduced state space ${\cal M}/G$.
In contrast to the method of co-moving frames, where one defines a mean phase
velocity of a relative periodic orbit, the method of connections is inherently
local. The two methods coincide for relative equilibria.
The meaning of the ‘method of connections’ in classical dynamics is clearest
in the work of Shapere and Wilczek: Shapere and Wilczek (1989, 2006) one can
observe a swimmer (or our dancer) from a fixed slice frame, or bring her back
to observe only the shape-changing dynamics, no drifting. Left to herself, she
will reemerge in the same pose someplace else: that shift is called a
‘geometrical phase’, which -while accruing it is the whole point of swimming-
has not played any role in our discussion of symmetry reduction. Conversely,
most gauge choices in quantum field theory are covariant, and while that
suffices to regularize path integrals, the method of slices says that this is
no symmetry reduction at all, and it yields no insight into the geometry of
nonlinear flows.
Symmetry reduction in dynamics (including classical field theories such as the
Navier-Stokes equations) closely parallels the reduction of gauge symmetry in
quantum field theories. There, the freedom of choosing moving frames shown in
Fig. 5 is called ‘gauge freedom’ and a particular prescription for choosing a
representative from each gauge orbit is called ‘gauge fixing’. Just like the
slice hyperplanes of Fig. 7 may intersect a group orbit many times, a gauge
fixing submanifold may not intersect a gauge orbit, or it may intersect it
more than once (‘Gribov ambiguity’). Gribov (1978); Vandersickel and Zwanziger
(2012) In this context a chart is called a ‘Gribov’ or ‘fundamental modular’
region and its border is called a ‘Gribov horizon’ (a convex manifold in the
space of gauge fields). The Gribov region is compact and bounded by the Gribov
horizon. Within a Gribov region the ‘Faddeev-Popov operator’ (analogue of the
group orbit tangent vector) is strictly positive, while on the Gribov horizon
it has at least one vanishing eigenvalue.
## VII Conclusions
As turbulent flow evolves, every so often we catch a glimpse of a familiar
structure. For any finite spatial resolution and time, the flow follows
unstable coherent structures belonging to an alphabet of representative
states, here called ‘templates’. However, in the presence of symmetries, near
recurrences can be identified only if shifted both in time and space.
In the method of sections (along time direction) and slices (along spatial
symmetry directions), the identification of physically nearby states is
achieved by cutting group orbits with a finite set of hyperplanes, one for
each continuous parameter, with each time trajectory and group orbit of
symmetry-equivalent points represented by a single point. The method of slices
is akin to (but distinct from) cutting across trajectories by means of
sections. Both methods reduce continuous symmetries: one sections the
continuous-time trajectories, the other slices the layers of the onion formed
by group-orbits. Both are triggered by analogous conditions: oriented piercing
of the section and oriented piercing of the slice. Just as a Poincaré section
goes bad, the slice hyperplane goes bad the moment transversality is lost. A
slice, however, is emphatically _not_ a Poincaré section: as the first step in
a reduction of dynamics, a slice replaces a trajectory by a continuous
symmetry-reduced trajectory, whereas in the next step a Poincaré section
replaces a _continuous_ time trajectory by a _discrete_ sequence of points.
The main lesson of the visual tour undertaken above is that if a dynamical
problem has a continuous symmetry, the symmetry _must_ be reduced before any
detailed analysis of the flow’s state space geometry can take place. So far,
this has only been achieved for transitionally turbulent numerical pipe flows,
Willis, Cvitanović, and Avila (2012) resulting in the discovery of the first
relative periodic orbits embedded in turbulence. In the future, it should be
the first step in the analysis of any turbulent data, numerical or
experimental. Once symmetry reduction is achieved, all solutions of a
turbulent flow can be plotted together: all symmetry-equivalent states are
represented by a single point, families of solutions are mapped to a single
solution, relative equilibria become equilibria, relative periodic orbits
become periodic orbits, and most importantly, the analysis of the global
dynamical system in terms of invariant solutions and their stable/unstable
manifolds can now commence.
###### Acknowledgements.
This article addresses the questions asked after the talk given at Kyoto 2011
IUTAM Symposium on ‘50 Years of Chaos: Applied and Theoretical’. We are
indebted to S. Flynn, S. Froehlich, J. Greensite, S.A. Solla, R. Wilczak and
A.P. Willis for inspiring discussions. P.C. thanks G. Robinson, Jr. for
support, Max-Planck-Institut für Dynamik und Selbstorganisation, Göttingen for
hospitality, and the Niedersächsischen Knackwurst and Bayerische Hefeweizen
for making it all possible. P.C. was partly supported by NSF grant DMS-0807574
and 2009 Forschungspreis der Alexander von Humboldt-Stiftung. D.B. thanks M.F.
Schatz and was supported by NSF grant CBET-0853691.
## References
* Faisst and Eckhardt (2003) H. Faisst and B. Eckhardt, “Traveling waves in pipe flow,” Phys. Rev. Lett. 91, 224502 (2003).
* Wedin and Kerswell (2004) H. Wedin and R. R. Kerswell, “Exact coherent structures in pipe flow,” J. Fluid Mech. 508, 333–371 (2004).
* Hof _et al._ (2004) B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M. Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R. R. Kerswell, and F. Waleffe, “Experimental observation of nonlinear traveling waves in turbulent pipe flow,” Science 305, 1594–1598 (2004).
* Kerswell (2005) R. R. Kerswell, “Recent progress in understanding the transition to turbulence in a pipe,” Nonlinearity 18, R17–R44 (2005).
* Gibson, Halcrow, and Cvitanović (2008) J. F. Gibson, J. Halcrow, and P. Cvitanović, “Visualizing the geometry of state-space in plane Couette flow,” J. Fluid Mech. 611, 107–130 (2008), arXiv:0705.3957.
* Rowley and Marsden (2000) C. W. Rowley and J. E. Marsden, “Reconstruction equations and the Karhunen-Loéve expansion for systems with symmetry,” Physica D 142, 1–19 (2000).
* Beyn and Thümmler (2004) W.-J. Beyn and V. Thümmler, “Freezing solutions of equivariant evolution equations,” SIAM J. Appl. Dyn. Syst. 3, 85–116 (2004).
* Siminos and Cvitanović (2011) E. Siminos and P. Cvitanović, “Continuous symmetry reduction and return maps for high-dimensional flows,” Physica D 240, 187–198 (2011).
* Froehlich and Cvitanović (2011) S. Froehlich and P. Cvitanović, “Reduction of continuous symmetries of chaotic flows by the method of slices,” Comm. Nonlinear Sci. and Numerical Simulation 17, 2074–2084 (2011), arXiv:1101.3037.
* Kerswell (2012) R. R. Kerswell, “Misunderstanding the turbulence in a pipe,” (2012), in preparation.
* Willis, Cvitanović, and Avila (2012) A. P. Willis, P. Cvitanović, and M. Avila, “Revealing the state space of turbulent pipe flow by symmetry reduction,” (2012), arXiv:1203.3701, submitted to J. Fluid Mech.
* Rössler (1976) O. E. Rössler, “An equation for continuous chaos,” Phys. Lett. A 57, 397 (1976).
* Cvitanović _et al._ (2012) P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, _Chaos: Classical and Quantum_ (Niels Bohr Inst., Copenhagen, 2012) ChaosBook.org.
* Gibson and Cvitanović (2012) J. F. Gibson and P. Cvitanović, “Movies of plane Couette,” (2012), ChaosBook.org/tutorials.
* Gibbon and McGuinness (1982) J. D. Gibbon and M. J. McGuinness, “The real and complex Lorenz equations in rotating fluids and lasers,” Physica D 5, 108–122 (1982).
* Fowler, Gibbon, and McGuinness (1982) A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, “The complex Lorenz equations,” Physica D 4, 139–163 (1982).
* Siminos (2009) E. Siminos, _Recurrent Spatio-temporal Structures in Presence of Continuous Symmetries_ , Ph.D. thesis, School of Physics, Georgia Inst. of Technology, Atlanta (2009),
ChaosBook.org/projects/theses.html.
* Shapere and Wilczek (2006) A. Shapere and F. Wilczek, “Geometry of self-propulsion at low Reynolds number,” J. Fluid Mech. 198, 557 (2006).
* Fels and Olver (1998) M. Fels and P. J. Olver, “Moving coframes: I. A practical algorithm,” Acta Appl. Math. 51, 161–213 (1998).
* Fels and Olver (1999) M. Fels and P. J. Olver, “Moving coframes: II. Regularization and theoretical foundations,” Acta Appl. Math. 55, 127–208 (1999).
* Olver (1999) P. J. Olver, _Classical Invariant Theory_ (Cambridge Univ. Press, Cambridge, 1999).
* Cvitanović, Davidchack, and Siminos (2010) P. Cvitanović, R. L. Davidchack, and E. Siminos, “On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain,” SIAM J. Appl. Dyn. Syst. 9, 1–33 (2010), arXiv:0709.2944.
* Stephani _et al._ (2009) H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, _Exact Solutions of Einstein’s Field Equations_ (Cambridge U. Press, Cambridge, 2009).
* Gibson, Halcrow, and Cvitanović (2009) J. F. Gibson, J. Halcrow, and P. Cvitanović, “Equilibrium and traveling-wave solutions of plane Couette flow,” J. Fluid Mech. 638, 243–266 (2009), arXiv:0808.3375.
* Mostow (1957) G. D. Mostow, “Equivariant embeddings in Euclidean space,” Ann. Math. 65, 432–446 (1957).
* Palais (1961) R. S. Palais, “On the existence of slices for actions of non-compact Lie groups,” Ann. Math. 73, 295–323 (1961).
* Guillemin and Sternberg (1990) V. Guillemin and S. Sternberg, _Symplectic Techniques in Physics_ (Cambridge Univ. Press, Cambridge, 1990).
* Golubitsky, Stewart, and Schaeffer (1988) M. Golubitsky, I. Stewart, and D. G. Schaeffer, _Singularities and Groups in Bifurcation Theory, vol. II_ (Springer, New York, 1988).
* Viswanath (2007) D. Viswanath, “Recurrent motions within plane Couette turbulence,” J. Fluid Mech. 580, 339–358 (2007), arXiv:physics/0604062.
* Duguet, Pringle, and Kerswell (2008) Y. Duguet, C. C. T. Pringle, and R. R. Kerswell, “Relative periodic orbits in transitional pipe flow,” Phys. Fluids 20, 114102 (2008), arXiv:0807.2580.
* Mellibovsky and Eckhardt (2011) F. Mellibovsky and B. Eckhardt, “Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow,” J. Fluid Mech. 670, 96–129 (2011).
* Ruelle (1973) D. Ruelle, “Bifurcations in presence of a symmetry group,” Arch. Rational Mech. Anal. 51, 136–152 (1973).
* Armbruster, Guckenheimer, and Holmes (1988) D. Armbruster, J. Guckenheimer, and P. Holmes, “Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry,” Physica D 29, 257–282 (1988).
* Gilmore and Letellier (2007) R. Gilmore and C. Letellier, _The Symmetry of Chaos_ (Oxford Univ. Press, Oxford, 2007).
* Gatermann (2000) K. Gatermann, _Computer Algebra Methods for Equivariant Dynamical Systems_ (Springer, New York, 2000).
* Marsden and Weinstein (1974) J. E. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” Rep. Math. Phys. 5, 121–30 (1974).
* Abraham and Marsden (1978) R. Abraham and J. E. Marsden, _Foundations of Mechanics_ (Benjamin-Cummings, Reading, Mass., 1978).
* Morrison and Greene (1980) P. J. Morrison and J. M. Greene, “Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics,” Phys. Rev. Lett. 45, 790–794 (1980), see also Phys. Rev. Lett. 48, 569 (1982).
* Smale (1970) S. Smale, “Topology and mechanics, I.” Invent. Math. 10, 305–331 (1970).
* Rowley _et al._ (2003) C. W. Rowley, I. G. Kevrekidis, J. E. Marsden, and K. Lust, “Reduction and reconstruction for self-similar dynamical systems,” Nonlinearity 16, 1257–1275 (2003).
* Shapere and Wilczek (1989) A. Shapere and F. Wilczek, “Gauge kinematics of deformable bodies,” Amer. J. Physics 57, 514–518 (1989).
* Gribov (1978) V. N. Gribov, “Quantization of nonabelian gauge theories,” Nucl. Phys. B139, 1 (1978).
* Vandersickel and Zwanziger (2012) N. Vandersickel and D. Zwanziger, “The Gribov problem and QCD dynamics,” (2012), arXiv:1202.1491.
|
arxiv-papers
| 2012-09-21T20:38:38 |
2024-09-04T02:49:35.402617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Predrag Cvitanovic, Daniel Borrero-Echeverry, Keith M. Carroll, Bryce\n Robbins, and Evangelos Siminos",
"submitter": "Daniel Borrero",
"url": "https://arxiv.org/abs/1209.4915"
}
|
1209.4939
|
# Searches for Lepton Flavour Violation and Lepton Number Violation in Hadron
Decays
P. Seyfert on behalf of the LHCb Collaboration Physikalisches Institut,
Heidelberg University, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
###### Abstract
In the Standard Model of particle physics, lepton flavour and lepton number
are conserved quantities although no fundamental symmetry demands their
conservation. I present recent results of searches for lepton flavour and
lepton number violating hadron decays measured at the $\PB$ factories and
LHCb.
In addition, the LHCb collaboration has recently performed a search for the
lepton flavour violating decay $\Ptauon\to\Pmuon\Pmuon\APmuon$. The obtained
upper exclusion limit, that has been presented in this talk for the first
time, is of the same order of magnitude as those observed at the $\PB$
factories. This is the first search for a lepton flavour violating $\Ptau$
decay at a hadron collider.
## I Introduction
In the Standard Model both lepton number as well as lepton flavour are
conserved quantities SM1 ; SM2 . Since both can be broken in extensions of the
standard model, observation of either of them would be a clear sign for new
physics.
Results of the search for lepton number violation (LNV) and lepton flavour
violation (LFV) in decays of hadrons are presented. These comprise the $\PB$
decay modes $\PBplus\to h^{+}\ell^{+}\ell^{\prime-}$, $\PBplus\to
h^{-}\ell^{+}\ell^{\prime+}$, and the corresponding $\PD$ modes $\PDplus\to
h^{+}\ell^{+}\ell^{\prime-}$ and $\PDplus\to h^{-}\ell^{+}\ell^{\prime+}$. The
final state meson $h$ may hereby either be a stable meson (pions or kaons) or
in case of $\PB$ decays also a $\PD$ meson. Additionally to these modes, the
first limit on the branching fraction $\PBplus\to\PDzero\Ppiplus\Pmuon\Pmuon$
is presented as well as a new result on the search for LFV in the decay of
$\Ptau$ leptons at the LHC.
Throughout this document charge conjugate decays are implied.
### I.1 Lepton Number Violation
Numerous models without lepton number conservation have been proposed, see
majorana for an overview. Similar to the fundamental diagram in the
neutrinoless double beta decay, any neutrinoless hadron decay with two same
sign leptons in the final state probes the existence of Majorana neutrinos. Of
the two lowest order diagrams for LNV in meson decays, one can go through an
on-shell neutrino, while the other contains a virtual neutrino (Fig. 1). In
$\PBplus$ decays one of them is Cabbibo favoured depending on the final state.
Thus for Majorana neutrino masses in the accessible mass range (up to
$\unit{5140}{\MeVovercsq}$) the modes
$\PBplus\to\PDminus\Pleptonplus\Pleptonplus$ and
$\PBplus\to\PDstar^{-}\Pleptonplus\Pleptonplus$ are more sensitive and also
provide a mass measurement. Beyond the accessible mass range other final
states ($\Ppiminus\Pleptonplus\Pleptonplus$,
$\PDsminus\Pleptonplus\Pleptonplus$) are more sensitive.
(a) (b)
Figure 1: Lowest order diagrams for LNV in meson decays, involving on-shell or
virtual Majorana neutrinos. Depending on the individual quark flavours either
of them can be Cabbibo favoured. Reproduced from lhcb-Blnv2 .
(a) (b)
Figure 2: Constraints on charged lepton couplings $V_{\Pe 4}$ and $V_{\Pmu 4}$
to a fourth heavy Majorana neutrino from 2009 as a function of the mass
$m_{4}$majorana .
In the framework of LNV through a fourth neutrino $N$ with a large Majorana
mass, an observation of LNV not only provides information about the mass
$m_{4}$ of the fourth neutrino, but also on the $\PW N\ell$ coupling strength
$|V_{\Plepton 4}|$. A compilation of different exclusion limits is shown in
Fig. 2. For the coupling to the muon the strongest constraints come from kaon
physics.
Complementary to the modes with one meson in the final state, it has been
suggested in 1108.6009 to also consider
$\PBplus\to\PDzero\Ppiplus\Pmuon\Pmuon$ with the diagram shown in Fig. 3.
Until 2012 no limit on the branching fraction of this decay has been measured.
Figure 3: Tree level Feynman diagram for the decay
$\PBminus\to\PDzero\Ppiplus\Pmuon\Pmuon$.
### I.2 Lepton Flavour Violation
In contrast to violation of lepton number, lepton flavour violation has been
observed experimentally in the neutrino sector. Through loop diagrams,
neutrino oscillation can also enter the charged sector as illustrated in Fig.
4 – the predicted rates however are immeasurable small, suppressed by powers
of $m^{2}_{\Pneutrino}/m^{2}_{\PW}$ lfvcalc .
Two examples how to introduce sizeable lepton flavour violation are multi
Higgs extensions by means of new scalar particles (see e. g. d44_1461 ) as in
the diagram in Fig. LABEL:fig:phis or by means of heavy neutrinos as
introduced in low scale seesaw models (e. g. D73_074011 ) which couple to
electrons and muons as shown in Fig. LABEL:fig:hadlfvbsmtwo. Other ways to
embed LFV in the standard model are given e. g. in leptoquark .
Figure 4: Feynman diagram for lepton flavour violating meson decays in the
Standard Model with neutrino oscillation.
(a) (b)
Figure 5: Examples for introduction of lepton flavour violating in meson
decays
Particularly interesting about LFV in $\PB$ decays compared to $\PD$ decays is
that the $\PB$ mass is high enough to produce a $\APtauon,\Pmuon$ pair in the
final state. For new physics introduced in a Higgs coupling, this final state
is most sensitive due to the high masses, and thereby Higgs couplings of the
leptons involved.
(a) (b)
Figure 6: Feynman diagrams for $\Ptauon\to\Pmuon\Pmuon\APmuon$ in different
models.
Similarly the decay $\Ptauon\to\Pmuon\Pmuon\APmuon$ is not entirely forbidden,
but neutrino oscillation at loop level alone cannot bring the branching
fraction to an observable level. As presented in renga the strongest limits
on LFV in lepton decays come from the $\Pmuon\to\Pelectron\Pphoton$ mode, the
search for $\Ptauon\to\Pmuon\Pmuon\APmuon$ is particularly interesting because
some new physics models (e. g. Littlest Higgs littlesthiggs ), as in Fig. 6,
have enhanced lepton flavour violating couplings to heavy leptons (favouring
$\Ptau$ over $\Pmu$ decays) and do not involve photon couplings and therefore
enhance the three lepton final state over the $\Plepton\Pphoton$ final state
tau23mutheory . Moreover to identify the character of new physics, a search in
both $\Plepton\to\Plepton^{\prime}\Pphoton$ and
$\Plepton\to\Plepton^{\prime}\Plepton^{\prime}\Plepton^{\prime}$ must be
performed.
## II Experimental Results
The study of rare decays naturally needs large event samples, which, for $\PB$
and $\PD$ mesons, is available at the $\PB$ factories and at the LHC. The most
stringent constraints on LFV and LNV in modes involving electrons come from
BaBar and Belle, while muonic final states are now best constrained by recent
LHCb measurements.
### II.1 Limits on Lepton Number Violation
Decays of $\PBplus$, $\PDplus$, and $\PKplus$ mesons were used to search for
Majorana neutrinos of different masses. The mass difference of the decaying
meson and the final state lepton is the upper limit on the mass of the on-
shell neutrino which can be produced. Since the neutrino mass is the invariant
mass of the final state meson-lepton pair, the sum of their rest masses is the
lower limit on the accessible mass range.
The strongest limits on the lepton coupling $|V_{\Pe 4}|^{2}$ and $|V_{\Pmu
4}|^{2}$ to a fourth neutrino are in the low neutrino mass region between
$\unit{140}{\MeVovercsq}$ and $\unit{353}{\MeVovercsq}$ coming from searches
for the decays $\PKplus\to\APelectron\APelectron\Ppiminus$ and
$\PKplus\to\APmuon\APmuon\Ppiminus$ respectively. Couplings down to
$|V_{\Plepton 4}|^{2}\lesssim 10^{-8}$ are thereby ruled out in the most
sensitive range.
The currently most stringent limits on LNV in charm decays and thereby higher
neutrino masses have been obtained by the BaBar collaboration babar-charm
shown in Tab. LABEL:tab:babar-charm.
The extension of the search range to higher masses is only possible in $\PB$
decays, the enormous production cross section in hadron collisions makes the
LHC the optimal place for searches for LNV in $\PB$ decays. LHCb recently
provided new results on the on-shell modes $\PBplus\to\Ppiminus\APmuon\APmuon$
and $\PBplus\to\PDsminus\APmuon\APmuon$, as well as the virtual modes
$\PBplus\to\PDplus\APmuon\APmuon$ and $\PBplus\to\PD^{*-}\APmuon\APmuon$ lhcb-
Blnv2 .
Limits on the branching fraction are hereby set as a function of the neutrino
mass for the on-shell modes. For comparison, assuming a flat distribution of
the decay products in phase space, the observed branching fraction is shown in
Tab. LABEL:tab:BLNV along with the modes which are sensitive to virtual
Majorana neutrinos and previous measurements.
The first search for $\PBplus\to\PDzero\Ppiminus\APmuon\APmuon$ has been
performed by LHCb lhcb-Blnv2 and showed no excess over the background. Since
this channel involves an on-shell Majorana neutrino, the limit is given as a
function of the neutrino mass as well.
Table 1: Current limits on lepton number violating charm (a) and bottom (b)
meson decays.
channel | limit | |
---|---|---|---
$\mathcal{B}(\PDplus\to\Ppiminus\APelectron\APelectron)$ | $<1.9\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDplus\to\Ppiminus\APmuon\APmuon)$ | $<2.0\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDplus\to\Ppiminus\APmuon\APelectron)$ | $<2.0\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDsplus\to\Ppiminus\APelectron\APelectron)$ | $<4.1\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDsplus\to\Ppiminus\APmuon\APmuon)$ | $<14\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDsplus\to\Ppiminus\APmuon\APelectron)$ | $<8.4\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDplus\to\PKminus\APelectron\APelectron)$ | $<0.9\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDplus\to\PKminus\APmuon\APmuon)$ | $<10\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDplus\to\PKminus\APmuon\APelectron)$ | $<1.9\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDsplus\to\PKminus\APelectron\APelectron)$ | $<5.2\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDsplus\to\PKminus\APmuon\APmuon)$ | $<13\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PDsplus\to\PKminus\APmuon\APelectron)$ | $<6.1\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PLambdac\to\APproton\APelectron\APelectron)$ | $<2.7\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PLambdac\to\APproton\APmuon\APmuon)$ | $<9.4\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
$\mathcal{B}(\PLambdac\to\APproton\APmuon\APelectron)$ | $<16\times 10^{-6}$ | @$90\,\%$ CL | babar-charm BaBar
(a)
channel | limit | | |
---|---|---|---|---
$\mathcal{B}(\PBminus\to\Ppi^{+}\Pelectron\Pelectron)$ | $<2.3\times 10^{-8}$ | @$90\,\%$ CL | babar-Blnv | BaBar
$\mathcal{B}(\PBminus\to\PK^{+}\Pelectron\Pelectron)$ | $<3.0\times 10^{-8}$ | @$90\,\%$ CL | babar-Blnv | BaBar
$\mathcal{B}(\PBminus\to\PK^{*+}\Pelectron\Pelectron)$ | $<2.8\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv | CLEO
$\mathcal{B}(\PBminus\to\Prho^{+}\Pelectron\Pelectron)$ | $<2.6\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv | CLEO
$\mathcal{B}(\PBminus\to\PD^{+}\Pelectron\Pelectron)$ | $<2.6\times 10^{-6}$ | @$90\,\%$ CL | belle-Blnv | Belle
$\mathcal{B}(\PBminus\to\PD^{+}\Pelectron\Pmuon)$ | $<1.8\times 10^{-6}$ | @$90\,\%$ CL | belle-Blnv | Belle
$\mathcal{B}(\PBminus\to\Ppi^{+}\Pmuon\Pmuon)$ | $<1.3\times 10^{-8}$ | @$95\,\%$ CL | lhcb-Blnv2 | LHCb
$\mathcal{B}(\PBminus\to\PK^{+}\Pmuon\Pmuon)$ | $<5.4\times 10^{-7}$ | @$95\,\%$ CL | lhcb-Blnv | LHCb
$\mathcal{B}(\PBminus\to\PK^{*+}\Pmuon\Pmuon)$ | $<4.4\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv | CLEO
$\mathcal{B}(\PBminus\to\Prho^{+}\Pmuon\Pmuon)$ | $<5.0\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv | CLEO
$\mathcal{B}(\PBminus\to\PD^{+}\Pmuon\Pmuon)$ | $<6.9\times 10^{-7}$ | @$95\,\%$ CL | lhcb-Blnv2 | LHCb
$\mathcal{B}(\PBminus\to\PD^{*+}\Pmuon\Pmuon)$ | $<2.4\times 10^{-6}$ | @$95\,\%$ CL | lhcb-Blnv2 | LHCb
$\mathcal{B}(\PBminus\to\PDs^{+}\Pmuon\Pmuon)$ | $<5.8\times 10^{-7}$ | @$95\,\%$ CL | lhcb-Blnv2 | LHCb
$\mathcal{B}(\PBminus\to\PDzero\Ppiplus\Pmuon\Pmuon)$ | $<1.5\times 10^{-6}$ | @$95\,\%$ CL | lhcb-Blnv2 | LHCb
(b)
LHCb also provides the strongest limits on $|V_{\mu 4}|$ up to the $\PBplus$
mass considering these results come from $\PBplus\to\Ppiminus\APmuon\APmuon$,
shown in Fig. 7.
Figure 7: Limit on $|V_{\mu 4}|^{2}$ from $\PBminus\to\Ppi^{+}\Pmuon\Pmuon$
measured by LHCb lhcb-Blnv2 . Figure 8: Relative reconstruction efficiency as
a function of the Majorana neutrino lifetime. The branching fraction limits
from lhcb-Blnv2 have been computed for the assumption of infinitively short
lifetimes ($100\,\%$ relative efficiency). For longer lifetimes, the
reconstruction efficiency decreases and the observed limit has to be scaled
down. The efficiencies are given for (a) $\PBplus\to\Ppiminus\APmuon\APmuon$,
(b) $\PBplus\to\PDsminus\APmuon\APmuon$, and (c)
$\PBplus\to\PDzero\Ppiminus\APmuon\APmuon$.
A natural way to search for a lepton number violating decay is to search for
same sign leptons from a common vertex. Thereby the analysis’ implications on
an intermediate on-shell neutrino are only drawn correctly if the lifetime of
the neutrino is short enough not to degrade the reconstruction. To estimate
how the observed limits are to be understood in models with long lived heavy
neutrinos, LHCb also provides the relative reconstruction efficiency as a
function of the neutrino lifetime, shown in Fig. 8.
### II.2 Limits on Lepton Flavour Violation
The tightest constraints on lepton flavour violating processes in charm decays
have been found by the BaBar collaboration, listed in Tab. LABEL:tab:babar-
charmlfv. For bottom decays, Tab. LABEL:tab:babar-bottomlfv shows the recent
results, involving $\Ptau$ leptons in the final state. Details are given in
giovanni . These results improved the limit on the energy scale at which LFV
can occur LFV_energyscale significantly.
The implication for new physics is that the energy scale for LFV effective
operators is pushed up from $\unit{2.2}{}$ to $\unit{11}{}$ or from
$\unit{2.6}{}$ to $\unit{15}{}$ for the $\Pbottom\to\Pdown$ and the
$\Pbottom\to\Pstrange$ transition respectively LFV_energyscale .
Table 2: Limits on lepton flavour violating hadron decays at $90\,\%$
confidence level. All listed limits from the BaBar collaboration.
channel | limit |
---|---|---
$\mathcal{B}(\PDplus\to\Ppiplus\APmuon\Pelectron)$ | $<3.6\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDplus\to\Ppiplus\APelectron\Pmuon)$ | $<2.9\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\Ppiplus\APmuon\Pelectron)$ | $<20\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\Ppiplus\APelectron\Pmuon)$ | $<12\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDplus\to\PKplus\APmuon\Pelectron)$ | $<2.8\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDplus\to\PKplus\APelectron\Pmuon)$ | $<1.2\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\PKplus\APmuon\Pelectron)$ | $<9.7\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\PKplus\APelectron\Pmuon)$ | $<14\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PLambdac\to\Pproton\APmuon\Pelectron)$ | $<19\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PLambdac\to\Pproton\APelectron\Pmuon)$ | $<9.9\times 10^{-6}$ | babar-charm
(a)
channel | limit |
---|---|---
$\mathcal{B}(\PBplus\to\PKplus\Ptauon\APmuon)$ | $<4.5\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\PKplus\APtauon\Pmuon)$ | $<2.8\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\PKplus\Ptauon\APelectron)$ | $<4.3\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\PKplus\APtauon\Pelectron)$ | $<1.5\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\Ptauon\APmuon)$ | $<6.2\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\APtauon\Pmuon)$ | $<4.5\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\Ptauon\APelectron)$ | $<7.4\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\APtauon\Pelectron)$ | $<2.0\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\Pmu^{\pm}\Pe^{\mp})$ | $<1.7\times 10^{-7}$ | babar-Blfv
(b)
The most recent result in this talk is the limit on LFV in
$\Ptauon\to\Pmuon\Pmuon\APmuon$ achieved by LHCb tau23muCONF . The hadron
collider environment introduces special experimental challenges compared to
the $\PB$ factories.
#### $\Ptau$ tag
At the $\PB$ factories, $\Ptau$ are produced in pairs. A clean event selection
therefore is to look at events with four tracks – three from the signal
candidate and one from a standard model one prong $\Ptau$ decay. At the LHC
the main source for $\Ptau$ is the leptonic $\PDsminus\to\Ptauon\APnut$ decay
tau23muCONF .
#### Normalisation
The $\Ptau$ tag automatically provides the number of produced $\Ptau$ which
enter the analysis. Since the main production mode for $\Ptau$ at the LHC does
not provide any further charged particles, the number of $\Ptau$ entering the
analysis is not directly accessible. A normalisation to allowed $\Ptau$ decays
is not possible since they are indistinguishable from more abundant $\PDplus$
decays with $\Ppizero$ in the decay chain.
#### Background
Having no production tag, background from events without $\Ptau$, such as
$\PB$ and $\PD$ cascade decays, is more severe in the LHCb analysis than for
the $\PB$ factories.
The main advantage of LHCb however is the huge production cross section for
$\Ptau$ from $\PDs$ decays. Considering the charm and bottom production cross
sections measured by LHCb LHCb-CONF-2010-013 ; Bprod and the known
semileptonic branching fractions pdg , about $8\times 10^{10}$ $\Ptau$ leptons
have been produced at LHCb in 2011 compared to a total of $10^{9}$ $\Ptau$
pairs at the $\PB$ factories.
The analysis strategy of tau23muCONF is similar to other rare decay searches
at LHCb. A loose cut based selection is applied to get a processable data
sample. All events passing this selection are classified in a three
dimensional likelihood space. The discriminating variables are the invariant
mass of the $\Ptauon\to\Pmuon\Pmuon\APmuon$ candidate, a multivariate
classifier $\mathcal{M}_{\text{3body}}$ for the three body decay properties
(geometry, displacement, track quality, isolation, and kinematics), and a
multivariate classifier for the particle identification
$\mathcal{M}_{\text{PID}}$ (combining information from muon stations, RICH
detectors, and the calorimeter signature). The latter classifiers use boosted
decision trees bdt with adaptive boosting bdt_boosted as implemented by TMVA
tmva .
(a) (b)
Figure 9: Distribution of signal events in the two multivariate likelihoods
for signal (blue / solid) and background (red / dashed).
(a) (b)
Figure 10: Invariant mass distribution for (a) simulated signal candidates and
(b) observed events in the two highest $\mathcal{M}_{\text{3body}}$ and
$\mathcal{M}_{\text{PID}}$ bins. The background fit
($\PDsplus\to\Peta\APmuon\Pnu$ in green / dotted; combinatorial in red /
dashed; combined in blue / solid) is shown in the range which is used for the
fit.
The signal efficiency of the multivariate classifiers as well as the invariant
mass resolution come from simulation and are calibrated on a control channel –
$\PDsplus\to\Ppiplus\Pphi(\APmuon\Pmuon)$ in the case for the three body
classifier and the invariant mass and $\PBplus\to\PJpsi(\APmuon\Pmuon)\PKplus$
for the particle identification.
The $\PDsplus\to\Pphi\Ppiplus$ calibration channel also serves as a
normalisation, since the branching fractions
$\mathcal{B}(\PDsplus\to\Pphi\Ppiplus)$ and
$\mathcal{B}(\PDsplus\to\APtauon\Pnut)$ are known – yielding the number of
$\Ptau$ which have been produced in $\PDs$ decays. To determine the fraction
of $\Ptau$ from $\PDs$ decays, $f(\PDs)$, the bottom and charm cross sections
measured by LHCb LHCb-CONF-2010-013 ; Bprod , as well as the branching
fractions of charm and bottom hadrons to $\Ptau$ are used. Hereby most of the
systematic uncertainties (e. g. luminosity measurement, reconstruction
efficiencies) cancel, i. e. $f(\PDs)$ is more accurately known than the
inclusive $\Ptau$ production cross section. Contributions from gauge bosons or
Drell-Yan processes have been evaluated to be negligible.
Using the above normalisation as well as the efficiencies for selection,
reconstruction, and trigger the branching fraction can be written as follows:
$\displaystyle\mathcal{B}(\Ptauon\to\Pmuon\Pmuon\APmuon)$
$\displaystyle=\frac{\mathcal{B}(\PDsplus\to\Pphi(\APmuon\Pmuon)\Ppiplus)}{\mathcal{B}(\PDsplus\to\APtauon\Pnut)}\times
f(\PDs)$
$\displaystyle\quad\times\frac{\varepsilon_{\text{norm}}}{\varepsilon_{\text{sig}}}\frac{N_{\Ptauon\to\Pmuon\Pmuon\APmuon}}{N_{\PDsplus\to\Pphi(\APmuon\Pmuon)\Ppiplus}}$
where $\varepsilon_{\text{norm}}$ is the total efficiency to trigger,
reconstruct and select the normalisation decay and $\varepsilon_{\text{sig}}$
is the total efficiency for the signal channel.
The dimuon decay of the $\Pphi$ is chosen to provide similar trigger and
particle identification properties compared to the signal being sought for.
The non resonant contribution from $\PDsplus\to\APmuon\Pmuon\Ppiplus$ decays
was found to be below $2\,\%$.
The three dimensional likelihood space is subdivided into 150 bins (five for
$\mathcal{M}_{\text{3body}}$ and $\mathcal{M}_{\text{PID}}$, and six for the
invariant mass) as shown in Fig. 9 and LABEL:fig:massbinning. The signal
efficiency for each bin is evaluated from the calibration channels and the
background in each bin is estimated from the sidebands in the invariant mass.
The background consists mainly of two components. Firstly combinatorial
background which is modelled by an exponential and secondly by
$\PDsplus\to\Peta(\APmuon\Pmuon\Pphoton)\APmuon\Pnum$ decays. This physical
background is not discriminated in the current analysis by either
$\mathcal{M}_{\text{3body}}$ or $\mathcal{M}_{\text{PID}}$ as it has the same
behaviour as the signal in all input quantities. Rejecting this decay will be
subject of future improvements. It is modelled by an exponential multiplied by
a second order polynomial for which all shape parameters have been fixed on
simulated events. The normalisation is left free in the final fit within one
standard deviation from the expected yield which is determined using the
normalisation channel and the branching fractions
$\mathcal{B}(\PDsplus\to\Peta\APmuon\Pnum),\mathcal{B}(\Peta\to\APmuon\Pmuon\Pphoton)$,
and $\mathcal{B}(\PDsplus\to\Pphi\Ppiplus)$. For illustration the invariant
mass distribution and the combined fit for the combination of the two highest
$\mathcal{M}_{\text{3body}}$ and two highest $\mathcal{M}_{\text{PID}}$ bins
is shown in Fig. LABEL:fig:unblinded.
For the final limit, all bins are combined using the CLs method CLs1 ; CLs2 .
Table 3: Limits on the branching fraction for $\Ptauon\to\Pmuon\Pmuon\APmuon$ obtained by different experiments. collaboration | limit | |
---|---|---|---
Belle | $<2.1\times 10^{-8}$ | @$90\,\%$ CL | tau23muBELLE
BaBar | $<3.3\times 10^{-8}$ | @$90\,\%$ CL | tau23muBABAR
LHCb | $<6.3\times 10^{-8}$ | @$90\,\%$ CL | tau23muCONF
The observed limits at $90\,\%$ confidence level is $6.3\times 10^{-8}$, in
agreement with the expected limit for the absence of a signal ($8.2\times
10^{-8}$). Tab. 3 shows the comparison to the limits from BaBar and Belle.
## III Conclusion
Hadron decays measured at the $\PB$ factories and at the LHC provide an
excellent and abundant probe to search for LNV and LFV. So far no signal has
been observed and only lower limits for the branching fractions are given. The
$\PB$ factories have achieved high sensitivity and ruled out branching
fractions to the level of $10^{-5}$. The huge production cross section of
$\PB$ mesons at the LHC furthermore enabled LHCb to improve the limits on LNV
in $\PB$ decays further. Finally, the first LFV $\Ptau$ decay search performed
at a hadron collider has been performed.
## References
* (1) S. Weinberg. A Model of Leptons. Phys. Rev. Lett., 19:1264–1266, Nov 1967.
* (2) S. L. Glashow, J. Iliopoulos, and L. Maiani. Weak Interactions with Lepton-Hadron Symmetry. Phys. Rev. D, 2:1285–1292, Oct 1970.
* (3) A. Atre, T. Han, S. Pascoli, and B. Zhang. The search for heavy Majorana neutrinos. Journal of High Energy Physics, 2009(05):030, 2009.
* (4) R. Aaij et al. Searches for Majorana neutrinos in ${B}^{\mathbf{-}}$ decays. Phys. Rev. D, 85:112004, Jun 2012, doi:10.1103/PhysRevD.85.112004.
* (5) N. Quintero, G. Lopez Castro, and D. Delepine. Lepton number violation in top quark and neutral B meson decays. Phys.Rev., D84:096011, 2011, 1108.6009.
* (6) X.-G. He, G. Valencia, and Y. Wang. Lepton flavor violating $\tau{}$ and $b$ decays and heavy neutrinos. Phys. Rev. D, 70:113011, Dec 2004.
* (7) M. Sher and Y. Yuan. Rare $b$ decays, rare $\tau{}$ decays, and grand unification. Phys. Rev. D, 44:1461–1472, Sep 1991.
* (8) T. Fujihara, S. K. Kang, C. S. Kim, D. Kimura, and T. Morozumi. Low scale seesaw model and lepton flavor violating rare $b$ decays. Phys. Rev. D, 73:074011, Apr 2006.
* (9) S. Davidson, D. C. Bailey, and B. A. Campbell. Model independent constraints on leptoquarks from rare processes. Z.Phys., C61:613–644, 1994, hep-ph/9309310.
* (10) F. Renga. Searches for LFV and LNV in charged lepton decays. In Zhao et al. fpcp2012 . FPCP2012-41.
* (11) M. Blanke, A. J. Buras, B. Duling, S. Recksiegel, and C. Tarantino. FCNC Processes in the Littlest Higgs Model with T-Parity: a 2009 Look. Acta Phys.Polon., B41:657–683, 2010, 0906.5454.
* (12) M. Blanke, A. J. Buras, B. Duling, A. Poschenrieder, and C. Tarantino. Charged Lepton Flavour Violation and (g-2)(mu) in the Littlest Higgs Model with T-Parity: A Clear Distinction from Supersymmetry. JHEP, 0705:013, 2007, hep-ph/0702136.
* (13) J. P. Lees et al. Searches for rare or forbidden semileptonic charm decays. Phys. Rev. D, 84:072006, Oct 2011.
* (14) J. P. Lees et al. Search for lepton-number violating processes in ${B}^{+}\rightarrow{}{h}^{-}{l}^{+}{l}^{+}$ decays. Phys. Rev. D, 85:071103, Apr 2012, doi:10.1103/PhysRevD.85.071103.
* (15) K. W. Edwards et al. Search for lepton-flavor-violating decays of $B$ mesons. Phys. Rev. D, 65:111102, Jun 2002, doi:10.1103/PhysRevD.65.111102.
* (16) O. Seon et al. Search for lepton-number-violating ${B}^{\mathbf{+}}\mathbf{\rightarrow{}}{D}^{\mathbf{-}}{\ell{}}^{\mathbf{+}}{\ell{}}^{\prime{}\mathbf{+}}$ decays. Phys. Rev. D, 84:071106, Oct 2011, doi:10.1103/PhysRevD.84.071106.
* (17) R. Aaij et al. Search for Lepton Number Violating Decays ${B}^{+}\rightarrow{}{\pi{}}^{-}{\mu{}}^{+}{\mu{}}^{+}$ and ${B}^{+}\rightarrow{}{K}^{-}{\mu{}}^{+}{\mu{}}^{+}$. Phys. Rev. Lett., 108:101601, Mar 2012, doi:10.1103/PhysRevLett.108.101601.
* (18) G. Marchiori. CP violation in other channels and rare decays. In Zhao et al. fpcp2012 . FPCP2012-25.
* (19) D. Black, T. Han, H.-J. He, and M. Sher. $\tau{}-\mu{}$ flavor violation as a probe of the scale of new physics. Phys. Rev. D, 66:053002, Sep 2002.
* (20) J. P. Lees et al. Search for the decay modes ${B}^{\mathbf{\pm{}}}\rightarrow{}{h}^{\mathbf{\pm{}}}\tau{}\ell{}$. Phys. Rev. D, 86:012004, Jul 2012, 1204.2852.
* (21) B. Aubert et al. Search for the Rare Decay $B\rightarrow{}\pi{}{l}^{+}{l}^{-}$. Phys. Rev. Lett., 99:051801, Jul 2007.
* (22) R. Aaij et al. Search for the lepton flavour violating decay $\Ptau^{-}\to\Pmu^{+}\Pmu^{-}\Pmu^{-}$. May 2012. LHCb-CONF-2012-015.
* (23) Prompt charm production in $pp$ collisions at $\sqrt{s}$ = 7 TeV. Dec 2010. LHCb-CONF-2010-013.
* (24) R. Aaij et al. Measurement of b hadron production fractions in 7 TeV pp collisions. Phys.Rev., D85:032008, 2012, 1111.2357.
* (25) K. Nakamura et al. Review of particle physics. J.Phys., G37:075021, 2010.
* (26) L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. Wadsworth International Group, Belmont, California, 1984.
* (27) Y. Freund and R. E. Schapire. Experiments with a new boosting algorithm, pages 209–217. ACM Press, New York, 1996.
* (28) A. Hoecker, P. Speckmayer, J. Stelzer, J. Therhaag, E. von Toerne, and H. Voss. TMVA: Toolkit for Multivariate Data Analysis. PoS, ACAT:040, 2007, physics/0703039.
* (29) T. Junk. Confidence Level Computation for Combining Searches with Small Statistics. Nucl. Instrum. Meth., A434:435–443, 1999, hep-ex/9902006.
* (30) A. L. Read. Presentation of search results: The CL(s) technique. J. Phys., G28:2693–2704, 2002.
* (31) K. Hayasaka et al. Search for lepton-flavor-violating $\Ptau$ decays into three leptons with 719 million produced pairs. Physics Letters B, 687(2–3):139 – 143, 2010.
* (32) J. P. Lees et al. Limits on $\tau{}$ lepton-flavor violating decays into three charged leptons. Phys. Rev. D, 81:111101, Jun 2010.
* (33) L. J. Hall and S. Raby. Complete supersymmetric SO(10) model. Phys. Rev. D, 51:6524–6531, Jun 1995.
* (34) P. Fileviez Perez and M. B. Wise. Breaking Local Baryon and Lepton Number at the TeV Scale. JHEP, 1108:068, 2011, 1106.0343.
* (35) L. Sun. $\Pbottom\to\Pstrange\Pphoton$. In Zhao et al. fpcp2012 . FPCP2012-4.
* (36) Z.-G. Zhao, J.-X. Lu, and Q. Wang, editors. Proceedings of the 10th International Conference on Flavor Physics and CP Violation, 2012.
* (37) A. Zee. A theory of lepton number violation and neutrino Majorana masses. Physics Letters B, 93(4):389 – 393, 1980.
* (38) Y. Amhis et al. Averages of $\Pbottom$-hadron, $\Pcharm$-hadron, and $\Ptau$-lepton properties as of early 2012. 2012, 1207.1158.
|
arxiv-papers
| 2012-09-21T23:50:35 |
2024-09-04T02:49:35.415947
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paul Seyfert",
"submitter": "Paul Seyfert",
"url": "https://arxiv.org/abs/1209.4939"
}
|
1209.4963
|
# Study of $\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$ via $\eta_{c}$
exclusive decays
M. Ablikim1, M. N. Achasov5, O. Albayrak3, D. J. Ambrose39, F. F. An1, Q.
An40, J. Z. Bai1, Y. Ban27, J. Becker2, J. V. Bennett17, M. Bertani18A, J. M.
Bian38, E. Boger20,a, O. Bondarenko21, I. Boyko20, R. A. Briere3, V. Bytev20,
X. Cai1, O. Cakir35A, A. Calcaterra18A, G. F. Cao1, S. A. Cetin35B, J. F.
Chang1, G. Chelkov20,a, G. Chen1, H. S. Chen1, J. C. Chen1, M. L. Chen1, S. J.
Chen25, X. Chen27, Y. B. Chen1, H. P. Cheng14, Y. P. Chu1, D. Cronin-
Hennessy38, H. L. Dai1, J. P. Dai1, D. Dedovich20, Z. Y. Deng1, A. Denig19, I.
Denysenko20,b, M. Destefanis43A,43C, W. M. Ding29, Y. Ding23, L. Y. Dong1, M.
Y. Dong1, S. X. Du46, J. Fang1, S. S. Fang1, L. Fava43B,43C, F. Feldbauer2, C.
Q. Feng40, R. B. Ferroli18A, C. D. Fu1, J. L. Fu25, Y. Gao34, C. Geng40, K.
Goetzen7, W. X. Gong1, W. Gradl19, M. Greco43A,43C, M. H. Gu1, Y. T. Gu9, Y.
H. Guan6, A. Q. Guo26, L. B. Guo24, Y. P. Guo26, Y. L. Han1, F. A. Harris37,
K. L. He1, M. He1, Z. Y. He26, T. Held2, Y. K. Heng1, Z. L. Hou1, H. M. Hu1,
T. Hu1, G. M. Huang15, G. S. Huang40, J. S. Huang12, X. T. Huang29, Y. P.
Huang1, T. Hussain42, C. S. Ji40, Q. Ji1, Q. P. Ji26,c, X. B. Ji1, X. L. Ji1,
L. L. Jiang1, X. S. Jiang1, J. B. Jiao29, Z. Jiao14, D. P. Jin1, S. Jin1, F.
F. Jing34, N. Kalantar-Nayestanaki21, M. Kavatsyuk21, W. Kuehn36, W. Lai1, J.
S. Lange36, C. H. Li1, Cheng Li40, Cui Li40, D. M. Li46, F. Li1, G. Li1, H. B.
Li1, J. C. Li1, K. Li10, Lei Li1, Q. J. Li1, S. L. Li1, W. D. Li1, W. G. Li1,
X. L. Li29, X. N. Li1, X. Q. Li26, X. R. Li28, Z. B. Li33, H. Liang40, Y. F.
Liang31, Y. T. Liang36, G. R. Liao34, X. T. Liao1, B. J. Liu1, C. L. Liu3, C.
X. Liu1, C. Y. Liu1, F. H. Liu30, Fang Liu1, Feng Liu15, H. Liu1, H. H. Liu13,
H. M. Liu1, H. W. Liu1, J. P. Liu44, K. Y. Liu23, Kai Liu6, P. L. Liu29, Q.
Liu6, S. B. Liu40, X. Liu22, Y. B. Liu26, Z. A. Liu1, Zhiqiang Liu1, Zhiqing
Liu1, H. Loehner21, G. R. Lu12, H. J. Lu14, J. G. Lu1, Q. W. Lu30, X. R. Lu6,
Y. P. Lu1, C. L. Luo24, M. X. Luo45, T. Luo37, X. L. Luo1, M. Lv1, C. L. Ma6,
F. C. Ma23, H. L. Ma1, Q. M. Ma1, S. Ma1, T. Ma1, X. Y. Ma1, Y. Ma11, F. E.
Maas11, M. Maggiora43A,43C, Q. A. Malik42, Y. J. Mao27, Z. P. Mao1, J. G.
Messchendorp21, J. Min1, T. J. Min1, R. E. Mitchell17, X. H. Mo1, C. Morales
Morales11, C. Motzko2, N. Yu. Muchnoi5, H. Muramatsu39, Y. Nefedov20, C.
Nicholson6, I. B. Nikolaev5, Z. Ning1, S. L. Olsen28, Q. Ouyang1, S.
Pacetti18B, J. W. Park28, M. Pelizaeus37, H. P. Peng40, K. Peters7, J. L.
Ping24, R. G. Ping1, R. Poling38, E. Prencipe19, M. Qi25, S. Qian1, C. F.
Qiao6, X. S. Qin1, Y. Qin27, Z. H. Qin1, J. F. Qiu1, K. H. Rashid42, G. Rong1,
X. D. Ruan9, A. Sarantsev20,d, B. D. Schaefer17, J. Schulze2, M. Shao40, C. P.
Shen37,e, X. Y. Shen1, H. Y. Sheng1, M. R. Shepherd17, X. Y. Song1, S.
Spataro43A,43C, B. Spruck36, D. H. Sun1, G. X. Sun1, J. F. Sun12, S. S. Sun1,
Y. J. Sun40, Y. Z. Sun1, Z. J. Sun1, Z. T. Sun40, C. J. Tang31, X. Tang1, I.
Tapan35C, E. H. Thorndike39, D. Toth38, M. Ullrich36, G. S. Varner37, B.
Wang9, B. Q. Wang27, D. Wang27, D. Y. Wang27, K. Wang1, L. L. Wang1, L. S.
Wang1, M. Wang29, P. Wang1, P. L. Wang1, Q. Wang1, Q. J. Wang1, S. G. Wang27,
X. L. Wang40, Y. D. Wang40, Y. F. Wang1, Y. Q. Wang29, Z. Wang1, Z. G. Wang1,
Z. Y. Wang1, D. H. Wei8, J. B. Wei27, P. Weidenkaff19, Q. G. Wen40, S. P.
Wen1, M. Werner36, U. Wiedner2, L. H. Wu1, N. Wu1, S. X. Wu40, W. Wu26, Z.
Wu1, L. G. Xia34, Z. J. Xiao24, Y. G. Xie1, Q. L. Xiu1, G. F. Xu1, G. M. Xu27,
H. Xu1, Q. J. Xu10, X. P. Xu32, Z. R. Xu40, F. Xue15, Z. Xue1, L. Yan40, W. B.
Yan40, Y. H. Yan16, H. X. Yang1, Y. Yang15, Y. X. Yang8, H. Ye1, M. Ye1, M. H.
Ye4, B. X. Yu1, C. X. Yu26, H. W. Yu27, J. S. Yu22, S. P. Yu29, C. Z. Yuan1,
Y. Yuan1, A. A. Zafar42, A. Zallo18A, Y. Zeng16, B. X. Zhang1, B. Y. Zhang1,
C. Zhang25, C. C. Zhang1, D. H. Zhang1, H. H. Zhang33, H. Y. Zhang1, J. Q.
Zhang1, J. W. Zhang1, J. Y. Zhang1, J. Z. Zhang1, S. H. Zhang1, X. J. Zhang1,
X. Y. Zhang29, Y. Zhang1, Y. H. Zhang1, Y. S. Zhang9, Z. P. Zhang40, Z. Y.
Zhang44, G. Zhao1, H. S. Zhao1, J. W. Zhao1, K. X. Zhao24, Lei Zhao40, Ling
Zhao1, M. G. Zhao26, Q. Zhao1, Q. Z. Zhao9,f, S. J. Zhao46, T. C. Zhao1, X. H.
Zhao25, Y. B. Zhao1, Z. G. Zhao40, A. Zhemchugov20,a, B. Zheng41, J. P.
Zheng1, Y. H. Zheng6, B. Zhong1, J. Zhong2, Z. Zhong9,f, L. Zhou1, X. K.
Zhou6, X. R. Zhou40, C. Zhu1, K. Zhu1, K. J. Zhu1, S. H. Zhu1, X. L. Zhu34, Y.
C. Zhu40, Y. M. Zhu26, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, B. S. Zou1, J. H.
Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, P. R. China
2 Bochum Ruhr-University, 44780 Bochum, Germany
3 Carnegie Mellon University, Pittsburgh, PA 15213, USA
4 China Center of Advanced Science and Technology, Beijing 100190, P. R. China
5 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
6 Graduate University of Chinese Academy of Sciences, Beijing 100049, P. R.
China
7 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
8 Guangxi Normal University, Guilin 541004, P. R. China
9 GuangXi University, Nanning 530004,P.R.China
10 Hangzhou Normal University, Hangzhou 310036, P. R. China
11 Helmholtz Institute Mainz, J.J. Becherweg 45,D 55099 Mainz,Germany
12 Henan Normal University, Xinxiang 453007, P. R. China
13 Henan University of Science and Technology, Luoyang 471003, P. R. China
14 Huangshan College, Huangshan 245000, P. R. China
15 Huazhong Normal University, Wuhan 430079, P. R. China
16 Hunan University, Changsha 410082, P. R. China
17 Indiana University, Bloomington, Indiana 47405, USA
18 (A)INFN Laboratori Nazionali di Frascati, Frascati, Italy; (B)INFN and
University of Perugia, I-06100, Perugia, Italy
19 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, 55099
Mainz, Germany
20 Joint Institute for Nuclear Research, 141980 Dubna, Russia
21 KVI/University of Groningen, 9747 AA Groningen, The Netherlands
22 Lanzhou University, Lanzhou 730000, P. R. China
23 Liaoning University, Shenyang 110036, P. R. China
24 Nanjing Normal University, Nanjing 210046, P. R. China
25 Nanjing University, Nanjing 210093, P. R. China
26 Nankai University, Tianjin 300071, P. R. China
27 Peking University, Beijing 100871, P. R. China
28 Seoul National University, Seoul, 151-747 Korea
29 Shandong University, Jinan 250100, P. R. China
30 Shanxi University, Taiyuan 030006, P. R. China
31 Sichuan University, Chengdu 610064, P. R. China
32 Soochow University, Suzhou 215006, China
33 Sun Yat-Sen University, Guangzhou 510275, P. R. China
34 Tsinghua University, Beijing 100084, P. R. China
35 (A)Ankara University, Ankara, Turkey; (B)Dogus University, Istanbul,
Turkey; (C)Uludag University, Bursa, Turkey
36 Universitaet Giessen, 35392 Giessen, Germany
37 University of Hawaii, Honolulu, Hawaii 96822, USA
38 University of Minnesota, Minneapolis, MN 55455, USA
39 University of Rochester, Rochester, New York 14627, USA
40 University of Science and Technology of China, Hefei 230026, P. R. China
41 University of South China, Hengyang 421001, P. R. China
42 University of the Punjab, Lahore-54590, Pakistan
43 (A)University of Turin, Turin, Italy; (B)University of Eastern Piedmont,
Alessandria, Italy; (C)INFN, Turin, Italy
44 Wuhan University, Wuhan 430072, P. R. China
45 Zhejiang University, Hangzhou 310027, P. R. China
46 Zhengzhou University, Zhengzhou 450001, P. R. China
a also at the Moscow Institute of Physics and Technology, Moscow, Russia
b on leave from the Bogolyubov Institute for Theoretical Physics, Kiev,
Ukraine
c Nankai University, Tianjin,300071,China
d also at the PNPI, Gatchina, Russia
e now at Nagoya University, Nagoya, Japan
f Guangxi University,Nanning,530004,China
###### Abstract
The process $\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$ has been
studied with a data sample of $106\pm 4$ million $\psi(3686)$ events collected
with the BESIII detector at the BEPCII storage ring. The mass and width of the
$P$-wave charmonium spin-singlet state $h_{c}(^{1}P_{1})$ are determined by
simultaneously fitting distributions of the $\pi^{0}$ recoil mass for 16
exclusive $\eta_{c}$ decay modes. The results, $M(h_{c})=3525.31\pm
0.11~{}{\rm(stat.)}\pm 0.14~{}{\rm(syst.)}$ MeV/$c^{2}$ and
$\Gamma(h_{c})=0.70\pm 0.28\pm 0.22$ MeV, are consistent with and more precise
than previous measurements. We also determine the branching ratios for the 16
exclusive $\eta_{c}$ decay modes, five of which have not been measured
previously. New measurements of the $\eta_{c}$ line-shape parameters in the
$E1$ transition $h_{c}\to\gamma\eta_{c}$ are made by selecting candidates in
the $h_{c}$ signal sample and simultaneously fitting the hadronic mass spectra
for the 16 $\eta_{c}$ decay channels. The resulting $\eta_{c}$ mass and width
values are $M(\eta_{c})=2984.49\pm 1.16\pm 0.52$ MeV/$c^{2}$ and
$\Gamma(\eta_{c})=36.4\pm 3.2\pm 1.7$ MeV.
###### pacs:
14.40.Pq, 13.25.Gv, 12.38.Qk
## I INTRODUCTION
Studies of charmonium states have played an important role in understanding
Quantum Chromodynamics (QCD) because of their relative immunity from
complications like relativistic effects and the large value of the strong
coupling constant $\alpha_{s}$. In the QCD potential model Eichten:1978tg ,
the spin-independent one-gluon exchange part of the $c\bar{c}$ interaction has
been defined quite well by existing experimental data. The spin dependence of
the $c\bar{c}$ potential is not as well understood. Until recently, the only
well-measured hyperfine splitting was that for the $1S$ states of charmonium,
$\Delta M_{hf}(1S)=M(J/\psi)-M(\eta_{c})=116\pm 1$ MeV/$c^{2}$ ref:PDG_2012 .
In the past several years BaBar etac_prime_babar , Belle etac_prime_belle ,
CLEO etac_prime_cleo , and BESIII etac_prime_bes have succeeded in
identifying $\eta_{c}(2S)$ and have measured $\Delta
M_{hf}(2S)=M(\psi(3686))-M(\eta_{c}(2S))=47\pm 1$ MeV/$c^{2}$.
Of the charmonium states below $D\bar{D}$ threshold, the $h_{c}(1^{1}P_{1})$
is experimentally the least accessible because it cannot be produced directly
in $e^{+}e^{-}$ annihilation or in the electric-dipole transition of a
$J^{PC}=1^{--}$ charmonium state. Limited statistics and photon-detection
challenges also were major obstacles to the observation of $h_{c}$ in
charmonium transitions. The precise measurement of $h_{c}$ properties is
important because a comparison of its mass with the masses of the $3P$ states
($\chi_{cJ}$) provides much-needed information about the spin dependence of
the $c\bar{c}$ interaction. According to QCD potential models, a small spin-
spin interaction would lead to hyperfine splitting of the $P$ states that is
close to zero swanson .
The first evidence of the $h_{c}$ state was reported by the Fermilab E760
experiment ref:E760hc and was based on the process
$p\bar{p}\to\pi^{0}J/\psi$. This result was subsequently excluded by the
successor experiment E835 ref:E835hc , which investigated the same reaction
with a larger data sample. E835 also studied $p\bar{p}\to
h_{c}\to\gamma\eta_{c}$, in this case finding an $h_{c}$ signal. Soon after
this the CLEO collaboration observed the $h_{c}$ and measured its mass
ref:cleohc05 ; ref:cleohc08 by studying the decay chain
$\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$ in $e^{+}e^{-}$ collisions.
CLEO subsequently presented evidence for $h_{c}$ decays to multi-pion final
states ref:cleohc09 . Recently, the BESIII collaboration used inclusive
methods to make the first measurements of the absolute branching ratios
$\mathcal{B}(\psi(3686)\to\pi^{0}h_{c})=(8.4\pm 1.3\pm 1.0)\times 10^{-4}$ and
$\displaystyle\mathcal{B}(h_{c}\to\gamma\eta_{c})=(54.3\pm 6.7\pm 5.2)\%$
ref:bes3hc10 . CLEO has confirmed the BESIII results Ge:2011kq and also
observed $h_{c}$ in $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ at $\sqrt{s}=4170$ MeV,
demonstrating a new prolific source of $h_{c}$ CLEO:2011aa .
$\eta_{c}(1S)$ is the lowest-lying $S$-wave spin-singlet charmonium state.
Although it has been known for about thirty years Himel:1980dj , its resonance
parameters are still interesting. For a long time, measurements of the
$\eta_{c}$ width from B-factories and from charmonium transitions were
inconsistent ref:PDG_2012 . The discrepancies can be attributed to poor
statistics and inadequate consideration of interference between $\eta_{c}$
decays and non-resonant backgrounds. Besides, the $\eta_{c}$ line shape also
could be distorted by photon energy dependence in the $M1$(or $E1$)
transition, which will affect the resonance parameters measurement. Recent
studies by Belle, Babar, CLEO, and BESIII Vinokurova:2011dy ;
delAmoSanchez:2011bt ; Mitchell:2008aa ; BESIII:2011ab , with large data
samples and careful consideration of interference, obtained similar $\eta_{c}$
width and mass results in two-photon-fusion production and $\psi(3686)$
decays. The $h_{c}\to\gamma\eta_{c}$ transition can provide a new laboratory
to study $\eta_{c}$ properties. The $\eta_{c}$ line shape in the $E1$
transition $h_{c}\to\gamma\eta_{c}$ should not be as distorted as in other
charmonium decays, because non-resonant interfering backgrounds to the
dominant transition are small.
In this paper, we report new measurements of the mass and width of the $h_{c}$
and $\eta_{c}$, and of the branching ratios
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$ and $\mathcal{B}_{3}(\eta_{c}\to X_{i})$, via the sequential process
$\psi(3686)\to\pi^{0}h_{c},~{}h_{c}\to\gamma\eta_{c},~{}\eta_{c}\to X_{i}$. In
this reaction $X_{i}$ signifies 16 exclusive hadronic final states:
$p\bar{p}$, $2(\pi^{+}\pi^{-})$, $2(K^{+}K^{-})$, $K^{+}K^{-}\pi^{+}\pi^{-}$,
$p\bar{p}\pi^{+}\pi^{-}$, $3(\pi^{+}\pi^{-})$, $K^{+}K^{-}2(\pi^{+}\pi^{-})$,
$K^{+}K^{-}\pi^{0}$, $p\bar{p}\pi^{0}$, $K^{0}_{S}K^{\pm}\pi^{\mp}$,
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$, $\pi^{+}\pi^{-}\eta$,
$K^{+}K^{-}\eta$, $2(\pi^{+}\pi^{-})\eta$, $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$, and
$2(\pi^{+}\pi^{-})\pi^{0}\pi^{0}$. Here $K_{S}^{0}$ is reconstructed in its
$\pi^{+}\pi^{-}$ decays, and $\eta$ in its $\gamma\gamma$ final state. The
data sample of $\psi(3686)$ events was collected with the BESIII detector at
the BEPCII $e^{+}e^{-}$ storage ring.
The remainder of this paper is structured as follows: Sect. II describes the
experiment and data sample; Sect. III presents the event selection and
background analysis; Sect. IV discusses the extraction of $h_{c}$ and
$\eta_{c}$ results; Sect. V describes the estimation of systematic
uncertainties; and Sec. VI provides a summary and discussion of the results.
## II Experiment and Data Sample
BEPCII is a two-ring $e^{+}e^{-}$ collider designed for a peak luminosity of
$10^{33}$ cm${}^{-2}s^{-1}$ at a beam current of 0.93 A per beam. The
cylindrical core of the BESIII detector consists of a helium-gas-based drift
chamber (MDC) for charged-particle tracking and particle identification by
d$E$/d$x$, a plastic scintillator time-of-flight system (TOF) for additional
particle identification, and a 6240-crystal CsI(Tl) Electromagnetic
Calorimeter (EMC) for electron identification and photon detection. These
components are all enclosed in a superconducting solenoidal magnet providing a
1.0-T magnetic field. The solenoid is supported by an octagonal flux-return
yoke with resistive-plate-counter muon detector modules (MUC) interleaved with
steel. The geometrical acceptance for charged tracks and photons is $93\%$ of
$4\pi$, and the resolutions for charged-track momentum and photon energy at 1
GeV are $0.5\%$ and $2.5\%$, respectively. More details on the features and
capabilities of BESIII are provided in Ref. ref:bes3 .
The data sample for this analysis consists of 156.4 pb-1 of $e^{+}e^{-}$
annihilation data collected at a center-of-mass energy of 3.686 GeV, the peak
of the $\psi(3686)$ resonance. By measuring the production of multihadronic
events we determine the number of $\psi(3686)$ decays in the sample to be
$(1.06\pm 0.04)\times 10^{8}$, where the uncertainty is dominated by
systematics ref:psiptotnumber . An additional 42 $\rm pb^{-1}$ of data were
collected at a center-of-mass energy of 3.65 GeV to determine non-resonant
continuum background contributions.
The optimization of the event selection and the estimation of physics
backgrounds are performed with simulated Monte Carlo (MC) samples. A
GEANT4-based Agostinelli:2002hh ; Allison:2006ve detector simulation package
is used to model the detector response. Signal and background processes are
generated with specialized models that have been packaged and customized for
BESIII ref:bes3gen . The $\psi(3686)$ resonance is generated by KKMC ref:kkmc
, and EvtGen ref:evtgen is used to model events for
$\psi(3686)\to\pi^{0}h_{c}$ and for exclusive backgrounds in $\psi(3686)$
decays. An inclusive sample (100 million events) is used to simulate hadronic
background processes. Known $\psi(3686)$ decay modes are generated with
EvtGen, using branching ratios set to world-average values ref:PDG_2012 . The
remaining $\psi(3686)$ decay modes are generated by LUNDCHARM ref:bes3gen ,
which is based on JETSET Sjostrand:1993yb and tuned for the charm-energy
region. The decays $\psi(3686)\to\pi^{0}h_{c}$ are excluded from this sample.
The $\psi(3686)\to\pi^{0}h_{c}$ events are generated with an $h_{c}$ mass of
$3525.28$ MeV/$c^{2}$ and a width equal to that of the $\chi_{c1}$ (0.9 MeV).
The $E1$ transition $h_{c}\to\gamma\eta_{c}$ is generated with an angular
distribution in the $h_{c}$ rest frame of $1+\cos^{2}\theta^{*}$, where
$\theta^{*}$ is the angle of the E1 photon with respect to the beam direction
in the $h_{c}$ rest frame. Multi-body $\eta_{c}$ decays are generated
according to phase space.
## III Event selection and background analysis
For $\psi(3686)\to\pi^{0}h_{c}$, $h_{c}\to\gamma\eta_{c}$, the expected
$\pi^{0}$ momentum is $P_{\pi^{0}}\simeq 84$ MeV/$c$, and the $E1$ transition
photon emitted in $h_{c}\to\gamma\eta_{c}$ has an expected energy of
$E(\gamma_{\rm E1})\simeq 503$ MeV in the $h_{c}$ rest frame. We select signal
candidates by demanding consistency with these expectations. For the selected
candidates we then compute the $\pi^{0}$ recoil mass. Fitting the distribution
of this recoil mass for the full event sample gives the results for the
$h_{c}$ resonance parameters and signal yields.
Charged tracks in BESIII are reconstructed from MDC hits within a polar-angle
($\theta$) acceptance range of $|\cos\theta|<0.93$. To optimize the momentum
measurement, we require that these tracks be reconstructed to pass within 10
cm of the interaction point in the beam direction and within 1 cm in the plane
perpendicular to the beam. Tracks used in reconstructing $K^{0}_{S}$ decays
are exempted from these requirements.
A vertex fit constrains charged tracks to a common production vertex, which is
updated on a run-by-run basis. For each charged track, time-of-flight and
d$E$/d$x$ information is combined to compute particle identification (PID)
confidence levels for the pion, kaon, and proton hypotheses. The track is
assigned to the particle type with the highest confidence level.
Electromagnetic showers are reconstructed by clustering EMC crystal energies.
Efficiency and energy resolution are improved by including energy deposits in
nearby TOF counters. A photon candidate is defined as a shower with an energy
deposit of at least 25 MeV in the “barrel” region ($|\cos\theta|<0.8$), or of
at least 50 MeV in the “end-cap” region ($0.86<|\cos\theta|<0.92$). Showers at
angles intermediate between the barrel and the end-cap are not well measured
and are rejected. An additional requirement on the EMC hit timing suppresses
electronic noise and energy deposits unrelated to the event.
A candidate $\pi^{0}$($\eta$) is reconstructed from pairs of photons with an
invariant mass in the range $|M_{\gamma\gamma}-m_{\pi^{0}}|<15\,\rm MeV/c^{2}$
($|M_{\gamma\gamma}-m_{\eta}|<15\,\rm MeV/c^{2}$) ref:PDG_2012 . A one-
constraint (1-C) kinematic fit is performed to improve the energy resolution,
with the $M(\gamma\gamma)$ constrained to the known $\pi^{0}$($\eta$) mass.
We reconstruct $K^{0}_{S}\to\pi^{+}\pi^{-}$ candidates using pairs of
oppositely charged tracks with an invariant mass in the range
$|M_{\pi\pi}-m_{K^{0}_{S}}|<20\,\rm MeV/c^{2}$, where $m_{K^{0}_{S}}$ is the
known $K^{0}_{S}$ mass ref:PDG_2012 . To reject random $\pi^{+}\pi^{-}$
combinations, a secondary-vertex fitting algorithm is employed to impose the
kinematic constraint between the production and decay vertices
ref:ks0-reconstruction . Accepted $K^{0}_{S}$ candidates are required to have
a decay length of at least twice the vertex resolution.
The $\eta_{c}$ candidate is reconstructed in 16 exclusive decay modes, and the
event is accepted or rejected based on consistency with the
$h_{c}\to\gamma\eta_{c}$ hypothesis. Specifically, the reconstructed mass
$M(\eta_{c})$ is required to be between 2.900 GeV/$c^{2}$ and 3.050
GeV/$c^{2}$, and the transition-photon energy is required to be between 0.450
GeV and 0.550 GeV. Events passing this selection are subjected to a
4-constraint (4-C) kinematic fit to take advantage of energy-momentum
conservation. Because of differing signal/background characteristics, we
individually optimize requirements on $\chi^{2}_{4C}$, the $\chi^{2}$ of the
4-C fit, for the 16 $\eta_{c}$ channels. If multiple $\eta_{c}$ candidates are
found in an event, the one with the smallest value of $\chi^{2}=\chi^{2}_{\rm
4C}+\chi^{2}_{\rm 1C}+\chi^{2}_{\rm pid}+\chi^{2}_{\rm vertex}$ is accepted,
where $\chi^{2}_{\rm 1C}$ is the $\chi^{2}$ of the 1-C fit of the
$\pi^{0}$($\eta$), $\chi^{2}_{\rm pid}$ is the PID $\chi^{2}$ summation for
all charged tracks included in the $h_{c}$ candidate, and $\chi^{2}_{\rm
vertex}$ is the $\chi^{2}$ of the $K_{S}^{0}$ vertex fit. If there is no
$\pi^{0}/\eta$ ($K_{S}^{0}$) in an event, the corresponding $\chi^{2}_{\rm
1C}$ ($\chi^{2}_{\rm vertex}$) is set to zero.
Based on studies of the inclusive MC sample, we identified several background
processes with potential to reduce the precision of measurements made with
specific $\eta_{c}$ exclusive channels because of sizable low-energy $\pi^{0}$
production. The processes and suppression procedures are as follows:
* •
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$
The mass $M_{X}$ of the system recoiling against the $\pi^{+}\pi^{-}$ in
$\psi(3686)\to\pi^{+}\pi^{-}X$ is calculated and the candidate is rejected if
$M_{X}$ is within $\pm 12$ MeV/$c^{2}$ of the known $J/\psi$ mass.
* •
$\psi(3686)\to\pi^{0}\pi^{0}J/\psi$
The mass $M_{X}$ of the system recoiling against the $\pi^{0}\pi^{0}$ in
$\psi(3686)\to\pi^{0}\pi^{0}X$ is calculated and the candidate is rejected if
$M_{X}$ is within $\pm 15$ MeV/$c^{2}$ of the known $J/\psi$ mass for all
$\eta_{c}$ final states except $\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$.
For this mode the lower $\pi^{0}$ momentum leads to recoil masses near 3.1
GeV/$c^{2}$, so the exclusion window is narrowed to $\pm 10$ MeV/$c^{2}$.
* •
$\psi(3686)\to\gamma\chi_{c2}$
A candidate is rejected if it includes a $\pi^{0}$ for which either daughter
photon has an energy within $\pm 5$ MeV of that expected for the $\psi(3686)$
radiative transition to $\chi_{c2}$ (128 MeV).
* •
$E1$ photon candidates that are $\pi^{0}$ decay products
A candidate is rejected if its $E1$ photon can be combined with another photon
in the event to form a $\pi^{0}$ within a mass window of $\pm 10$ MeV/$c^{2}$.
* •
$\pi^{0}$ candidates that are from $\eta\to\pi^{+}\pi^{-}\pi^{0}$
Masses $M(\pi^{+}\pi^{-}\pi^{0})$ are calculated for all possible combinations
in the event and the candidate is rejected if any combination has a mass
within $\pm 15$ MeV/$c^{2}$ of the known $\eta$ mass.
Decisions about whether to apply a requirement to a particular $\eta_{c}$ mode
and the optimization of the $\chi^{2}_{4C}$ and PID requirements were made on
a channel-by-channel basis. The figure-of-merit used was
$\mathcal{S}=N_{S}/\sqrt{N_{S}+N_{B}}$, where $N_{S}$ is the number of signal
and $N_{B}$ the number of background candidates. PDG values ref:PDG_2012 are
used for the input $\eta_{c}$ branching ratios, and for channels not tabulated
by the PDG we estimate branching ratios based on conjugate channels or other
similar modes. The optimized selection criteria are listed in Table 1, in
which the $N(p),~{}N(\pi)~{}{\rm and}~{}N(K)$ denote the numbers of identified
protons, pions and kaons in a event.
Table 1: Event-selection requirements for each exclusive channel.
Mode | $\chi^{2}_{\rm~{}4C}$ | PID | $\pi^{+}\pi^{-}J/\psi$ veto | $\pi^{0}\pi^{0}J/\psi$ veto | $\gamma\chi_{c2}$ veto | $\pi^{0}$ veto for $E1$ photon | $\eta\to\pi^{+}\pi^{-}\pi^{0}$ veto
---|---|---|---|---|---|---|---
$p\bar{p}$ | 30 | $N(p)\geq 1$ | no | no | yes | no | no
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 60 | $N(\pi)\geq 3$ | yes | yes | yes | yes | yes
$K^{+}K^{-}K^{+}K^{-}$ | 60 | $N(K)\geq 3$ | no | no | no | yes | no
$K^{+}K^{-}\pi^{+}\pi^{-}$ | 40 | $N(K)\geq 2,N(\pi)\geq 0$ | yes | yes | yes | yes | yes
$p\bar{p}\pi^{+}\pi^{-}$ | 30 | $N(p)\geq 2,N(\pi)\geq 0$ | yes | yes | yes | yes | yes
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 50 | $N(\pi)\geq 4$ | yes | yes | no | yes | yes
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 70 | $N(K)\geq 2,N(\pi)\geq 2$ | yes | no | no | no | no
$K^{+}K^{-}\pi^{0}$ | 50 | $N(K)\geq 1$ | no | yes | no | no | no
$p\bar{p}\pi^{0}$ | 40 | $N(p)\geq 1$ | no | yes | yes | yes | no
$K^{0}_{S}K^{\pm}\pi^{\mp}$ | 70 | $-$ | no | no | no | no | yes
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | 50 | $-$ | no | no | yes | no | no
$\pi^{+}\pi^{-}\eta$ | 50 | $-$ | no | no | no | yes | no
$K^{+}K^{-}\eta$ | 70 | $N(K)\geq 1$ | no | no | yes | yes | no
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\eta$ | 30 | $-$ | yes | no | no | yes | no
$\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 40 | $-$ | yes | yes | yes | yes | yes
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 60 | $-$ | yes | yes | no | yes | no
The $\pi^{0}$ recoil mass spectra for events passing these requirements show
clear $h_{c}$ signals in the expected range, as can be seen in Fig. 1. No
peaking backgrounds in the signal region are found in the 100-million-event
inclusive MC sample, in the continuum data sample taken at $\sqrt{s}=3.65$
GeV, or in $\eta_{c}$-candidate-mass side-band distributions.
Figure 1: The $\pi^{0}$ recoil mass spectrum in
$\psi(3686)\to\pi^{0}{h}_{c},{h}_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$
summed over the 16 final states $X_{i}$. The dots with error bars represent
the $\pi^{0}$ recoil mass spectrum in data. The solid line shows the total fit
function and the dashed line is the background component of the fit.
## IV EXTRACTION OF YIELDS AND RESONANCE PARAMETERS
We obtain the $h_{c}$ mass, width and branching ratios from simultaneous fits
to the $\pi^{0}$ recoil mass distributions for the 16 exclusive $\eta_{c}$
decay modes. Here only 1-C kinematic fits with $\pi^{0}$ mass hypothesis are
used to improve the energy resolution. The 4C-fits used in event selection are
not used in the $\pi^{0}$ recoil mass reconstruction, because the energy
resolution of the signal $\pi^{0}$ in 4C-fits is not as good as in the
1C-fits, according to a MC study. From the same data sample we also determine
the $\eta_{c}$ resonance parameters by fitting the 16 invariant-mass spectra
of the hadronic system accompanying the transition photon in
$h_{c}\to\gamma\eta_{c}$.
### IV.1 Fitting the $h_{c}$ signal
To extract the $h_{c}$ resonant parameters and the yield for each $\eta_{c}$
decay channel, the 16 $\pi^{0}$ recoil mass distributions are fitted
simultaneously with a binned maximum likelihood method. A Breit-Wigner
function convolved with the instrumental resolution is used to describe the
signal shape. An efficiency correction is not needed because of the small
$h_{c}$ width and the good $\pi^{0}$ mass resolution. The resolution function
is channel-dependent and is obtained from MC simulation. The parameters
$M(h_{c})$ and $\Gamma(h_{c})$ of the Breit-Wigner function are constrained to
be the same for all 16 channels, which is essential for the decay modes with
low statistics. For the recoil mass fit to each channel, the background shape
is obtained from the $\eta_{c}$ mass side bands (2300$-$2700, 3070$-$3200
MeV/$c^{2}$), and the signal and the background normalizations for each mode
are allowed to float. The summed and mode-by-mode fit results are shown in
Figs. 1 and 2, respectively. The $\chi^{2}$ per degree of freedom for this fit
is 1.60, where sparsely populated bins are combined so that there are at least
seven counts per bin in the $\chi^{2}$ calculation. The parameters of the
$h_{c}$ resonance are determined to be $M(h_{c})=3525.31\pm 0.11$ MeV/$c^{2}$
and $\Gamma(h_{c})=0.70\pm 0.28$ MeV, where the errors are statistical only.
Figure 2: The simultaneously fitted $\pi^{0}$ recoil mass spectra in
$\psi(3686)\to\pi^{0}{h}_{c},{h}_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$
for the 16 final states $X_{i}$.
The MC-determined selection efficiency $\epsilon_{i}$ and yield $N_{i}$ for
each $\eta_{c}$ decay mode are listed in Table 2.
Table 2: MC-determined efficiencies $\epsilon_{i}$ and yields $N_{i}$ for $\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c},\eta_{c}\to X_{i}$, where $X_{i}$ refers to the 16 final states . Mode | $\epsilon_{i}(\%)$ | $N_{i}$
---|---|---
$p\bar{p}$ | 22.2 | 15.3 $\pm$ 4.5
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 12.6 | 100.3 $\pm$ 11.3
$K^{+}K^{-}K^{+}K^{-}$ | 6.6 | 6.6 $\pm$ 2.6
$K^{+}K^{-}\pi^{+}\pi^{-}$ | 8.7 | 38.4 $\pm$ 7.0
$p\bar{p}\pi^{+}\pi^{-}$ | 7.8 | 19.0 $\pm$ 5.4
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 5.4 | 50.5 $\pm$ 9.0
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 2.7 | 10.3 $\pm$ 4.9
$K^{+}K^{-}\pi^{0}$ | 11.4 | 54.9 $\pm$ 9.2
$p\bar{p}\pi^{0}$ | 8.9 | 14.4 $\pm$ 4.6
$K^{0}_{S}K^{\pm}\pi^{\mp}$ | 8.9 | 107.1 $\pm$ 11.8
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | 3.4 | 43.3 $\pm$ 8.0
$\pi^{+}\pi^{-}\eta$ | 4.3 | 32.9 $\pm$ 6.7
$K^{+}K^{-}\eta$ | 3.0 | 6.7 $\pm$ 3.2
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\eta$ | 1.9 | 38.6 $\pm$ 7.6
$\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 5.5 | 118.4 $\pm$ 12.8
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 2.2 | 175.2 $\pm$ 17.3
Total | - | 831.9$\pm$35.0
Based on these numbers, we can calculate the product branching ratios
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$. The branching ratio for $\eta_{c}\to X_{i}$ for each of the 16 final
states $X_{i}$ can then be obtained by combining our measurements with
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})=(4.36\pm
0.42)\times 10^{-4}$, the average of two recent measurements by CLEO
ref:cleohc08 and BESIII ref:bes3hc10 . These branching ratios, with both
statistical and systematic errors, are presented in Section VI.
### IV.2 Measurement of $\eta_{c}$ resonance parameters
In addition to determining the $h_{c}$ resonance parameters, we can also
measure the $\eta_{c}$ mass and width with the same event sample. The decay
chain $h_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$ is reconstructed and
kinematically fitted in the 16 $\eta_{c}$ final states $X_{i}$. For candidates
with satisfactory kinematic fits, we use the resulting track and photon
momenta to compute the hadronic mass. We populate distributions of this
hadronic mass by removing our previous $E1$ photon-energy and $M(\eta_{c})$
requirements and selecting candidates inside a $\pi^{0}$ recoil mass window of
$\pm 5\,\rm MeV/c^{2}$ around the $h_{c}$ mass, keeping all other criteria
unchanged.
The line shape for the $\eta_{c}$ signal for these fits is parameterized as
$(E_{\gamma}^{3}\times BW(m)\times f_{d}(E_{\gamma}))\otimes R_{i}(m)$, where
$BW(m)$ is the Breit-Wigner function for $\eta_{c}$ as a function of the
invariant mass $m$ of the decay products for each channel,
$E_{\gamma}(m)=\frac{M(h_{c})^{2}-m^{2}}{2M(h_{c})}$ is the energy of the
transition photon in the rest frame of $h_{c}$, and $f_{d}(E_{\gamma})$ is a
function that damps the divergent tail due to the $E_{\gamma}^{3}$ factor,
which incorporates the energy dependence of the $E1$ matrix element and the
phase-space factor. $R_{i}(m)$ is the signal resolution function for the $i$th
decay mode, which is parameterized by double Gaussians to account for the
distorting effects of the kinematic fit and detector smearing. The damping
function that we use was introduced by the KEDR collaboration ref:kedretac :
$f_{d}(E_{\gamma})=\frac{E_{0}^{2}}{E_{\gamma}E_{0}+(E_{\gamma}-E_{0})^{2}},$
where $E_{0}=E_{\gamma}(m_{\eta_{c}})$ is the $E1$-transition-photon peak
energy. The $\eta_{c}$-candidate hadronic invariant mass spectra from low and
high side bands in the $h_{c}$ mass (3500$-$3515, 3535$-$3550 MeV/$c^{2}$) are
used to obtain the background functions for the $\eta_{c}$ mass fit. To
mitigate the effects of bin-to-bin fluctuations, these side-band mass spectra
are smoothed before fitting. A toy MC study was performed to test the effect
of the smoothing and it was demonstrated to be a robust procedure that does
not systematically distort the fit results. The channel-by-channel signal and
background normalizations are free parameters determined by the fit.
We ignore the effect of interference between the signal and background, which
was considered in the previous measurement of $\psi(3686)\to\gamma\eta_{c}$
BESIII:2011ab , because the branching ratio of $h_{c}\to\gamma\eta_{c}$ is
about $50\%$ (branching ratio of $M1$ transition $\psi(3686)\to\gamma\eta_{c}$
is about $0.3\%$). The radiative decay of $h_{c}\to\gamma 0^{-}$ should be the
same level of $\psi(3686)\to\gamma 0^{-}$, in this case, the non-$\eta_{c}$
intensity in $h_{c}$ is much smaller than that for
$\psi(3686)\to\gamma\eta_{c}$.
Figs. 3 and 4 show the hadronic-mass-fit results. The $\eta_{c}$ mass and
width are determined to be $M(\eta_{c})=2984.49\pm 1.16$ MeV/$c^{2}$ and
$\Gamma(\eta_{c})=36.4\pm 3.2$ MeV, where the errors are statistical. The
$\chi^{2}$ per degree of freedom for this fit is 1.52, using the same
$\chi^{2}$ calculation method to accommodate low-statistics bins as for the
fit to the $\pi^{0}$ recoil mass spectrum.
(a)
(b)
Figure 3: (a) The hadronic mass spectrum in
$\psi(3686)\to\pi^{0}{h}_{c},{h}_{c}\to\gamma\eta_{c}$, $\eta_{c}\to X_{i}$
summed over the 16 final states $X_{i}$. The dots with error bars represent
the hadronic mass spectrum in data. The solid line shows the total fit
function and the dashed line is the background component of the fit. (b) The
background-subtracted hadronic mass spectrum with the signal shape overlaid.
Figure 4: The simultaneously fitted hadronic mass spectra for the 16
$\eta_{c}$ decay channels.
## V Systematic Uncertainties
### V.1 $h_{c}$ parameter measurements
The systematic uncertainties for the $M(h_{c})$ and $\Gamma(h_{c})$
measurements are summarized in Table 3. All sources are treated as
uncorrelated, so the total systematic uncertainty is obtained by summing them
in quadrature. The following subsections describe the procedures and
assumptions that led to these estimates of the uncertainties.
Table 3: The systematic errors for the $h_{c}$ mass and width measurements. Sources | $\Delta M_{h_{c}}$ (MeV$/c^{2}$) | $\Delta\Gamma_{h_{c}}$ (MeV)
---|---|---
Energy calibration | 0.13 | 0.07
Signal shape | 0.00 | 0.06
Fitting range | 0.04 | 0.16
Binning | 0.02 | 0.01
Background shape | 0.01 | 0.08
Background veto | 0.01 | 0.08
Kinematic fit | 0.03 | 0.03
Mass of $\psi(3686)$ | 0.03 | 0.02
Total | 0.14 | 0.22
#### V.1.1 Energy calibration
The potential inconsistency of the photon-energy measurement between data and
MC is evaluated by studying
$\psi(3686)\to\gamma\chi_{c1,2}~{}(~{}\chi_{c1,2}\to\gamma
J/\psi,J/\psi\to\mu^{+}\mu^{-}$) for photons with low energy and radiative
Bhabha events for photons with high energy. Discrepancies of $0.4\%$ in the
energy scale and $4\%$ in the energy resolution between data and MC are found.
We vary the photon response accordingly and take the changes in the results as
the estimated systematic error. For the $M(h_{c})$ measurement, besides the
above studies, the reconstructed photon position and error matrix are taken
into account as additional sources of uncertainty.
#### V.1.2 Signal shape
The uncertainty associated with the $h_{c}$ signal shape in the $\pi^{0}$
recoil mass spectrum includes contributions from the photon line shape and the
1-C kinematic fit. We estimate these by determining the changes in results
after reasonable adjustments in the photon response. The photon-energy
resolution is estimated with the control sample
$\psi(3686)\to\gamma\chi_{c2}$. As above, the energy resolution in data is
found to be about 4% worse than in the MC simulation. We correct for this
discrepancy by adding single-Gaussian smearing to the energy of the $\pi^{0}$
daughter photons and then using the alternative $\pi^{0}$ shape to redo the
fit. The changes in results are assigned as the systematic errors.
#### V.1.3 Fitting range and binning
The systematic uncertainties due to the fitting of the $\pi^{0}$ recoil mass
spectrum are evaluated by varying the fitting range and the bin size in the
fit. The spreads of results obtained with the alternative assumptions are used
to assign the systematic errors.
#### V.1.4 Background shape
To estimate the uncertainty associated with the side-band method for assigning
background function shapes, we use an ARGUS function Albrecht:1990am as an
alternative background description for each channel and record the changes in
the fit results.
#### V.1.5 Background veto
The systematic uncertainties associated with the requirements to suppress
background are estimated by varying the excluded ranges.
#### V.1.6 Kinematic fit
Systematic uncertainties caused by the kinematic fit are studied by tuning the
tracking parameters and error matrices of charged tracks and photons based on
the data. Control samples of $J/\psi\to\phi f_{0}(980),\phi\to
K^{+}K^{-},f_{0}(980)\to\pi^{+}\pi^{-}$, and $\psi(3686)\to\gamma\chi_{cJ}$
are used for this purpose Ablikim:2012pg . Channel-by-channel changes of
$M(\eta_{c})$ and $\Gamma(\eta_{c})$ are calculated after the tuning and then
averaged by yields and taken as systematic errors.
#### V.1.7 $\psi(3686)$ mass
The systematic uncertainties of the $M(h_{c})$ and $\Gamma(h_{c})$
determinations associated with the uncertainty in the $\psi(3686)$ mass are
estimated to be 0.03 MeV/$c^{2}$ and 0.02 MeV, respectively. These are found
by shifting $M_{\psi(3686)}$ by one standard deviation according to the PDG
value ref:PDG_2012 and redetermining the results.
### V.2 $\eta_{c}$ branching ratio measurements
The systematic errors in the $\eta_{c}$ branching ratio measurements are
listed in Tables 4. All sources are treated as uncorrelated, so the total
systematic uncertainty is obtained by summing them in quadrature. The
following subsections describe the procedures and assumptions that led to the
estimates of these uncertainties.
Table 4: The systematic errors (in %) in the $\eta_{c}$ branching ratio
measurements of the $\eta_{c}$ exclusive decay channels.
Sources | $p\bar{p}$ | $2(\pi^{+}\pi^{-})$ | $2(K^{+}K^{-})$ | $K^{+}K^{-}\pi^{+}\pi^{-}$ | $p\bar{p}\pi^{+}\pi^{-}$ | $3(\pi^{+}\pi^{-})$ | $K^{+}K^{-}2(\pi^{+}\pi^{-})$ | $K^{+}K^{-}\pi^{0}$
---|---|---|---|---|---|---|---|---
N($\psi(3686)$) | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0
Tracking | 4.0 | 8.0 | 8.0 | 8.0 | 8.0 | 12.0 | 12.0 | 4.0
PID ($K^{0}_{S}$) | 2.0 | 6.0 | 6.0 | 4.0 | 4.0 | 8.0 | 8.0 | 2.0
Photon eff | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0
Fit range | 2.2 | 1.2 | 2.6 | 2.9 | 1.5 | 5.3 | 3.3 | 2.7
Bkg shape | 10.3 | 2.5 | 4.7 | 0.9 | 0.3 | 0.2 | 3.5 | 2.8
Signal shape | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3
KmFit eff. | 7.0 | 6.3 | 7.0 | 8.8 | 10.8 | 7.3 | 4.2 | 2.0
Bkg veto | 5.9 | 5.5 | 1.1 | 0.6 | 3.1 | 2.3 | 5.2 | 1.7
Cross feed | 0.0 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.4
$\eta_{c}$ decay models | 0.0 | 2.1 | 3.7 | 0.6 | 2.5 | 0.0 | 3.0 | 4.6
$\eta_{c}$ line shape | 0.7 | 0.8 | 0.6 | 0.9 | 0.6 | 0.6 | 0.6 | 0.7
Sum | 15.7 | 14.8 | 14.9 | 14.1 | 15.7 | 18.0 | 17.8 | 10.6
Sources | $p\bar{p}\pi^{0}$ | $K^{0}_{S}K^{\pm}\pi^{\mp}$ | $K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | $\pi^{+}\pi^{-}\eta$ | $K^{+}K^{-}\eta$ | $2(\pi^{+}\pi^{-})\eta$ | $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | $2(\pi^{+}\pi^{-}\pi^{0})$
---|---|---|---|---|---|---|---|---
N($\psi(3686)$) | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0
Tracking | 4.0 | 8.0 | 12.0 | 4.0 | 4.0 | 8.0 | 4.0 | 8.0
PID ($K^{0}_{S}$) | 2.0 | 1.0 | 1.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0
Photon eff | 5.0 | 3.0 | 3.0 | 5.0 | 5.0 | 5.0 | 7.0 | 7.0
Fit range | 7.7 | 2.1 | 1.5 | 0.6 | 6.0 | 1.8 | 0.6 | 2.0
Bkg shape | 0.1 | 4.7 | 4.7 | 0.1 | 5.9 | 0.8 | 3.3 | 1.6
Signal shape | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3
KmFit eff. | 6.8 | 6.8 | 7.3 | 2.0 | 1.2 | 6.7 | 2.4 | 2.4
Bkg veto | 3.7 | 0.7 | 2.8 | 11.8 | 5.4 | 14.7 | 12.8 | 5.5
Cross feed | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.3 | 0.0
$\eta_{c}$ decay models | 5.8 | 2.5 | 5.2 | 5.5 | 8.1 | 0.0 | 0.1 | 0.5
$\eta_{c}$ line shape | 0.6 | 0.6 | 0.8 | 0.8 | 0.7 | 0.7 | 0.7 | 0.7
Sum | 14.8 | 13.2 | 17.0 | 15.4 | 15.3 | 19.4 | 16.4 | 13.3
#### V.2.1 Tracking and photon detection
The uncertainty in the tracking efficiency is 2% per track and the uncertainty
due to photon detection is 1% per photon Ablikim:2011kv . MC studies
demonstrate that the trigger efficiency for signal events is almost 100%, so
that the associated uncertainty in the results is negligible.
#### V.2.2 PID and $K^{0}_{S}$ reconstruction
The systematic uncertainties due to kaon and pion identifications are
determined to be 2% in Ref. Ablikim:2011kv . We choose $J/\psi\to
K^{*0}K^{0}_{S},K^{*0}\to K\pi$ to evaluate the efficiency of $K^{0}_{S}$
reconstruction. The $1\%$ difference between data and MC is assigned as the
systematic error due to this source.
#### V.2.3 Kinematic fitting
The systematic errors associated with kinematic fitting are estimated by using
the control samples of $\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ with $J/\psi$ decay
to hadronic final states, which have similar event topology as
$\psi(3686)\to\pi^{0}h_{c},h_{c}\to\gamma\eta_{c}$. The average efficiency
difference between data and MC, with the same $\chi^{2}$ requirements in the
$h_{c}$ selection, is taken as the systematic uncertainty.
#### V.2.4 Cross-feed
To evaluate the effect of cross-feed among the 16 signal modes, we use samples
of 50,000 MC events per mode. We find that $\eta_{c}\to
2(\pi^{+}\pi^{-}),\eta_{c}\to K^{+}K^{-}\pi^{0}$ and
$\eta_{c}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ are contaminated by $\eta_{c}\to
K^{0}_{S}K^{\pm}\pi^{\mp}$ with levels of $2.5\%$, $1.4\%$, and $1.3\%$,
respectively. These numbers are assigned as the systematic errors associated
with cross-feed. For other channels, this contamination is found to be
negligible.
#### V.2.5 $\eta_{c}$ decay models
We use phase space to simulate $\eta_{c}$ decays in our analysis. To estimate
the systematic uncertainty due to neglecting intermediate states in these
decays, we extract invariant masses of $\eta_{c}$ daughter particles from
$\psi(3686)\to\gamma\eta_{c},\eta_{c}\to X_{i}$. We analyze MC samples
generated according to these invariant masses. To illustrate, Fig. 5 shows the
invariant-mass distribution comparison between the data and MC for the decay
mode $\eta_{c}\to K^{0}_{S}K^{\pm}\pi^{\mp}$. In addition, for channels with
low statistics and well-understood intermediate states, MC samples with these
intermediate states were generated according to the relative branching ratios
given by PDG. The spreads of the efficiencies obtained from the phase-space
and alternative MC are taken as the systematic errors.
Figure 5: The dots show the mass spectra for
$\psi(3686)\to\gamma\eta_{c},\eta_{c}\to K^{0}_{S}K^{\pm}\pi^{\mp}$ in data,
and the solid lines are the corresponding mass spectra from the MC simulation.
#### V.2.6 $\eta_{c}$ line shape
Because of the $\eta_{c}$ mass window requirement in our event selection, the
line shape of $\eta_{c}$ could be a source of systematic error in the
measurement. We vary the input $\eta_{c}$ resonant parameters by one standard
deviation to estimate the uncertainty due to this source.
### V.3 $\eta_{c}$ parameter measurements
Systematic errors for the $M(\eta_{c})$ and $\Gamma(\eta_{c})$ measurements
are summarized in Table 5. All sources are treated as uncorrelated, so the
total systematic uncertainty is obtained by summing in quadrature. The
following subsections describe the procedures and assumptions that led to the
estimates of these uncertainties.
Table 5: The systematic errors for $\eta_{c}$ parameter measurements. Sources | M($\eta_{c}$) (MeV$/c^{2}$) | $\Gamma(\eta_{c})$ (MeV)
---|---|---
Background shape | $0.36$ | $1.45$
Fitting range | $0.03$ | $0.33$
Resolution description | $0.10$ | $0.02$
Mass-dependent efficiencies | $0.11$ | $0.27$
Mass-dependent resolutions | $0.00$ | $0.01$
Kinematic fitting | $0.33$ | $0.76$
Fitting method | $0.11$ | $0.40$
Sum | $0.52$ | $1.74$
#### V.3.1 Background shape
Our standard background shape is the smoothed $h_{c}$ side-band shape. To
estimate the systematic uncertainty due to the background procedure, we change
the smoothing level and technique, and vary the $h_{c}$ side-band ranges. The
largest changes in results among these alternatives are assigned as the
systematic errors.
#### V.3.2 Fitting range
The systematic uncertainties due to the fitting range are estimated by
considering several alternatives to the standard fitting range of 2.3-3.2 $\rm
GeV/c^{2}$, 2.4-3.2 $\rm GeV/c^{2}$, 2.5-3.2 $\rm GeV/c^{2}$, 2.6-3.2 $\rm
GeV/c^{2}$, and 2.3-3.15 $\rm GeV/c^{2}$. The systematic uncertainties are
assigned to be the largest differences between the standard fit results and
those from the alternative ranges.
#### V.3.3 Resolution description
In order to estimate the systematic uncertainties associated with the
detector-resolution description, we use MC signal shapes obtained by setting
the $\eta_{c}$ width to zero as alternatives to double Gaussians. The changes
in fit results between these two methods provide the systematic errors.
#### V.3.4 Mass-dependent efficiency and resolution
Since the $\eta_{c}$ signal spreads over a sizable mass range, the
uncertainties due to the use of mass-independent efficiencies and resolutions
need to be estimated. Mass-dependent efficiencies and resolutions are
determined from MC simulation and used as an alternative to the default
assumption, and the resulting differences are taken to be the systematic
errors.
#### V.3.5 Kinematic fitting
The method to evaluate the systematic errors due to the kinematic fitting
procedure and momentum measurement is the same as that in the measurement of
the $h_{c}$ parameters.
#### V.3.6 Fitting method
Because we use the smoothed side-band shape to describe the background, the
potential for bias due to the smoothing technique must be considered. This was
investigated with a toy MC study. We start with a signal sample for each of
the 16 channels selected from our standard MC to have the same statistics as
data. A corresponding background sample for each channel is constructed from
the mass side bands in data. The hadronic-mass distributions for these samples
are then treated with a variety of smoothing procedures and fitted. The ranges
in the fit results are used to set the systematic errors from this source.
## VI SUMMARY AND DISCUSSION
In summary, we have studied the process $\psi(3686)\to\pi^{0}h_{c}$ followed
by $h_{c}\to\gamma\eta_{c}$ with an exclusive-reconstruction technique. Using
a sample of 106 million $\psi(3686)$ decays we have obtained new measurements
of the mass and width of the $h_{c}$ and $\eta_{c}$ charmonium resonances, and
of the branching ratios for 16 exclusive $\eta_{c}$ hadronic decay modes.
The total yield of events, measured by fitting the $\pi^{0}$ recoil mass
spectrum, is $832\pm 35$ events, where the error is statistical only. With
these events we measure the mass and width of the $h_{c}$:
$\displaystyle M(h_{c})=3525.31\pm 0.11\pm 0.14\,\rm{MeV}/c^{2},~{}and$
$\displaystyle\Gamma(h_{c})=0.70\pm 0.28\pm 0.22\,\rm{MeV},$
where the first errors are statistical and the second are systematic. These
results are consistent with the results of a previous inclusive measurement by
BESIII ref:bes3hc10 :
$\displaystyle M(h_{c})=3525.40\pm 0.13\pm 0.18\,\rm{MeV}/c^{2},~{}and$
$\displaystyle\Gamma(h_{c})<1.44\,\rm{MeV}~{}(at~{}90\%~{}confidence~{}level).$
The branching-ratio results
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$ and $\mathcal{B}_{3}(\eta_{c}\to X_{i})$ are given in Table 6, quoted
with the statistical and systematic errors of this measurement and, for
$\mathcal{B}_{3}$, an additional systematic error associated with the input
branching-ratio product
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})$.
Most of our $\mathcal{B}_{3}(\eta_{c}\to X_{i})$ branching-fraction results
are consistent with PDG values ref:PDG_2012 , and several branching fractions
are measured for the first time.
Table 6:
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})\times\mathcal{B}_{3}(\eta_{c}\to
X_{i})$ and $\mathcal{B}_{3}(\eta_{c}\to X_{i})$ with systematic errors. The
third errors in $\mathcal{B}_{3}$ measurement are systematic errors due to
uncertainty of
$\mathcal{B}_{1}(\psi(3686)\to\pi^{0}h_{c})\times\mathcal{B}_{2}(h_{c}\to\gamma\eta_{c})$.
$X_{i}$ | $\mathcal{B}_{1}\times\mathcal{B}_{2}\times\mathcal{B}_{3}~{}(\times 10^{-6})$ | $\mathcal{B}_{3}~{}(\%)$ | $\mathcal{B}_{3}$ in PDG $(\%)$
---|---|---|---
$p\bar{p}$ | 0.65 $\pm$ 0.19 $\pm$ 0.10 | 0.15$\pm$0.04$\pm$0.02$\pm$0.01 | 0.141$\pm$0.017
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 7.51 $\pm$ 0.85 $\pm$ 1.11 | 1.72$\pm$0.19$\pm$0.25$\pm$0.17 | 0.86$\pm$0.13
$K^{+}K^{-}K^{+}K^{-}$ | 0.94 $\pm$ 0.37 $\pm$ 0.14 | 0.22$\pm$0.08$\pm$0.03$\pm$0.02 | 0.134$\pm$0.032
$K^{+}K^{-}\pi^{+}\pi^{-}$ | 4.16 $\pm$ 0.76 $\pm$ 0.59 | 0.95$\pm$0.17$\pm$0.13$\pm$0.09 | 0.61$\pm$0.12
$p\bar{p}\pi^{+}\pi^{-}$ | 2.30 $\pm$ 0.65 $\pm$ 0.36 | 0.53$\pm$0.15$\pm$0.08$\pm$0.05 | <1.2 (at 90% C.L.)
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | 8.82 $\pm$ 1.57 $\pm$ 1.59 | 2.02$\pm$0.36$\pm$0.36$\pm$0.19 | 1.5$\pm$0.50
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{-}\pi^{-}$ | 3.60 $\pm$ 1.71 $\pm$ 0.64 | 0.83$\pm$0.39$\pm$0.15$\pm$0.08 | 0.71$\pm$0.29
$K^{+}K^{-}\pi^{0}$ | 4.54 $\pm$ 0.76 $\pm$ 0.48 | 1.04$\pm$0.17$\pm$0.11$\pm$0.10 | 1.2$\pm$0.1
$p\bar{p}\pi^{0}$ | 1.53 $\pm$ 0.49 $\pm$ 0.23 | 0.35$\pm$0.11$\pm$0.05$\pm$0.03 | –
$K^{0}_{S}K^{\pm}\pi^{\mp}$ | 11.35 $\pm$ 1.25 $\pm$ 1.50 | 2.60$\pm$0.29$\pm$0.34$\pm$0.25 | 2.4$\pm$0.2
$K^{0}_{S}K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ | 12.01 $\pm$ 2.22 $\pm$ 2.04 | 2.75$\pm$0.51$\pm$0.47$\pm$0.27 | –
$\pi^{+}\pi^{-}\eta$ | 7.22 $\pm$ 1.47 $\pm$ 1.11 | 1.66$\pm$0.34$\pm$0.26$\pm$0.16 | 4.9$\pm$1.8
$K^{+}K^{-}\eta$ | 2.11 $\pm$ 1.01 $\pm$ 0.32 | 0.48$\pm$0.23$\pm$0.07$\pm$0.05 | <1.5 (at 90% C.L.)
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\eta$ | 19.17 $\pm$ 3.77 $\pm$ 3.72 | 4.40$\pm$0.86$\pm$0.85$\pm$0.42 | –
$\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 20.31 $\pm$ 2.20 $\pm$ 3.33 | 4.66$\pm$0.50$\pm$0.76$\pm$0.45 | –
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ | 75.13 $\pm$ 7.42 $\pm$ 9.99 | 17.23$\pm$1.70$\pm$2.29$\pm$1.66 | –
Combining our measurement of $M(h_{c})$ with the previously-determined mass of
the centroid of the ${}^{3}P_{J}$ states leads to
$\displaystyle\Delta~{}M_{hf}\equiv\langle
M(1^{3}P)\rangle-M(1^{1}P_{1})=-0.01\pm 0.11~{}(\rm stat.)\pm 0.15~{}(\rm
syst.)\,\rm{MeV/c^{2}},$ (1)
consistent with the lowest-order expectation that the $1P$ hyperfine splitting
is zero.
The line shape of $\eta_{c}$ was also studied from the $E1$ transition
$h_{c}\to\gamma\eta_{c}$, and the measured resonance parameters are:
$\displaystyle M(\eta_{c})=2984.49\pm 1.16\pm 0.52\,\rm{MeV/c^{2}},and$
$\displaystyle\Gamma(\eta_{c})=36.4\pm 3.2\pm 1.7\,\rm{MeV}.$
These results are consistent with the recent BESIII results from
$\psi(3686)\to\gamma\eta_{c}$ BESIII:2011ab :
$\displaystyle M(\eta_{c})=2984.3\pm 0.6\pm 0.6\,\rm{MeV/c^{2}},and$
$\displaystyle\Gamma(\eta_{c})=32.0\pm 1.2\pm 1.0\,\rm{MeV};$
and B-factory results from $\gamma\gamma\to\eta_{c}$ and $B$ decays
Vinokurova:2011dy ; delAmoSanchez:2011bt . Because of the larger $\psi(3686)$
data sample that will be coming from BESIII and the advantage of negligible
interference effects, we expect that $h_{c}\to\gamma\eta_{c}$ will provide the
most reliable determinations of the $\eta_{c}$ resonance parameters in the
future.
## VII ACKNOWLEDGMENTS
The BESIII collaboration thanks the staff of BEPCII and the computing center
for their hard efforts. This work is supported in part by the Ministry of
Science and Technology of China under Contract No. 2009CB825200; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 10745001,
10625524, 10821063, 10825524, 10835001, 10935007, 11125525; Joint Funds of the
National Natural Science Foundation of China under Contracts Nos. 11079008,
11179007, 10979058; the Chinese Academy of Sciences (CAS) Large-Scale
Scientific Facility Program; CAS under Contracts Nos. KJCX2-YW-N29,
KJCX2-YW-N45; 100 Talents Program of CAS; Istituto Nazionale di Fisica
Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; U. S. Department of Energy under Contract Nos. DE-
FG02-04ER41291, DE-FG02-91ER40682, DE-FG02-94ER40823; U.S. National Science
Foundation; University of Groningen (RuG); the Helmholtzzentrum fuer
Schwerionenforschung GmbH (GSI), Darmstadt; and WCU Program of National
Research Foundation of Korea under Contract No. R32-2008-000-10155-0.
## References
* (1) E. Eichten et al., Phys. Rev. D 17, 3090 (1978) [Erratum-ibid. D 21, 313 (1980)].
* (2) J. Beringer et al. [Particle Data Group], Phys. Rev. D86, 010001 (2012).
* (3) B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 72, 031101 (2005).
* (4) K. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 98, 082001 (2007).
* (5) D. M. Asner et al. [CLEO Collaboration], Phys. Rev. Lett. 92, 142001 (2004).
* (6) M. Ablikim et al., arXiv:1205.5103v2 [hep-ex].
* (7) See, for example, E. S. Swanson, Phys. Rep. 429, 243 (2006), and references therein.
* (8) T. A. Armstrong et al. [E760 Collaboration], Phys. Rev. Lett. 69, 2337 (1992).
* (9) M. Andreotti et al. [E835 Collaboration], Phys. Rev. D 72, 032001 (2005).
* (10) J. L. Rosner et al. [CLEO Collaboration], Phys. Rev. Lett. 95, 102003 (2005).
* (11) S. Dobbs et al. [CLEO Collaboration], Phys. Rev. Lett. 101, 182003 (2008).
* (12) G. S. Adams et al. [CLEO Collaboration], Phys. Rev. D 80, 051106 (2009).
* (13) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 104, 132002 (2010).
* (14) J. Y. Ge et al. [CLEO Collaboration], Phys. Rev. D 84, 032008 (2011).
* (15) T. K. Pedlar et al. [CLEO Collaboration], Phys. Rev. Lett. 107, 041803 (2011).
* (16) T. Himel et al., Phys. Rev. Lett. 45, 1146 (1980).
* (17) A. Vinokurova et al. [Belle Collaboration], Phys. Lett. B 706, 139 (2011).
* (18) P. del Amo Sanchez et al. [BABAR Collaboration], Phys. Rev. D 84, 012004 (2011).
* (19) R. E. Mitchell et al. [CLEO Collaboration], Phys. Rev. Lett. 102, 011801 (2009) [Erratum-ibid. 106, 159903 (2011)].
* (20) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 108, 222002 (2012).
* (21) M. Ablikim et al. [BESIII Collaboration], Nucl. Instrum. Meth. A 614, 345 (2010).
* (22) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 81, 052005 (2010).
* (23) S. Agostinelli et al. [GEANT4 Collaboration], Nucl. Instrum. Meth. A 506, 250 (2003); Geant4 version: v09-03p0; Physics List simulation engine: BERT; Physics List engine packaging library: PACK 5.5.
* (24) J. Allison et al., IEEE Trans. Nucl. Sci. 53, 270 (2006).
* (25) R. G. Ping, Chinese Physics C 32, 8 (2008).
* (26) S. Jadach , B. F. L. Ward and Z. Was, Comp. Phys. Commun. 130, 260 (2000); S. Jadach, B. F. L. Ward and Z. Was, Phys. Rev. D 63, 113009 (2001).
* (27) D.J. Lange, Nucl. Instrum. Meth. A 462, 152 (2001).
* (28) T. Sjostrand, Comput. Phys. Commun. 82, 74 (1994).
* (29) M. Xu, et. al., Chinese Physics C 33, 428 (2009).
* (30) V.V. Anashin et al., arXiv:1012.1694 [hep-ex].
* (31) H. Albrecht et al. [ARGUS Collaboration], Phys. Lett. B 241, 278 (1990).
* (32) M. Ablikim et al. [BESIII Collaboration], arXiv:1208.4805 [hep-ex].
* (33) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 83, 112005 (2011).
|
arxiv-papers
| 2012-09-22T06:58:54 |
2024-09-04T02:49:35.423352
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The BESIII Collaboration: M. Ablikim, M. N. Achasov, O. Albayrak, D.\n J. Ambrose, F. F. An, Q. An, J. Z. Bai, Y. Ban, J. Becker, J. V. Bennett, M.\n Bertani, J. M. Bian, E. Boger, O. Bondarenko, I. Boyko, R. A. Briere, V.\n Bytev, X. Cai, O. Cakir, A. Calcaterra, G. F. Cao, S. A. Cetin, J. F. Chang,\n G. Chelkov, G. Chen, H. S. Chen, J. C. Chen, M. L. Chen, S. J. Chen, X. Chen,\n Y. B. Chen, H. P. Cheng, Y. P. Chu, D. Cronin-Hennessy, H. L. Dai, J. P. Dai,\n D. Dedovich, Z. Y. Deng, A. Denig, I. Denysenko, M. Destefanis, W. M. Ding,\n Y. Ding, L. Y. Dong, M. Y. Dong, S. X. Du, J. Fang, S. S. Fang, L. Fava, F.\n Feldbauer, C. Q. Feng, R. B. Ferroli, C. D. Fu, J. L. Fu, Y. Gao, C. Geng, K.\n Goetzen, W. X. Gong, W. Gradl, M. Greco, M. H. Gu, Y. T. Gu, Y. H. Guan, A.\n Q. Guo, L. B. Guo, Y. P. Guo, Y. L. Han, F. A. Harris, K. L. He, M. He, Z. Y.\n He, T. Held, Y. K. Heng, Z. L. Hou, H. M. Hu, T. Hu, G. M. Huang, G. S.\n Huang, J. S. Huang, X. T. Huang, Y. P. Huang, T. Hussain, C. S. Ji, Q. Ji, Q.\n P. Ji, X. B. Ji, X. L. Ji, L. L. Jiang, X. S. Jiang, J. B. Jiao, Z. Jiao, D.\n P. Jin, S. Jin, F. F. Jing, N. Kalantar-Nayestanaki, M. Kavatsyuk, W. Kuehn,\n W. Lai, J. S. Lange, C. H. Li, Cheng Li, Cui Li, D. M. Li, F. Li, G. Li, H.\n B. Li, J. C. Li, K. Li, Lei Li, Q. J. Li, S. L. Li, W. D. Li, W. G. Li, X. L.\n Li, X. N. Li, X. Q. Li, X. R. Li, Z. B. Li, H. Liang, Y. F. Liang, Y. T.\n Liang, G. R. Liao, X. T. Liao, B. J. Liu, C. L. Liu, C. X. Liu, C. Y. Liu, F.\n H. Liu, Fang Liu, Feng Liu, H. Liu, H. H. Liu, H. M. Liu, H. W. Liu, J. P.\n Liu, K. Y. Liu, Kai Liu, P. L. Liu, Q. Liu, S. B. Liu, X. Liu, Y. B. Liu, Z.\n A. Liu, Zhiqiang Liu, Zhiqing Liu, H. Loehner, G. R. Lu, H. J. Lu, J. G. Lu,\n Q. W. Lu, X. R. Lu, Y. P. Lu, C. L. Luo, M. X. Luo, T. Luo, X. L. Luo, M. Lv,\n C. L. Ma, F. C. Ma, H. L. Ma, Q. M. Ma, S. Ma, T. Ma, X. Y. Ma, Y. Ma, F. E.\n Maas, M. Maggiora, Q. A. Malik, Y. J. Mao, Z. P. Mao, J. G. Messchendorp, J.\n Min, T. J. Min, R. E. Mitchell, X. H. Mo, C. Morales Morales, C. Motzko, N.\n Yu. Muchnoi, H. Muramatsu, Y. Nefedov, C. Nicholson, I. B. Nikolaev, Z. Ning,\n S. L. Olsen, Q. Ouyang, S. Pacetti, J. W. Park, M. Pelizaeus, H. P. Peng, K.\n Peters, J. L. Ping, R. G. Ping, R. Poling, E. Prencipe, M. Qi, S. Qian, C. F.\n Qiao, X. S. Qin, Y. Qin, Z. H. Qin, J. F. Qiu, K. H. Rashid, G. Rong, X. D.\n Ruan, A. Sarantsev, B. D. Schaefer, J. Schulze, M. Shao, C. P. Shen, X. Y.\n Shen, H. Y. Sheng, M. R. Shepherd, X. Y. Song, S. Spataro, B. Spruck, D. H.\n Sun, G. X. Sun, J. F. Sun, S. S. Sun, Y. J. Sun, Y. Z. Sun, Z. J. Sun, Z. T.\n Sun, C. J. Tang, X. Tang, I. Tapan, E. H. Thorndike, D. Toth, M. Ullrich, G.\n S. Varner, B. Wang, B. Q. Wang, D. Wang, D. Y. Wang, K. Wang, L. L. Wang, L.\n S. Wang, M. Wang, P. Wang, P. L. Wang, Q. Wang, Q. J. Wang, S. G. Wang, X. L.\n Wang, Y. D. Wang, Y. F. Wang, Y. Q. Wang, Z. Wang, Z. G. Wang, Z. Y. Wang, D.\n H. Wei, J. B. Wei, P. Weidenkaff, Q. G. Wen, S. P. Wen, M. Werner, U.\n Wiedner, L. H. Wu, N. Wu, S. X. Wu, W. Wu, Z. Wu, L. G. Xia, Z. J. Xiao, Y.\n G. Xie, Q. L. Xiu, G. F. Xu, G. M. Xu, H. Xu, Q. J. Xu, X. P. Xu, Z. R. Xu,\n F. Xue, Z. Xue, L. Yan, W. B. Yan, Y. H. Yan, H. X. Yang, Y. Yang, Y. X.\n Yang, H. Ye, M. Ye, M. H. Ye, B. X. Yu, C. X. Yu, H. W. Yu, J. S. Yu, S. P.\n Yu, C. Z. Yuan, Y. Yuan, A. A. Zafar, A. Zallo, Y. Zeng, B. X. Zhang, B. Y.\n Zhang, C. Zhang, C. C. Zhang, D. H. Zhang, H. H. Zhang, H. Y. Zhang, J. Q.\n Zhang, J. W. Zhang, J. Y. Zhang, J. Z. Zhang, S. H. Zhang, X. J. Zhang, X. Y.\n Zhang, Y. Zhang, Y. H. Zhang, Y. S. Zhang, Z. P. Zhang, Z. Y. Zhang, G. Zhao,\n H. S. Zhao, J. W. Zhao, K. X. Zhao, Lei Zhao, Ling Zhao, M. G. Zhao, Q. Zhao,\n Q. Z. Zhao, S. J. Zhao, T. C. Zhao, X. H. Zhao, Y. B. Zhao, Z. G. Zhao, A.\n Zhemchugov, B. Zheng, J. P. Zheng, Y. H. Zheng, B. Zhong, J. Zhong, Z. Zhong,\n L. Zhou, X. K. Zhou, X. R. Zhou, C. Zhu, K. Zhu, K. J. Zhu, S. H. Zhu, X. L.\n Zhu, Y. C. Zhu, Y. M. Zhu, Y. S. Zhu, Z. A. Zhu, J. Zhuang, B. S. Zou, J. H.\n Zou",
"submitter": "Aiqiang Guo",
"url": "https://arxiv.org/abs/1209.4963"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.